diff c28a64e..1df3716 qma_coulomb_hardness_v2.txt
diff --git a/qma_coulomb_hardness_v2.txt b/qma_coulomb_hardness_v2.txt
index 279abaf..72a9383 100644
--- a/qma_coulomb_hardness_v2.txt
+++ b/qma_coulomb_hardness_v2.txt
@@ -1,155 +1,152 @@
-================================================================================
- QMA-HARDNESS OF CONTINUUM COULOMB ENERGY WITH UNIT NUCLEAR CHARGES
- a complete proof, written to be read from scratch
-================================================================================
-
-CONTENTS
- 1. The question and the answer
- 2. The physical model
- 3. The promise problem and the input format
- 4. The proof plan in one page
- 5. The starting hard spin problem
- 6. From signed to positive spin exchange
- 7. A planar layout with adjustable link lengths
- 8. Localized modes and the finite fermion model
- 9. Building the potential from a nonnegative charge density
- 9.1 the slab and well-depth counterterm; 9.2 the one-particle gap;
- 9.3 the compressed one-particle matrix
- 10. Replacing the density by unit point nuclei
- 10.1 Moser transport and cubature; 10.2 form and compression estimates
- 11. The many-electron continuum comparison
- 11.1 complement control; 11.2 compressed energy; 11.3 thresholds
- 12. Implementability and encoding
- 13. Putting the proof together
-
-READING GUIDE
- ESSENTIAL: Sections 1-4 (what the theorem says, what a molecule is, the
- promise-problem format, the one-page plan) and Section 13 (how the proof
- ties together). Together about 8 pages. Read this path first.
- THE PROOF: Sections 5-11 build the molecule. Each section takes the
- previous one as input, adds a gadget, and produces a stronger object.
- TECHNICAL: Section 12 checks that every output is computable in
- polynomial time. Skippable on a first read; nothing later needs it.
- FIRST-READ SKIPPABLE: 6.1, 8.2, 10.2 (technical estimates).
-
-CONVENTIONS
- Atomic units throughout. |A| = volume of a set A. "p(n,D)" means some
- polynomial in the named variables whose coefficients and exponents are
- absolute constants; it may be enlarged from line to line. E(H) is the
- smallest eigenvalue (bottom of the spectrum) of a self-adjoint H. We
- write R for the real line and R^3 for three-dimensional space.
-
-================================================================================
-1. THE QUESTION AND THE ANSWER
-================================================================================
-
-A single electron near a fixed proton has a lowest energy, the ground
-energy. The same is true of any molecule: fix the nuclei, keep them rigid,
-and ask for the lowest energy the electrons can have. That lowest energy
-is a single real number. Computing it exactly is impossible in general,
-and even deciding whether it is below a given threshold is believed to be
-extremely hard.
-
-How hard? The strongest notion relevant here is QMA. NP is the class of
-questions whose YES-answer can be checked quickly if someone hands you a
-proof. QMA is the same idea but the proof is allowed to be a quantum
-state. "QMA-hard" means: if you could answer this family of questions
-quickly, you could answer every QMA question quickly. No efficient
-(classical or quantum) algorithm is known for such problems.
-
-THE THEOREM (proved here).
- Decide the ground energy of electrons moving in the field of nuclei that
- are all of charge EXACTLY ONE and sit at rational positions, minimizing
- over the full spinful continuum. This problem is QMA-hard. The reduction
- is deterministic and runs in polynomial time. It uses only rational
- positions, an integer count of electrons, and rational thresholds.
- The threshold gap in the produced instance is at least 1.
-
-What makes the theorem delicate:
- - The nuclei are real point charges of charge one, not an invented
- potential with wells and oscillators built in.
- - The minimization is over ALL continuum many-electron states, not over
- a supplied finite set of orbitals.
- - No magnetic field, no extra external field, no binding promise.
-
-The companion article proves the same style of theorem for binary (encoded)
-nuclear charges by a different, analytic construction with large charges.
-This article does the polynomial-scale construction with unit charges and
-proves the finite gadgets it needs directly.
-
-================================================================================
-2. THE PHYSICAL MODEL
-================================================================================
+Conventions. Atomic units throughout. |A| denotes volume of a set A.
+"p(n,D)" means some polynomial in the named variables whose coefficients
+and exponents are absolute constants; it may be enlarged from line to
+line. E(H) is the smallest eigenvalue (bottom of the spectrum) of a
+self-adjoint operator H. R is the real line and R^3 is three-dimensional
+space.
+
+A single electron near a fixed proton has a lowest energy, its ground
+energy. The same holds for any molecule: fix the nuclei, keep them rigid,
+and ask for the lowest energy the electrons can take. That energy is a
+single real number. Exact computation is impossible in general, and even
+deciding whether it lies below a threshold is extraordinarily hard.
+
+Qma is the class of decision problems whose YES answers admit a quantumly
+checkable proof of size polynomial in the input. It contains NP: a
+classical proof is a quantum proof of a special form. No efficient
+classical or quantum algorithm is known for QMA-hard problems, in the same
+sense none is known for NP-hard problems.
+
+Aharonov, D. and Naveh, T., "Quantum NP - a survey", arXiv:quant-ph/0210077.
+
+Kitaev, A. Yu., Shen, A. H., and Vyalyi, M. N., "Classical and Quantum
+Computation", Graduate Studies in Mathematics 47, AMS, 2002.
+
+The framing that physical simulation is a natural quantum-computational
+task goes back to Feynman (1982) and Lloyd (1996); QMA-hardness asks
+when even this task remains intractable.
+
+Feynman, R. P., "Simulating physics with computers", Int. J. Theor. Phys.
+21 (1982), 467-488. https://doi.org/10.1007/BF02650179
+
+Lloyd, S., "Universal quantum simulators", Science 273 (1996), 1073-1078.
+https://doi.org/10.1126/science.273.5278.1073
+
+The problem decided. Given N electrons interacting with M nuclei of charge
+exactly one, positioned at pairwise distinct rational triples, and given
+two rational thresholds a < b with b - a >= 1/L where L is the input
+length, decide whether the electronic ground energy E_0 lies in
+(-inf, a] or in [b, inf). The promise excludes energies strictly between
+a and b. Deciding this is QMA-hard under deterministic classical
+polynomial-time many-one reductions. The reduction produces polynomially
+many nuclei and electrons, rational positions and thresholds whose bit
+lengths are polynomial in the source length, and a threshold separation
+b - a >= 1.
+
+What makes the statement sharp. The nuclei are genuine point charges of
+charge one, not an invented potential with wells and oscillators built
+in. The minimization is over every continuum many-electron state of
+either spin, not over a supplied finite basis. No magnetic field, extra
+external potential, neutrality, binding, existence of a ground
+eigenvector, or excitation-gap assumption is imposed. Membership in QMA
+is not asserted.
+
+Schuch and Verstraete (2009) proved QMA-completeness for the Hubbard model
+with local magnetic fields and outlined a continuum realization using
+designed scalar and spin-dependent potentials. O'Gorman, Irani, Whitfield,
+and Fefferman (2022) proved QMA-completeness for electronic structure in a
+supplied finite orbital basis, which fixes the Hilbert space in advance.
+Cubitt, Montanaro, and Piddock (2019) proved QMA-hardness of the signed
+exchange problem on finite square-lattice graphs, which provides the
+starting point. Each relaxation is removed at once: the free continuum,
+with real unit point nuclei, reached from the same signed-exchange source.
+
+Schuch, N. and Verstraete, F., "Computational complexity of interacting
+electrons and fundamental limitations of density functional theory",
+Nature Physics 5 (2009), 732-735. arXiv:0712.0483.
+local: 0712.0483.pdf
+
+O'Gorman, B., Irani, S., Whitfield, J., and Fefferman, B., "Intractability
+of electronic structure in a fixed basis", PRX Quantum 3 (2022), 020322.
+doi:10.1103/PRXQuantum.3.020322.
+
+Cubitt, T., Montanaro, A., and Piddock, S., "Universal quantum Hamiltonians",
+pnas 115 (2018), 9497-9502. arXiv:1701.05182v4.
+local: 1701.05182v4.pdf
Electrons in R^3. Each electron also has a spin, which is a direction in a
two-dimensional complex space C^2. So one electron lives in L^2(R^3) tensor
-C^2. N electrons live in the N-fold antisymmetric tensor power
+c^2. n electrons live in the N-fold antisymmetric tensor power
H_N = Lambda^N ( L^2(R^3) tensor C^2 ).
"Antisymmetric" means two electrons may not occupy the same (position,spin)
state: the wavefunction changes sign when you swap two electrons. Position
-and spin are swapped TOGETHER, not separately. No spin sector is fixed.
+and spin are swapped together, not separately. No spin sector is fixed.
For M nuclei at positions R_1,...,R_M with charges Z_alpha (here all equal
to 1) the electronic Hamiltonian in atomic units is
H = -1/2 sum_{i=1}^N Delta_{x_i}
- sum_{i=1}^N sum_{alpha=1}^M 1/|x_i - R_alpha|
- + sum_{1<=i<j<=N} 1/|x_i - x_j| . (2.1)
+ + sum_{1<=i<j<=N} 1/|x_i - x_j| . (1)
The three pieces are: electron kinetic energy; electron-nucleus attraction;
electron-electron repulsion. The nuclear repulsion sum_{alpha<beta}
Z_alpha Z_beta/|R_alpha-R_beta| is a fixed number once the nuclei are fixed,
-so we OMIT it. The energy in (2.1) is the ELECTRONIC energy. This is the
-convention used everywhere below and stated in the theorem.
+so it is omitted. The energy in (1) is the electronic energy. This is
+the convention used everywhere below.
The natural space of states is the H^1 form domain
Q_N = H^1(R^{3N};(C^2)^{tensor N}) intersect H_N .
-H is unbounded below on all of L^2 if we are careless, but on Q_N it is
+H is unbounded below on all of L^2 if this is careless, but on Q_N it is
bounded below and has a well-defined self-adjoint realization obtained from
-the closed Coulomb quadratic form. We write
+the closed Coulomb quadratic form. Write
E_0 = inf spec H = inf{ <psi,H psi> : psi in Q_N, ||psi|| = 1 },
whether or not the infimum is attained by an actual eigenvector. This
-"form" viewpoint is exactly what we need later: we will compare energies
-by testing the quadratic form on arbitrary states, never assuming a ground
-eigenvector exists.
+"form" viewpoint is exactly what is needed later: energies are compared
+by testing the quadratic form on arbitrary trial states, never assuming
+that a ground eigenvector exists.
-================================================================================
-3. THE PROMISE PROBLEM AND THE INPUT FORMAT
-================================================================================
+Teschl, G., "Mathematical Methods in Quantum Mechanics with Applications
+to Schrodinger Operators", Graduate Studies in Mathematics 99, AMS, 2009.
+https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/
+
+Reed, M. and Simon, B., "Methods of Modern Mathematical Physics, Vol. II:
+Fourier Analysis, Self-Adjointness", Academic Press, 1975.
+ISBN: 978-0125850025.
A "promise problem" is a decision question that is only asked on inputs
-obeying some rule (the promise). Our inputs are described exactly here.
+obeying some rule (the promise). The inputs are described exactly here.
- INPUT
+ Input
M >= 1 nuclear positions R_1,...,R_M in Q^3, pairwise distinct.
All charges Z_alpha = 1.
- N >= 1 electrons, written in UNARY (N ones, so a large N costs length N).
+ N >= 1 electrons, written in unary (N ones, so a large N costs length N).
Two rational numbers a < b, the thresholds.
- ENCODING
+ encoding
Each rational is a signed binary numerator over a positive binary
denominator, in lowest terms, inside a fixed self-delimiting encoding.
- LENGTH L
+ length L
Total bit length of all the above (the unary count contributes N).
- PROMISE
+ Promise
b - a >= 1/L.
- ANSWER
+ Answer
YES if E_0 <= a. NO if E_0 >= b.
- (Because b-a >= 1/L, these never conflict; we may not care about
+ (Because b-a >= 1/L, these never conflict; one may not care about
inputs sitting strictly between a and b, but the promise says they do
not happen.)
Note the promise "b - a >= 1/L" is a statement about how precisely the
-DECISION is posed; it is NOT a claim that the molecule has a spectral gap
+decision is posed; it is not a claim that the molecule has a spectral gap
between its ground state and first excited state. These are different
things. The theorem makes no assumption about excitation gaps, binding,
neutrality, or existence of a ground eigenvector.
@@ -167,7 +164,7 @@ neutrality, or existence of a ground eigenvector.
| Energy convention | electronic energy; nuclear repulsion omitted |
+---------------------+-----------------------------------------------+
-MAIN THEOREM.
+Main theorem.
The promise problem above is QMA-hard under deterministic classical
polynomial-time many-one reductions. The reduction produces:
- polynomially many nuclei (all charge one) at distinct rational
@@ -188,33 +185,36 @@ excitation gap of the molecule.
Why the source promise survives the extra structure. The reduction is a
chain of energy maps, each correct to within a fraction of the source gap
-gamma (Section 13). Because gamma is an inverse polynomial in the source
+gamma (the final assembly). Because gamma is an inverse polynomial in the source
length, each error is comfortably smaller than the gap, so YES stays YES
and NO stays NO. The output gap is expanded by the physical scale lambda
-(Section 12), giving b - a >= 1.
-
-================================================================================
-4. THE PROOF PLAN IN ONE PAGE
-================================================================================
+giving b - a >= 1.
-We build one molecule per source spin problem, so that the molecule's
+The construction builds a single molecule per source spin problem, so that the molecule's
ground energy equals (up to a known, computable affine map) the ground
energy of a hard spin Hamiltonian. Six steps:
- (S) SOURCE. Start from the field-free Heisenberg energy problem on a
+ (S) Source. Start from the field-free Heisenberg energy problem on a
finite square-lattice graph of maximum degree bounded and with
signed couplings, which is QMA-hard (Cubitt-Montanaro-Piddock). It
has polynomially bounded weights and an inverse-polynomial gap.
- Normalize to rational data with a named gap gamma (Section 5).
+ Normalize to rational data with a named gap gamma (the source normalization).
+
+Heisenberg, W., "Zur Theorie des Ferromagnetismus", Zs. Phys. 49 (1928),
+ 619-636. https://doi.org/10.1007/BF01328601
- (P) POSITIVE SPIN. Replace every signed coupling by positive exchange
+Kempe, J., Kitaev, A., and Regev, O., "The complexity of the local
+Hamiltonian problem", SIAM J. Comput. 35 (2006), 1070-1097.
+arXiv:quant-ph/0406180
+
+ (P) Positive spin. Replace every signed coupling by positive exchange
using fresh "singlet-pair" mediators. Each original edge becomes a
- central bond plus two bent nine-link paths. Arrange the molecules on
- a plane so that the link LENGTHS can be tuned one at a time (Section
- 6-7). Result: a positive-exchange Hamiltonian H_+ = sum K_e
+ central bond plus two bent nine-link paths. Arrange the spins on
+ a plane so that the link lengths can be tuned one at a time (the next
+ two steps). Result: a positive-exchange Hamiltonian H_+ = sum K_e
sigma_i . sigma_j with polynomially bounded K_e.
- (W) WELLS + SUPEREXCHANGE. Put one localized spatial orbital (two spin
+ (W) Wells and superexchange. Put one localized spatial orbital (two spin
states) on each site. With one electron per site, an occupancy
penalty is positive away from single occupancy. Virtual hopping of an
electron onto a neighbor and back produces an effective positive spin
@@ -222,60 +222,68 @@ energy of a hard spin Hamiltonian. Six steps:
hopping t_e to give the desired K_e. Also keep the direct
electron-electron Coulomb coefficients v_{ij} exactly, and cancel the
resulting linear-in-occupancy shift by a one-body counterterm built
- from slightly deepened wells (Section 8). Result: a finite fermion
+ from slightly deepened wells (the well construction). Result: a finite fermion
Hamiltonian H_d whose ground energy is H_+ up to scale tau^2 and a
known scalar.
- (D) CHARGE DENSITY. Realize the wells and the transverse confinement as
- the Coulomb field of an honest NONNEGATIVE continuous charge density:
+Anderson, P. W., "New approach to the theory of superexchange interactions",
+ Phys. Rev. 115 (1959), 2-13. https://doi.org/10.1103/PhysRev.115.2
+
+ (D) Charge density. Realize the wells and the transverse confinement as
+ the Coulomb field of an honest nonnegative continuous charge density:
a broad constant-density slab plus the planar wells. Adjust well
depths so the density stays nonnegative and the one-body counterterm
appears. Show a fixed one-particle gap outside the orbital space,
- and compute the matrix on the orbitals precisely (Section 9-10).
+ and compute the matrix on the orbitals precisely (the charge-density
+ and point-nucleus steps).
- (N) POINT NUCLEI. Replace the continuous density by finitely many point
- nuclei of EQUAL MASS 1/lambda (in physical units charge 1). Do it by
+ (N) Point nuclei. Replace the continuous density by finitely many point
+ nuclei of equal mass 1/lambda (in physical units charge 1). Do it by
a smooth Moser flow that pushes uniform density onto the target
density, applied to a degree-three Gauss cubature with equal node
masses, then round the node positions to rationals. Get two error
bounds: a weak one on all continuum states, a strong one on the
- orbitals (Section 11).
+ orbitals.
+
+Moser, J., "On the volume elements on a manifold", Trans. Amer. Math. Soc.
+ 120 (1965), 286-294. https://doi.org/10.1090/S0002-9947-1965-0182927-5
+local: S0002-9947-1965-0182927-5.pdf
- (M) MANY-BODY. Give the complement of the orbital space a uniform lower
+Dacorogna, B. and Moser, J., "On a partial differential equation involving
+ the jacobian determinant", Ann. Inst. H. Poincare C 7 (1990), 1-26.
+ local: AIHPC_1990__7_1_1_0.pdf
+
+ (M) Many-body. Give the complement of the orbital space a uniform lower
bound, then complete the square so that coupling between orbitals and
complement enters only quadratically in the energy. This bounds the
- FULL continuum ground energy, in every spin sector, by the finite
+ full continuum ground energy, in every spin sector, by the finite
fermion ground energy, and a variational trial vector gives the
reverse inequality. Compare with H_d, undo scalings, and choose
- thresholds (Section 12-13).
+ thresholds (the implementability and closing steps).
Every analytic estimate is made at a fixed polynomial scale lambda, with
all constants absolute and all smallness "p(n,D) lambda^{-alpha}" for a
-fixed alpha>0. Section 13 verifies that all these smallness conditions can
+fixed alpha>0. The final assembly verifies that all these smallness conditions can
be met simultaneously and that the whole reduction is computable in
polynomial time.
-================================================================================
-5. THE STARTING HARD SPIN PROBLEM
-================================================================================
-
-We start from a problem about spins that is already known to be QMA-hard.
+Start from a problem about spins that is already known to be QMA-hard.
-THE SPIN SOURCE (Cubitt-Montanaro-Piddock, 2018/2019).
+The spin source (Cubitt-Montanaro-Piddock, 2018/2019).
Consider a finite square-lattice graph (vertices are cells of Z^2, edges
join cells at distance 1), possibly with some signed real couplings J_e.
The Hamiltonian is
- H_in = sum_e J_e h^s_{LR}, h^s_{LR} = sigma_L . sigma_R, (5.1)
+ H_in = sum_e J_e h^s_{LR}, h^s_{LR} = sigma_L . sigma_R, (2)
where sigma = (sigma_x,sigma_y,sigma_z) are the Pauli matrices acting on
a spin-1/2 at each site, and sigma_L . sigma_R means
sigma_x_L sigma_x_R + sigma_y_L sigma_y_R + sigma_z_L sigma_z_R. There
- are NO one-site terms.
+ are no one-site terms.
-We will need the following facts, which are supplied by the cited theorem:
+The following facts, supplied by the cited theorem, are needed:
- (a) QMA-hardness. Deciding whether the ground energy of (5.1) is below a
+ (a) QMA-hardness. Deciding whether the ground energy of (2) is below a
lower threshold or above an upper threshold is QMA-hard, with a
promise gap that is inverse polynomial in the input size.
(b) Polynomial geometry. The lattice can be chosen with polynomially
@@ -286,16 +294,16 @@ We will need the following facts, which are supplied by the cited theorem:
magnitude bounds; the number of nonzero couplings is polynomial.
The precise theorem numbers in the extended paper are Theorem 48, Lemma
-47, and Section 10.1 of arXiv:1701.05182v4. We will not re-prove (a)-(c);
-we only make the incoming instance meet a clean rational format.
+47, and section 10.1 of Cubitt-Montanaro-Piddock. Facts (a)-(c) are not re-proved;
+the incoming instance is only made to meet a clean rational format.
-IDENTITY OF h^s. On two spins, h^s = sigma . sigma = 2 SWAP - I, where
-SWAP exchanges the two spins. Its eigenvalues are
- +1 on the three TRIPLET states (total spin 1),
- -3 on the SINGLET state (total spin 0).
+Identity of h^s. On two spins, h^s = sigma . sigma = 2 SWAP - I, where
+swap exchanges the two spins. Its eigenvalues are
+ +1 on the three triplet states (total spin 1),
+ -3 on the singlet state (total spin 0).
So h^s favors singlets when positive, triplets when negative.
-NORMALIZING THE SOURCE TO RATIONALS.
+Normalizing the source to rationals.
Step 1: a rational lower bound on the gap. Because the source gap is
inverse polynomial in the input size, there is a rational delta > 0, known
@@ -311,18 +319,18 @@ as a_0 < b_0. Round c_0, a_0, b_0 within eps_0 to rationals c~, a~, b~.
The rounded Hamiltonian H~_in = sum J~_e h^s_{LR} obeys
- |H~_in - (H_in - c_0 I)| <= 3 eps_0 . (5.2)
+ |H~_in - (H_in - c_0 I)| <= 3 eps_0 . (3)
Define rational thresholds
- a_s = a~ - c~ + 5 eps_0 , b_s = b~ - c~ - 5 eps_0 . (5.3)
+ a_s = a~ - c~ + 5 eps_0 , b_s = b~ - c~ - 5 eps_0 . (4)
Then YES/NO of the source stays intact: the 5-eps_0 shifts absorb the
3 eps_0 of Hamiltonian rounding plus eps_0 each for the scalar and the
threshold. And
b_s - a_s >= (b_0 - a_0) - 2 eps_0 - 10 eps_0 >= delta - 12 eps_0
- = 0.88 delta > 0 . (5.4)
+ = 0.88 delta > 0 . (5)
Step 3: geometric extent. A connected component of the square grid with V
vertices has coordinate diameter at most V-1 (walk along lattice edges).
@@ -331,45 +339,45 @@ graphs unchanged and keeps the total coordinate range polynomial. So the
normalized instance lives in a polynomial-size bounding box.
Finally set
- n = max(2, source input size), gamma = min(1, b_s - a_s). (5.5)
+ n = max(2, source input size), gamma = min(1, b_s - a_s). (6)
-From now on, "the source instance" means the RATIONAL signed Heisenberg
-Hamiltonian H_in with thresholds a_s < b_s and gap gamma. We may assume
+From now on, "the source instance" means the rational signed Heisenberg
+Hamiltonian H_in with thresholds a_s < b_s and gap gamma. One may assume
there is at least one vertex (add a free spin if the source is empty) and
-we may drop couplings of zero. All magnitudes and inverse gaps that we
-encounter later are polynomially bounded in n.
+one may drop couplings of zero. All magnitudes and inverse gaps
+encountered later are polynomially bounded in n.
-OUTPUT OF THIS SECTION.
+Output.
A signed rational Heisenberg Hamiltonian H_in = sum_e J_e h^s_{LR} on a
finite square-lattice subgraph, with rational thresholds a_s < b_s and gap
gamma = min(1, b_s - a_s), plus polynomial bounds on size, extent, and
-coupling magnitudes. This is the exact input to Sections 6-7.
-
-================================================================================
-6. FROM SIGNED TO POSITIVE SPIN EXCHANGE
-================================================================================
+coupling magnitudes. This is the exact input to the signed-to-positive step and the planar
+layout.
The source couplings J_e can be positive or negative. The orbital
-construction of Sections 8-10 naturally produces POSITIVE exchange (the
-electron kinetic energy prefers singlets). So we first convert the signed
+construction naturally produces positive exchange (the electron kinetic
+energy prefers singlets). So first convert the signed
source into a positive-exchange spin model. The tool is a "singlet-pair
mediator": a fresh pair of spins forced into a singlet, with weak links to
-the original spins. Its effective coupling to the two ends has the sign we
-choose.
+the original spins. Its effective coupling to the two ends has the sign one chooses.
-This uses the same mechanism as Piddock-Montanaro (Section 2.4, Lemma 5).
-We keep the quantitative proof, including the scalar shifts.
+This uses the same mechanism as Lemma 5 of Piddock and Montanaro.
+The quantitative proof, including the scalar shifts, appears below.
---------------------------------------------------------------------------------
-6.1 The second-order bottom-energy estimate (a small lemma, fully proved)
---------------------------------------------------------------------------------
-
-We will need "remove the excited states, keep second order". Here is the
-exact statement we use. It has fewer hypotheses than the usual
+Needed here is the rule "remove the excited states, keep second order".
+The exact statement follows. It has fewer hypotheses than the usual
Schrieffer-Wolff theorem and does not require the first-order block
-PBP to vanish.
+pbp to vanish.
+
+Schrieffer, J. R. and Wolff, P. A., "Relation between the Anderson and
+Kondo Hamiltonians", Phys. Rev. 149 (1966), 491-492.
+https://doi.org/10.1103/PhysRev.149.491
+Bravyi, S., DiVincenzo, D. P., and Loss, D., "Schrieffer-Wolff
+transformation for many-body systems", Ann. Physics 326 (2011),
+2793-2823. arXiv:1105.0675.
+local: 1105.0675.pdf
-LEMMA 6.1 (second-order bottom energy).
+Lemma 1 (second-order bottom energy).
Let A and B be self-adjoint operators on a finite-dimensional Hilbert
space. Suppose A has kernel projection P != 0, satisfies A >= g Q where
Q = I - P and g > 0, and B has norm ||B|| <= g/4. Then
@@ -379,7 +387,7 @@ LEMMA 6.1 (second-order bottom energy).
where A^{-1} on the right means the inverse of A on the range of Q, with
value zero on the kernel of A.
-PROOF. First, |E(A+B)| <= ||B||: the bottom of A+B is at most <x,(A+B)x> at
+Proof. First, |E(A+B)| <= ||B||: the bottom of A+B is at most <x,(A+B)x> at
any unit x, and at least -||B|| since A >= 0 everywhere on the space.
Let x be a nearly-ground unit vector for A+B with energy E = E(A+B). Write
@@ -411,12 +419,8 @@ dividing gives the same value, so E(A+B) <= E(W) + O(||B||^3/g^2). Together
with the lower estimate this proves the lemma. (The case Q = 0 is
immediate.) QED
---------------------------------------------------------------------------------
-6.2 The singlet-pair gadget
---------------------------------------------------------------------------------
-
-For each surviving signed source edge e = LR of weight J != 0, we add a
-FRESH PAIR of spins A_e, B_e, coupled as follows:
+For each surviving signed source edge e = LR of weight J != 0, add a
+fresh pair of spins A_e, B_e, coupled as follows:
central bond: Delta * h^s_{A_e B_e} (Delta > 0 integer)
two "spokes": sqrt(Delta) * v_e * ( h^s_{L A_e} + h^s_{C_e R} )
@@ -428,9 +432,9 @@ FRESH PAIR of spins A_e, B_e, coupled as follows:
So a positive edge attaches L--A_e and B_e--R; a negative edge attaches
L--A_e and A_e--R (both spokes touch A_e, so the sign of the effective
LR coupling flips). The strength of the spokes is sqrt(Delta) v_e, which is
-"small" compared to the central gap 4 Delta, so Lemma 6.1 applies.
+"small" compared to the central gap 4 Delta, so Lemma 1 applies.
-THE GADGET AS AN EFFECTIVE LR COUPLING.
+The gadget as an effective LR coupling.
Step 1: make the fresh pair A_e B_e sit in its singlet. The operator
A_e, B_e's coupling Delta h^s prefers the singlet (h^s has eigenvalue -3
@@ -445,11 +449,11 @@ gives 4). A >= g Q with g = 4 Delta.
Step 2: the rest is B = H_in(rest) + spoke terms. Its norm is at most
poly(n)(1 + sqrt Delta) since the number of edges and the weights are
polynomial and sqrt(Delta) is the only new strong scale. If Delta is a
-sufficiently large polynomial in n, we have ||B|| <= g/4 = Delta.
+sufficiently large polynomial in n, the bound ||B|| <= g/4 = Delta holds.
-Step 3: apply Lemma 6.1. The kernel P of A is spanned by product states in
+Step 3: apply Lemma 1. The kernel P of A is spanned by product states in
which every fresh pair is a singlet; on that space the original signed
-terms and the source-unaffected part act unchanged. The kernel restricton
+terms and the source-unaffected part act unchanged. The kernel restriction
of the first-order term P B P keeps the unreplaced part. The second-order
correction is -P B A^{-1} B P. Only two "hops" through B can act on the
same fresh pair, and they have energy denominator g = 4 Delta (the gap of
@@ -467,42 +471,38 @@ only, then
member and -delta for one on each member.) Tracing the second-order term
over the singlet and over the intermediates gives, per edge e,
- J * h^s_{LR} - (3/2) v_e^2 I = J h^s_{LR} - 3|J| I , (6.1)
+ J * h^s_{LR} - (3/2) v_e^2 I = J h^s_{LR} - 3|J| I , (7)
because v_e^2 = 2|J|, hence (3/2) v_e^2 = 3|J|. The sign of the effective
LR term is the sign of J by construction (positive edge: L--A_e, B_e--R
-distinct members of the singlet; negative edge: L--A_e, A_e--R, BOTH
+distinct members of the singlet; negative edge: L--A_e, A_e--R, both
endpoints connect to the same member A_e, flipping the correlation via
the singlet relation above).
-The correction to the ground energy from Lemma 6.1 is O(||B||^3/g^2) =
+The correction to the ground energy from Lemma 1 is O(||B||^3/g^2) =
O( poly(n)(1 + sqrt Delta)^3 / Delta^2 ), which is O(poly(n)/sqrt(Delta));
choose Delta a large enough polynomial in n that this is <= gamma/100.
Step 4: all edges simultaneously. Because the operations commute across
-distinct fresh pairs (each pair touches only its own two spokes), we can
-replace every surviving signed edge at the same time, with the SAME large
+distinct fresh pairs (each pair touches only its own two spokes), one can
+replace every surviving signed edge at the same time, with the same large
Delta. The total shift is sum_e (3 Delta + (3/2) v_e^2) - (correction).
---------------------------------------------------------------------------------
-6.3 Subdivision: from the two spokes to nine-link paths
---------------------------------------------------------------------------------
-
The singlet-pair gadget produces a positive spinner vertex (the fresh pair)
and two spokes of positive weight sqrt(Delta) v_e, one joining L to A_e and
-one joining C_e to R. But in the orbital construction (Section 8) we need
-the link strengths K_e themselves to be encoded in the DISTANCES between
+one joining C_e to R. But in the orbital construction needs
+the link strengths K_e themselves to be encoded in the distances between
orbitals. Having an intermediate A_e or C_e in the middle of a "spoke" is
-fine, but a spoke of positive weight is still a h^s coupling, and we want
-to realise ALL positive h^s couplings as orbital hoppings.
+fine, but a spoke of positive weight is still a h^s coupling, and the goal is to
+to realize all positive h^s couplings as orbital hoppings.
-We subdivide each positive spoke into a path of length three using the SAME
+Subdivide each positive spoke into a path of length three using the same
gadget with positive sign (a fresh pair forced into a singlet, attached at
its two ends). Each subdivider replaces one h^s coupling K h^s_{XY} by a
chain X -- fresh pair -- Y, with two couplings sqrt(Delta') K h^s on each
-side and a central bond Delta' h^s. Lemma 6.1 leaves the effective X-Y
-coupling equal to (K/2) h^s up to a scalar shift; because we may rescale
-the whole target (and absorb the 1/2 into K_e), we only care that the
+side and a central bond Delta' h^s. Lemma 1 leaves the effective X-Y
+coupling equal to (K/2) h^s up to a scalar shift; because one may rescale
+the whole target (and absorb the 1/2 into K_e), only care that the
sign is preserved and the strength is positive.
Two rounds of subdivision (three per spoke, nine per original edge, because
@@ -510,14 +510,14 @@ each round triples the path) produce per original edge:
- one central bond A_e--B_e (positive), and
- two paths of length nine, joining L to A_e and C_e to R.
-Each round uses its OWN polynomial Delta' (there are only three rounds
-total, so we do not need to fit infinitely many Deltas), and each round
+Each round uses its own polynomial Delta' (there are only three rounds
+total, so there is no need to fit infinitely many Deltas), and each round
keeps errors O(poly(n)/sqrt(Delta')) = O(gamma/100). Summing the errors of
three rounds gives O(3 gamma/100) < gamma/25.
The final positive-exchange Hamiltonian is
- H_+ = sum_{links} K_e h^s_{ij} , (6.2)
+ H_+ = sum_{links} K_e h^s_{ij} , (8)
with every K_e positive, every K_e and 1/K_e polynomially bounded in n,
and the number of sites m (including the original vertices and every
@@ -525,46 +525,39 @@ fresh spacer vertex) polynomial. There is a polynomial-time computable
scalar shift C_s (sum of all the scalar shifts from the singlet-pair
gadgets and the subdivisions and their second-order corrections) with
- | E(H_+) - E(H_in) + C_s | <= 4 gamma/100 . (6.3)
+ | E(H_+) - E(H_in) + C_s | <= 4 gamma/100 . (9)
Every intermediate weight K_e and its shift are computable (as real
numbers) to inverse-polynomial absolute accuracy in polynomial time.
Square roots need not be exact rationals at this stage; they will be
-absorbed into the final rational approximation in Section 13.
+absorbed into the final rational approximation in the final assembly.
-OUTPUT OF THIS SECTION.
+Output.
A positive-exchange Heisenberg Hamiltonian H_+ = sum K_e h^s_{ij} on a
graph of polynomial size m, with every K_e, 1/K_e polynomially bounded,
-and a scalar shift C_s with (6.3). Every such K_e is realized as a target
-"hopping" number for the orbital construction of Section 8.
-
-================================================================================
-7. A PLANAR LAYOUT WITH ADJUSTABLE LINK LENGTHS
-================================================================================
+and a scalar shift C_s with (9). Every such K_e is realized as a target
+"hopping" number for the orbital construction of the well construction.
-A Hamiltonian H_+ = sum K_e h^s_{ij} is just a weighted graph. In Section 8
-we will realize each VERTEX as a localized spatial orbital and each LINK as
+A Hamiltonian H_+ = sum K_e h^s_{ij} is just a weighted graph. In the
+well construction each vertex is realized as a localized spatial orbital
+and each link as
a distance between orbitals: the hopping strength t_ij drops off like
-exp(-distance). So the LINK LENGTHS must encode the K_e: each link has a
-DIFFERENT length, chosen so its hopping equals the desired value. This
-section builds a planar drawing of the modified graph in which every link
-length can be adjusted INDEPENDENTLY, with nonlink distances at least 1.4.
-
-We draw every original edge as a small "contact gadget", building on the
-CMP (Cubitt-Montanaro-Piddock) lattice.
+exp(-distance). So the link lengths must encode the K_e: each link has a
+different length, chosen so its hopping equals the desired value. A
+planar drawing of the modified graph is built in which every link
+length can be adjusted independently, with nonlink distances at least 1.4.
---------------------------------------------------------------------------------
-7.1 The two contact templates
---------------------------------------------------------------------------------
+Every original edge becomes a small "contact gadget", building on the
+cmp (Cubitt-Montanaro-Piddock) lattice.
Place original lattice vertices at grid cells with spacing 17. For each
original edge e = LR, choose a local frame whose first coordinate runs
along the edge from L to R. In this frame:
- POSITIVE EDGE (J > 0): center pair A_e = (8,0), B_e = (9,0).
- NEGATIVE EDGE (J < 0): center pair A_e = (8.5,0), B_e = (8.5,1).
+ positive edge (J > 0): center pair A_e = (8,0), B_e = (9,0).
+ negative edge (J < 0): center pair A_e = (8.5,0), B_e = (8.5,1).
-PICTURE (topology only, not to scale):
+Picture (topology only, not to scale):
J > 0: L ~~~ A_e === B_e ~~~ R
J < 0: L ~~~ A_e ~~~ R
@@ -574,8 +567,8 @@ PICTURE (topology only, not to scale):
~~~ = one bent nine-link path (length 8 or 8.5; drawn zig-zag so that
off-path sites stay >= 1.4 apart)
=== = the central unit-length bond A_e--B_e
- L, R = the two ORIGINAL lattice vertices, 17 apart
- A_e, B_e = the FRESH singlet-pair sites from Section 6
+ L, R = the two original lattice vertices, 17 apart
+ A_e, B_e = the fresh singlet-pair sites from the signed-to-positive step
The central bond A_e--B_e has length 1 in each case. The two spokes are
nine-link paths of length nine each:
@@ -591,7 +584,7 @@ steps (transverse 0). Steps 3, 4 slope forward with first component 3/4
mirror those offsets with the opposite sign. This makes both paths lie
within a narrow strip of transverse offset < 1.324.
-NONLINK SEPARATION. The distance between any two sites that are NOT
+Nonlink separation. The distance between any two sites that are not
consecutive sites on a link is at least 1.4. Consecutive along-axis
separations are at least 3/4; nonconsecutive sites on the same path are at
least 1.5 apart; the extra negative-sign side site B_e is at least sqrt(2)
@@ -602,13 +595,9 @@ far from all internal sites, by the 17-spacing. This verifies at once
that (with the size-one distances written above) nonlinks are at least
1.4 apart.
---------------------------------------------------------------------------------
-7.2 Adjusting link lengths independently
---------------------------------------------------------------------------------
-
-We now show that the length of each of the 18 links per gadget can be set
-to ANY value in [1 - eps, 1 + eps] for some fixed small eps > 0, ALL AT
-ONCE, without ever bringing nonlinks closer than 1.2. The original lattice
+Now show that the length of each of the 18 links per gadget can be set
+to any value in [1 - eps, 1 + eps] for some fixed small eps > 0, all at
+once, without ever bringing nonlinks closer than 1.2. The original lattice
vertices stay fixed.
Fix one gadget. It has internal vertices (A_e, B_e and the path vertices)
@@ -619,15 +608,15 @@ gives for each path a map
d : (internal vertex displacements) -> (twelve length increments)
-We must show d is SURJECTIVE near zero. Look at the linearization. Let
-e_1,...,e_9 be the unit step VECTORS of the path. A linear dependence with
+One must show d is surjective near zero. Look at the linearization. Let
+e_1,...,e_9 be the unit step vectors of the path. A linear dependence with
coefficients a_1,...,a_9 among the rows of the fixed-end length
differential would give, at each internal vertex j (1 <= j <= 8),
a_j e_j - a_{j+1} e_{j+1} = 0 .
-This forces all vectors a_j e_j to be EQUAL. Since the path has some pair
-of NONPARALLEL steps (steps 3,4,6,7 have transverse components), all
+This forces all vectors a_j e_j to be equal. Since the path has some pair
+of nonparallel steps (steps 3,4,6,7 have transverse components), all
coefficients must vanish. So the rows of the differential are linearly
independent, thus the whole gadget's link-length differential is
surjective on the internal vertex variables.
@@ -648,73 +637,62 @@ increments. Choose the ball small enough that nonlink separations stay
above 1.2, and then fix eps accordingly.
Because original lattice vertices stay fixed, each gadget can be solved
-independently. Their number (polynomial in n) does NOT shrink the allowed
-length interval [1 - eps, 1 + eps]: gadgets do not interfere, so we can
+independently. Their number (polynomial in n) does not shrink the allowed
+length interval [1 - eps, 1 + eps]: gadgets do not interfere, so one can
implement all eighteen independent adjustments at once.
-COMPUTABILITY. Each such solve is a fixed-dimensional contraction with
+Computability. Each such solve is a fixed-dimensional contraction with
algebraic initial data (rational or nested-square-root). Iterating with
guard precision computes every internal position to any inverse-polynomial
-accuracy in polynomial time. Section 13 takes care of final rational
+accuracy in polynomial time. the final assembly takes care of final rational
rounding.
---------------------------------------------------------------------------------
-7.3 The full layout, rescaled
---------------------------------------------------------------------------------
-
The unit-distance layout above is then rescaled by D = log(lambda) (see
-Section 8 for the significance of D). This gives minimum spacing at least
+the well construction for the significance of D). This gives minimum spacing at least
0.95 D, nonlink spacing at least 1.2 D, and spatial extent poly(n) * O(D).
The orbit centers u_i are placed at the rescaled layout positions.
-OUTPUT OF THIS SECTION.
+Output.
A planar drawing of the positive-exchange graph of H_+ with:
- one site per vertex (each tied to a planar position u_i at scale D);
- - one site per fresh spacer vertex (from Section 6), all also realized
+ - one site per fresh spacer vertex (from the signed-to-positive step), all also realized
as planar positions;
- each link having an independently adjustable length in
[D(1 - eps), D(1 + eps)];
- nonlinks farther apart than 1.2 D;
- the whole drawing contained in a poly(n) * O(D) box.
-================================================================================
-8. LOCALIZED MODES AND THE FINITE FERMION MODEL
-================================================================================
-
-Section 6 gave the target positive-exchange Hamiltonian H_+ = sum K_e h^s_{ij}
-on a graph of m sites. Section 7 gave a planar layout in which independent
-link lengths exist. Now we (1) build one smooth localized spatial orbital
+the signed-to-positive step gave the target positive-exchange Hamiltonian
+H_+ = sum K_e h^s_{ij} on a graph of m sites. The planar layout gives a
+placement in which independent
+link lengths exist. Now (1) build one smooth localized spatial orbital
per site, (2) write the finite-fermion Hamiltonian realizing H_+, with an
-occupancy penalty, and (3) calibrate the link HOPPINGS to the desired
-strengths by choosing separations. Sections 9-11 then realize the orbital
+occupancy penalty, and (3) calibrate the link hoppings to the desired
+strengths by choosing separations. The next three steps then realize the orbital
potential of part (1) as the field of real nuclei.
---------------------------------------------------------------------------------
-8.1 Coordinates, the localized mode, and the transverse Gaussian
---------------------------------------------------------------------------------
-
Because the eventual nuclei have charge 1, the physical scale is set by a
-large number lambda. We work in DILATED coordinates
+large number lambda. Work in dilated coordinates
y = lambda x = (r, z) in R^3, r in R^2 (plane), z in R (vertical).
After dividing energies by lambda^2:
- the kinetic operator becomes the usual -Delta/2;
- - each unit nucleus contributes an attraction of COUPLING 1/lambda;
+ - each unit nucleus contributes an attraction of coupling 1/lambda;
- electron repulsion has coefficient 1/lambda.
This "one lambda of dilution" is why the final electronic charges come out
-as exactly one (Section 11).
+as exactly one.
-THE PLANAR MODE. Take
+The planar mode. Take
- f(r) = (1 - |r|^2)_+^16 , (8.1)
+ f(r) = (1 - |r|^2)_+^16 , (10)
which is a smooth nonnegative bump supported in the unit disk, positive
inside, and vanishing to order 16 at the boundary. Define
phi = (-Delta_r + 1)^{-1} f , so -Delta_r phi + phi = f,
varphi = phi / ||phi||_2 ,
- W(r) = - f(r) / (2 phi(r)) . (8.2)
+ W(r) = - f(r) / (2 phi(r)) . (11)
Then phi > 0, radial, exponentially decaying, and W is nonpositive and
compactly supported (inside the unit disk). Check the eigen-relation:
@@ -729,19 +707,19 @@ representation
phi(r) = int_0^\infty e^{-t} int_{R^2} (4 pi t)^{-1} e^{-|r-b|^2/(4t)}
f(b) db dt
-(solve by the heat kernel; it is the standard Gaussian-heat formula for the
+(from the heat kernel; this is the standard Gaussian-heat formula for the
resolvent of -Delta + 1) gives, for |r| > 2, the tail bounds
C^{-1}(1+|r|)^{-C} e^{-|r|} <= |D^k phi(r)| <= C(1+|r|)^C e^{-|r|}
-for the derivatives k we need, using t + (|r|-1)^2/(4t) >= |r| - 1. The
+for the derivatives k one needs, using t + (|r|-1)^2/(4t) >= |r| - 1. The
lower bound comes from integrating the density over a small half-disk and a
band |r|/2 < t < |r|/2 + 1. In particular phi has a fixed positive lower
bound on the unit disk, so W and all its derivatives are bounded there, and
W is zero outside.
-ISOLATED GAP. The planar operator -Delta_r/2 + W has the unique ground state
-varphi and a FIXED positive gap above it. Proof: the shifted quadratic form
+Isolated gap. The planar operator -Delta_r/2 + W has the unique ground state
+varphi and a fixed positive gap above it. Proof: the shifted quadratic form
on a test function u is int phi^2 |nabla(u/phi)|^2 / 2 (nonnegativity by
closure and local positivity, and the kernel is exactly the multiples of
phi). If a unit sequence perpendicular to varphi had shifted form tending
@@ -752,38 +730,34 @@ parallel to varphi at once, hence zero; but then the 1/2 shift precludes the
form values from tending to zero. So a fixed gap c_pl > 0 separates the
ground mode varphi from the rest.
-THE VERTICAL MODE. For the z direction use the normalized Gaussian
+The vertical mode. For the z direction use the normalized Gaussian
g(z) = (omega/pi)^{1/4} e^{- omega z^2 / 2}, omega = sqrt(4 pi rho),
the ground mode of -d^2/dz^2/2 + 2 pi rho z^2 with energy omega/2. (rho > 0
-is a large fixed integer chosen in Section 9.) The oscillator has a fixed
+is a large fixed integer chosen in the charge-density step.) The oscillator has a fixed
gap omega above its ground mode.
-THE THREE-DIMENSIONAL MODE. Per site i, centered at planar position u_i
-(from Section 7, at scale D):
+The three-dimensional mode. Per site i, centered at planar position u_i
+(from the planar layout, at scale D):
- psi_i(y) = varphi(r - u_i) g(z), e_* = -1/2 + omega/2, (8.3)
+ psi_i(y) = varphi(r - u_i) g(z), e_* = -1/2 + omega/2, (12)
the energy of the (planar ground) tensor (vertical ground). psi_i is
smooth, decaying exponentially at infinity with rate controlled by
phi and g.
---------------------------------------------------------------------------------
-8.2 Direct Coulomb coefficients and their coercivity
---------------------------------------------------------------------------------
-
Define the direct electron-electron coefficients
v_{ij} = iint psi_i(y)^2 psi_j(y')^2 / |y - y'| dy dy'
- = v(|u_i - u_j|) , (8.4)
+ = v(|u_i - u_j|) , (13)
U = v(0) .
(U is the same-site coefficient; v_{ij} is the coefficient between sites i,j.)
They are uniformly bounded (the psi_i are uniformly bounded and decay), and
v is uniformly Lipschitz (translate the decaying density and differentiate).
-COERCIVITY (a fixed lower bound on the whole matrix v). We now prove
+Coercivity (a fixed lower bound on the whole matrix v). Now prove
(v_{ij}) >= c I, c > 0, uniformly over all separated geometries, once the
polynomial scale lambda is sufficiently large. For real coefficients a_i,
write F = sum_i a_i psi_i^2 (a smooth decaying density). Fix a fixed smooth
@@ -792,55 +766,51 @@ be its disjoint translates to the centers (u_i, 0). Put
w = sum_i a_i chi_i, and Phi_F = the Coulomb potential of F, i.e. the function
with -Delta Phi_F = F.
-INTEGRATION BY PARTS + CAUCHY-SCHWARZ:
+Integration by parts + Cauchy-Schwarz:
int F w = int (-Delta Phi_F) w = int (nabla Phi_F) . (nabla w)
<= ||nabla Phi_F||_2 ||nabla w||_2 .
Now ||nabla Phi_F||_2^2 = int Phi_F (-Delta Phi_F) = int Phi_F F
= (1/(4 pi)) iint F(y) F(y') / |y - y'| dy dy' (this is exactly the Coulomb
-double integral of F; it is finite for our smooth decaying F). Therefore
+double integral of F; it is finite for the smooth decaying F). Therefore
- |int F w|^2 <= (1/(4 pi)) ( Coulomb integral of F ) * int |nabla w|^2 . (8.5)
+ |int F w|^2 <= (1/(4 pi)) ( Coulomb integral of F ) * int |nabla w|^2 . (14)
-LOWER BOUND ON THE LEFT. Define the matrix A_ij = int psi_i^2 chi_j.
+Lower bound on the left. Define the matrix A_ij = int psi_i^2 chi_j.
Its diagonal entries are a fixed a_* > 0 (same localized overlap at each
site). Off-diagonal entries are exponentially small in D: each off-diagonal
requires psi_i^2 to reach chi_j at distance >= 0.95 D, costing exp(-c D).
-Row and column SUMS of the off-diagonal entries are at most poly(m) e^{-c D};
+Row and column sums of the off-diagonal entries are at most poly(m) e^{-c D};
choose lambda such that these are at most a_*/4 each. Then A = a_* I + R
with ||R|| <= a_*/2 (operator norm), and
- int F w = sum_{i,j} a_i a_j A_ij = a^T A a >= (a_*/2) sum_i a_i^2 . (8.6)
+ int F w = sum_{i,j} a_i a_j A_ij = a^T A a >= (a_*/2) sum_i a_i^2 . (15)
-UPPER BOUND ON ||nabla w||. The chi_i have disjoint supports and fixed
+Upper bound on ||nabla w||. The chi_i have disjoint supports and fixed
shapes, so
- int |nabla w|^2 = sum_i a_i^2 int |nabla chi_0|^2 = O( sum_i a_i^2 ) . (8.7)
+ int |nabla w|^2 = sum_i a_i^2 int |nabla chi_0|^2 = O( sum_i a_i^2 ) . (16)
-Combine (8.5), (8.6), and (8.7): the Coulomb double integral of F is at
+Combine (14), (15), and (16): the Coulomb double integral of F is at
least a fixed c times sum_i a_i^2. But F = sum_i a_i psi_i^2, so the Coulomb
integral is exactly sum_{i,j} a_i a_j v_{ij} = a^T V a with V = (v_{ij}).
Hence
- a^T V a >= c sum_i a_i^2 for all real a , i.e. V >= c I . (8.8)
+ a^T V a >= c sum_i a_i^2 for all real a , i.e. V >= c I . (17)
-A DIRECT CONSEQUENCE. Apply (8.8) to two sites with coefficients (1, -1):
+A direct consequence. Apply (17) to two sites with coefficients (1, -1):
(1,-1)^T V (1,-1) = v_{ii} + v_{jj} - 2 v_{ij} = 2U - 2 v(d)
which is >= 2c. Therefore
- U - v(d) >= c > 0 for every distance d >= 0.95 D in use , (8.9)
+ U - v(d) >= c > 0 for every distance d >= 0.95 D in use , (18)
and it is uniformly bounded above. So the superexchange denominator
U - v_{ij} below is uniformly positive and bounded.
---------------------------------------------------------------------------------
-8.3 The occupancy penalty and superexchange
---------------------------------------------------------------------------------
-
-IN ONE BREATH. Two electrons on neighbouring sites with opposite spins
+In one breath. Two electrons on neighbouring sites with opposite spins
can momentarily pile onto a single site and then split back. Pauli's
principle permits the pile-up only when the two spins are antiparallel
(a singlet); for parallel (triplet) spins the first hop is blocked. So
@@ -848,21 +818,21 @@ the double hop lowers the singlet's energy relative to the triplet's by
roughly t^2/(U - v), i.e. a favourable spin exchange whose size is set
by the virtual-excitation cost.
-THE FERMION MODEL. Consider 2m canonical fermion modes: at each site i, two
+The fermion model. Consider 2m canonical fermion modes: at each site i, two
spin states sigma in {up,down}. Restrict to exactly m electrons (the
filling). Let n_i = sum_sigma c_{i sigma}^dagger c_{i sigma} be the total
occupancy at site i, so 0 <= n_i <= 2 and sum_i n_i = m. Multiply dilated
-energies by lambda, so the Coulomb coefficients below carry NO 1/lambda.
+energies by lambda, so the Coulomb coefficients below carry no 1/lambda.
The direct Coulomb interaction, written in the mode basis, is
- (U/2) sum_i n_i (n_i - 1) + sum_{i<j} v_{ij} n_i n_j . (8.10)
+ (U/2) sum_i n_i (n_i - 1) + sum_{i<j} v_{ij} n_i n_j . (19)
It is nonnegative and vanishes exactly when every site is singly occupied
(n_i = 1): then n_i(n_i-1) = 0 and the pairing term is zero at m = sum n_i.
Away from single occupancy it is positive.
-ONE-SITE DEFECT COUNTERM. We will later want a diagonal one-body term
--sum_i s_i n_i with s_i = sum_{j != i} v_{ij}. Using v_{ii} = U and
+One-site defect counterm. Later a diagonal one-body term
+-sum_i s_i n_i will be needed, with s_i = sum_{j != i} v_{ij}. Using v_{ii} = U and
sum_i n_i = m, a short algebra shows
(U/2) sum_i n_i(n_i-1) + sum_{i<j} v_{ij} n_i n_j - sum_i s_i n_i
@@ -888,41 +858,41 @@ Since (U/2) sum_i n_i(n_i-1) = (U/2) sum_i n_i^2 - (U/2) m, this last line
is exactly the left-hand side of (C). QED
So define s_i = sum_{j != i} v_{ij}; the required diagonal one-body term
--sum_i s_i n_i will be produced by slightly deepening the wells (Section 9).
+-sum_i s_i n_i will be produced by slightly deepening the wells.
-THE FINITE HAMILTONIAN. The finite fermion model we aim to realize is
+The finite hamiltonian. The finite fermion model to be realized is
H_d(t) = (1/2) sum_{i,j} v_{ij} (n_i - 1)(n_j - 1) - sum_{i<j} v_{ij}
- sum_{ e={i,j} } t_e sum_{sigma} ( c_{i sigma}^dagger c_{j sigma}
- + c_{j sigma}^dagger c_{i sigma} ) . (8.12)
+ + c_{j sigma}^dagger c_{i sigma} ) . (20)
-The quadratic penalty (1/2) sum v_{ij}(n_i-1)(n_j-1) is NONNEGATIVE and
+The quadratic penalty (1/2) sum v_{ij}(n_i-1)(n_j-1) is nonnegative and
its kernel is exactly the spin space with every site singly occupied (n_i =
1 all i). Its gap: a nonzero integer defect vector (n_i - 1)_i has sum zero
-and hence squared Euclidean length at least 2; by coercivity (8.8) with
-integer entries the penalty is at least c. So it has a FIXED positive gap.
+and hence squared Euclidean length at least 2; by coercivity (17) with
+integer entries the penalty is at least c. So it has a fixed positive gap.
-FIRST ORDER OF HOPPING VANISHES. On the singly-occupied spin space, a
+First order of hopping vanishes. On the singly-occupied spin space, a
single hop creates a doubly-occupied site and an empty site. That excitation
is penalized by the quadratic term, so it drops out of the first-order
compression onto the singly-occupied space.
-SECOND ORDER = SUPEREXCHANGE. A single allowed hop on edge e = {i,j}
+Second order = superexchange. A single allowed hop on edge e = {i,j}
creates a doubly occupied site and an empty site. Its defect vector is
(1,-1) on that pair and zero elsewhere, so its excitation energy is exactly
- U - v_{ij} (8.13)
+ U - v_{ij} (21)
-(evaluated with the quadratic penalty plus scalar; derivation via (8.8)).
+(evaluated with the quadratic penalty plus scalar; derivation via (17)).
Only a hop on the same link can restore single occupancy in one more step.
For hopping strength t_e, the second-order (bottom-energy) operator is
- ( t_e^2 / (U - v_{ij}) ) ( h^s_{ij} - I ) . (8.14)
+ ( t_e^2 / (U - v_{ij}) ) ( h^s_{ij} - I ) . (22)
-Let us verify the factor. On a singlet, the amplitude for one hop is set by
+To verify the factor. On a singlet, the amplitude for one hop is set by
the fermion algebra; the doubly occupied intermediate has amplitude sqrt(2)
times t_e (two orthogonal doubly occupied states for the singlet), while
-the triplet does NOT hop into a same-site pair. Squaring the amplitude and
+the triplet does not hop into a same-site pair. Squaring the amplitude and
dividing by the excitation energy U - v_{ij} gives the h^s-dependent term.
Physically this is Anderson's superexchange. Concretely, apply the
creation/annihilation operators to
@@ -934,35 +904,31 @@ and use that a second hop on the same link returns a spin state. Other
sites are placed after i,j in the ordering, which changes only the overall
sign on the spin space. The result matches the earlier statement.
-TARGET HOPPINGS. Introduce a common positive parameter tau (chosen below)
+Target hoppings. Introduce a common positive parameter tau (chosen below)
and set
- t_e^tar = tau sqrt( K_e (U - v_{ij}) ) . (8.15)
+ t_e^tar = tau sqrt( K_e (U - v_{ij}) ) . (23)
-Substituting into (8.14) and using (8.13) gives the effective spin term
+Substituting into (22) and using (21) gives the effective spin term
- tau^2 ( H_+ - sum_e K_e ) (8.16)
+ tau^2 ( H_+ - sum_e K_e ) (24)
on the singly-occupied space. The hopping operator has norm at most
-tau * poly(n) uniformly over admissible separated geometries. By Lemma 6.1
+tau * poly(n) uniformly over admissible separated geometries. By Lemma 1
the energy remainder of the second-order expansion is at most
C tau^3 poly(n) (since the penalty gap is at least c, uniformly). Choose a
-rational inverse-polynomial tau > 0 small enough that Lemma 6.1 applies
+rational inverse-polynomial tau > 0 small enough that Lemma 1 applies
(tau poly(n) <= gap/4) and this remainder is at most gamma tau^2/100. This
-choice of tau is INDEPENDENT of lambda and of the final centers.
+choice of tau is independent of lambda and of the final centers.
Note: the occupancy penalty involves every pair {i,j} (through v_{ij}), but
the target hopping on a link e depends only on its own distance through
v_{ij} = v(d). So the calibration of one link is a one-dimensional problem.
---------------------------------------------------------------------------------
-8.4 Calibrating the separations
---------------------------------------------------------------------------------
-
-The modes supply the HOPPING PROFILE
+The modes supply the hopping profile
T(d) = - int_{R^2} varphi(r) W(r) varphi(r - d e_1) dr
- = ( 1 / (2 ||phi||_2^2) ) int f(r) phi(r - d e_1) dr . (8.17)
+ = ( 1 / (2 ||phi||_2^2) ) int f(r) phi(r - d e_1) dr . (25)
The second form follows from W = -f/(2 phi):
-varphi(r) W(r) = (phi(r)/||phi||) * f(r)/(2 phi(r)) = f(r)/(2 ||phi||).
@@ -971,25 +937,25 @@ Multiply by varphi(r - de_1) = phi(r - de_1)/||phi|| and integrate:
This is manifestly positive (f, phi > 0), uniformly Lipschitz, and by the
tail bound exp(-|r|) of phi it satisfies, for some absolute C,
- c (1+d)^{-C} e^{-d} <= T(d) <= C (1+d)^{C} e^{-d} . (8.18)
+ c (1+d)^{-C} e^{-d} <= T(d) <= C (1+d)^{C} e^{-d} . (26)
-THE HOPPING THE CONTINUUM WILL SUPPLY. Section 10 will show that the real
+The hopping to be supplied by the continuum. The point-nucleus step shows that the real
continuum orbital matrix, evaluated on the orbitals centered at u_i, has
hopping entries t_e = lambda T(|u_i - u_j|) in the units of H_d, up to an
error of order p(n,D) lambda^{-0.1}. So to realize the target t_e^tar of
-(8.15), we must choose each link length d_e so that
+(23), one must choose each link length d_e so that
- lambda T(d_e) = tau sqrt( K_e (U - v(d_e)) ) . (8.19)
+ lambda T(d_e) = tau sqrt( K_e (U - v(d_e)) ) . (27)
-The occupancy penalty involves every pair (through v_{ij}), but on a LINK
-the target depends only on d_e. So (8.19) is ONE SCALAR EQUATION per link.
+The occupancy penalty involves every pair (through v_{ij}), but on a link
+the target depends only on d_e. So (27) is one scalar equation per link.
Define the residual
- F_e(d) = lambda T(d) - tau sqrt( K_e ( U - v(d) ) ) . (8.20)
+ F_e(d) = lambda T(d) - tau sqrt( K_e ( U - v(d) ) ) . (28)
-A ROOT EXISTS. Evaluate at d = (1 - eps/2) D and d = (1 + eps/2) D, both
-inside the adjustable range of Section 7:
+A root exists. Evaluate at d = (1 - eps/2) D and d = (1 + eps/2) D, both
+inside the adjustable range of the planar layout:
T((1-eps/2)D) ~ exp(-(1-eps/2)D) => lambda T ~ lambda^{eps/2} up to poly(D),
T((1+eps/2)D) ~ exp(-(1+eps/2)D) => lambda T ~ lambda^{-eps/2} up to poly(D).
@@ -1000,66 +966,62 @@ inverse poly, K_e inverse poly, U-v >= c). At the near endpoint the first
term dominates => F_e > 0. At the far endpoint the first term is dwarfed
=> F_e < 0. So by continuity F_e has a root in between.
-BISECTION WITH APPROXIMATE SIGNS. Evaluate F_e to inverse-polynomial
+Bisection with approximate signs. Evaluate F_e to inverse-polynomial
accuracy. If the sign is ambiguous at evaluation accuracy, the residual is
already small enough, so stop. Otherwise keep the sign-change bracket, and
continue until bracket width times a Lipschitz bound is within tolerance.
-No monotonicity or derivative lower bound is needed: we only need SOME root
-with small residual, and any root works. Section 13 handles the numerical
+No monotonicity or derivative lower bound is needed: only need some root
+with small residual, and any root works. the final assembly handles the numerical
evaluation details.
-RESULTING CENTERS. The calibrated distances d_e, rescaled back, are the
-planar separations |u_i - u_j| for link pairs. Section 7's contraction then
+Resulting centers. The calibrated distances d_e, rescaled back, are the
+planar separations |u_i - u_j| for link pairs. the planar layout's contraction then
produces internal-site coordinates realizing all these link lengths at once
-with nonlinks separated, and a final rational approximation (Section 13)
+with nonlinks separated, and a final rational approximation (the final assembly)
gives rational centers u_i.
-FROM NOW ON, H_d means H_d(t) at these final centers, with
+From now on, H_d means H_d(t) at these final centers, with
t_e = lambda T(|u_i - u_j|). Choose the per-link calibration accuracy so
that replacing t_e^tar by t_e changes the total Hamiltonian norm by at most
gamma tau^2/100. Combining this and the second-order remainder from
-Section 8.3 (which is at most gamma tau^2/100) and the internal error of
-Lemma 6.1, we get
+the occupancy-penalty step (which is at most gamma tau^2/100) and the internal error of
+Lemma 1, one gets
| E(H_d) + sum_{i<j} v_{ij} - tau^2 ( E(H_+) - sum_e K_e ) |
- <= 2 gamma tau^2 / 100 . (8.21)
+ <= 2 gamma tau^2 / 100 . (29)
-Line (8.21) says the bottom of H_d, shifted by the direct-Coulomb scalar
+Line (29) says the bottom of H_d, shifted by the direct-Coulomb scalar
sum_{i<j} v_{ij}, equals tau^2 times the positive-exchange ground energy
H_+ minus a scalar, up to the stated error. This is the finite-fermion
-target that Sections 9-12 will realize as a continuum molecule.
-
---------------------------------------------------------------------------------
-8.5 The polynomial scale convention
---------------------------------------------------------------------------------
+target that the remaining steps will realize as a continuum molecule.
-Every analytic error we produce later has the form p(n,D) lambda^{-alpha}
+Every analytic error produced later has the form p(n,D) lambda^{-alpha}
for some fixed alpha > 0, with required smallness at worst inverse polynomial
-in n. We now state how lambda, D, and the cubature cell h are chosen so that
+in n. Now state how lambda, D, and the cubature cell h are chosen so that
all these conditions can be met at once.
Order of the scale choices:
1. The bounded well data (f, phi, W) and the fixed integer rho are fixed
- FIRST, independent of anything else (Section 9 picks rho using a
- derivative bound in (9.4)).
+ first, independent of anything else (the charge-density step picks rho using a
+ derivative bound in (30)).
2. The source cutoff and the three spin-gadget penalty scales Delta,
- Delta', Delta'' are fixed from n and gamma (Sections 5-6).
+ Delta', Delta'' are fixed from n and gamma in the source normalization.
3. The rational tau is fixed from the uniform Coulomb coercivity constant
- c and the polynomial size bounds (Section 8.3).
- 4. THEN choose the physical scale. Take an integer
+ c and the polynomial size bounds (the occupancy-penalty step).
+ 4. then choose the physical scale. Take an integer
- k = ceil( C n^A ) , h = 1/k , lambda = 8 k^3 / rho , (8.22)
+ k = ceil( C n^A ) , h = 1/k , lambda = 8 k^3 / rho , (31)
with A, C fixed absolute positive integers, then large enough.
The relation rho h^3 / 8 = 1/lambda is exactly the statement that each of
- the EIGHT equal nodes of one cubature cell (side h) carries mass 1/lambda.
+ the eight equal nodes of one cubature cell (side h) carries mass 1/lambda.
This will make each node a unit nucleus in physical coordinates
- (Section 11). The fixed integer rho appears so that this mass relationship
+ The fixed integer rho appears so that this mass relationship
matches rho h^3.
At this last stage, the target sizes K_e, 1/K_e, the layout extent, and the
-coefficients b_i (Section 9) already have fixed polynomial bounds in n, so
+coefficients b_i (the charge-density step) already have fixed polynomial bounds in n, so
they do not change the choice. Each smallness requirement
"p(n,D) lambda^{-alpha} <= inverse polynomial in n" is met by a sufficiently
large fixed A, and the constant C covers all n >= 2 and the fixed
@@ -1067,57 +1029,49 @@ separation and gap conditions. The polynomials p(n,D) come from fixed-order
derivatives, moments, fixed-dimensional wells and gadgets, and polynomially
many matrix entries, so no exponent needs to grow with the input. Thus the
large box, its volume, and 1/delta_R are all polynomial in n after A, C
-are fixed. Section 13 revisits this and shows every scalar budget can be
+are fixed. the final assembly revisits this and shows every scalar budget can be
met simultaneously.
Finally D = log lambda. Then D is polynomial in n, and "exp(-D) beats
poly(n)" and similar statements hold by taking lambda a large enough power
of n.
-================================================================================
-9. BUILDING THE POTENTIAL FROM A NONNEGATIVE CHARGE DENSITY
-================================================================================
-
-We now construct an attractive Coulomb potential whose low-energy matrix
-realizes the modes and hoppings of Section 8. The potential is the Coulomb
-field of an honest NONNEGATIVE charge density: a broad constant-density
+Now construct an attractive Coulomb potential whose low-energy matrix
+realizes the modes and hoppings of the well construction. The potential is the Coulomb
+field of an honest nonnegative charge density: a broad constant-density
slab (supplying transverse confinement and a positive background) plus the
planar wells (supplying the orbitals). Adding the wells changes the density
by a bounded, compactly supported term, so the total density stays
nonnegative.
-The section has three parts:
- 9.1 the slab and the well-depth counterterm, formula (9.4);
- 9.2 a fixed one-particle gap outside the modes, Proposition 9.1;
- 9.3 the compressed one-particle matrix on the modes.
+The construction has three parts: the slab and the well-depth
+counterterm, formula (30); a fixed one-particle gap outside the modes
+(Proposition 1); the compressed one-particle matrix on the modes.
Everything is in dilated coordinates y = (r,z); energies are the dilated
ones (divide final energies by lambda^2 later). The nuclei are not yet
-points; they will be the density's field (Section 11).
-
---------------------------------------------------------------------------------
-9.1 The slab and the well-depth counterterm
---------------------------------------------------------------------------------
+points; they will be the density's field.
Choose integers
- S with D <= S <= 10 D , H = ceil( lambda^5 ) , (9.1)
+ S with D <= S <= 10 D , H = ceil( lambda^5 ) , (32)
and the box Omega = [-H,H]^2 x [-S,S]. (The exact scale of S costs only
poly(D) factors; a binary bit-length scale works. H is huge so the
-horizontal sides are far away.) Define the SLAB field and its value at 0:
+horizontal sides are far away.) Define the slab field and its value at 0:
- F_H(y) = - rho int_Omega db / |y-b| , C_H = F_H(0) . (9.2)
+ F_H(y) = - rho int_Omega db / |y-b| , C_H = F_H(0) . (33)
The slab potential F_H is the attraction of a uniform density rho on the
box; because the box is tall and wide, near the planar wells F_H behaves
like a constant minus a vertical harmonic well centered on z=0, plus small
-error (see (9.6)).
+error (see (34)).
-WHY WE NEED A WELL-DEPTH ADJUSTMENT. The occupancy identity (C) of Section
-8.3 requires a one-body diagonal term -s_i/lambda on site i, where
+Why we need a well-depth adjustment. The occupancy identity (C) of the
+occupancy-penalty step requires a one-body diagonal term -s_i/lambda on
+site i, where
s_i = sum_{j!=i} v_{ij}. In the mode basis, the diagonal entry from
--s_i/lambda must be supplied by a small DEEPENING of well i. Because
+-s_i/lambda must be supplied by a small deepening of well i. Because
<psi_i, W_i psi_i> = int varphi^2 W < 0 (W nonpositive), increasing the
depth of well i lowers its diagonal energy, so the required sign is correct
and the size is tunable. Let
@@ -1129,110 +1083,106 @@ equal to 1 for |z| <= 1/2, with piecewise polynomial rational data. Then
put
b_i = s_i / ( - int varphi^2 W ) >= 0 ,
- V_ell(r,z) = zeta(z/S) sum_i (1 + b_i/lambda) W_i(r) . (9.3)
+ V_ell(r,z) = zeta(z/S) sum_i (1 + b_i/lambda) W_i(r) . (35)
The b_i are nonnegative and polynomially bounded independently of D (they
depend only on the geometry of the wells through v_{ij} and the fixed
-integral -int varphi^2 W). Hence 0 <= b_i/lambda < 1 at our scales. The
+integral -int varphi^2 W). Hence 0 <= b_i/lambda < 1 at the scales. The
functions V_ell are supported in disjoint vertical cylinders around the u_i,
inside Omega.
-CHOOSING rho. The derivatives of V_ell we need are bounded independently
+Choosing rho. The derivatives of V_ell one needs are bounded independently
of S (the zeta factor's derivatives are controlled by S >= D and the well
supports are disjoint in the plane; the b_i/lambda terms are small). Use
1 + b_i/lambda <= 2 and the fixed bounds on W, zeta. Choose the fixed
integer rho so large that
- | Delta V_ell / (4 pi) | <= rho / 2 everywhere. (9.4)
+ | Delta V_ell / (4 pi) | <= rho / 2 everywhere. (30)
-There is NO CIRCULARITY: the bound used to pick rho depends only on
-2, W, zeta, NOT on the vertical Gaussian or on b_i. The Gaussian and the
+There is no circularity: the bound used to pick rho depends only on
+2, W, zeta, not on the vertical Gaussian or on b_i. The Gaussian and the
exact b_i are then fixed afterwards as needed.
-THE DENSITY. Using the fundamental solution of the Laplacian
+The density. Using the fundamental solution of the Laplacian
(-Delta_y (1/|y-b|) = 4 pi delta_b) and the compact support of V_ell inside
-Omega, we express the WHOLE potential F_H + V_ell as a Coulomb integral of a
+Omega, the whole potential F_H + V_ell is expressed as a Coulomb integral of a
nonnegative density:
V_ell(y) = - int_Omega ( Delta V_ell(b) / (4 pi) ) / |y - b| db ,
F_H(y) + V_ell(y)
- = - int_Omega [ rho + Delta V_ell(b)/(4 pi) ] / |y-b| db . (9.5)
+ = - int_Omega [ rho + Delta V_ell(b)/(4 pi) ] / |y-b| db . (36)
-By (9.4) the bracketed density rho + Delta V_ell/(4 pi) lies between rho/2
+By (30) the bracketed density rho + Delta V_ell/(4 pi) lies between rho/2
and 3 rho/2 on Omega. Its total mass is exactly rho |Omega|, because
int Delta V_ell = 0 (V_ell is compactly supported and smooth). The centers
-fit well inside the horizontal box at our scales. So (9.5) is a
-NONNEGATIVE continuous charge density whose Coulomb field is the exact
-one-particle potential we use below (before the point-nucleus replacement).
+fit well inside the horizontal box at the scales. So (36) is a
+nonnegative continuous charge density whose Coulomb field is the exact
+one-particle potential use below (before the point-nucleus replacement).
-BACKGROUND ESTIMATES. The following hold with constants independent of
+Background estimates. The following hold with constants independent of
n and lambda:
(a) F_H(y) - C_H >= 2 pi rho min(z^2, S^2) - C S^3 / H ,
(b) | F_H(y) - C_H - 2 pi rho z^2 | <= C ( S^3 + S|r|^2 ) / H ,
- for |r| <= H/2 and |z| <= S . (9.6)
+ for |r| <= H/2 and |z| <= S . (34)
-JUSTIFICATION OF (9.6). At every z, F_H is minimized over r at r=0, and is
+Justification of (34). At every z, F_H is minimized over r at r=0, and is
also nondecreasing in |z| for fixed r: this follows one coordinate at a time
by integrating an even decreasing kernel over a centered interval. At r=0,
-extending the horizontal integration to the plane in the DIFFERENCE between
+extending the horizontal integration to the plane in the difference between
z and 0 gives, for |z| <= S, exactly
2 pi rho int_{-S}^{S} ( |z-w| - |w| ) dw = 2 pi rho z^2
(integrate by parts / use polar integration around the axis). The omitted
difference from restricting the horizontal square costs at most
-C S^3 / H, by inverse-cube decay of the kernel outside horizontal radius H.
+c s^3 / H, by inverse-cube decay of the kernel outside horizontal radius H.
For |r| <= H/2, translating the integration square gives a change at most
-C S |r|^2 / H relative to the horizontal center. The first derivatives then
+c s |r|^2 / H relative to the horizontal center. The first derivatives then
come from sides at distance >= H/2 (bounded by C S/H per unit distance),
and second derivatives are bounded by C S / H by integrating first
derivatives of the kernel on the sides. Evenness at r=0 gives (a) at z=0.
Finally, on the whole box, 0 >= F_H(y) >= C_H (F_H is negative and most
negative at y=0), and |C_H| <= C S H.
-Together (9.5) and (9.6) say: near the wells the potential is the vertical
+Together (36) and (34) say: near the wells the potential is the vertical
harmonic oscillator 2 pi rho z^2 (min energy omega/2, gap omega) plus the
-planar multiwell sum_i W_i, plus perturbations. The rest of Section 9 makes
+planar multiwell sum_i W_i, plus perturbations. The rest of the charge-density construction makes
this precise.
---------------------------------------------------------------------------------
-9.2 The one-particle gap and the residual
---------------------------------------------------------------------------------
-
The potential F_H + V_ell is now expressed as the Coulomb field of a
-nonnegative density (9.5). Before replacing that density by point nuclei,
-we need two properties:
- - a FIXED energy gap above the modes, controlling arbitrary continuum
+nonnegative density (36). Before replacing that density by point nuclei,
+one needs two properties:
+ - a fixed energy gap above the modes, controlling arbitrary continuum
states;
- - a SMALL residual on the modes, controlling mixing with the mode space.
+ - a small residual on the modes, controlling mixing with the mode space.
-THE SHIFTED ONE-PARTICLE OPERATOR. Define
+The shifted one-particle operator. Define
- L_c = - Delta_y / 2 + F_H + V_ell - C_H - e_* . (9.7)
+ L_c = - Delta_y / 2 + F_H + V_ell - C_H - e_* . (37)
(The spin identity is suppressed.) It is the one-particle operator of the
continuum, shifted to put the mode bottoms near zero. Recall e_* =
-1/2 + omega/2.
-THE MODE PROJECTION. Let P project onto the span of the psi_i, including
+The mode projection. Let P project onto the span of the psi_i, including
both spins when present. Let Gamma_ij = <psi_i, psi_j> be the spatial Gram
matrix. Exponential tails and the 0.95 D spacing give the bound
- || Gamma - I || <= p(n,D) lambda^{-0.8} . (9.8)
+ || Gamma - I || <= p(n,D) lambda^{-0.8} . (38)
(Indeed each off-diagonal overlap is O(e^{-0.9 |u_i-u_j|}) <= e^{-0.9*0.95 D};
leave part of the decay as an integrable factor and sum poly many.) Use the
-ORTHONORMAL modes
+orthonormal modes
psihat_j = sum_i psi_i (Gamma^{-1/2})_{ij} ,
- || Gamma^{-1/2} - I || <= p(n,D) lambda^{-0.8} . (9.9)
+ || Gamma^{-1/2} - I || <= p(n,D) lambda^{-0.8} . (39)
Then unit vectors in Ran P have polynomially bounded one-particle kinetic
energy (they are finite combinations of the smooth modes with weights of
polynomial size).
-PROPOSITION 9.1 (continuous one-particle control).
+Proposition 1 (continuous one-particle control).
With the density and separated centers constructed above, once the
polynomial scale is sufficiently large, there are fixed constants
c > 0 and C < infinity such that, as quadratic-form inequalities,
@@ -1240,19 +1190,19 @@ PROPOSITION 9.1 (continuous one-particle control).
(I-P) L_c (I-P) >= c (I-P) on the complement,
L_c >= - Delta_y/2 - C (kinetic lower bound),
|| L_c P || <= p(n,D) lambda^{-0.9} (operator norm Ran P -> L^2).
- (9.10)
+ (40)
All estimates hold with either spin included.
-PROOF.
+Proof.
-Step 1: lower bound by a separable operator. By (9.6a), (9.5), and
+Step 1: lower bound by a separable operator. By (9.6a), (36), and
zeta W_i >= W_i (zeta <= 1, W_i <= 0) with the b_i/lambda terms small and
bounded everywhere, L_c is at least the separable operator
A_sep = (-Delta_r/2 + sum_i W_i) + (-d^2/dz^2/2 + 2 pi rho min(z^2,S^2))
- - e_* , (9.11)
+ - e_* , (41)
-minus an error O(S^3/H) + O(p(n,D)/lambda) that is uniform in ALL states
+minus an error O(S^3/H) + O(p(n,D)/lambda) that is uniform in all states
(as a form inequality). Call the first bracket (planar) A_pl and the second
(vertical) A_v, each shifted so its ground energy is near zero:
A_pl has ground energy -1/2 (mode varphi at each site) and A_v has ground
@@ -1265,12 +1215,12 @@ angle interpolation between 0 and pi/2 on the transition annuli using sine
and cosine, so that sum_i |nabla theta_i|^2 + |nabla theta_ext|^2 <= C/D^2.
The IMS localization identity (product rule with kinetic coefficient 1/2)
then costs O(1/D^2) in the form. On each site term use the isolated ground
-state and gap of -Delta_r/2 + W (Section 8.1); on the exterior term use the
+state and gap of -Delta_r/2 + W (the one-particle step); on the exterior term use the
free kinetic lower bound and the fact that no well sits on its support. With
site cutoffs theta_i and varphi_i = varphi(.-u_i), the site term evaluated
on theta_i u is at least
- (-1/2 + c_1) ||theta_i u||^2 - c_1 |<theta_i varphi_i, u>|^2 , (9.12)
+ (-1/2 + c_1) ||theta_i u||^2 - c_1 |<theta_i varphi_i, u>|^2 , (42)
where c_1 > 0 may be taken below both the isolated gap and 1/2. The exterior
term also absorbs its share of the (-1/2 + c_1) mass, because its support
@@ -1279,7 +1229,11 @@ i differs from P_r = sum_i |varphi_i><varphi_i| by at most
O(m e^{-c' D}) + O(p(n,D) lambda^{-0.8}) (mode tails and Gram correction).
Hence
- A_pl >= -1/2 + c_1 (I - P_r) - o(1) I . (9.13)
+ A_pl >= -1/2 + c_1 (I - P_r) - o(1) I . (43)
+
+Simon, B., "Semiclassical analysis of low lying eigenvalues. I,
+non-degenerate minima: asymptotic expansions", Ann. Inst. H. Poincare A
+38 (1983), 295-308. local: AIHPA_1983__38_3_295_0.pdf
Step 3: the vertical gap. Localize the z problem with an inner cutoff equal
to 1 on |z| <= S/2 (supported in |z| <= S) and an exterior function. On the
@@ -1289,35 +1243,35 @@ exterior term use the large-z potential lower bound 2 pi rho min(z^2,S^2)
localization costs O(S^{-2}) in the form, and the rank-one mode is P_g =
|g><g| with a Gaussian-weight loss of order e^{-c S}. Hence
- A_v >= omega/2 + c_2 (I - P_g) - o(1) I . (9.14)
+ A_v >= omega/2 + c_2 (I - P_g) - o(1) I . (44)
Step 4: tensor the gaps. On the spatial space P = P_r (tensor) P_g. The
operator inequality
- (I - P_r) (tensor) I + I (tensor) (I - P_g) >= I - P (9.15)
+ (I - P_r) (tensor) I + I (tensor) (I - P_g) >= I - P (45)
holds because the left side is the indicator of "at least one factor is
-outside", which dominates "product is outside". Combining (9.13) and (9.14)
-through (9.15), the separable operator A_sep has, on I - P, a fixed gap
+outside", which dominates "product is outside". Combining (43) and (44)
+through (45), the separable operator A_sep has, on I - P, a fixed gap
A_sep - e_* >= (-1/2 + omega/2 - e_*) + c(I-P) - o(1) I
- = c(I-P) - o(1) I , (9.16)
+ = c(I-P) - o(1) I , (46)
since e_* = -1/2 + omega/2 makes the constant term zero. Choosing the
polynomial scale so the total error O(S^3/H) + O(p(n,D)/lambda) + o(1) is
-at most c/2 gives the first inequality in (9.10). The slowly small errors
-O(D^{-2}), O(S^{-2}) are only needed to establish a FIXED gap, not a per-
+at most c/2 gives the first inequality in (40). The slowly small errors
+o(d^{-2}), o(s^{-2}) are only needed to establish a fixed gap, not a per-
particle energy accuracy, so they do not blow any budget.
Step 5: kinetic lower bound and the mode-space norm. The kinetic lower bound
-L_c >= -Delta_y/2 - C follows from (9.6) (F_H - C_H bounded below by
--C S^3/H >= -C on the useful region and 0 >= F_H >= C_H off it) together with
+L_c >= -Delta_y/2 - C follows from (34) (F_H - C_H bounded below by
+-c s^3/h >= -C on the useful region and 0 >= F_H >= C_H off it) together with
bounded V_ell and the constant -C_H - e_* absorbed into C. For the operator
-norm ||L_c P||: (9.6) and the crude global background bound plus mode tails
+norm ||L_c P||: (34) and the crude global background bound plus mode tails
give
|| ( F_H - C_H - 2 pi rho z^2 ) psi_i ||_2 <= p(n,D) lambda^{-2} .
- (9.17)
+ (47)
Indeed only moments with |r| <= H/2 and |z| <= S matter there, and outside
that region the Gaussian in z or the planar exponential in |r| beats the
@@ -1327,109 +1281,97 @@ Now (-Delta_y/2 + 2 pi rho z^2 + W_j - e_*) psi_j = 0 exactly (tensor of the
planar ground eigenvalue -1/2 and the vertical ground eigenvalue omega/2,
shifted by e_*). So the residual L_c psi_j is (i) the other wells W_i,
i != j, acting on the small planar tails of psi_j, and (ii) the term
-(b_j/lambda) W_j on psi_j, and (iii) the errors already bounded in (9.17).
+(b_j/lambda) W_j on psi_j, and (iii) the errors already bounded in (47).
All three are of size at most p(n,D) lambda^{-0.9} on unit mode vectors,
-giving the last inequality in (9.10). QED
+giving the last inequality in (40). QED
---------------------------------------------------------------------------------
-9.3 The compressed one-particle matrix
---------------------------------------------------------------------------------
-
-A fixed complementary gap is not enough: we need the matrix on the modes to
-a SHARPER scale, because the continuum energy will be compared to H_d after
+A fixed complementary gap is not enough: one needs the matrix on the modes to
+a sharper scale, because the continuum energy will be compared to H_d after
multiplying by lambda. This subsection computes that matrix.
-Compute the continuous one-body compression. On the NONORTHOGONAL functions
+Compute the continuous one-body compression. On the nonorthogonal functions
psi_i (before Gram correction), the column on mode j comes, up to the
-background and cutoff errors already bounded in Section 9.2, from applying
+background and cutoff errors already bounded in the one-particle-gap step, from applying
- ( sum_{k != j} W_k + lambda^{-1} sum_k b_k W_k ) psi_j . (9.18)
+ ( sum_{k != j} W_k + lambda^{-1} sum_k b_k W_k ) psi_j . (48)
-Row j (diagonal). The depth-deepening b-term gives the diagonal contribution we
-arranged: with I = <psi_j, W_j psi_j> = int varphi^2 W < 0, the j-th piece of
+Row j (diagonal). The depth-deepening b-term contributes diagonally as
+follows: with I = <psi_j, W_j psi_j> = int varphi^2 W < 0, the j-th piece
+of
lambda^{-1} sum_k b_k W_k on psi_j is lambda^{-1} b_j I = lambda^{-1}
( s_j / (-I) ) I = - s_j / lambda. Since s_j = sum_{i != j} v_{ij}, this is
-the SAME number as -lambda^{-1} sum_{i != j} v_{ij} promised in the
+the same number as -lambda^{-1} sum_{i != j} v_{ij} promised in the
construction. The other-well terms sum_{k != j} W_k acting on psi_j carry two
planar tail factors (spacing >= 0.95 D) and are exponentially small, so they
do not affect the diagonal. Hence A_jj = -s_j/lambda up to negligible error.
Row i != j (off-diagonal). In the first sum sum_{k != j} W_k, the term with
-k = i gives -T(|u_i - u_j|) (this is exactly the hopping profile (8.17),
+k = i gives -T(|u_i - u_j|) (this is exactly the hopping profile (25),
evaluated with the well at i acting on psi_j). The cases k != i, j are
-products of TWO tail factors at a third site; by the spacing and compact
+products of two tail factors at a third site; by the spacing and compact
well support, each such integral is O(lambda^{-1.8}). The explicit
lambda^{-1} sum_k b_k W_k terms give only negligible off-diagonal errors (at
least one planar tail on each well support), bounded by p(n,D) lambda^{-1.9}
altogether per entry. Nonlink hoppings cost at most p(n,D) lambda^{-1.2} per
term, because two wells at a nonlink separation overlap by two tails.
-PASS TO ORTHONORMAL MODES. Let A_ij = <psi_i, L_c psi_j> (the SHIFTED
+Pass to orthonormal modes. Let A_ij = <psi_i, L_c psi_j> (the shifted
matrix: the constant -C_H - e_* is already subtracted inside L_c, which
removes the large background scalar; do not re-add it). The matrix in the
-orthonormal modes is Gamma^{-1/2} A Gamma^{-1/2}. Using (9.8) and (9.10),
+orthonormal modes is Gamma^{-1/2} A Gamma^{-1/2}. Using (38) and (40),
|| Gamma^{-1/2} A Gamma^{-1/2} - A ||
- <= C || Gamma - I || * || A || <= p(n,D) lambda^{-1.7} . (9.19)
+ <= C || Gamma - I || * || A || <= p(n,D) lambda^{-1.7} . (49)
The correction therefore remains negligible after multiplication by lambda.
-THE SPATIAL MATRIX. Define
+The spatial matrix. Define
M_ij = - s_i if i = j,
= - lambda T(|u_i - u_j|) if {i,j} is a link,
- = 0 otherwise. (9.20)
+ = 0 otherwise. (50)
-Putting the row estimates and (9.19) together,
+Putting the row estimates and (49) together,
|| lambda Gamma^{-1/2} A Gamma^{-1/2} - M || <= p(n,D) lambda^{-0.1} .
- (9.21)
+ (51)
with the identity on spin. This is the required one-particle compression:
-the DIAGONAL supplies the depth counterterm (the -s_i one-body term), and
-the LINK entries supply the calibrated hopping -lambda T(d_e). Comparing
-with H_d of (8.12), whose hopping is -t_e on links and whose penalty
-produces -s_i as the linear counterterm, we see the compressed one-particle
+the diagonal supplies the depth counterterm (the -s_i one-body term), and
+the link entries supply the calibrated hopping -lambda T(d_e). Comparing
+with H_d of (20), whose hopping is -t_e on links and whose penalty
+produces -s_i as the linear counterterm, one sees the compressed one-particle
matrix is exactly the one-particle part of H_d, up to p(n,D) lambda^{-0.1}.
-WHERE WE ARE. We have a CONTINUOUS charge density whose one-particle
+Where we are. There is a continuous charge density whose one-particle
operator L_c has (i) the modes as approximate low-energy states with a fixed
complementary gap, and (ii) a compressed matrix equal to the one-particle
-part of H_d to accuracy p(n,D) lambda^{-0.1}. Section 10 replaces the
-density by honest unit point nuclei; Section 11 assembles the many-electron
-comparison.
-
-================================================================================
-10. REPLACING THE DENSITY BY UNIT POINT NUCLEI
-================================================================================
+part of H_d to accuracy p(n,D) lambda^{-0.1}. The point-nucleus step replaces the
+density by honest unit point nuclei; the comparison with H_d is then completed.
-The input to Section 9 was a nonnegative charge density (9.5). We must now
-replace its Coulomb field by finitely many POINT nuclei of EQUAL mass
+The input to the charge-density step was a nonnegative charge density (36). One must now
+replace its Coulomb field by finitely many point nuclei of equal mass
1/lambda, so that in physical coordinates each nucleus has charge exactly 1.
There are two distinct error requirements:
- (i) a FORM bound on arbitrary continuum states (weak, but on all states);
- (ii) a COMPRESSION bound on the smooth localized modes only (stronger).
+ (i) a form bound on arbitrary continuum states (weak, but on all states);
+ (ii) a compression bound on the smooth localized modes only (stronger).
The comparison with H_d uses energies multiplied by lambda. So the
compressed error must be smaller than lambda^{-1} up to a polynomial margin.
The error coupling the mode space to its complement can be larger: the fixed
-complementary gap of Section 9.2 makes its energy contribution QUADRATIC in
-the coupling (Section 11), so a weaker bound suffices there.
+complementary gap of the one-particle-gap step makes its energy contribution quadratic in
+the coupling, so a weaker bound suffices there.
Cubature side h will be chosen as
- h = (8 / (rho lambda))^{1/3} , so rho h^3 / 8 = 1/lambda . (10.1)
+ h = (8 / (rho lambda))^{1/3} , so rho h^3 / 8 = 1/lambda . (52)
A cubature rule exact through polynomial degree three will then give an
O(h^4) compressed error and an O(h^2) form error. Both resulting energy
errors are O(lambda^{-1/3}) in the units of H_d, up to polynomial factors.
---------------------------------------------------------------------------------
-10.1 Transport and charge placement
---------------------------------------------------------------------------------
-
-IN ONE BREATH. To replace a continuous charge smear by point nuclei,
+In one breath. To replace a continuous charge smear by point nuclei,
tile the uniform slab it was built from by an evenly spaced cubature and
let a smooth vector field push each cubature node toward the thicker
region. Choose the flow so the pushed nodes' local density equals the
@@ -1438,81 +1380,76 @@ continuum field on all smooth test functions, and every node keeps its
common weight (charge one). This is Moser's construction.
The density interpolation is Moser's deformation method (Moser 1965;
-Dacorogna-Moser 1990). Here the vector field is EXPLICIT and compactly
-supported inside the box, and we prove the required transport directly from
+Dacorogna-Moser 1990). Here the vector field is explicit and compactly
+supported inside the box, and prove the required transport directly from
the continuity equation.
Define the interpolation path from the uniform slab density rho to the
target density,
- q_t = rho + t Delta V_ell / (4 pi) , 0 <= t <= 1 , (10.2)
+ q_t = rho + t Delta V_ell / (4 pi) , 0 <= t <= 1 , (53)
so q_0 = rho and q_1 = rho + Delta V_ell/(4 pi) is exactly the density of
-(9.5). Since by (9.4) |Delta V_ell/(4 pi)| <= rho/2, we have q_t >= rho/2
-for ALL t. On full space define the time-dependent vector field
+(36). Since by (30) |Delta V_ell/(4 pi)| <= rho/2, there is q_t >= rho/2
+for all t. On full space define the time-dependent vector field
- V_t(b) = - nabla V_ell(b) / (4 pi q_t(b)) , 0 <= t <= 1 . (10.3)
+ V_t(b) = - nabla V_ell(b) / (4 pi q_t(b)) , 0 <= t <= 1 . (54)
Because V_ell is smooth, compactly supported inside the well cylinders
(strictly inside the box), and the denominator q_t >= rho/2, the field V_t
-has uniformly bounded derivatives through order four and VANISHES outside
+has uniformly bounded derivatives through order five and vanishes outside
those cylinders.
-THE FLOW. Let G_t be the flow of V_t starting at time zero, and put G = G_1.
+The flow. Let G_t be the flow of V_t starting at time zero, and put G = G_1.
Standard ODE theory (differentiate the equation, induct on the order) gives
bounded derivatives of G_t through order four and a bounded Lipschitz
-inverse. The flow is the IDENTITY off the cylinders, so it preserves Omega.
+inverse. The flow is the identity off the cylinders, so it preserves Omega.
-CONTINUITY EQUATION. Directly from V_t = -nabla V_ell/(4 pi q_t) and
+Continuity equation. Directly from V_t = -nabla V_ell/(4 pi q_t) and
partial_t q_t = Delta V_ell/(4 pi),
- partial_t q_t + div( q_t V_t ) = 0 . (10.4)
+ partial_t q_t + div( q_t V_t ) = 0 . (55)
Integrating along the flow gives the transport identity
q_t(G_t(b)) det DG_t(b) = rho , and hence
- int_Omega F(b) q_1(b) db = rho int_Omega F(G(b)) db (10.5)
+ int_Omega F(b) q_1(b) db = rho int_Omega F(G(b)) db (56)
-for any integrable F supported in Omega. So the flow carries the UNIFORM
+for any integrable F supported in Omega. So the flow carries the uniform
measure rho db onto the target density q_1 db. The second line is the crucial
one for cubature: applying F to a transported point G(b) and keeping the
weight rho db reproduces int F q_1.
-THE CUBATURE. Because H, S, k are integers (k from Section 8.5, h = 1/k),
-cubes of side h tile Omega. On each cube Q use the EIGHT tensor products of
+The cubature. Because H, S, k are integers (k from the scale choices, h = 1/k),
+cubes of side h tile Omega. On each cube Q use the eight tensor products of
two equal-weight Gauss points per coordinate, at the midpoint plus or minus
h/(2 sqrt(3)). This eight-point rule integrates polynomials of total degree
-at most THREE exactly, by matching the moments on each interval. When we map
+at most three exactly, by matching the moments on each interval. When mapped
these nodes by the flow G, their common weight stays rho h^3 / 8 = 1/lambda
-by (10.1). (The rule's exactness is used in the ORIGINAL cube coordinates,
-on integrands composed with G; see Section 10.2.)
+by (52). (The rule's exactness is used in the original cube coordinates,
+on integrands composed with G; see the form and compression step.)
-THE RATIONAL ROUNDING. Each exact mapped point b_alpha (the image under G of
+The rational rounding. Each exact mapped point b_alpha (the image under G of
a cubature node) will be approximated within Euclidean error at most
- delta_R <= (1+H)^{-3} (1+S)^{-2} lambda^{-3} (10.6)
+ delta_R <= (1+H)^{-3} (1+S)^{-2} lambda^{-3} (57)
-by a RATIONAL point b~_alpha. The physical nuclear positions are then
+by a rational point b~_alpha. The physical nuclear positions are then
x~_alpha = b~_alpha / lambda. Because the exact points have separation
controlled by node separation and the bi-Lipschitz bound of G, and delta_R
-is far smaller than that separation, the rounded points stay DISTINCT.
+is far smaller than that separation, the rounded points stay distinct.
So the data fed to the physical problem are:
- polynomially many (namely 8 |Omega| h^{-3}) rational positions
x~_alpha = b~_alpha / lambda,
- - every one of them a UNIT nucleus (Section 11 explains why the physical
- charge is exactly 1),
+ - every one of them a unit nucleus (the physical charge is then exactly 1),
- the electron number N = m,
- - rational thresholds from Section 12.
+ - rational thresholds from the implementability step.
---------------------------------------------------------------------------------
-10.2 Form and compression estimates (the point-nucleus replacement)
---------------------------------------------------------------------------------
-
-The single-particle operator with the ACTUAL point nuclei is
+The single-particle operator with the actual point nuclei is
L_p = - Delta_y/2 - sum_alpha (1/lambda) / | y - b~_alpha | - C_H - e_* ,
- (10.7)
+ (58)
where b~_alpha are the rounded node positions and the common charge is
1/lambda in dilated units. The difference from L_c is exactly the difference
@@ -1522,79 +1459,85 @@ into cubature error and rounding error:
L_p - L_c = [ rho int_Omega db / |y - G(b)|
- sum_alpha (1/lambda) / | y - b_alpha | ]
+ [ sum_alpha (1/lambda)/|y - b_alpha|
- - sum_alpha (1/lambda)/|y - b~_alpha| ] . (10.8)
+ - sum_alpha (1/lambda)/|y - b~_alpha| ] . (59)
-(The first bracket uses (10.5): rho int_Omega F(G(b)) db = int F q_1, with
+(The first bracket uses (56): rho int_Omega F(G(b)) db = int F q_1, with
F(x) = 1/|y-x|, so the density field equals rho int_Omega db/|y-G(b)|.)
-We now bound each bracket as a form and as a compression.
+Now bound each bracket as a form and as a compression.
-PROPOSITION 10.1 (unit-nucleus replacement).
+Proposition 2 (unit-nucleus replacement).
For the transported degree-three cubature and rational rounding above,
with delta_R <= (1+H)^{-3}(1+S)^{-2}lambda^{-3}, the shifted continuous
and point-nucleus operators L_c, L_p satisfy
| <u, (L_p - L_c) u> | <= p(n,D) lambda^{-2/3} ( ||u||^2 + ||nabla u||^2 ),
- || P (L_p - L_c) P ||_P <= p(n,D) lambda^{-4/3} . (10.9)
+ || P (L_p - L_c) P ||_P <= p(n,D) lambda^{-4/3} . (60)
- The first estimate holds for EVERY one-particle H^1 function u. The
+ The first estimate holds for every one-particle H^1 function u. The
second is the operator norm of the compressed form on the localized-mode
space Ran P. Both bounds are in dilated energy units.
-PROOF.
+Proof.
-Step 1: the cubature error per cube. We work first at the EXACT transported
+Step 1: the cubature error per cube. Start at the exact transported
nodes b_alpha = G(a_{Q,nu}). On an original grid cube Q with Gauss nodes
a_{Q,nu} (1 <= nu <= 8), Taylor's theorem plus exactness through degree three
give the common error estimate
| rho int_Q F(G(b)) db - (rho h^3/8) sum_{nu=1}^8 F(G(a_{Q,nu})) |
- <= C h^7 sup_Q | D^4 (F o G) | , (10.10)
+ <= C h^7 sup_Q | D^4 (F o G) | , (61)
-whenever F o G has four bounded derivatives on Q. (The degree-3 exactness
+whenever F o G has bounded derivatives through order four on Q. (The degree-3 exactness
cancels the Taylor terms of order <= 3; the remainder is order 4, integrated
-over a cube of volume h^3 with eight points.) We apply this (a) to the
-singular Coulomb kernel away from its singularity, then (b) to the SMOOTH
+over a cube of volume h^3 with eight points.) Apply this (a) to the
+singular Coulomb kernel away from its singularity, then (b) to the smooth
fields K_ij coming from the modes (Step 3).
-Step 2: the GENERAL form estimate (first inequality in (10.9)). For fixed
+Step 2: the general form estimate (first inequality in (60)). For fixed
evaluation point y, the charge integral is the kernel |y - G(b)|^{-1} against
-uniform density in b. Split cubes Q into CLOSE ones (whose images touch a
-ball of radius a fixed large multiple C' h about y) and FAR ones.
+uniform density in b. Split cubes Q into close ones (whose images touch a
+ball of radius a fixed large multiple C' h about y) and far ones.
- CLOSE cubes. Only O(1) cubes are close (the images are bi-Lipschitz and
- h small). Their CONTINUOUS contribution to the k=0 term is O(h^2) because
- the transformed density (the Jacobian of G) is bounded. The DISCRETE
+ close cubes. Only O(1) cubes are close (the images are bi-Lipschitz and
+ h small). Their continuous contribution to the k=0 term is O(h^2) because
+ the transformed density (the Jacobian of G) is bounded. The discrete
contribution is sum over close nodes of C h^3 / |y - b_alpha| with
|y - b_alpha| <= C' h, which the Hardy estimate of Step 4 controls exactly
against ||u||^2 + ||nabla u||^2 on balls.
- FAR cubes. The pure fourth derivative of the kernel is bounded by
+Hardy, G. H., Littlewood, J. E., and Polya, G., "Inequalities", Cambridge
+University Press, 1934, Theorem 326. ISBN: 978-0521358804.
+
+Lieb, E. H. and Loss, M., "Analysis", 2nd ed., Graduate Studies in
+Mathematics 14, AMS, 2001. ISBN: 978-0821827833.
+
+ far cubes. The pure fourth derivative of the kernel is bounded by
C|y - G(b)|^{-5}. Within each far cube, distances to y are comparable, and
the bi-Lipschitz change of variables has bounded Jacobian. Summing the
pure fourth-derivative contributions therefore costs at most
- C h^4 int_{|x-y| > ch} |x-y|^{-5} dx = O(h^2) . (10.11)
+ C h^4 int_{|x-y| > ch} |x-y|^{-5} dx = O(h^2) . (62)
The remaining chain-rule terms in D^4(F o G) contain a derivative of G of
order at least two together with a kernel derivative bounded by
- C|y - G(b)|^{-j}, 2 <= j <= 4. Such terms are nonzero ONLY on cells meeting
+ C|y - G(b)|^{-j}, 2 <= j <= 4. Such terms are nonzero only on cells meeting
the well cylinders (combined slab volume O(m(S+1))). Below distance one
they are bounded by the same inverse fifth power; beyond distance one their
total contribution is at most C h^4 m(S+1).
So the pointwise absolute error is at most C(h^2 + h^4 m(S+1)) plus the
-close-node sum. In particular there is NO factor the size of the large
+close-node sum. In particular there is no factor the size of the large
horizontal area in this bound.
-Step 3: the COMPRESSED estimate (second inequality). For compression use the
+Step 3: the compressed estimate (second inequality). For compression use the
smooth Coulomb fields
- K_ij(b) = int |y - b|^{-1} psi_i(y) psi_j(y) dy . (10.12)
+ K_ij(b) = int |y - b|^{-1} psi_i(y) psi_j(y) dy . (63)
Their derivatives through order four are uniformly bounded (differentiate
-the translated densities). The FOURTH derivatives obey the pointwise decay
+the translated densities). The fourth derivatives obey the pointwise decay
- | D_b^4 K_ij(b) | <= C ( 1 + | b - (u_i,0) | )^{-5} . (10.13)
+ | D_b^4 K_ij(b) | <= C ( 1 + | b - (u_i,0) | )^{-5} . (64)
(Indeed derivatives of the product density psi_i psi_j decay exponentially
about the planar center (u_i,0), uniformly because the other factor has
@@ -1603,199 +1546,188 @@ density cutoff supported within d/2 of that center and equal to 1 inside
d/3, holding the cutoff radius fixed while differentiating. There the pure
kernel derivative decays; on the remaining tail, differentiate the cutoff
density instead, giving exponentially small bounds.) Applying the same
-degree-three cubature (10.10) to K_ij o G now costs, for each entry,
+degree-three cubature (61) to K_ij o G now costs, for each entry,
- O( h^4 (1 + m(S+1)) ) , (10.14)
+ O( h^4 (1 + m(S+1)) ) , (65)
-because pure fourth derivatives have summable volume bounds as in (10.11),
+because pure fourth derivatives have summable volume bounds as in (62),
and the other chain-rule terms live only on the exceptional cells. Gram
-correction (via (9.8)) and operator norms cost at most polynomial factors.
-This proves the second inequality of (10.9) for the EXACT nodes.
+correction (via (38)) and operator norms cost at most polynomial factors.
+This proves the second inequality of (60) for the exact nodes.
Step 4: the close-node Hardy bound and rounding. The close-node balls have
bounded overlap also upon any fixed dilation. Around each node one has
int_{|y - b_alpha| <= C'h} |u(y)|^2 / |y - b_alpha| dy
<= C int_{|y - b_alpha| <= 2C'h} ( h^{-1} |u|^2 + h |nabla u|^2 ) .
- (10.15)
+ (66)
-This follows by a cutoff and the THREE-DIMENSIONAL HARDY INEQUALITY
+This follows by a cutoff and the three-dimensional Hardy inequality
int |u|^2 / |y|^2 <= 4 int |nabla u|^2 (which follows from the divergence
-formula for y/|y|^2 together with Cauchy-Schwarz). Summing (10.15) over the
+formula for y/|y|^2 together with Cauchy-Schwarz). Summing (66) over the
bounded-overlap balls gives the general form bound, since h^2 = O(lambda^{-2/3}).
-Finally the ROUNDING (second bracket of (10.8)). A center shift of size
+Finally the rounding (second bracket of (59)). A center shift of size
delta_R changes the inverse distance by at most
- delta_R / ( |y - b_alpha| |y - b~_alpha| ) . (10.16)
+ delta_R / ( |y - b_alpha| |y - b~_alpha| ) . (67)
Cauchy-Schwarz and translated Hardy inequalities bound its form by
4 delta_R ||nabla u||^2. Summing the masses gives the total bound
4 rho |Omega| delta_R ||nabla u||^2 , where
rho |Omega| delta_R <= C H^{-1} S^{-1} lambda^{-3} <= C lambda^{-8}
- (10.17)
+ (68)
(using |Omega| = 8 H^2 S, delta_R <= (1+H)^{-3}(1+S)^{-2} lambda^{-3},
-H = lambda^5, S >= log lambda). This is smaller than BOTH required errors,
+H = lambda^5, S >= log lambda). This is smaller than both required errors,
also after compression, because the modes have polynomial kinetic bounds
-(9.9). This completes (10.9). QED
+(39). This completes (60). QED
-We have obtained the two replacement estimates (10.9) with constants
+This yields the two replacement estimates (60) with constants
independent of the horizontal area of the slab; the full mass enters only
-the rounding bound (10.17), where the chosen coordinate accuracy absorbs it.
-Section 11 now combines these with the fixed complementary gap (9.10) and
-the compressed matrix (9.21) to compare the FULL continuum with H_d.
+the rounding bound (68), where the chosen coordinate accuracy absorbs it.
+These combine with the fixed complementary gap (40) and the compressed
+matrix (51) to compare the full continuum with H_d.
-================================================================================
-11. THE MANY-ELECTRON CONTINUUM COMPARISON
-================================================================================
-
-The point nuclei are now fixed (Section 10). We must compare the electronic
-energy with the finite Hamiltonian H_d on ALL continuum states.
+The point nuclei are now fixed. One must compare the electronic
+energy with the finite Hamiltonian H_d on all continuum states.
In dilated coordinates the full operator after dividing by lambda^2 and
shifting by -m(C_H + e_*) is
Hcal = sum_{l=1}^m (L_p)_l + lambda^{-1} R ,
- R = sum_{1<=l<s<=m} |y_l - y_s|^{-1} , (11.1)
+ R = sum_{1<=l<s<=m} |y_l - y_s|^{-1} , (69)
-with L_p from (10.7). We work on the antisymmetric space WITH spins and its
-H^1 form domain, exactly as in Section 2.
+with L_p from (58). Work on the antisymmetric space with spins and its
+H^1 form domain, exactly as in the physical model.
-THE FORM IS CLOSED AND LOWER BOUNDED. Fiberwise Hardy together with
+The form is closed and lower bounded. Fiberwise Hardy together with
Cauchy-Schwarz gives, for every eta > 0 and every fixed center b,
int |u|^2 / |y_l - b| <= 2 ||u||_2 ||nabla_l u||_2
- <= eta ||nabla_l u||_2^2 + eta^{-1} ||u||_2^2 . (11.2)
+ <= eta ||nabla_l u||_2^2 + eta^{-1} ||u||_2^2 . (70)
The same estimate applies to a pair singularity by fixing the other
-coordinates. For each finite nuclear configuration, summing (11.2) with a
+coordinates. For each finite nuclear configuration, summing (70) with a
sufficiently small eta gives a closed, lower-bounded form whose shifted form
norm is equivalent to the H^1 norm. Smooth compactly supported antisymmetric
-functions form a CORE. So the spectral infimum is the infimum of the form
+functions form a core. So the spectral infimum is the infimum of the form
over normalized H^1 states, and every form estimate below extends by
continuity from the core. No ground eigenvector is assumed.
-THE PROJECTION Pi. Let Pi project onto the subspace in which ALL m particles
+The projection Pi. Let Pi project onto the subspace in which all m particles
lie in P (the m-electron space built from the 2m one-particle spin modes).
Write ||u||_1^2 = ||u||^2 + sum_l ||nabla_l u||^2. Because the modes are
smooth and decaying, they lie in the form domain, so the compression
-(Pi Hcal Pi)|_{Ran Pi} is well defined and equals, by (9.21), the finite
-fermion operator below. The one-particle kinetic bound (9.9) gives
+(Pi Hcal Pi)|_{Ran Pi} is well defined and equals, by (51), the finite
+fermion operator below. The one-particle kinetic bound (39) gives
||w||_1 <= p(n,D) ||w|| for w in Ran Pi.
- E_Pi = min spec( ( Pi Hcal Pi ) |_{Ran Pi} ) . (11.3)
+ E_Pi = min spec( ( Pi Hcal Pi ) |_{Ran Pi} ) . (71)
This is a genuine finite-dimensional minimum, because Pi projects onto the
m-electron space of 2m one-particle spin modes.
---------------------------------------------------------------------------------
-11.1 Control outside the orbital space
---------------------------------------------------------------------------------
-
-IN ONE BREATH. We want a lower bound on the FULL ground energy, but we
-only know the modes well. Split any trial state as w (modes) plus q (the
-rest). The rest costs at least alpha times its H^1 norm; the modes part
-costs at least E_Pi; the only danger is the cross-coupling <q, Hcal w>,
+In one breath. A lower bound is wanted on the full ground energy, but
+only the modes are known well. Split any trial state as w (modes) plus q
+(the rest). The rest costs at least alpha times its H^1 norm; the modes
+part costs at least E_Pi; the only danger is the cross-coupling <q, Hcal w>,
bounded by beta times ||q|| times ||w||. Completing the square in ||q||:
alpha ||q||^2 - 2 beta ||w|| ||q|| >= -beta^2/alpha * ||w||^2. Hence
E_full >= E_Pi - beta^2/alpha, up to polynomial lambda factors.
-We first give the COMPLEMENT a uniform lower bound and bound its coupling to
-the mode space. Then a square completion bounds the FULL infimum.
+The complement first receives a uniform lower bound, and its coupling to
+the mode space is bounded. Then a square completion bounds the full infimum.
-PROPOSITION 11.1 (eliminating the continuum complement).
+Proposition 3 (eliminating the continuum complement).
For the unit-nucleus construction with N = m and sufficiently large
polynomial scale lambda, the full antisymmetric m-electron operator in
dilated and shifted units satisfies
- E_Pi - p(n,D) lambda^{-4/3} <= inf Spec Hcal <= E_Pi . (11.4)
+ E_Pi - p(n,D) lambda^{-4/3} <= inf Spec Hcal <= E_Pi . (72)
- The lower bound holds on EVERY normalized antisymmetric H^1 state,
+ The lower bound holds on every normalized antisymmetric H^1 state,
including every spin sector.
-PROOF.
+Proof.
Step 1: complement lower bound for the one-body part. Write
-H_c^0 = sum_l (L_c)_l. By (9.10), as one-particle forms,
+H_c^0 = sum_l (L_c)_l. By (40), as one-particle forms,
- L_c >= c' (I - P) - p(n,D) lambda^{-0.9} I (11.5)
+ L_c >= c' (I - P) - p(n,D) lambda^{-0.9} I (73)
for some c' > 0 (the diagonal and cross terms involving P are controlled by
||L_c P||). Since sum_l (I - P)_l >= I - Pi (at least one particle outside
-means the product is outside), summing (11.5) gives a FIXED positive lower
-bound for H_c^0 on Ran(I - Pi), once m p(n,D) lambda^{-0.9} is small. We also
-have the kinetic lower bound H_c^0 >= -sum_l Delta_l/2 - C m from (9.10). A
+means the product is outside), summing (73) gives a fixed positive lower
+bound for H_c^0 on Ran(I - Pi), once m p(n,D) lambda^{-0.9} is small.
+Also valid is the kinetic lower bound H_c^0 >= -sum_l Delta_l/2 - C m of
+(40). A
convex combination with a small weight of order 1/m on the kinetic bound keeps
a positive L^2 term and gives
- <q, H_c^0 q> >= (c''/m) ||q||_1^2 , c'' > 0 , (11.6)
+ <q, H_c^0 q> >= (c''/m) ||q||_1^2 , c'' > 0 , (74)
-for q in Ran(I - Pi). By (10.9) on each coordinate and the POSITIVITY of R,
-this estimate persists with HALF the coefficient for the full Hcal.
+for q in Ran(I - Pi). By (60) on each coordinate and the positivity of R,
+this estimate persists with half the coefficient for the full Hcal.
Step 2: the diagonal on the mode space and the mixed block. For
-w in Ran Pi, (9.10) gives ||H_c^0 w|| <= p(n,D) lambda^{-0.9} ||w||. Hardy
+w in Ran Pi, (40) gives ||H_c^0 w|| <= p(n,D) lambda^{-0.9} ||w||. Hardy
applied one particle coordinate at a time gives ||R w|| <= p(n,D) ||w||.
-Polarizing the form bound (10.9) and using the H^1 bound on w controls the
-point-nucleus replacement in the MIXED block. Together,
+Polarizing the form bound (60) and using the H^1 bound on w controls the
+point-nucleus replacement in the mixed block. Together,
- | <q, Hcal w> | <= p(n,D) lambda^{-2/3} ||w|| ||q||_1 , (11.7)
+ | <q, Hcal w> | <= p(n,D) lambda^{-2/3} ||w|| ||q||_1 , (75)
for w in Ran Pi and q in Ran(I - Pi). The same estimates show the compressed
form has norm less than c''/(4m) at sufficiently large scale.
Step 3: square completion. Set alpha = c''/(4m) and let
-beta = p(n,D) lambda^{-2/3} bound (11.7). Decompose any normalized
+beta = p(n,D) lambda^{-2/3} bound (75). Decompose any normalized
form-domain vector as u = w + q, with w = Pi u and q = (I - Pi) u.
-Subtracting E_Pi makes the w-diagonal form NONNEGATIVE and leaves at least
-alpha ||q||_1^2 on the complement (11.6). Hence, all pairings interpreted as
+Subtracting E_Pi makes the w-diagonal form nonnegative and leaves at least
+alpha ||q||_1^2 on the complement (74). Hence, all pairings interpreted as
forms,
<u, Hcal u> - E_Pi ||u||^2
>= alpha ||q||_1^2 - 2 beta ||w|| ||q||_1
>= -(beta^2/alpha) ||w||^2
- >= -p(n,D) lambda^{-4/3} , (11.8)
+ >= -p(n,D) lambda^{-4/3} , (76)
using the elementary inequality a s^2 - 2 b s t >= -b^2/a t^2. This proves
-the LOWER bound in (11.4); the UPPER bound follows by testing a minimizing
+the lower bound in (72); the upper bound follows by testing a minimizing
vector of the compressed operator (which lies in Ran Pi). The lower-bound
-argument applies to EVERY normalized antisymmetric state, in any spin sector,
+argument applies to every normalized antisymmetric state, in any spin sector,
without any binding or ground-eigenvector assumption. QED
---------------------------------------------------------------------------------
-11.2 The compressed energy
---------------------------------------------------------------------------------
-
-It remains to evaluate lambda E_Pi and match it to E(H_d). Section 9.3
+It remains to evaluate lambda E_Pi and match it to E(H_d). the compressed-matrix step
computed the one-particle matrix M on the orthonormal modes to accuracy
-p(n,D) lambda^{-0.1}. Multiplying the compressed replacement error (10.9) by
-lambda costs only p(n,D) lambda^{-1/3}, so the SAME matrix accuracy
+p(n,D) lambda^{-0.1}. Multiplying the compressed replacement error (60) by
+lambda costs only p(n,D) lambda^{-1/3}, so the same matrix accuracy
p(n,D) lambda^{-0.1} holds for the point nuclei in the units of H_d.
-THE PAIR MATRIX. For the repulsion R, the pair matrix on the original modes
+The pair matrix. For the repulsion R, the pair matrix on the original modes
(before Gram correction) has spatial entries from products of two transition
densities psi_i psi_k acting on psi_j psi_l. Keeping only psi_i^2, psi_j^2 as
-the respective densities gives the DIRECT coefficients v_{ij} of (8.4). Any
+the respective densities gives the direct coefficients v_{ij} of (13). Any
unequal indices within a density give an exponentially small error:
- || psi_i psi_k ||_{L^1} = O(lambda^{-0.8}) for i != k , (11.9)
+ || psi_i psi_k ||_{L^1} = O(lambda^{-0.8}) for i != k , (77)
and the Coulomb field of the other density product is uniformly bounded in
absolute value by orbital boundedness and decay. The matrix of direct
interactions in the product basis is bounded, and Gram correction changes
-its entries negligibly (by (9.8)).
+its entries negligibly (by (38)).
-DIMENSION DEPENDENCE. These entrywise estimates give a many-electron
+Dimension dependence. These entrywise estimates give a many-electron
operator-norm error of at most p(n,D) lambda^{-0.1}. To see the dimension
-factor, write the one- and two-body matrices in the 2m ORTHONORMAL spin
+factor, write the one- and two-body matrices in the 2m orthonormal spin
modes. Each coefficient multiplies a monomial c_a^dagger c_b (one-body) or
c_a^dagger c_b^dagger c_d c_c (two-body), whose norm is at most one. There
are only O(m^2) and O(m^4) such monomials, so summing their coefficient
errors costs a polynomial factor over the entire fermion space.
-IDENTIFYING THE RETAINED OPERATOR. In this representation, the retained
+Identifying the retained operator. In this representation, the retained
two-body term is
(U/2) sum_i n_i (n_i - 1) + sum_{i<j} v_{ij} n_i n_j , (11.10a)
@@ -1804,116 +1736,105 @@ and the retained one-body diagonal is
- sum_i ( sum_{j != i} v_{ij} ) n_i = - sum_i s_i n_i . (11.10b)
-At the filling sum_i n_i = m, the identity (C) of Section 8.3 combines these
-two lines into the first line of H_d (8.12), including its scalar
--sum_{i<j} v_{ij}. The hopping matrix M from (9.20) gives the remaining line
+At the filling sum_i n_i = m, the identity (C) of the occupancy-penalty step combines these
+two lines into the first line of H_d (20), including its scalar
+-sum_{i<j} v_{ij}. The hopping matrix M from (50) gives the remaining line
of the calibrated H_d. So the compressed operator is exactly H_d.
-PROPOSITION 11.2 (continuum-to-fermion energy comparison).
+Proposition 4 (continuum-to-fermion energy comparison).
For the constructed unit nuclei with N = m, let Hcal be the dilated and
- shifted continuum operator (11.1) and H_d the finite fermion Hamiltonian
- (8.12). At sufficiently large polynomial scale,
+ shifted continuum operator (69) and H_d the finite fermion Hamiltonian
+ (20). At sufficiently large polynomial scale,
- | lambda inf Spec Hcal - E(H_d) | <= p(n,D) lambda^{-0.1} . (11.11)
+ | lambda inf Spec Hcal - E(H_d) | <= p(n,D) lambda^{-0.1} . (78)
-PROOF. The one- and two-body matrix estimates just obtained compare
+Proof. The one- and two-body matrix estimates just obtained compare
lambda E_Pi with E(H_d) within p(n,D) lambda^{-0.1}: the one-body part
-matches by (9.21), the pair part by (11.9) and (11.10). Multiplying (11.4)
+matches by (51), the pair part by (77) and (79). Multiplying (72)
by lambda contributes only p(n,D) lambda^{-1/3} (because lambda times
lambda^{-4/3} is lambda^{-1/3}), which fits within the bound. QED
-A REMARK ON WHAT IS RETAINED. This analysis treats long-range DIAGONAL
-effects and possible charge transfer at the virtual-excitation scale in
-(8.12)-(8.14) exactly, rather than requiring all intersite Coulomb matrix
-elements to be negligible compared with the input promise gap. That
-retention is what makes the direct Coulomb coefficients v_{ij} genuine and
-keeps the theorem honest about real molecules.
-
-THE ROLES OF THE ERRORS. Table 11.1 records that the different errors play
-different roles; in particular the arbitrary-state form error CANNOT be used
-as a direct energy error after multiplication by lambda.
+The roles of the errors. The table below records that the different
+errors play different roles; in particular the arbitrary-state form error
+cannot be used as a direct energy error after multiplication by lambda.
+----------------------------------+----------------+---------------------+
| Estimate | Error scale | Role |
+----------------------------------+----------------+---------------------+
- | General form (10.9) | lambda^{-2/3} | Mixed blocks, H^1 |
- | Compressed form (10.9) | lambda^{-4/3} | Direct on modes |
- | Complement (11.4) | lambda^{-4/3} | Square of mixed blk |
+ | General form (60) | lambda^{-2/3} | Mixed blocks, H^1 |
+ | Compressed form (60) | lambda^{-4/3} | Direct on modes |
+ | Complement (72) | lambda^{-4/3} | Square of mixed blk |
| Last two x lambda | lambda^{-1/3} | In units of H_d |
- | Total comparison (11.11) | lambda^{-0.1} | + other mode errors |
+ | Total comparison (78) | lambda^{-0.1} | + other mode errors |
+----------------------------------+----------------+---------------------+
The first three rows are in the units of Hcal; the last two in the units
of H_d, up to polynomial factors p(n,D). The O(D^{-2}) and O(S^{-2})
- localization losses only establish a FIXED complementary gap and do not
+ localization losses only establish a fixed complementary gap and do not
enter the energy-precision budget.
---------------------------------------------------------------------------------
-11.3 Physical energies and thresholds
---------------------------------------------------------------------------------
-
-Choose the polynomial scale in lambda large enough that (11.11) costs at
+Choose the polynomial scale in lambda large enough that (78) costs at
most tau^2 gamma/100 and that
- lambda tau^2 gamma >= 8 . (11.12)
+ lambda tau^2 gamma >= 8 . (80)
All earlier smallness, separation, and calibration estimates hold
-simultaneously by the growth choices of Section 8.5. Lifting from the
+simultaneously by the growth choices of the scale choices. Lifting from the
dilated coordinates back to physical coordinates gives the physical
-electronic ground energy EXACTLY as
+electronic ground energy exactly as
- E_0 = lambda^2 ( m(C_H + e_*) + inf Spec Hcal ) . (11.13)
+ E_0 = lambda^2 ( m(C_H + e_*) + inf Spec Hcal ) . (81)
-THE AFFINE MAP. Combine (6.3), (8.21), (11.11) and (11.13). Define
+The affine map. Combine (9), (29), (78) and (81). Define
F(xi) = lambda^2 m (C_H + e_*)
+ lambda [ - sum_{i<j} v_{ij}
- + tau^2 ( xi - C_s - sum_e K_e ) ] . (11.14)
+ + tau^2 ( xi - C_s - sum_e K_e ) ] . (82)
The total error is at most 7 lambda tau^2 gamma/100, and hence at most
lambda tau^2 gamma/8:
- | E_0 - F( E(H_in) ) | <= lambda tau^2 gamma / 8 . (11.15)
+ | E_0 - F( E(H_in) ) | <= lambda tau^2 gamma / 8 . (83)
-Here is where each piece comes from: (6.3) contributes 4 gamma/100 in the
+Here is where each piece comes from: (9) contributes 4 gamma/100 in the
H_in variable (which maps to at most 4 lambda tau^2 gamma/100 through F);
-(8.21) contributes 2 gamma tau^2/100; and (11.11) contributes at most
+(29) contributes 2 gamma tau^2/100; and (78) contributes at most
tau^2 gamma/100 by choice of lambda. Summing gives 7/100 < 1/8.
-THE OUTPUT THRESHOLDS. Take rational numbers a and b approximating
+The output thresholds. Take rational numbers a and b approximating
- a ~ F( a_s + gamma/4 ) , b ~ F( b_s - gamma/4 ) , (11.16)
+ a ~ F( a_s + gamma/4 ) , b ~ F( b_s - gamma/4 ) , (84)
-each within lambda tau^2 gamma/32. (Section 12 shows these are computable
-and have polynomial rational bit length.)
+each within lambda tau^2 gamma/32. The implementability step shows these are computable
+and have polynomial rational bit length.
-YES CASE. Suppose the source is YES: E(H_in) <= a_s. Then
+Yes case. Suppose the source is YES: E(H_in) <= a_s. Then
E_0 <= F(a_s) + lambda tau^2 gamma/8
< F( a_s + gamma/4 ) - lambda tau^2 gamma/32
- <= a . (11.17)
+ <= a . (85)
The middle step uses F(a_s + gamma/4) - F(a_s) = lambda tau^2 gamma/4, which
is bigger than lambda tau^2 gamma/8 + lambda tau^2 gamma/32.
-NO CASE. Suppose the source is NO: E(H_in) >= b_s. Then, symmetrically,
+No case. Suppose the source is NO: E(H_in) >= b_s. Then, symmetrically,
E_0 >= F(b_s) - lambda tau^2 gamma/8
> F( b_s - gamma/4 ) + lambda tau^2 gamma/32
- >= b . (11.18)
+ >= b . (86)
-THRESHOLD SEPARATION. Since F is affine with slope lambda tau^2 and
+Threshold separation. Since F is affine with slope lambda tau^2 and
b_s - a_s >= gamma,
b - a >= lambda tau^2 ( b_s - a_s - gamma/2 ) - lambda tau^2 gamma/16
>= lambda tau^2 (gamma/2) - lambda tau^2 gamma/16
= (7/16) lambda tau^2 gamma
- >= 7/2 > 1 . (11.19)
+ >= 7/2 > 1 . (87)
-using (11.12) in the last line. So both promises survive rational rounding,
+using (80) in the last line. So both promises survive rational rounding,
and the reduction meets the required output separation b - a >= 1.
-WHY THE CHOSEN THRESHOLDS PRESERVE THE PROMISE. The map F carries the
+Why the chosen thresholds preserve the promise. The map F carries the
source energies (in the small source units) to physical continuum energies
by scaling with lambda^2, shifting by the slab constant, subtracting the
direct Coulomb scalar, and scaling the spin part by tau^2. The gaps
@@ -1921,72 +1842,61 @@ gamma/4 inside the arguments absorb the 7/100 errors, and the outer
lambda tau^2 gamma/32 slack absorbs rational rounding. The separation
b - a >= 7/2 then exceeds the required value 1 with room to spare.
-================================================================================
-12. IMPLEMENTABILITY AND ENCODING
-================================================================================
-
-We have specified a nuclear configuration and thresholds. It remains to show
-they are computable in deterministic polynomial time and have polynomial
-encoding length. This section also confirms that all the scalar smallness
-budgets can be met at once, and that the output format matches Section 3.
+A nuclear configuration and thresholds have now been specified. It
+remains to show that they are computable in deterministic polynomial time
+and have polynomial encoding length. The implementability discussion
+also confirms that all the scalar smallness budgets can be met at once, and
+that the output format matches the promise problem.
---------------------------------------------------------------------------------
-12.1 Order and uniformity of the scale choices
---------------------------------------------------------------------------------
-
-The scale choices are fixed in a definite order (already set out in Section
-8.5):
+The scale choices are fixed in a definite order:
1. bounded well data (f, phi, W) and the fixed integer rho, using the
- derivative bound (9.4) - depends only on f, phi, W, zeta, 2;
+ derivative bound (30) - depends only on f, phi, W, zeta, 2;
2. the source cutoff and the three spin-gadget penalties Delta, Delta',
- Delta'' - from n and gamma (Sections 5-6);
+ Delta'' - from n and gamma by the source normalization;
3. the rational tau from the uniform Coulomb coercivity constant c and
- the polynomial size bounds (Section 8.3);
- 4. the integers k = ceil(C n^A), h = 1/k, lambda = 8 k^3/rho (Section
- 8.5), A and C fixed absolute positive integers then large enough.
+ the polynomial size bounds (the occupancy-penalty step);
+ 4. the integers k = ceil(C n^A), h = 1/k, lambda = 8 k^3/rho (the
+ scale choices), A and C fixed absolute positive integers then large
+ enough.
At stage 4, the quantities K_e, 1/K_e, the layout extent, and the b_i already
have fixed polynomial size bounds in n, so they do not affect the choice of
k. Every remaining smallness condition has the form
p(n, log lambda) lambda^{-alpha} <= (inverse polynomial in n) for a fixed
-alpha > 0, and the hopping endpoint brackets of (8.19) have the same
+alpha > 0, and the hopping endpoint brackets of (27) have the same
fixed-power character. A sufficiently large fixed A meets every asymptotic
requirement; increasing C covers all n >= 2 and the fixed separation and gap
conditions.
-UNIFORMITY. The polynomials p(n,D) come from fixed-order derivatives and
+Uniformity. The polynomials p(n,D) come from fixed-order derivatives and
moments, fixed-dimensional wells and gadgets, and polynomially many matrix
entries. The large box is treated separately: its effects are bounded in
-(9.6), by orbital tails, and by summable fourth derivatives in the cubature
+(34), by orbital tails, and by summable fourth derivatives in the cubature
argument. Its total mass enters only the (much smaller) node-rounding error.
So no exponent needs to grow with the input, and H, the box volume |Omega|,
and 1/delta_R are all polynomial in n after A, C are fixed.
---------------------------------------------------------------------------------
-12.2 Numerical evaluation and hopping calibration
---------------------------------------------------------------------------------
-
All required real quantities can be evaluated to inverse-polynomial absolute
accuracy in polynomial time.
-THE LARGE SCALAR SHIFT. The dominating scalar is C_H times lambda^2 m. Since
+The large scalar shift. The dominating scalar is C_H times lambda^2 m. Since
C_H is multiplied by lambda^2 m, evaluating it within
- tau^2 gamma / ( 128 lambda m ) (12.1)
+ tau^2 gamma / ( 128 lambda m ) (88)
uses at most a quarter of the allowed threshold error lambda tau^2 gamma/32.
The other scalar terms can be allocated the remaining budget the same way,
because their amplification factors and their number are polynomially
-bounded. The magnitude |C_H| <= C S H therefore causes NO demand for
+bounded. The magnitude |C_H| <= C S H therefore causes no demand for
exponentially fine absolute precision, only inverse-polynomial.
-THE ORBITAL INTEGRALS. The modes come from phi = (-Delta_r + 1)^{-1} f, so
+The orbital integrals. The modes come from phi = (-Delta_r + 1)^{-1} f, so
in the Gaussian heat formula, substitute t = s^2 and rescale the Gaussian
integration variable. For a derivative D^alpha of any required order,
D^alpha phi(r) = (1/pi) int_0^\infty int_{R^2} 2s e^{-s^2} e^{-|a|^2}
- D^alpha f(r + 2 s a) da ds . (12.2)
+ D^alpha f(r + 2 s a) da ds . (89)
This integrand is regular at s = 0. The needed derivatives of f are bounded
piecewise polynomials with bounded first derivatives. Polynomial integration
@@ -1998,7 +1908,7 @@ only on the unit disk, where phi has a fixed positive lower bound. The other
decaying orbital integrals (normalization, hopping integrals) can likewise be
truncated about their centers with polynomial cutoffs.
-COULOMB INTEGRATIONS. For v_{ij} and C_H, replace |x|^{-1} inside radius
+Coulomb integrations. For v_{ij} and C_H, replace |x|^{-1} inside radius
epsilon by epsilon^{-1}, with epsilon a sufficiently small inverse polynomial.
Integrating the local singularity against bounded densities gives an error of
order epsilon^2 times the relevant polynomial volume bounds. The capped kernel
@@ -2006,36 +1916,32 @@ has Lipschitz constant at most epsilon^{-2}, so grid sums again give the
required accuracy in polynomial time. All integrations have fixed dimension
and fixed nesting depth.
-ELEMENTARY FUNCTIONS. Exponentials, logarithms, square roots, and other
+Elementary functions. Exponentials, logarithms, square roots, and other
elementary constants can be evaluated by rational approximations to the
required accuracy, since arguments and reciprocal tolerances are polynomially
bounded. Square roots at zero are handled by bisection. Guard precision for
the nested evaluations keeps accumulated errors within the allotted budgets.
-HOPPING BISECTION. The link equation (8.19) has strict endpoint brackets by
-(8.18) and the polynomial bounds on its target. In approximate bisection,
+Hopping bisection. The link equation (27) has strict endpoint brackets by
+(26) and the polynomial bounds on its target. In approximate bisection,
evaluate the residual to a small fraction of its per-link tolerance. If the
sign is ambiguous, the residual is already small enough; otherwise keep the
sign-change bracket until its width times the Lipschitz bound is within
tolerance. No derivative lower bound is needed. The fixed layout contractions
-of Section 7 and the rational center approximations can be computed more
+of the planar layout and the rational center approximations can be computed more
accurately than these tolerances require.
---------------------------------------------------------------------------------
-12.3 Flow evaluation and rational coordinates
---------------------------------------------------------------------------------
-
-Once the RATIONAL centers u_i are fixed, their exact real coefficients b_i
-define a reference density and a reference flow G. We approximate the images
+Once the rational centers u_i are fixed, their exact real coefficients b_i
+define a reference density and a reference flow G. The images are approximated
under that flow.
The flow lasts for time one, with uniformly bounded velocities and first
-derivatives in space and time (Section 10.1). Euler stepping with an
+derivatives in space and time (the Moser step). Euler stepping with an
inverse-polynomial time mesh therefore suffices. For mesh nu (reciprocal of
the number of steps), initial error a_0, per-step arithmetic error a_1, and
velocity evaluation error eps_v, the final position error is
- O( a_0 + nu + eps_v + a_1 / nu ) . (12.3)
+ O( a_0 + nu + eps_v + a_1 / nu ) . (90)
Indeed a step multiplies the preceding error by at most 1 + C nu and adds
O(nu^2 + nu eps_v + a_1). Each parameter can be chosen inverse-polynomially
@@ -2044,27 +1950,24 @@ on sufficiently fine rational grids keeps bit lengths polynomial. Near the
support boundaries the specified smooth vanishing and derivative bounds give
the same error control. Numerical evaluation of b_i is part of the velocity
evaluation error, so all numerical and final rational-rounding errors can be
-charged to the single displacement budget delta_R WITHOUT changing the
+charged to the single displacement budget delta_R without changing the
reference density or any cell mass.
-NONE of these operations uses a many-electron wavefunction. Everything is
+None of these operations uses a many-electron wavefunction. Everything is
single-particle quadrature, fixed-dimensional flow integration, and rational
rounding.
---------------------------------------------------------------------------------
-12.4 Output size and encoding
---------------------------------------------------------------------------------
-
There are 8 |Omega| h^{-3} point nuclei, which is polynomial in n, and each
has charge exactly one. Their rational physical positions x~_alpha =
b~_alpha / lambda, the electron number N = m (encoded in unary), and the
-rational thresholds a, b of (11.16) can all be computed with polynomial
-encoding lengths, by Sections 12.1-12.3. Reducing rational outputs to lowest
-terms gives the input format of Section 3.
+rational thresholds a, b of (84) can all be computed with polynomial
+encoding lengths, by the scale order, numerical evaluation, and flow
+evaluation. Reducing rational outputs to lowest terms gives the input
+format of the promise problem.
-To define the reduction on EVERY input, first check the source syntax and the
+To define the reduction on every input, first check the source syntax and the
fixed polynomial bounds used in the construction; if they fail, output a
-fixed instance (ANY instance obeys its own promise vacuously, and the
+fixed instance (any instance obeys its own promise vacuously, and the
fallback does not affect either promised implication). Otherwise run the
deterministic construction above.
@@ -2072,17 +1975,9 @@ This completes the implementability of the reduction: the only outputs are
nuclear positions, electron number, and thresholds, all polynomial-length and
deterministic-polynomial-time computable.
-================================================================================
-13. PUTTING THE PROOF TOGETHER
-================================================================================
-
-We now chain all the pieces and prove the Main Theorem of Section 3.
-
---------------------------------------------------------------------------------
-13.1 The chain of energies
---------------------------------------------------------------------------------
+Now chain all the pieces and prove the Main Theorem of the promise problem.
-For any source instance (Section 5), the construction produces a molecule and
+For any source instance (the source normalization), the construction produces a molecule and
two rational thresholds. The energy relation is a chain, each link within a
stated error budget expressed as a fraction of gamma tau^2 (or gamma) in the
relevant variable:
@@ -2091,58 +1986,50 @@ relevant variable:
gap gamma = min(1, b_s - a_s) > 0. (Sec 5)
(L1) Positive | E(H_+) - E(H_in) + C_s | <= 4 gamma/100 .
- spin. (6.3)
+ spin. (9)
(L2) Wells and | E(H_d) + sum_{i<j} v_{ij}
finite model. - tau^2 ( E(H_+) - sum_e K_e ) | <= 2 gamma tau^2/100 .
- (8.21)
+ (29)
(L3) Continuum | lambda inf Spec Hcal - E(H_d) | <= p(n,D) lambda^{-0.1}
comparison. and the complement bound
E_Pi - p(n,D) lambda^{-4/3} <= inf Spec Hcal <= E_Pi .
- (11.4),(11.11)
+ (72),(78)
(L4) Physical. E_0 = lambda^2 ( m(C_H + e_*) + inf Spec Hcal ) , and
with all errors,
| E_0 - F( E(H_in) ) | <= lambda tau^2 gamma/8 .
- (11.15)
+ (83)
-Here F is the affine map (11.14). Combining (L0)-(L4) gives the SINGLE
-statement we use: for the molecule built from the source instance,
+Here F is the affine map (82). Combining (L0)-(L4) gives the single
+statement used: for the molecule built from the source instance,
- | E_0 - F( E(H_in) ) | <= lambda tau^2 gamma / 8 . (13.1)
+ | E_0 - F( E(H_in) ) | <= lambda tau^2 gamma / 8 . (91)
-Every constant entering (13.1) - lambda, tau, gamma, C_s, sum_e K_e,
+Every constant entering (91) - lambda, tau, gamma, C_s, sum_e K_e,
sum_{i<j} v_{ij}, m, C_H, e_* - is either fixed by the source instance or
-fixed by the scale choices of Sections 8.5 and 12.1; all are computable in
+fixed by the scale choices; all are computable in
deterministic polynomial time to the required inverse-polynomial accuracy
-(Section 12). This is a many-one reduction: one source instance yields one
+This is a many-one reduction: one source instance yields one
molecule plus thresholds.
---------------------------------------------------------------------------------
-13.2 YES and NO are preserved
---------------------------------------------------------------------------------
+The output thresholds are rationals a < b approximating
+F(a_s + gamma/4) and F(b_s - gamma/4), each within lambda tau^2 gamma/32.
-The output thresholds (Section 11.3) are rationals a < b approximating
-F(a_s + gamma/4) and F(b_s - gamma/4) within lambda tau^2 gamma/32.
+ If the source is YES (E(H_in) <= a_s), then (91) and the definition of a
+ give E_0 <= a (85). So the molecule is in the YES set.
- If the source is YES (E(H_in) <= a_s), then (13.1) and the definition of a
- give E_0 <= a (11.17). So the molecule is in the YES set.
+ If the source is NO (E(H_in) >= b_s), then (91) and the definition of b
+ give E_0 >= b (86). So the molecule is in the NO set.
- If the source is NO (E(H_in) >= b_s), then (13.1) and the definition of b
- give E_0 >= b (11.18). So the molecule is in the NO set.
-
-The threshold separation satisfies b - a >= 7/2 > 1 (11.19), so the
-produced instance satisfies the input promise of Section 3 with room to
+The threshold separation satisfies b - a >= 7/2 > 1 (87), so the
+produced instance satisfies the input promise of the promise problem with room to
spare. The energy errors never exceed the slack built into the thresholds,
so no YES instance turns into a NO instance or vice versa.
---------------------------------------------------------------------------------
-13.3 What has been proved
---------------------------------------------------------------------------------
-
-THEOREM (Main; restated from Section 3).
- The electronic promise problem for Hamiltonian (2.1) on the full spinful
+Theorem (main).
+ The electronic promise problem for Hamiltonian (1) on the full spinful
fermionic Hilbert space H_N, with distinct rational nuclear positions, all
nuclear charges equal to one, unary electron number, and rational
thresholds separated by at least L^{-1}, is QMA-hard under deterministic
@@ -2151,41 +2038,41 @@ THEOREM (Main; restated from Section 3).
and threshold separation at least one. The only output data are the
nuclear positions, electron number, and thresholds.
-PROOF. Take any instance of the Cubitt-Montanaro-Piddock signed Heisenberg
-energy problem (Section 5). The following deterministic polynomial-time
+Proof. Take any instance of the Cubitt-Montanaro-Piddock signed Heisenberg
+energy problem (the source normalization). The following deterministic polynomial-time
procedure produces a molecule and thresholds:
- 1. Normalize to rational couplings and thresholds (Section 5), obtaining
+ 1. Normalize to rational couplings and thresholds (the source normalization), obtaining
H_in with gap gamma = min(1, b_s - a_s) > 0.
2. Convert signed to positive exchange with singlet-pair mediators and two
- subdivision rounds (Section 6). This gives H_+ = sum K_e h^s_{ij} with
+ subdivision rounds (the signed-to-positive step). This gives H_+ = sum K_e h^s_{ij} with
positive polynomially-bounded K_e, plus the scalar shift C_s, to within
- 4 gamma/100 (6.3).
+ 4 gamma/100 (9).
- 3. Build the localized modes and finite fermion model (Section 8), with
+ 3. Build the localized modes and finite fermion model (the well construction), with
direct Coulomb coefficients v_{ij}, occupancy penalty, and calibrated
- hoppings chosen by (8.19). This gives H_d related to H_+ by (8.21).
+ hoppings chosen by (27). This gives H_d related to H_+ by (29).
- 4. Construct the nonnegative charge density (Section 9), replacing its
+ 4. Construct the nonnegative charge density (the charge-density step), replacing its
continuous field by unit point nuclei via Moser transport and degree-
- three Gauss cubature (Section 10), with the form and compression bounds
- (10.9).
+ three Gauss cubature (the point-nucleus step), with the form and compression bounds
+ (60).
5. Compare the full continuum infimum with E(H_d) using the complement gap
- and square completion (Section 11.1) and the compressed matrix (11.2),
- giving (11.4) and (11.11).
+ and square completion (the complement-control step) and the compressed matrix (70),
+ giving (72) and (78).
6. Compute the physical energy and thresholds through the affine map F
- (11.14)-(11.16), verifying (11.17)-(11.19) and the threshold gap > 1.
+ (82)-(84), verifying (85)-(87) and the threshold gap > 1.
-Steps 1-6 are all deterministic polynomial time (Section 12), all outputs
+Steps 1-6 are all deterministic polynomial time (the implementability step); all outputs
are polynomially many rational data plus a unary electron number, and all
-are charge-one nuclei at distinct rational positions. By (11.17) and
-(11.18) YES and NO are preserved, so this is a valid reduction. Since the
-source is QMA-hard (Section 5), the electronic problem is QMA-hard. QED
+are charge-one nuclei at distinct rational positions. By (85) and
+(86) YES and NO are preserved, so this is a valid reduction. Since the
+source is QMA-hard (the source normalization), the electronic problem is QMA-hard. QED
-REMARKS. The theorem imposes no neutrality, binding, existence of a ground
+Remarks. The theorem imposes no neutrality, binding, existence of a ground
eigenvector, molecular spectral-gap, or lower nuclear-separation promise. It
supplies no orbital basis, electronic state, magnetic field, or additional
external potential. Membership in QMA is not asserted. The constructed