diff b1f9bc4..56bc728 spin_one_periodic_haldane_v2.txt
diff --git a/spin_one_periodic_haldane_v2.txt b/spin_one_periodic_haldane_v2.txt
index aa0a662..b4bb7ac 100644
--- a/spin_one_periodic_haldane_v2.txt
+++ b/spin_one_periodic_haldane_v2.txt
@@ -30,7 +30,7 @@ maketitle
Abstract:
We prove a uniform positive spectral gap for the antiferromagnetic
spin-one Heisenberg chain on even periodic rings, in the
-$liminf$ (even-periodic) formulation of the spin-one Haldane
+liminf (even-periodic) formulation of the spin-one Haldane
conjecture. The proof combines a spatial transfer operator, a
purity bootstrap that doubles length and inverse temperature
together, and two finite, exactly verified initial data sets.
@@ -46,51 +46,51 @@ for the pure bilinear model on even periodic rings, with bounds that
are uniform in the ring length.
-Let $S^x,S^y,S^zin M_3(C)$ be self-adjoint spin-one matrices with
+Let S^x,S^y,S^zin M_3(C) be self-adjoint spin-one matrices with
[
[S^alpha,S^beta]=itextstylesum_etaε_{alphabetaeta}S^eta,
- qquad sum_alpha(S^alpha)^2=2I_3 .
+ sum_alpha(S^alpha)^2=2I_3 .
]
-For even $Lge4$ put $cH_L=(C^3)^{ ⊗ L}$, and on it
+For even Lge4 put cH_L=(C^3)^{ ⊗ L}, and on it
H_L=sum_{j=0}^{L-1}sum_{alpha=x,y,z}S_j^alpha S_{j+1}^alpha,
- qquad S_L^alpha=S_0^alpha .
+ S_L^alpha=S_0^alpha .
-Let $E_0(L)$ be the least energy, $E_1(L)$ the next distinct energy,
-$gamma_L=E_1(L)-E_0(L)$, and $Delta_1=liminf_{Lto∞,
-L even}gamma_L$. Then the ground state of $H_L$ is unique
-for every even $Lge60$, and
+Let E_0(L) be the least energy, E_1(L) the next distinct energy,
+gamma_L=E_1(L)-E_0(L), and Delta_1=liminf_{Lto∞,
+L even}gamma_L. Then the ground state of H_L is unique
+for every even Lge60, and
gamma_L&>frac4{105}log(80)/(79) && (Lge60 even),
gamma_L&>(log 20)/(784) && (Lge2304 even),
-so $Delta_1gelog(20)/784>0$. The same conclusions hold for any
+so Delta_1gelog(20)/784>0. The same conclusions hold for any
simultaneously similar spin-one realization satisfying the same
commutator and Casimir normalizations.
-The next distinct energy always exists: in the $S^z$ basis the
-all-zero and all-$+1$ product vectors have energy expectations $0$
-and $L$, so $H_L$ is not scalar. Because $H_L$ is self-adjoint and
+The next distinct energy always exists: in the S^z basis the
+all-zero and all-+1 product vectors have energy expectations 0
+and L, so H_L is not scalar. Because H_L is self-adjoint and
non-scalar in finite dimension, its full spectrum has a unique least
-eigenvalue and a next distinct one; $gamma_L$ is the difference.
+eigenvalue and a next distinct one; gamma_L is the difference.
## How the proof works
-A shifted partition function $Z_L(b)=Tr e^{-b(H_L+aLI)}$ can be
+A shifted partition function Z_L(b)=Tr e^{-b(H_L+aLI)} can be
read in two ways at once. Physically it is a sum of positive Gibbs
weights, one per physical eigenvector. Spatially
-(Section~sec:transfer) it is a moment $Tr(X_b^L)$ of a
-self-adjoint, Hilbert--Schmidt operator $X_b$; for even $L$ this is a
+(Section~sec:transfer) it is a moment Tr(X_b^L) of a
+self-adjoint, Hilbert--Schmidt operator X_b; for even L this is a
sum of nonnegative weights, one per spatial eigenvalue. One object,
two probability distributions.
-For a probability distribution $(p_i)$ call $sum_i p_i^2$ its
+For a probability distribution (p_i) call sum_i p_i^2 its
purity; purity near one means one weight carries almost all
the mass. Squaring and renormalizing the weights turns purity defect
-$1-P$ into roughly its square (Lemma~lem:purity). Squaring
+1-P into roughly its square (Lemma~lem:purity). Squaring
physical weights doubles the inverse temperature, while squaring
spatial weights doubles the length. Exact cancellation identities
couple the two improvements (Lemma~lem:rounded-update), so
@@ -102,8 +102,8 @@ Gibbs distribution bounds each excited-to-ground weight ratio, hence
the energy gap.
Two finite initializations supply the seeds. One begins at length
-$60$ and inverse temperature $105/4$, the other at length $2304$ and
-inverse temperature $784$ with a smaller defect, giving the stronger
+60 and inverse temperature 105/4, the other at length 2304 and
+inverse temperature 784 with a smaller defect, giving the stronger
bound. Both rest on short-chain thermal traces
(Appendices~sec:thermal120--sec:thermal72) and one
explicit family of periodic matrix-product trial vectors
@@ -123,9 +123,9 @@ given interaction.
The Affleck--Kennedy--Lieb--Tasaki construction gave a rigorous
gapped spin-one chain with biquadratic interactions
-[cite]. Its bond interaction is $h+tfrac13h^2$ up
-to a scalar, with $h=mathbf S_jbfS_jcdotmathbf S_bfS_{j+1}$, whereas
-(eq:main-H) uses $h$ itself. Yarotsky proved that a unique
+[cite]. Its bond interaction is h+tfrac13h^2 up
+to a scalar, with h=mathbf S_jbfS_jcdotmathbf S_bfS_{j+1}, whereas
+(eq:main-H) uses h itself. Yarotsky proved that a unique
ground state and a uniform
periodic gap persist under sufficiently small translation-invariant
finite-range perturbations of the AKLT point
@@ -133,10 +133,10 @@ cite[arXiv v1, Theorems~1 and~3]{Yarotsky2006}; this controls a
neighborhood, not a gapped path to the pure bilinear point.
For the pure chain, White and Huse's DMRG study estimated a bulk gap
-of $0.41050(2)$ and effective spin-one-half end modes
+of 0.41050(2) and effective spin-one-half end modes
[cite]; neutron scattering found experimental evidence
in the nearly one-dimensional spin-one antiferromagnet
-$CsNiCl_3$ [cite]. Batista and Ortiz proposed a
+CsNiCl_3 [cite]. Batista and Ortiz proposed a
spin--fermion proof [cite]; its gap step uses a
first-order perturbative reduction without a uniform remainder, so it
does not close the argument. Tasaki proved a nontrivial
@@ -176,148 +176,148 @@ We use self-adjoint spin-one matrices throughout; the non-self-adjoint
case reduces to this by the next lemma.
[Representation]
-Let $S^x,S^y,S^zin M_3(C)$ satisfy
+Let S^x,S^y,S^zin M_3(C) satisfy
[
[S^alpha,S^beta]=itextstylesum_gammaε_{alphabetagamma}S^gamma,
- qquad sum_alpha(S^alpha)^2=2I .
+ sum_alpha(S^alpha)^2=2I .
]
They are simultaneously similar to the standard self-adjoint spin-one
matrices; if they are already self-adjoint, the similarity is unitary.
The tensor-power similarity preserves the full spectrum of every
-periodic $H_n$.
+periodic H_n.
Proof.
-Put $S^pm=S^xpm iS^y$. The assumptions give
-
- [S^z,S^pm]=pm S^pm,qquad
- S^-S^+=2I-(S^z)^2-S^z,qquad S^+S^-=2I-(S^z)^2+S^z .
-
-Take an $S^z$-eigenvector $v$ of eigenvalue $m$ with maximal real part.
-Then $S^+v=0$, so $m=1$ or $m=-2$. The case $m=-2$ would permit
-indefinite lowering: a last nonzero lowered vector $w$ has weight
-$-2-k$, while $S^+S^-w=0$ forces weight $-1$ or $2$, contradicting
-finite dimension. Hence $m=1$.
-The vectors $v$, $S^-v/sqrt2$, $(S^-)^2v/2$ are nonzero by
-(eq:ladder-identities) and have distinct weights $1,0,-1$; they
-form a basis, and there is no weight $-2$, so they are the whole
-standard spin-one module with ladder coefficients $sqrt2$. When the
-matrices are self-adjoint and $norm v=1$ the vectors are orthonormal.
-Conjugating each tensor factor conjugates every term of $H_n$, so the
+Put S^pm=S^xpm iS^y. The assumptions give
+
+ [S^z,S^pm]=pm S^pm,
+ S^-S^+=2I-(S^z)^2-S^z, S^+S^-=2I-(S^z)^2+S^z .
+
+Take an S^z-eigenvector v of eigenvalue m with maximal real part.
+Then S^+v=0, so m=1 or m=-2. The case m=-2 would permit
+indefinite lowering: a last nonzero lowered vector w has weight
+-2-k, while S^+S^-w=0 forces weight -1 or 2, contradicting
+finite dimension. Hence m=1.
+The vectors v, S^-v/sqrt2, (S^-)^2v/2 are nonzero by
+(eq:ladder-identities) and have distinct weights 1,0,-1; they
+form a basis, and there is no weight -2, so they are the whole
+standard spin-one module with ladder coefficients sqrt2. When the
+matrices are self-adjoint and norm v=1 the vectors are orthonormal.
+Conjugating each tensor factor conjugates every term of H_n, so the
spectrum is unchanged.
-In the weight basis $|s>$, $sin{-1,0,1}$, the generator is
-$S^z|s>=s|s>$ with ladder entries $sqrt2$. In the
+In the weight basis |s>, sin{-1,0,1}, the generator is
+S^z|s>=s|s> with ladder entries sqrt2. In the
Cartesian basis indexed by axes, the real matrices
[
- (S^alpha)_{ij}=-iε_{alpha i j},qquad G_alpha=iS^alpha
+ (S^alpha)_{ij}=-iε_{alpha i j}, G_alpha=iS^alpha
]
-obey the same relations; each $G_alpha$ is real of operator norm one.
-For $h=sum_alpha S^alpha ⊗ S^alpha$ contraction of the epsilon
+obey the same relations; each G_alpha is real of operator norm one.
+For h=sum_alpha S^alpha ⊗ S^alpha contraction of the epsilon
tensors gives
- -h=sum_alpha G_alpha ⊗ G_alpha,qquad
- h=F-|Ω><Ω|,quad
+ -h=sum_alpha G_alpha ⊗ G_alpha,
+ h=F-|Ω><Ω|,
Ω=sum_{i=1}^3|ii>,
-where $F(u ⊗ v)=v ⊗ u$. Hence $h$ has eigenvalues
-$-2,-1,1$ and $hle I$.
+where F(u ⊗ v)=v ⊗ u. Hence h has eigenvalues
+-2,-1,1 and hle I.
## Twists and partition functions
-Number the sites $0,...,n-1$ with $nge4$. A proper rotation
-$gin SO(3)$ is unitary on $C^3$. Define
+Number the sites 0,...,n-1 with nge4. A proper rotation
+gin SO(3) is unitary on C^3. Define
- H_n(g)=sum_{j=0}^{n-2}h_{j,j+1}+h_{n-1,0}(g),qquad
+ H_n(g)=sum_{j=0}^{n-2}h_{j,j+1}+h_{n-1,0}(g),
h_{n-1,0}(g)=sum_alpha S_{n-1}^alpha,(gS^alpha g^{-1})_0 ,
-so only the last bond is conjugated, on its site-$0$ factor; $H_n(g)$
-is self-adjoint and $H_n(I)=H_n$. With the shift
+so only the last bond is conjugated, on its site-0 factor; H_n(g)
+is self-adjoint and H_n(I)=H_n. With the shift
- a=(700741)/(500000),qquad
- Z_n(b,g)=Trexp[-b(H_n(g)+anI)],qquad Z_n(b)=Z_n(b,I),
+ a=(700741)/(500000),
+ Z_n(b,g)=Trexp[-b(H_n(g)+anI)], Z_n(b)=Z_n(b,I),
-all $Z_n(b,g)$ are strictly positive, and the scalar shift $an$ will
+all Z_n(b,g) are strictly positive, and the scalar shift an will
cancel from every purity ratio and from the final gap.
We need the diagonal rotations
[
- P_x=diag(1,-1,-1),quad
- P_y=diag(-1,1,-1),quad
+ P_x=diag(1,-1,-1),
+ P_y=diag(-1,1,-1),
P_z=diag(-1,-1,1),
]
-$D_2={I,P_x,P_y,P_z}$, and the cyclic rotation
-$C:e_xmapsto e_ymapsto e_zmapsto e_x$; write $P=P_x$. By a global
-cyclic rotation the three nonidentity elements of $D_2$ have equal
+D_2={I,P_x,P_y,P_z}, and the cyclic rotation
+C:e_xmapsto e_ymapsto e_zmapsto e_x; write P=P_x. By a global
+cyclic rotation the three nonidentity elements of D_2 have equal
partition functions. After a global rotation and a change to the
-weight basis, $P$ and $C$ act as $exp(-itheta S^z)$ with
-$theta=pi$ and $theta=2pi/3$.
+weight basis, P and C act as exp(-itheta S^z) with
+theta=pi and theta=2pi/3.
-For later use with explicit matrices, a word $x=(x_0,...,x_{n-1})$
-in the weight basis has diagonal entry $sum_j x_{j-1}x_j$ (indices
-mod $n$); each allowed step changing two neighboring letters by
-opposite units contributes $1$, and a seam step from column $y$ to row
-$x$ contributes the phase
+For later use with explicit matrices, a word x=(x_0,...,x_{n-1})
+in the weight basis has diagonal entry sum_j x_{j-1}x_j (indices
+mod n); each allowed step changing two neighboring letters by
+opposite units contributes 1, and a seam step from column y to row
+x contributes the phase
exp!(itheta(y_0-x_0)),
-since $gS^pm g^{-1}=e^{mp itheta}S^pm$. These formulas tie the
-finite matrices below to the physical $H_n(g)$ in
+since gS^pm g^{-1}=e^{mp itheta}S^pm. These formulas tie the
+finite matrices below to the physical H_n(g) in
(eq:twisted-hamiltonian).
## A uniform energy bound
[Energy bounds]
-For every $nge4$ and $gin SO(3)$, $-(3n)/(2)Ile H_n(g)le nI$.
+For every nge4 and gin SO(3), -(3n)/(2)Ile H_n(g)le nI.
Proof.
-The upper bound is $hle I$ plus unitary conjugation of the seam. For
-the lower bound consider $T=-h_{12}-h_{23}$ on three open sites in
+The upper bound is hle I plus unitary conjugation of the seam. For
+the lower bound consider T=-h_{12}-h_{23} on three open sites in
Cartesian coordinates. By (eq:cartesian-bond) an unequal pair
-swaps with coefficient $-1$, an equal pair changes to either other equal
-pair with coefficient $1$, and there are no diagonal terms. Give each
-three-letter word a positive weight $q$ according to its equality
+swaps with coefficient -1, an equal pair changes to either other equal
+pair with coefficient 1, and there are no diagonal terms. Give each
+three-letter word a positive weight q according to its equality
pattern:
{c|c|c}
-word type & $q$ & absolute weighted row sum hline
-$aaa$ & $4$ & $3+3+3+3=12$
-$aab$ or $abb$ & $3$ & $4+3+2=9$
-$aba$ & $2$ & $3+3=6$
-$abc$ & $2$ & $2+2=4$
+word type & q & absolute weighted row sum
+aaa & 4 & 3+3+3+3=12
+aab or abb & 3 & 4+3+2=9
+aba & 2 & 3+3=6
+abc & 2 & 2+2=4
-Every row sum is at most $3q$. With $D$ the diagonal of these weights,
-$D^{-1}TD$ has absolute row sums at most three, hence spectral radius
-at most three; since $T$ is self-adjoint and similar to this matrix,
-$h_{12}+h_{23}ge-3I$.
+Every row sum is at most 3q. With D the diagonal of these weights,
+D^{-1}TD has absolute row sums at most three, hence spectral radius
+at most three; since T is self-adjoint and similar to this matrix,
+h_{12}+h_{23}ge-3I.
An open triad containing the seam is unitarily equivalent to an
untwisted triad (delete the seam bond and rotate all sites on one side,
using rotation invariance of the internal interaction), so the bound
-holds for every consecutive pair of bonds of $H_n(g)$. Summing the $n$
+holds for every consecutive pair of bonds of H_n(g). Summing the n
triad inequalities counts each bond twice and gives
-$H_n(g)ge-(3n)/(2)I$.
+H_n(g)ge-(3n)/(2)I.
-The trial inputs of Appendix~sec:trials give $E_0(n)+an<0$ at
-$n=72,120$, hence $Z_n(b)>1$ for every $b>0$; those lower bounds are
+The trial inputs of Appendix~sec:trials give E_0(n)+an<0 at
+n=72,120, hence Z_n(b)>1 for every b>0; those lower bounds are
paired with finite thermal upper bounds in Section~sec:initialization.
# Partition functions as spatial moments
-We now construct a self-adjoint spatial operator whose $L$th moment is
+We now construct a self-adjoint spatial operator whose Lth moment is
the physical partition function (eq:shift). This is what turns
even spatial powers into positive weights. The operator itself need
not be positive.
-Let $mathcal GcalGsubset SO(3)$ be the group of proper signed permutation
-matrices (it contains $D_2$ and $C$). For $ginmathcal G$calG$ write
+Let mathcal GcalGsubset SO(3) be the group of proper signed permutation
+matrices (it contains D_2 and C). For ginmathcal GcalG write
[
g e_alpha=ε_g(alpha)e_{sigma_g(alpha)},
- qquad ε_g(alpha)in{-1,1}.
+ ε_g(alpha)in{-1,1}.
]
Rotation covariance of the Cartesian matrices gives
@@ -325,18 +325,18 @@ Rotation covariance of the Cartesian matrices gives
[Spatial transfer]
-For each $b>0$ there are a Hilbert space $mathcal K_b$,calK_b$, a compact
-self-adjoint operator $X_b$, and a unitary representation
-$gmapsto U_g$ of $mathcal G$calG$ commuting with $X_b$, such that
+For each b>0 there are a Hilbert space mathcal K_b,calK_b, a compact
+self-adjoint operator X_b, and a unitary representation
+gmapsto U_g of mathcal GcalG commuting with X_b, such that
- Z_n(b,g)=Tr(X_b^nU_g)qquad(nge4, ginmathcal G).calG).
+ Z_n(b,g)=Tr(X_b^nU_g)(nge4, ginmathcal G).calG).
-Both $X_b$ and the representation commute with complex conjugation, and
-$X_b$ is Hilbert--Schmidt, so $X_b^k$ is trace class for every integer
-$kge2$. If $lambda_i(b)$ are the nonzero eigenvalues, counted with
+Both X_b and the representation commute with complex conjugation, and
+X_b is Hilbert--Schmidt, so X_b^k is trace class for every integer
+kge2. If lambda_i(b) are the nonzero eigenvalues, counted with
multiplicity, then
- sum_i|lambda_i(b)|^k<∞ (kge2),qquad
+ sum_i|lambda_i(b)|^k<∞ (kge2),
Z_n(b)=sum_ilambda_i(b)^n (nge4).
@@ -344,55 +344,55 @@ multiplicity, then
Proof.
Histories and kernel. A bond history is a finite ordered list
[
- x=((t_1,alpha_1),...,(t_k,alpha_k)),quad
+ x=((t_1,alpha_1),...,(t_k,alpha_k)),
0<t_1<...<t_k<b, alpha_jin{x,y,z},
]
-including the empty list. Let $Ω_b$ be the disjoint union of
+including the empty list. Let Ω_b be the disjoint union of
these labeled simplexes, with Lebesgue measure on each and unit mass on
the empty history. The measure is finite:
mu(Ω_b)=sum_{kge0}(3^kb^k)/(k!)=e^{3b},
- qquad mathcal K_b=L^2(Ω_b,calK_b=L^2(Ω_b,mu;C).
-
-For two histories $x,y$, merge their labeled times increasingly and let
-$k_b(x,y)$ be the trace of the resulting time-ordered product of the
-$G_alpha$ (empty pair: the identity, trace three; time collisions: a
-null event, set kernel zero). Since $norm{G_alpha}=1$ the kernel is
-real with $|k_b(x,y)|le3$, and exchanging $x,y$ keeps the merged
-product unchanged, so $k_b(y,x)=k_b(x,y)$. The integral operator
-$K_b$ is thus self-adjoint and Hilbert--Schmidt with
-$norm{K_b}_{HS}^2le9e^{6b}$. Set $X_b=e^{-ab}K_b$; compact
+ mathcal K_b=L^2(Ω_b,calK_b=L^2(Ω_b,mu;C).
+
+For two histories x,y, merge their labeled times increasingly and let
+k_b(x,y) be the trace of the resulting time-ordered product of the
+G_alpha (empty pair: the identity, trace three; time collisions: a
+null event, set kernel zero). Since norm{G_alpha}=1 the kernel is
+real with |k_b(x,y)|le3, and exchanging x,y keeps the merged
+product unchanged, so k_b(y,x)=k_b(x,y). The integral operator
+K_b is thus self-adjoint and Hilbert--Schmidt with
+norm{K_b}_{HS}^2le9e^{6b}. Set X_b=e^{-ab}K_b; compact
self-adjoint spectral theory gives real square-summable eigenvalues,
-and boundedness gives $sum_i|lambda_i|^k<∞$ for $kge2$.
+and boundedness gives sum_i|lambda_i|^k<∞ for kge2.
-Rotations. The permutation $sigma_g$ acts on labels preserving
-times and measure. Put $s_g(x)=prod_{alphain x}ε_g(alpha)$
-and $(U_g f)(x)=s_g(sigma_g^{-1}x)f(sigma_g^{-1}x)$. The cocycle rule
-$s_{gh}(x)=s_g(sigma_hx)s_h(x)$ shows these real unitaries form a
+Rotations. The permutation sigma_g acts on labels preserving
+times and measure. Put s_g(x)=prod_{alphain x}ε_g(alpha)
+and (U_g f)(x)=s_g(sigma_g^{-1}x)f(sigma_g^{-1}x). The cocycle rule
+s_{gh}(x)=s_g(sigma_hx)s_h(x) shows these real unitaries form a
representation. Rotating every matrix inside a site trace, via
(eq:rotation-covariance), gives
-$k_b(sigma_gx,sigma_gy)=s_g(x)s_g(y)k_b(x,y)$, so $U_g$ commutes with
-$K_b$, $X_b$; the kernel of $K_bU_g$ is
+k_b(sigma_gx,sigma_gy)=s_g(x)s_g(y)k_b(x,y), so U_g commutes with
+K_b, X_b; the kernel of K_bU_g is
(K_bU_g)(x,y)=k_b(x,sigma_gy)s_g(y).
-The cycle trace. For $nge2$ both $K_b^{n-1}$ and $K_bU_g$ are
-Hilbert--Schmidt (the iterated kernel of $K_b^{n-1}$ is bounded because
-$k_b$ is bounded and $mu$ finite). The Hilbert--Schmidt product trace
-identity $Tr(AB)=int a(x,y)b(y,x),dmu x,dmu y$ and
+The cycle trace. For nge2 both K_b^{n-1} and K_bU_g are
+Hilbert--Schmidt (the iterated kernel of K_b^{n-1} is bounded because
+k_b is bounded and mu finite). The Hilbert--Schmidt product trace
+identity Tr(AB)=int a(x,y)b(y,x),dmu x,dmu y and
(eq:twisted-kernel) give
Tr(K_b^nU_g)=int_{Ω_b^n}(prod_{j=0}^{n-2}k_b(x_j,x_{j+1}))
k_b(x_{n-1},sigma_gx_0)s_g(x_0),dmu^n ,
with all iterated integrals absolutely integrable, so no diagonal
-formula for $K_b$ is needed.
+formula for K_b is needed.
-Identification. Expand $exp[-bH_n(g)]$ in its ordered-time
+Identification. Expand exp[-bH_n(g)] in its ordered-time
series. By (eq:cartesian-bond) each event picks a bond and one of
-three labels and inserts the corresponding two $G$ factors; at the seam
-the site-$0$ factor is rotated. At total degree $k$ there are $(3n)^k$
+three labels and inserts the corresponding two G factors; at the seam
+the site-0 factor is rotated. At total degree k there are (3n)^k
choices each of tensor norm at most one, so absolute convergence holds:
3^nsum_{kge0}((3nb)^k)/(k!)=3^ne^{3nb}.
@@ -401,28 +401,28 @@ We may regroup by bond; interleavings of bond histories partition the
global time simplex up to null collisions, and at each site the tensor
trace is the kernel on the two incident histories in their original
chronological order. For an explicit seam check, call its history
-$e_0$ and the others $e_1,...,e_{n-1}$ around the ring. The physical
+e_0 and the others e_1,...,e_{n-1} around the ring. The physical
integrand is
[
s_g(e_0)k_b(sigma_ge_0,e_1)k_b(e_1,e_2)... k_b(e_{n-1},e_0),
]
which equals the right side of (eq:cycle-integral) after setting
-$x_0=e_0,x_1=e_{n-1},...,x_{n-1}=e_1$ (spatial enumeration reversed,
+x_0=e_0,x_1=e_{n-1},...,x_{n-1}=e_1 (spatial enumeration reversed,
chronological products untouched). Thus
-$Tr e^{-bH_n(g)}=Tr(K_b^nU_g)$, and multiplying by $e^{-abn}$ gives
+Tr e^{-bH_n(g)}=Tr(K_b^nU_g), and multiplying by e^{-abn} gives
(eq:spatial-trace); the compact trace formula gives
(eq:spatial-moments).
-For even $nge4$, (eq:spatial-trace) supplies the probabilities
+For even nge4, (eq:spatial-trace) supplies the probabilities
- p_i=(|lambda_i(b)|^n)/(Z_n(b)),qquad
- sum_i p_i=1,qquad
+ p_i=(|lambda_i(b)|^n)/(Z_n(b)),
+ sum_i p_i=1,
sum_i p_i^2=frac{Z_{2n}(b)}{Z_n(b)^2}.
Odd moments are also useful for polynomial estimates, but evenness is
essential here. The spatial eigenvalues are distinct from the physical
-energies of $H_n$.
+energies of H_n.
## Rotation sectors
@@ -430,44 +430,44 @@ Small twisted partition functions split the spatial spectrum into a few
lists with known multiplicities; this later isolates a dominant weight.
[Sector moments]
-Fix $b>0$ and let $X_b,U_g$ be as in Proposition~prop:transfer.
+Fix b>0 and let X_b,U_g be as in Proposition~prop:transfer.
There are real eigenvalue lists (with multiplicity) whose integer
-moments $I_k,O_k,N_k$ converge absolutely for $kge2$, with
+moments I_k,O_k,N_k converge absolutely for kge2, with
Z_n(b)&=I_n+2O_n+3N_n,
Z_n(b,P)&=I_n+2O_n-N_n,
Z_n(b,C)&=I_n-O_nqquad(nge4).
-The $I$ list is the restriction to the joint invariant subspace of
-$D_2$ and $C$; the $O$ list occurs twice and the $N$ list three times
+The I list is the restriction to the joint invariant subspace of
+D_2 and C; the O list occurs twice and the N list three times
in the full spatial list. All three moments are nonnegative at even
-order, and $J_n=I_n+2O_n$ is the moment on the $D_2$-invariant subspace.
+order, and J_n=I_n+2O_n is the moment on the D_2-invariant subspace.
Proof.
-The four $D_2$ characters define orthogonal projections
-$Pi_chi=frac14sum_{gin D_2}chi(g)U_g$ commuting with $X_b$ and
-summing to the identity. Conjugation by $U_C$ permutes the three
+The four D_2 characters define orthogonal projections
+Pi_chi=frac14sum_{gin D_2}chi(g)U_g commuting with X_b and
+summing to the identity. Conjugation by U_C permutes the three
nontrivial character spaces cyclically, so their restrictions are
-unitarily equivalent; write $N_k$ for the common moment. In the
-trivial-character space, split by the eigenvalues $1,omega,baromega$
-of $U_C$ ($omega=e^{2pi i/3}$); these reduce $X_b$, complex
+unitarily equivalent; write N_k for the common moment. In the
+trivial-character space, split by the eigenvalues 1,omega,baromega
+of U_C (omega=e^{2pi i/3}); these reduce X_b, complex
conjugation exchanges the last two spaces and preserves each real
-eigenvalue of $X_b$ with multiplicity, so their common moment is $O_k$
-and the first space gives $I_k$. Absolute convergence is
+eigenvalue of X_b with multiplicity, so their common moment is O_k
+and the first space gives I_k. Absolute convergence is
(eq:spatial-moments).
The decomposition gives (eq:sector-total). On the three
-nontrivial spaces a fixed $U_P$ has signs $+1,-1,-1$ and is the identity
-on the trivial space, giving (eq:sector-pi). The operator $U_C$
+nontrivial spaces a fixed U_P has signs +1,-1,-1 and is the identity
+on the trivial space, giving (eq:sector-pi). The operator U_C
permutes the nontrivial spaces (zero contribution) and has coefficients
-$1,omega,baromega$ on the remaining three with
-$omega+baromega=-1$, giving (eq:sector-cyclic). The $J_n$
+1,omega,baromega on the remaining three with
+omega+baromega=-1, giving (eq:sector-cyclic). The J_n
identity is the decomposition of the trivial space, with nonnegativity
at even order.
-The invariant $I$ space need not be one-dimensional; the point is that
+The invariant I space need not be one-dimensional; the point is that
every weight outside it occurs at least twice in the full list, which
identifies a unique largest weight without discarding any sector.
@@ -483,46 +483,46 @@ By Proposition~prop:transfer, each partition function is both a
physical Gibbs trace (counting every physical eigenvector with
multiplicity) and, at even lengths, a nonnegative spatial moment
(counting absolute powers of every spatial eigenvalue). Define, for
-even $nge4$,
+even nge4,
- S(n,b)=frac{Z_{2n}(b)}{Z_n(b)^2},qquad
+ S(n,b)=frac{Z_{2n}(b)}{Z_n(b)^2},
T(n,b)=(Z_n(2b))/(Z_n(b)^2).
-The shift $aL$ cancels from both ratios. First, the inequality behind
+The shift aL cancels from both ratios. First, the inequality behind
their improvement under squaring.
[Squaring probabilities]
-Let $(w_i)$ be a finite or countable family of nonnegative numbers
-with $sum_iw_i=1$ and purity $P=sum_iw_i^2in(0,1]$. The normalized
-squares $w_i^2/P$ have purity $P'$ with
+Let (w_i) be a finite or countable family of nonnegative numbers
+with sum_iw_i=1 and purity P=sum_iw_i^2in(0,1]. The normalized
+squares w_i^2/P have purity P' with
- 1-P'le f(1-P),qquad f(u)=(u^2)/(2(1-u)^2) (0le u<1),
+ 1-P'le f(1-P), f(u)=(u^2)/(2(1-u)^2) (0le u<1),
-and $f$ is increasing. Consequently
+and f is increasing. Consequently
- 0<S(n,b),T(n,b)le1,quad
- 1-S(2n,b)le f(1-S(n,b)),quad
+ 0<S(n,b),T(n,b)le1,
+ 1-S(2n,b)le f(1-S(n,b)),
1-T(n,2b)le f(1-T(n,b)).
Proof.
-Some weight is positive, so $P>0$, and $w_i^2le w_i$ gives $Ple1$.
+Some weight is positive, so P>0, and w_i^2le w_i gives Ple1.
Nonnegativity allows rearrangement of all countable sums, so
[
1-P'=frac{2sum_{i<j}w_i^2w_j^2}{P^2}
lefrac{2(sum_{i<j}w_iw_j)^2}{P^2}
=((1-P)^2)/(2P^2),
]
-and $f'(u)=u/(1-u)^3ge0$. For $S(n,b)$ use the spatial
-probabilities $|lambda_i(b)|^n/Z_n(b)$ of
-(eq:spatial-probabilities): squaring them changes $n$ to $2n$.
-For $T(n,b)$ list all physical eigenvalues $ε_k(n)$ with
+and f'(u)=u/(1-u)^3ge0. For S(n,b) use the spatial
+probabilities |lambda_i(b)|^n/Z_n(b) of
+(eq:spatial-probabilities): squaring them changes n to 2n.
+For T(n,b) list all physical eigenvalues ε_k(n) with
multiplicity and use
w_k(n,b)=frac{e^{-b(ε_k(n)+an)}}{Z_n(b)} .
-Their purity is $T(n,b)$, and squaring them changes $b$ to $2b$.
+Their purity is T(n,b), and squaring them changes b to 2b.
## The coupled update
@@ -539,27 +539,27 @@ Both denominators are at most one, and the following form lets the two
initial bounds differ and be rounded upward.
[Coupled update]
-Suppose $nge4$ even, $b>0$, and $p,qin[0,1)$ with
-$1-S(n,b)le p$, $1-T(2n,b)le q$. If $p_+,q_+in[0,1)$ obey
+Suppose nge4 even, b>0, and p,qin[0,1) with
+1-S(n,b)le p, 1-T(2n,b)le q. If p_+,q_+in[0,1) obey
- p_+ge f(1-(1-p)^2(1-q)),qquad
+ p_+ge f(1-(1-p)^2(1-q)),
q_+ge f(1-(1-q)^2(1-p_+)),
-then $1-S(2n,2b)le p_+$ and $1-T(4n,2b)le q_+$.
+then 1-S(2n,2b)le p_+ and 1-T(4n,2b)le q_+.
Proof.
Equation (eq:space-time-cancel) gives
-$S(n,2b)ge(1-p)^2(1-q)$; spatial squaring in Lemma~lem:purity
-and monotonicity of $f$ prove the first bound. Next
-(eq:time-space-cancel) gives $T(4n,b)ge(1-q)^2(1-p_+)$, and
-temporal squaring proves the second. Every input of $f$ lies in $[0,1)$
+S(n,2b)ge(1-p)^2(1-q); spatial squaring in Lemma~lem:purity
+and monotonicity of f prove the first bound. Next
+(eq:time-space-cancel) gives T(4n,b)ge(1-q)^2(1-p_+), and
+temporal squaring proves the second. Every input of f lies in [0,1)
because the product subtracted from one is positive and at most one.
In words: round the first output up, feed it into the second, round
-that up. The pair based at $(n,b)$ becomes a pair based at $(2n,2b)$
-whose physical member sits at length $4n$; neither bound has to
+that up. The pair based at (n,b) becomes a pair based at (2n,2b)
+whose physical member sits at length 4n; neither bound has to
decrease.
## Quadratic decay and intermediate lengths
@@ -568,40 +568,40 @@ Once both defects are small one common bound suffices. Set
G(u)=frac12((1-u)^{-1}+(1-u)^{-2}+(1-u)^{-3})^2,
-increasing on $[0,1)$, with
+increasing on [0,1), with
f(1-(1-u)^3)=G(u)u^2 .
[Uniform gap from two initial purities]
-For every even $Lge4$ let $H_L$ be a non-scalar finite-dimensional
-self-adjoint Hamiltonian, and suppose that $ainR$ and, for each
-$b>0$, a real finite or countable list $(lambda_i(b))$ satisfy
+For every even Lge4 let H_L be a non-scalar finite-dimensional
+self-adjoint Hamiltonian, and suppose that ainR and, for each
+b>0, a real finite or countable list (lambda_i(b)) satisfy
[
- sum_i|lambda_i(b)|^4<∞,qquad
+ sum_i|lambda_i(b)|^4<∞,
Z_L(b):=Tr e^{-b(H_L+aLI)}=sum_i|lambda_i(b)|^L
- quad(Lge4 even).
+ (Lge4 even).
]
-Define $S,T$ by (eq:purities) and $G$ by
-(eq:bootstrap-coefficient). Let $n_0ge4$ even, $b_0>0$, and let
-$u_0,C$ satisfy
+Define S,T by (eq:purities) and G by
+(eq:bootstrap-coefficient). Let n_0ge4 even, b_0>0, and let
+u_0,C satisfy
- 0<u_0<tfrac16,quad G(u_0)le C,quad r:=Cu_0<1,quad C(1-3u_0)>3,
+ 0<u_0<tfrac16, G(u_0)le C, r:=Cu_0<1, C(1-3u_0)>3,
with seeds
- S(n_0,b_0)ge1-u_0,qquad T(2n_0,b_0)ge1-u_0 .
+ S(n_0,b_0)ge1-u_0, T(2n_0,b_0)ge1-u_0 .
-For $jge0$ set
+For jge0 set
- n_j=2^jn_0,quad b_j=2^jb_0,quad u_j=C^{-1}r^{2^j}.
+ n_j=2^jn_0, b_j=2^jb_0, u_j=C^{-1}r^{2^j}.
Then
- S(n_j,b_j)ge1-u_j,qquad T(2n_j,b_j)ge1-u_j,
+ S(n_j,b_j)ge1-u_j, T(2n_j,b_j)ge1-u_j,
-and for every even $Lge n_0$ the physical ground state is unique with
+and for every even Lge n_0 the physical ground state is unique with
gap
gamma_L>(log(1/r))/(b_0).
@@ -609,43 +609,43 @@ gap
Proof.
-Doubling. By definition $u_{j+1}=Cu_j^2$, $0<u_{j+1}<u_jle
-u_0$. Suppose both defects at $(n_j,b_j)$ are at most $u=u_j$. The
+Doubling. By definition u_{j+1}=Cu_j^2, 0<u_{j+1}<u_jle
+u_0. Suppose both defects at (n_j,b_j) are at most u=u_j. The
first bound in Lemma~lem:rounded-update is at most
[
f(1-(1-u)^3)=G(u)u^2le Cu^2=u_{j+1}le u .
]
-Use $p_+=u_{j+1}$ in the second update; its input is at most
-$1-(1-u)^3$, so its output is also at most $Cu^2=u_{j+1}$. Induction
+Use p_+=u_{j+1} in the second update; its input is at most
+1-(1-u)^3, so its output is also at most Cu^2=u_{j+1}. Induction
gives (eq:dyadic-purities).
-Intermediate even lengths. For a nonnegative list $(a_i)$ with
-finite positive $p$th-power sum and $qge p>0$, normalization gives
-$sum a_i^qle(sum a_i^p)^{q/p}$ (the normalized $p$th powers are
-probabilities, whose $q/p$ powers sum to at most one). Applied at the
-two temperatures, whenever $nle Lle2n$ with both even,
+Intermediate even lengths. For a nonnegative list (a_i) with
+finite positive pth-power sum and qge p>0, normalization gives
+sum a_i^qle(sum a_i^p)^{q/p} (the normalized pth powers are
+probabilities, whose q/p powers sum to at most one). Applied at the
+two temperatures, whenever nle Lle2n with both even,
[
- Z_L(b)le Z_n(b)^{L/n},qquad
+ Z_L(b)le Z_n(b)^{L/n},
Z_L(2b)ge Z_{2n}(2b)^{L/(2n)},
]
which combine to
T(L,b)ge[T(2n,b)S(n,b)^2]^{L/(2n)} .
-The bracket lies in $(0,1]$ and the exponent in $[1/2,1]$. For
-$n=n_j$, $b=b_j$:
+The bracket lies in (0,1] and the exponent in [1/2,1]. For
+n=n_j, b=b_j:
T(L,b_j)ge(1-u_j)^3ge1-3u_j>tfrac12
- qquad(n_jle Lle2n_j, L even).
+ (n_jle Lle2n_j, L even).
The physical spectrum. The largest weight in
(eq:gibbs-weights) is at least its purity, since
-$sum_kw_k^2le(max_kw_k)sum_kw_k$, hence $>1/2$. These weights count
+sum_kw_k^2le(max_kw_k)sum_kw_k, hence >1/2. These weights count
all physical eigenvectors, so two ground-state eigenvectors would give
two equal maximal weights, impossible: the ground state is unique.
-Since $H_L$ is non-scalar and self-adjoint in finite dimension, a next
-distinct energy exists. Let $w_0$ be the ground weight and $w_1$ the
+Since H_L is non-scalar and self-adjoint in finite dimension, a next
+distinct energy exists. Let w_0 be the ground weight and w_1 the
weight of an eigenvector at that next energy. Then
e^{-b_jgamma_L}
@@ -657,8 +657,8 @@ weight of an eigenvector at that next energy. Then
using the shift cancellation and (eq:bootstrap-hypotheses).
-Taking logarithms and $b_j=2^jb_0$ gives (eq:bootstrap-gap) on
-$n_jle Lle2n_j$; the dyadic intervals cover every even $Lge n_0$.
+Taking logarithms and b_j=2^jb_0 gives (eq:bootstrap-gap) on
+n_jle Lle2n_j; the dyadic intervals cover every even Lge n_0.
## The two parameter choices
@@ -666,39 +666,39 @@ $n_jle Lle2n_j$; the dyadic intervals cover every even $Lge n_0$.
We will prove the seeds (eq:bootstrap-seeds) for
{c|c|c|c|c}
- n_0&b_0&u_0&C&r=Cu_0 hline
+ n_0&b_0&u_0&C&r=Cu_0
60&105/4&1/8&79/10&79/80
2304&784&1/100&5&1/20
The scalar hypotheses are checked directly. For the first row,
-$G(1/8)=913952/117649<79/10$ and
-$3/[C(1-3u_0)]=48/79<1$. For the second, $1-(1-u)^3le3u$ gives for
-$0<ule1/100$
+G(1/8)=913952/117649<79/10 and
+3/[C(1-3u_0)]=48/79<1. For the second, 1-(1-u)^3le3u gives for
+0<ule1/100
[
- G(u)lefrac9{2(1-3u)^2}le(45000)/(9409)<5,qquad
+ G(u)lefrac9{2(1-3u)^2}le(45000)/(9409)<5,
frac3{C(1-3u_0)}=(60)/(97)<1 .
]
-So the two rows give the bounds $(4/105)log(80/79)$ and $(log20)/784$
+So the two rows give the bounds (4/105)log(80/79) and (log20)/784
of (eq:bootstrap-gap), once their seeds hold. The second row will
come from six applications of Lemma~lem:rounded-update to bounds
-based at $(36,49/4)$. Everything left is finite: establish the two seed
+based at (36,49/4). Everything left is finite: establish the two seed
inequalities without losing the margins these updates need.
# Two finite initializations
-The bootstrap needs a spatial purity at length $n$ and a physical
-purity at length $2n$, at the same inverse temperature. We build two
-such seed pairs. The first starts at length $60$; the second reaches
-length $2304$ and gives the stronger eventual gap. Each seed rests on
+The bootstrap needs a spatial purity at length n and a physical
+purity at length 2n, at the same inverse temperature. We build two
+such seed pairs. The first starts at length 60; the second reaches
+length 2304 and gives the stronger eventual gap. Each seed rests on
a few short-chain partition functions and one periodic trial vector.
## Thermal data and moment bounds
-Write $J_n=I_n+2O_n$ for the $D_2$-invariant moment, $I_n$ for the
-moment also fixed by $C$, and $N_n$ for one nontrivial $D_2$ sector
-(Lemma~lem:sectors). Since each $O_n$ occurs twice and each
-$N_n$ three times,
+Write J_n=I_n+2O_n for the D_2-invariant moment, I_n for the
+moment also fixed by C, and N_n for one nontrivial D_2 sector
+(Lemma~lem:sectors). Since each O_n occurs twice and each
+N_n three times,
J_n&=frac{Z_n+3Z_n(P)}4,& N_n&=frac{Z_n-Z_n(P)}4,
@@ -709,21 +709,21 @@ $N_n$ three times,
all at one common inverse temperature.
[Finite thermal inputs]
-For $a=700741/500000$ the tables below give centers of the shifted
-$Z_n(b,g)$ in units $10^{-6}$, with absolute error below $10^{-5}$ at
-$b=21/2$ and below $30cdot10^{-6}$ at $b=49/4$:
+For a=700741/500000 the tables below give centers of the shifted
+Z_n(b,g) in units 10^{-6}, with absolute error below 10^{-5} at
+b=21/2 and below 30cdot10^{-6} at b=49/4:
[
{c|rrr}
&multicolumn{3}{c}{b=21/2}
- n&g=1&g=P&g=C hline
+ n&g=1&g=P&g=C
6&8945303&346734&939987
8&3741281&567167&984108
10&2327547&721090&1002613
-qquad
+
{c|rr}
&multicolumn{2}{c}{b=49/4}
- n&g=1&g=P hline
+ n&g=1&g=P
4&124885379&65757
5&2925&2960742
6&12870614&286599
@@ -742,49 +742,49 @@ tables by rational enclosures of the physical matrix exponentials with
explicit error bounds; they are rigorous, not numerical estimates. The
other finite input is Lemma~lem:trial-inputs:
- E_0(72)<-72a,qquad E_0(120)<-120a,
+ E_0(72)<-72a, E_0(120)<-120a,
-so $Z_{72}(b)>1$ and $Z_{120}(b)>1$ for every $b>0$ (one Boltzmann
+so Z_{72}(b)>1 and Z_{120}(b)>1 for every b>0 (one Boltzmann
factor already exceeds one).
Two elementary lemmas convert the tables into spectral bounds.
[Capped masses]
-Let $(t_i)$ be finite or countable with $0le t_ile h$ and
-$sum_it_ile m$, where $h>0, mge0$. For $rge1$ put
+Let (t_i) be finite or countable with 0le t_ile h and
+sum_it_ile m, where h>0, mge0. For rge1 put
- mathcal C(m,h,r)=qh^r+(m-qh)^r,calC(m,h,r)=qh^r+(m-qh)^r,qquad q=lfloor m/hrfloor.
+ mathcal C(m,h,r)=qh^r+(m-qh)^r,calC(m,h,r)=qh^r+(m-qh)^r, q=lfloor m/hrfloor.
-Then $sum_it_i^rlemathcal C(m,h,r)$;calC(m,h,r)$; without the cap,
-$sum_it_i^rle m^r$.
+Then sum_it_i^rlemathcal C(m,h,r);calC(m,h,r); without the cap,
+sum_it_i^rle m^r.
Proof.
For a finite family at fixed total mass, move mass from the smaller of
-two coordinates in $(0,h)$ to the larger until one reaches $0$ or $h$;
-convexity of $xmapsto x^r$ shows the sum of $r$th powers does not
-decrease. Repetition leaves at most one coordinate in $(0,h)$, giving
+two coordinates in (0,h) to the larger until one reaches 0 or h;
+convexity of xmapsto x^r shows the sum of rth powers does not
+decrease. Repetition leaves at most one coordinate in (0,h), giving
(eq:capped-function) at the actual mass; the right side is
continuous and nondecreasing in the mass, so bounded by its value at
-$m$. Take increasing unions of finite subfamilies for the countable
-case. The uncapped bound follows from $t_i^rle t_i(sum_jt_j)^{r-1}$.
+m. Take increasing unions of finite subfamilies for the countable
+case. The uncapped bound follows from t_i^rle t_i(sum_jt_j)^{r-1}.
[Polynomial filter]
-Let $εge0$, $v>0$, and let $(lambda_i)$ be a real eigenvalue
-list with moments $M_j=sum_ilambda_i^j$ converging absolutely, and
-$|M_j-c_j|leε$ for the exponents occurring in a polynomial
-$F(x)=sum_jf_jx^jge0$ on $R$. If
+Let εge0, v>0, and let (lambda_i) be a real eigenvalue
+list with moments M_j=sum_ilambda_i^j converging absolutely, and
+|M_j-c_j|leε for the exponents occurring in a polynomial
+F(x)=sum_jf_jx^jge0 on R. If
[
B=sum_jf_jc_j+εsum_j|f_j|
quadandquad F(x)>B for |x|ge v,
]
-then every $|lambda_i|<v$.
+then every |lambda_i|<v.
Proof.
-Absolute convergence gives $sum_iF(lambda_i)le B$; nonnegativity
-means a single $|lambda_i|ge v$ would contradict this.
+Absolute convergence gives sum_iF(lambda_i)le B; nonnegativity
+means a single |lambda_i|ge v would contradict this.
All moments converge by Proposition~prop:transfer, and the
@@ -795,204 +795,204 @@ moment error (odd moments included, with signs kept in the filter).
## The initialization at length sixty
-At inverse temperature $b=105/4$, $S(60,b)>7/8$ and $T(120,b)>7/8$.
+At inverse temperature b=105/4, S(60,b)>7/8 and T(120,b)>7/8.
Proof.
-Work at $b_0=21/2$. From the $b_0=21/2$ table and
+Work at b_0=21/2. From the b_0=21/2 table and
(eq:sector-inversion) the tenth moments obey
[
- m_I=(1042655)/(10^6),quad m_O=(40041)/(10^6),quad
+ m_I=(1042655)/(10^6), m_O=(40041)/(10^6),
m_N=(401625)/(10^6),
]
-with margins $19/(12cdot10^6)$, $7/(12cdot10^6)$,
-$3/(4cdot10^6)$ over centers plus $10^{-5}$. Apply
+with margins 19/(12cdot10^6), 7/(12cdot10^6),
+3/(4cdot10^6) over centers plus 10^{-5}. Apply
Lemma~lem:polynomial-filter to
[
- F_I(x)=x^6(-2+10x^2)^2,qquad F_N(x)=x^6(-25+100x^2)^2 .
+ F_I(x)=x^6(-2+10x^2)^2, F_N(x)=x^6(-25+100x^2)^2 .
]
-For a filter $x^6(c_0+c_1x^2)^2$ the summed upper bound is the centered
-combination of moments $6,8,10$ with coefficients $c_0^2,2c_0c_1,c_1^2$
-plus $10^{-5}(|c_0|+|c_1|)^2$. Rational evaluation gives
+For a filter x^6(c_0+c_1x^2)^2 the summed upper bound is the centered
+combination of moments 6,8,10 with coefficients c_0^2,2c_0c_1,c_1^2
+plus 10^{-5}(|c_0|+|c_1|)^2. Rational evaluation gives
[
- |lambda|<U:=(100245)/(100000) (I sector),qquad
+ |lambda|<U:=(100245)/(100000) (I sector),
|lambda|<V:=(88643)/(100000) (N sector),
]
since the filter at the endpoint exceeds its bound by more than
-$0.0301$ ($I$) and $0.3463$ ($N$), and each filter increases with $|x|$
-beyond its endpoint ($c_1>0$, $c_0+c_1x^2>0$ there).
+0.0301 (I) and 0.3463 (N), and each filter increases with |x|
+beyond its endpoint (c_1>0, c_0+c_1x^2>0 there).
-The inequalities $U^{10}<m_I<2U^{10}$ and $V^{10}<m_N<2V^{10}$ allow the
-two-piece (eq:capped-function). For $k=30,120$ set
+The inequalities U^{10}<m_I<2U^{10} and V^{10}<m_N<2V^{10} allow the
+two-piece (eq:capped-function). For k=30,120 set
[
- A_k=U^k+(m_I-U^{10})^{k/10},quad B_k=m_O^{k/10},quad
+ A_k=U^k+(m_I-U^{10})^{k/10}, B_k=m_O^{k/10},
C_k=V^k+(m_N-V^{10})^{k/10},
]
-bounding $I_k,O_k,N_k$. These are even moments, hence nonnegative, so
+bounding I_k,O_k,N_k. These are even moments, hence nonnegative, so
- Z_k(b_0)le A_k+2B_k+3C_k,qquad
+ Z_k(b_0)le A_k+2B_k+3C_k,
Z_k(b_0,P),Z_k(b_0,C)le A_k+2B_k .
-Put $m=A_{30}+2B_{30}+3C_{30}$, $p=A_{30}+2B_{30}$,
-$q=A_{120}+2B_{120}+3C_{120}$, and
+Put m=A_{30}+2B_{30}+3C_{30}, p=A_{30}+2B_{30},
+q=A_{120}+2B_{120}+3C_{120}, and
[
- D=(1450)/(1000),quad E=(1202)/(1000),quad
- tau=(1225)/(1000),quadtau_0=(1264)/(1000),quad
+ D=(1450)/(1000), E=(1202)/(1000),
+ tau=(1225)/(1000),quadtau_0=(1264)/(1000),
R_q=(684)/(10000).
]
The scalar estimates are
- m^5<D^2,quad p^5<E^2,quad qge1,quad (q-1)^5<R_q^2,quad
- (D+11E)/(12)letau,quad (D+3E)/(4)letau_0,
+ m^5<D^2, p^5<E^2, qge1, (q-1)^5<R_q^2,
+ (D+11E)/(12)letau, (D+3E)/(4)letau_0,
-with positive margins exceeding $0.001068$, $0.000433$,
-$0.0000472$ in the first, second and fourth strict inequalities; all
+with positive margins exceeding 0.001068, 0.000433,
+0.0000472 in the first, second and fourth strict inequalities; all
numbers are rational and stated above.
-Raise the inverse temperature to $b=(5/2)b_0$. For positive Boltzmann
-factors $z_i$, $sum_i z_i^{5/2}le(sum_iz_i)^{5/2}$, so
+Raise the inverse temperature to b=(5/2)b_0. For positive Boltzmann
+factors z_i, sum_i z_i^{5/2}le(sum_iz_i)^{5/2}, so
(eq:short-moment-upper)--(eq:seed60-scalars) give
[
- Z_{30}(b)le D,qquad Z_{30}(b,P),Z_{30}(b,C)le E ;
+ Z_{30}(b)le D, Z_{30}(b,P),Z_{30}(b,C)le E ;
]
-by (eq:sector-inversion) the $I$ and $J$ portions of the thirtieth
-spatial moment are at most $tau$ and $tau_0$.
-
-Let $w_i=|lambda_i(X_b)|^{30}$, so $sum_iw_ile D$, while
-(eq:trial-thermal-consequence) gives $sum_iw_i^4=Z_{120}(b)>1$.
-Since $D(0.88)^3<1$, some $w_0>0.88$ exists, and it is unique because
-$2(0.88)>D$. It lies in $I$: an $O$ weight occurs at least twice and an
-$N$ weight at least three times (counting multiplicities, including
+by (eq:sector-inversion) the I and J portions of the thirtieth
+spatial moment are at most tau and tau_0.
+
+Let w_i=|lambda_i(X_b)|^{30}, so sum_iw_ile D, while
+(eq:trial-thermal-consequence) gives sum_iw_i^4=Z_{120}(b)>1.
+Since D(0.88)^3<1, some w_0>0.88 exists, and it is unique because
+2(0.88)>D. It lies in I: an O weight occurs at least twice and an
+N weight at least three times (counting multiplicities, including
equal moduli of opposite-sign spatial eigenvalues).
-To sharpen $w_0$, suppose $0le tle0.998$ and $w_0ge t$. Let $x$ be
-the remaining $I$ mass, $y$ the full $O$-pair mass, $z$ the full $N$-triple
-mass; their squared weights sum to at most $x^2+y^2/2+z^2/3$ with
+To sharpen w_0, suppose 0le tle0.998 and w_0ge t. Let x be
+the remaining I mass, y the full O-pair mass, z the full N-triple
+mass; their squared weights sum to at most x^2+y^2/2+z^2/3 with
[
- 0le xletau-t,quad 0le y,quad
- x+yletau_0-t,quad zle D-t-x-y .
+ 0le xletau-t, 0le y,
+ x+yletau_0-t, zle D-t-x-y .
]
-The divisors use identical repeated lists. Since $D>tau_0$ the last
+The divisors use identical repeated lists. Since D>tau_0 the last
bound is nonnegative on the whole polygon, and the convex quadratic
-$x^2+y^2/2+(D-t-x-y)^2/3$ attains its maximum at a vertex:
+x^2+y^2/2+(D-t-x-y)^2/3 attains its maximum at a vertex:
R(t)=max_{(x,y)inmathcal V_t}calV_t
- (x^2+(y^2)/(2)+((D-t-x-y)^2)/(3)),quad
+ (x^2+(y^2)/(2)+((D-t-x-y)^2)/(3)),
mathcal V_t=calV_t={(0,0),(tau-t,0),(tau-t,tau_0-tau),(0,tau_0-t)}.
-Thus the remaining squared weights sum to at most $R(t)$, their fourth
-powers to at most $R(t)^2$. Exact evaluation gives
+Thus the remaining squared weights sum to at most R(t), their fourth
+powers to at most R(t)^2. Exact evaluation gives
[
{c|ccc}
- t&0.88&0.99&0.998 hline
+ t&0.88&0.99&0.998
R(t)&0.1359&0.0721&0.068404
]
and
[
- 1-0.99^4-R(0.88)^2=0.02093518,qquad
+ 1-0.99^4-R(0.88)^2=0.02093518,
1-0.998^4-R(0.99)^2=0.002777621984.
]
-If $w_0le0.99$ the first would contradict $sum_iw_i^4>1$, and the
-second then excludes $w_0le0.998$. Keeping the dominant contribution
+If w_0le0.99 the first would contradict sum_iw_i^4>1, and the
+second then excludes w_0le0.998. Keeping the dominant contribution
in the fourth moment,
[
S(60,b)=(sum_iw_i^4)/((sum_iw_i^2)^2)
ge(1+(R(0.998))/(0.998^2))^{-2}>0.8756>frac78 .
]
-For the physical purity at length $120$, at temperature $b_0$: the
-largest Boltzmann factor $z_0$ exceeds one (trial bound) while the sum
-of all factors is at most $q$. After raising to the power $5/2$ the
-ratio of remaining mass to $z_0^{5/2}$ is at most $(q-1)^{5/2}<R_q$,
+For the physical purity at length 120, at temperature b_0: the
+largest Boltzmann factor z_0 exceeds one (trial bound) while the sum
+of all factors is at most q. After raising to the power 5/2 the
+ratio of remaining mass to z_0^{5/2} is at most (q-1)^{5/2}<R_q,
so
[
T(120,b)ge(1+R_q)^{-2}>0.8760>frac78 .
]
This last step concerns physical Boltzmann factors, separate from the
-spatial weights used for $S$.
+spatial weights used for S.
## The initialization giving the stronger gap
-At inverse temperature $b=784$,
+At inverse temperature b=784,
[
- 1-S(2304,b)le(84513)/(10^7)<frac1{100},qquad
+ 1-S(2304,b)le(84513)/(10^7)<frac1{100},
1-T(4608,b)le(27493)/(5cdot10^6)<frac1{100}.
]
Proof.
-We isolate one large $J$ eigenvalue, bound the rest, get seeds at
-length $36$, then apply six rounded updates to reach $2304$.
+We isolate one large J eigenvalue, bound the rest, get seeds at
+length 36, then apply six rounded updates to reach 2304.
-Start at $b_0=49/4$; the $b_0=49/4$ table gives centers for $J_n,N_n$
-via (eq:sector-inversion) with error at most $ε=30cdot10^{-6}$.
+Start at b_0=49/4; the b_0=49/4 table gives centers for J_n,N_n
+via (eq:sector-inversion) with error at most ε=30cdot10^{-6}.
Use
[
- F_J(x)=x^4(7+17x-280x^2-185x^3+1000x^4)^2,quad
+ F_J(x)=x^4(7+17x-280x^2-185x^3+1000x^4)^2,
F_N(x)=x^4(12-394x^2+1000x^4)^2,
]
-of degree $12$, so the table supplies every needed moment. The summed
+of degree 12, so the table supplies every needed moment. The summed
upper bounds of Lemma~lem:polynomial-filter are
[
- B_J=318604.54848175,qquad B_N=48169.880699 .
+ B_J=318604.54848175, B_N=48169.880699 .
]
-Set $v_J=100139/100000$, $v_N=872/1000$. Direct comparison gives
+Set v_J=100139/100000, v_N=872/1000. Direct comparison gives
[
- F_J(v_J)-B_J>180,quad F_J(-v_J)-B_J>496888,quad
+ F_J(v_J)-B_J>180, F_J(-v_J)-B_J>496888,
F_N(pm v_N)-B_N>654 .
]
-To cover the whole exterior rays, write $Q(x)=7+17x-280x^2-185x^3+1000x^4$;
-for $yge0$,
+To cover the whole exterior rays, write Q(x)=7+17x-280x^2-185x^3+1000x^4;
+for yge0,
[
Q(1+y)&=559+2902y+5165y^2+3815y^3+1000y^4,
Q(-1-y)&=895+3978y+6275y^2+4185y^3+1000y^4.
]
-both positive and increasing. Likewise $12-394r^2+1000r^4$ is positive
-and increasing for $rge v_N$ (value at $v_N$ positive, derivative
-$4r(1000r^2-197)>0$). Hence every $J$ eigenvalue has modulus below
-$v_J$ and every $N$ eigenvalue below $v_N$.
+both positive and increasing. Likewise 12-394r^2+1000r^4 is positive
+and increasing for rge v_N (value at v_N positive, derivative
+4r(1000r^2-197)>0). Hence every J eigenvalue has modulus below
+v_J and every N eigenvalue below v_N.
-The twelfth-moment upper bounds are $m_J=1.0657205$, $m_N=0.2783155$.
-Put $ell=999/1000$ and, for $k=36,72$,
+The twelfth-moment upper bounds are m_J=1.0657205, m_N=0.2783155.
+Put ell=999/1000 and, for k=36,72,
- B(k)=3mathcal C(m_N,v_N^calC(m_N,v_N^{12},k/12),quad
+ B(k)=3mathcal C(m_N,v_N^calC(m_N,v_N^{12},k/12),
R_A(k)=B(k)+(m_J-ell^{12})^{k/12}
-(with $m_J>ell^{12}$). If all $J$ eigenvalues had modulus at most
-$ell$, Lemma~lem:cappedmass would give
-$Z_{72}(b_0)lemathcal C(m_J,calC(m_J,ell^{12},6)+B(72)<1$, with strict margin
-above $0.0693$; the trial inequality contradicts this. So some
-$lambda_0$ in $J$ has $|lambda_0|>ell$; removing it leaves twelfth
-moment at most $m_J-ell^{12}$ in $J$, and with the three $N$ sectors
-$R_A(k)$ bounds the remaining full $k$th moment. Thus
+(with m_J>ell^{12}). If all J eigenvalues had modulus at most
+ell, Lemma~lem:cappedmass would give
+Z_{72}(b_0)lemathcal C(m_J,calC(m_J,ell^{12},6)+B(72)<1, with strict margin
+above 0.0693; the trial inequality contradicts this. So some
+lambda_0 in J has |lambda_0|>ell; removing it leaves twelfth
+moment at most m_J-ell^{12} in J, and with the three N sectors
+R_A(k) bounds the remaining full kth moment. Thus
1-S(36,b_0)&le p_0:=1-(1+frac{R_A(36)}{ell^{36}})^{-2},
1-T(72,b_0)&le q_0:=1-(v_J^{72}+R_A(72))^{-2}.
-The first retains $lambda_0^{72}$; the second uses $Z_{72}(2b_0)>1$ and
-$Z_{72}(b_0)le v_J^{72}+R_A(72)$. Numerically $p_0<0.047916$,
-$q_0<0.181521$, but the updates use their exact rational definitions.
+The first retains lambda_0^{72}; the second uses Z_{72}(2b_0)>1 and
+Z_{72}(b_0)le v_J^{72}+R_A(72). Numerically p_0<0.047916,
+q_0<0.181521, but the updates use their exact rational definitions.
Apply Lemma~lem:rounded-update six times, rounding each new defect
-up to a multiple of $10^{-7}$ and using the rounded spatial defect in
-the physical update. With $f(u)=u^2/[2(1-u)^2]$ and
-$lceil xrceil_7=10^{-7}lceil10^7xrceil$:
+up to a multiple of 10^{-7} and using the rounded spatial defect in
+the physical update. With f(u)=u^2/[2(1-u)^2] and
+lceil xrceil_7=10^{-7}lceil10^7xrceil:
[
p_{j+1}=bigllceil f(1-(1-p_j)^2(1-q_j))bigrrceil_7,
- quad
+
q_{j+1}=bigllceil f(1-(1-q_j)^2(1-p_{j+1}))bigrrceil_7 .
]
Exact rational evaluation gives
[
{c|cc}
- j&p_j&q_j hline
+ j&p_j&q_j
1&151249/2500000&433453/2500000
2&343303/5000000&1632381/10000000
3&44601/625000&361773/2500000
@@ -1001,9 +1001,9 @@ Exact rational evaluation gives
6&84513/10000000&27493/5000000
]
-Each update lies in $[0,1)$, where $f$ increases, so upward rounding
+Each update lies in [0,1), where f increases, so upward rounding
preserves the bounds. Length and inverse temperature have multiplied by
-$2^6$: $36cdot64=2304$ and $(49/4)cdot64=784$. The last row proves
+2^6: 36cdot64=2304 and (49/4)cdot64=784. The last row proves
the proposition.
@@ -1011,29 +1011,29 @@ the proposition.
[Periodic inputs for boundary estimates]
-Let $X_b$ be the transfer operator of Proposition~prop:transfer
-with $a=700741/500000$, and put
+Let X_b be the transfer operator of Proposition~prop:transfer
+with a=700741/500000, and put
[
- b_0=49/4,quad b_1=49/2,quad ell=999/1000,
- U=100139/100000,quad V=872/1000,quad m_J=1.0657205,quad m_N=.2783155 .
+ b_0=49/4, b_1=49/2, ell=999/1000,
+ U=100139/100000, V=872/1000, m_J=1.0657205, m_N=.2783155 .
]
-At $b=b_0$ the full spatial list is the $D_2$-invariant list $J$ plus
-three identical lists $N$. Every $J$ eigenvalue has modulus below $U$
-and the sum of its twelfth powers is at most $m_J$; every $N$ eigenvalue
-has modulus below $V$ and each $N$ list's twelfth powers sum to at most
-$m_N$; some $J$ eigenvalue has modulus above $ell$. Moreover
-$1-S(72,b_1)le151249/2500000$. For $n=72,120$, $E_0(n)+na<0$, so a
-physical ground Boltzmann factor of $Z_n(b)$ exceeds one for every
-$b>0$.
+At b=b_0 the full spatial list is the D_2-invariant list J plus
+three identical lists N. Every J eigenvalue has modulus below U
+and the sum of its twelfth powers is at most m_J; every N eigenvalue
+has modulus below V and each N list's twelfth powers sum to at most
+m_N; some J eigenvalue has modulus above ell. Moreover
+1-S(72,b_1)le151249/2500000. For n=72,120, E_0(n)+na<0, so a
+physical ground Boltzmann factor of Z_n(b) exceeds one for every
+b>0.
Proof.
-The decomposition is Lemma~lem:sectors with $J=I+2O$; the filters,
+The decomposition is Lemma~lem:sectors with J=I+2O; the filters,
twelfth-moment bounds and dominant eigenvalue are proved at the start of
Proposition~prop:seed2304. The first row of the rounded update
-table gives the display at $(72,49/2)$; the variational assertion is
+table gives the display at (72,49/2); the variational assertion is
Lemma~lem:trial-inputs.
@@ -1045,35 +1045,35 @@ those proofs are part of the argument.
Proof. [Proof of Theorem~thm:main]
Proposition~prop:transfer identifies each partition function with
-a moment of a self-adjoint spatial transfer operator, so $S$ and $T$ of
-Section~sec:bootstrap are well defined. The $H_L$ are self-adjoint
+a moment of a self-adjoint spatial transfer operator, so S and T of
+Section~sec:bootstrap are well defined. The H_L are self-adjoint
and non-scalar, so the bootstrap criterion applies.
Proposition~prop:seed60 gives
-$S(60,105/4)>7/8$ and $T(120,105/4)>7/8$. Apply
+S(60,105/4)>7/8 and T(120,105/4)>7/8. Apply
Proposition~prop:bootstrap with
-$(n_0,b_0,u_0,C)=(60,105/4,1/8,79/10)$; its parameter inequalities hold
-by Section~sec:bootstrap, and $Cu_0=79/80$. Hence the ground
-state is unique and $gamma_L>frac4{105}log(80/79)$ for every even
-$Lge60$, proving (eq:gap60).
-
-Proposition~prop:seed2304 gives both defects at most $1/100$ at
-the pair with $(n_0,b_0)=(2304,784)$. Apply the same theorem with
-$u_0=1/100$, $C=5$; since $Cu_0=1/20$,
+(n_0,b_0,u_0,C)=(60,105/4,1/8,79/10); its parameter inequalities hold
+by Section~sec:bootstrap, and Cu_0=79/80. Hence the ground
+state is unique and gamma_L>frac4{105}log(80/79) for every even
+Lge60, proving (eq:gap60).
+
+Proposition~prop:seed2304 gives both defects at most 1/100 at
+the pair with (n_0,b_0)=(2304,784). Apply the same theorem with
+u_0=1/100, C=5; since Cu_0=1/20,
[
- gamma_L>(log20)/(784)qquad(Lge2304 even),
+ gamma_L>(log20)/(784)(Lge2304 even),
]
proving (eq:gap2304). The limit inferior over even lengths gives
-$Delta_1gelog(20)/784>0$. Finally, Lemma~lem:representation
+Delta_1gelog(20)/784>0. Finally, Lemma~lem:representation
carries every spectral conclusion to any simultaneously similar spin-one
realization, via the tensor-power similarity.
The bounds become uniform on all even lengths as follows: the
-second bound covers $Lge2304$, and the finitely many even $L$ with
-$4le Lle2302$ are finite-dimensional self-adjoint Hamiltonians, each
+second bound covers Lge2304, and the finitely many even L with
+4le Lle2302 are finite-dimensional self-adjoint Hamiltonians, each
with a positive gap above its full ground sector. This yields
-$inf_{Lge4,,L even}gamma_L>0$, though it gives neither a
+inf_{Lge4,,L even}gamma_L>0, though it gives neither a
numerical value nor simplicity at those small lengths.
## An infinite-volume consequence
@@ -1081,89 +1081,89 @@ numerical value nor simplicity at those small lengths.
The finite-volume estimate also controls local excitations in a
thermodynamic limit, using the limiting argument of
cite[arXiv v4, End Matter, Definition~11 and Theorem~12]{Tasaki2025}.
-A local operator on the spin chain $Z$ acts on finitely many sites; a
-state $omega$ is a linear functional on local operators with
-$omega(I)=1$, $omega(A^*A)ge0$. Put
-$[H,A]=sum_{jinZ}[mathbf S_jbfS_jcdotmathbf S_bfS_{j+1},A]$, only finitely
+A local operator on the spin chain Z acts on finitely many sites; a
+state omega is a linear functional on local operators with
+omega(I)=1, omega(A^*A)ge0. Put
+[H,A]=sum_{jinZ}[mathbf S_jbfS_jcdotmathbf S_bfS_{j+1},A], only finitely
many terms nonzero.
For the self-adjoint Hamiltonians (eq:main-H), the normalized
ground states of the even periodic rings have a subsequence, as
-$Lto∞$, converging on every local operator to a state $omega$;
+Lto∞, converging on every local operator to a state omega;
each such limit obeys
omega(A^*[H,A])ge(log20)/(784)(omega(A^*A)-|omega(A)|^2)
-for every local operator $A$.
+for every local operator A.
Proof.
-Place the even ring of length $L$ on ${-L/2,...,L/2-1}$ with its
+Place the even ring of length L on {-L/2,...,L/2-1} with its
closing bond at the endpoints. Compactness of the density matrices on
each finite interval and a diagonal subsequence give convergence on all
local operators; the compatible limits form a positive normalized
-functional $omega$. For $Lge2304$, Theorem~thm:main gives a
-unique ground state $psi_L$ and gap above $delta=log(20)/784$.
-Writing $omega_L(A)=<psi_L,Apsi_L>$, the spectral
-inequality on the orthogonal complement of $psi_L$ gives
+functional omega. For Lge2304, Theorem~thm:main gives a
+unique ground state psi_L and gap above delta=log(20)/784.
+Writing omega_L(A)=<psi_L,Apsi_L>, the spectral
+inequality on the orthogonal complement of psi_L gives
[
omega_L(A^*[H_L,A])gedelta(omega_L(A^*A)-|omega_L(A)|^2).
]
-For fixed local $A$ and large $L$, the closing bond commutes with $A$ and
-$[H_L,A]=[H,A]$; passing to the local limit proves the claim.
+For fixed local A and large L, the closing bond commutes with A and
+[H_L,A]=[H,A]; passing to the local limit proves the claim.
-In particular $omega$ is a ground state (no local perturbation lowers
-its energy). For $omega(A)=0$, (eq:local-limit-gap) is the
+In particular omega is a ground state (no local perturbation lowers
+its energy). For omega(A)=0, (eq:local-limit-gap) is the
locally unique gapped property of
cite[arXiv v4, Definition~11]{Tasaki2025}, with gap at least
-$log(20)/784$. The argument does not identify limits of different
+log(20)/784. The argument does not identify limits of different
subsequences, nor uniqueness among all infinite-volume ground states.
appendix
-# Thermal calculation at inverse temperature $21/2$
+# Thermal calculation at inverse temperature 21/2
-We prove the $b=21/2$ table of Lemma~lem:thermal-inputs. The
-calculation uses the physical Hamiltonians at $n=6,8,10$ with twists
-$1,P,C$; all arithmetic is integral or rational, complex numbers being
-written in $Z[omega]$ with $omega=e^{2pi i/3}$.
+We prove the b=21/2 table of Lemma~lem:thermal-inputs. The
+calculation uses the physical Hamiltonians at n=6,8,10 with twists
+1,P,C; all arithmetic is integral or rational, complex numbers being
+written in Z[omega] with omega=e^{2pi i/3}.
## Finite matrices and their columns
-Fix $b=21/2$ and one length, write $H=H_n(g)$. The three twists have
-angles $theta=0,pi,2pi/3$ about the weight-basis axis. Set
+Fix b=21/2 and one length, write H=H_n(g). The three twists have
+angles theta=0,pi,2pi/3 about the weight-basis axis. Set
- W=I-(H_n(g)+(3n/2)I)/(16),qquad B=16W .
+ W=I-(H_n(g)+(3n/2)I)/(16), B=16W .
-Lemma~lem:energy gives $norm Wle1$, $norm Ble16$. In a fixed
+Lemma~lem:energy gives norm Wle1, norm Ble16. In a fixed
total-weight sector the vectors are words
-$xin{-1,0,1}^n$ of fixed $sum_jx_j$, the diagonal of $B$ is
+xin{-1,0,1}^n of fixed sum_jx_j, the diagonal of B is
[
B_{x,x}=16-(3n)/(2)-sum_{j=0}^{n-1}x_{j-1}x_j ,
]
-and for each $j$ the move $y_j=x_j-delta$, $y_k=x_k+delta$
-($k=j+1bmod n$, $delta=pm1$, letters in ${-1,0,1}$) has entry $-1$,
-except at the seam $j=n-1,k=0$ where it is $-e^{ithetadelta}$ (the
-phase (eq:seam-phase)). On a coordinate $u+vomega$ the cyclic
+and for each j the move y_j=x_j-delta, y_k=x_k+delta
+(k=j+1bmod n, delta=pm1, letters in {-1,0,1}) has entry -1,
+except at the seam j=n-1,k=0 where it is -e^{ithetadelta} (the
+phase (eq:seam-phase)). On a coordinate u+vomega the cyclic
twist acts by
- -omega(u+vomega)=v+(v-u)omega,qquad
+ -omega(u+vomega)=v+(v-u)omega,
-omega^{-1}(u+vomega)=(u-v)+uomega,
so every matrix-vector product is integer arithmetic in two coordinates.
-Only some columns are needed. Let $s(x)=(x_{n-1},x_0,...,x_{n-2})$.
-The seam formula gives $H_{s(x),s(y)}=e^{itheta(y_{n-1}-x_{n-1})}H_{x,y}$,
-so $|x>mapsto e^{-itheta x_{n-1}}|s(x)>$ commutes with $H$.
-Reversal of the word and global negation of letters both send $H$ to its
+Only some columns are needed. Let s(x)=(x_{n-1},x_0,...,x_{n-2}).
+The seam formula gives H_{s(x),s(y)}=e^{itheta(y_{n-1}-x_{n-1})}H_{x,y},
+so |x>mapsto e^{-itheta x_{n-1}}|s(x)> commutes with H.
+Reversal of the word and global negation of letters both send H to its
complex conjugate, and composing either permutation with conjugation
gives an antiunitary symmetry. Each symmetry preserves the norm of the
-corresponding column of any real exponential of $H$. In every sector of
+corresponding column of any real exponential of H. In every sector of
nonnegative total weight take the lexicographically least word in each
orbit under rotations and reversal, give it the orbit size as
multiplicity, and double for strictly positive total weight to include
-the opposite sector. These multiplicities sum to $3^n$ and reproduce
+the opposite sector. These multiplicities sum to 3^n and reproduce
the full Frobenius norm of the exact exponential. No equivariance of the
rounded columns is needed: each computed representative is compared with
its own exact column at its own multiplicity.
@@ -1172,110 +1172,110 @@ its own exact column at its own multiplicity.
Write
[
- E=e^{84(W-I)}=e^{-(b/2)(H_n(g)+(3n/2)I)},qquad x_n=(3/2-a)bn,
+ E=e^{84(W-I)}=e^{-(b/2)(H_n(g)+(3n/2)I)}, x_n=(3/2-a)bn,
]
-so $Z_n(b,g)=e^{x_n}norm{E}_F^2$ (Frobenius norm). For $0le jle256$
+so Z_n(b,g)=e^{x_n}norm{E}_F^2 (Frobenius norm). For 0le jle256
let
- p_j=frac{84^j/j!}{sum_{l=0}^{256}84^l/l!},qquad
+ p_j=frac{84^j/j!}{sum_{l=0}^{256}84^l/l!},
P(W)=sum_{j=0}^{256}p_jW^j .
-These are mean-$84$ Poisson probabilities conditioned to $le256$; for
-a mean-$84$ Poisson variable $V$,
+These are mean-84 Poisson probabilities conditioned to le256; for
+a mean-84 Poisson variable V,
[
Pr(V>256)lefrac{mathbb E(2^V)}bbE(2^V){2^{257}}=frac{e^{84}}{2^{257}}
<frac{27^{28}}{2^{257}}<2^{-117},
]
-and since $norm{W^j}le1$ conditioning changes the operator average by
+and since norm{W^j}le1 conditioning changes the operator average by
at most twice this probability (Fox--Glynn coefficient argument
cite[Proposition~1]{FoxGlynn1988}):
norm{P(W)-E}<2^{-116}.
-Put $Q=2^{56}$, $q_j=lfloor Qp_jrfloor$. From $z=0$, for
-$j=256,...,0$:
+Put Q=2^{56}, q_j=lfloor Qp_jrfloor. From z=0, for
+j=256,...,0:
z←fl(Bz/16)+q_je_r ,
-with $fl$ the floor on each real coordinate (on $u,v$
-separately for $u+vomega$). The ideal suffix at step $j$ is
-$Qsum_{l=j}^{256}p_lW^{l-j}e_r$ of norm at most $Q$, and one
-coordinate floor adds less than $2√(d_0)$ in a sector of dimension
-$d_0$ while the coefficient floor adds at most one; contraction of $W$
+with fl the floor on each real coordinate (on u,v
+separately for u+vomega). The ideal suffix at step j is
+Qsum_{l=j}^{256}p_lW^{l-j}e_r of norm at most Q, and one
+coordinate floor adds less than 2√(d_0) in a sector of dimension
+d_0 while the coefficient floor adds at most one; contraction of W
bounds the error at every stage by
257(2√(3^n)+1)le125159.
-The zero coefficients above $q_{172}$ may be skipped without changing
-this count (all $257$ steps are still charged).
+The zero coefficients above q_{172} may be skipped without changing
+this count (all 257 steps are still charged).
-The repeated computed-column list, divided by $Q$ and multiplied by
-$e^{x_n/2}$, differs from $√(Z_n(b,g))$ by at most
+The repeated computed-column list, divided by Q and multiplied by
+e^{x_n/2}, differs from √(Z_n(b,g)) by at most
e^{x_n/2}√(3^n)(frac{257(2√(3^n)+1)}Q+2^{-116})
<4cdot10^{-7},
-because $√(3^n)le243$, $x_n/2<6$, $e^{x_n/2}<3^6$, and
-$3^6cdot243(125159/2^{56}+2^{-116})<4cdot10^{-7}$.
+because √(3^n)le243, x_n/2<6, e^{x_n/2}<3^6, and
+3^6cdot243(125159/2^{56}+2^{-116})<4cdot10^{-7}.
## Exact evaluation by packed integers
-For efficiency the recurrence runs on batches of at most $192$ columns,
-storing a digit list $y_0,...,y_{w-1}$ as
-$pack(y)=sum_{t=0}^{w-1}y_t2^{64t}$ (arbitrary precision).
+For efficiency the recurrence runs on batches of at most 192 columns,
+storing a digit list y_0,...,y_{w-1} as
+pack(y)=sum_{t=0}^{w-1}y_t2^{64t} (arbitrary precision).
Integer linear arithmetic preserves this identity before the floor even
-with carries. For either coefficient, $|u|,|v|le2|u+vomega|$, so by
-(eq:thermal120-column-error) every coordinate of $Bz$ is at most
+with carries. For either coefficient, |u|,|v|le2|u+vomega|, so by
+(eq:thermal120-column-error) every coordinate of Bz is at most
32(Q+125159)=2305843009217699040<2^{63}.
-Let $s_w=sum_{t=0}^{w-1}2^{64t}$ and
-$bias=2^{63}s_w$, $mask=(2^{60}-1)s_w$,
-$bias16=2^{59}s_w$. For $Y=pack(y)$ obeying
+Let s_w=sum_{t=0}^{w-1}2^{64t} and
+bias=2^{63}s_w, mask=(2^{60}-1)s_w,
+bias16=2^{59}s_w. For Y=pack(y) obeying
(eq:thermal120-lane-bound), the operation
[
texttt{(((Y+bias)>{}>4)&mask)-bias16}
]
-equals $pack((lfloor y_t/16rfloor)_t)$. Adding the
-bias makes every base-$2^{64}$ digit lie in $[0,2^{64})$; the global
+equals pack((lfloor y_t/16rfloor)_t). Adding the
+bias makes every base-2^{64} digit lie in [0,2^{64}); the global
shift moves four low bits of each higher lane into the four high bits of
the previous lane; the mask deletes exactly those crossing bits, keeping
-$lfloor(y_t+2^{63})/16rfloor$ in lane $t$; subtracting $2^{59}$ gives
-$lfloor y_t/16rfloor$, also for negative $y_t$. Hence the packed
+lfloor(y_t+2^{63})/16rfloor in lane t; subtracting 2^{59} gives
+lfloor y_t/16rfloor, also for negative y_t. Hence the packed
recurrence equals (eq:thermal120-horner). Final coordinates obey
-$2(Q+125159)<2^{63}$, so biasing and unpacking recovers every integer
-exactly; for $u+vomega$ the squared modulus is $u^2-uv+v^2$, and summing
+2(Q+125159)<2^{63}, so biasing and unpacking recovers every integer
+exactly; for u+vomega the squared modulus is u^2-uv+v^2, and summing
these integers with representative multiplicities gives the integer
-$t_{n,g}$.
+t_{n,g}.
## Exact totals and trace enclosures
The integer recurrence gives:
{ccl}
-toprule
-$n$&$g$&$t_{n,g}$
-midrule
-6&$1$&93637037510293158533967385845216
-6&$P$&3629519998654831111446492683663
-6&$C$&9839536903588146388838689926305
-8&$1$&4947327904487991279560670127283
-8&$P$&749999795795292705866802820621
-8&$C$&1301346729026243689610427021457
-10&$1$&388818111664728958208623998490
-10&$P$&120458568770462253326886628639
-10&$C$&167487042402286114495695779656
-bottomrule
+
+n&g&t_{n,g}
+
+6&1&93637037510293158533967385845216
+6&P&3629519998654831111446492683663
+6&C&9839536903588146388838689926305
+8&1&4947327904487991279560670127283
+8&P&749999795795292705866802820621
+8&C&1301346729026243689610427021457
+10&1&388818111664728958208623998490
+10&P&120458568770462253326886628639
+10&C&167487042402286114495695779656
+
Enclose the scalar factor by
[
- L_n=sum_{j=0}^{100}(x_n^j)/(j!),qquad
+ L_n=sum_{j=0}^{100}(x_n^j)/(j!),
U_n=L_n+2frac{x_n^{101}}{101!}.
]
-Since $0<x_n<12$ and later terms have ratio at most $x_n/102<1/2$,
-$L_n<e^{x_n}<U_n$. For each center $c_{n,g}$ of the $b=21/2$ table,
+Since 0<x_n<12 and later terms have ratio at most x_n/102<1/2,
+L_n<e^{x_n}<U_n. For each center c_{n,g} of the b=21/2 table,
exact rational comparison gives
c_{n,g}-2cdot10^{-6}
@@ -1283,55 +1283,55 @@ exact rational comparison gives
<frac{U_nt_{n,g}}{Q^2}
<c_{n,g}+2cdot10^{-6}.
-These put every scaled norm below $4$, so by
+These put every scaled norm below 4, so by
(eq:thermal120-scaled-error) replacing its square by the true trace
-costs less than $(8+4cdot10^{-7})4cdot10^{-7}$. Together with
+costs less than (8+4cdot10^{-7})4cdot10^{-7}. Together with
(eq:thermal120-rational-enclosure),
[
|Z_n(21/2,g)-c_{n,g}|
<2cdot10^{-6}+(8+4cdot10^{-7})4cdot10^{-7}<10^{-5}.
]
-This proves the $b=21/2$ table.
+This proves the b=21/2 table.
-# Thermal calculation at inverse temperature $49/4$
+# Thermal calculation at inverse temperature 49/4
-We prove the $b=49/4$ table, using integer approximations to columns of a
+We prove the b=49/4 table, using integer approximations to columns of a
physical matrix exponential. Throughout
[
- b=(49)/(4),qquad J=280,qquad Q=2^{55},qquad 4le nle12,
+ b=(49)/(4), J=280, Q=2^{55}, 4le nle12,
]
-with twists the identity and rotation by $pi$ about $z$ (the latter has
-the same partition function as $P$ by a global rotation).
+with twists the identity and rotation by pi about z (the latter has
+the same partition function as P by a global rotation).
## The physical matrix and its rational approximation
-In the weight basis a word $w=(w_0,...,w_{n-1})$, $w_iin{-1,0,1}$,
+In the weight basis a word w=(w_0,...,w_{n-1}), w_iin{-1,0,1},
has conserved total weight. On each weight block set
- R=I-(frac32nI+H)/(16),qquad D=32R ,
+ R=I-(frac32nI+H)/(16), D=32R ,
an integer matrix with diagonal
-$D_{w,w}=32-3n-2sum_{i=0}^{n-1}w_{i-1}w_i$, an entry $-2$ for an allowed
-opposite unit step at letters $i-1,i$, and $+2$ on the twisted seam
-$i=0$ (indices mod $n$). Lemma~lem:energy gives
+D_{w,w}=32-3n-2sum_{i=0}^{n-1}w_{i-1}w_i, an entry -2 for an allowed
+opposite unit step at letters i-1,i, and +2 on the twisted seam
+i=0 (indices mod n). Lemma~lem:energy gives
[
- 0letfrac32nI+Hletfrac52nI,qquad
+ 0letfrac32nI+Hletfrac52nI,
spec(R)⊆[1-5n/32,1]⊆[-tfrac78,1],
]
-so $R$ is a self-adjoint contraction and
+so R is a self-adjoint contraction and
[
F=exp![-tfrac b2(tfrac32nI+H)]=exp[98(R-I)] .
]
On the full physical space (direct sum over weight blocks),
- Z_n(b,g)=e^{x_n}norm{F}_F^2,qquad x_n=bn(tfrac32-a).
+ Z_n(b,g)=e^{x_n}norm{F}_F^2, x_n=bn(tfrac32-a).
Define the conditioned Poisson probabilities
- p_j=frac{v_j}{sum_{ell=0}^Jv_ell},quad
- v_j=98^j(J!)/(j!),quad 0le jle J .
+ p_j=frac{v_j}{sum_{ell=0}^Jv_ell},
+ v_j=98^j(J!)/(j!), 0le jle J .
-Since $norm Rle1$, discarding the Poisson tail and renormalizing
+Since norm Rle1, discarding the Poisson tail and renormalizing
changes the exponential by at most twice the tail probability
(Fox--Glynn):
[
@@ -1339,37 +1339,37 @@ changes the exponential by at most twice the tail probability
le2e^{-98}sum_{j=281}^∞(98^j)/(j!)
le2^{-280}e^{98}<2^{-84}<Q^{-1},
]
-using $2^jge2^{281}$ in the tail and $e^{-98}sum_{jge0}196^j/j!=e^{98}$,
-then $e<4$.
+using 2^jge2^{281} in the tail and e^{-98}sum_{jge0}196^j/j!=e^{98},
+then e<4.
## Integer columns and their error
-For a unit basis column $e_r$, start at $X=0$ and for $j=J,...,0$ do
+For a unit basis column e_r, start at X=0 and for j=J,...,0 do
- X←Bigllfloor(DX)/(32)Bigrrfloor+c_je_r,qquad
+ X←Bigllfloor(DX)/(32)Bigrrfloor+c_je_r,
c_j=lfloor Qp_jrfloor,
with coordinatewise floor; all quantities are integers, and since
-$c_j=0$ for $j>191$ the initial zero steps may be omitted. Compare with
-the ideal suffix $Y_j=Qsum_{ell=j}^Jp_ell R^{ell-j}e_r$,
-$norm{Y_j}_2le Q$. In a block of dimension $d_0$ a vector floor adds
-less than $√(d_0)$ and a coefficient floor less than one, so
-contraction bounds the accumulated error by $281(√(d_0)+1)$. With
+c_j=0 for j>191 the initial zero steps may be omitted. Compare with
+the ideal suffix Y_j=Qsum_{ell=j}^Jp_ell R^{ell-j}e_r,
+norm{Y_j}_2le Q. In a block of dimension d_0 a vector floor adds
+less than √(d_0) and a coefficient floor less than one, so
+contraction bounds the accumulated error by 281(√(d_0)+1). With
- d_n=lfloor√(3^n)rfloor+1,quad
- E_n=281(d_n+1)+1,quad mathcal E_n=d_nE_n,calE_n=d_nE_n,
+ d_n=lfloor√(3^n)rfloor+1,
+ E_n=281(d_n+1)+1, mathcal E_n=d_nE_n,calE_n=d_nE_n,
every computed column obeys
norm{X-QFe_r}_2le E_n .
-The bound charges all $281$ ideal steps (including the omitted ones).
+The bound charges all 281 ideal steps (including the omitted ones).
-The recursion is exactly signed-$64$-bit arithmetic: every coordinate is
-at most $Q+281(d_n+1)<2Q$ with $d_nle730$; any row of $D$ has absolute
-sum at most $|32-3n|+2n+4nle76$. Hence every product and partial sum is
-below $76cdot2Q=5476377146882523136<2^{63}-1$, and division by $32$ plus
-$c_jle Q$ stays below $207165582859042816<2^{63}-1$. Starting at $X=0$
+The recursion is exactly signed-64-bit arithmetic: every coordinate is
+at most Q+281(d_n+1)<2Q with d_nle730; any row of D has absolute
+sum at most |32-3n|+2n+4nle76. Hence every product and partial sum is
+below 76cdot2Q=5476377146882523136<2^{63}-1, and division by 32 plus
+c_jle Q stays below 207165582859042816<2^{63}-1. Starting at X=0
the bounds establish exactness of the next integer operation, and
contraction establishes the next coordinate bound; the induction does not
assume the overflow-freeness it proves. Final squares and sums use
@@ -1382,29 +1382,29 @@ under cyclic shifts and reversal and give the representative twice the
orbit size (this includes the uncomputed negative-weight block); in the
zero-weight block include digit negation in the orbit and use its size
without a factor of two. Take the lexicographically least word,
-handle periodic words by set cardinalities; the multiplicities $nu_r$
-satisfy $sum_rnu_r=3^n$.
+handle periodic words by set cardinalities; the multiplicities nu_r
+satisfy sum_rnu_r=3^n.
Symmetries: reversal fixes the seam as an unordered bond and global
negation preserves every entry. For the twisted chain a bare shift moves
-the seam; left rotation by $q$ moves it to $(k-1,k)$, $k=-qbmod n$, and
+the seam; left rotation by q moves it to (k-1,k), k=-qbmod n, and
the diagonal unitary
[
- G_k|w>=(-1)^{sum_{j=0}^{k-1}w_j}|w>,quad G_0=I,
+ G_k|w>=(-1)^{sum_{j=0}^{k-1}w_j}|w>, G_0=I,
]
flips the hopping signs exactly at the two boundary bonds of that arc,
-restoring the seam. The resulting signed shift commutes with $H$, so
+restoring the seam. The resulting signed shift commutes with H, so
exact exponential columns at rotated words have equal norms; the
untwisted chain needs no correction.
-Let $X_r$ be the output of (eq:thermal72-horner) at representative
-$r$ and
+Let X_r be the output of (eq:thermal72-horner) at representative
+r and
W_{n,g}=sum_rnu_rsum_wX_r(w)^2 .
Rounding need not preserve the signed symmetries, so form abstract direct
-sums repeating each exact (resp. computed) representative column $nu_r$
-times: the first has squared norm $Q^2norm{F}_F^2$, the second $W_{n,g}$.
+sums repeating each exact (resp. computed) representative column nu_r
+times: the first has squared norm Q^2norm{F}_F^2, the second W_{n,g}.
By (eq:thermal72-column-error),
|sqrt{W_{n,g}}-Qnorm{F}_F|le√(3^n)E_n<mathcal E_n,calE_n,
@@ -1416,13 +1416,13 @@ columns.
Equations (eq:thermal72-coefficients),
(eq:thermal72-horner), (eq:thermal72-total) give (second
-column: no twist; third: $pi$ twist):
+column: no twist; third: pi twist):
begingroup
footnotesize
{r|rr}
-(n)&(W_{n,I})&(W_{n,P}) hline
+(n)&(W_{n,I})&(W_{n,P})
4&1298092848946206529211955779616040&683496640248050713756440676071
5&9095379218684502088743241789&9205948219074287936685289758037
6&11971284263768914176431179720622&266573596119930013833919176645
@@ -1436,43 +1436,43 @@ footnotesize
endgroup
-Turn these integers into the $b=49/4$ table. Let
+Turn these integers into the b=49/4 table. Let
[
- r_{n,g}=lfloorsqrt{W_{n,g}}rfloor,quad
- s_n=sum_{j=0}^{100}(x_n^j)/(j!),quad
+ r_{n,g}=lfloorsqrt{W_{n,g}}rfloor,
+ s_n=sum_{j=0}^{100}(x_n^j)/(j!),
t_n=frac{x_n^{100}}{100!}.
]
-Since $0le x_n<15$, later omitted terms have ratio at most $15/101$ and
-the tail is at most $15t_n/86<t_n$, so $s_nle e^{x_n}le s_n+t_n$.
+Since 0le x_n<15, later omitted terms have ratio at most 15/101 and
+the tail is at most 15t_n/86<t_n, so s_nle e^{x_n}le s_n+t_n.
Equations (eq:thermal72-trace) and (eq:thermal72-frobenius)
give
- L_{n,g}=frac{s_nmax(0,r_{n,g}-mathcal E_n)^2}calE_n)^2{Q^2},quad
- U_{n,g}=frac{(s_n+t_n)(r_{n,g}+1+mathcal E_n)^2}calE_n)^2{Q^2},quad
+ L_{n,g}=frac{s_nmax(0,r_{n,g}-mathcal E_n)^2}calE_n)^2{Q^2},
+ U_{n,g}=frac{(s_n+t_n)(r_{n,g}+1+mathcal E_n)^2}calE_n)^2{Q^2},
L_{n,g}le Z_n(b,g)le U_{n,g},
all rational/integer with the integer square root fixed by
-$r^2le W<(r+1)^2$. For the integer centers $C_{n,g}$ of the $b=49/4$
+r^2le W<(r+1)^2. For the integer centers C_{n,g} of the b=49/4
table, direct evaluation yields
[
- L_{n,g}>frac{C_{n,g}-30}{10^6},qquad
- U_{n,g}<frac{C_{n,g}+30}{10^6}qquad(4le nle12, g=I,P),
+ L_{n,g}>frac{C_{n,g}-30}{10^6},
+ U_{n,g}<frac{C_{n,g}+30}{10^6}(4le nle12, g=I,P),
]
-each strict inequality with margin above $13/10^6$. This gives a
+each strict inequality with margin above 13/10^6. This gives a
reproducible finite verification of all eighteen enclosures and proves the
-$b=49/4$ table.
+b=49/4 table.
# Two variational inputs
The initializations need a lower bound on the untwisted partition function
-at lengths $72$ and $120$. One fixed triple of integer matrices defines
+at lengths 72 and 120. One fixed triple of integer matrices defines
periodic matrix-product vectors at both lengths with energy per bond below
-$-a=-700741/500000$. A block decomposition reduces the computation to
-powers of integer matrices of order at most $162$.
+-a=-700741/500000. A block decomposition reduces the computation to
+powers of integer matrices of order at most 162.
## Norm and energy contractions
-For $Nge4$ and real $Dtimes D$ matrices $A_s$ ($s=-1,0,1$) define, in
+For Nge4 and real Dtimes D matrices A_s (s=-1,0,1) define, in
the orthonormal spin-one weight basis,
psi_N(s_0,...,s_{N-1})=Tr(A_{s_0}... A_{s_{N-1}}),
@@ -1480,7 +1480,7 @@ the orthonormal spin-one weight basis,
the periodic matrix-product representation of
cite[Section~2.1, Equations~(2)--(3)]{PerezGarcia2007}. Put
- B_{st}&=A_sA_t,quad T=sum_{s=-1}^1A_s ⊗ A_s,
+ B_{st}&=A_sA_t, T=sum_{s=-1}^1A_s ⊗ A_s,
O&=sum_{s,t=-1}^1st,B_{st} ⊗ B_{st}
+sum_{substack{s=-1,0t=0,1}}
@@ -1491,46 +1491,46 @@ cite[Section~2.1, Equations~(2)--(3)]{PerezGarcia2007}. Put
All traces are ordinary, unnormalized.
-Let $Nge4$, $Dge1$ and $A_{-1},A_0,A_1in M_D(R)$; define
-$psi_N,T,O$ by (eq:trial-vector)--(eq:trial-insertion) and,
-for the periodic spin-one Hamiltonian $H_N$, set
-$V_N=Tr(T^N), U_N=Tr(T^{N-2}O)$. Then
-$norm{psi_N}^2=V_N$ and
-$<psi_N,H_Npsi_N>=NU_N$. In particular, if $V_N>0$ and
-$U_N<-aV_N$, then $E_0(N)<-Na$ and $Z_N(b)>1$ for every $b>0$.
+Let Nge4, Dge1 and A_{-1},A_0,A_1in M_D(R); define
+psi_N,T,O by (eq:trial-vector)--(eq:trial-insertion) and,
+for the periodic spin-one Hamiltonian H_N, set
+V_N=Tr(T^N), U_N=Tr(T^{N-2}O). Then
+norm{psi_N}^2=V_N and
+<psi_N,H_Npsi_N>=NU_N. In particular, if V_N>0 and
+U_N<-aV_N, then E_0(N)<-Na and Z_N(b)>1 for every b>0.
Proof.
-Expanding $T^N$ and using $Tr(X ⊗ Y)=Tr(X)Tr(Y)$, the sum of
-squares of the (real) amplitudes (eq:trial-vector) is $V_N$; no
-self-adjointness or positivity of $T$ is assumed. In the weight basis a
-bond has diagonal $st$ on $(s,t)$ and off-diagonal $1$ from $(s,t)$ to
-$(s+1,t-1)$ for $s=-1,0$, $t=0,1$ (from
-$h=S^z ⊗ S^z+(S^+ ⊗ S^-+S^- ⊗ S^+)/2$ and ladder entries
-$sqrt2$). Inserting this bond into the two amplitude layers gives
-exactly $O$; summing the other indices gives $U_N$ for one bond, and
-cyclicity of the amplitude trace makes $psi_N$ translation invariant, so
-every bond including the seam contributes equally. For $V_N>0$ the
-variational principle gives $E_0(N)le NU_N/V_N<-Na$, so the shifted
-ground energy is negative and its term in $Tr e^{-b(H_N+NaI)}$ exceeds
-one for every $b>0$.
+Expanding T^N and using Tr(X ⊗ Y)=Tr(X)Tr(Y), the sum of
+squares of the (real) amplitudes (eq:trial-vector) is V_N; no
+self-adjointness or positivity of T is assumed. In the weight basis a
+bond has diagonal st on (s,t) and off-diagonal 1 from (s,t) to
+(s+1,t-1) for s=-1,0, t=0,1 (from
+h=S^z ⊗ S^z+(S^+ ⊗ S^-+S^- ⊗ S^+)/2 and ladder entries
+sqrt2). Inserting this bond into the two amplitude layers gives
+exactly O; summing the other indices gives U_N for one bond, and
+cyclicity of the amplitude trace makes psi_N translation invariant, so
+every bond including the seam contributes equally. For V_N>0 the
+variational principle gives E_0(N)le NU_N/V_N<-Na, so the shifted
+ground energy is negative and its term in Tr e^{-b(H_N+NaI)} exceeds
+one for every b>0.
## One integer matrix triple
-Set $D=26$, index eight types $alpha=1,...,8$ by
-$(ell_1,...,ell_8)=(1,1,1,1,3,3,3,5)$, and order virtual labels
-$(alpha,d)$ first by $alpha$ then by
-$d=-ell_alpha,-ell_alpha+2,...,ell_alpha$. The matrices $A_s$
-use the diagonal parameters $D_alpha$ of Table~tab:trial-diagonal
-and the parameters $P_{alphabeta}$ of Table~tab:trial-pairs,
-defined only when $ell_beta=ell_alpha+2$.
+Set D=26, index eight types alpha=1,...,8 by
+(ell_1,...,ell_8)=(1,1,1,1,3,3,3,5), and order virtual labels
+(alpha,d) first by alpha then by
+d=-ell_alpha,-ell_alpha+2,...,ell_alpha. The matrices A_s
+use the diagonal parameters D_alpha of Table~tab:trial-diagonal
+and the parameters P_{alphabeta} of Table~tab:trial-pairs,
+defined only when ell_beta=ell_alpha+2.
[ht]
centering
{c|rrrrrrrr}
-$alpha$&1&2&3&4&5&6&7&8 hline
-$D_alpha$&$-289$&$-84$&$-9$&33&$-22$&27&71&$-11$
+alpha&1&2&3&4&5&6&7&8
+D_alpha&-289&-84&-9&33&-22&27&71&-11
Caption: Diagonal parameters, common to both lengths.
@@ -1538,78 +1538,78 @@ Caption: Diagonal parameters, common to both lengths.
[ht]
centering
-{c|r@{qquad}c|r@{qquad}c|r}
-$(alpha,beta)$&$P_{alphabeta}$&
-$(alpha,beta)$&$P_{alphabeta}$&
-$(alpha,beta)$&$P_{alphabeta}$ hline
-$(1,5)$&$-11$&$(2,7)$&$-49$&$(4,6)$&9
-$(1,6)$&$-35$&$(3,5)$&$-6$&$(4,7)$&$-32$
-$(1,7)$&$-28$&$(3,6)$&$-25$&$(5,8)$&1
-$(2,5)$&$-23$&$(3,7)$&$-20$&$(6,8)$&2
-$(2,6)$&16&$(4,5)$&$-34$&$(7,8)$&$-16$
+{c|r@{}c|r@{}c|r}
+(alpha,beta)&P_{alphabeta}&
+(alpha,beta)&P_{alphabeta}&
+(alpha,beta)&P_{alphabeta}
+(1,5)&-11&(2,7)&-49&(4,6)&9
+(1,6)&-35&(3,5)&-6&(4,7)&-32
+(1,7)&-28&(3,6)&-25&(5,8)&1
+(2,5)&-23&(3,7)&-20&(6,8)&2
+(2,6)&16&(4,5)&-34&(7,8)&-16
-Caption: Parameters joining virtual types whose $ell$ differ by $2$.
+Caption: Parameters joining virtual types whose ell differ by 2.
-Entry rules. Row $(alpha,d)$, column $(beta,e)$, with $ell=ell_alpha$,
-$k=ell_beta$, $sin{-1,0,1}$. The entry is zero unless
+Entry rules. Row (alpha,d), column (beta,e), with ell=ell_alpha,
+k=ell_beta, sin{-1,0,1}. The entry is zero unless
- e=d+2s,qquad |k-ell|le2,qquad (k=ell⇒alpha=beta).
+ e=d+2s, |k-ell|le2, (k=ell⇒alpha=beta).
For an allowed entry set
[
p=D_alpha,& k=ell, P_{alphabeta},& k=ell+2,
-P_{betaalpha},& k=ell-2,
]
-$r=(d+ell)/2$, and the entry $pf$ where
+r=(d+ell)/2, and the entry pf where
{c|ccc}
- &s=-1&s=0&s=1 hline
+ &s=-1&s=0&s=1
k=ell+2&2(ell+1-r)(ell+2-r)&2(r+1)(ell-r+1)&(r+1)(r+2)
k=ell&2(ell-r+1)&d&-(r+1)
k=ell-2&2&-2&1
-lists the values of $f$. These rules do not depend on $N$.
+lists the values of f. These rules do not depend on N.
## Finite contractions and variational bounds
-Evaluate the contractions at $N=72$ and $N=120$. The support rule
+Evaluate the contractions at N=72 and N=120. The support rule
(eq:trial-support) makes this a short block calculation: on paired
-virtual labels $((alpha,d),(beta,e))$ both $T$ and $O$ preserve $d-e$
-(the two layers shift by the same $2s$ for $T$, and by the same total
-physical weight $s+t$ for $O$, also in each off-diagonal term). The
+virtual labels ((alpha,d),(beta,e)) both T and O preserve d-e
+(the two layers shift by the same 2s for T, and by the same total
+physical weight s+t for O, also in each off-diagonal term). The
difference blocks are exhaustive with dimensions
{c|rrrrrrrrrrr}
-d-e&-10&-8&-6&-4&-2&0&2&4&6&8&10 hline
+d-e&-10&-8&-6&-4&-2&0&2&4&6&8&10
dim&1&8&32&80&136&162&136&80&32&8&1
,
-summing to $676=26^2$, each block once.
+summing to 676=26^2, each block once.
-The whole integer computation: form $A_s$, $B_{st}$, $T$, $O$ from the
+The whole integer computation: form A_s, B_{st}, T, O from the
tables and (eq:trial-transfer), (eq:trial-insertion),
(eq:trial-support), (eq:trial-entry); restrict to each
-difference block giving $T_delta,O_delta$; compute powers by
+difference block giving T_delta,O_delta; compute powers by
[
- P_0=I,quad P_{2j}=P_j^2 (jge1),quad P_{2j+1}=P_j^2T_delta (jge0),
+ P_0=I, P_{2j}=P_j^2 (jge1), P_{2j+1}=P_j^2T_delta (jge0),
]
then accumulate
- U_N=sum_deltasum_{i,j}(P_{N-2})_{ij}(O_delta)_{ji},quad
+ U_N=sum_deltasum_{i,j}(P_{N-2})_{ij}(O_delta)_{ji},
V_N=sum_deltasum_{i,j}(P_{N-2})_{ij}(T_delta^2)_{ji},
-using only arbitrary-precision integers. For $N=72$ the power is $70$;
-for $N=120$ use $T^{118}=T^{64}T^{32}T^{16}T^4T^2$ and
-$T^{120}=T^{64}T^{32}T^{16}T^8$ within each block. The resulting exact
+using only arbitrary-precision integers. For N=72 the power is 70;
+for N=120 use T^{118}=T^{64}T^{32}T^{16}T^4T^2 and
+T^{120}=T^{64}T^{32}T^{16}T^8 within each block. The resulting exact
integer comparisons are
- V_N>0,qquad -c_NV_Nle10^{12}U_N<(-c_N+1)V_N,
- qquad
+ V_N>0, -c_NV_Nle10^{12}U_N<(-c_N+1)V_N,
+
{c|r}
- N&c_N hline
+ N&c_N
72&1401482112601
120&1401482105174
.
@@ -1617,14 +1617,14 @@ integer comparisons are
For the untwisted periodic spin-one Heisenberg Hamiltonian and
-$a=700741/500000$, $E_0(72)<-72a$ and $E_0(120)<-120a$; consequently
-$Z_{72}(b)>1$ and $Z_{120}(b)>1$ for every $b>0$.
+a=700741/500000, E_0(72)<-72a and E_0(120)<-120a; consequently
+Z_{72}(b)>1 and Z_{120}(b)>1 for every b>0.
Proof.
For each row of (eq:trial-integer-results),
-$(-c_N+1)/10^{12}<-700741/500000$, so the corresponding explicit matrices
-have $V_N>0$ and $U_N/V_N<-a$; apply Lemma~lem:trial-contraction.
+(-c_N+1)/10^{12}<-700741/500000, so the corresponding explicit matrices
+have V_N>0 and U_N/V_N<-a; apply Lemma~lem:trial-contraction.
The support rule places these trial vectors in total weight zero, but the
@@ -1642,10 +1642,10 @@ bibitem[Affleck and Lieb(1986)]{AffleckLieb1986} I. Affleck and E.~H. Lieb, A pr
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+bibitem[White and Huse(1993)]{WhiteHuse1993} S.~R. White and D.~A. Huse, Numerical renormalization-group study of low-lying eigenstates of the antiferromagnetic S=1 Heisenberg chain, Phys. Rev. B 48, 3844--3852 (1993).
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