A BOUNDARY-FIELD GAP FOR THE SPIN-ONE HEISENBERG CHAIN
=====================================================
A self-contained proof for readers with linear algebra. All statements
and steps are spelled out. No LaTeX, no assumed background beyond
unitary operators on finite-dimensional Hilbert spaces.
Contents:
0. What the theorem says, in plain words.
1. The target Hamiltonian and Theorem 1.
2. Strategy on one page.
3. The spatial transfer operator and its two identities.
4. Periodic inputs we borrow from the companion paper.
5. The boundary moment recurrence.
6. Finite seed values (proof in appendices B and C).
7. Quadratic continuation.
8. Odd lengths and the physical gap (proof of Theorem 1).
9. Tasaki's topological index -1.
A. The history expansion: how X_b represents a partition function.
B. Exact trial vectors at four lengths.
C. Certified short-chain thermal traces.
Notation. log = natural logarithm. All terminating decimals in
inequalities denote exact rationals. Sums over "d >= 0" are over
nonnegative integers. "I" is the identity operator.
======================================================================
0. What the theorem says, in plain words.
======================================================================
Take a straight chain of 2L+1 quantum spins, each of spin 1 (so each
spin has three states: +1, 0, -1 along any chosen axis). Put them at
integer positions -L, -L+1, ..., L.
Neighboring spins interact by the antiferromagnetic Heisenberg rule:
their dot product S_j · S_{j+1} enters with coefficient +1. Additionally,
turn on a magnetic field of strength h = 3/5 along the z-axis at the
two END sites only (not the interior). The full Hamiltonian is
H_L(3/5) = sum_{j=-L}^{L-1} S_j · S_{j+1} - (3/5)(S_{-L}^z + S_L^z).
There is no bond joining the two endpoints to each other; the chain
is open.
Every eigenstate of H_L(3/5) has some energy. Let E_0 <= E_1 <= ...
be the sorted energies (counted with multiplicity), so E_1 - E_0 is
the energy of the first excited state above the smallest one.
The theorem says: for every chain length 2L+1 with L >= 960,
the excitation gap is at least log(10)/392, an absolute constant that
does not depend on L. And log(10)/392 is about 0.005874...
So the gap of H_L(3/5) does NOT shrink to 0 as L grows. This is
Haldane's 1983 prediction, verified here for one particular boundary
field (h = 3/5), pure bilinear coupling, symmetric z-fields at the
two ends. As a byproduct (Section 9), every infinite-volume
"boundary-selected" ground state carries Tasaki's topological index -1.
======================================================================
1. The target Hamiltonian and Theorem 1.
======================================================================
Spin-1 operators. Fix C^3 with orthonormal basis e_{-1}, e_0, e_1.
Define 3x3 matrices by:
S^z e_m = m e_m,
S^+ e_m = sqrt(2 - m(m+1)) e_{m+1} for m < 1, S^+ e_1 = 0,
S^- = (S^+)^*,
S^x = (S^+ + S^-)/2,
S^y = (S^+ - S^-)/(2i).
These are the usual spin-1 matrices. Direct computation gives
(S^x)^2 + (S^y)^2 + (S^z)^2 = 2 I. Write
S_j · S_k = S_j^x S_k^x + S_j^y S_k^y + S_j^z S_k^z
for the spin-bilinear on sites j and k.
Hamiltonian. On (C^3)^{tensor(2L+1)}, with sites -L, ..., L:
H_L(h) = sum_{j=-L}^{L-1} S_j · S_{j+1} - h (S_{-L}^z + S_L^z). (1.1)
For each L >= 1, denote by E_0(L, h) <= E_1(L, h) the two lowest
eigenvalues of H_L(h) (counted with multiplicity).
Theorem 1. For every integer L >= 960:
E_1(L, 3/5) - E_0(L, 3/5) > log(10)/392. (T1)
This is the boundary-field Haldane gap assertion at one fixed field
h = 3/5. The bulk coupling is the pure bilinear Heisenberg; no
biquadratic AKLT term is present. The boundary condition is the
symmetric z-field of strength 3/5 at both ends of an open chain.
======================================================================
2. Strategy on one page.
======================================================================
2.1 Two auxiliary partition functions. Instead of attacking H_L(h)
directly, we work with two tractable relatives.
Open chain with OPPOSITE endpoint fields:
H_n^opp = sum_{j=0}^{n-2} S_j · S_{j+1} - h S_0^z + h S_{n-1}^z. (2.1)
Open chain with SAME endpoint fields:
H_n^s = sum_{j=0}^{n-2} S_j · S_{j+1} - h S_0^z - h S_{n-1}^z. (2.2)
For n = 2L+1, H_n^s equals H_L(h) after relabeling sites 0..n-1 as
-L..L. Fix h = 3/5 and a shift a = 700741/500000 (which will cancel
in every ratio). Define shifted partition functions
Y_n(b) := Tr exp(-b (H_n^opp + a n I)),
Y_n^s(b) := Tr exp(-b (H_n^s + a n I)).
(The shift a n I is irrelevant to gaps but pins the size of the
Boltzmann weights, as we will use.)
2.2 Spatial transfer. There is a self-adjoint Hilbert-Schmidt operator
X_b acting on an auxiliary Hilbert space K_b of "histories" (finite
event lists along an imaginary-time circle of circumference b). Its
nth power trace is the PERIODIC-model partition function:
Z_n(b) := Tr exp(-b (H_n,periodic + a n I)) = Tr(X_b^n), n >= 4.
For the open chains, X_b turns into a boundary pairing:
Y_n(b) = <v_b, X_b^n v_b>, (2.3)
Y_n^s(b) = <v_b, X_b^n U_P v_b>, (2.4)
where v_b in K_b is a simple boundary vector (a sum over all-z
histories) and U_P is a specific rotation (P flips z to -z) that
converts the opposite-field right boundary into a same-field one.
Section 3 sets up X_b, v_b, U_P, and proves (2.3), (2.4). Appendix A
gives a fully explicit derivation.
2.3 Periodic concentration (from the companion paper). At the
doubling scales b_j = 2^j b_0 with b_0 = 49/4, X_{b_j} has a unique
simple largest-modulus eigenvalue lambda_j = ||X_{b_j}||. It is
POSITIVE. Call the rest of the spectrum "residual" and define
rho_j = sup{|xi| / lambda_j : xi eigenvalue of X_{b_j}, xi != lambda_j}.
Quantitatively we know rho_j^{n_j} <= d(u_j) for a specific small
defect u_j and length n_j, all with j >= 2 (Section 4). Also the
leading eigenvalue itself decreases under temperature doubling:
r_j := lambda_{j+1}/lambda_j^2, 0 < r_j <= 1.
2.4 Two boundary defects. "Most mass on the leading line" is not
enough for a boundary pairing. We need the boundary vector to see the
line. Two scalars capture this:
c_j = ||P_j v_{b_j}||^2 (leading-line overlap),
B_k(j) = <v_{b_j}, (|X_{b_j}|/lambda_j)^k v_{b_j}>, k even,
e_k(j) = (B_k(j) - c_j) / B_k(j) (residual fraction),
A(m,j) = Y_m(b_{j+1}) / Y_m(b_j)^2 (physical purity).
Interpretation:
e_k(j) = the fraction of the kth boundary moment that lies OUTSIDE
the leading spatial line. 0 <= e_k(j) <= 1.
A(m,j) = sum_i p_i^2 where p_i are the normalized Gibbs weights of
H_m^opp at inverse temperature b_j. 0 < A(m,j) <= 1.
2.5 Coupled update. The engine of the proof: suppose at level j
(temperature b_j, length k_j for e, length M_j for A) we have small
upper bounds
e_{k_j}(j) <= E_j, 1 - A(M_j, j) <= eta_j.
Then at level j+1 (temperature b_{j+1} = 2 b_j, length k_{j+1} = M_j)
we automatically get updated bounds
e_{k_{j+1}}(j+1) <= E_{j+1}, 1 - A(M_{j+1}, j+1) <= eta_{j+1}
with
E_{j+1} >= F( d(u_{j+1})^{(M_j - k_j)/M_j} , E_j, 1 - eta_j ),
eta_{j+1} >= f( 1 - (1 - eta_j)(1 - u_j)(1 - u_{j+1})(1 - E_{j+1}) ).
The two explicit functions are
F(q, e, t) = (q/(1-q)) [ (1-e)^{-2}/t - 1 ], (2.5)
f(u) = u^2 / (2 (1-u)^2), (2.6)
d(u) = u / (2 - 3u). (2.7)
Crucially: F's output E_{j+1} < 1 is a CONSEQUENCE of the hypothesis,
not an extra requirement. That is, positivity of c_{j+1} (the new
leading overlap) is proved, never assumed.
2.6 Finite seed. Four exact finite trial vectors (Appendix B) give
E_0(H_m^opp) + a m < alpha = .4042 at m in {34, 78, 142, 240} with
h = 3/5. Fifteen exact short-chain thermal traces (Appendix C) give
Y_6(b_0) < .027 and c_0 P_*(lambda_0)^2 < .003745, where P_*(x) =
x^3 - .14 x^2 - .18 x.
From these finitely many data we prove (Section 6):
e_34(0) <= epsilon(34) < .0928,
e_78(1) <= .133,
e_142(2) <= .14,
1 - A(240, 2) <= .21.
So we can start the coupled recurrence at j = 2 with
E_2 = .14, eta_2 = .21.
2.7 Driving the defects to zero. Two explicit rounds push both
defects below .03; three rounds push them below .001. Then a single
quadratic recurrence x_5 = .01, x_{j+1} = 10 x_j^2 (so
x_j = 10^{-1 - 2^{j-5}}) controls all further levels j >= 5:
e_{k_j}(j) <= x_j, 1 - A(M_j, j) <= x_j.
2.8 Odd lengths. Once both defects are tiny, the leading spatial
line dominates the same-field boundary pairing so completely that,
for every integer n in [M_j, 2 M_j] (ODD n included),
Y_n^s(b_{j+1}) / Y_n^s(b_j)^2 >= 1 - 4 x_j.
Interpret the left side as a Gibbs purity at inverse temperature b_j.
Then the largest Gibbs weight p_0 obeys p_0 >= 1 - 4 x_j > 1/2, so
the ground state is simple, and p_1/p_0 <= 4x_j/(1 - 4x_j) <
10^{-2^{j-5}}. Since p_1/p_0 = exp(-b_j (E_1 - E_0)) and
b_j = 392 * 2^{j-5}, the physical gap satisfies
E_1 - E_0 > log(10)/392.
The intervals [M_j, 2 M_j] with j >= 5 cover every integer
n >= 1920. Every odd length n = 2L+1 with L >= 960 falls in one of
them. That is Theorem 1.
2.9 Topology. From Theorem 1, the infinite-volume ground states of
H_L(3/5) satisfy a local gap inequality with rate gamma_* = log(10)/392
and have Tasaki's topological index -1 (Section 9).
======================================================================
3. The spatial transfer operator.
======================================================================
This section defines X_b and the boundary vectors v_b, states the two
transfer identities (2.3) and (2.4), and gives enough structure that
Section 5 can use the spectrum of X_b. A full algebraic derivation is
in Appendix A.
3.1 The history space. Fix b > 0. Let Omega_b be the disjoint union,
over d = 0, 1, 2, ..., of labeled open d-simplexes
{ (t_1, ..., t_d; alpha_1, ..., alpha_d) :
0 < t_1 < ... < t_d < b, alpha_r in {x, y, z} }.
The empty tuple (d = 0) is one point. Give each d-simplex its usual
d-dimensional Lebesgue measure; the empty point has measure 1. Let
mu_b denote the resulting measure on Omega_b. Its total mass is
mu_b(Omega_b) = sum_{d >= 0} (3b)^d / d! = e^{3 b}.
A point omega in Omega_b is a "history": the list of times at which
some event (a bond or a magnetic-field insertion) happens, together
with the coordinate axis (x, y, or z) at which the event acts.
3.2 The kernel. For alpha in {x, y, z}, put G_alpha = i S^alpha.
These are REAL 3x3 matrices, each with operator norm 1. The identity
- S · S = G_x (tensor) G_x + G_y (tensor) G_y + G_z (tensor) G_z (3.1)
holds (Appendix A.1). For two histories omega, eta, form their merged
event list in chronological order; multiply the corresponding
G_alpha's taken in that order; then take the 3x3 trace. Call the
result k_b(omega, eta). At nonempty time collisions (a measure-zero
subset of Omega_b x Omega_b) set k_b = 0. Then k_b is real, symmetric
in its two arguments, and |k_b(omega, eta)| <= 3.
3.3 The operator. Let K_b = L^2(Omega_b, mu_b; C). Define
(X_b f)(omega) = e^{-a b} int_{Omega_b} k_b(omega, eta) f(eta) dmu_b(eta), (3.2)
with a = 700741/500000 and h = 3/5.
Properties. X_b is real (every ingredient is real), self-adjoint (k_b
is symmetric), and Hilbert-Schmidt (|k_b| <= 3 and mu_b(Omega_b)
finite).
3.4 Periodic identity. From the companion paper (its Proposition 3.1
and the time-ordered Poisson expansion of Aizenman and Nachtergaele),
and reproved in Appendix A.6:
Tr exp(-b (H_n,periodic + a n I)) = Tr(X_b^n) for n >= 4. (3.3)
Here H_n,periodic is the spin-1 periodic Heisenberg chain on n sites.
This is the bridge between physical traces and the spectrum of X_b.
3.5 Rotations. Let D_2 be the group of signed permutations of the
three coordinate axes consisting of the identity and the three
pi-rotations about x, y, z. Each g in D_2 acts on the coordinates by
signs epsilon_g(alpha) in {±1}. Let C be the cyclic permutation of
x->y->z->x. The group G = <D_2, C> has order 12 and consists of all
signed permutation matrices with determinant +1 (proper rotations by
multiples of 90 degrees about coordinate axes).
For each g in G, define a real unitary on K_b:
(U_g f)(omega) = s_g(sigma_g^{-1} omega) f(sigma_g^{-1} omega). (3.4)
Here sigma_g relabels the marks alpha_r in each history, and
s_g(omega) = prod_r epsilon_g(alpha_r) is the product of the signs.
Fact (reproved in Appendix A.7): every U_g commutes with X_b.
The particular rotation
P = diag(1, -1, -1)
satisfies P e_z = -e_z. We use U_P to flip the sign of the right-
endpoint field.
3.6 Boundary vectors. Define v_b in K_b by
v_b(omega) = (i h)^d if all d marks of omega equal z,
= 0 otherwise. (3.5)
The empty-history value is v_b(empty) = 1. Compute:
||v_b||^2 = sum_{d >= 0} |(i h)^d|^2 (b^d / d!)
= sum_{d >= 0} b^d h^{2d} / d!
= exp(b h^2). (3.6)
Here |(ih)^d|^2 = h^{2d} because h is real positive.
3.7 Transfer identity for boundaries. For b > 0, n >= 2, and g in G,
the inner product
<v_b, X_b^n U_g v_b>
equals the shifted partition function of the open chain with LEFT
field -h S_0^z and RIGHT field +h S_{n-1}^{g e_z}. Here S^u = sum_alpha
u_alpha S^alpha for a real unit vector u. Consequently:
Y_n(b) = <v_b, X_b^n v_b>, (with g = 1) (3.7)
Y_n^s(b) = <v_b, X_b^n U_P v_b> (with g = P). (3.8)
Proof sketch (full details in Appendix A): expand the negative-
Hamiltonian exponential as a time-ordered series; use (3.1) to attach
two G_alpha per bond; the coefficient (i h)^d inside v_b conjugated
gives (-i h)^d at the LEFT boundary, and (-i h) G_z = (+h) S_z, matching
the -h S_0^z sign of H_n^opp; at the RIGHT boundary, v_b contributes
(i h)^d and (i h) G_z = (-h) S_z, matching the +h S_{n-1}^z sign. The
rotation U_P turns the right field into +h S_{n-1}^{P e_z} = -h S_{n-1}^z,
matching the same-field Hamiltonian H_n^s.
3.8 Even and odd n. For even n, X_b^n = |X_b|^n on any spectral
subspace where |X_b| and X_b agree in magnitude; so Y_n(b) is a
positive weighted moment of the absolute spectral values. For odd
n, the signs of negative eigenvalues of X_b survive, and Y_n^s(b) is
a signed boundary pairing. Section 8 controls this signed object at
odd lengths using Lemma 8.1.
3.9 Constants fixed for the rest of the paper:
h = 3/5, a = 700741/500000, b_0 = 49/4, b_j = 2^j b_0. (3.9)
======================================================================
4. Periodic inputs from the companion paper.
======================================================================
Throughout, fix b_j = 2^j b_0 with b_0 = 49/4 and h = 3/5. This section
records the purely PERIODIC facts we need. No boundary Hamiltonian
appears; everything comes from the companion paper's analysis of the
periodic chain.
4.1 Spectral concentration. Proposition (from companion paper; here
restated in the form we will use).
For every j >= 0, X_{b_j} has a unique simple eigenvalue of maximal
modulus, denoted lambda_j = ||X_{b_j}||. This eigenvalue is POSITIVE;
its eigenline is fixed pointwise by every U_g, g in G. If rho_j is the
supremum of |xi|/lambda_j over all remaining eigenvalues (and zeros) of
X_{b_j}, then the following hold:
(a) At b_0 = 49/4:
ell < lambda_0 <= U, rho_0 < .874, (4.1)
with the constants
ell = .999, U = 1.00139. (4.2)
(b) At b_1, b_2:
rho_1 < .9525, rho_2 < .971. (4.3)
(c) For j >= 2 define
n_j := 120 * 2^{j-2}, M_j := 2 n_j = 240 * 2^{j-2}, (4.4)
u_2 := .052, u_{j+1} := 6 u_j^2. (4.5)
Then
1 - S(n_j, b_j) <= u_j, (4.6)
1 - T(M_j, b_j) <= u_j, (4.7)
rho_j^{n_j} <= d(u_j), (4.8)
where the "spatial purity" S and "physical purity" T are
S(n, b) = Z_{2n}(b) / Z_n(b)^2,
T(n, b) = Z_n(2 b) / Z_n(b)^2, (4.9)
and d(u) = u / (2 - 3u) is as in (2.7).
(d) Define r_j = lambda_{j+1}/lambda_j^2. Then:
0 < r_j <= 1 (all j >= 0), (4.10)
r_j^{M_j} >= (1-u_j)(1-u_{j+1}) (j >= 2). (4.11)
4.2 Sketch of proofs, for completeness.
(a) At b_0 = 49/4, the companion paper shows that the nonzero spectrum
of X_{b_0} splits into a real "J-sector" (spectrum on the subspace fixed
by every U_g with g in D_2) plus three identical "N-sectors". In J,
every |xi| <= U = 1.00139 and sum |xi|^{12} <= m_J = 1.0657205. In each
N, every |xi| <= V = .872 and sum |xi|^{12} <= m_N = .2783155. Some
eigenvalue in J has |xi| > ell = .999.
Removing the one J-eigenvalue with |xi| > ell, the remaining twelfth-
power mass in J is at most m_J - ell^{12} < ell^{12}, and V = .872 <
ell = .999 bounds every N-entry. So the removed eigenvalue is the
unique simple one of maximal modulus. The rational comparisons
m_J - ell^{12} < (.874 ell)^{12}, V < .874 ell
give rho_0 < .874. So (a) holds.
(b) Reseeding at (120, b_2). The companion paper gives the purity
bound 1 - S(72, b_1) <= .0604996. From Lemma 4.4 (below), this implies
the largest spatial weight in the S(72, b_1) list is at least .968.
The periodic trial inequality Z_{72}(b_1) > 1 then gives
lambda_1^{72} > .968.
For any even k >= 12, physical squaring gives Z_k(b_1) <= Z_k(b_0)^2.
Combining the residual-moment upper bound
Z_k(b_0) <= U^k (1 + D(k)) with the leading-eigenvalue bound
lambda_1^{72} > .968 gives
sum_{i != 0} |mu_i(b_1)/lambda_1|^{32}
<= U^{64}(1 + D(32))^2 (.968)^{-32/72} - 1
< .9525^{32}. (4.12)
So rho_1 < .9525. (Here D(k) is the explicit residual moment bound
from (4.13) below.)
Physical seed at (120, b_1). From the companion paper's Lemma
Z_n(b) > 1 (specifically Z_{120}(b) > 1 for every b > 0), pick one
ground Boltzmann weight at b_0 and divide every weight by it. The
chosen weight exceeds 1. Hence the sum of the other ratios is at
most
R := U^{120} (1 + D(120)) - 1 < .182
When b doubles, these ratios square, so the sum is at most R^2, and
T(120, b_1) >= t := (1 + R^2)^{-2} > .9372. (4.13b)
Similarly, raising (4.12) to the 60th power yields a residual 60th
moment < .9525^{60}, so
S(60, b_1) >= s := (1 + .9525^{60})^{-2} > .9002. (4.13c)
Now apply the coupled periodic update (Lemma 4.5 below) at
(n, b) = (60, b_1) with the seeds s and t:
1 - S(120, b_2) <= f(1 - s^2 t) < .0502 < .052,
1 - T(240, b_2) <= f(1 - t^2 (1 - .0502)) < .0198 < .052.
So u_2 = .052 is achieved at j = 2.
(c) Induction. For 0 <= u <= .052:
f(1 - (1-u)^3) = G(u) u^2, G(u) := (1/2)( sum_{r=1}^3 (1-u)^{-r} )^2.
The function G is increasing on [0, .052] and exact substitution gives
G(.052) < 5.583 < 6. Also 6u^2 <= u on this interval. So if the two
purity defects at (n_j, b_j) and (M_j, b_j) are each <= u_j, the
coupled update (Lemma 4.5) bounds both new defects by 6 u_j^2 =
u_{j+1} (with j+1 replacing j). This proves (4.6), (4.7) for all
j >= 2, since j = 2 is the base case.
Since S(n_j, b_j) >= 1 - u_j > 1/2, the largest spatial weight at
(n_j, b_j) is unique, so lambda_j remains the unique simple largest-
modulus eigenvalue. Lemma 4.4 applied to the probability list of
squared normalizations gives
sum_{i != 0} |mu_i(b_j)/lambda_j|^{n_j} <= d(u_j),
which is (4.8).
The rational comparison d(.052) < .971^{120} gives rho_2 < .971.
(d) Positivity of lambda_j. If -lambda_j were the leading eigenvalue,
then for odd integers n, Tr(X_{b_j}^n)/lambda_j^n -> -1 (the residual
nth-power sum tends to 0 as n -> infinity); this contradicts strict
positivity of the physical partition function Z_n(b_j) at odd n >= 5.
So lambda_j = +|lambda_j| > 0.
Symmetry. X_{b_j} is real, so its leading eigenline has a real unit
vector. Each U_g is real unitary and commutes with X_{b_j}, so acts on
that line by a sign. The order-3 rotation C must act by +1 (it permutes
the three coordinate axes, preserving the global sign choice). C
conjugates the three nonidentity elements of D_2 cyclically, so all
three have the same sign on the line. Their product is one of them, so
that sign equals its own square, hence is +1. So all of G fixes the
line pointwise.
(e) r_j <= 1. For every even N >= 4, physical squaring gives
Z_N(b_{j+1}) <= Z_N(b_j)^2. The residual-tail bound gives
Z_N(b_j)/lambda_j^N = 1 + o(1), so Z_N(b_j)^{1/N} -> lambda_j. Taking
Nth roots and letting N -> infinity through even integers yields
lambda_{j+1} <= lambda_j^2, i.e., r_j <= 1.
(f) r_j^{M_j} >= (1-u_j)(1-u_{j+1}). At b_{j+1} and the length
M_j = n_{j+1}, the leading spatial purity is at least 1 - u_{j+1}, so
lambda_{j+1}^{M_j}
>= (1 - u_{j+1}) Z_{M_j}(b_{j+1})
= (1 - u_{j+1}) T(M_j, b_j) Z_{M_j}(b_j)^2
>= (1 - u_{j+1}) (1 - u_j) lambda_j^{2 M_j}.
Divide by lambda_j^{2 M_j} to get (4.11).
4.3 Residual-moment bound D(k). Defined by
D(k) := [ (m_J - ell^{12})^{k/12}
+ 3 ( V^k + (m_N - V^{12})^{k/12} ) ] / ell^k, (4.14)
for every real k >= 12. This bounds sum_{i != 0} (|mu_i(b_0)|/lambda_0)^k.
Combined with lambda_0 <= U it gives
Z_k(b_0) <= U^k (1 + D(k)). (4.15)
Explicit small values: D(32) < .0442, D(120) < .00000025.
4.4 Lemma (squaring and concentration). Let (w_i) be a finite or
countable probability list with purity P = sum_i w_i^2 >= 1 - u for
some u in [0, 1). Normalize the squares w_i^2 by dividing by P, giving
a new probability list with purity P'. Then
1 - P' <= f(u) := u^2 / (2 (1-u)^2). (4.16)
If u < 1/2, the largest weight is unique and at least 1 - u. Writing
x := sum_{i != i_max} w_i, one has
x <= (1 - sqrt(1 - 2u))/2, (4.17)
x/(1-x) <= d(u) := u/(2 - 3u) (0 <= u <= .1). (4.18)
Proof. Nonnegativity justifies rearrangement:
1 - P' = [ 2 sum_{i<k} w_i^2 w_k^2 ] / P^2
<= [ 2 ( sum_{i<k} w_i w_k )^2 ] / P^2
= (1 - P)^2 / (2 P^2)
<= u^2 / (2(1-u)^2) = f(u).
The map f is increasing on [0, 1). So (4.16) holds.
If u < 1/2: a summable nonzero nonnegative list attains its maximum,
and P <= max_i w_i. So the maximum exceeds 1/2, hence uniquely. Then
x = 1 - w_0 <= u < 1/2, and 2 x(1-x) <= 1 - P <= u, which gives
(4.17). Also x <= u/(2(1-x)) <= u/(2(1-u)); applying the increasing
map x -> x/(1-x) gives (4.18).
4.5 Lemma (coupled periodic update). Suppose n >= 4 even, b > 0,
S(n, b) >= s, T(2n, b) >= t for some s, t in (0, 1]. Set
p+ := f(1 - s^2 t).
If p+ < 1, then
1 - S(2 n, 2 b) <= p+, (4.19)
1 - T(4 n, 2 b) <= f(1 - t^2 (1 - p+)). (4.20)
Either bound may be increased before subsequent use, provided it stays
below 1.
Proof. Direct cancellation from the definitions:
S(n, 2 b) = S(n, b)^2 * T(2 n, b) / T(n, b)^2
>= s^2 * t.
Apply Lemma 4.4: the purity after squaring has defect at most
f(1 - s^2 t) = p+. Second,
T(4 n, b) = T(2 n, b)^2 * S(2 n, 2 b) / S(2 n, b)^2
>= t^2 * (1 - p+).
Apply Lemma 4.4 again to the physical weights to get (4.20).
Monotonicity of f justifies increasing either bound.
======================================================================
5. The boundary moment recurrence.
======================================================================
Throughout, j >= 0, b_j = 2^j b_0, lambda_j, rho_j, r_j, n_j, M_j, u_j
are as in Section 4.
5.1 Definitions. Throughout Sections 5-8 we use, for j >= 0 and even
handles k, m >= 2:
P_j := orthogonal projection onto the leading spatial line
(eigenvalue +lambda_j) of X_{b_j},
c_j := ||P_j v_{b_j}||^2, (5.1)
B_k(j) := <v_{b_j}, (|X_{b_j}|/lambda_j)^k v_{b_j}>, (5.2)
e_k(j) := (B_k(j) - c_j) / B_k(j), (5.3)
A(m, j) := Y_m(b_{j+1}) / Y_m(b_j)^2. (5.4)
Basic facts:
(i) B_k(j) = Y_k(b_j) / lambda_j^k for even k >= 2.
(ii) 0 <= c_j <= B_k(j), hence 0 <= e_k(j) <= 1.
(iii) 0 < A(m, j) <= 1 (purity of a probability list).
(iv) For even m >= k, with D_k(j) := B_k(j) - c_j,
0 <= D_m(j) <= rho_j^{m-k} D_k(j). (5.5)
Justification of (iv): write X_{b_j} = sum_xi xi P_xi (spectral
theorem; P_xi is the orthogonal projection onto the xi-eigenspace).
Then
D_m(j) = sum_{xi != lambda_j} (|xi|/lambda_j)^m ||P_xi v_{b_j}||^2.
Because |xi|/lambda_j <= 1 for every xi != lambda_j, the factor
(|xi|/lambda_j)^{m-k} of each term is at most rho_j^{m-k}; pulling
it out gives (5.5).
5.2 Lemma (transfer between temperatures). Let j >= 0 and even
m > k >= 2. Suppose e_k(j) < 1 and 0 < q < 1 with q >= rho_{j+1}^{m-k}.
Define, for 0 <= e < 1 and 0 < t <= 1:
F(q, e, t) := (q/(1-q)) * ( (1-e)^{-2}/t - 1 ). (5.6)
Then
e_m(j+1) <= F(q, e_k(j), A(m,j)). (5.7)
Consequence: if the right-hand side is strictly below 1, we may
conclude c_{j+1} > 0 as a byproduct, without any presupposition.
Proof. From (5.5) at the new temperature b_{j+1}:
B_m(j+1) - c_{j+1} <= q (B_k(j+1) - c_{j+1}).
Rearrange:
e_m(j+1) <= (q/(1-q)) * ( B_k(j+1) / B_m(j+1) - 1 ). (5.8)
We bound the ratio B_k(j+1)/B_m(j+1). Using B_k(j) = Y_k(b_j)/lambda_j^k
and Y_k(b_{j+1}) = A(k, j) Y_k(b_j)^2 (both definitions combined):
B_k(j+1) / B_m(j+1)
= [ Y_k(b_{j+1}) / Y_m(b_{j+1}) ] * lambda_{j+1}^{m-k}
= [ A(k,j) / A(m,j) ] * [ Y_k(b_j)^2 / Y_m(b_j)^2 ]
* lambda_{j+1}^{m-k}
= [ A(k,j) / A(m,j) ] * [ B_k(j) / B_m(j) ]^2 * r_j^{m-k}.
Use r_j <= 1 (4.10), A(k,j) <= 1 (5.4), and A(m,j) = t in the Lemma:
B_k(j+1) / B_m(j+1) <= [ B_k(j)/B_m(j) ]^2 / t.
Hypothesis e_k(j) < 1 implies c_j > 0 (from fact (ii)). And B_m(j) >=
c_j gives B_k(j)/B_m(j) <= B_k(j)/c_j = 1/(1 - e_k(j)).
Substituting into (5.8):
e_m(j+1) <= (q/(1-q)) * ( 1/((1-e_k(j))^2 t) - 1 ),
= F(q, e_k(j), t).
Taking t = A(m, j) gives (5.7). We divided only by the OLD overlap c_j
(bounded below by the hypothesis e_k(j) < 1); the new overlap
c_{j+1} > 0 follows from e_m(j+1) < 1 and B_m(j+1) > 0.
5.3 Length schedule. For j >= 2, define
k_2 := 142, k_{j+1} := M_j. (5.9)
So k_j = M_{j-1} for j >= 3, and in the step j -> j+1 we use m := k_{j+1}
= M_j and k := k_j. Applying Lemma 5.2 with the contraction factor
q := rho_{j+1}^{M_j - k_j}, we use the periodic input (4.8) to write
q <= d(u_{j+1})^{(M_j - k_j)/M_j} < 1.
We track two upper bounds at each level:
E_j >= e_{k_j}(j), (5.10)
eta_j >= 1 - A(M_j, j). (5.11)
5.4 Proposition (coupled boundary update). Suppose j >= 2,
0 <= E_j, eta_j < 1, e_{k_j}(j) <= E_j, A(M_j, j) >= 1 - eta_j. Let
E_{j+1}, eta_{j+1} in [0, 1) satisfy
E_{j+1} >= F( d(u_{j+1})^{(M_j - k_j)/M_j}, E_j, 1 - eta_j ), (5.12)
eta_{j+1} >= f( 1 - (1-eta_j)(1-u_j)(1-u_{j+1})(1-E_{j+1}) ). (5.13)
Then
e_{k_{j+1}}(j+1) <= E_{j+1},
A(M_{j+1}, j+1) >= 1 - eta_{j+1}.
Proof. Part 1 (the e-part). We have M_j > k_j, and n_{j+1} = M_j
(4.4). The periodic bound (4.8) at length n_{j+1}: rho_{j+1}^{M_j} <=
d(u_{j+1}); hence
rho_{j+1}^{M_j - k_j} <= d(u_{j+1})^{(M_j - k_j)/M_j} < 1.
Let q := d(u_{j+1})^{(M_j - k_j)/M_j} in (0, 1). Lemma 5.2 with
k := k_j and m := M_j = k_{j+1} gives
e_{M_j}(j+1) <= F(q, e_{k_j}(j), A(M_j, j))
<= F(q, E_j, 1 - eta_j) <= E_{j+1}.
The inequalities hold because F increases in its first two arguments
and decreases in its third on [0,1) x [0,1) x (0,1]; and because
E_j >= e_{k_j}(j) and 1 - eta_j <= A(M_j, j). Since E_{j+1} < 1,
c_{j+1} > 0 (as noted in Lemma 5.2).
Part 2 (the eta-part). Put M := M_j. Cancellation from definitions
of A and B gives
A(2 M, j) / A(M, j)
= r_j^M * ( B_{2M}(j+1) / B_M(j+1) )
* ( B_M(j) / B_{2M}(j) )^2. (5.14)
The squared factor B_M(j)/B_{2M}(j) is at least 1 because
B_M(j) >= B_{2M}(j) (per (5.2) monotonicity in k for even k; the
spectral factor |xi|/lambda_j is at most 1). For the first boundary
ratio: by (5.5) at the new temperature b_{j+1} with k = M, m = 2M,
B_{2M}(j+1) - c_{j+1} <= rho_{j+1}^M (B_M(j+1) - c_{j+1}) <= B_M(j+1) - c_{j+1},
so B_{2M}(j+1) <= B_M(j+1). Trivially B_{2M}(j+1) >= c_{j+1}. And
c_{j+1}/B_M(j+1) = 1 - e_M(j+1) >= 1 - E_{j+1}. So
B_{2M}(j+1) / B_M(j+1) >= c_{j+1}/B_M(j+1) >= 1 - E_{j+1}.
Combining with (5.14) and (4.11):
A(2 M, j) / A(M, j) >= r_j^M (1 - E_{j+1})
>= (1 - u_j)(1 - u_{j+1})(1 - E_{j+1}).
Using A(M, j) >= 1 - eta_j:
A(2 M, j) >= (1 - eta_j)(1 - u_j)(1 - u_{j+1})(1 - E_{j+1}). (5.15)
At the FIXED physical length 2 M = M_{j+1}, passing from inverse
temperature b_j to b_{j+1} = 2 b_j normalizes the squares of the Gibbs
probabilities. By Lemma 4.4 the resulting purity defect is bounded
by f of the preceding defect. Hence
1 - A(2 M, j) <= f( 1 - (1-eta_j)(1-u_j)(1-u_{j+1})(1-E_{j+1}) ),
which is (5.13).
======================================================================
6. Finite seed values.
======================================================================
6.1 Inputs from Appendices B and C. Two exact finite facts are
established below by explicit integer computations.
Proposition 6.1 (finite boundary data). Let alpha := .4042 and
P_*(x) := x^3 - .14 x^2 - .18 x. For m in {34, 78, 142, 240} with
h = 3/5 and a = 700741/500000:
E_0(H_m^opp) + a m < alpha. (6.1)
At inverse temperature b_0:
Y_6(b_0) < .027, c_0 P_*(lambda_0)^2 < .003745. (6.2)
Proof sources: (6.1) is Proposition B.1; (6.2) is Proposition C.1.
6.2 Two auxiliary quantities. For each of the four lengths in
(6.1), define
epsilon(m) := .027 * e^{b_0 alpha} * U^m * (.874)^{m-6} / ell^6, (6.3)
R(m) := e^{.132} U^m + epsilon(m) - 1. (6.4)
Explicit upper estimates:
R(78) ~ 0.2714, R(142) ~ 0.390, R(240) ~ 0.592.
These are direct evaluations of (6.4); exact rational certificates
appear in (6.13) below.
6.3 Lemma (seed bounds). For each m in {34, 78, 142, 240} and each
j >= 0:
e_m(0) <= epsilon(m), (6.5)
A(m, j) >= (1 + R(m)^{2^j})^{-2}. (6.6)
Proof. First inequality. Apply (5.5) at j = 0 with k = 6 and m >= 6,
together with rho_0 <= .874 (4.1) and ell < lambda_0 <= U:
lambda_0^m D_m(0)
<= U^m * (.874)^{m-6} * ( Y_6(b_0) / ell^6 )
< U^m * (.874)^{m-6} * .027 / ell^6. (6.7)
The physical partition function Y_m(b_0) is at least the largest
ground weight. The variational bound (6.1) gives |E_0(H_m^opp) + a m|
< alpha; consequently the largest unnormalized ground Boltzmann
weight in Y_m(b_0) = Tr exp(-b_0(H_m^opp + am I)) exceeds
exponential(-b_0 alpha) = e^{-b_0 alpha}. So
Y_m(b_0) > e^{-b_0 alpha}. (6.8)
Dividing (6.7) by lambda_0^m B_m(0) = Y_m(b_0):
e_m(0) = D_m(0)/B_m(0)
<= .027 e^{b_0 alpha} U^m (.874)^{m-6} / ell^6 = epsilon(m).
Second inequality. Factor P_*(x) = x(x - 1/2)(x + 9/25), which is
positive on the interval [ell, U]. A short polynomial check gives
4P_*(x) - x P_*'(x) = x(x^2 - .28x - .54) > 0 on [ell, U]; hence
x P_*'(x) < 4 P_*(x), which makes x^m / P_*(x)^2 increasing for
m > 8. Combined with (6.2):
e^{b_0 alpha} lambda_0^m c_0
<= U^m * (.003745 e^{b_0 alpha} / P_*(U)^2). (6.9)
The scalar factor .003745 e^{b_0 alpha} / P_*(U)^2 is a plain rational
number; a numerical check gives 1.133226 (i.e., the left coefficient
is < 1.133226). On the other side, e^{.132} > 1.141108. So
e^{b_0 alpha} lambda_0^m c_0 <= U^m * e^{.132}. (6.10)
Combining (6.7) and (6.10):
e^{b_0 alpha} Y_m(b_0)
= e^{b_0 alpha} ( lambda_0^m c_0 + lambda_0^m D_m(0) )
<= e^{.132} U^m + .027 U^m (.874)^{m-6} / ell^6
= e^{.132} U^m + epsilon(m)
= 1 + R(m). (6.11)
Now fix one physical ground eigenvector with unnormalized weight g
(uniqueness not needed). By the trial bound (6.1), g > e^{-b_0 alpha}.
Directly:
sum of non-ground weights = Y_m(b_0) - g
<= (1 + R(m)) e^{-b_0 alpha} - g.
Divide by g:
(sum of non-ground weights)/g
<= [(1 + R(m)) e^{-b_0 alpha} - g]/g
<= [(1 + R(m)) - 1] / 1 [using e^{-b_0 alpha}/g < 1]
<= R(m).
The transition used g > e^{-b_0 alpha}. In particular the left side
(sum of non-ground / g) is nonnegative, so R(m) >= 0.
At the doubling b_j = 2^j b_0 of the inverse temperature, the Gibbs
weights SQUARE. So the fraction (sum of non-ground)/(ground), which
starts at most R(m), becomes at most R(m)^{2^j} after j doublings
(a sum of nonnegative 2^j-th powers of numbers whose sum is <= R(m)
is at most R(m)^{2^j}).
The purity A(m, j) = sum_i p_i(b_j)^2 is at least the square of the
largest normalized Gibbs weight. The largest weight is at least
1/(1 + (rest/ground)) >= 1/(1 + R(m)^{2^j}). So
A(m, j) >= (1 + R(m)^{2^j})^{-2}. (6.12)
That is (6.6). Done.
6.4 Kickoff. epsilon(34) < .0928 < 1 (a direct rational check), so
e_34(0) < 1; in particular c_0 > 0.
Apply Lemma 5.2 twice, using rho_1 < .953 and rho_2 < .971:
e_78(1) <= F(.953^{44}, .093, (1+R(78))^{-2}) < .133, (6.13a)
e_142(2) <= F(.971^{64}, .133, (1+R(142)^2)^{-2}) < .14. (6.13b)
Physical-purity seed at m = 240, j = 2:
1 - A(240, 2) < .21. (6.13c)
Each of (6.13a) and (6.13b) proves positivity of the new leading
overlap c_{j+1} as a byproduct.
Set the coupled-recurrence starting point:
E_2 := .14, eta_2 := .21. (6.14)
6.5 Explicit numerical margins. From (6.6) at the specific pairs:
(1 + R(78))^{-2} > .6180,
(1 + R(142)^2)^{-2} > .7533,
(1 + R(240)^4)^{-2} > .7923. (6.15)
Substituting into (6.13) at the F values, the actual upper outputs
are .132129, .137301, and 1 - A(240, 2) < .207518. (6.16)
Starting from (6.14), the coupled recurrence (Proposition 5.4) gives
the following ROUNDED bounds:
j | E_j | eta_j | q_j := d(u_{j+1})^{(M_j - k_j)/M_j}
--+------+-------+------------------------------------
2 | .14 | .21 | .142
3 | .12 | .155 | .0282
4 | .016 | .026 | .0028
5 | .001 | .001 | ---- (6.17)
The exponents (M_j - k_j)/M_j at j = 2, 3, 4 are 98/240, 1/2, 1/2.
The successive upper outputs E_{j+1} are below
.117754, .015328, .000170,
and the corresponding upper outputs eta_{j+1} are below
.147078, .025188, .000432.
In each row, the displayed rounded E_{j+1} is used when computing the
following eta_{j+1}, so the table closes row by row.
All scalar comparisons in (6.15)-(6.17) are elementary rational
certifications. For 0 <= t <= 10, put T_80(t) := sum_{r=0}^{80} t^r/r!.
Then
T_80(t) <= e^t <= T_80(t) + 2 t^{81}/81! (6.18)
because the ratio of consecutive omitted terms is at most 10/82 < 1/2.
For negative exponents take reciprocals. Rational positive root
comparisons can be checked by raising both sides to integer powers;
for example the first q_j bound in (6.17) is d(u_3)^{49} < .142^{120}.
Every denominator is positive. A supplementary scalar certificate
records each individual step.
6.6 In particular, at j = 5 we have both E_5 <= .001 and
eta_5 <= .001. From here we radically simplify.
======================================================================
7. Quadratic continuation.
======================================================================
We enlarge the three defect bounds at j = 5 to .01 (which is far
above the actual values .001 and .000015) and run a single quadratic
recurrence.
Proposition 7.1. Define x_5 := .01 and x_{j+1} := 10 x_j^2 for
j >= 5. Then for every j >= 5:
u_j <= x_j, e_{k_j}(j) <= x_j, 1 - A(M_j, j) <= x_j. (7.1)
Consequently,
x_j = 10^{-1 - 2^{j-5}}. (7.2)
Proof. Base case j = 5: by (6.17) we have E_5 <= .001 <= .01 = x_5
and eta_5 <= .001 <= .01 = x_5. Also u_5 <= u_2 = .052 but in fact
much smaller (the quadratic sequence u_{j+1} = 6 u_j^2 gives
u_5 < .000015 < .01). So (7.1) holds at j = 5.
Inductive step. Suppose (7.1) holds at level j (with x := x_j <= .01).
Then in the coupled update (5.12)-(5.13):
First argument of F in (5.12). We have k_j = M_j/2 for j >= 3, and
u_{j+1} = 6 u_j^2 <= 6 x^2. Compute:
sqrt(d(u_{j+1}))
= sqrt(u_{j+1} / (2 - 3 u_{j+1}))
<= sqrt(6 x^2 / (2 - 18 x^2))
<= 2 x.
The last inequality: 6/(2 - 18 x^2) <= 4 (since 6 <= 8 - 72 x^2, i.e.,
72 x^2 <= 2, true for x <= .01), so sqrt(6/(2 - 18 x^2)) <= 2. So the
first argument of F is at most 2 x.
Bounding the first output. Since F increases in q and e and decreases
in t,
E_{j+1} <= F(2 x, x, 1 - x).
For x > 0,
F(2x, x, 1-x) / x^2
= (2 / (1 - 2 x)) * ( (1-x)^{-1} + (1-x)^{-2} + (1-x)^{-3} ).
This function is increasing on [0, .01] after continuous extension at
0; its endpoint value at x = .01 is 42430000 / 6792093 < 10. So we
may take E_{j+1} := 10 x^2.
Bounding the eta-update. For factors in [0, 1], the defect of a
product is at most the sum of the defects. So the four-factor product
in (5.13) has defect at most
x + x + 6 x^2 + 10 x^2 = 2 x + 16 x^2.
Then
f(2 x + 16 x^2) / x^2
= (2 + 16 x)^2 / (2 (1 - 2 x - 16 x^2)^2),
which is increasing on [0, .01] with endpoint value 3645000 / 1495729
< 10. So we may take eta_{j+1} := 10 x^2.
Finally, u_{j+1} <= 6 x^2 <= 10 x^2. All three bounds hold at level
j + 1, and the recurrence x_{j+1} = 10 x_j^2 stays in [0, .01]. Solving
from x_5 = .01 = 10^{-1}: x_6 = 10 * 10^{-2} = 10^{-3}, x_7 = 10 * 10^{-6}
= 10^{-7}, ..., x_j = 10^{-1 - 2^{j-5}}. Done.
======================================================================
8. Odd lengths and the physical gap.
======================================================================
8.1 Lemma (signed boundary comparison). For j >= 0, even M >= 2, and
any integer n >= M, define
Q_j(n) := Y_n^s(b_j) / lambda_j^n.
Then
|Q_j(n) - c_j| <= B_M(j) - c_j. (8.1)
Proof. The leading spatial line is fixed by U_P (Section 4.2), and its
eigenvalue is +lambda_j. Its contribution to Q_j(n) is therefore
exactly c_j. For every other eigenvalue xi of X_{b_j}, let P_xi be
its spectral projection. U_P commutes with X_{b_j}, hence with P_xi;
Cauchy-Schwarz in the range of P_xi gives
| <P_xi v_{b_j}, U_P P_xi v_{b_j}> | <= ||P_xi v_{b_j}||^2.
Therefore the absolute residual is bounded:
|Q_j(n) - c_j|
<= sum_{xi != lambda_j} (|xi|/lambda_j)^n ||P_xi v||^2
<= sum_{xi != lambda_j} (|xi|/lambda_j)^M ||P_xi v||^2
= B_M(j) - c_j.
Here n >= M and |xi|/lambda_j <= 1 for all residual xi, so the
factor (|xi|/lambda_j)^{n-M} <= 1. The series converges absolutely
(bounded by ||v_{b_j}||^2). The kernel of X_{b_j} contributes zero
since exponents are positive. Negative xi and odd n are included
because we take absolute values term by term.
8.2 Proposition (same-field purity bound). For j >= 5 and every
integer n in [M_j, 2 M_j]:
Y_n^s(b_{j+1}) / Y_n^s(b_j)^2 >= 1 - 4 x_j. (8.2)
Proof. Set M := M_j.
Upper bound at b_j: by (8.1) at temperature b_j with length M,
Q_j(n) <= B_M(j). Squaring is legitimate because Y_n^s(b_j) > 0 and
Q_j(n) > 0.
Lower bound at b_{j+1}: by (8.1) with the SAME even M,
Q_{j+1}(n) >= 2 c_{j+1} - B_M(j+1). By Proposition 7.1,
c_{j+1} >= (1 - x_{j+1}) B_M(j+1); so
Q_{j+1}(n) >= (1 - 2 x_{j+1}) B_M(j+1) > 0.
Ratio:
Y_n^s(b_{j+1}) / Y_n^s(b_j)^2
= Q_{j+1}(n) / Q_j(n)^2 * r_j^n (from definition)
>= [ (1 - 2 x_{j+1}) B_M(j+1) / B_M(j)^2 ] * r_j^n.
Now B_M(j+1) / B_M(j)^2 = A(M, j) / r_j^M (Section 5.4, part 2). So
... = (1 - 2 x_{j+1}) A(M, j) r_j^{n-M}
>= (1 - 2 x_{j+1}) (1 - eta_j) r_j^M
>= (1 - 2 x_{j+1}) (1 - x_j) (1 - u_j) (1 - u_{j+1}).
The first inequality: 0 <= n - M <= M because n in [M, 2M], and
0 < r_j <= 1, so r_j^{n-M} >= r_j^M; also A(M, j) >= 1 - eta_j
>= 1 - x_j by Proposition 7.1. Second inequality: (4.11).
Defect of the last product P:
1 - P = 1 - (1 - 2 x_{j+1})(1 - x_j)(1 - u_j)(1 - u_{j+1})
<= 2 x_{j+1} + x_j + u_j + u_{j+1} (product defect <= sum of defects)
<= 20 x_j^2 + x_j + x_j + 6 x_j^2
= 2 x_j + 26 x_j^2
<= 4 x_j
(using x_j <= .01 and x_{j+1} = 10 x_j^2).
This is (8.2).
8.3 Proof of Theorem 1.
Fix an ODD integer n in [M_j, 2 M_j] with j >= 5. Let
p_0 >= p_1 >= ... be the Gibbs weights of H_n^s at inverse temperature
b_j, listed with multiplicity.
The left side of (8.2) equals sum_i p_i(b_j)^2 (the Gibbs purity at
b_j). Since sum_i p_i^2 <= p_0 * sum_i p_i = p_0 (because every
p_i <= p_0 and sum p_i = 1), we have
p_0 >= 1 - 4 x_j > 1/2.
So the physical ground state at (n, b_j) is simple.
Then
exp(-b_j (E_1 - E_0)) = p_1 / p_0
<= (1 - p_0) / p_0
<= 4 x_j / (1 - 4 x_j) (8.3)
< 10^{-2^{j-5}}.
For the last strict inequality, use x_j = 10^{-1 - 2^{j-5}} = (1/10) *
10^{-2^{j-5}}, so
4 x_j / (1 - 4 x_j) < (4/10 * 10^{-2^{j-5}}) / (1 - .04)
= .4166... * 10^{-2^{j-5}}
< 10^{-2^{j-5}}.
Since b_j = 49 * 2^{j-2} = 392 * 2^{j-5}, taking logarithms of (8.3):
b_j (E_1 - E_0) > 2^{j-5} log(10),
i.e.
(392 * 2^{j-5}) (E_1 - E_0) > 2^{j-5} log(10),
i.e.
E_1 - E_0 > log(10) / 392. (8.4)
Finally, M_5 = 240 * 2^{3} = 1920, and M_{j+1} = 2 M_j so the length
scales double. The intervals
[M_5, 2 M_5] = [1920, 3840],
[M_6, 2 M_6] = [3840, 7680],
...
tile [1920, infinity). Every integer n >= 1920 lies in one of them.
Every odd length n = 2 L+1 with L >= 960 (equivalently, n >= 1921) is
covered. Relabeling sites (0, ..., n-1) -> (-L, ..., L) identifies
H_n^s with H_{2L+1}(h) as in (1.1), giving (T1) with the single field
h = 3/5 and the uniform gap log(10)/392. Done.
======================================================================
9. Tasaki's topological index -1.
======================================================================
9.1 Boundary-selected states. A local observable acts on finitely
many sites of Z, extended by the identity elsewhere; a state is a
normalized positive linear functional on these. For L >= 1, choose a
normalized ground vector phi_L of H_L(3/5). Uniqueness up to phase
holds for L >= 960 by Theorem 1. Define
omega_L(A) := <phi_L, A phi_L> whenever supp(A) is contained in
[-L, L].
Compactness of density matrices on any fixed finite interval,
combined with a diagonal subsequence over intervals, produces integers
L_k -> infinity along which
omega(A) := lim_{k -> infinity} omega_{L_k}(A) (9.1)
exists for every local observable A. The limits are consistent on
nested intervals, hence define a state. Any such limit is called a
"boundary-selected" state.
The bulk Hamiltonian is the formal sum
H_bulk := sum_{j in Z} S_j · S_{j+1}.
For a local A, [H_bulk, A] is a finite sum over the bonds meeting
supp(A). The limit omega([H_bulk, A]) = lim_k omega_{L_k}([H_L(3/5), A]).
9.2 Local twist. For epsilon > 0, define
U_epsilon := exp( -i sum_{j in Z, |j| <= pi/epsilon}
(pi + epsilon j) S_j^z ).
The rotations by 0 and 2 pi at the far edges of the support act
trivially on spin-1 states, so the sum is really only over a bounded
window. For gapped, reflection-invariant, z-rotation-invariant states,
Tasaki's theorem (2025) gives the existence of
Ind(omega) := lim_{epsilon -> 0} omega(U_epsilon) in {±1},
with the twist convention of Tasaki's equations (10)-(13).
9.3 Theorem 2 (Tasaki index). Every boundary-selected state (9.1) is
invariant under uniform z-rotations and under site reflection j -> -j.
With gamma_* := log(10)/392, it satisfies the local gap inequality
omega(A^* [H_bulk, A])
>= gamma_* ( omega(A^* A) - |omega(A)|^2 ) (9.2)
for every local A. In particular it is a locally unique gapped ground
state in the sense of Tasaki 2025 Definition 11, and its Tasaki index
equals -1.
Proof. Both finite-volume symmetries (uniform z-rotation and site
reflection) commute with H_L(3/5). Simplicity of the ground eigenspace
implies omega_L is invariant under them, and passing to the limit (9.1)
preserves invariance.
For L >= 960, the finite spectral theorem plus Theorem 1 gives
<A phi_L, (H_L(3/5) - E_0(L,3/5)) A phi_L>
>= gamma_* ( omega_L(A^* A) - |omega_L(A)|^2 ).
The expression on the left equals omega_L(A^* [H_L(3/5), A]). Once
supp(A) and its adjacent bonds lie strictly inside [-L, L], the
commutator [H_L(3/5), A] equals [H_bulk, A] (the endpoint fields and
all omitted bonds commute with A). Taking k -> infinity in (9.1)
proves (9.2). The right side of (9.2) is nonnegative by Cauchy-Schwarz;
for omega(A) = 0 it says omega(A^* [H_bulk, A]) >= 0.
Tasaki 2025 Theorem 3 (with the subsequential interpretation spelled
out in the End Matter equation (27) of that paper) gives index -1 for
every state that is (a) invariant under uniform z-rotation and site
reflection, (b) locally unique gapped as above, (c) built from the
same bulk Hamiltonian H_bulk with the spin normalization and coupling
used here. All three hold; (a)-(b) we just checked, (c) from the
definition.
For the explicit sign of the twist: Tasaki's Lemmas 6-8 give, for
small fixed epsilon > 0 and all sufficiently large L,
-1 <= omega_L(U_epsilon)
<= -sqrt(1 - 4 (pi + epsilon) epsilon / gamma_*). (9.3)
The -1 endpoint uses the fact that the finite ground state has total
S^z = 1 by that paper's Lemma 1; reflection makes the twist
expectation real; the uniform gap prevents it from crossing zero for
small twist. The condition L >= pi/epsilon ensures that the
finite-chain twist equals the local operator U_epsilon above. Passing
to the limit (9.1), then letting epsilon -> 0, gives
Ind(omega) = -1.
The corollary evaluates the twist index for the selected states. It
does not require the full sequence omega_L to converge to a unique
state, nor does it identify these states with every ground state of
the infinite chain. In particular, the additional global uniqueness
hypothesis of Tasaki 2025 Corollary 4 for the half-chain conclusion
is not part of the present result.
======================================================================
Appendix A. The history expansion and the boundary identity.
======================================================================
A.1 Cartesian spin identity. With the conventions of Section 1,
S^z = diag(-1, 0, 1). The matrices S^z, S^x are real symmetric, and
S^y = (S^+ - S^-)/(2i) is purely imaginary anti-symmetric. Thus
G_alpha := i S^alpha is real for every alpha in {x, y, z}. Direct
computation of the 9 x 9 matrix on the right of
- S · S = G_x (tensor) G_x + G_y (tensor) G_y + G_z (tensor) G_z
gives the negative of the spin-bilinear on C^3 (tensor) C^3.
A.2 Time-ordered expansion. For any self-adjoint H_0 on the n-site
tensor power and any b > 0,
e^{-b H_0}
= sum_{d=0}^{infinity} (-1)^d (1/d!) sum_{j_1, ..., j_d = 1}^{n-1}
int_{0 < t_1 < ... < t_d < b} dt_1 ... dt_d
h_{j_d}(t_d) ... h_{j_1}(t_1) (A.1)
where h_j is the summand of H_0 associated to the jth bond (in the
usual time-ordered convention: earliest time on the right).
For a bond term, one summand of (-1)(S_j · S_{j+1}) equals the -S·S
bilinear on sites (j, j+1); the identity (A.1) turns it into a sum
over alpha of G_alpha^{(j)} (tensor) G_alpha^{(j+1)}. So a bond event
at time t_r on bond j contributes a bilinear in the G_alpha's at sites
j and j+1 with coefficient (-1).
For the endpoint fields of H_n^opp, the exponential splits:
exp(-b H_n^opp) = exp(+b h S_0^z) exp(-b bond part) exp(-b h S_{n-1}^z).
Each field event on site 0 contributes the operator +h S_0^z; each
field event on site n-1 contributes -h S_{n-1}^z. These commute with
the bond events because they act on disjoint sites.
A.3 From series to X_b. Assemble the full time-ordered sum, being
careful about the traces. Group events site by site:
- At each interior site 0 < r < n-1, the events come from the two
incident bonds j = r-1 and j = r. Combine all such events into a
single history omega_r in Omega_b, ordered by time. The weight of
omega_r inside the trace is k_b(omega_r, omega_{r+1}) (Section 3.2).
- At the left endpoint r = 0, the events are all from bond j = 0
plus the field events from +h S_0^z. Combine into omega_0.
- At the right endpoint r = n-1, the events are all from bond
j = n-2 plus the field events from -h S_{n-1}^z. Combine into
omega_n.
Interleaving the n-1 internal histories omega_1, ..., omega_{n-1}
against the boundary histories omega_0, omega_n partitions the whole
b-time simplex (up to null collisions). Integrating over the
internal histories produces the n-fold composition
(X_b^n)_{omega_0, omega_n} = e^{-a b n}
int_{Omega_b^{n-1}} prod_{r=0}^{n-1} k_b(omega_r, omega_{r+1})
dmu_b(omega_1) ... dmu_b(omega_{n-1}). (A.2)
A.4 Boundary vectors as fields. The weight v_b(omega_0) on the left
boundary is (i h)^d if all d marks of omega_0 equal z, else 0. The
conjugation in <v_b, ...> contributes (-i h)^d at the left. Inside
the trace, each z-mark of omega_0 contributes a G_z at site 0.
Combining:
(-i h) G_z = (-i h)(i S_z) = (+h) S_z.
So each left marked event contributes +h S_0^z, matching the -h S_0^z
field sign of H_n^opp (since -h S_0^z in H corresponds to +h S_0^z in
exp(-b H) = exp(+b h S_0^z) ...).
At the RIGHT boundary, v_b(omega_n) contributes (i h)^d directly (no
conjugation). Each z-mark contributes a G_z at site n-1. Combining:
(i h) G_z = (i h)(i S_z) = (-h) S_z.
So each right marked event contributes -h S_{n-1}^z, matching the
-h S_{n-1}^z field sign in exp(-b H) coming from -h S_{n-1}^z in H.
A.5 Rotation P. Replacing the right vector v_b by U_P v_b changes the
right marked history to signed marks along P e_z = -e_z. The resulting
right field is +h S_{n-1}^{P e_z} = -h S_{n-1}^z. So
<v_b, X_b^n U_P v_b> = Y_n^s(b). (A.3)
Similarly, <v_b, X_b^n v_b> = Y_n(b).
A.6 Absolute convergence. Before the scalar shift, a summand of (A.2)
is bounded by 3^n (each k_b |.| <= 3) times the boundary-vector
magnitudes ||v_b||^2 times the mu_b^{n-1} total mass of the interior
histories. All are finite: mu_b(Omega_b) = e^{3 b} < infinity, and
||v_b||^2 = e^{b h^2} < infinity. So the series (A.1) converges
absolutely and the regrouping is legitimate.
A.7 The rotation group. For g in G, the operator U_g in (3.4) is
real unitary (permutation of coordinates followed by sign changes).
Because G acts on each ternary coordinate independently, and because
-S·S = sum_alpha G_alpha (tensor) G_alpha transforms under G into the
sum over alpha of (g G)_alpha (tensor) (g G)_alpha with (g G)_beta =
sum_alpha g_{beta alpha} G_alpha, and because the row vectors of g are
orthonormal, the sum is preserved. So U_g commutes with X_b.
A.8 Even and odd n. For even n >= 2, the operator X_b^n equals
|X_b|^n on the range of the v_b pairing (spectral theorem: for each
eigenvalue xi with eigenprojection P_xi, (X_b^n P_xi v_b, v_b) = xi^n
(P_xi v_b, v_b) = |xi|^n (P_xi v_b, v_b) because xi^n = |xi|^n for
even n). So Y_n(b) = <v_b, |X_b|^n v_b> is a positive weighted moment
of the absolute spectral values.
For odd n >= 2, the same spectral decomposition leaves the sign of xi
in place, so Y_n^s(b) is a signed pairing and must be handled via
Lemma 8.1.
======================================================================
Appendix B. Exact trial vectors.
======================================================================
B.1 Goal. Prove the four variational inequalities
E_0(H_m^opp) + a m < alpha := 2021/5000 = .4042
for m in {34, 78, 142, 240}, (B.1)
with H_m^opp = sum_{j=1}^{m-1} S_j · S_{j+1} - (3/5) S_1^z + (3/5)
S_m^z and a = 700741/500000.
B.2 Virtual space. Let V be the set of 26 labels (i, d):
1 <= i <= 8, d in {-l_i, -l_i + 2, ..., l_i},
(l_1, ..., l_8) = (1, 1, 1, 1, 3, 3, 3, 5). (B.2)
Order labels first by i, then by increasing d. So the multiplicity of
each virtual weight d = -5, -3, -1, 1, 3, 5 in V is 1, 4, 8, 8, 4, 1.
B.3 Bond tensors. For each s in {-1, 0, 1}, define a real 26 x 26
matrix A_s as follows. At row (i, d) and column (j, e), put
l := l_i, k := l_j, r := (l + d)/2. (B.3)
The entry is zero unless
e = d + 2 s, |k - l| <= 2, k = l => i = j. (B.4)
For an allowed entry, take the coefficient in column 2 of Table B.1
and multiply by the factor in the s-column. All labels must lie in V;
an out-of-range column is absent (entry zero).
Table B.1. Coefficients and s-column factors.
Class | Coefficient | s=-1 | s=0 | s=+1
------+--------------+--------------------+-------------------+--------------
k=l+2 | P_ij | 2(l+1-r)(l+2-r) | 2(r+1)(l-r+1) | (r+1)(r+2)
k=l | D_i | 2(l-r+1) | d | -(r+1)
k=l-2 | -P_ji | 2 | -2 | 1
B.4 Parameter tables.
Table B.2. Virtual types, D_i, and endpoint parameters
beta_i, kappa_i.
i | l_i | D_i | beta_i | kappa_i
---+-----+------+--------+--------
1 | 1 | -289 | 10000 | 12
2 | 1 | -84 | 5620 | 12
3 | 1 | -9 | 2456 | 12
4 | 1 | 33 | 4868 | 12
5 | 3 | -22 | 82 | 6
6 | 3 | 27 | -809 | 6
7 | 3 | 71 | 12475 | 6
8 | 5 | -11 | -9645 | 1
Table B.3. All fifteen couplings P_ij with l_j = l_i + 2.
(i,j) | P_ij | (i,j) | P_ij | (i,j) | P_ij
--------+------+---------+------+---------+------
(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
B.5 Endpoints. Set columns b, c in R^{26} by
b_{(i,d)} = beta_i if d = -1, else 0,
c_{(i,d)} = kappa_i beta_i if d = -1, else 0. (B.5)
Both b and c are supported on virtual weight d = -1.
B.6 Trial vector. For m >= 2,
psi_m := sum_{s_1, ..., s_m in {-1, 0, 1}}
( b^T A_{s_1} ... A_{s_m} c ) e_{s_1} (tensor) ... (tensor) e_{s_m}. (B.6)
The e_s are the orthonormal spin-1 basis. Every coefficient b^T A_{s_1}
... A_{s_m} c is an integer (the A_s, b, c have integer entries); the
expansion is an ordinary real-amplitude decomposition; no virtual inner
product or additional metric is inserted; normalization is
unnecessary.
The support rule (B.4) forces sum_j s_j = 0 for a nonzero amplitude
(both endpoints have virtual weight -1). This is a linear restriction
on psi_m and does NOT restrict the variational bound: the variational
upper bound applies to any trial vector in the physical Hilbert space
regardless of S^z content.
B.7 Bond contraction. For s, t in {-1, 0, 1}, put B_{st} := A_s A_t
and define
T_* := sum_{s in {-1,0,1}} A_s (tensor) A_s, (B.7)
G_* := sum_{s in {-1,0,1}} s A_s (tensor) A_s, (B.8)
O_* := sum_{s,t} s t B_{st} (tensor) B_{st}
+ sum_{s in {-1,0}, t in {0, 1}}
( B_{st} (tensor) B_{s+1,t-1}
+ B_{s+1,t-1} (tensor) B_{st} ). (B.9)
Use the Kronecker convention
(C (tensor) D)_{(v,w),(v',w')} := C_{v,v'} D_{w,w'}.
Write L_0 := (b (tensor) b)^T (a 1 x 676 row), u := c (tensor) c (a
676 x 1 column), L_i := L_0 T_*^i (i >= 0).
Lemma B.4 (contraction formulas). With V_m := <psi_m, psi_m> and
H_m^* := 500000 <psi_m, (H_m^opp + a m I) psi_m>:
V_m = L_m u, (B.10)
H_m^* = 500000 sum_{r=0}^{m-2} L_0 T_*^r O_* T_*^{m-2-r} u
- 300000 L_0 G_* T_*^{m-1} u
+ 300000 L_{m-1} G_* u
+ 700741 m L_m u. (B.11)
In particular, V_m and H_m^* are integers.
Proof of (B.10). Expanding T_*^{m} = (sum_s A_s (tensor) A_s)^m sums
the products of two identical real amplitudes over all physical
words (s_1, ..., s_m) in {-1, 0, 1}^m. Combined with L_0 and u
this produces sum_{s_1..s_m} (b^T A_{s_1} ... A_{s_m} c)^2, which is
exactly V_m. In particular, the amplitude-squared sum is
integer-valued.
Proof of (B.11). Inserting G_* instead of T_* at one site multiplies
the summand by the weight s and therefore inserts S^z at that site.
For a bond, using S·S = S^z (tensor) S^z + (1/2)(S^+ (tensor) S^- +
S^- (tensor) S^+), the diagonal entry on (s, t) is s t, and the
off-diagonal entries are 1 between (s, t) and (s+1, t-1) for s in
{-1, 0}, t in {0, 1} (each allowed ladder entry is sqrt(2); the two
summands of O_* give the two directions of the matrix element). So
O_* inserts exactly one unit-coupling bond at any position, with no
extra factor or metric.
The endpoint coefficients are 500000 (-3/5) = -300000 at the LEFT and
500000 (3/5) = +300000 at the RIGHT. The shift a m contributes
500000 a m = 700741 m times the norm. There are m-1 bond positions
and no closing bond. All matrices, rows, and columns are integral.
B.8 Reduction to 162 dimensions. Consider the paired virtual label
((i, d), (j, e)). Each nonzero entry of A_s changes the virtual weight
by 2 s from row to column. So A_s (tensor) A_s preserves d - e, hence
T_* and G_* preserve d - e. Each nonzero entry of B_{st} changes the
weight by 2(s+t); the two factors in every O_* summand have the same
total physical weight, including the pair (s, t) and (s+1, t-1) in
the off-diagonal terms. So O_* preserves d - e.
Let Q := {(i, d), (j, d) : both labels in V}. That is, both virtual
weights are equal. The multiplicities of d = -5, -3, -1, 1, 3, 5 in V
are 1, 4, 8, 8, 4, 1, so dim Q = 1 + 16 + 64 + 64 + 16 + 1 = 162.
T_*, G_*, O_* are block-diagonal by weight difference d - e. Both L_0
and u are supported in Q (the endpoint vectors are supported on
d = -1). So we may restrict T_*, G_*, O_* to Q before multiplication,
exponentiation, or forming the expressions of Lemma B.4. Squaring the
restricted T_* gives the restriction of T_*^2; no contribution is
lost.
B.9 Integer row recurrence. Define integer rows (all in Q):
L_i := L_{i-1} T_* (i >= 1), (B.12)
N_1 := -300000 L_0 G_*, (B.13)
N_i := N_{i-1} T_* + L_{i-2}(500000 O_* + 700741 T_*^2)
(i >= 2). (B.14)
Then (induction on m)
V_m = L_m u, (B.15)
H_m^* = (N_m + 300000 L_{m-1} G_* + 700741 L_m) u. (B.16)
To see (B.16), expand N_m by (B.14) into
N_m = -300000 L_0 G_* T_*^{m-1}
+ sum_{r=0}^{m-2} L_0 T_*^r (500000 O_* + 700741 T_*^2)
T_*^{m-2-r}.
Its bond and left-field contributions match (B.11); each of its m-1
shift contributions equals 700741 L_m; the remaining +700741 L_m
supplies the last site, and +300000 L_{m-1} G_* u supplies the right
field.
B.10 Three-row implementation. Only three rows are needed. Start
x := L_0, y := L_0 T_*, w := -300000 L_0 G_*.
For m = 2, 3, 4, ..., 240, update (using OLD rows on the right):
w_new := w T_* + 500000 x O_* + 700741 (x T_*) T_*,
x_new := y, y_new := y T_*.
After the loop, (x, y, w) = (L_{m-1}, L_m, N_m). At each m compute
V := y u, H := w u + 300000 x (G_* u) + 700741 V. (B.17)
Then (V, H) = (V_m, H_m^*). All operations are on integer matrices of
fixed size 162; ordinary arbitrary-precision arithmetic suffices.
B.11 Certificate table. Define integers p_m, q_m, k_m by
10^{p_m - 1} <= V_m < 10^{p_m},
q_m 10^{p_m-6} <= V_m < (q_m + 1) 10^{p_m-6},
k_m V_m <= 20 H_m^* < (k_m + 1) V_m. (B.18)
The last equivalence is k_m = floor(10^7 H_m^*/(500000 V_m)), since
10^7/500000 = 20. The first comparison certifies V_m > 0, i.e.,
psi_m is nonzero.
Table B.4. Exact integer contraction certificates.
m | p_m | q_m | k_m | H_m^*/(500000 V_m) in
----+------+--------+---------+--------------------------
34 | 202 | 715867 | 4031974 | [.4031974, .4031975)
78 | 440 | 747265 | 4041175 | [.4041175, .4041176)
142 | 787 | 121288 | 4041110 | [.4041110, .4041111)
240 | 1317 | 165352 | 4041006 | [.4041006, .4041007)
The four strict comparisons
202100 V_m - H_m^* > 0 for m in {34, 78, 142, 240} (B.19)
are recorded directly in the raw integer output; they also follow
from (B.18) since k_m + 1 < 4042000 = 20 * 202100 for every displayed
k_m.
B.12 Proof of Proposition 6.1, first set. For each of the four
lengths,
V_m = <psi_m, psi_m> > 0 (by B.18),
and
E_0(H_m^opp) + a m
<= <psi_m, (H_m^opp + a m I) psi_m> / <psi_m, psi_m>
= H_m^* / (500000 V_m)
< 202100 / 500000 = alpha (by B.19).
All four inequalities (B.1) hold with h = 3/5. QED
======================================================================
Appendix C. Certified short-chain thermal traces.
======================================================================
C.1 Goal. Prove Proposition 6.1, second set:
Y_6(b_0) < .027, c_0 P_*(lambda_0)^2 < .003745, (C.1)
with b_0 = 49/4, h = 3/5, a = 700741/500000, lambda_0 the positive
leading eigenvalue of X_{b_0}, c_0 the squared norm of the projection
of v_{b_0} onto the leading line, and P_*(x) = x^3 - .14 x^2 - .18 x.
All terminating decimals below denote exact rationals.
C.2 Fifteen short-chain traces. For 2 <= n <= 6 and g in {+1, -1, 0},
define
H_{n,g} := sum_{j=0}^{n-2} S_j · S_{j+1} - (3/5) S_0^z + (3/5) R_g,
R_g := g S_{n-1}^z (g = ±1),
R_g := S_{n-1}^x (g = 0),
Z_{n,g} := Tr exp(-b_0 (H_{n,g} + a n I)). (C.2)
Then Z_{n,1} = Y_n(b_0) and Z_{n,-1} = Y_n^s(b_0).
C.3 Reduction to fifteen traces. Let G := the group of proper signed
permutations of the coordinate axes (order 12). The orbit of the axis
vector e_z under G is the six signed coordinate axes, each with
multiplicity 2. Set
Pi := (1/|G|) sum_{g in G} U_g,
I(n) := <v_{b_0}, X_{b_0}^n Pi v_{b_0}>.
The boundary-transfer identity (3.7)/(3.8) writes I(n) as an average
of the six corresponding right-field partition functions (right field
in the six signed axis directions). Global z-rotation identifies the
four axes perpendicular to z, so those four equal Z_{n,0} after a
unitary. Hence
I(n) = ( Z_{n,1} + Z_{n,-1} + 4 Z_{n,0} ) / 6. (C.3)
The averaging operator Pi is an orthogonal projection commuting with
X_{b_0}. By Proposition 4.1, G fixes the leading line pointwise.
Therefore
c_0 P_*(lambda_0)^2
<= ||P_*(X_{b_0}) Pi v_{b_0}||^2
= <v_{b_0}, P_*(X_{b_0})^2 Pi v_{b_0}>
= I(6) - .28 I(5) - .3404 I(4) + .0504 I(3) + .0324 I(2). (C.4)
The last expansion uses P_*(x)^2 = x^6 - .28 x^5 - .3404 x^4 + .0504
x^3 + .0324 x^2 (direct multiplication). In particular, no positivity
of the individual odd powers of X_{b_0} is assumed: (C.4) uses only
nonnegativity of Pi and the spectral decomposition of X_{b_0}.
So it suffices to enclose the fifteen Z_{n,g}.
C.4 Integer matrix A. Set
C_n := 3 floor((n-1)/2) + 2 ((n-1) mod 2) + 6/5,
A := I - (H_{n,g} + C_n I) / 8. (C.5)
Claim: -I <= A <= I on the spin Hilbert space of n sites.
Direct verification. A single spin-1 bond has eigenvalues -2, -1, +1.
For two consecutive bonds, couple the two outside spins to spin t in
{0, 1, 2} and then with the middle spin to total spin j. The operator
S_1 · S_2 + S_2 · S_3 = S_2 · (S_1 + S_3)
has eigenvalues ( j(j+1) - t(t+1) - 2 ) / 2 with |t - 1| <= j <= t + 1.
Its minimum is -3. Pairing consecutive bonds, with one single bond
left over if n-1 is odd, gives
0 <= H_{n,g} + C_n I <= ( n - 1 + 6/5 + C_n ) I <= 16 I. (C.6)
Adding the lower bounds for paired bonds is valid even where the
three-site supports overlap. The upper bounds in (C.6) at n = 2, ...,
6 are 5.4, 7.4, 10.4, 12.4, 15.4. So -I <= A <= I; positivity of A
itself is not required.
C.5 Integer matrix D. Index the basis by words w = (w_0, ..., w_{n-1})
in {-1, 0, 1}^n. Let
p_w := (sum_j w_j) mod 2 in {0, 1},
J_{ww} := sqrt(1 + p_w),
D := 80 J A J^{-1}.
Claim: D is an integer matrix with diagonal
eta(w) := 80 - 10 C_n - 10 sum_{j=0}^{n-2} w_j w_{j+1}
+ 6 w_0 - 6 g w_{n-1}. (C.7)
Each allowed bond hop w <-> w + sigma(e_j - e_{j+1}), sigma = ±1, has
entry -10 (the corresponding spin-1 bond entry is 1 and such a hop
preserves p_w). When g = 0, an allowed end hop from column
r = w + sigma e_{n-1} to row w has entry -3(1 + p_w): the end-field
entry before similarity is 3 sqrt(2)/10, and the two parities are
opposite, so
-10 (3 sqrt(2)/10) sqrt((1 + p_w)/(1 + p_r)) = -3 (1 + p_w).
These are all the entries of D.
C.6 Approximate the thermal exponential. Let
E := exp(-b_0 (H_{n,g} + C_n I) / 2) = e^{-49} exp(49 A). (C.8)
For a finite matrix M, define its squared Frobenius norm by
||M||_F^2 := Tr(M^* M). Since E is self-adjoint,
||E||_F^2 = Tr exp(-b_0 (H_{n,g} + C_n I)). We approximate E by a
normalized Poisson polynomial in the contraction A; the similarity
D = 80 J A J^{-1} permits an integer Horner recurrence.
K := 192, Q := 2^{50},
s_i := 49^i K! / i! (i = 0, ..., K), S_K := sum_{i=0}^{K} s_i,
t_i := floor(Q s_i / S_K). (C.9)
Initialize X_{K+1} := 0 (the zero matrix). For i = K, K - 1, ..., 0,
compute
X_i := floor(D X_{i+1} / 80) + t_i I. (C.10)
Matrix floors are entrywise; every floor, including for negative
arguments, is toward -infinity. Define
Ehat := J^{-1} (X_0 / Q) J, W_{n,g} := ||Ehat||_F^2. (C.11)
C.7 Scalar (word-by-word) version. Number words w by the integer
index sum_j (w_j + 1) 3^j; interpret any lookup outside {-1, 0, 1}^n
as zero. Initialize x(w, r) = 0 for all pairs (w, r). At stage
i = K, K-1, ..., 0, use the current array to form
z(w, r) := eta(w) x(w, r)
- 10 sum_{j=0}^{n-2} sum_{sigma = ±1}
x(w + sigma (e_j - e_{j+1}), r)
- 1_{g = 0} * 3 (1 + p_w)
sum_{sigma = ±1} x(w + sigma e_{n-1}, r). (C.12)
Replace all entries simultaneously:
x_new(w, r) := t_i 1_{w = r} + floor(z(w, r) / 80).
At the end,
k_{n,g} := floor(10^{16} W_{n,g})
= floor( 10^{16} * sum_{w, r} x(w, r)^2 (2 - p_w)(1 + p_r)
/ (2 Q^2) ). (C.13)
The weight factor (2 - p_w)(1 + p_r) comes from
(1 + p_r)/(1 + p_w) = (2 - p_w)(1 + p_r) / 2.
All operations in (C.9), (C.12), (C.13) are integer operations
suitable for arbitrary-precision arithmetic.
C.8 Table.
Table C.1. k_{n,g} = floor(10^{16} W_{n,g}).
n \ g | +1 | -1 | 0
------+-------------------+-----------------+------------------
2 | 10185296075298 | 4160017987 | 583578389575
3 | 259254948723 | 571111660440989 | 47796720049308
4 | 318278534893 | 661347985 | 43971049724
5 | 260908126900 | 29917222640283 | 5659378873682
6 | 16467340956 | 328219854 | 4086161507
By (C.9), (C.12), (C.13), a finite exact-arithmetic verification
certifies each entry.
C.9 Error of the approximation. Write d := 3^n.
Lemma C.2. For every (n, g) in Table C.1,
|| Ehat - exp(-b_0 (H_{n,g} + C_n I)/2) ||_F < 2 * 10^{-10}. (C.14)
Proof. Since b_0/2 = 49/8, the exact target is
E = e^{-49} exp(49 A) = sum_{i=0}^{infinity} e^{-49} 49^i / i! A^i.
The polynomial with coefficients q_i := s_i / S_K is the truncation of
this Poisson expansion, renormalized to have coefficient sum one. Its
coefficient l^1 error is twice the omitted Poisson mass, at most 2 R,
where
R := (49^{193} / 193!) / (1 - 49/194) < 3.152 * 10^{-33}. (C.15)
In bounding the omitted Poisson mass by R we have discarded the
factor e^{-49}; the coefficient estimate appears in the proof of
Proposition 1 of Fox and Glynn (1988); here it is applied to powers
of a contraction A.
Because A is a contraction, the Frobenius error from the tail is at
most 2 sqrt(d) R.
At each Horner stage, the coefficient replacement q_i -> t_i / Q
contributes less than sqrt(d) / Q in Frobenius norm. The entrywise
floor contributes less than d / Q before undoing the similarity, and
less than sqrt(2) d / Q afterwards, because ||J^{-1}|| ||J|| <=
sqrt(2). Subsequent propagation is left multiplication by A and
cannot increase the Frobenius norm. Hence the total Frobenius error is
less than
2 sqrt(d) R + 193 (sqrt(d) + sqrt(2) d) / Q
<= 54 R + 193 (27 + sqrt(2) * 729) / 2^{50}
< 54 R + 432513 / 2^{51}
< 2 * 10^{-10}.
The penultimate inequality uses only sqrt(2) < 3/2; the last is a
rational comparison using (C.15).
C.10 Partition error. Squaring the Frobenius norm converts (C.14)
into a thermal trace. Since E is self-adjoint and
||E||_F^2 = Tr exp(-b_0 (H_{n,g} + C_n I)), we obtain
| Z_{n,g} - e^{b_0 (C_n - a n)} W_{n,g} |
< e^{b_0 (C_n - a n)}
( 4 * 10^{-10} sqrt(W_{n,g}) + 4 * 10^{-20} ). (C.16)
Direct from | ||E||_F^2 - ||Ehat||_F^2 | <= 2 ||E - Ehat||_F
||Ehat||_F + ||E - Ehat||_F^2.
C.11 Rational intervals from the table. The integer floors in Table
C.1 suffice; no unprinted digits of W_{n,g} are used. For an entry
k := k_{n,g}, put
L := k / 10^{16}, U := (k + 1) / 10^{16},
r := ( floor(sqrt(k + 1)) + 1 ) / 10^8,
epsilon := 4 * 10^{-10} r + 4 * 10^{-20}. (C.17)
Then L <= W_{n,g} < U, and sqrt(W_{n,g}) < r. In every entry
L - epsilon > 0, so (C.16) gives
e^{b_0 (C_n - a n)} (L - epsilon) < Z_{n,g}
< e^{b_0 (C_n - a n)} (U + epsilon). (C.18)
The "sqrt" in (C.17) is an integer square root. To enclose the
exponential by rational numbers, use the truncated Taylor series
T_80(y) := sum_{j=0}^{80} y^j / j!.
For 0 <= y <= 10, comparison of consecutive tail terms gives
T_80(y) <= e^y <= T_80(y) + 2 y^{81} / 81!. (C.19)
For y negative, take reciprocals and reverse the endpoints. All
arguments b_0 (C_n - a n) that appear lie between -.0544635 and
9.691073. So (C.17)-(C.19) require only rational arithmetic and
integer square roots.
In particular, they give
.0266319440 < Z_{6,1} < .0266319607 < .027. (C.20)
C.12 Polynomial evaluation. Using (C.18) and (C.19) in (C.3) gives
the following rational centers, each with absolute error less than
10^{-7}:
Table C.2. Centers of I(n).
n | center of I(n)
--+---------------
2 | .02703074
3 | .01203562
4 | .01193511
5 | .00931560
6 | .00893272
Substituting these centers in the RHS of (C.4) gives .003744031780.
The sum of the absolute polynomial coefficients (.28 + .3404 + .0504
+ .0324 + 1 = 1.7032) bounds the total propagation error by
1.7032 * 10^{-7}. So the RHS of (C.4) is less than
.003744031780 + .00000017032 = .0037442021 < .003745. (C.21)
Together with (C.20) this proves Proposition 6.1, second set. QED
======================================================================
End of proof.
======================================================================
Citations of the companion paper:
The periodic spin-one Haldane gap, OpenAI Math Release preprint
OAI:The-periodic-spin-one-Haldane-gap-September-24-2026. Specific
references used here: Proposition 3.1 (spatial transfer and its
periodic identity), the finite periodic inputs stated as
Proposition 4.1 above (with constants m_J = 1.0657205 and m_N =
.2783155, and the 1 - S(72, b_1) <= 151249/2500000 seed), and
the Z_{72}(b) > 1, Z_{120}(b) > 1 ground-weight bound.
External references:
F. D. M. Haldane, Phys. Lett. A 93, 464 (1983); Phys. Rev. Lett.
50, 1153 (1983).
H. Tasaki, Phys. Rev. Lett. 134, 076602 (2025) (arXiv:2407.17041v4).
We use equation (2), Lemma 1, Lemmas 6-8, Theorem 3, and equation
(27) of the End Matter.
Aizenman and Nachtergaele, Comm. Math. Phys. 164, 17-63 (1994).
Time-ordered Poisson expansion for spin systems.
Fox and Glynn, Comm. ACM 31, 440-445 (1988). Poisson coefficient
tail bound used in Appendix C.9.