A boundary-field gap for the spin-one Heisenberg chain

1 Introduction

Equal positive magnetic fields at the endpoints of an odd open spin-one Heisenberg chain select a unique finite-volume ground state [20, Lemma 1]. The remaining question is whether the excitation gap stays positive as the chain grows. We prove a quantitative gap for one fixed field in the pure bilinear model. This is the boundary formulation used by Tasaki to connect the Haldane gap problem to a nontrivial topological index [20].

In the orthonormal basis e_m, m=-1,0,1, of \C^3, let

and put S^-=(S^+)^*, S^x=(S^++S^-)/2, and S^y=(S^+-S^-)/(2i). Thus \sum_\alpha(S^\alpha)^2=2\Id. For L\ge1 and h>0, consider the operator on (\C^3)^{\otimes(2L+1)} given by

There is no bond between the endpoints. Write E_0(L,h)\le E_1(L,h) for its two lowest eigenvalues, counted with multiplicity. We address the fixed-boundary-field gap assertion: for some single h>0, the difference E_1(L,h)-E_0(L,h) stays bounded away from zero as L\to\infty.

Context and prior work. Haldane’s prediction, formulated in 1981 and published in 1983, distinguishes integer and half-odd-integer isotropic antiferromagnetic Heisenberg chains, with a positive bulk gap for positive integer spin [8, 9, 10]. For spin one-half, Lieb, Schultz and Mattis had already constructed excitations of the periodic chain whose energy above the ground state vanishes with its length [13, Appendix B]. The Affleck–Kennedy–Lieb–Tasaki construction gave a rigorous gapped spin-one model [1, 2]. Its interaction contains a biquadratic term: with q=\mathbf S_j\cdot\mathbf S_{j+1}, it is proportional to q+q^2/3 up to an additive constant. Our Hamiltonian has the pure bilinear interaction q and fixed endpoint fields.

The endpoint condition matters even when the bulk interaction is fixed. White and Huse numerically characterized the effective spin-one-half degrees of freedom at free ends [21]. To estimate bulk properties, including the gap, they added physical spin-one-half endpoint sites and adjusted their couplings. Magnetic fields on the original spin-one endpoints give a different finite-volume model, whose gap requires its own boundary estimates.

Tasaki formulated precisely this boundary problem in his work on the topological character of the spin-one Heisenberg ground state [20]. Equation (2) of that paper is the Hamiltonian (1.1); Lemma 1 proves uniqueness for every L\ge1 and h>0. Assumption 2 asks for a positive gap uniform in all sufficiently large L, at one fixed h>0. Theorem 1.1 resolves this fixed-boundary-field Haldane gap assertion positively and establishes that assumption. Section 7 then derives Tasaki index -1 for every boundary-selected subsequential local limit.

The topological viewpoint developed through several distinct descriptions. Den Nijs and Rommelse introduced nonlocal string order [4], and Kennedy and Tasaki connected it to hidden symmetry breaking [11]. Pollmann, Turner, Berg and Oshikawa described symmetry protection through matrix-product states and the entanglement spectrum [18]; Ogata extended the bond-centered reflection index to infinite-chain states beyond the matrix-product description [15]. Tasaki’s local-twist index [19] instead uses the twist expectation studied by Nakamura and Todo [14]. The boundary-state theorem in [20] uses uniform z rotations and site-centered reflection. These invariances imply invariance under site reflection combined with a global \pi rotation about z, the symmetry studied by Fuji, Pollmann and Oshikawa [7]. Our consequence concerns Tasaki’s index in this precise symmetry setting; it does not identify the different diagnostics above.

Methodological inputs. For frustration-free chains, Lemm and Mozgunov proved finite-size criteria involving both interior and boundary-segment gaps [12]. Here the boundary information comes from weighted spatial moments and a coupled Gibbs-purity recurrence. The finite trial vectors use the matrix-product framework of finitely correlated states [5] and its finite-chain formulation [17]; they certify energy upper bounds that initialize the recurrence.

The continuous-history representation is an instance of the time-ordered Poisson expansion of Aizenman and Nachtergaele [3]. The particular spatial transfer operator and finite periodic estimates used here come from [16], which proves a uniform gap on even periodic rings. We retain that source’s spin normalization, coupling, and energy shift, and state explicitly the transfer and finite-input results used below. The endpoint pairings and all estimates specific to the open chain are proved here.

Proof strategy. For each inverse temperature b>0, a partition function of the periodic model is a trace of a power of a compact self-adjoint spatial operator X_b. For the open chain with opposite endpoint fields, the corresponding partition function is instead a boundary pairing \langle v_b,X_b^n v_b\rangle, where n is the number of sites. At even n, this is a positive weighted moment of the absolute spatial eigenvalues. A rotation of the right boundary vector produces the same-field Hamiltonian in (1.1).

The periodic estimates show that one spatial eigenvalue dominates sufficiently high even moments. For the open chain, however, the contribution of each eigenspace is weighted by its overlap with v_b, and this vector changes with temperature. Periodic concentration therefore leaves a boundary-overlap problem. We solve it by contracting the residual boundary-weighted moments while tracking the physical Gibbs purity (the sum of the squared Gibbs probabilities) of the opposite-field chain. These two estimates improve together when both length and inverse temperature double, without assuming a nonzero boundary overlap at the next temperature. Once the leading spatial line dominates the boundary pairing, its rotation invariance controls the same-field expression at every odd length in a full dyadic interval. The purity defect then decreases at a rate exponential in the inverse temperature, which yields a uniform physical gap.

Sections 2 and 3 establish the transfer representation and the periodic estimates in the form needed here. Section 4 proves the boundary recurrence. Section 5 initializes and iterates it, and Section 6 proves Theorem 1.1 at all the required odd lengths. Section 7 applies the resulting gap to the boundary-selected infinite-volume states. Appendices A and B give complete finite certificates for the boundary initialization. All terminating decimals used in inequalities denote exact rationals.

2 Spatial transfer with endpoint fields

We first put the periodic and open chains in the same transfer representation. The boundary vectors, including their complex phases, will distinguish the physical endpoint fields. The time-ordered bond expansion has a general formulation in [3, arXiv v1, Section 2.2, Equations (2.10)–(2.12)]. We prove the required boundary identity directly with the specified kernel and field coefficients.

2.1 Hamiltonians and shifted partition functions

From now on set

It is convenient to label an n-site chain by 0,\ldots,n-1. Write S_j\cdot S_k=\sum_\alpha S_j^\alpha S_k^\alpha and define

For n=2L+1, H_n^{\rm s} is (1.1) after relabeling. Our partition functions are ordinary traces over the entire physical Hilbert space:

The shift is per site, also for open chains. It will cancel from all physical Gibbs weight ratios.

2.2 The spatial operator

We use the concrete spatial-transfer operator in [16, Proposition 3.1]. Its auxiliary space records ordered bond events in imaginary time; each physical site contributes a kernel depending on its two incident histories. Use the unitarily equivalent Cartesian spin matrices (S^\alpha)_{rs}=-i\varepsilon_{\alpha rs} and set G_\alpha=iS^\alpha. Then G_\alpha is real, \norm{G_\alpha}=1, and

A history is an ordered list \omega=((t_1,\alpha_1),\ldots,(t_d,\alpha_d)) with 0<t_1<\cdots<t_d<b and \alpha_r\in\{x,y,z\}; the empty list is allowed. Give the disjoint union \Omega_b of these labeled simplexes Lebesgue measure on each simplex and mass one at the empty history. Its measure \mu_b has total mass e^{3b}.

For two histories, merge their events in chronological order and take the trace of the resulting product of G matrices, with increasing time from left to right. Denote this real kernel by k_b(\omega,\eta). Set it to zero at nonempty time collisions, a null set, and use trace 3 for the two empty histories. The kernel is symmetric: exchanging the two histories leaves their merged list unchanged. In particular, symmetry does not reverse a product of noncommuting matrices. Moreover |k_b|\le3. On \mathcal K_b=L^2(\Omega_b,\mu_b;\C) define

It is real, self-adjoint, and Hilbert–Schmidt. The cited transfer Proposition identifies its periodic moments:

where \mu_i(b) lists its nonzero eigenvalues with multiplicity. This exact operator, with the shift in (2.1), is the one used for the periodic inputs in Section 3.

Rotations will let us change one endpoint field and, later, show that the leading spatial line is fixed. Let D_2 consist of the identity and the three diagonal rotations by \pi, let C cyclically permute the coordinate axes, and put \mathcal G=\langle D_2,C\rangle. This is a group of twelve proper signed permutations. If ge_\alpha=\epsilon_g(\alpha) e_{\sigma_g(\alpha)}, write \sigma_g\omega for relabeling the marks and s_g(\omega)=\prod_r\epsilon_g(\alpha_r). The operators

form a real unitary representation and commute with X_b. This follows either directly by rotating every matrix in the kernel, or from the same Proposition. In particular, fix P=\diag(1,-1,-1), so Pe_z=-e_z.

2.3 Boundary vectors

Define v_b\in\mathcal K_b by

The empty-history value is one. The simplex volumes give \norm{v_b}^2=\sum_{d\ge0}b^dh^{2d}/d!=e^{bh^2}. The inner product below is conjugate linear in its first argument.

For even lengths, the opposite-field expression is the positive moment Y_n(b)=\langle v_b,|X_b|^n v_b\rangle. At odd lengths, the powers of X_b retain the signs of its eigenvalues. We first concentrate the even opposite-field moments. The resulting leading-line estimate will then control the signed same-field pairing at odd lengths; Figure 1 distinguishes the two boundary pairings.

3 Periodic concentration estimates

The boundary argument requires quantitative information about the spatial spectrum, including the change of its leading eigenvalue when the inverse temperature doubles. We extract that information from finite periodic-chain estimates in [16]. The extraction is given in detail because the periodic gap theorem alone does not provide these boundary estimates. Throughout, terminating decimals denote exact rational numbers, and eigenvalue lists count multiplicities.

For even n\ge4, put

The first is the purity of the spatial probabilities |\mu_i(b)|^n/Z_n(b), where (\mu_i(b)) is the nonzero spectrum of X_b; the second is the purity of the physical Gibbs probabilities. Both lie in (0,1]. Doubling n squares the first probability list, whereas doubling b squares the second.

For later reference we also record explicitly the interaction of the two periodic purities. This is [16, Lemma 4.2]; we include its short proof by partition-function cancellation.

We next state the finite inputs gathered in the companion [16, Corollary 5.6]. Its shift, spin normalization and unit bilinear coupling are exactly those used here. In particular, we use an intermediate row in the initialization proof, not merely its final conclusion.

No boundary Hamiltonian occurs in Proposition 3.3. The finite periodic thermal and variational certificates supporting it are given in the cited source. We now deduce all the spatial estimates that will be used for boundaries. The residual ratios will contract the boundary-weighted tails. The upper and lower bounds comparing successive leading eigenvalues will control the change of temperature in the boundary concentration and physical-purity estimates, respectively. Finally, positivity and rotation invariance of the leading line will control the same-field pairing at odd lengths.

4 Concentration of boundary-weighted moments

We now pass from unweighted spatial traces to the boundary pairing for the opposite-field chain. We will control two quantities together: the fraction of an even boundary moment outside the leading spatial line, and the defect of the physical Gibbs purity. The boundary coefficients depend on temperature. We compare the weighted moments across temperatures through physical partition-function identities, while contracting the residual spatial powers at a fixed temperature.

Use the simple positive leading eigenvalue \lambda_j and residual ratio \rho_j from Section 3. Let P_j be the orthogonal projection onto its line, and put

We use k,m even and at least two. Then B_k(j)=Y_k(b_j)/\lambda_j^k>0, so e_k(j) is defined even if c_j=0. Also 0\le e_k(j)\le1 and 0<A(m,j)\le1. Thus e_k(j) is the residual fraction of a spatial boundary moment, whereas A(m,j) is the physical Gibbs purity of the opposite-field chain at inverse temperature b_j.

The normalized absolute spatial eigenvalues are at most one. Thus B_m(j)\le B_k(j) when m\ge k. More precisely, if D_k(j)=B_k(j)-c_j, the spectral theorem gives

The powers act on the residual spectrum; no upper bound for the total boundary norm is needed in this comparison.

To couple this residual estimate to physical purity, we use a shorter reference length for the boundary defect. Recall M_j=240\,2^{j-2} and u_j from Proposition 3.4. For j\ge2 let

We track upper bounds E_j on e_{k_j}(j) and \eta_j on 1-A(M_j,j). The difference M_j-k_j supplies residual contraction at the next temperature. Its output at length k_{j+1}=M_j then controls the ratio B_{2M_j}(j+1)/B_{M_j}(j+1) needed to improve physical purity. The notation \eta_j denotes only this physical purity defect.

5 Finite initialization and quadratic improvement

We now initialize the boundary recurrence and show that it continues indefinitely. The sixth boundary moment starts a sequence of residual estimates at (34,b_0), (78,b_1), and (142,b_2). A physical-purity estimate at (240,b_2) then starts the coupled update. The inputs are four variational energies and thermal moments on at most six sites.

5.1 Starting residual and physical-purity bounds

The thermal estimates bound the residual and leading contributions to Y_m(b_0), while the variational estimate supplies a lower bound for one ground weight. Keep \ell,U from the periodic seed and define, for each of the four lengths in (5.1),

In particular \epsilon(34)<.0928<1, proving c_0>0. The estimates \rho_1\le.953 and \rho_2\le.971 now allow two applications of Lemma 4.1:

The bound at m=240, j=2 gives 1-A(240,2)<.21. Thus the induction of Proposition 4.2 starts with

Both new leading overlaps have also been proved positive by the strict defect bounds; no overlap was assumed at the next temperature.

5.2 Explicit finite comparisons

We record numerical margins to make the initialization reproducible. The three lower bounds from Lemma 5.2 used above are

Using the full expressions in (5.3) gives upper outputs below .132129, .137301, and .207518 in (5.5), (5.6), and 1-A(240,2) respectively. The lower bounds in (5.8) are rounded for display.

Starting from (5.7), the following rounded bounds satisfy Proposition 4.2:

The exponents (M_j-k_j)/M_j are 98/240, 1/2, 1/2. The successive upper outputs for E_{j+1} are below .117754,.015328,.000170; those for \eta_{j+1} are below .147078,.025188,.000432. In computing the latter outputs one uses the displayed rounded E_{j+1}, so each line closes with the next.

For clarity, all the scalar comparisons here and in the periodic initialization have an elementary rational verification. For 0\le t\le10, put T_{80}(t)=\sum_{r=0}^{80}t^r/r!. Then

Indeed the ratio of consecutive omitted terms is at most 10/82<1/2. For negative exponents use reciprocal intervals. Rational positive root comparisons can be checked by raising both sides to their integer powers; for example the first q bound in (5.9) is d(u_3)^{49}<.142^{120}. Every denominator is positive. These rules, exact rational input values, and the displayed formulas suffice to verify the finite comparisons without floating-point assumptions. The supplementary scalar certificate records all of them.

5.3 Indefinite continuation

We now enlarge the three defect bounds at j=5 to .01 so that a single quadratic recurrence controls all later levels.

6 Same fields, odd lengths, and the physical gap

The estimates obtained so far concern even moments of the opposite-field chain. We now use the rotation-invariant leading line to control the same-field partition function at every odd length in a dyadic interval.

One concentrated even moment therefore controls the normalized same-field pairing Q_j(n) at every longer length, including odd lengths. To compare two temperatures, we use the common even reference length M_j=k_{j+1}. The lower bound for r_j^{M_j} will cover the interval M_j\le n\le2M_j.

7 The boundary-selected infinite-volume state

The gap just proved supplies the hypothesis of Tasaki’s topological-index theorem. We formulate the consequence for subsequential local limits, so that no convergence or uniqueness of the infinite-volume state is presupposed.

A local observable is a matrix acting on finitely many sites of \mathbb Z, extended by the identity on the other sites. A state on these observables is a normalized positive linear functional. Write \phi_L for a normalized ground vector of H_L(3/5) and set \omega_L(A)=\langle\phi_L,A\phi_L\rangle whenever the support of A lies in [-L,L]. For L\ge960 the vector is unique up to phase by Theorem 1.1, so this functional is well defined. Compactness of the density matrices on each finite interval, followed by a diagonal subsequence over the intervals, gives integers L_k\to\infty for which

exists for every local observable. The limits are consistent on nested intervals and therefore define a state. We call any such state boundary-selected.

The bulk Hamiltonian is the formal sum H_{\rm bulk}=\sum_{j\in\mathbb Z}S_j\cdot S_{j+1}. For a local A, its commutator [H_{\rm bulk},A] is the finite sum of the commutators with bonds meeting the support of A. To specify the index, for \varepsilon>0 define the local twist

The rotations by 0 and 2\pi outside this interval act trivially for spin one. For the symmetric gapped states considered below, Tasaki’s theorem guarantees the existence of the index \operatorname{Ind}(\omega)=\lim_{\varepsilon\downarrow0} \omega(U_\varepsilon)\in\{-1,1\}; we use precisely the twist convention in [20, Equations (10)–(13)].

The corollary evaluates the twist index for the selected states. It does not require the full sequence \omega_L to converge or identify these states with every ground state of the infinite chain. In particular, the additional global uniqueness hypothesis of the half-chain conclusion in [20, Corollary 4] is not part of our result.

A Certified short-chain thermal moments

This appendix proves the two short-chain estimates used to initialize the boundary argument. Throughout, h=3/5, a=700741/500000, and b_0=49/4. All terminating decimals below denote exact rational numbers.

The first bound controls the residual contribution to Y_m(b_0) in Lemma 5.2; the second controls the leading contribution. We first reduce both bounds to fifteen short-chain partition functions, then enclose these traces using matrices of order at most 3^6=729.

A.1 Reduction to fifteen thermal traces

For 2\le n\le6 and g\in\{1,-1,0\}, define

Thus Z_{n,1}=Y_n(b_0) and Z_{n,-1}=Y_n^{\rm s}(b_0). The first bound in Proposition A.1 concerns Z_{6,1}. For the second, average the right-field direction over the rotation group.

Let \mathcal G be the rotation group generated by the diagonal half-turns and the cyclic permutation of the coordinate axes, and let

The group has order 12, and the orbit of z is the six signed coordinate axes, each with the same multiplicity. The boundary transfer identity therefore writes I(n) as the average of the six corresponding right-field partition functions. Global rotations about z identify the four perpendicular orientations. Hence

The averaging operator \Pi is an orthogonal projection commuting with X_{b_0}. By Proposition 3.4, \mathcal G fixes the leading line pointwise. Therefore

In particular, no positivity of the individual odd powers of X_{b_0} is being assumed.

It therefore suffices to enclose Z_{n,g} for 2\le n\le6 and g\in\{1,-1,0\}. The remaining subsections provide these enclosures and evaluate (A.4).

A.2 An integer matrix for the thermal approximation

Set

The operator A is a self-adjoint contraction. Here is a direct verification with the spin normalization in the theorem. A single bond has eigenvalues -2,-1,1. For two consecutive bonds, couple the two outside spins to spin t\in\{0,1,2\} and then couple this with the middle spin to total spin j. The operator

has eigenvalues [j(j+1)-t(t+1)-2]/2, with |t-1|\le j\le t+1. Its minimum is -3. Pairing consecutive bonds, with one single bond left over if necessary, and using the total boundary-field norm 6/5 gives

Adding the lower bounds for the paired bonds is valid even where the three-site supports overlap. In order n=2,\ldots,6, the displayed upper bounds are 5.4,7.4,10.4,12.4,15.4. Thus -\Id\le A\le\Id; positivity of A itself is not required.

Index the spin-weight basis by words w=(w_0,\ldots,w_{n-1}) in \{-1,0,1\}^n, and let

The matrix D is integral. Indeed, its diagonal is

Each allowed bond hop w\leftrightarrow w+\sigma(\mathbf e_j-\mathbf e_{j+1}), \sigma=\pm1, has entry -10: the corresponding entry of the spin-one bond is 1, and such a hop preserves p_w. When g=0, an allowed end hop from column r=w+\sigma\mathbf e_{n-1} to row w has entry -3(1+p_w). To see this coefficient, the end-field entry before similarity is 3\sqrt2/10, and the two parities are opposite, so

These are all the entries of D.

A.3 The exact recurrence

The recurrence approximates the half-temperature exponential

For a finite matrix M, write \|M\|_{\rm F}^2=\Tr(M^*M) for its squared Frobenius norm. Since E is self-adjoint, \|E\|_{\rm F}^2 is the thermal trace \Tr\exp[-b_0(H_{n,g}+C_n\Id)]. We approximate E by a normalized Poisson polynomial in the contraction A; the similarity D=80JAJ^{-1} permits an integer Horner recurrence.

Put

Starting with the zero matrix \mathcal X_{K+1}, compute

Matrix floors are entrywise, and every floor, including one with a negative argument, is toward -\infty. Define

For completeness, the following scalar version specifies the recurrence without matrix construction. Number words, if desired, by \sum_j(w_j+1)3^j, and interpret a lookup outside \{-1,0,1\}^n as zero. Initialize x(w,r)=0 for all pairs of words. At stage i=K,\ldots,0, use the current array to form

Replace all entries simultaneously. At the end, the required table entry is the integer quotient

The weight follows from (1+p_r)/(1+p_w)=(2-p_w)(1+p_r)/2. All operations in (A.7), (A.10), and (A.11) are integer operations; they may be performed with arbitrary-precision integers.

The resulting fifteen entries are

Equations (A.7), (A.10), and (A.11) provide a finite exact-arithmetic verification of every entry.

A.4 Error of the matrix approximation

Squaring the Frobenius norm converts this estimate into a thermal trace. In fact, E is self-adjoint and \|E\|_{\rm F}^2=\Tr\exp[-b_0(H_{n,g}+C_n\Id)]. Consequently

This follows from |\|E\|_{\rm F}^2-\|\widehat E\|_{\rm F}^2| \le2\|E-\widehat E\|_{\rm F}\|\widehat E\|_{\rm F} +\|E-\widehat E\|_{\rm F}^2.

A.5 Rational intervals from the printed table

The displayed integer floors suffice to verify the needed estimates; no unprinted digits of W_{n,g} are used. For an entry k=k_{n,g}, put

Then L\le W_{n,g}<U and \sqrt{W_{n,g}}<r. In every entry L-\varepsilon>0, so (A.15) gives

The square root in (A.16) is just an integer square root. To enclose the remaining exponential by rational numbers, set

For 0\le y\le10, comparison of consecutive tail terms gives

For a negative argument, take reciprocals and reverse the endpoints. All arguments b_0(C_n-an) here lie between -.0544635 and 9.691073. Thus (A.16)–(A.18) require only rational arithmetic and integer square roots. In particular they give

Using (A.17) and (A.18) in (A.3) gives the following rational centers, each with absolute error less than 10^{-7}:

Substitution of these centers in the right-hand side of (A.4) gives .003744031780. The sum of the absolute polynomial coefficients is 1.7032, so the total possible error is less than 1.7032\cdot10^{-7}. The right-hand side of (A.4) is consequently less than

Together with (A.19), this proves Proposition A.1.

B Exact trial vectors for opposite fields

We prove the four variational inequalities used to initialize the boundary argument. Throughout this appendix,

and the unshifted opposite-field Hamiltonian on m sites is

The bulk matrices below are those of the periodic variational construction in [16, Section “Two variational inputs”]. We give their complete integer specification and the new endpoint vectors, so the present finite estimates can be reproduced without importing a variational bound for open chains.

The proof consists of a fixed integer matrix-product vector, an exact contraction for its norm and energy, and four integer comparisons. No spectral approximation enters the calculation.

B.1 The complete tensor and endpoint data

Let \mathcal V have the 26 labels

Order them first by i and then by increasing d. Three real 26-by-26 matrices A_s, s\in\{-1,0,1\}, are specified as follows. At row (i,d) and column (j,e) put \ell=\ell_i, k=\ell_j, and r=(\ell+d)/2. The entry is zero unless

For an allowed entry, multiply the coefficient in the second column of Table 1 by the factor in its s column. The parameters D_i and P_{ij} appear in Tables 2 and 3. The latter parameter is defined only when \ell_j=\ell_i+2. All labels in these rules must belong to (B.3); an out-of-range column is absent.

Define columns \mathbf b,\mathbf c\in\mathbb R^{26} by

All their parameters are included in Table 2.

For each m\ge2, the physical vector is

This is the open-boundary matrix-product form of [17, Section 3.1, Equation (11)], with endpoint rows and columns absorbed into the first and last matrices. The vectors e_{-1},e_0,e_1 are the orthonormal spin basis fixed in the statement of the problem. In particular, the coefficients in (B.6) are ordinary real amplitudes: no virtual inner product or physical basis weights are inserted. Normalization of the matrices A_s is unnecessary. The support rule forces a nonzero amplitude to have \sum_j s_j=0, because both endpoints have virtual weight -1. This restriction on the trial vector does not restrict the variational upper bound on the smallest eigenvalue of the full Hamiltonian.

The following tables complete the integer specification of these matrices and endpoints.

B.2 Norm, fields, and the bond insertion

For s,t\in\{-1,0,1\}, put B_{st}=A_sA_t and define

We use the usual Kronecker convention (C\otimes D)_{(v,w),(v',w')}=C_{v,v'}D_{w,w'}. Write

B.3 The invariant 162-dimensional contraction

All contractions in Lemma B.2 can be carried out in a single 162-dimensional space. On a paired virtual label ((i,d),(j,e)), consider the difference d-e. Each nonzero entry of A_s changes virtual weight by 2s from row to column. Consequently every summand A_s\otimes A_s of T_* or G_* preserves this difference. A nonzero entry of B_{st} changes weight by 2(s+t). The two factors in every tensor summand of O_* have the same total physical weight, including the pair (s,t) and (s+1,t-1) in the off-diagonal terms. Each such summand also preserves d-e.

Let \mathcal K_0 be the coordinate subspace with paired labels

The multiplicities of the virtual weights d=-5,-3,-1,1,3,5 are 1,4,8,8,4,1, respectively; hence

The preceding argument shows that T_*,G_*,O_* are block diagonal by weight difference. Both L_0 and u are supported in \mathcal K_0, because the two endpoint vectors are supported on d=-1. We may therefore restrict these three matrices to (B.12) before multiplying them, taking powers, or forming the expressions of Lemma B.2. In particular, squaring the restricted T_* gives the restriction of T_*^2; there is no contribution through a discarded state. From now on the same symbols denote these restricted integer matrices and endpoint tensors, ordered by the inherited lexicographic order.

B.4 An exact row recurrence

The energy sum in (B.11) need not be evaluated separately at every bond. Define integer rows by

Then

To verify the latter identity, induction on m in (B.14) gives

Its bond contributions and left field are exactly those in (B.11). Each of its m-1 shift contributions equals 700741L_m. The final term 700741L_m in (B.15) supplies the remaining site shift, and the other final term supplies the right field with the required positive sign. This proves (B.15).

For specificity, the whole contraction can be implemented by keeping just three rows. Begin with

For each m=2,3,\ldots,240, make the following updates, whose right-hand sides all use the old rows:

After this iteration, (x,y,w)=(L_{m-1},L_m,N_m). At a required length compute

Thus (V,H)=(V_m,H_m^*). The initialization realizes the same invariant at m=1, which proves the indexing of the algorithm. Ordinary exact integer matrix multiplication suffices throughout; sparsity can reduce the work but changes no operation or comparison.

B.5 The integer comparisons

Table 4 gives the results of (B.17)–(B.18). The integers p_m, q_m, and k_m in the table have the following exact meanings:

The last comparison is equivalent to k_m=\lfloor10^7H_m^*/(500000V_m)\rfloor, since 10^7/500000=20. The first comparison in particular certifies that the vector \psi_m is nonzero.

All four strict comparisons

follow as well. They are recorded directly in the full integer output, and they also follow from the last line of (B.19), since every displayed k_m+1 is less than 4042000=20\cdot202100.

The accompanying computational files make the integer comparison fully reproducible. The program verification/trial_verify.py constructs the matrices from the data in this appendix and executes the row recurrence with arbitrary-precision integers. The file verification/trial-exact-matrices.json contains every entry of A_s, the endpoint vectors, the restricted paired labels, and the matrices T_*,G_*,O_*. The file verification/trial-exact-results.json contains the full integers V_m, H_m^*, and 202100V_m-H_m^* for each row, rather than only their shortened display in Table 4. The equations and tables of this appendix specify the calculation independently of these files.