Spin-one chains: purities, Haldane gaps, and boundary fields

Source and verification

This article explains family 268 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Spin chains and a uniform spectral gap

A finite quantum spin chain is a tensor product of small Hilbert spaces, with a Hamiltonian that couples neighboring factors. For spin one, each site has space ${\mathbb{C}}^3$. In the basis $e_{-1},e_0,e_1$, define

$$\begin{array}{c}{S}^z{e}_m=m{e}_m,\\ S^{+}{e}_m=\sqrt{2-m(m+1)}{e}_{m+1},\\ (m<1),\\ S^{+}{e}_1=0,S^{-}=(S^{+}{)}^{\ast },\\ S^x=(S^{+}+S^{-})/2,\\ S^y=(S^{+}-S^{-})/(2i).\end{array}$$

These self-adjoint matrices satisfy $(S^x{)}^2+(S^y{)}^2+(S^z{)}^2=2I$. Write $S_j^{\alpha }$ for $S^{\alpha }$ acting at site $j$, and ${\mathbf{S}}_j\cdot {\mathbf{S}}_k={\sum }_{\alpha }{S}_j^{\alpha }{S}_k^{\alpha }$.

On a periodic ring with even length $L\ge 4$, the pure bilinear antiferromagnetic Heisenberg Hamiltonian, with coupling one, is

$$H_L=\sum _{j=0}^{L-1}{\mathbf{S}}_j\cdot {\mathbf{S}}_{j+1},S_L^{\alpha }=S_0^{\alpha }.$$

Let $E_0(L)$ be its lowest energy and $E_1(L)$ the next distinct energy. Its gap is ${\gamma }_L=E_1(L)-E_0(L)$. Each non-scalar finite-dimensional Hamiltonian has a positive gap in this sense. A uniform gap requires a lower bound that persists as the number of sites grows.

Family 268 contains a manuscript claiming such a bound for even periodic rings and a companion treating odd open chains with specified endpoint magnetic fields. Their boundary conditions and spectral conventions must be kept separate. The prerequisites are tensor products, the finite-dimensional spectral theorem, compact self-adjoint operators, elementary probability, and matrix exponentials. (OpenAI, 2026) (OpenAI, 2026a)

Claimed periodic bounds

The ground state is unique for every even $L\ge 60$. For these lengths, ${\gamma }_L>(4/105)\log (80/79)$. For every even $L\ge 2304$, the stronger bound ${\gamma }_L>\log (20)/784$ holds. Consequently the even-length lower limit of the gaps is at least $\log (20)/784$. (OpenAI, 2026b)

The interaction here is just ${\mathbf{S}}_j\cdot {\mathbf{S}}_{j+1}$. The well-known gapped AKLT model includes an additional quadratic term in this bond operator. A gap for that different interaction, or for a neighborhood of it, does not by itself prove the stated result at the pure bilinear point.

🏷️ Two probability distributions from one partition function

At inverse temperature $b>0$, introduce the shifted partition function

$$\begin{array}{c}{Z}_L(b)=\mathrm{Tr}{e}^{-b(H_L+aLI)},\\ a=\frac{700741}{500000}.\end{array}$$

The scalar shift helps organize finite estimates but cancels from normalized Gibbs probabilities and spectral gaps. If ${\epsilon }_k(L)$ lists all energy eigenvalues with multiplicity, the physical probabilities are

$$w_k=\frac{e^{-b({\epsilon }_k(L)+aL)}}{Z_L(b)}.$$

The manuscript constructs a compact self-adjoint spatial operator $X_b$ with square-summable eigenvalues ${\lambda }_i(b)$ such that

$$\begin{array}{rl}{Z}_L(b)& =\mathrm{Tr}{X}_b^L\\ & =\sum _i|{\lambda }_i(b){|}^L.\end{array}$$

This identity is used for even $L\ge 4$. The spatial operator need not be positive. Even powers make the last terms nonnegative, giving a second probability distribution $|{\lambda }_i(b){|}^L/Z_L(b)$. This is why parity enters the argument.

The transfer construction can be described concretely. In Cartesian spin coordinates, $G_{\alpha }=i{S}^{\alpha }$ is real with norm one, and the negative bond interaction is ${\sum }_{\alpha }{G}_{\alpha }\otimes G_{\alpha }$. A bond history is a finite ordered list of times in $(0,b)$, each labeled by $x,y,z$. Give each ordered time simplex its Lebesgue measure and the empty list mass one. The total history measure is ${\sum }_{k\ge 0}(3b{)}^k/k!=e^{3b}$.

For two histories, merge their events chronologically and take the trace of the resulting product of $G$ matrices. This kernel is real, bounded in absolute value by three, and symmetric: swapping the two histories leaves their merged chronological list unchanged. It does not reverse the order of noncommuting matrices. The resulting integral operator, multiplied by $e^{-ab}$, is self-adjoint and Hilbert–Schmidt. Expanding the physical exponential in ordered bond events and regrouping the absolutely convergent series gives the spatial trace identity. This analytic identification is an essential input beyond the finite computations. (OpenAI, 2026b)

For a probability distribution, its purity is the sum of squared probabilities. Define

$$\begin{array}{rl}S(n,b)& =\frac{Z_{2n}(b)}{Z_n(b{)}^2},\\ T(n,b)& =\frac{Z_n(2b)}{Z_n(b{)}^2}.\end{array}$$

$S$ is the spatial purity; $T$ is the physical Gibbs purity. Both lie in $(0,1]$. Doubling length squares the spatial weights; doubling inverse temperature squares the physical weights.

🏷️ Squaring probabilities and coupling the purities

Let $p_i\ge 0$ sum to one, and put $P={\sum }_i{p}_i^2$. The normalized squares $p_i^2/P$ have purity $P^{\prime }$. Direct expansion gives

$$\begin{array}{rl}1-P^{\prime }& =\frac{2{\sum }_{i<j}{p}_i^2{p}_j^2}{P^2}\\ & \le \frac{(1-P{)}^2}{2P^2}.\end{array}$$

The inequality uses ${\sum }_{i<j}{p}_i^2{p}_j^2\le ({\sum }_{i<j}{p}_i{p}_j{)}^2$ and $2{\sum }_{i<j}{p}_i{p}_j=1-P$. Nonnegative summands justify the same calculation for countable distributions. Thus a purity defect $u=1-P$ improves to at most

$$f(u)=\frac{u^2}{2(1-u{)}^2},0\le u<1.$$

For small $u$, this is a quadratic improvement. However, separately squaring the two distributions changes different parameters. The connection between them comes from exact cancellation of partition functions:

$$\begin{array}{rl}S(n,2b)& =S(n,b{)}^2\frac{T(2n,b)}{T(n,b{)}^2},\\ T(4n,b)& =T(2n,b{)}^2\frac{S(2n,2b)}{S(2n,b{)}^2}.\end{array}$$

The denominators are at most one. If $S(n,b)\ge 1-p$ and $T(2n,b)\ge 1-q$, one may first bound

$$\begin{array}{rl}{p}_{+}& =f(1-(1-p{)}^2(1-q)),\\ q_{+}& =f(1-(1-q{)}^2(1-p_{+})).\end{array}$$

Provided these bounds remain below one, they control the defects of $S(2n,2b)$ and $T(4n,2b)$. The first follows by cancellation and spatial squaring; the second follows by the other cancellation and physical squaring. Either output may be rounded upward before the next use.

To obtain a single recurrence, set

$$G(u)=\frac{1}{2}{\left[\sum _{k=1}^3(1-u{)}^{-k}\right]}^2.$$

Then $f(1-(1-u{)}^3)=G(u)u^2$. If $G(u_0)\le C$ and $C{u}_0<1$, the common defect bound can decrease by $u_{j+1}=C{u}_j^2$. With $r=C{u}_0$,

$$\begin{array}{c}{n}_j=2^j{n}_0,b_j=2^j{b}_0,\\ u_j=C^{-1}{r}^{2^j}.\end{array}$$

The error decays exponentially in the inverse temperature, not merely with the number of updates.

🏷️ Intermediate lengths and the physical energy gap

A result only at lengths $n_j$ would leave gaps in the sequence of systems. For nonnegative spatial eigenvalue magnitudes, monotonicity of power-sum norms gives, whenever $n\le L\le 2n$ and the lengths are even,

$$T(L,b)\ge {\left[T(2n,b)S(n,b{)}^2\right]}^{L/(2n)}.$$

Indeed, $Z_L(b)\le Z_n(b{)}^{L/n}$ and $Z_L(2b)\ge Z_{2n}(2b{)}^{L/(2n)}$. Dividing gives the inequality. At the $j$th scale it yields $T(L,b_j)\ge (1-u_j{)}^3\ge 1-3u_j$ throughout the interval.

The largest physical Gibbs probability is at least its purity. If $u_0<1/6$, it therefore exceeds $1/2$. Since the probability list counts individual eigenvectors, two ground eigenvectors would supply two equal maximal probabilities, which is impossible. This proves simplicity rather than assuming it.

For the ground weight $w_0$ and a weight $w_1$ at the next distinct energy,

$$e^{-b_j{\gamma }_L}=\frac{w_1}{w_0}\le \frac{3u_j}{1-3u_j}.$$

If in addition $C(1-3u_0)>3$, the last quantity is strictly less than $r^{2^j}$. Dividing its logarithm by $b_j=2^j{b}_0$ gives ${\gamma }_L>\log (1/r)/b_0$. The dyadic intervals cover every even $L\ge n_0$.

The two finite initializations in the manuscript are:

Starting length and inverse temperature	Common defect and recurrence constant	Resulting lower bound
$n_0=60$, $b_0=105/4$	$u_0=1/8$, $C=79/10$	$(4/105)\log (80/79)$
$n_0=2304$, $b_0=784$	$u_0=1/100$, $C=5$	$\log (20)/784$

For example, $G(1/8)=913952/117649<79/10$. For the second row, $G(u)\le 9/[2(1-3u{)}^2]<5$ on $0\le u\le 1/100$. Both rows satisfy the remaining strict conditions. Establishing their initial spatial and physical purities is the finite-certificate part of the argument. (OpenAI, 2026b)

🏷️ Thermal certificates and variational vectors

The thermal inputs are rigorously enclosed traces of short twisted chains. The twists rotate the spin coordinates across one seam; they help separate spatial moments before applying polynomial filters. This finite separation is used as an input here, without developing an abstract symmetry classification.

One initialization uses chains of lengths $6,8,10$ at $b=21/2$ with three twists; the other uses lengths $4$ through $12$ at $b=49/4$ with two twists. The trace calculations use polynomial approximations to matrix exponentials with explicit errors, rather than relying on numerical eigenvalue estimates.

For example, if $W$ is a self-adjoint contraction, then $e^{\lambda (W-I)}$ is the average of $W^K$ for a Poisson variable $K$ of mean $\lambda$. Conditioning on $K\le J$ gives a polynomial with rational coefficients when $\lambda$ is rational. Its operator-norm error is at most twice the discarded tail probability: both conditional averages have norm at most one. In one table $\lambda =84$, $J=256$, and the exponential-moment bound gives a tail smaller than $2^{-117}$. The source combines this truncation estimate with explicit integer-arithmetic error bounds and converts column errors into trace intervals.

An upper bound on ground energy supplies the complementary lower bound on a partition function. Real integer matrices $A_{-1},A_0,A_1$ define a periodic matrix-product vector by

$$\begin{array}{rl}& {\psi }_N(s_0,\dots ,s_{N-1})\\ & =\mathrm{Tr}(A_{s_0}\cdots A_{s_{N-1}}).\end{array}$$

With $R={\sum }_s{A}_s\otimes A_s$, expansion gives $\Vert {\psi }_N{\Vert }^2=\mathrm{Tr}{R}^N$. A two-site insertion matrix similarly computes one bond energy; cyclicity makes all $N$ bonds contribute equally. The supplied integer data verify $E_0(N)<-aN$ at $N=72,120$, so the shifted ground Boltzmann factor exceeds one and $Z_N(b)>1$. These are finite variational inequalities, not claims that the trial vectors are exact ground states. Combining them with the thermal intervals and elementary moment inequalities supplies the two purity seeds. (OpenAI, 2026b)

🏷️ Endpoint fields require an additional argument

For the companion, use $2L+1$ sites labeled $-L,\dots ,L$ and impose the same field at both ends:

$$\begin{array}{rl}{H}_L(h)=& \sum _{j=-L}^{L-1}{\mathbf{S}}_j\cdot {\mathbf{S}}_{j+1}\\ & -h(S_{-L}^z+S_L^z),\\ & h=3/5.\end{array}$$

There is no closing bond. Here $E_0\le E_1$ are the two lowest eigenvalues counted with multiplicity, so a positive difference itself implies uniqueness. The companion claims $E_1-E_0>\log (10)/392$ for every $L\ge 960$, that is, every odd chain with at least 1921 sites. It does not assert the same bound for arbitrary endpoint fields or a field-free open chain. (OpenAI, 2026a)

The spatial transfer representation becomes a boundary pairing. For an $n$-site chain with opposite endpoint fields, its shifted partition function is $Y_n(b)=⟨v_b,X_b^n{v}_b⟩$. A rotation $U$ that flips the right field gives the same-field expression $Y_n^{\mathrm{s}}(b)=⟨v_b,X_b^{n}U{v}_b⟩$.

At even powers, the opposite-field pairing has positive spectral weights proportional to the squared boundary overlaps. A dominant eigenvalue in the periodic trace need not have a useful overlap with $v_b$. Moreover, $v_b$ changes with temperature. Consequently the periodic gap theorem alone cannot supply the companion result.

The source proves that the spatial leading eigenvalue ${\lambda }_j>0$ is simple and its line is fixed by the relevant rotations. Write $c_j$ for the squared projection of $v_{b_j}$ onto that line and, for even $k$,

$$\begin{array}{c}{B}_k(j)=⟨v_{b_j},(|X_{b_j}|/{\lambda }_j{)}^k{v}_{b_j}⟩,\\ e_k(j)=1-\frac{c_j}{B_k(j)}.\end{array}$$

The residual fraction $e_k(j)$ remains defined even if $c_j=0$. If all residual eigenvalues have relative magnitude at most ${\rho }_j<1$, then increasing the power from $k$ to $m$ contracts their contribution by ${\rho }_j^{m-k}$.

Across temperatures, combine this contraction with the physical purity $A(m,j)=Y_m(2b_j)/Y_m(b_j{)}^2$. For even $m>k\ge 2$, assuming the old defect satisfies $e_k(j)<1$, the key recurrence has the form

$$\begin{array}{c}{e}_m(j+1)\le F(q,e_k(j),A(m,j)),\\ F(q,e,t)=\frac{q}{1-q}\left[\frac{1}{(1-e{)}^{2}t}-1\right].\end{array}$$

where ${\rho }_{j+1}^{m-k}\le q<1$. Only the old overlap is divided by. A new upper bound below one proves that the next overlap is positive, so the argument does not assume the nonvanishing it needs to propagate.

Thermal moments on at most six sites and four integer trial vectors initialize a coupled recurrence for this boundary defect and the opposite-field Gibbs purity. Once both are small, a common bound satisfies $x_{j+1}=10x_j^2$, starting with $x_5=1/100$. Rotation invariance of the leading line controls the signed same-field pairing, including negative residual eigenvalues at odd powers. The resulting same-field purity is at least $1-4x_j$ throughout each length interval. With $b_j=3922^{j-5}$, the same ground-to-excited probability-ratio argument gives the claimed gap. (OpenAI, 2026a)

🏷️ Local limits and the boundary-selected index

A local observable acts on finitely many sites. Compactness of its finite-volume density matrices gives subsequences of ground-state expectations converging to a state $\omega$. A finite-volume gap $\delta$ and a unique ground vector imply

$$\begin{array}{rl}& \omega (A^{\ast }[H,A])\\ & \ge \delta (\omega (A^{\ast }A)-|\omega (A){|}^2)\end{array}$$

in such a limit. To see this, apply the finite spectral inequality to the component of $A\psi$ orthogonal to the ground vector. For fixed local $A$, sufficiently distant endpoint fields or the periodic closing bond commute with it. The commutator therefore stabilizes before taking the limit. This proves a local excitation gap without identifying all subsequential limits or all infinite-chain ground states.

The boundary companion also invokes Tasaki’s index theorem. Its index is the limiting expectation of a slowly varying local spin rotation: the angle rises from zero to $2\pi$ across a long interval, and the interval length then tends to infinity. Under the specified reflection and axial-rotation symmetries and the proven boundary-field gap, the imported theorem gives index $-1$ for every subsequential limit selected by these boundary fields. The finite-volume limit is taken before the twist scale tends to infinity. This is not a claim of global uniqueness among all infinite-volume states. (OpenAI, 2026a)

🏷️ Finite verification and its limits

For the boundary companion, we ran all three supplied programs with assertions enabled. They passed 67 exact scalar comparisons and equalities, four variational certificates, five direct physical-space comparisons, and all 15 thermal recurrences with both thermal conclusions. The three trial output files match the stored files byte-for-byte. Every remaining certificate data field also matches after removing explicitly inspected timing, Python-version, and operating-system metadata. The README’s literal comparison rejects some of those environment differences, so we do not report that literal comparison as passing.

The complete periodic verifier also passed from a fresh output directory with assertions enabled. It recomputed all 27 thermal totals and their rational enclosures, both full trial-state norm and energy contractions at lengths 72 and 120, the common trial matrices and a four-site physical cross-check, and every scalar, rounding, initialization, and packing check. All 17 subprocesses exited successfully. The wrapper checked its source manifest and compared the complete exact integers and rational intervals with the supplied evidence; this was the full recomputation, not a smoke test or a check of stored certificates alone.

The finite programs check the stated scalar inequalities, thermal enclosures, and variational contractions. Their successful execution does not replace the analytic spatial-transfer identity, the purity recurrences, the boundary-overlap argument, or the application of the external index theorem. Our source review checks selected mechanisms and the explicit deductions above, rather than independently verifying both complete research proofs.

🏷️ Key steps and proof outlines

Short moments first cap every spatial eigenvalue

The finite thermal tables supply moments of the spatial spectrum, but the bootstrap needs concentration at much larger powers. Two elementary optimizations make that passage rigorous.

First, let $M_j={\sum }_i{\lambda }_i^j$, with certified intervals $|M_j-c_j|\le ϵ$. For a polynomial $F(x)={\sum }_j{f}_j{x}^j$ nonnegative on the real line,

$$\sum _{i}F({\lambda }_i)\le B:=\sum _j{f}_j{c}_j+ϵ\sum _j|f_j|.$$

If $F(x)>B$ whenever $|x|\ge v$, then every eigenvalue satisfies $|{\lambda }_i|<v$. Each term of the sum is nonnegative, so even one offending eigenvalue contradicts the upper bound. For example, the source uses polynomials of the form $x^6(c_0+c_1{x}^2{)}^2$, built from just three moments. The absolute coefficient sum propagates the interval error, including coefficients of either sign.

Second, suppose nonnegative masses satisfy $t_i\le h$ and ${\sum }_i{t}_i\le m$. For $r\ge 1$, convexity gives

$$\sum _i{t}_i^r\le q{h}^r+(m-qh{)}^r,q=\lfloor m/h\rfloor .$$

To prove this, transfer mass from a smaller partially filled coordinate to a larger one until one becomes zero or reaches $h$. The sum of powers increases, leaving at most one partially filled coordinate. Finite truncations prove the countable version.

Apply these two lemmas to powers of the spatial eigenvalues in the source’s invariant subspaces. They turn short moments into upper bounds on long moments. A variational vector supplies a complementary lower bound $Z_{120}(b)>1$. In the first initialization, masses $w_i=|{\lambda }_i{|}^{30}$ satisfy ${\sum }_i{w}_i\le 1.45$ and ${\sum }_i{w}_i^4>1$. If all $w_i\le 0.88$, then

$$\sum _i{w}_i^4\le 0.88^3\sum _i{w}_i<1,$$

a contradiction. Moreover, two weights exceeding $0.88$ cannot fit within total mass $1.45$. This identifies a unique dominant weight, after which the remainder estimates yield the initial purities. It explains how the short-chain certificate actually feeds the infinite sequence of lengths (OpenAI, 2026b).

The boundary recurrence proves its next nonzero overlap

Here is the derivation behind the companion’s function $F(q,e,t)$. At the new temperature, residual contraction gives

$$B_m(j+1)-c_{j+1}\le q(B_k(j+1)-c_{j+1}).$$

Rearrange this without dividing by $c_{j+1}$:

$$e_m(j+1)\le \frac{q}{1-q}\left(\frac{B_k(j+1)}{B_m(j+1)}-1\right).$$

Let $r_j={\lambda }_{j+1}/{\lambda }_j^2$. The partition-function definitions give the exact identity

$$\frac{B_k(j+1)}{B_m(j+1)}=\frac{A(k,j)}{A(m,j)}{\left(\frac{B_k(j)}{B_m(j)}\right)}^2{r}_j^{m-k}.$$

The periodic analysis supplies $r_j\le 1$, and physical purity gives $A(k,j)\le 1$. At the old temperature, the hypothesis $e_k(j)<1$ ensures $c_j>0$, so

$$\frac{B_k(j)}{B_m(j)}\le \frac{B_k(j)}{c_j}=\frac{1}{1-e_k(j)}.$$

Substitution gives the stated recurrence. An output strictly below one then forces $c_{j+1}>0$. The argument therefore establishes the overlap needed for the next iteration as part of the iteration itself. This is the additional mechanism that allows periodic spectral concentration to control a changing boundary vector (OpenAI, 2026a).

🏷️ Connections

• Area laws and tensor networks explains consequences of a uniform gap and the role of tensor representations; the present family seeks to establish a gap for a specific interaction.
• Unbiased bases and exact certificates likewise separates finite certificate computations from the mathematical reductions that make them relevant to a continuous or unbounded problem.

🏷️ Manuscripts and verification scope

• The periodic spin-one Haldane gap — pinned manuscript and companion source.
• A boundary-field gap for the spin-one Heisenberg chain — pinned manuscript and companion source.

No family scope page was present at lean/docs/268.md in the pinned snapshot. This does not establish that all related formalization code is absent. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. A boundary-field gap for the spin-one Heisenberg chain.

🐻  OpenAI 2026b. The periodic spin-one Haldane gap.