Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2610.34330

On the uniqueness of translation-invariant ground states of the $S=1$ Heisenberg and $J$--$D$ antiferromagnetic chains

Hal Tasaki

math-phcond-mat.stat-mechquant-phSubmitted Oct 25, 2026• Announced Oct 27, 2026• 0 citations• Revised Oct 27, 2026 (v2)
v1 · Oct 25, 2026v2 · Oct 27, 2026
arXivHTMLPDF

Abstract

Using the analytic machinery and numerical inputs of OpenAI's proofs of the Haldane gap, we prove uniqueness of the translation-invariant ground state of the infinite $S=1$ Heisenberg antiferromagnetic chain and of the $J$-$D$ model for sufficiently small $|D|$. Uniqueness is among all translation-invariant ground states in the operator-algebraic sense, without assuming a gap or spin-rotation symmetry for a competing state. The proof combines concentration of the spatial transfer spectrum, an exponentially small low-temperature free-energy correction, and a Golden--Thompson comparison for separated local observables. The unique translation-invariant state is pure, locally unique and gapped, and continuous in $D$. The ground states of even periodic chains and odd open chains with equal positive endpoint fields converge to this state along their full sequences, uniformly in $D$, with a stretched-exponential local rate. Every choice of ground states of open chains without endpoint magnetic fields has the same local limit. These conclusions do not establish uniqueness among arbitrary nontranslation-invariant ground states of the infinite chain. The author formulated the problems and proposed the core strategies. ChatGPT Astra made substantial contributions to developing the proofs and preparing the manuscript.

Comments
46 pages, 2 figures. Version 2: The title has been changed. We prove uniqueness among translation-invariant ground states of the infinite chain, a substantially stronger result than the boundary-condition independence established in version 1

Related papers

$SU_2(\mathbb{C})$ symmetry in quantum spin chain ground states and Haldane's conjecture

Anilesh Mohari

In this paper, we prove that any translation and $SU_2(\IC)$-invariant pure state of $\IM=\otimes_{k \in \IZ}\!M^{(k)}_d(\IC)$, that is also real, lattice symmetric and reflection positive with a certain twist $r_0 \in U_d(\IC)$, is finitely correlated and its two-point spatial correlation function decays exponentially whenever $d$ is an odd integer. In particular, the Heisenberg iso-spin anti-ferromagnetic integer spin model admits unique low temperature limiting ground state and its spatial correlation function decays exponentially. The unique low temperature limiting ground state of the Hamiltonian is determined by the unique solution to Clebsch-Gordon inter-twinning isometry between two representations of $SU_2(\IC)$.

arXiv:2112.13976•Dec 28, 2021•0 citations

The equivalence of topological indices for $\mathrm{U}(1)\rtimes\mathbb{Z}_2$-symmetric quantum spin chains

Hal Tasaki

Symmetry-protected topological (SPT) phases in quantum spin chains are distinguished by topological indices. Ogata's operator-algebraic indices apply to a broad range of symmetry classes without requiring continuous symmetry, whereas the elementary twist index of Tasaki requires $\mathrm{U}(1)$ symmetry. We prove their equivalence in the common setting of integer-spin chains with on-site $\mathrm{U}(1)\rtimes\mathbb{Z}_2$ symmetry and a symmetric locally-unique gapped ground state, without assuming translation invariance or a matrix product representation. A charge-fluctuation estimate yields a strong-limit construction of half-chain rotations with their phases fixed. An exact group commutator identity then identifies the Ogata index with the limit of local twist expectations and gives an explicit error bound. This provides a computable formula for the index when a lower bound on the gap and the relevant ground-state expectations are available. Combining the equivalence with a recent boundary-field gap theorem of OpenAI and our earlier twist-index calculation, we obtain a spin-rotation-invariant locally-unique gapped ground state of the infinite $S=1$ antiferromagnetic Heisenberg chain with both indices equal to $-1$, without assuming infinite-volume uniqueness. The author formulated the problem and proposed the basic strategy, while ChatGPT Astra made substantial contributions to the development of the proof.

math-phCross-listedarXiv:2609.34119•Sep 29, 2026•0 citations

Ground State of the S = 1 Antiferromagnetic Heisenberg Chain Is Topologically Nontrivial if Gapped

Hal Tasaki

Under the widely accepted but unproven assumption that the one-dimensional S=1 antiferromagnetic Heisenberg model has a unique gapped ground state, we prove that the model belongs to a nontrivial symmetry-protected topological phase. In other words, we rigorously rule out the possibility that the model has a unique gapped ground state that is topologically trivial. To be precise, we assume that the models on open finite chains with a boundary magnetic field have unique ground states with a uniform gap and prove that the ground state of the infinite chain has a nontrivial topological index. This further implies the presence of a gapless edge excitation in the model on the half-infinite chain and the existence of a topological phase transition in the model that interpolates between the Heisenberg chain and the trivial model.

arXiv:2407.17041•Physical Review Letters•Feb 19, 2025•4 citations

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

OpenAI

We prove a uniform spectral gap for odd open spin-one antiferromagnetic Heisenberg chains with the same fixed magnetic field at both endpoints. The field h=3/5 gives a gap greater than \log(10)/392 on every chain of 2L+1 sites with L ≥ 960. Tasaki's index theorem then gives index −1 for every subsequential local limit of these boundary-selected ground states.

•Oct 6, 2026•0 citations

Pure gapped ground states of spin chains are short-range entangled

Wojciech De Roeck, Martin Fraas, Bruno de O. Carvalho

We consider spin chains with a finite range Hamiltonian. For reasons of simplicity, the chain is taken to be infinitely long. A ground state is said to be a unique gapped ground state if its GNS Hamiltonian has a unique ground state, separated by a gap from the rest of the spectrum. By combining some powerful techniques developed in the last years, we prove that each unique gapped ground state is short-range entangled: It can be mapped into a product state by a finite time evolution map generated by a Hamiltonian with exponentially quasi-local interaction terms. This claim makes precise the common belief that one-dimensional gapped systems are topologically trivial in the bulk.

math-phCross-listedarXiv:2511.14699•Nov 19, 2025•0 citations

Sign structure and ground-state properties for a spin−S t−J chain

Qing-Rui Wang, Peng Ye

The antiferromagnetic Heisenberg spin chain of odd spin $S$ is in the Haldane phase with several defining physical properties, such as thermodynamical ground-state degeneracy, symmetry-protected edge states, and nonzero string order parameter. If nonzero hole concentration $\ensuremath{\delta}$ and hole hopping energy $t$ are considered, the spin chain is replaced by a $\text{spin}\ensuremath{-}S$ $t\ensuremath{-}J$ chain. The motivation of this paper is to generalize the discussions of the Haldane phase to the doped spin chain. The first result of this paper is that, for the model considered here, the ${\mathbb{Z}}_{2}$ sign structure in the usual Ising basis can be totally removed by two consecutive unitary transformations consisting of a spatially local one and a nonlocal one. Direct from the sign structure, the second result of this paper is that the Marshall theorem and the Lieb-Mattis theorem for pure spin systems are generalized to the $t\ensuremath{-}J$ chain for arbitrary $S$ and $\ensuremath{\delta}$. A corollary of the theorem provides us with the ground-state degeneracy in the thermodynamic limit. The third result of this paper is about the phase diagram. We show that the defining properties of the Haldane phase survive in the small $t/J$ limit. The large $t/J$ phase supports a gapped spin sector with similar properties (ground-state degeneracy, edge state, and string order parameter) of the Haldane chain, although the charge sector is gapless.

arXiv:1310.6496•Physical Review B•Jul 9, 2014•8 citations
On the uniqueness of translation-invariant ground states of the S=1S=1S=1 Heisenberg and JJJ--DDD antiferromagnetic chains
Using the analytic machinery and numerical inputs of OpenAI's proofs of the Haldane gap, we prove uniqueness of the translation-invariant ground state of the infinite S=1S=1S=1 Heisenberg antiferromagnetic chain and of the JJJ-DDD model for sufficiently small ∣D∣|D|∣D∣. Uniqueness is among all translation-invariant ground states in the operator-algebraic sense, without assuming a gap or spin-rotation symmetry for a competing state. The proof combines concentration of the spatial transfer spectrum, an exponentially small low-temperature free-energy correction, and a Golden--Thompson comparison for separated local observables. The unique translation-invariant state is pure, locally unique and gapped, and continuous in DDD. The ground states of even periodic chains and odd open chains with equal positive endpoint fields converge to this state along their full sequences, uniformly in DDD, with a stretched-exponential local rate. Every choice of ground states of open chains without endpoint magnetic fields has the same local limit. These conclusions do not establish uniqueness among arbitrary nontranslation-invariant ground states of the infinite chain.
The author formulated the problems and proposed the core strategies. ChatGPT Astra made substantial contributions to developing the proofs and preparing the manuscript.
SU2(C)SU_2(\mathbb{C})SU2​(C) symmetry in quantum spin chain ground states and Haldane's conjecture
In this paper, we prove that any translation and $SU_2(\IC)$-invariant pure state of $\IM=\otimes_{k \in \IZ}\!M^{(k)}_d(\IC)$, that is also real, lattice symmetric and reflection positive with a certain twist $r_0 \in U_d(\IC)$, is finitely correlated and its two-point spatial correlation function decays exponentially whenever ddd is an odd integer. In particular, the Heisenberg iso-spin anti-ferromagnetic integer spin model admits unique low temperature limiting ground state and its spatial correlation function decays exponentially. The unique low temperature limiting ground state of the Hamiltonian is determined by the unique solution to Clebsch-Gordon inter-twinning isometry between two representations of $SU_2(\IC)$.
The equivalence of topological indices for U(1)⋊Z2\mathrm{U}(1)\rtimes\mathbb{Z}_2U(1)⋊Z2​-symmetric quantum spin chains
Symmetry-protected topological (SPT) phases in quantum spin chains are distinguished by topological indices. Ogata's operator-algebraic indices apply to a broad range of symmetry classes without requiring continuous symmetry, whereas the elementary twist index of Tasaki requires U(1)\mathrm{U}(1)U(1) symmetry. We prove their equivalence in the common setting of integer-spin chains with on-site U(1)⋊Z2\mathrm{U}(1)\rtimes\mathbb{Z}_2U(1)⋊Z2​ symmetry and a symmetric locally-unique gapped ground state, without assuming translation invariance or a matrix product representation. A charge-fluctuation estimate yields a strong-limit construction of half-chain rotations with their phases fixed. An exact group commutator identity then identifies the Ogata index with the limit of local twist expectations and gives an explicit error bound. This provides a computable formula for the index when a lower bound on the gap and the relevant ground-state expectations are available. Combining the equivalence with a recent boundary-field gap theorem of OpenAI and our earlier twist-index calculation, we obtain a spin-rotation-invariant locally-unique gapped ground state of the infinite S=1S=1S=1 antiferromagnetic Heisenberg chain with both indices equal to −1-1−1, without assuming infinite-volume uniqueness. The author formulated the problem and proposed the basic strategy, while ChatGPT Astra made substantial contributions to the development of the proof.
Under the widely accepted but unproven assumption that the one-dimensional S=1 antiferromagnetic Heisenberg model has a unique gapped ground state, we prove that the model belongs to a nontrivial symmetry-protected topological phase. In other words, we rigorously rule out the possibility that the model has a unique gapped ground state that is topologically trivial. To be precise, we assume that the models on open finite chains with a boundary magnetic field have unique ground states with a uniform gap and prove that the ground state of the infinite chain has a nontrivial topological index. This further implies the presence of a gapless edge excitation in the model on the half-infinite chain and the existence of a topological phase transition in the model that interpolates between the Heisenberg chain and the trivial model.
We consider spin chains with a finite range Hamiltonian. For reasons of simplicity, the chain is taken to be infinitely long. A ground state is said to be a unique gapped ground state if its GNS Hamiltonian has a unique ground state, separated by a gap from the rest of the spectrum. By combining some powerful techniques developed in the last years, we prove that each unique gapped ground state is short-range entangled: It can be mapped into a product state by a finite time evolution map generated by a Hamiltonian with exponentially quasi-local interaction terms. This claim makes precise the common belief that one-dimensional gapped systems are topologically trivial in the bulk.
The antiferromagnetic Heisenberg spin chain of odd spin SSS is in the Haldane phase with several defining physical properties, such as thermodynamical ground-state degeneracy, symmetry-protected edge states, and nonzero string order parameter. If nonzero hole concentration $\ensuremath{\delta}$ and hole hopping energy ttt are considered, the spin chain is replaced by a $\text{spin}\ensuremath{-}S$ $t\ensuremath{-}J$ chain. The motivation of this paper is to generalize the discussions of the Haldane phase to the doped spin chain. The first result of this paper is that, for the model considered here, the Z2{\mathbb{Z}}_{2}Z2​ sign structure in the usual Ising basis can be totally removed by two consecutive unitary transformations consisting of a spatially local one and a nonlocal one. Direct from the sign structure, the second result of this paper is that the Marshall theorem and the Lieb-Mattis theorem for pure spin systems are generalized to the $t\ensuremath{-}J$ chain for arbitrary SSS and $\ensuremath{\delta}$. A corollary of the theorem provides us with the ground-state degeneracy in the thermodynamic limit. The third result of this paper is about the phase diagram. We show that the defining properties of the Haldane phase survive in the small t/Jt/Jt/J limit. The large t/Jt/Jt/J phase supports a gapped spin sector with similar properties (ground-state degeneracy, edge state, and string order parameter) of the Haldane chain, although the charge sector is gapless.