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.
Comments
17 pages. Version 2 incorporates recent results from OpenAI. In addition to the index-equivalence theorem, we prove the existence of a locally-unique gapped ground state of the infinite S=1 Heisenberg antiferromagnetic chain with both indices equal to -1
The equivalence of topological indices for U(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) symmetry. We prove their equivalence in the common setting of integer-spin chains with on-site U(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=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.
Abstract We introduce an index for symmetry-protected topological (SPT) phases of infinite fermionic chains with an on-site symmetry given by a finite group G. This index takes values in Z2×H1(G,Z2)×H2(G,U(1)p) with a generalised Wall group law under stacking. We show that this index is an invariant of the classification of SPT phases. When the ground state is translation invariant and has reduced density matrices with uniformly bounded rank on finite intervals, we derive a fermionic matrix product representative of this state with on-site symmetry.
We study symmetric gapped ground states of one-dimensional spin S chains with a continuous exponential symmetry U(1)⋊Z2 with integer base q, where the U(1) charge has a modulated unit of charge qL−j decaying exponentially with site j. The simplest model is a quantum breakdown model that can be viewed as a q-deformation of the XXZ Haldane spin chain. We show that the topology of their symmetric ground states crucially depends on S and q. For integer S, odd q allows a nontrivial Z2 Haldane-type symmetry-protected topological (SPT) phase with protected edge modes, while even q allows no nontrivial SPT phase. For half-odd integer S, odd q prohibits any unique symmetric gapped ground state due to a Lieb-Schultz-Mattis obstruction, whereas even q enforces a novel symmetric gapped phase which we name as the ``half-SPT" phase. Unconventionally, the half-SPT phase only has a symmetry protected edge mode on one edge, and has half of the string orders of an SPT phase. Our results uncover a new type of symmetry-protected gapped phases beyond existing classification schemes.
We study S=1 quantum spin systems on the infinite chain with short ranged Hamiltonians that have a certain rotational and discrete symmetry. We define a Z_{2} index for any gapped unique ground state, and prove that it is invariant under a smooth deformation. By using the index, we provide the first rigorous proof of the existence of a "topological" phase transition, which cannot be characterized by any conventional order parameters, between the Affleck-Kennedy-Lieb-Tasaki (AKLT) model and trivial models. This rigorously establishes that the AKLT model is in a nontrivial symmetry protected topological phase.
We present a rigorous but elementary index theory for a class of one-dimensional systems of interacting (and possibly disordered) fermions with U(1)⋊Z2 symmetry defined on the infinite chain. The class includes the Su–Schrieffer–Heeger (SSH) model [Su et al., “Solitons in polyacetylene,” Phys. Rev. Lett. 42, 1698 (1979); Su et al., “Soliton excitations in polyacetylene,” Phys. Rev. B 22, 2099 (1983); and Asbóth et al., A Short Course on Topological Insulators: Band-Structure Topology and Edge States in One and Two Dimensions, Lecture Notes in Physics (Springer, 2016)] as a special case. For any locally unique gapped (fixed-charge) ground state of a model in the class, we define a Z2 index in terms of the sign of the expectation value of the local twist operator. We prove that the index is topological in the sense that it is invariant under continuous modification of models in the class with a locally unique (fixed-charge) gapped ground state. This establishes that any path of models in the class that connects the two extreme cases of the SSH model must go through a phase transition. Our rigorous Z2 classification is believed to be optimal for the class of models considered here. We also show an interesting duality of the index and prove that any topologically nontrivial model in the class has a gapless edge excitation above the ground state when defined on the half-infinite chain. The results extend to other classes of models, including the extended Hubbard model. Our strategy to focus on the expectation value of local unitary operators makes the theory intuitive and conceptually simple. This paper also contains a careful discussion about the notion of unique gapped ground states of a particle system on the infinite chain. (There are two lecture videos in which the main results of this paper are discussed [H. Tasaki, “Rigorous index theory for one-dimensional interacting topological insulators: A brief introduction,” online lecture (21:41), November, 2021, seehttps://www.gakushuin.ac.jp/~881791/OL/#Index1DTI2021S and https://youtu.be/ypGVb3eYrpg and H. Tasaki, “Rigorous index theory for one-dimensional interacting topological insulators: With a pedagogical introduction to the topological phase transition in the SSH model,” online lecture (49:07), November, 2021, see https://www.gakushuin.ac.jp/~881791/OL/#Index1DTI2021L and https://youtu.be/yxZYOevV2Y].
This thesis studies a pair of symmetry protected topological (SPT) phases which arise when considering one-dimensional quantum spin systems possessing a natural orthogonal group symmetry. Particular attention is given to a family of exactly solvable models whose ground states admit a matrix product state description and generalize the AKLT chain. We call these models ``SO(n) AKLT chains'' and the phase they occupy the ``SO(n) Haldane phase''. We present new results describing their ground state structure and, when n is even, their peculiar O(n)-to-SO(n) symmetry breaking. We also prove that these states have arbitrarily large correlation and injectivity length by increasing n, but all have a 2-local parent Hamiltonian, in contrast to the natural expectation that the interaction range of a parent Hamiltonian should diverge as these quantities diverge. We extend Ogata's definition of an SPT index for a split state for a finite symmetry group G to an SPT index for a compact Lie group G. We then compute this index, which takes values in the second Borel group cohomology H2(SO(n),U(1)), at a single point in each of the SPT phases. The two points have different indices, confirming the two SPT phases are indeed distinct. Chapter 1 contains an introduction with a detailed overview of the contents of this thesis, which includes several chapters of background information before presenting new results in Chapter 7 and Chapter 8.
On the uniqueness of translation-invariant ground states of the S=1 Heisenberg and J--D 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=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.