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arXiv:2609.34119

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

Hal Tasaki

math-phcond-mat.stat-mechcond-mat.str-elquant-phSubmitted Sep 28, 2026• Announced Sep 29, 2026• 0 citations
v1 · Sep 28, 2026v2 · Oct 10, 2026
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Abstract

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 when a lower bound on the gap and the relevant ground-state expectations are available. As an application, a previous twist-index calculation determines the Ogata index to be $(-1)^S$ for the antiferromagnetic Heisenberg chain under explicit uniqueness and boundary-field gap assumptions. The author formulated the problem and proposed the basic strategy, while ChatGPT made substantial contributions to the development of the proof.

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Symmetry protected topological orders and the group cohomology of their symmetry group

Xie Chen, Zheng‐Cheng Gu, Zheng-Xin Liu +1 more

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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 when a lower bound on the gap and the relevant ground-state expectations are available. As an application, a previous twist-index calculation determines the Ogata index to be (−1)S(-1)^S(−1)S for the antiferromagnetic Heisenberg chain under explicit uniqueness and boundary-field gap assumptions.
The author formulated the problem and proposed the basic strategy, while ChatGPT 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)\mathbb {Z}_2 \times H^1(G,\mathbb {Z}_2) \times H^2(G, U(1)_{\mathfrak {p}})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 SSS chains with a continuous exponential symmetry U(1)⋊Z2U(1)\rtimes\mathbb Z_2U(1)⋊Z2​ with integer base qqq, where the U(1) charge has a modulated unit of charge qL−jq^{L-j}qL−j decaying exponentially with site jjj. The simplest model is a quantum breakdown model that can be viewed as a qqq-deformation of the XXZ Haldane spin chain. We show that the topology of their symmetric ground states crucially depends on SSS and qqq. For integer SSS, odd qqq allows a nontrivial Z2\mathbb Z_2Z2​ Haldane-type symmetry-protected topological (SPT) phase with protected edge modes, while even qqq allows no nontrivial SPT phase. For half-odd integer SSS, odd qqq prohibits any unique symmetric gapped ground state due to a Lieb-Schultz-Mattis obstruction, whereas even qqq 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)SO(n)SO(n) AKLT chains'' and the phase they occupy the ``SO(n)SO(n)SO(n) Haldane phase''. We present new results describing their ground state structure and, when nnn is even, their peculiar O(n)O(n)O(n)-to-SO(n)SO(n)SO(n) symmetry breaking. We also prove that these states have arbitrarily large correlation and injectivity length by increasing nnn, 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 GGG to an SPT index for a compact Lie group GGG. We then compute this index, which takes values in the second Borel group cohomology H2(SO(n),U(1))H^2(SO(n),U(1))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.
Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry GGG. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phases which is protected by SO(3)\mathit{SO}(3)SO(3) spin rotation symmetry. The topological insulator is another example of SPT phases which are protected by U(1)U(1)U(1) and time-reversal symmetries. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: Distinct ddd-dimensional bosonic SPT phases with on-site symmetry GGG (which may contain antiunitary time-reversal symmetry) can be labeled by the elements in H1+d[G,UT(1)]{\mathcal{H}}^{1+d}[G,{U}_{T}(1)]H1+d[G,UT​(1)], the Borel (1+d)(1+d)(1+d)-group-cohomology classes of GGG over the GGG module UT(1){U}_{T}(1)UT​(1). Our theory, which leads to explicit ground-state wave functions and commuting projector Hamiltonians, is based on a new type of topological term that generalizes the topological $\ensuremath{\theta}$ term in continuous nonlinear $\ensuremath{\sigma}$ models to lattice nonlinear $\ensuremath{\sigma}$ models. The boundary excitations of the nontrivial SPT phases are described by lattice nonlinear $\ensuremath{\sigma}$ models with a nonlocal Lagrangian term that generalizes the Wess-Zumino-Witten term for continuous nonlinear $\ensuremath{\sigma}$ models. As a result, the symmetry GGG must be realized as a non-on-site symmetry for the low-energy boundary excitations, and those boundary states must be gapless or degenerate. As an application of our result, we can use ${\mathcal{H}}^{1+d}[U(1)\ensuremath{\rtimes}{Z}_{2}^{T},{U}_{T}(1)]$ to obtain interacting bosonic topological insulators (protected by time reversal Z2T{Z}_{2}^{T}Z2T​ and boson number conservation), which contain one nontrivial phase in one-dimensional (1D) or 2D and three in 3D. We also obtain interacting bosonic topological superconductors (protected by time-reversal symmetry only), in term of H1+d[Z2T,UT(1)]{\mathcal{H}}^{1+d}[{Z}_{2}^{T},{U}_{T}(1)]H1+d[Z2T​,UT​(1)], which contain one nontrivial phase in odd spatial dimensions and none for even dimensions. Our result is much more general than the above two examples, since it is for any symmetry group. For example, we can use ${\mathcal{H}}^{1+d}[U(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{T},{U}_{T}(1)]$ to construct the SPT phases of integer spin systems with time-reversal and U(1)U(1)U(1) spin rotation symmetry, which contain three nontrivial SPT phases in 1D, none in 2D, and seven in 3D. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: $({G}_{H},{G}_{\ensuremath{\Psi}},{\mathcal{H}}^{1+d}[{G}_{\ensuremath{\Psi}},{U}_{T}(1)])$, where GH{G}_{H}GH​ is the symmetry group of the Hamiltonian and ${G}_{\ensuremath{\Psi}}$ the symmetry group of the ground states.