We define a Z2-valued topological and gauge invariant associated with any one-dimensional, translation-invariant topological insulator that satisfies either particle–hole symmetry or chiral symmetry. The invariant can be computed from the Berry phase associated with a suitable basis of Bloch functions that is compatible with the symmetries. We compute the invariant in the Su–Schrieffer–Heeger model for chiral symmetric insulators and in the Kitaev model for particle–hole symmetric insulators. We show that in both cases, the Z2 invariant predicts the existence of zero-energy boundary states for the corresponding truncated models.
We define a Z2-valued topological and gauge invariant associated with any one-dimensional, translation-invariant topological insulator that satisfies either particle–hole symmetry or chiral symmetry. The invariant can be computed from the Berry phase associated with a suitable basis of Bloch functions that is compatible with the symmetries. We compute the invariant in the Su–Schrieffer–Heeger model for chiral symmetric insulators and in the Kitaev model for particle–hole symmetric insulators. We show that in both cases, the Z2 invariant predicts the existence of zero-energy boundary states for the corresponding truncated models.
We introduce Z2-valued bulk invariants for symmetry-protected topological phases in 2+1-dimensional driven quantum systems. These invariants adapt the W3 invariant, expressed as a sum over degeneracy points of the propagator, to the respective symmetry class of the Floquet-Bloch Hamiltonian. The bulk-boundary correspondence that holds for each invariant relates a nonzero value of the bulk invariant to the existence of symmetry-protected topological boundary states. To demonstrate this correspondence we apply our invariants to a chiral Harper, time-reversal Kane-Mele, and particle-hole symmetric graphene model with periodic driving, where they successfully predict the appearance of boundary states that exist despite the trivial topological character of the Floquet bands. Especially for particle-hole symmetry, the combination of the W3 and the Z2 invariants allows us to distinguish between weak and strong topological phases.
We investigate the topological properties of one-dimensional weakly interacting topological insulators using bosonization. To do that we study the topological edge states that emerge at the edges of a model realized by a strong impurity or at the boundary between topologically distinct phases. In the bosonic model, the edge states are manifested as degenerate bosonic kinks at the boundaries. We first illustrate this idea on the example of the interacting Su-Schrieffer-Heeger (SSH) chain. We compute the localization length of the edge states as the width of an edge soliton that occurs in the SSH model in the presence of a strong impurity. Next, we examine models of two capacitively coupled SSH chains that can be either identical or in distinct topological phases. We find that weak Hubbard interaction reduces the ground-state degeneracy in the topological phase of identical chains. We then prove that, similarly to the noninteracting model, the degeneracy of the edge states in the interacting case is protected by chiral symmetry. We then study topological insulators built from two SSH chains with interchain hopping that represent models of different chiral symmetric universality classes. We demonstrate in bosonic language that the topological index of a weakly coupled model is determined by the type of interchain coupling, invariant under one of two possible chiral symmetry operators. Finally, we show that a general one-dimensional model in a phase with topological index $\ensuremath{\nu}$ is equivalent at low energies to a theory of at least $\ensuremath{\nu}$ SSH chains. We illustrate this idea on the example of an SSH model with longer-range hopping.
Discrete-time quantum walks have been shown to simulate all known topological phases in one and two dimensions. Being periodically driven quantum systems, their topological description, however, is more complex than that of closed Hamiltonian systems. We map out the topological phases of the particle-hole symmetric one-dimensional discrete-time quantum walk. We find that there is no chiral symmetry in this system: its topology arises from the particle-hole symmetry alone. We calculate the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological invariant in a simple way that is consistent with a general definition for one-dimensional periodically driven quantum systems. These results allow for a transparent interpretation of the edge states on a finite lattice via the the bulk-boundary correspondence. We find that the bulk Floquet operator does not contain all the information needed for the topological invariant. As an illustration to this statement, we show that in the split-step quantum walk, the edges between two bulks with the same Floquet operator can host topologically protected edge states.
We explore one-dimensional fermionic symmetry-protected topological (SPT) phases related by the crystalline equivalence principle. In particular, we study charge-conserving many-body topological phases of fermions protected respectively by chiral and reflection symmetries. While the classifications of the two crystalline-equivalent SPT phases are identical, their topological properties and phase structures can be very different, depending on the microscopic details. Specifically, we consider certain extensions of the Su-Schrieffer-Heeger model, with and without interactions, that preserve both chiral and reflection symmetries, and explicitly compute the many-body topological invariants based on the systems' ground states. The phase structures determined by these topological invariants align perfectly with the many-body spectra of deformations among the models. As expected, gapped deformations exist only when all the topological invariants remain unchanged. Moreover, we show that decomposable systems -- those that can be decomposed into local and decoupled subsystems -- can be topologically characterized by real-space quantum numbers directly associated with the symmetries. For reflection-symmetric systems, these quantum numbers are related to the many-body topological invariants via a bulk-center correspondence, which can be justified using the Atiyah-Hirzebruch spectral sequence in generalized homology theory. Finally, we discuss the role of transition symmetry in the many-body topologies of these SPT phases.
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].
We establish a quantitative bulk-boundary correspondence in interacting topological insulators by relating many-body topological invariants to characteristic degeneracy structures in the entanglement spectrum. Focusing on generalized Su-Schrieffer-Heeger chains with higher winding number, we construct a gauge-invariant many-body winding invariant based on Pancharatnam geometric phases that remains well defined in the presence of interactions. We show that this invariant uniquely determines the low-lying entanglement-spectrum degeneracy, which exhibits a universal 4ν scaling with the winding number ν, providing a concrete formulation of bulk-boundary correspondence beyond single-particle topology. Using exact diagonalization, we demonstrate the robustness of this correspondence under interactions and symmetry-preserving disorder, and identify inversion symmetry as a minimal protecting symmetry that stabilizes both the quantization of the invariant and the associated entanglement degeneracies. Our results unify geometric-phase invariants and entanglement diagnostics within a many-body framework and provide a route to identifying interacting topological phases beyond band theory.