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arXiv:2411.19268

Crystalline-equivalent topological phases of many-body fermionic systems in one dimension

Chen-Shen Lee, Ken Shiozaki, Chang-Tse Hsieh

Published Nov 28, 2024• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2411.19268

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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 framework for investigating the effects of interactions on crystalline symmetry-protected topological (SPT) phases. Within this framework, one can establish a direct connection between the equivalence classes of free-fermion systems and their corresponding interacting classes. A central component of this framework is the Atiyah-Hirzebruch spectral sequence, which provides a systematic way to represent crystalline SPT phases as SPT phases with internal symmetries on subspaces. We demonstrate the application of this approach through examples in various dimensions: 1d systems with U(1) and reflection symmetry, 2d systems with U(1) and C_{n} rotation symmetry, and 3d systems with U(1) and inversion symmetry.
We study the classification of interacting fermionic symmetry-protected topological (SPT) phases with both rotation symmetry and Abelian internal symmetries in one, two, and three dimensions. By working out this classification, on the one hand, we demonstrate the recently proposed correspondence principle between crystalline topological phases and those with internal symmetries through explicit block-state constructions. We find that for the precise correspondence to hold it is necessary to change the central extension structure of the symmetry group by the Z2{\mathbb{Z}}_{2}Z2​ fermion parity. On the other hand, we uncover new classes of intrinsically fermionic SPT phases that are only enabled by interactions, both in 2D and 3D. Several 3D Majorana-type fermionic SPT phases are identified. Moreover, several new instances of Lieb-Schultz-Mattis-type theorems for Majorana-type fermionic SPT phases are obtained and we discuss their interpretations from the perspective of bulk-boundary correspondence.
Abstract We propose that symmetry-protected topological (SPT) phases with crystalline symmetry are formulated by an equivariant generalized homology hnG(X)h^G_n(X)hnG​(X) over a real space manifold X with G a crystalline symmetry group. The Atiyah–Hirzebruch spectral sequence unifies various notions in crystalline SPT phases, such as the layer construction, higher-order SPT phases, and Lieb–Schultz–Mattis-type theorems. This formulation is applicable to not only free fermionic systems but also interacting systems with arbitrary onsite and crystal symmetries.
Topological insulators and superconductors are subsets of (almost) noninteracting symmetry protected topological (SPT) phases. More generally, SPT phases represent equivalence classes of (equilibrium) gapped quantum states with short-range entanglement that are protected by certain symmetries. Two quantum states are “equivalent” if they can be adiabatically deformed into one another without closing the bulk gap. Thus, quantum states belonging to different SPT phases cannot be smoothly deformed into each other without a phase transition, as long as the underlying symmetry is preserved. However, upon removing the symmetry all SPT states can be adiabatically deformed to a trivial product state (e.g., an atomic insulator) without undergoing a phase transition. In this chapter, we generalize our discussion of (almost) noninteracting topological states by removing some of the explicit and implicit constraints underlying the classification and properties of topological insulators and superconductors discussed in Chapter 3 . In Section 4.1 , we enlarge the type of symmetries that protect the topological phases by supplementing internal symmetries (e.g., time-reversal and particle-hole symmetries) with crystalline symmetries (e.g., mirror and rotation symmetries) and discuss topological crystalline insulators and superconductors , a subsets of SPT phases protected by internal and crystalline symmetries. In Section 4.2 , we remove the requirement of having a finite (bulk) gap and discuss Weyl semimetals – crystalline systems characterized by topologically protected bulk band degeneracy points separated in momentum space, which support low energy excitations that are massless chiral fermions, i.e., Weyl fermions. Finally, in Section 4.3 , we remove the condition of thermodynamic equilibrium and consider topological quantum states in periodically driven systems, focusing on the so-called Floquet topological insulators , which exhibit a richer topological structure than their static counterparts. We note that aspects regarding the effects of particle-particle interactions, including the emergence of topological order in systems with long-range entanglement, will be addressed in Chapter 5 . We also point out that the areas summarized in this chapter have grown tremendously in recent years and continue to grow; even a brief mentioning of all recent developments would be impossible within the constraints of a book chapter. Instead, we point out the key elements that determine the nontrivial topology of the quantum systems mentioned above, discuss specific manifestations of their topological properties (e.g., the presence of protected gapless states), and provide representative examples.
We develop a theoretical framework for the classification and construction of symmetry protected topological (SPT) phases, which are a special class of zero-temperature phases of strongly interacting gapped quantum many-body systems that exhibit topological properties. The framework unifies various proposals for the classification of SPT phases, including the group (super-)cohomology proposal, the (spin-)cobordism proposal, the Freed-Hopkins proposal, and the Kitaev proposal. The power of the framework is demonstrated in a number of applications: (1) the classification and construction of 3D fermionic SPT phases in Wigner-Dyson classes A and AII with glide symmetry, (2) the classification and construction of 3D bosonic SPT phases with space-group symmetries for all 230 space groups, (3) the derivation of a Mayer-Vietoris sequence relating the classification of SPT phases with and without reflection symmetry, and (4) an interpretation of the structure of general crystalline SPT phases via the Atiyah-Hirzebruch spectral sequence. The framework is based on the idea that short-range entangled states form what are known as Ω-spectra in the sense of algebraic topology, and models the classification of SPT phases by what are called generalized cohomology theories.
We develop a theoretical framework for the classification and construction of symmetry protected topological (SPT) phases, which are a special class of zero-temperature phases of strongly interacting gapped quantum many-body systems that exhibit topological properties. The framework unifies various proposals for the classification of SPT phases, including the group (super-)cohomology proposal, the (spin-)cobordism proposal, the Freed-Hopkins proposal, and the Kitaev proposal. The power of the framework is demonstrated in a number of applications: (1) the classification and construction of 3D fermionic SPT phases in Wigner-Dyson classes A and AII with glide symmetry, (2) the classification and construction of 3D bosonic SPT phases with space-group symmetries for all 230 space groups, (3) the derivation of a Mayer-Vietoris sequence relating the classification of SPT phases with and without reflection symmetry, and (4) an interpretation of the structure of general crystalline SPT phases via the Atiyah-Hirzebruch spectral sequence. The framework is based on Kitaev's idea that short-range entangled states form what are known as ΩΩΩ-spectra in the sense of algebraic topology, and models the classification of SPT phases by what are called generalized cohomology theories.