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arXiv:2206.01110 · Physical review. B./Physical review. B

Generalized Thouless pumps in (1+1) -dimensional interacting fermionic systems

Shuhei Ohyama, Ken Shiozaki, Masatoshi Sato

Published Oct 17, 2022• 10 citations
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Abstract

The Thouless pump is a phenomenon in which $U(1)$ charges are pumped from an edge of a fermionic system to another edge. The Thouless pump has been generalized in various dimensions and for various charges. In this paper we investigate the generalized Thouless pumps of fermion parity in both trivial and nontrivial phases of $(1+1)$-dimensional interacting fermionic short-range entangled (SRE) states. For this purpose, we use matrix product states (MPSs). MPSs describe many-body systems in $(1+1)$ dimensions and can characterize SRE states algebraically. We prove fundamental theorems for fermionic MPSs (fMPSs) and use them to investigate the generalized Thouless pumps. We construct nontrivial pumps in both the trivial and nontrivial phases, and we show the stability of the pumps against interactions. Furthermore, we define topological invariants for the generalized Thouless pumps in terms of fMPSs and establish consistency with existing results. These are invariants of the family of SRE states that are not captured by the higher dimensional Berry curvature. We also argue a relation between the topological invariants of the generalized Thouless pump and the twist of the $K$ theory in the Donovan-Karoubi formulation.

DOI
10.1103/physrevb.106.165115

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The Thouless pump is a phenomenon in which U(1)U(1)U(1) charges are pumped from an edge of a fermionic system to another edge. The Thouless pump has been generalized in various dimensions and for various charges. In this paper we investigate the generalized Thouless pumps of fermion parity in both trivial and nontrivial phases of (1+1)(1+1)(1+1)-dimensional interacting fermionic short-range entangled (SRE) states. For this purpose, we use matrix product states (MPSs). MPSs describe many-body systems in (1+1)(1+1)(1+1) dimensions and can characterize SRE states algebraically. We prove fundamental theorems for fermionic MPSs (fMPSs) and use them to investigate the generalized Thouless pumps. We construct nontrivial pumps in both the trivial and nontrivial phases, and we show the stability of the pumps against interactions. Furthermore, we define topological invariants for the generalized Thouless pumps in terms of fMPSs and establish consistency with existing results. These are invariants of the family of SRE states that are not captured by the higher dimensional Berry curvature. We also argue a relation between the topological invariants of the generalized Thouless pump and the twist of the KKK theory in the Donovan-Karoubi formulation.
We define and study analogs of the Thouless charge pump for many-body gapped systems in dimension DDD. We show how to attach a topological invariant to a DDD-dimensional family of such systems, provided all of them have an on-site U(1)U(1)U(1) symmetry. For a large class of families we argue that this topological invariant is an integer. In the case of gapped systems of free fermions in two dimensions, the invariant can be expressed in terms of the curvature of the Bloch-Berry connection. We also obtain a new formula for the Thouless charge pump in 1d which involves only static linear response and is analogous to the Streda formula for Hall conductivity.
We extend the formalism of matrix product states (MPS) to describe one-dimensional gapped systems of fermions with both unitary and antiunitary symmetries. Additionally, systems with orientation-reversing spatial symmetries are considered. The short-ranged entangled phases of such systems are classified by three invariants, which characterize the projective action of the symmetry on edge states. We give interpretations of these invariants as properties of states on the closed chain. The relationship between fermionic MPS systems at a renormalization group fixed point and equivariant algebras is exploited to derive a group law for the stacking of fermionic phases. The result generalizes known classifications to symmetry groups that are nontrivial extensions of fermion parity and time-reversal.
Nonlinear parameter-gauge coupling approach to generalization of generalized Thouless pumps and −1-1−1-form anomaly
We study the nontrivial topology of the parameter space of general U(1)U(1)U(1)-symmetric fermionic non-degenerately gapped system and its consequences on the transport properties in arbitrary dimensions. By a nonlinear parameter-gauge topological response theory, we find that such nontrivial topology can impose quantization constraints on the charge transport in the presence of background fluxes or, more generally, instantons in general dimensions and our result generalizes the Thouless pump and its higher dimensional generalizations. We also show that these nontrivial transport properties are related to an unconventional quantum anomaly, which generalizes −1-1−1-form anomalies. This anomaly imposes non-perturbative ingappabilities of various types of spatial interfaces or time-dependent system evolution.
Thouless pumping represents a powerful concept to probe quantized topological invariants in quantum systems. We explore this mechanism in a generalized Rice-Mele Fermi-Hubbard model characterized by the presence of competing onsite and intersite interactions. Contrary to recent experimental and theoretical results, showing a breakdown of quantized pumping induced by the onsite repulsion, we prove that sufficiently large intersite interactions allow for an interaction-induced recovery of Thouless pumps. Our analysis further reveals that the occurrence of stable topological transport at large interactions is connected to the presence of a spontaneous bond-order-wave in the ground-state phase diagram of the model. Finally, we discuss a concrete experimental setup based on ultracold magnetic atoms in an optical lattice to realize the newly introduced Thouless pump. Our results provide a new mechanism to stabilize Thouless pumps in interacting quantum systems.
We consider symmetry protected topological (SPT) phases with finite non-invertible symmetry C\mathcal{C}C in 1+1d. In particular, we investigate interfaces and parameterized families of them within the framework of matrix product states. After revealing how to extract the C\mathcal{C}C-SPT invariant, we identify the algebraic structure of symmetry operators acting on the interface of two C\mathcal{C}C-SPT phases. By studying the representation theory of this algebra, we show that there must be a degenerate interface mode between different C\mathcal{C}C-SPT phases. This result generalizes the bulk-boundary correspondence for ordinary SPT phases. We then propose the classification of one-parameter families of C\mathcal{C}C-SPT states based on the explicit construction of invariants of such families. Our invariant is identified with a non-abelian generalization of the Thouless charge pump, which is the pump of a local excitation within a C\mathcal{C}C-SPT phase. Finally, by generalizing the results for one-parameter families of SPT phases, we conjecture the classification of general parameterized families of general gapped phases with finite non-invertible symmetries in both 1+1d and higher dimensions.
We study state-sum constructions of GGG-equivariant spin topological quantum field theory (TQFTs) and their relationship to matrix product states. In the Neveu-Schwarz, Ramond, and twisted sectors, states of the TQFT are generalized matrix product states. Our results are applied to the classification of fermionic short-range-entangled phases with a unitary symmetry GGG to determine the group law on the set of such phases. Interesting subtleties appear when the total symmetry group is a nontrivial extension of GGG by fermion parity.