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arXiv:2112.15533 · Physical review. B./Physical review. B

Exactly solvable lattice models for interacting electronic insulators in two dimensions

Qing-Rui Wang, Yang Qi, Chen Fang, Meng Cheng, Zheng‐Cheng Gu

Published Sep 6, 2023• 6 citations
arXivHTMLPDF

Abstract

Topological insulators in two-dimensional systems are well-understood within the context of free-fermion systems. Nevertheless, comprehending their strongly interacting counterparts presents a significant challenge. Here, the authors developed an innovative lattice model based on $U(1{)}_{f}$ fermionic charge decorations. The model furnishes a rigorous mathematical framework for delving into fermionic symmetry-protected topological phases that conserve $U(1{)}_{f}$ charge, among other types of symmetries. Notably, this research expands the existing theoretical landscape by presenting an exactly solvable Hamiltonian with a finite local Hilbert space and a sophisticated classification scheme for continuous symmetries.

DOI
10.1103/physrevb.108.l121104

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Topological insulators in two-dimensional systems are well-understood within the context of free-fermion systems. Nevertheless, comprehending their strongly interacting counterparts presents a significant challenge. Here, the authors developed an innovative lattice model based on U(1)fU(1{)}_{f}U(1)f​ fermionic charge decorations. The model furnishes a rigorous mathematical framework for delving into fermionic symmetry-protected topological phases that conserve U(1)fU(1{)}_{f}U(1)f​ charge, among other types of symmetries. Notably, this research expands the existing theoretical landscape by presenting an exactly solvable Hamiltonian with a finite local Hilbert space and a sophisticated classification scheme for continuous symmetries.
Topology is routinely used to understand the physics of electronic insulators. However, for strongly interacting electronic matter, such as Mott insulators, a comprehensive topological characterization is still lacking. When their ground state only contains short-range entanglement and does not break symmetries spontaneously, they generically realize crystalline fermionic symmetry-protected topological phases (cFSPTs), supporting gapless modes at the boundaries or at the lattice defects. Here, we provide an exhaustive classification of cFSPTs in two dimensions with U ( 1 ) charge-conservation and spinful time-reversal symmetries, namely, those generically present in spin-orbit coupled insulators, for any of the 17 wallpaper groups. It has been shown that the classification of cFSPTs can be understood from appropriate real-space decorations of lower-dimensional subspaces, and we expose how these relate to the Wyckoff positions of the lattice. We find that all nontrivial one-dimensional decorations require electronic interactions. Furthermore, we provide model Hamiltonians for various decorations, and discuss the signatures of cFSPTs. This classification paves the way to further explore topological interacting insulators, providing the backbone information in generic model systems and ultimately in experiments. Published by the American Physical Society 2024
Topology is routinely used to understand the physics of electronic insulators. However, for strongly interacting electronic matter, such as Mott insulators, a comprehensive topological characterization is still lacking. When their ground state only contains short-range entanglement and does not break symmetries spontaneously, they generically realize crystalline fermionic symmetry-protected topological phases (cFSPTs), supporting gapless modes at the boundaries or at the lattice defects. Here, we provide an exhaustive classification of cFSPTs in two dimensions with charge-conservation and spinful time-reversal symmetries, namely, those generically present in spin-orbit coupled insulators, for any of the 17 wallpaper groups. It has been shown that the classification of cFSPTs can be understood from appropriate real-space decorations of lower-dimensional subspaces, and we expose how these relate to the Wyckoff positions of the lattice. We find that all nontrivial one-dimensional decorations require electronic interactions. Furthermore, we provide model Hamiltonians for various decorations, and discuss the signatures of cFSPTs. This classification paves the way to further explore topological interacting insulators, providing the backbone information in generic model systems and ultimately in experiments.
Symmetry-protected topological (SPT) phases (e.g., the well known topological insulators) are a class of energetically gapped condensed matter systems that exhibit interesting topological properties only in the presence of certain global symmetries. While a great understanding of noninteracting fermionic SPT phases has been achieved recently, interacting SPT phases, in particular those that exist only in strongly interacting systems, are much less understood in general. Here, the authors study strongly interacting fermionic SPT phases in two spatial dimensions with finite Abelian unitary symmetries and provide a potentially complete classification of them.
We propose a general construction of commuting projector lattice models for 2D and 3D topological phases enriched by U(1) symmetry, with finite-dimensional Hilbert space per site. The construction starts from a commuting projector model of the topological phase and decorates U(1) charges to the state space in a consistent manner. We show that all 2D U(1) symmetry-enriched topological phases, which allow gapped boundaries without breaking the symmetry, can be realized through our construction. We also construct a large class of 3D topological phases with U(1) symmetry fractionalized on particles or loop excitations.
Strong interactions can give rise to new fermionic symmetry-protected topological phases which have no analogs in free fermion systems. As an example, we have systematically studied a spinless fermion model with U(1)U(1)U(1) charge conservation and time-reversal symmetry on a three-leg ladder using density-matrix renormalization group. In the noninteracting limit, there are no topological phases. Turning on interactions, we found two gapped phases. One is trivial and is adiabatically connected to a band insulator, while another one is a nontrivial symmetry-protected topological phase resulting from strong interactions.
In this paper, we present an exactly solvable model for two-dimensional topological superconductors with helical Majorana edge modes protected by time-reversal symmetry. Our construction is based on the idea of decorated domain walls and makes use of the Kasteleyn orientation on a two-dimensional lattice, which was used for the construction of the symmetry protected fermion phase with Z2{Z}_{2}Z2​ symmetry by Tarantino et al. and Ware et al. By decorating the time-reversal domain walls with spinful Majorana chains, we are able to construct a commuting projector Hamiltonian with zero correlation length ground state wave function that realizes a strongly interacting version of the two-dimensional topological superconductor. From our construction, it can be seen that the ${T}^{2}=\ensuremath{-}1$ transformation rule for the fermions is crucial for the existence of such a nontrivial phase; with T2=1{T}^{2}=1T2=1, our construction does not work.