Exactly solvable lattice models for interacting electronic insulators in two dimensions

TL;DR

This paper constructs exactly solvable lattice models for 2D interacting electronic insulators with symmetry, providing a classification framework and extending to crystalline topological insulators, advancing understanding of fermionic SPT phases.

Contribution

It introduces a lattice model construction and classification for 2D interacting electronic insulators with symmetry, including crystalline cases, using U(1) charge decoration ideas.

Findings

Constructed exactly solvable lattice models for 2D electronic insulators.

Classified 2D interacting electronic insulators with various symmetries.

Extended the framework to crystalline topological insulators.

Abstract

In the past decade, tremendous efforts have been made towards understanding fermionic symmetry protected topological (FSPT) phases in interacting systems. Nevertheless, for systems with continuum symmetry, e.g., electronic insulators, it is still unclear how to construct an exactly solvable model with a finite dimensional Hilbert space in general. In this paper, we give a lattice model construction and classification for 2D interacting electronic insulators. Based on the physical picture of $\mathrm{U(1)}_f$-charge decorations, we illustrate the key idea by considering the well known 2D interacting topological insulator. Then we generalize our construction to an arbitrary 2D interacting electronic insulator with symmetry $G_f=\mathrm{U(1)}_f \rtimes_{\rho_1,\omega_2} G$, where $\mathrm{U(1)}_f$ is the charge conservation symmetry and $\rho_1, \omega_2$ are additional data which fully characterize the group structure of $G_f$. Finally we study more examples, including the full interacting classification of 2D crystalline topological insulators.