Fermionic tensor networks for higher order topological insulators from charge pumping

TL;DR

This paper develops a method using charge pumping and fermionic tensor networks to construct higher-dimensional topological insulator states, enabling efficient numerical simulations of complex topological phases.

Contribution

It introduces a systematic approach to generate (d+1)-dimensional tensor network states from d-dimensional models for topological insulators, including second-order phases.

Findings

Constructed a 2D PEPS for a Chern insulator from an MPS of the SSH model.

Built a 3D TNS for a chiral hinge TI from a 2D PEPS of the quadrupole model.

Identified gapped Hamiltonians interpolating between trivial and topological phases with constant bond dimension.

Abstract

We apply the charge pumping argument to fermionic tensor network representations of d-dimensional topological insulators (TIs) to obtain tensor network states for (d+1)-dimensional TIs. We exemplify the method by constructing a two-dimensional projected entangled pair states (PEPS)for a Chern insulator starting from a matrix product state (MPS) in d=1 describing pumping in the Su-Schrieffer-Heeger (SSH) model. In extending the argument to second-order TIs, we build a three-dimensional TNS for a chiral hinge TI from a PEPS in d=2 for the obstructed atomic insulator (OAI) of the quadrupole model. The (d+1)-dimensional TNSs obtained in this way have a constant bond dimension inherited from the d-dimensional TNSs in all but one spatial direction, making them candidates for numerical applications. From the d-dimensional models, we identify gapped next-nearest neighbour Hamiltonians interpolating between the trivial and OAI phases of the fully dimerized SSH and quadrupole models, whose ground states are given by an MPS and a PEPS with a constant bond dimension equal to 2, respectively.