Tensor network trial states for chiral topological phases in two dimensions and a no-go theorem in any dimension

TL;DR

This paper constructs tensor network states for chiral topological phases in two dimensions using Gaussian Grassmann integrals, proves the absence of gapped short-range parent Hamiltonians in any dimension, and discusses implications for Wannier functions.

Contribution

It introduces a method to represent chiral topological states as tensor network states and proves a no-go theorem for gapped parent Hamiltonians in any dimension.

Findings

Constructed tensor network states for chiral fermion phases.

Proved all short-range quadratic parent Hamiltonians are gapless.

Showed non-existence of compactly-supported Wannier functions for these band structures.

Abstract

Trial wavefunctions that can be represented by summing over locally-coupled degrees of freedom are called tensor network states (TNSs); they have seemed difficult to construct for two-dimensional topological phases that possess protected gapless edge excitations. We show it can be done for chiral states of free fermions, using a Gaussian Grassmann integral, yielding $p_x \pm i p_y$ and Chern insulator states, in the sense that the fermionic excitations live in a topologically non-trivial bundle of the required type. We prove that any strictly short-range quadratic parent Hamiltonian for these states is gapless; the proof holds for a class of systems in any dimension of space. The proof also shows, quite generally, that sets of compactly-supported Wannier-type functions do not exist for band structures in this class. We construct further examples of TNSs that are analogs of fractional (including non-Abelian) quantum Hall phases; it is not known whether parent Hamiltonians for these are also gapless.