Topology and Phases in Fermionic Systems

TL;DR

This paper explores topological obstructions in fermionic systems and demonstrates that combining two conjugate systems can eliminate these obstructions, enabling matrix product state representations and boundary condition insensitivity.

Contribution

It introduces a method to overcome topological obstructions in free fermionic systems by using two conjugate copies, facilitating ground state representations.

Findings

Overcoming topological obstructions with conjugate system pairs

Ground states can be expressed as matrix product states

Ground state density matrices are insensitive to boundary conditions

Abstract

There can exist topological obstructions to continuously deforming a gapped Hamiltonian for free fermions into a trivial form without closing the gap. These topological obstructions are closely related to obstructions to the existence of exponentially localized Wannier functions. We show that by taking two copies of a gapped, free fermionic system with complex conjugate Hamiltonians, it is always possible to overcome these obstructions. This allows us to write the ground state in matrix product form using Grassman-valued bond variables, and show insensitivity of the ground state density matrix to boundary conditions.