Symmetries and boundary theories for chiral Projected Entangled Pair States

TL;DR

This paper explores the topological properties of two-dimensional chiral fermionic PEPS, revealing how symmetries lead to boundary chiral modes and a universal entropy correction, with implications for understanding topological phases.

Contribution

It demonstrates the topological nature of chiral PEPS, constructs boundary theories from virtual symmetries, and links these to chiral edge modes and entropy corrections.

Findings

PEPS exhibit topological properties linked to virtual symmetries.

Boundary theories reveal chiral modes arising from these symmetries.

Universal correction to the area law for the zero Rényi entropy is identified.

Abstract

We investigate the topological character of lattice chiral Gaussian fermionic states in two dimensions possessing the simplest descriptions in terms of projected entangled-pair states (PEPS). They are ground states of two different kinds of Hamiltonians. The first one, $\mathcal H_\mathrm{ff}$, is local, frustration-free, and gapless. It can be interpreted as describing a quantum phase transition between different topological phases. The second one, $\mathcal H_\mathrm{fb}$ is gapped, and has hopping terms scaling as $1/r^3$ with the distance $r$. The gap is robust against local perturbations, which allows us to define a Chern number for the PEPS. As for (non-chiral) topological PEPS, the non-trivial topological properties can be traced down to the existence of a symmetry in the virtual modes that are used to build the state. Based on that symmetry, we construct string-like operators acting on the virtual modes that can be continuously deformed without changing the state. On the torus, the symmetry implies that the ground state space of the local parent Hamiltonian is two-fold degenerate. By adding a string wrapping around the torus one can change one of the ground states into the other. We use the special properties of PEPS to build the boundary theory and show how the symmetry results in the appearance of chiral modes, and a universal correction to the area law for the zero R\'{e}nyi entropy.