Conformal invariance of chiral edge theories

TL;DR

This paper demonstrates that chiral edge theories of gapped topological phases are conformal field theories, linking bulk symmetries to edge excitations, and rules out certain non-unitary theories as gapped phases.

Contribution

It establishes the conformal invariance of chiral edge theories under specific conditions and connects bulk symmetries to affine Lie algebra structures, also ruling out some non-unitary theories.

Findings

Edge theories with same-direction propagation are conformal field theories.

Bulk symmetries induce affine Lie algebra structures in edge theories.

Certain trial wavefunctions cannot describe gapped phases due to mismatch in edge excitation counts.

Abstract

The low-energy effective quantum field theory of the edge excitations of a fully-gapped bulk topological phase corresponding to a local interaction Hamiltonian must be local and unitary. Here it is shown that whenever all the edge excitations propagate in the same direction with the same velocity, it is a conformal field theory. In particular, this is the case in the quantum Hall effect for model "special Hamiltonians", for which the ground state, quasihole, and edge excitations can be found exactly as zero-energy eigenstates, provided the spectrum in the interior of the system is fully gapped. In addition, other conserved quantities in the bulk, such as particle number and spin, lead to affine Lie algebra symmetries in the edge theory. Applying the arguments to some trial wavefunctions related to non-unitary conformal field theories, it is argued that the Gaffnian state and an infinite number of others cannot describe a gapped topological phase because the numbers of edge excitations do not match any unitary conformal field theory.