Galois Conjugates of Topological Phases

TL;DR

This paper explores Galois conjugates of topological phases, showing that non-Hermitian models retain a form of topological protection, but their ground states cannot be realized as ground states of Hermitian Hamiltonians with topological order.

Contribution

It proves that Galois conjugated topological phases cannot be obtained as ground states of Hermitian Hamiltonians satisfying Lieb-Robinson bounds.

Findings

Non-Hermitian Galois conjugates retain a generalized code property.

Hermitian Hamiltonians cannot realize these non-unitary topological phases.

The Gaffnian wave function cannot be a gapped fractional quantum Hall ground state.

Abstract

Galois conjugation relates unitary conformal field theories (CFTs) and topological quantum field theories (TQFTs) to their non-unitary counterparts. Here we investigate Galois conjugates of quantum double models, such as the Levin-Wen model. While these Galois conjugated Hamiltonians are typically non-Hermitian, we find that their ground state wave functions still obey a generalized version of the usual code property (local operators do not act on the ground state manifold) and hence enjoy a generalized topological protection. The key question addressed in this paper is whether such non-unitary topological phases can also appear as the ground states of Hermitian Hamiltonians. Specific attempts at constructing Hermitian Hamiltonians with these ground states lead to a loss of the code property and topological protection of the degenerate ground states. Beyond this we rigorously prove that no local change of basis can transform the ground states of the Galois conjugated doubled Fibonacci theory into the ground states of a topological model whose Hermitian Hamiltonian satisfies Lieb-Robinson bounds. These include all gapped local or quasi-local Hamiltonians. A similar statement holds for many other non-unitary TQFTs. One consequence is that the "Gaffnian" wave function cannot be the ground state of a gapped fractional quantum Hall state.