Galois conjugates of topological phases

Galois conjugation relates unitary conformal field theories and topological quantum field theories (TQFTs) to their nonunitary 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 nonunitary 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 quasilocal Hamiltonians. A similar statement holds for many other nonunitary TQFTs. One consequence is that these nonunitary TQFTs do not describe physical realizations of topological phases. In particular, this implies that the “Gaffnian” wave function can not be the ground state of a gapped fractional quantum Hall state.

Figure 1

Edge labeling for a plaquette of the honeycomb lattice.

Figure 2

The $F$ symbol.

Figure 3

The $t$-deformation parameters of Fibonacci and Yang-Lee anyons correspond to different primitive roots of unity.

Figure 4

Scaling of the finite-size gap $\Delta \left(L\right)$ (in units of $J_p$) with linear system size for the Hermitian projector model $H^{\mathrm{Herm}}$ on two different lattice geometries: the honeycomb lattice with $L\times W$ plaquettes (a) and two-leg ladder systems of length $L$ (b). This figure can be reproduced using the VisTrails[33] workflows Figs. 4a and 4b included in the Supplementary Material.[37]

Figure 5

Edge labeling for a plaquette of the ladder lattice.

Figure 6

Ground-state degeneracy splitting of the non-Hermitian doubled Yang-Lee model when perturbed by a string tension $(\theta \ne 0)$. This figure can be reproduced using the VisTrails[33] workflow Fig. 6 included in the Supplementary Material.[37]

Figure 7

Ground-state degeneracy splitting of the Hermitian model $H^{\mathrm{Herm}}$, the counterpart to the DYL model, when perturbed by a string tension $(\theta \ne 0)$ (a). The slope of the splitting around the unperturbed model $(\theta =0)$ is given in the inset (a) for different system sizes $L$. (b) Shows the low-energy spectrum, which clearly shows that the degeneracy at $\theta =0$ is due to a level crossing. This figure can be reproduced using the VisTrails[33] workflows Figs. 7a and 7b included in the Supplementary Material.[37]

Figure 8

The low-energy spectra of the doubled Yang-Lee model (a) and its Hermitian counterpart (b) for a wide range of coupling parameters. Data shown are for a ladder of length $L=8$. This figure can be reproduced using the VisTrails[33] workflows Figs. 8a and 8b included in the Supplementary Material.[37]