Edge-entanglement spectrum correspondence in a nonchiral topological phase and Kramers-Wannier duality

TL;DR

This work extends the edge-ES correspondence beyond chiral topological phases by analyzing the Wen-plaquette model with ${\mathbb{Z}}_2$ order. Using exact solutions on a full cylinder and a microscopic Schrieffer-Wolff perturbative approach, it shows that while the unperturbed model exhibits an exact edge-ES correspondence within each topological sector, generic local perturbations destroy it. Remarkably, when the Wen-plaquette model is treated as a symmetry-enriched topological phase with translational invariance along the edge, both edge and entanglement Hamiltonians become Kramers-Wannier self-dual and flow to the $c = 1/2$ Ising CFT, yielding a finite domain where edge-ES holds. The results provide a concrete, microscopic route to edge-ES in non-chiral topological phases and suggest broader applicability to SETs and higher ${\mathbb{Z}}_N$ orders.

Abstract

In a system with chiral topological order, there is a remarkable correspondence between the edge and entanglement spectra: the low-energy spectrum of the system in the presence of a physical edge coincides with the lowest part of the entanglement spectrum (ES) across a virtual cut of the system, up to rescaling and shifting. In this paper, we explore whether the edge-ES correspondence extends to nonchiral topological phases. Specifically, we consider the Wen-plaquette model which has Z_2 topological order. The unperturbed model displays an exact correspondence: both the edge and entanglement spectra within each topological sector a (a = 1,...,4) are flat and equally degenerate. Here, we show, through a detailed microscopic calculation, that in the presence of generic local perturbations: (i) the effective degrees of freedom for both the physical edge and the entanglement cut consist of a spin-1/2 chain, with effective Hamiltonians H_edge^a and H_ent.^a, respectively, both of which have a Z_2 symmetry enforced by the bulk topological order; (ii) there is in general no match between their low energy spectra, that is, there is no edge-ES correspondence. However, if supplement the Z_2 topological order with a global symmetry (translational invariance along the edge/cut), i.e. by considering the Wen-plaquette model as a symmetry enriched topological phase (SET), then there is a finite domain in Hamiltonian space in which both H_edge^a and H_ent.^a realize the critical Ising model, whose low-energy effective theory is the c = 1/2 Ising CFT. This is achieved because the presence of the global symmetry implies that both Hamiltonians, in addition to being Z_2 symmetric, are Kramers-Wannier self-dual. Thus, the bulk topological order and the global translational symmetry of the Wen-plaquette model as a SET imply an edge-ES correspondence at least in some finite domain in Hamiltonian space.

Figures (1)

• Figure 1: (Color online.) Semi-infinite cylinder that terminates in the $x$-direction, with the periodic direction along $y$. The grey plaquette and white plaquette operators are shown as blue and red circles respectively. The boundary operators (BO) can be thought of as half of the plaquette operators in the bulk (acting on the white ($p$) or grey (${\tilde{p}}$) plaquettes). The green ellipses on the edge correspond to the virtual, boundary spin degree of freedom. The blue BO acts on a single virtual spin, while the red BO acts on a pair of nearest-neighbor virtual spins.