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

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

Wen Wei Ho, Lukasz Cincio, Heidar Moradi, Davide Gaiotto, and Guifre Vidal

Phys. Rev. B 91, 125119 – Published 11 March 2015

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 into two parts, up to rescaling and shifting. This correspondence is believed to be due to the existence of protected gapless edge modes. In this paper, we explore whether the edge-entanglement spectrum correspondence extends to nonchiral topological phases, where there are no protected gapless edge modes. Specifically, we consider the Wen-plaquette model, which is equivalent to the Kitaev toric code model and has $Z_{2}$ topological order (quantum double of $Z_{2})$ . The unperturbed Wen-plaquette model displays an exact correspondence: both the edge and entanglement spectra within each topological sector $a (a = 1, \dots, 4)$ 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- $1 / 2)$ spin 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 the low-energy spectra of $H_{edge}^{a}$ and $H_{ent}^{a}$ , 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/entanglement 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 the effective degrees of freedom of both the edge and entanglement cut are governed by Kramers-Wannier self-dual Hamiltonians, in addition to them being $Z_{2}$ symmetric, which is imposed by the topological order. Thus, by considering the Wen-plaquette model as a SET, the topological order in the bulk together with the translation invariance of the perturbations along the edge/cut imply an edge-ES correspondence at least in some finite domain in Hamiltonian space.

Received 14 December 2014
Revised 16 February 2015

DOI:https://doi.org/10.1103/PhysRevB.91.125119

Authors & Affiliations

Wen Wei Ho, Lukasz Cincio, Heidar Moradi, Davide Gaiotto, and Guifre Vidal

Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5 Canada

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Vol. 91, Iss. 12 — 15 March 2015

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Images

Figure 1
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.
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Figure 2
(Left) Gluing two semi-infinite cylinders of circumference $L_{y}$ together. The $L$ semi-infinite cylinder is colored red, the $R$ semi-infinite cylinder is colored blue, while the plaquette terms belonging to the strip are colored green. The entanglement cut is naturally taken to be the divisor through the strip so that the system is bipartitioned into the $L$ and $R$ semi-infinite cylinders. A checkerboard coloring has been made on the infinite cylinder, and the $L_{y} / 2$ effective virtual spin- $1 / 2$ degrees of freedom are denoted by red and blue ellipses living on the boundaries of the semi-infinite cylinders. (Right) The mapping, Eq. (3), gives rise to an effective Hamiltonian acting on the spin-ladder system, with each spin chain having $L_{y} / 2$ sites, which were previously the red and blue ellipses. The red and blue plaquette operators have been mapped to $z$ -rung and $x$ -rung operators respectively. The effective Hamiltonian's ground states correspond to the ground states of the infinite cylinder.
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Figure 3
Illustration of the terms appearing in the first line of Eq. (26). The Schrieffer-Wolff transformation rotates the perturbed Hamiltonian into a Hamiltonian that is block diagonal in the old eigenspaces (so that $P_{α}^{a} = I_{α}^{a})$ , and adds small dynamics due to the perturbation on top of each space $[H_{α}^{a} ≃ O (ε)]$ .
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Figure 4
Rescaled and shifted entanglement spectra in each topological sector $a$ for Wen-plaquette model on an $L_{y} = 20$ (even), infinite cylinder, for $ε h_{X} = - 0.01$ , $ε h_{Y} = - 0.015$ , and $ε h_{Z} = - 0.02$ . The data points are marked by blue crosses. We have also plotted the $c = 1 / 2$ Ising CFT spectra in each sector for comparison (scaling dim. versus momenta), with the identification that entanglement energy equals scaling dimension. They are given by the green squares and circles. Squares and circles represent conformal towers labeled by different primaries at the bottom of the towers, and double circles denote twofold degenerate states. (a) The ES of the ground state $| I 〉$ with the identity anyonic flux through the cylinder. This corresponds to the $(\tilde{Q}, Q) = (+ 1, + 1)$ sector of the TFIM, whose low-energy effective theory is the $c = 1 / 2$ Ising CFT in the sector that the identity primary $1$ and its descendants (green circles), and energy density primary $ε$ & its descendants (green squares) belong to. The two points labeled $T$ are the holomorphic and antiholomorphic stress-energy tensors. (b) ES of $| e 〉$ , the state with the electric anyonic flux. This gives the $(\tilde{Q}, Q) = (+ 1, - 1)$ sector of the TFIM. This corresponds to the sector of Ising CFT which the spin primary $σ$ and its descendants belong to (green circles). (c) ES of $| m 〉$ , the state with the magnetic anyonic flux. This gives the $(\tilde{Q}, Q) = (- 1, + 1)$ sector of the TFIM, which in turn corresponds to the sector of the Ising CFT (with a $D_{ε}$ defect insertion [46, 47, 48]) which the disorder primary $μ$ and its descendants belong to (green circles). The spectra of the conformal families from $σ$ and $μ$ conincide, and so do the entanglement spectra. (d) ES of $| ɛ 〉$ , the state with the fermionic anyonic flux. This gives the $(\tilde{Q}, Q) = (- 1, - 1)$ sector of the TFIM. This corresponds to the sector of the Ising CFT (with a $D_{ε}$ defect insertion), which the two Majorana fermion primaries $ψ, \bar{ψ}$ and their descendants belong to (green squares and circles).
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Figure 5
Rescaled and shifted entanglement spectra in each topological sector for the perturbed Wen-plaquette model on an $L_{y} = 19$ (odd), infinite cylinder. Shown also are the $c = 1 / 2$ CFT spectra with a conformal duality defect $D_{σ}$ (see Fig. 4 for details on labels). (a) The ES corresponds to the spectrum of the duality-twisted Ising model with primary fields $(0, 1 / 16), (1 / 2, 1 / 16)$ . (b) The ES corresponds to the spectrum of the duality-twisted Ising model with primary fields $(1 / 16, 0), (1 / 16, 1 / 2)$ .
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