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Physical Review B

Mutual Chern-Simons Landau-Ginzburg theory for continuous quantum phase transition of Z2 topological order

Su-Peng Kou, Jing Yu, Xiao-Gang Wen

Published Sep 3, 2009• 16 citations
Publisher

Abstract

In this paper, we develop an effective mutual Chern-Simons Landau-Ginzburg (MCSLG) theory to describe the continuous topological quantum phase transition (TQPT). In particular, we consider the TQPT between a spin-polarized phase (a state without topological order) and a ${Z}_{2}$ topologically ordered state. The TQPT is not induced by spontaneous symmetry breaking. Instead the ${Z}_{2}$ topological order is broken down by the condensation of ${Z}_{2}$ charged quasiparticles. By generalizing the hierarchy theory of fractional quantum Hall effect to ${Z}_{2}$ topological order, we show that the TQPT belongs to the universal class of three-dimensional Ising phase transition. In the end, we applied the MCSLG theory to the toric code model.

DOI
10.1103/physrevb.80.125101

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In this paper, we develop an effective mutual Chern-Simons Landau-Ginzburg (MCSLG) theory to describe the continuous topological quantum phase transition (TQPT). In particular, we consider the TQPT between a spin-polarized phase (a state without topological order) and a Z2{Z}_{2}Z2​ topologically ordered state. The TQPT is not induced by spontaneous symmetry breaking. Instead the Z2{Z}_{2}Z2​ topological order is broken down by the condensation of Z2{Z}_{2}Z2​ charged quasiparticles. By generalizing the hierarchy theory of fractional quantum Hall effect to Z2{Z}_{2}Z2​ topological order, we show that the TQPT belongs to the universal class of three-dimensional Ising phase transition. In the end, we applied the MCSLG theory to the toric code model.
In this paper, we develop an effective mutual Chern-Simons Landau-Ginzburg (MCSLG) theory to describe the continuous topological quantum phase transition (TQPT). In particular, we consider the TQPT between a spin-polarized phase (a state without topological order) and a Z[subscript 2] topologically ordered state. The TQPT is not induced by spontaneous symmetry breaking. Instead the [subscript 2]topological order is broken down by the condensation of [subscript 2] charged quasiparticles. By generalizing the hierarchy theory of fractional quantum Hall effect to [subscript 2] topological order, we show that the TQPT belongs to the universal class of three-dimensional Ising phase transition. In the end, we applied the MCSLG theory to the toric code model.
We study several different Z2{Z}_{2}Z2​ topological ordered states in frustrated spin systems. The effective theories for those different Z2{Z}_{2}Z2​ topological orders all have the same form---a Z2{Z}_{2}Z2​ gauge theory which can also be written as a mutual $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ Chern-Simons theory. However, we find that the different Z2{Z}_{2}Z2​ topological orders are reflected in different projective realizations of lattice symmetry in the same effective mutual Chern-Simons theory. This result is obtained by comparing the ground-state degeneracy, the ground-state quantum numbers, the gapless edge state, and the projective symmetry group of quasiparticles calculated from the slave-particle theory and from the effective mutual Chern-Simons theories. Our study reveals intricate relations between topological order and symmetry.
We demonstrate that two toric code layers on the square lattice coupled by an Ising interaction display two distinct phases with intrinsic topological order. The second-order quantum phase transition between the weakly coupled ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ and the strongly coupled Z2{\mathbb{Z}}_{2}Z2​ topological order can be described by the condensation of bosonic quasiparticles from both sides and belongs to the three-dimensional Ising∗{\mathrm{Ising}}^{*}Ising∗ universality class. This can be shown by an exact duality transformation to the transverse-field Ising model on the square lattice, which builds on the existence of an extensive number of local Z2{\mathbb{Z}}_{2}Z2​ conserved parities. These conserved quantities correspond to the product of two adjacent star operators on different layers. Notably, we show that the low-energy effective model derived about the limit of large Ising coupling is given by an effective single-layer toric code in terms of the conserved quantities of the Ising toric code bilayer. The two topological phases are further characterized by the topological entanglement entropy which serves as a nonlocal order parameter.
In this paper we study the quantum phase transition between a quantum state with topological order and that without topological order (TQPT) in the transverse Wen-plaquette model. By mapping the transverse Wen-plaquette model onto the d =1 Ising model in a transverse field, some exact results of the TQPT are obtained. The transverse Wen-plaquette model undergoes a continuous TQPT which is characterized in terms of the expectation values of string operators. In particular, the duality relationship between open-string and closed-string is explored —the expectation value of closed-string operator and that of open-string operator play the role of "order parameter" and "disorder order parameter" of the topological order, respectively. At the critical point the "condensation" of both Z 2 vortex and Z 2 charge indicates that the TQPT is a new type of phase transition beyond the Landau-Ginzburg-Wilson paradigm.
The competition between fractionalized spin-liquid states and magnetically ordered phases is an important paradigm in frustrated magnetism. Spin-orbit coupled Mott insulators with Ising-like magnetic anisotropies, such as Kitaev materials, are a particularly rich playground to explore this competition. In this work, we use effective field theory methods to show that a direct quantum phase transition can occur in two-dimensional (2D) Ising spin systems between a topologically ordered chiral spin liquid and a phase with magnetic long-range order. Such a transition can be protected by lattice symmetries and is described by a theory of massless Majorana fields coupled to non-Abelian SO(N)\text{SO}(N)SO(N) gauge fields with a Chern-Simons term. We further show that Euclidean Majorana zero modes bound to Z2{\mathbb{Z}}_{2}Z2​ monopole-instantons in the emergent non-Abelian gauge field are key to understanding spontaneous symmetry breaking in the ordered phase.
We investigate the quantum phase diagram of the KKK-layer Ising toric code corresponding to KKK layers of two-dimensional toric codes coupled by Ising interactions. While for small Ising interactions the system displays Z2K{\mathbb{Z}}_{2}^{K}Z2K​ topological order originating from the toric codes in each layer, the system shows Z2{\mathbb{Z}}_{2}Z2​ topological order in the high-Ising limit. The latter is demonstrated for general KKK by deriving an effective low-energy model in KthK\mathrm{th}Kth-order degenerate perturbation theory, which is given as an effective anisotropic single-layer toric code in terms of collective pseudospins 1/2 referring to the two ground states of isolated Ising chain segments. For the specific cases K=3K=3K=3 and K=4K=4K=4 we apply high-order series expansions to determine the gap series in the low- and high-Ising limit. Extrapolation of the elementary energy gaps gives convincing evidence that the ground-state phase diagram consists of a single quantum critical point and our findings suggest a quantum phase transition in the 3D Ising* universality class for both KKK separating both types of topological order, which is consistent with former findings for the bilayer Ising toric code.