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arXiv:1601.06506 · Physical Review A

Quantum phase transition from Z2×Z2 to Z2 topological order

Mohammad Hossein Zarei

Published Apr 4, 2016• 8 citations
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Abstract

Although the topological order is known as a quantum order in quantum many-body systems, it seems that there is not a one-to-one correspondence between topological phases and quantum phases. As a well-known example, it has been shown that all one-dimensional (1D) quantum phases are topologically trivial [X. Chen et al., Phys. Rev. B 83, 035107 (2011)]. By such facts, it seems a challenging task to understand when a quantum phase transition between different topological models necessarily reveals different topological classes of them. In this paper, we make an attempt to consider this problem by studying a phase transition between two different quantum phases which have a universal topological phase. We define a Hamiltonian as interpolation of the toric code model with ${Z}_{2}$ topological order and the color code model with ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ topological order on a hexagonal lattice. We show such a model is exactly mapped to many copies of 1D quantum Ising model in transverse field by rewriting the Hamiltonian in a new complete basis. Consequently, we show that the universal topological phase of the color code model and the toric code model reflects in the 1D nature of the phase transition. We also consider the expectation value of Wilson loops by a perturbative calculation and show that behavior of the Wilson loop captures the nontopological nature of the quantum phase transition. The result on the point of phase transition also shows that the color code model is strongly robust against the toric code model.

DOI
10.1103/physreva.93.042306

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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.
We study the phase transition from two different topological phases to the ferromagnetic phase by focusing on points of the phase transition. To this end, we present a detailed mapping from such models to the Ising model in a transverse field. Such a mapping is derived by rewriting the initial Hamiltonian in a new basis so that the final model in such a basis has a well-known approximated phase transition point. Specifically, we consider the toric codes and the color codes on various lattices with Ising perturbation. Our results provide a useful table to compare the robustness of the topological codes and to explicitly show that the robustness of the topological codes depends on triangulation of their underlying lattices.
A quantum Ising model in a transverse field is of the simplest quantum many-body systems used for studying universal properties of quantum phase transitions. Interestingly, it is well known that such phase transitions can be mapped to topological phase transitions in toric code models. Therefore, one can expect that well-known properties of the transverse Ising model are used for characterizing topological phase transitions in toric code models. In this paper, we consider the magnetization of the Ising model and show, while it is a local order parameter, it is mapped to a nonlocal order parameter on the topological side. Consequently, we introduce a string order parameter for topological phase transitions in toric code models defined on different lattices with different dimensions in the presence of a uniform magnetic field. Since such a string order parameter is dual of the Ising order parameter with well-known properties, it leads to a topological (nonlocal) characterization of phase transition in the toric code model in uniform field. Our results show that although topological phase transitions do not follow a symmetry-breaking mechanism, it might still be possible to use concepts of symmetry-breaking phase transitions for topological ones by finding suitable mappings.
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.
Distinguishing different topologically ordered phases and characterizing phase transitions between them is a difficult task due to the absence of local order parameters. In this paper, we use a combination of analytical and numerical approaches to distinguish two such phases and characterize a phase transition between them. The ``toric code'' and ``double semion'' models are simple lattice models exhibiting Z2{\mathbb{Z}}_{2}Z2​ topological order. Although both models express the same topological ground state degeneracies and entanglement entropies, they are distinct phases of matter because their emergent quasiparticles obey different statistics. For a 1D model, we tune a phase transition between these two phases and obtain an exact solution to the entire phase diagram, finding a second-order Ising$\ifmmode\times\else\texttimes\fi{}$Ising transition. We then use exact diagonalization to study the 2D case and find indications of a first-order transition. We show that the quasiparticle statistics provides a robust indicator of the distinct topological orders throughout the whole phase diagram.
Quantum phase transitions between different topologically ordered phases exhibit rich structures and are generically challenging to study in microscopic lattice models. In this paper, we propose a tensor-network solvable model that allows us to tune between different symmetry enriched topological (SET) phases. Concretely, we consider a decorated two-dimensional toric code model for which the ground state can be expressed as a two-dimensional tensor-network state with bond dimension D=3D=3D=3 and two tunable parameters. We find that the time-reversal (TR) symmetric system exhibits three distinct phases: (i) an SET toric code phase in which anyons transform nontrivially under TR, (ii) a toric code phase in which TR does not fractionalize, and (iii) a topologically trivial phase that is adiabatically connected to a product state. We characterize the different phases using the topological entanglement entropy and a membrane order parameter that distinguishes the two SET phases. Along the phase boundary between the SET toric code phase and the toric code phase, the model has an enhanced U(1)U(1)U(1) symmetry and the ground state is a quantum critical loop gas wavefunction whose squared norm is equivalent to the partition function of the classical O(2)O(2)O(2) model. By duality transformations, this tensor-network solvable model can also be used to describe transitions between SET double-semion phases and between ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T}$ symmetry protected topological phases in two dimensions.