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arXiv:1503.05957 · Physical Review B

Quantum phase transition in the Z3 Kitaev-Potts model

Razieh Mohseninia, Saeed S. Jahromi, Laleh Memarzadeh, Vahid Karimipour

Published Jun 5, 2015• 11 citations
arXivHTMLPDF

Abstract

The stability of the topological order phase induced by the ${\mathbb{Z}}_{3}$ Kitaev model, which is a candidate for fault-tolerant quantum computation, against the local order phase induced by the three-state Potts model is studied. We show that the low-energy sector of the Kitaev-Potts model is mapped to the Potts model in the presence of transverse magnetic field. Our study relies on two high-order series expansions based on continuous unitary transformations in the limits of small and large Potts couplings as well as mean-field approximation. Our analysis reveals that the topological phase of the ${\mathbb{Z}}_{3}$ Kitaev model breaks down to the Potts model through a first-order phase transition. We capture the phase transition by analysis of the ground-state energy, one-quasiparticle gap, and geometric measure of entanglement.

DOI
10.1103/physrevb.91.245110

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The stability of the topological order phase induced by the Z3{\mathbb{Z}}_{3}Z3​ Kitaev model, which is a candidate for fault-tolerant quantum computation, against the local order phase induced by the three-state Potts model is studied. We show that the low-energy sector of the Kitaev-Potts model is mapped to the Potts model in the presence of transverse magnetic field. Our study relies on two high-order series expansions based on continuous unitary transformations in the limits of small and large Potts couplings as well as mean-field approximation. Our analysis reveals that the topological phase of the Z3{\mathbb{Z}}_{3}Z3​ Kitaev model breaks down to the Potts model through a first-order phase transition. We capture the phase transition by analysis of the ground-state energy, one-quasiparticle gap, and geometric measure of entanglement.
We investigate the low-energy spectral properties and robustness of the topological phase of color code, which is a quantum spin model for the aim of fault-tolerant quantum computation, in the presence of a uniform magnetic field or Ising interactions, using high-order series expansion and exact diagonalization. In a uniform magnetic field, we find first-order phase transitions in all field directions. In contrast, our results for the Ising interactions unveil that for strong enough Ising couplings, the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological phase of color code breaks down to symmetry broken phases by first- or second-order phase transitions.
Topologically ordered quantum systems have robust physical properties, such as quasiparticle statistics, and ground-state degeneracy, which do not depend on the microscopic details of the Hamiltonian. We consider topological phase transitions under a deformation such as an effective string tension on a Z3{\mathbb{Z}}_{3}Z3​ topological state. This is studied in terms of the gauge-symmetry-preserved quantum state renormalization group, first proposed by He, Moradi, and Wen [Phys. Rev. B 90, 205114 (2014)]. In this approach modular matrices SSS and TTT can be obtained and used as order parameters to characterize the topological properties of the phase and determine phase transitions. From a mapping to a classical two-dimensional Potts model on the square lattice, the critical string tension, at which the transition to a topologically trivial phase takes place, can be obtained analytically and agrees with the numerically determined value. Such a transition can be generalized to a ZN{\mathbb{Z}}_{N}ZN​ topological model under a string tension and determined in the same way. With different deformations, the Z3{\mathbb{Z}}_{3}Z3​ topological phase can also be driven to a critical phase, which contains, in the large deformation limit, the wave function analogous to the Rokhsar-Kivelson point in the quantum dimer model in one case and the fully packed loop model in another case.
We investigate the quantum robustness of the topological order in the toric code on the honeycomb lattice in the presence of a uniform parallel field. For a field in z z -direction, the low-energy physics is in the flux-free sector and can be mapped to the transverse-field Ising model on the honeycomb lattice. One finds a second-order quantum phase transition in the 3D Ising ^\star ⋆ universality class for both signs of the field. The same is true for a postive field in x x -direction where an analogue mapping in the charge-free sector yields a ferromagnetic transverse-field Ising model on the triangular lattice and the phase transition is still 3D Ising ^\star ⋆ . In contrast, for negative x x -field, the charge-free sector is mapped to the highly frustrated antiferromagnetic transverse-field Ising model on the triangular lattice which is known to host a quantum phase transition in the 3D XY ^\star ⋆ universality class. Further, the charge-free sector does not always contain the low-energy physics for negative x x -fields and a first-order phase transition to the polarized phase in the charge-full sector takes place at larger negative field values. We quantify the location of this transition by comparing quantum Monte Carlo simulations and high-field series expansions. The full extension of the topological phase in the presence of x x - and z z -fields is determined by perturbative linked-cluster expansions using a full graph decomposition. Extrapolating the high-order series of the charge and the flux gap allows to estimate critical exponents of the gap closing. This analysis indicates that the topological order breaks down by critical lines of 3D Ising ^\star ⋆ and 3D XY ^\star ⋆ type with interesting potential multi-critical crossing points. All findings for the toric code on the honeycomb lattice can be transferred exactly to the toric code on the triangular lattice.
The robustness of the topological color code, which is a class of error-correcting quantum codes, is investigated under the influence of a uniform magnetic field on the honeycomb lattice. Our study relies on two high-order series expansions using perturbative continuous unitary transformations in the limit of low and high fields, exact diagonalization, and a classical approximation. We show that the topological color code in a single parallel field is isospectral to the Baxter-Wu model in a transverse field on the triangular lattice. It is found that the topological phase is stable up to a critical field beyond which it breaks down to the polarized phase by a first-order phase transition. The results also suggest that the topological color code is more robust than the toric code in the parallel magnetic field.
We study the stability of topological order against local perturbations by considering the effect of a magnetic field on a spin model--the toric code--which is in a topological phase. The model can be mapped onto a quantum loop gas where the perturbation introduces a bare loop tension. When the loop tension is small, the topological order survives. When it is large, it drives a continuous quantum phase transition into a magnetic state. The transition can be understood as the condensation of "magnetic" vortices, leading to confinement of the elementary "charge" excitations. We also show how the topological order breaks down when the system is coupled to an Ohmic heat bath and relate our results to error rates for topological quantum computations.
We investigate the phase diagram of a bilayer Kitaev honeycomb model with Ising interlayer interactions, deriving effective models via perturbation theory and performing Majorana mean-field theory calculations. We show that a diverse array of magnetic and topological phase transitions occur, depending on the direction of the interlayer Ising interaction and the relative sign of Kitaev interactions. When two layers have the same sign of the Kitaev interaction, a first-order transition from a Kitaev spin liquid to a magnetically ordered state takes place. The magnetic order points along the Ising axis and it is (anti)ferromagnetic for (anti)ferromagnetic Kitaev interactions. However, when two layers have opposite signs of the Kitaev interaction, we observe a notable weakening of magnetic ordering tendencies and the Kitaev spin liquid survives up to a remarkably larger interlayer exchange. Our mean-field analysis suggests the emergence of an intermediate gapped Z2{\mathbb{Z}}_{2}Z2​ spin-liquid state, which eventually becomes unstable upon vison condensation. The confined phase is described by a highly frustrated ${120}^{\ensuremath{\circ}}$ compass model. We furthermore use perturbation theory to study the model with the Ising axis pointing along the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ axis or lying in the xyxyxy plane. In both cases, our analysis reveals the formation of one-dimensional Ising chains, which remain decoupled in perturbation theory, resulting in a subextensive ground-state degeneracy. Our results highlight the interplay between topological order and magnetic ordering tendencies in bilayer quantum spin liquids.