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arXiv:0709.2276

Exact results of the quantum phase transition for the topological order

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

Published Sep 14, 2007• 2 citations
arXivHTMLPDF

Abstract

In this paper a duality between the d=2 Wen-plaquette model in a transverse field and the d=1 Ising model in a transverse field is used to learn the nature of the quantum phase transition (QPT) between a spin-polarized phase and a topological ordered state with the string-net condensation. The QPT is not induced by spontaneous symmetry breaking and there are no conventional Landau-type local order parameters. Instead the string-like non-local order parameters are introduced to describe the QPT. In particular, the duality character between the open-string and closed string for the QPT are explored.

DOI
10.48550/arxiv.0709.2276

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In this paper a duality between the d=2 Wen-plaquette model in a transverse field and the d=1 Ising model in a transverse field is used to learn the nature of the quantum phase transition (QPT) between a spin-polarized phase and a topological ordered state with the string-net condensation. The QPT is not induced by spontaneous symmetry breaking and there are no conventional Landau-type local order parameters. Instead the string-like non-local order parameters are introduced to describe the QPT. In particular, the duality character between the open-string and closed string for the QPT are explored.
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 nature of phase transitions involving the questions why and how phase transitions take place has not been sufficiently touched in the literature. In contrast, the current attention to certain extent still focus on the description of critical phenomena and the classification of the associated phase transition along with the Ginzburg-Landau-Wilson paradigm, where the key issue is to identify phenomenologically order parameters and related symmetries. This brings the question to topological phase transitions (TPTs), where no obvious order parameter and the broken symmetry are identified. Here we present a unified pattern description of the second-order quantum phase transition (QPT) and TPT, both involved in the one-dimensional anisotropic quantum XY model in a transverse field, which contains the transverse Ising model (TIM) as a limit case. Away from the TIM, the XY model enters the ferromagnetic phase (marked by a second-order QPT or a direct TPT) as increasing ferromagentic exchange coupling, a series of TPTs occur, which are absent in the TIM. The TPTs behave like the first-order QPTs. In the isotropic and large exchange coupling cases, the ground state of the XY model is dominated by two topologically different vortices along positive and negative direction of the transverse field. We confirm the above conclusion by analyzing the energy contributions of the patterns to the ground state and calculating the ground state pattern occupations of the XY model. The results have been obtained in a unified and self-evident way and answer the questions why and how the QPT and TPTs take place in the XY model.
Applying the Jordan-Wigner transformation to spin-1/2 operators on special quasi-one dimensional paths, we show that the two-dimensional Kitaev model can be exactly mapped to a free Majorana fermion model without any redundant degrees of freedom. Via duality transformation, it can be further shown that the quantum phase transitions of the Kitaev model are described by non-local topological order parameters, which become Landau type local order parameters in the dual space. A closed relationship between conventional and topological quantum phase transitions is revealed, and the validity of conventional Landau phase transition theory with spontaneous symmetry breaking and local order parameters is also extended.
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 study the quench dynamics of the topological quantum phase transition in the two-dimensional transverse Wen-plaquette model, which has a phase transition from a Z2{Z}_{2}Z2​ topologically ordered to a spin-polarized state. By mapping the Wen-plaquette model onto a one-dimensional quantum Ising model, we calculate the expectation value of the plaquette operator Fi{F}_{i}Fi​ during a slowly quenching process from a topologically ordered state. A logarithmic scaling law of quench dynamics near the quantum phase transition is found, which is analogous to the well-known static critical behavior of the specific heat in the one-dimensional quantum Ising model.