arXiv:cond-mat/0402566 · Europhysics Letters (EPL)

Topological order in the (2+1)D compact lattice superconductor

Anders Vestergren, Jack Lidmar, T. H. Hansson

Published Dec 16, 2004• 9 citations

Abstract

We study topological aspects of a compact lattice superconductor, and show that the characteristic energy splitting, Δ, between almost degenerate ground states, is simply related to a novel order parameter , which is closely related to large Wilson loops. Using Monte Carlo methods, we study the scaling properties of close to the deconfining phase transition, and conclude that Δ ∼ e − L /ξ , where L is the size of the system, thus giving quantitative support to the vortex tunneling scenario proposed by Wen.

DOI: 10.1209/epl/i2004-10320-5

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arXiv:cond-mat/0402566 · Europhysics Letters (EPL)

Topological order in the (2+1)D compact lattice superconductor

Anders Vestergren, Jack Lidmar, T. H. Hansson

Published Dec 16, 2004• 9 citations

arXiv HTML PDF

Abstract

DOI: 10.1209/epl/i2004-10320-5

Related papers

Topological Order in the (2+1)D Compact Lattice Superconductor

Anders Vestergren, Jack Lidmar, T. H. Hansson

PACS. 74.20.-z – Theories and models of superconducting state. PACS. 11.15.Ha – Lattice gauge theory. PACS. 71.10.Hf – Non-Fermi-liquid ground states, electron phase diagrams and phase transitions in model systems. Abstract. – We study topological aspects of a compact lattice superconductor, and show that the characteristic energy splitting, ∆, between almost degenerate ground states, is simply related to a novel order parameter ˜ W, which is closely related to large Wilson loops. Using Monte Carlo methods, we study the scaling properties of ˜ W close to the deconfining phase transition, and conclude that ∆ ∼ e −L/ξ, where L is the size of the system, thus giving quantitative support to the vortex tunneling scenario proposed by Wen. The concept of topological order was first introduced to describe the ground states of the quantum Hall system [1]. Although the present interest in topological order to large extent derives from the quest for exotic non-Fermi liquid states of relevance for the high-Tc superconductors [2–6], the concept is of much more general interest. Both spin liquids [7,8], believed to be of relevance for frustrated magnets, and ordinary BCS superconductors with dynamicalPACS. 74.20.-z – Theories and models of superconducting state. PACS. 11.15.Ha – Lattice gauge theory. PACS. 71.10.Hf – Non-Fermi-liquid ground states, electron phase diagrams and phase transitions in model systems. Abstract. – We study topological aspects of a compact lattice superconductor, and show that the characteristic energy splitting, ∆, between almost degenerate ground states, is simply related to a novel order parameter ˜ W, which is closely related to large Wilson loops. Using Monte Carlo methods, we study the scaling properties of ˜ W close to the deconfining phase transition, and conclude that ∆ ∼ e −L/ξ, where L is the size of the system, thus giving quantitative support to the vortex tunneling scenario proposed by Wen. The concept of topological order was first introduced to describe the ground states of the quantum Hall system [1]. Although the present interest in topological order to large extent derives from the quest for exotic non-Fermi liquid states of relevance for the high-Tc superconductors [2–6], the concept is of much more general interest. Both spin liquids [7,8], believed to be of relevance for frustrated magnets, and ordinary BCS superconductors with dynamical

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Topological quantum phase transitions in topological superconductors

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In this paper we show that BF topological superconductors (insulators) exibit phase transitions between different topologically ordered phases characterized by different ground-state degeneracy on manifold with non-trivial topology. These phase transitions are induced by the condensation of (or lack of) topological defects. We concentrate on the (2+1)-dimensional case where the BF model reduces to a mixed Chern-Simons term and we show that the superconducting phase has a ground-state degeneracy k and not k 2 . When the symmetry is U (1) × U (1), namely when both gauge fields are compact, the mixed Chern-Simons model is not equivalent to the sum of two Chern-Simons terms with opposite chirality (even if naively diagonalizable) since the U (1) symmetry requires an ultraviolet regularization that makes the diagonalization impossible. We analyze this aspect using a lattice regularization, where the gauge fields become angular variables. In addition, we will show that the phase in which both gauge fields are compact is not allowed dynamically.In this paper we show that BF topological superconductors (insulators) exibit phase transitions between different topologically ordered phases characterized by different ground-state degeneracy on manifold with non-trivial topology. These phase transitions are induced by the condensation of (or lack of) topological defects. We concentrate on the (2+1)-dimensional case where the BF model reduces to a mixed Chern-Simons term and we show that the superconducting phase has a ground-state degeneracy k and not k 2 . When the symmetry is U (1) × U (1), namely when both gauge fields are compact, the mixed Chern-Simons model is not equivalent to the sum of two Chern-Simons terms with opposite chirality (even if naively diagonalizable) since the U (1) symmetry requires an ultraviolet regularization that makes the diagonalization impossible. We analyze this aspect using a lattice regularization, where the gauge fields become angular variables. In addition, we will show that the phase in which both gauge fields are compact is not allowed dynamically.

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Transition of a Z3 topologically ordered phase to trivial and critical phases

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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 ${\mathbb{Z}}_{3}$ 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 $S$ and $T$ 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 ${\mathbb{Z}}_{N}$ topological model under a string tension and determined in the same way. With different deformations, the ${\mathbb{Z}}_{3}$ 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.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 ${\mathbb{Z}}_{3}$ 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 $S$ and $T$ 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 ${\mathbb{Z}}_{N}$ topological model under a string tension and determined in the same way. With different deformations, the ${\mathbb{Z}}_{3}$ 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.

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\mathrm{U}(1)

matter fields coupled to an emergent

{\mathbb{Z}}_{2}

gauge field. Subsequently, we use a generalized Wilson-loop order parameter, the equal-time Fredenhagen-Marcu order parameter, to characterize the topological ordering in the different phases. Our computation provides evidence of the toric code phase both in the Mott-insulator and the charge-

2e

superconductor phases, while showing that the toric code phase disappears in the charge-

e

superconductor phase. In addition, we perturbatively analyze the influence of Cooper pair tunneling on the topological gap of the toric code in the limit of strong charging energy and show that the toric code phase is, in fact, stabilized by the Cooper pair tunneling. Our results are relevant for experimental realizations of the Majorana toric code.

log2

. Their robust quantization property is inferred from the underlying lattice gauge theory description of topological superconductors and is corroborated via exact solutions and numerical simulations. Transitions between topologically trivial and nontrivial phases are accompanied by scaling behavior, a hallmark of genuine order parameters, captured by entanglement critical exponents. These order parameters are experimentally measurable utilizing state-of-the-art techniques.

{\mathbb{Z}}_{3}

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

S

and

T

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

{\mathbb{Z}}_{N}

topological model under a string tension and determined in the same way. With different deformations, the

{\mathbb{Z}}_{3}

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.