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arXiv:cond-mat/0312587 · Physical Review Letters

Strong-Weak Coupling Self-Duality in the Two-Dimensional Quantum Phase Transition of p+ip Superconducting Arrays

Cenke Xu, Joel E. Moore

Published Jul 23, 2004• 115 citations
arXivHTMLPDF

Abstract

The 2D quantum phase transition that occurs in a square lattice of Josephson-coupled p +/- ip superconductors is an example of how four-body interactions in d = 2 reproduce nonperturbative effects caused by two-body interactions in d = 1. The ordered phase has an unconventional "bond order" of the local T-breaking variable. This problem can be analyzed using an exact self-duality; this duality in classical notation is the 3D generalization of the Kramers-Wannier duality of the 2D Ising model, and there are similar exact dualities in dimensions d > or = 3. We discuss the excitation spectrum and experimental signatures of the ordered and disordered phases, and the relationship between our model and previously studied behavior of 2D boson models with four-boson interactions.

DOI
10.1103/physrevlett.93.047003

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The 2D quantum phase transition that occurs in a square lattice of Josephson-coupled p +/- ip superconductors is an example of how four-body interactions in d = 2 reproduce nonperturbative effects caused by two-body interactions in d = 1. The ordered phase has an unconventional "bond order" of the local T-breaking variable. This problem can be analyzed using an exact self-duality; this duality in classical notation is the 3D generalization of the Kramers-Wannier duality of the 2D Ising model, and there are similar exact dualities in dimensions d > or = 3. We discuss the excitation spectrum and experimental signatures of the ordered and disordered phases, and the relationship between our model and previously studied behavior of 2D boson models with four-boson interactions.
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.
Using large-scale quantum Monte Carlo simulations of lattice bosonic models, we precisely investigate the effect of weak Josephson tunneling between 2D superfluid or superconducting layers. In the clean case, the Kosterlitz-Thouless transition immediately turns into 3DXY, with phase coherence and superflow in all spatial directions, and a strong enhancement of the critical temperature. However, when disorder is present, rare regions fluctuations can lead to an intermediate finite-temperature phase —the so-called sliding regime— where only 2D superflow occurs within the layers without any transverse superfluid coherence, while a true 3D Bose-Einstein condensate exists. Critical properties of such an unconventional regime are carefully investigated.
Duality places an important constraint on the renormalization group flows and the phase diagrams. For self-dual theories, the self-duality can be promoted as a symmetry, this leads to the multicriticalities. This work investigates a description of the deconfined quantum criticality, the Nf=2QED3{N}_{f}=2\phantom{\rule{4pt}{0ex}}{\mathrm{QED}}_{3}Nf​=2QED3​, as an example of self-dual theories, and its multicritical behavior under perturbative deformations. The multicriticality is described by the theory with Gross-Neveu couplings and falls in a different universality class than the standard deconfined quantum criticality. We systematically calculate the scaling dimensions of various operators in the 3D quantum electrodynamics with the Chern-Simons term and Gross-Neveu couplings by the large-NNN renormalization group analysis. Specifically, we find certain Drelativistic four-fermion interactions which correspond to the dimer-dimer interactions in the lattice model will drive the deconfined quantum criticality to the first-order transition, this result is consistent with previous numerical studies.
We have performed large-scale Monte Carlo simulations on a model describing a (2+1)-dimensional array of dissipative Josephson junctions. We find three distinct stable quantum phases of the system. The most ordered state features long-range spatial ordering in the phase $\ensuremath{\theta}$ of the superconducting order parameter, but temporal ordering only in spatial gradients $\ensuremath{\Delta}\ensuremath{\theta}$, not in $\ensuremath{\theta}$. Significantly, the most ordered state therefore does not have three-dimensional (3D) XYXYXY ordering. Rather, it features two-dimensional (2D) spin waves coexisting with temporally disordered phases $\ensuremath{\theta}$. There is also an intermediate phase featuring quasi-long-range spatial order in $\ensuremath{\theta}$ coexisting with a gas of instantons in $\ensuremath{\Delta}\ensuremath{\theta}$. We briefly discuss possible experimental signatures of such a state, which may be viewed as a local metal and a global superconductor. The most disordered state has phase disorder in all spatio-temporal directions, and may be characterized as a gas of proliferated vortices coexisting with a gas of $\ensuremath{\Delta}\ensuremath{\theta}$ instantons. The phase transitions between these phases are discussed. The transition from the most ordered state to the intermediate state is driven by proliferation of instantons in $\ensuremath{\Delta}\ensuremath{\theta}$. The transition from the intermediate state to the most disordered state is driven by the proliferation of spatial point vortices in the background of a proliferated $\ensuremath{\Delta}\ensuremath{\theta}$-instanton gas, and constitutes a Berezinskii-Kosterlitz-Thouless phase transition. The model also features a direct phase transition from the most ordered state to the most disordered state, and this transition is neither in the 2D XYXYXY nor in the 3D XYXYXY universality class. It comes about via a simultaneous proliferation of point vortices in two spatial dimensions and $\ensuremath{\Delta}\ensuremath{\theta}$ instantons, with a complicated interplay between them. The results are compared to, and differ in a fundamental way from, the results that are found in dissipative quantum rotor systems. The difference originates with the difference in the values that the fundamental degrees of freedom can take in the latter systems compared to dissipative Josephson junction arrays.
Abstract We review our recent results on Anderson localization in systems of two interacting particles coupled by contact interactions. Based on an exact mapping to an effective single-particle problem, we numerically investigate the occurrence of metal-insulator phase transitions for the pair in two-(2D) and three-dimensional (3D) disordered lattices. In two dimensions, we find that interactions cause an exponential enhancement of the pair localization length with respect to its single-particle counterpart, but do not induce a delocalization transition. In particular we show that previous claims of 2D interaction-induced Anderson transitions are the results of strong finite-size effects. In three dimensions we find that the pair undergoes a metal-insulator transition belonging to the same (orthogonal) universality class of the noninteracting model. We then explore the phase diagram in the space of energy E, disorder W and interaction strength U, which reveals a rich and counterintuitive structure, endowed with multiple metallic and insulating phases. We point out that this phenomenon originates from the molecular and scattering-like nature of the pair states available at given energy and disorder strength.
We study phase transitions of coupled two dimensional XY systems with spatial anisotropy and U(1)×Z2U(1) \times \mathbb{Z}_2U(1)×Z2​ symmetry, motivated by spinless bosonic atoms trapped in square optical lattice on the metastable first excited p−p-p−level orbitals with anisotropic hopping amplitudes. The phase transitions of the system are generally split into an Ising transition and an XY transition, but the sequence and the nature of the transitions depend on the ratio between the anisotropic couplings. In the isotropic limit the XY variables are expected to be disordered before the Ising variables when thermal or quantum fluctuations are turned on gradually. In the anisotropic limit with zero perpendicular hoppings, the finite temperature transition is a Kosterlitz-Thouless transition driven by proliferation of hybrid half vortices, and the zero temperature quantum phase transition is split into a bond order transition and a 3D XY transition, which can be driven by the condensation of either single vortices or half vortices. After the condensation of half vortices the resultant state is a Mott Insulator of paired bosons. A small perpendicular hopping JbJ_bJb​ leads to a 2D Ising transition at low temperature and a 2+1d quantum Ising transition with a small charging energy at zero temperature. Global phase diagrams for both classical and quantum phase transitions are drawn. The analytical results obtained in this work are expected to be checked both numerically and experimentally.