Abstract Topological phases have been a central focus of condensed-matter physics for over 50 years. Along with many experimental applications, they have provided much intellectual interest due to their characterization via some form of topological ordering, as opposed to the symmetry-breaking ordering of conventional continuous phase transitions. This distinction is most subtle in the case of the Berezinskii–Kosterlitz–Thouless (BKT) transition as its experimental realizations appear to break U(1) symmetry at low temperature. It also presents two further paradoxes: (i) its prototypical short-range interacting planar XY spin model behaves as an emergent long-range interacting electrolyte; (ii) its topological ordering is not accompanied by a topological nonergodicity within the BKT picture. This review paper addresses these three interconnected questions. We review a series of papers that demonstrate that U(1) symmetry is indeed broken, but within a broader framework than that traditionally used to characterize broken symmetry. We discuss recovery of this symmetry by breaking velocity-symmetry in a deterministic Markov process. We then expand on a modern field theory of the emergent electrolyte that maps directly from the spin field to an emergent lattice electric field governed by an augmented electrostatic Boltzmann distribution. This local model of electrolyte physics resolves both the short-range–long-range paradox and the question of topological nonergodicity—as in contrast with the BKT picture, it describes global topological defects and their nonergodic freezing by the topological ordering. It also connects the broken U(1) symmetry with the topological ordering, providing a comprehensive framework for broken symmetry at the transition. We introduce long-time topological stability as a measure of topological nonergodicity—within a general framework for weakly broken ergodicity.
Abstract Topological phases have been a central focus of condensed-matter physics for over 50 years. Along with many experimental applications, they have provided much intellectual interest due to their characterization via some form of topological ordering, as opposed to the symmetry-breaking ordering of conventional continuous phase transitions. This distinction is most subtle in the case of the Berezinskii–Kosterlitz–Thouless (BKT) transition as its experimental realizations appear to break U(1) symmetry at low temperature. It also presents two further paradoxes: (i) its prototypical short-range interacting planar XY spin model behaves as an emergent long-range interacting electrolyte; (ii) its topological ordering is not accompanied by a topological nonergodicity within the BKT picture. This review paper addresses these three interconnected questions. We review a series of papers that demonstrate that U(1) symmetry is indeed broken, but within a broader framework than that traditionally used to characterize broken symmetry. We discuss recovery of this symmetry by breaking velocity-symmetry in a deterministic Markov process. We then expand on a modern field theory of the emergent electrolyte that maps directly from the spin field to an emergent lattice electric field governed by an augmented electrostatic Boltzmann distribution. This local model of electrolyte physics resolves both the short-range–long-range paradox and the question of topological nonergodicity—as in contrast with the BKT picture, it describes global topological defects and their nonergodic freezing by the topological ordering. It also connects the broken U(1) symmetry with the topological ordering, providing a comprehensive framework for broken symmetry at the transition. We introduce long-time topological stability as a measure of topological nonergodicity—within a general framework for weakly broken ergodicity.
This thesis addresses the emergent electrostatics of two-dimensional, toroidal magnetic models that possess XY symmetry, providing a platform for novel investigations into the Berezinskii-Kosterlitz-Thouless (BKT) phase transition.The BKT transition drives the thermal dissociation of bound pairs of topological defects in many two-dimensional systems, including the two-dimensional XY model of magnetism. The XY model is closely analogous to the two-dimensional Coulomb gas, but can be simulated without computing the long-range interactions of the Coulombic system. This thesis elucidates this paradox by showing that Villain's approximation to the XY model is strictly equivalent to the Maggs-Rossetto (MR) electrostatic model when applied to the two-dimensional Coulomb gas.The mapping is used to probe the BKT transition through the application of the MR algorithm to the two-dimensional Coulomb gas. By simulating the Coulombic system, fluctuations in the winding of charges around the torus are shown to turn on at the BKT transition temperature. These topological-sector fluctuations in the electric field therefore signal the high-temperature phase of the transition.It is then shown that the effective critical exponent of Bramwell-Holdsworth (BH) theory can be measured in superfluid 4He films, which correspond to effective Coulomb gases in the limit of large but finite system size. With the Coulombic system taken as the base BKT system, it is inferred that BH theory is a general property of BKT systems.
The Berezinskii-Kosterlitz-Thouless (BKT) phase transition drives the unbinding of topological defects in many two-dimensional systems. In the two-dimensional Coulomb gas, it corresponds to an insulator-conductor transition driven by charge deconfinement. We investigate the global topological properties of this transition, both analytically and by numerical simulation, using a lattice-field description of the two-dimensional Coulomb gas on a torus. The BKT transition is shown to be an ergodicity breaking between the topological sectors of the electric field, which implies a definition of topological order in terms of broken ergodicity. The breakdown of local topological order at the BKT transition leads to the excitation of global topological defects in the electric field, corresponding to different topological sectors. The quantized nature of these classical excitations, and their strict suppression by ergodicity breaking in the low-temperature phase, afford striking global signatures of topological-sector fluctuations at the BKT transition. We discuss how these signatures could be detected in experiments on, for example, magnetic films and cold-atom systems.
The Berezinskii-Kosterlitz-Thouless (BKT) transition is the paradigmatic example of a topological phase transition without symmetry breaking, where a quasiordered phase, characterized by a power-law scaling of the correlation functions at low temperature, is disrupted by the proliferation of topological excitations above the critical temperature T_{BKT}. In this Letter, we consider the effect of long-range decaying couplings ∼r^{-2-σ} on the BKT transition. After pointing out the relevance of this nontrivial problem, we discuss the phase diagram, which is far richer than the corresponding short-range one. It features-for 7/4<σ<2-a quasiordered phase in a finite temperature range T_{c}<T<T_{BKT}, which occurs between a symmetry broken phase for T<T_{c} and a disordered phase for T>T_{BKT}. The transition temperature T_{c} displays unique universal features quite different from those of the traditional, short-range XY model. Given the universal nature of our findings, they may be observed in current experimental realizations in 2D atomic, molecular, and optical quantum systems.
The Berezinskii-Kosterlitz-Thouless (BKT) transition is an archetypal example of a topological phase transition, which is driven by the proliferation of vortices. In this Letter, we analyze the persistence of the BKT transition in the XY model under the influence of long-range algebraically decaying interactions of the form ∼1/r^{2+σ}. The model hosts a magnetized low temperature phase for sufficiently small σ. Crucially, in the presence of long-range interactions, spin waves renormalize the interaction between vortices, which stabilizes the BKT transition. As a result, we find that there is no direct transition from the magnetized to the disordered phase and that the BKT transition persists for arbitrary long-range exponents, which is distinct from previous results. The key methodological advance is the inclusion of coupling between spin wave and topological excitations. We do this by developing Landau-Peierls-type arguments and using higher-order renormalization group calculations, obtaining compatible results. We emphasize that Landau-Peierls-type arguments are a powerful tool for analyzing continuous spin models. We discuss the relevance of our findings for current Rydberg atom experiments, and highlight the importance of long-range couplings for other types of topological defects.
We investigate the nature of the phase transition occurring in a planar XY-model spin system with dipole–dipole interactions. It is demonstrated that a Berezinskii–Kosterlitz–Thouless (BKT) type of phase transition always takes place at a finite temperature separating the ordered (ferro) and the disordered (para) phases. The low-temperature phase corresponds to an ordered state with thermal fluctuations, composed of a 'gas' of bound vortex–antivortex pairs, which would, when considered isolated, be characterized by a constant vortex–antivortex attraction force which is due to the dipolar interaction term in the Hamiltonian. Using a topological charge model, we show that small bound pairs are easily polarized, and screen the vortex–antivortex interaction in sufficiently large pairs. Screening changes the linear attraction potential of vortices to a logarithmic one, and leads to the familiar pair dissociation mechanism of the BKT type phase transition. The topological charge model is confirmed by numerical simulations, in which we demonstrate that the transition temperature slightly increases when compared with the BKT result for short-range interactions.
The Berezinskii–Kosterlitz–Thouless (BKT) transition is the prototype of a phase transition driven by the formation and interaction of topological defects in two-dimensional (2D) systems. In typical models, these are vortices: above a transition temperature TBKT, vortices are free; below this transition temperature, they get confined. In this work, we extend the concept of BKT transition to quantum systems in two dimensions. In particular, we demonstrate that a zero-temperature quantum BKT phase transition driven by a coupling constant can occur in 2D models governed by an effective gauge field theory with a diverging dielectric constant. One particular example is that of a compact U(1) gauge theory with a diverging dielectric constant, where the quantum BKT transition is induced by non-relativistic, purely 2D magnetic monopoles, which can be viewed also as electric vortices. These quantum BKT transitions have the same diverging exponent z as the quantum Griffiths transition but are not related to disorder.