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arXiv:2207.14343 · Physical review. B./Physical review. B

Fractonic Berezinskii-Kosterlitz-Thouless transition from a renormalization group perspective

Kevin T. Grosvenor, Ruben Lier, Piotr Surówka

Published Jan 27, 2023• 8 citations
arXivHTMLPDF

Abstract

Proliferation of defects is a mechanism that allows for topological phase transitions. Such a phase transition is found in two dimensions for the XY model, which lies in the Berezinskii-Kosterlitz-Thouless (BKT) universality class. The transition point can be found using renormalization group analysis. We apply renormalization group arguments to determine the nature of BKT transitions for the three-dimensional plaquette-dimer model, which is a model that exhibits fractonic mobility constraints. We show that an important part of this analysis demands a modified dimensional analysis that changes the interpretation of scaling dimensions upon coarse-graining. Using this modified dimensional analysis, we compute the beta functions of the model and predict a finite critical value above which the fractonic phase melts, proliferating dipoles. Importantly, the transition point is found through a renormalization group analysis that accounts for the phenomenon of UV/IR mixing, characteristic of fractonic models.

DOI
10.1103/physrevb.107.045139

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Proliferation of defects is a mechanism that allows for topological phase transitions. Such a phase transition is found in two dimensions for the XY model, which lies in the Berezinskii-Kosterlitz-Thouless (BKT) universality class. The transition point can be found using renormalization group analysis. We apply renormalization group arguments to determine the nature of BKT transitions for the three-dimensional plaquette-dimer model, which is a model that exhibits fractonic mobility constraints. We show that an important part of this analysis demands a modified dimensional analysis that changes the interpretation of scaling dimensions upon coarse-graining. Using this modified dimensional analysis, we compute the beta functions of the model and predict a finite critical value above which the fractonic phase melts, proliferating dipoles. Importantly, the transition point is found through a renormalization group analysis that accounts for the phenomenon of UV/IR mixing, characteristic of fractonic models.
We consider close-packed tiling models of geometric objects---a mixture of hardcore dimers and plaquettes---as a generalization of the familiar dimer models. Specifically, on an anisotropic cubic lattice, we demand that each site be covered by either a dimer on a zzz link or a plaquette in the $x\text{\ensuremath{-}}y$ plane. The space of such fully packed tilings has an extensive degeneracy. This maps onto a fracton-type ``higher-rank electrostatics,'' which can exhibit a plaquette-dimer liquid and an ordered phase. We analyze this theory in detail, using height representations and T-duality to demonstrate that the concomitant phase transition occurs due to the proliferation of dipoles formed by defect pairs. The resultant critical theory can be considered as a fracton version of the Kosterlitz-Thouless transition. A significant new element is its UV-IR mixing, where the low-energy behavior of the liquid phase and the transition out of it is dominated by local (short-wavelength) fluctuations, rendering the critical phenomenon beyond the renormalization group paradigm.
It is argued that two-dimensional U(N) spin models for any N undergo a Berezinskii-Kosterlitz-Thouless (BKT)-like phase transition, similarly to the famous XY model. This conclusion follows from the Berezinskii-like calculation of the two-point correlation function in U(N) models, approximate renormalization group analysis, and numerical investigations of the U(2) model. It is shown, via Monte Carlo simulations, that the universality class of the U(2) model coincides with that of the XY model. Moreover, preliminary numerical results point out that two-dimensional SU(N) spin models with the fundamental and adjoint terms and N>4 exhibit two phase transitions of BKT type, similarly to Z(N) vector models.
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.
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.
We investigate the general features of the renormalization-group flow at the Berezinskii-Kosterlitz-Thouless (BKT) transition, providing a thorough quantitative description of the asymptotc critical behavior, including the multiplicative and subleading logarithmic corrections. For this purpose, we consider the RG flow of the sine-Gordon model around the renormalizable point which describes the BKT transition. We reduce the corresponding $\ensuremath{\beta}$ functions to a universal canonical form, valid to all perturbative orders. Then we determine the asymptotic solutions of the RG equations in various critical regimes: the infinite-volume critical behavior in the disordered phase, the finite-size scaling limit for homogeneous systems of finite size, and the trap-size scaling limit occurring in two-dimensional bosonic particle systems trapped by an external space-dependent potential.