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arXiv:2003.04909 · Nuclear Physics B

Topological phase transitions in four dimensions

Nicolò Defenu, Andrea Trombettoni, Dario Zappalà

Published Dec 28, 2020• 11 citations
arXivHTMLPDF

Abstract

We show that four-dimensional systems may exhibit a topological phase transition analogous to the well-known Berezinskii-Kosterlitz-Thouless vortex unbinding transition in two-dimensional systems. We study a suitable generalization of the sine-Gordon model in four dimensions and the renormalization group flow equation of its couplings, showing that the critical value of the frequency is the square of the corresponding value in 2D. The value of the anomalous dimension at the critical point is determined (η=1/32) and a conjecture for the universal jump of the superfluid stiffness (4/π2) presented.

DOI
10.1016/j.nuclphysb.2020.115295

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We show that four-dimensional systems may exhibit a topological phase transition analogous to the well-known Berezinskii-Kosterlitz-Thouless vortex unbinding transition in two-dimensional systems. We study a suitable generalization of the sine-Gordon model in four dimensions and the renormalization group flow equation of its couplings, showing that the critical value of the frequency is the square of the corresponding value in 2D. The value of the anomalous dimension at the critical point is determined (η=1/32) and a conjecture for the universal jump of the superfluid stiffness (4/π2) presented.
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.
A number of two-dimensional (2D) critical phenomena can be described in terms of the 2D sine-Gordon model. With bosonization, several 1D quantum systems can be transformed to the same model. However, the transition of the 2D sine-Gordon model, the Berezinskii-Kosterlitz-Thouless (BKT) transition, is essentially different from a second-order transition. The divergence of the correlation length is more rapid than any power law, and there are logarithmic corrections. These pathological features make it difficult to determine the BKT transition point and critical indices from finite-size calculations. In this paper we calculate correlation functions of this model using a real-space renormalization technique. It is found that several correlation functions, or eigenvalues of the corresponding transfer matrix for a finite system, become degenerate on the BKT line, including the logarithmic corrections. By the use of this degeneracy, which reflects the hidden SU(2) symmetry on the BKT line, it is possible to determine the BKT critical line with high precision from a small amount of data and to identify the universality class. In addition, new universal relations are found. These results shed light on the relation between Abelian and non-Abelian bosonization.
The Berezinskii-Kosterlitz-Thouless (BKT) phase transition is considered in the condition of lowest temperatures, when thermal fluctuations give place to quantum ones. For this goal, the critical dynamic of the Sine-Gordon model near the quantum critical point is considered. The approach based on the Keldysh-Schwinger technique of non-equilibrium dynamics description is used, as well as the method of taking into account the crossover from the thermal fluctuation regime to the quantum one in the renormalization group technique. For the system with low vortex concentration, it is shown that the BKT transition unavoidably occurs at a temperature above the crossover temperature from the thermal fluctuation regime to quantum one. As a result at small temperatures the critical exponent of the BKT transition, η=1/4η=1/4η=1/4, remains unchanged, however, the universal relation of the superfluid density jump adjusts a little from the well-known relation by Nelson and Kosterlitz.
The study of the Berezinskii-Kosterlitz-Thouless transition in two-dimensional ${|\ensuremath{\varphi}|}^{4}$ models can be performed in several representations, and the amplitude-phase (AP) Madelung parametrization is a natural way to study the contribution of density fluctuations to nonuniversal quantities. We introduce a functional renormalization group scheme in AP representation where amplitude fluctuations are integrated first to yield an effective sine-Gordon model with renormalized superfluid stiffness. By a mapping between the lattice XYXYXY and continuum ${|\ensuremath{\varphi}|}^{4}$ models, our method applies to both on equal footing. Our approach correctly reproduces the existence of a line of fixed points and of universal thermodynamics and it allows to estimate universal and nonuniversal quantities of the two models, finding good agreement with available Monte Carlo results. The presented approach is flexible enough to treat parameter ranges of experimental relevance.
The renormalization group equations for the sine-Gordon system with a short wave-length cutoff in two dimensions are derived by means of a momentum-shell recursion method. In a special limit they agree with the Kosterlitz renormalization group equations which describe the critical properties of the phase transitions of the two-dimensional (2-D) neutral Coulomb gas with single charges ±1 and of the 2-D isotropic X-Y model in the absence of an external field. The results, therefore, give rise to a possibility that certain classes of phase transitions in low dimensions can be described in a unified way with the sine-Gordon Hamiltonian.
Abstract We introduce and study the properties of a periodic model interpolating between the sine-and the sinh-Gordon theories in 1 + 1 dimensions. This model shows the peculiarities, due to the preservation of the functional form of their potential across RG flows, of the two limiting cases: the sine-Gordon, not having conventional order/magnetization at finite temperature, but exhibiting Berezinskii–Kosterlitz–Thouless (BKT) transition; and the sinh-Gordon, not having a phase transition, but being integrable. The considered interpolation, which we term as sn-Gordon model, is performed with potentials written in terms of Jacobi functions. The critical properties of the sn-Gordon theory are discussed by a renormalization-group approach. The critical points, except the sinh-Gordon one, are found to be of BKT type. Explicit expressions for the critical coupling as a function of the elliptic modulus are given.