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Physical review. B, Condensed matter

Critical and noncritical behavior of the Kosterlitz-Thouless-Berezinskii transition

Mark G. Friesen

Published Jan 1, 1996• 6 citations
Publisher

Abstract

The standard treatment of the Kosterlitz-Thouless-Berezinskii (KTB) transition is supplemented here by the inclusion of free-vortex screening effects. We obtain the high-density corrections to the KTB critical behavior. The size of the KTB critical region is extracted, and is found in most cases to be quite small. The tricritical point, separating second-order (KTB) phase transitions from first-order transitions, is also determined. We make predictions concerning similar KTB-like transitions, such as in the layered cuprate superconductors.

DOI
10.1103/physrevb.53.r514

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The standard treatment of the Kosterlitz-Thouless-Berezinskii (KTB) transition is supplemented here by the inclusion of free-vortex screening effects. We obtain the high-density corrections to the KTB critical behavior. The size of the KTB critical region is extracted, and is found in most cases to be quite small. The tricritical point, separating second-order (KTB) phase transitions from first-order transitions, is also determined. We make predictions concerning similar KTB-like transitions, such as in the layered cuprate superconductors.
We have performed a Wilson-Kogut renormalization-group study of Kosterlitz-Thouless-Berezinskii (KTB) layers coupled to one another by a Lawrence-Doniach-type term, in zero field. The recursion relations that we derive are valid when the interlayer coupling \ensuremath{\lambda} is weak and correctly reduce to the recursion relations for the KTB transition when \ensuremath{\lambda}=0. We find that there is crossover away from the two-dimensional KTB behavior and that the size of the upward shift in the transition temperature and the critical region are weakly dependent on \ensuremath{\lambda}. We apply our findings to layered high-temperature superconductors and conclude that their critical behavior at zero field is explained by our model.
Fundamental properties of the Kosterlitz-Thouless-Berezinskii (KTB) transition which occurs in systems in the universality class of the two-dimensional X-Y model are reviewed here with an emphasis on the real-space renormalization-group analysis used to derive the recursion relations. A derivation of the unique temperature dependence of the correlation length of this system will be presented and the signatures of the KTB transition in electronic transport measurements will be explained. Extensions of this model and technique to layered superconductors will also be presented. The size of the three-dimensional (3D) critical region, the nature of the 3D cross-over, and the effect of a finite electric current will be discussed.
Abstract In this paper, we thoroughly examined the Berezinskii-Kosterlitz-Thouless (BKT) phase transition in the two-dimensional XY model on the honeycomb lattice. To address its thermodynamical behavior, we combined standard numerical Monte Carlo simulations with the simulated annealing (SA) protocol and entropic simulations based on the Wang-Landau (WL) algorithm. The transition temperature was determined using the second (ϒ) and fourth-order (ϒ4) helicity modulus as the order parameter. Our best finite-size scaling analysis suggests T BKT = 0.575(8) from SA and T BKT = 0.576(3) from WL. These values deviate significantly from the expected theoretical value of 1 / 2 . We believe that this discrepancy suggests that the theoretical assumptions regarding the analytical calculation may need to be revisited. Additionally, we calculated the vortex density and the formation energy of the vortex-antivortex pairs, where the obtained vortex formation energy is 2μ = 5.80(12). Upon comparison with the square lattice, our results support the notion of instability of the honeycomb lattice to support the spin long-range order and give additional backing to the critical behavior we found.
We test an improved finite-size scaling method for reliably extracting the critical temperature T BKT of a Berezinskii–Kosterlitz–Thouless (BKT) transition. Using known single-parameter logarithmic corrections to the spin stiffness ρ s at T BKT in combination with the Kosterlitz–Nelson relation between the transition temperature and the stiffness, ρ s ( T BKT ) = 2 T BKT /π, we define a size-dependent transition temperature T BKT ( L 1 , L 2 ) based on a pair of system sizes L 1 , L 2 , e.g., L 2 = 2 L 1 . We use Monte Carlo data for the standard two-dimensional classical XY model to demonstrate that this quantity is well behaved and can be reliably extrapolated to the thermodynamic limit using the next expected logarithmic correction beyond the ones included in defining T BKT ( L 1 , L 2 ). For the Monte Carlo calculations we use GPU (graphical processing unit) computing to obtain high-precision data for L up to 512. We find that the sub-leading logarithmic corrections have significant effects on the extrapolation. Our result T BKT = 0.8935(1) is several error bars above the previously best estimates of the transition temperature, T BKT ≈ 0.8929. If only the leading log-correction is used, the result is, however, consistent with the lower value, suggesting that previous works have underestimated T BKT because of the neglect of sub-leading logarithms. Our method is easy to implement in practice and should be applicable to generic BKT transitions.
Besides the Berezinskii-Kosterlitz-Thouless phase transition, the two-dimensional generalized XY model, identified by a generalization parameter qqq (as proposed by Romano and Zagrebnov), can also support a first-order phase transition, starting from a critical value qc{q}_{c}qc​. However, the value of qc{q}_{c}qc​ at which this transition takes place is unknown. In this paper, we take two approaches to accurately determine the critical parameter qc{q}_{c}qc​. Furthermore, we show that the model is characterized by three distinct regions concerning both first-order and Berezinskii-Kosterlitz-Thouless phase transitions. Finally, the underlying mechanism governing such transitions is presented, along with an estimation of the critical temperatures.
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.