A new approach to the Berezinskii–Kosterlitz–Thouless transition in the two-dimensional Coulomb gas model is explored by Monte Carlo simulation and finite size scaling. The usual mapping of a neutral two-dimensional superconductor in zero magnetic field to a Coulomb gas leads to an unscreened logarithmic interaction between the vortices, and with periodic boundary conditions vortex configurations are always vorticity neutral with an equal number of plus and minus vortices. We demonstrate that relaxing the neutrality condition has certain advantages. It leads to non-neutral vortex configurations that can appear in real systems with open boundary conditions and permits calculation of the compressibility, which for thin film superconductors corresponds to the magnetic permeability. The vortex-number fluctuation has remarkable scaling properties at and below the Berezinskii–Kosterlitz–Thouless transition. The fugacity variable becomes dangerously irrelevant in the low-temperature phase and leads to a multiplicative scaling correction to the mean-square vortex-number fluctuation and to the magnetic permeability. This multiplicative correction strongly affects the scaling properties of the vorticity fluctuation at and below the transition. Consequences of these findings are demonstrated using Monte Carlo simulations. Inclusion of the next-higher order correction to scaling is found to play an important role in the analysis of numerical data for the vortex number fluctuation and permits accurate determination of the critical properties.
A new approach to the Berezinskii–Kosterlitz–Thouless transition in the two-dimensional Coulomb gas model is explored by Monte Carlo simulation and finite size scaling. The usual mapping of a neutral two-dimensional superconductor in zero magnetic field to a Coulomb gas leads to an unscreened logarithmic interaction between the vortices, and with periodic boundary conditions vortex configurations are always vorticity neutral with an equal number of plus and minus vortices. We demonstrate that relaxing the neutrality condition has certain advantages. It leads to non-neutral vortex configurations that can appear in real systems with open boundary conditions and permits calculation of the compressibility, which for thin film superconductors corresponds to the magnetic permeability. The vortex-number fluctuation has remarkable scaling properties at and below the Berezinskii–Kosterlitz–Thouless transition. The fugacity variable becomes dangerously irrelevant in the low-temperature phase and leads to a multiplicative scaling correction to the mean-square vortex-number fluctuation and to the magnetic permeability. This multiplicative correction strongly affects the scaling properties of the vorticity fluctuation at and below the transition. Consequences of these findings are demonstrated using Monte Carlo simulations. Inclusion of the next-higher order correction to scaling is found to play an important role in the analysis of numerical data for the vortex number fluctuation and permits accurate determination of the critical properties.
We revisit the scaling properties of the resistivity and the current-voltage characteristics at and below the Berezinskii-Kosterlitz-Thouless transition, both in zero and nonzero magnetic fields. The scaling properties are derived by integrating the renormalization group flow equations up to a scale where they can be reliably matched to simple analytic expressions. The vortex fugacity turns out to be dangerously irrelevant for these quantities below Tc, thereby altering the scaling behavior. We derive the possible crossover effects as the current, magnetic field, or system size is varied, and find a strong multiplicative logarithmic correction near Tc, all of which is necessary to account for when interpreting experiments and simulation data. Our analysis clarifies a longstanding discrepancy between the finite size dependence found in many simulations and the current-voltage characteristics of experiments. We further show that the logarithmic correction can be avoided by approaching the transition in a magnetic field, thereby simplifying the scaling analysis. We confirm our results by large-scale numerical simulations, and calculate the dynamic critical exponent z, for relaxational Langevin dynamics and for resistively and capacitively shunted Josephson junction dynamics.
We revisit the scaling properties of the resistivity and the current-voltage characteristics at and below the Berezinskii-Kosterlitz-Thouless transition, both in zero and nonzero magnetic field. The scaling properties are derived by integrating the renormalization group flow equations up to a scale where they can be reliably matched to simple analytic expressions. The vortex fugacity turns out to be dangerously irrelevant for these quantities below Tc, thereby altering the scaling behavior. We derive the possible crossover effects as the current, magnetic field or system size is varied, and find a strong multiplicative logarithmic correction near Tc, all which is necessary to account for when interpreting experiments and simulation data. Our analysis clarifies a longstanding discrepancy between the finite size dependence found in many simulations and the current-voltage characteristics of experiments. We further show that the logarithmic correction can be avoided by approaching the transition in a magnetic field, thereby simplifying the scaling analysis. We confirm our results by large scale numerical simulations, and calculate the dynamic critical exponent z, for relaxational Langevin dynamics and for resistively and capacitively shunted Josephson junction dynamics.
The Berezinskii-Kosterlitz-Thouless (BKT) mechanism describes universal vortex unbinding in many two-dimensional systems, including the paradigmatic XY model. However, most of these systems present a complex interplay between excitations at different length scales that complicates theoretical calculations of nonuniversal thermodynamic quantities. These difficulties may be overcome by suitably modifying the initial conditions of the BKT flow equations to account for noncritical fluctuations at small length scales. In this work, we perform a systematic study of the validity and limits of this two-step approach by constructing optimised initial conditions for the BKT flow. We find that the two-step approach can accurately reproduce the results of Monte Carlo simulations of the traditional XY model. To systematically study the interplay between vortices and spin-wave excitations, we introduce a modified XY model with increased vortex fugacity. We present large-scale Monte Carlo simulations of the spin stiffness and vortex density for this modified XY model and show that even at large vortex fugacity, vortex unbinding is accurately described by the nonperturbative functional renormalization group.
A numerical study of the neutral two-dimensional (2D) lattice Coulomb gas is performed to examine dynamic scaling, finite size effects, and the dynamic critical exponent z of the Kosterlitz-Thouless-Berezinskii transition. By studying large system sizes (L=100), we show z=2.0±0.2 using Fisher-Fisher-Huse (FFH) scaling. We also present evidence that the vortex correlation length is finite below the transition temperature, in contrast to conventional wisdom. Finally, we conclude that previous findings for a variety of experimental systems that z≃5.6 using FFH scaling indicates that finite size effects are ubiquitous in 2D superconductors and Josephson Junction Arrays.
In three research articles we have studied the critical properties of effective lattice models for strongly correlated electron systems by Monte Carlo simulations. A similar model is used in a fourth article for investigating thermal fluctuations of vortices in a rotating Bose–Einstein condensate. In the first part of this thesis we review the necessary background and introduce the models one by one. The last part is a collection of the papers.\n\nPaper I [1]: We consider the scaling of the mean square dipole moment in a plasma with logarithmic interactions in a two- and three-dimensional system. In both cases, we establish the existence of a low-temperature regime where the mean square dipole moment does not scale with system size and a hightemperature regime does scale with system size. Thus, there is a nonanalytic change in the polarizability of the system as a function of temperature, and hence a metal-insulator transition in both cases. The relevance of this transition in three dimensions to quantum phase transitions in 2 + 1-dimensional systems is briefly discussed.\n\nPaper II [2]: The existence of a discontinuity in the inverse dielectric constant of the two-dimensional Coulomb gas is demonstrated on purely numerical grounds. This is done by expanding the free energy in an applied twist and performing a finite-size scaling analysis of the coefficients of higher-order terms. The phase transition, driven by unbinding of dipoles, corresponds to the Kosterlitz-Thouless transition in the 2D XY model. The method developed is also used for investigating the possibility of a Kosterlitz-Thouless phase transition in a threedimensional system of point charges interacting with a logarithmic pair-potential, a system related to effective theories of low-dimensional strongly correlated systems. We also contrast the finite-size scaling of the fluctuations of the dipole moments of the two-dimensional Coulomb gas and the three-dimensional logarithmic system to those of the three-dimensional Coulomb gas.\n\nPaper III [3]: We perform large-scale Monte Carlo simulations on an effective gauge theory for an easy plane quantum anti-ferromagnet, including a Berry phase term that projects out the S = 1/2 sector. Without a Berry phase term, the model exhibits a phase transition in the 3DXY universality class associated with proliferation of gauge-charge neutral U(1) vortices. The instantons that eliminate the phase transition in the gauge-charged sector are cancelled by the Berry phases. The result is a first order phase transition. This gauge theory therefore does not exhibit deconfined criticality.\n\nPaper IV [4]: We perform Monte Carlo studies of vortices in three dimensions in a cylindrical confinement, with uniform and nonuniform density. The former is relevant to rotating 4He, the latter is relevant to a rotating trapped Bose condensate. In the former case we find dominant angular thermal vortex fluctuations close to the cylinder wall. For the latter case, a novel effect is that at low temperatures the vortex solid close to the center of the trap crosses directly over to a tension-less vortex tangle near the edge of the trap. At higher temperatures an intermediate tensionful vortex liquid located between the vortex solid and the vortex tangle, may exist.
Phase transitions and their critical scaling properties, especially in systems with disorder, are important both for our theoretical understanding of our environment, but also for their practical use in applications and materials in our everyday life. This thesis presents results from finite size scaling analysis of critical phenomena in systems with disorder, using high-precision Monte Carlo simulations and state of the art numerical methods. Specifically, theoretical models suitable for simulations in the presence of uncorrelated or correlated disorder are studied. Uncorrelated strong disorder, as present in the two dimensional gauge glass model to study the vortex glass phase of high temperature superconductors in an applied magnetic field is shown to lack a finite temperature phase transition. Further, results from dynamic quantities, such as resistance and autocorrelation functions, indicate the existence of two distinct diverging correlation times, one associated with local relaxation and one associated with vortex phase slips. Correlated disorder is studied both in the superfluid transition of helium-4 and in the anisotropic critical scaling of a transverse Meissner-like transition in an experimental setup of a high temperature superconductor. For the superfluid helium transition, it is shown that the presence of fractally correlated disorder presumably alters the universality class of the pure model. Also, a comparison with experimental data suggests that the critical scaling theory describing the heat capacity of helium-4 may need to be modified in the presence of the disorder. In the case of superconductors, analyzing experimental data from resistance measurements in a system with columnar defects together with an anisotropy in the applied magnetic field, reveals a fully anisotropic scaling regime. Finally, a data analysis is presented from simulations of a charged particle gas system in three dimensions, where the normal Coulomb interaction between charges is changed into a logarithmic interaction. Previous work indicates the possibility of a transition similar to the Kosterlitz-Thouless transition in certain two dimensional systems. On the contrary, our simulations seem to favor a system whose critical scaling behavior is consistent with a transition occurring only at zero critical temperature. Overall, disorder in the model systems studied leads to important modifications of the critical scaling properties of pure systems, and thereby also to possible changes of the corresponding universality classes. This results in interesting predictions with experimentally relevant consequences.