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arXiv:cond-mat/0412281 · Physical Review B

Instanton correlators and phase transitions in two- and three-dimensional logarithmic plasmas

Kjetil Børkje, Steinar Kragset, Asle Sudbø

Published Feb 14, 2005• 14 citations
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Abstract

The existence of a discontinuity in the inverse dielectric constant of the two-dimensional (2D) 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 three-dimensional 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.

DOI
10.1103/physrevb.71.085112

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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.
We present an exact derivation for the asymptotic large distance behavior of the spin two-point correlation function in the XY-model. This allows for the exact obtainment of the critical exponent η = 1/4 at the Kosterlitz-Thouless transition that occurs in this model and in the 2D neutral Coulomb gas and which has been previously obtained by scaling arguments. In order to do that, we use the language of sine-Gordon theory to obtain a Coulomb Gas description of the XY-model spin correlation function, which becomes identified with the soliton correlator of that theory. Using a representation in terms of bipolar coordinates we obtain an exact expression for the asymptotic large distance behavior of the relevant correlator at β 2 = 8π, which corresponds to the Kosterlitz-Thouless transition. The result is obtained by approaching this point from the plasma (high-temperature) phase of the gas. The vortex correlator of the XY-model is also obtained using the same procedure. KEY WORDS: soliton correlation functions, sine-Gordon theory, XY-model, 2D Coulomb gas, Kosterlitz-Thouless transition
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.
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 zzz of the Kosterlitz-Thouless-Berezinskii transition. By studying large system sizes (L=100L=100L=100), we show z=2.0±0.2z=2.0 \pm0.2z=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≃5.6z\simeq 5.6z≃5.6 using FFH scaling indicates that finite size effects are ubiquitous in 2D superconductors and Josephson Junction Arrays.
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.
In this Letter, we study finite-temperature phase transitions in a two-dimensional classical statistical model called ``dipole-conserving XY model''. Unlike the conventional XY model, the original phase field θθθ has no quasi-long-range order, conventional phase vortices have finite self-energy, and the standard helicity modulus vanishes identically. Analytic vortex energetics and Gaussian continuum theory show that Kosterlitz--Thouless (KT) criticality is instead controlled by phase-gradient (PG) vortices defined in the two compact dipole fields χα=a∂αθχ_α=a\partial_αθχα​=a∂α​θ, whose self-energies grow logarithmically with system size. We determine the phase diagram using parallel-tempering Metropolis Monte Carlo simulations and three diagnostics tailored to dipole conservation. KT finite-size scaling of dipole-field correlation-ratio crossings locates the critical temperatures; generalized helicity moduli defined through quadratic phase twists measure the stiffness of individual dipole channels; and PG-vortex densities obtained from plaquette winding numbers identify the proliferating vortex species. At isotropic couplings, the two PG-vortex species unbind simultaneously at a single KT transition. Spatial anisotropy separates their unbinding temperatures, yielding two KT transitions and an intermediate phase with quasi-long-range order in only one dipole channel. The transitions merge again when the mixed-derivative channel is removed. Our results establish the finite-temperature phase structure of the dipole-conserving XY model and identify thermal PG-vortex unbinding as a novel route to KT criticality in systems with higher-moment conservation.