The three-dimensional integer-valued lattice gauge theory, which is also known as a ``frozen superconductor,'' can be obtained as a certain limit of the Ginzburg-Landau theory of superconductivity, and is believed to be in the same universality class. It is also exactly dual to the three-dimensional XY model. We use this duality to demonstrate the practicality of recently developed methods for studying topological defects, and investigate the critical behavior of the phase transition using numerical Monte Carlo simulations of both theories. On the gauge theory side, we concentrate on the vortex tension and the penetration depth, which map onto the correlation lengths of the order parameter and the Noether current in the XY model, respectively. We show how these quantities behave near the critical point, and that the penetration depth exhibits critical scaling only very close to the transition point. This may explain the failure of superconductor experiments to see the inverted XY model scaling.
The three-dimensional integer-valued lattice gauge theory, which is also known as a ``frozen superconductor,'' can be obtained as a certain limit of the Ginzburg-Landau theory of superconductivity, and is believed to be in the same universality class. It is also exactly dual to the three-dimensional XY model. We use this duality to demonstrate the practicality of recently developed methods for studying topological defects, and investigate the critical behavior of the phase transition using numerical Monte Carlo simulations of both theories. On the gauge theory side, we concentrate on the vortex tension and the penetration depth, which map onto the correlation lengths of the order parameter and the Noether current in the XY model, respectively. We show how these quantities behave near the critical point, and that the penetration depth exhibits critical scaling only very close to the transition point. This may explain the failure of superconductor experiments to see the inverted XY model scaling.
We investigate the critical relaxational dynamics of the three-dimensional (3D) lattice Z_{N} gauge models with N=6 and N=8, whose equilibrium critical behavior at their topological transitions belongs to the inverted XY (IXY) universality class [this is also the universality class of the continuous transitions of the 3D lattice U(1) gauge Higgs models with a one-component complex scalar field], which is connected to the standard XY universality class by a nonlocal duality relation of the partition functions. Specifically, we consider the purely relaxational dynamics realized by a locally reversible Metropolis dynamics, as commonly used in Monte Carlo simulations. To determine the corresponding dynamic exponent z, we focus on the out-of-equilibrium critical relaxational flows arising from instantaneous quenches to the critical point, which are analyzed within an out-of-equilibrium finite-size scaling framework. We obtain the estimate z=2.59(3). A numerical analysis of the equilibrium critical dynamics give consistent, but less accurate, results. This dynamic exponent is expected to characterize the critical slowing down of the purely relaxational dynamics of all topological transitions that belong to the 3D IXY universality class. We note that this result implies that the critical relaxational dynamics of the 3D IXY universality class is slower than that of the standard 3D XY universality class, whose relaxational dynamic exponent z≈2.02 is significantly smaller, although they share the same length-scale critical exponent ν≈0.6717.
Using the scaling theory of quantum critical phenomena we explore the occurrence of universal critical behavior at the insulator-to-superconductor and superconductor-to-normal state transitions at zero temperature.Experimentally, these phase transitions are driven by doping and correspond to critical end points of the phase transition łine in the temperature-hole concentration plane.Provided that the order parameter is a complex scalar in two dimensions, and that the London relation between superfluid number density and magnetic penetration depth holds, the scaling theory predicts universal behavior close to the insulator-to-superconductor transition.In particular, transition temperature and zero temperature penetration depth are universally related and the sheet resistance adopts a universal value.These predictions agree remarkably well with available experimental data and provide useful constraints for a microscopic theory.
The deconfined phase of (2+1)-dimensional Zk gauge theory exhibits topological order, with e and m anyons that have a $2\ensuremath{\pi}/k$ braiding phase. Proliferating either e or m drives Higgs or confinement transitions, respectively. At the multicritical point where these transitions meet, the theory enjoys an additional duality symmetry that exchanges e and m anyons. This symmetry forces anyons with nontrivial braiding to close their gaps simultaneously, giving rise to a critical theory that mixes strong interactions with mutual statistics. We propose an effective $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ gauge theory with a mutual Chern-Simons term at level k to describe the vicinity of the multicritical point for $k\ensuremath{\ge}4$. The emergence of a global $\mathrm{U}{(1)}^{\mathrm{top}}\ifmmode\times\else\texttimes\fi{}\mathrm{U}{(1)}^{\mathrm{top}}$ symmetry at the critical point imposes powerful constraints on universal properties of the phase transition. In particular, we show that (1) the lattice magnetic flux operator embeds as a conserved U(1) current with protected scaling dimension; (2) the first-order line emanating from the critical point for k=2 disappears generically for sufficiently large k; (3) the correlation length exponent approaches that of the three-dimensional XY model with corrections of order 1/k2 in the large-k limit. These predictions can be tested in near-term numerical simulations and pave the way for a more general exploration of topological quantum criticality enriched with anyon-permuting symmetries.
We have studied the critical properties of three-dimensional U(1)-symmetric lattice gauge theories. The models apply to various physical systems such as insulating phases of strongly correlated electron systems as well as superconducting and superfluid states of liquid metallic hydrogen under extreme pressures. This thesis contains an introductory part and a collection of research papers of which seven are published works and one is submitted for publication.\n\nPaper I: Critical properties of the 2+1-dimensional compact abelian Higgs model with gauge charge q=2 are studied. We introduce a novel method of computing the third moment M3 of the action which allows us to extract correlation length and specific heat critical exponents ν and α without invoking hyperscaling. Finite-size scaling analysis of M3 yields the ratio (1+α)/ν and 1/ν separately. We find that α and ν vary along the critical line of the theory, which however exhibits a remarkable resilience of Z2 criticality. We conclude that the model is a fixed-line theory, which we propose to characterize the zero temperature quantum phase transition from a Mott-Hubbard insulator to a charge fractionalized insulator in two spatial dimensions.\n\nPaper II: Large scale Monte Carlo simulations are employed to study phase transitions in the three-dimensional compact abelian Higgs model in adjoint representations of the matter field, labeled by an integer q, for q=2,3,4,5. We also study various limiting cases of the model, such as the Zq lattice gauge theory, dual to the 3DZq spin model, and the 3D xy spin model which is dual to the Zq lattice gauge theory in the limit q → ∞. In addition, for benchmark purposes, we study the 2D square lattice 8-vertex model, which is exactly solvable and features non-universal critical exponents. The critical exponents α and ν are calculated from finite size scaling of the third moment of the action, and the method is tested thoroughly on models with known values for these exponents. We have found that for q=3, the three-dimensional compact abelian Higgs model exhibits a second order phase transition line which joins a first order phase transition line at a tricritical point. The results for q=2 in Paper I are reported with a higher lever of detail.\n\nPaper III: This paper is based on a talk by F. S. Nogueira in the Aachen HEP 2003 conference where a review of the results for the compact abelian Higgs model from Paper I and Paper II was presented, as well as the results for the q=1 case studied by F. S. Nogueira, H. Kleinert and A. Sudbø.\n\nPaper IV: We study the effects of a Chern-Simons (CS) term in the phase structure of two different abelian gauge theories in three dimensions. By duality transformations we show how the compact U(1) gauge theory with a CS term for certain values of the CS coupling can be written as a gas of vortex loops interacting through steric repulsion. This theory is known to exhibit a phase transition governed by proliferation of vortex loops. We also employ Monte Carlo simulations to study the non-compact U(1) abelian Higgs model with a CS term. Finite size scaling of the third moment of the action yields critical exponents α and ν that vary continuously with the strength of the CS term, and a comparison with available analytical results is made.\n\nPaper V: The critical properties of N-component Ginzburg-Landau theory are studied in d=2+1 dimensions. The model is dualized to a theory of N vortex fields interacting through a Coulomb and a screened potential. The model with N=2 shows two anomalies in the specific heat. From Monte Carlo simulations we calculate the critical exponents α and ν and the mass of the gauge field. We conclude that one anomaly corresponds to an inverted 3D xy fixed point, while the other corresponds to a 3D xy fixed point. There are N fixed points, namely one corresponding to an inverted 3D xy fixed point, and N-1corresponding to neutral 3D xy fixed points. Applications are briefly discussed.\n\nPaper VI: The phase diagram and critical properties of the N-component London superconductor are studied both analytically and through large-scale Monte-Carlo simulations in d=2+1 dimensions. The model with different bare phase stiffnesses for each flavor is a model of superconductivity which should arise out of metallic phases of light atoms under extreme pressure. A projected mixture of electronic and protonic condensates in liquid metallic hydrogen under extreme pressure is the simplest example, corresponding to N=2 with individually conserved matter fields. We compute critical exponents α and ν for N=2 and N=3. The results from Paper V are presented at a higher level of detail. For the arbitrary N case, there are N fixed points,namely one charged inverted 3D xy fixed point, and N-1 neutral 3D xy fixed points. We explicitly identify one charged vortex mode and N-1 neutral vortex modes. The model for N=2 and equal bare phase stiffnesses corresponds to a field theoretical description of an easy-plane quantum antiferromagnet. In this case, the critical exponents are computed and found to be non 3D xy values. Furthermore, we study the model in an external magnetic field, and find a novel feature, namely N-1 superfluid phases arising out of N charged condensates. In particular, for N=2 we point out the possibility of two novel types of field-induced phase transitions in ordered quantum fluids: i) A phase transition from a superconductor to a superfluid or vice versa, driven by tuning an external magnetic field. This identifies the superconducting phase of liquid metallic hydrogen as a novel quantum fluid. ii) A phase transition corresponding to a quantum fluid analogue of sublattice melting, where a composite field-induced Abrikosov vortex lattice is decomposed and disorders the phases of the constituent condensate with lowest bare phase stiffness. Both transitions belong to the 3D xy universality class.\n\nPaper VII: We consider the vortex superconductor with two individually conserved condensates in a finite magnetic field. The ground state is a lattice of cocentered vortices in both order parameters. We find two novel phase transitions when temperature is increased at fixed magnetic field. i) A "vortex sublattice melting" transition where vortices in the field with lowest phase stiffness ("light vortices") loose cocentricity with the vortices with large phase stiffness ("heavy vortices"), entering a liquid state (the structure factor of the light vortex sublattice vanishes continuously.) This transition is in the 3D xy universality class. ii) A first order melting transition of the lattice of heavy vortices in a liquid of light vortices.\n\nPaper VIII: We report on large-scale Monte Carlo simulations of a novel type of a vortex matter phase transition which should take place in a three dimensional two-component superconductor. We identify the regime where first, at a certain temperature a field-induced lattice of co-centered vortices of both order parameters melts, causing the system to loose superconductivity. In this state the two-gap system retains a broken composite symmetry and we observe that at a higher temperature it undergoes an extra phase transition where the disordered composite one-flux-quantum vortex lines are "ionized" into a "plasma" of constituent fractional flux vortex lines in individual order parameters. This is the hallmark of the superconductor-to-superfluid-to-normal fluid phase transitions projected to occur in e.g. liquid metallic hydrogen.
We consider the scaling behavior in the critical domain of superconductors at zero external magnetic field. The first part of the paper is concerned with the Ginzburg-Landau model in the zero-magnetic-field Meissner phase. We discuss the scaling behavior of the superfluid density and we give an alternative proof of Josephson's relation for a charged superfluid. This proof is obtained as a consequence of an exact renormalization group equation for the photon mass. We obtain Josephson's relation directly in the form ${\ensuremath{\rho}}_{s}\ensuremath{\sim}{t}^{\ensuremath{\nu}};$ that is, we do not need to assume that the hyperscaling relation holds. Next, we give an interpretation of a recent experiment performed in thin films of ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}.$ We argue that the measured mean-field-like behavior of the penetration depth exponent ${\ensuremath{\nu}}^{\ensuremath{'}}$ is possibly associated with a nontrivial critical behavior and we predict the exponents $\ensuremath{\nu}=1$ and $\ensuremath{\alpha}=\ensuremath{-}1$ for the correlation length and specific heat, respectively. In the second part of the paper we discuss the scaling behavior in the continuum dual Ginzburg-Landau model. After reviewing lattice duality in the Ginzburg-Landau model, we discuss the continuum dual version by considering a family of scalings characterized by a parameter $\ensuremath{\zeta}$ introduced such that ${m}_{h,0}^{2}\ensuremath{\sim}{t}^{\ensuremath{\zeta}},$ where mh,0 is the bare mass of the magnetic induction field. We discuss the difficulties in identifying the renormalized magnetic induction mass with the photon mass. We show that the only way to have a critical regime with ${\ensuremath{\nu}}^{\ensuremath{'}}=\ensuremath{\nu}\ensuremath{\approx}2/3$ is having $\ensuremath{\zeta}\ensuremath{\approx}4/3,$ that is, with mh,0 having the scaling behavior of the renormalized photon mass.
It is argued that the penetration depth and the correlation length at the critical point of the 3D superconductor diverge with the same critical exponents, as follows from the general scaling arguments and from the independent calculations in both Ginzburg-Landau and its dual theory. The incorrect prediction of Kiometzis, Kleinert and Schakel (KKS) that this is is not so, is the result of the faulty treatment of the dual theory, in which two, instead of one, coupling constants are tuned to reach the critical point. The recent paper by the present author criticized by KKS in cond-mat/9702159 differs on this point from KKS, and consequently finds the expected relation between the divergence of the two lengths in the critical region.