## Criticality and novel quantum liquid phases in Ginzburg--Landau theories with compact and non-compact gauge fields

##### Abstract

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.
Paper 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.
Paper 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.
Paper 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.
Paper 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.
Paper 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.
Paper 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.
Paper 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.
Paper 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.

##### Has parts

Sudbø, Asle; Smørgrav, Eivind; Smiseth, Jo; Nogueira, Flavio S.; Hove, Joakim. Criticality in the (2+1)-Dimensional Compact Higgs Model and Fractionalized Insulators. .Smiseth, Jo; Smørgrav, Eivind; Nogueira, Flavio S.; Hove, Joakim; Sudbø, Asle. Phase Structure of d=2+1 Compact Lattice Gauge Theories and the Transition from Mott Insulator to Fractionalized Insulator. .

Nogueira, Flavio S.; Smiseth, Jo; Smørgrav, Eivind; Sudbø, Asle. Compact U(1) gauge theories in 2+1 dimensions and the physics of low dimensional insulating materials. .

Smørgrav, Eivind; Smiseth, Jo; Sudbø, Asle; Nogueira, Flavio S.. Phase structure of Abelian Chern-Simons gauge theories. .

Smiseth, Jo; Smørgrav, Eivind; Sudbø, Asle. Critical properties of the N-color London model. .

Smiseth, Jo; Smørgrav, Eivind; Babaev, Egor; Sudbø, Asle. Field- and temperature induced topological phase transitions in the three-dimensional N-component London superconductor. .

Smørgrav, Eivind; Smiseth, Jo; Babaev, Egor; Sudbø, Asle. Vortex sub-lattice melting in a two-component superconductor. .

Smørgrav, Eivind; Smiseth, Jo; Babaev, Egor; Sudbø, Asle. Observation of a metallic superfluid in a numerical experiment. .