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arXiv:cond-mat/0207501 · Physical Review Letters

Criticality in the ( 2 + 1 )-Dimensional Compact Higgs Model and Fractionalized Insulators

Asle Sudbø, E. Smørgrav, J. Smiseth, Flavio S. Nogueira, J. Hove

Published Nov 7, 2002• 37 citations
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Abstract

We use a novel method of computing the third moment ${M}_{3}$ of the action of the ($2+1$)-dimensional compact Higgs model in the adjoint representation with $q=2$ to extract correlation length and specific heat exponents $\ensuremath{\nu}$ and $\ensuremath{\alpha}$ without invoking hyperscaling. Finite-size scaling analysis of ${M}_{3}$ yields the ratios $(1+\ensuremath{\alpha})/\ensuremath{\nu}$ and $1/\ensuremath{\nu}$ separately. We find that $\ensuremath{\alpha}$ and $\ensuremath{\nu}$ vary along the critical line of the theory, which however exhibits a remarkable resilience of ${Z}_{2}$ criticality. We propose this novel universality class to be that of the quantum phase transition from a Mott-Hubbard insulator to a charge-fractionalized insulator in two spatial dimensions.

DOI
10.1103/physrevlett.89.226403

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We use a novel method of computing the third moment M3{M}_{3}M3​ of the action of the (2+12+12+1)-dimensional compact Higgs model in the adjoint representation with q=2q=2q=2 to extract correlation length and specific heat exponents \ensuremathν\ensuremath{\nu}\ensuremathν and \ensuremathα\ensuremath{\alpha}\ensuremathα without invoking hyperscaling. Finite-size scaling analysis of M3{M}_{3}M3​ yields the ratios (1+\ensuremathα)/\ensuremathν(1+\ensuremath{\alpha})/\ensuremath{\nu}(1+\ensuremathα)/\ensuremathν and 1/\ensuremathν1/\ensuremath{\nu}1/\ensuremathν separately. We find that \ensuremathα\ensuremath{\alpha}\ensuremathα and \ensuremathν\ensuremath{\nu}\ensuremathν vary along the critical line of the theory, which however exhibits a remarkable resilience of Z2{Z}_{2}Z2​ criticality. We propose this novel universality class to be that of the quantum phase transition from a Mott-Hubbard insulator to a charge-fractionalized insulator in two spatial dimensions.
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 study the filling-controlled metal-insulator transition in the two-dimensional Hubbard model near half-filling with the use of zero temperature quantum Monte Carlo methods. In the metallic phase, the compressibility behaves as κ∝∣μ−μc∣−0.58±0.08κ\propto |μ- μ_c|^{-0.58\pm0.08}κ∝∣μ−μc​∣−0.58±0.08 where μcμ_cμc​ is the critical chemical potential. In the insulating phase, the localization length follows ξl∝∣μ−μc∣−νlξ_l \propto |μ- μ_c|^{-ν_l}ξl​∝∣μ−μc​∣−νl​ with νl=0.26±0.05ν_l = 0.26 \pm 0.05νl​=0.26±0.05. Under the assumption of hyperscaling, the compressibility data leads to a correlation length exponent νκ=0.21±0.04ν_κ= 0.21 \pm 0.04νκ​=0.21±0.04. Our results show that the exponents νκν_κνκ​ and νlν_lνl​ agree within statistical uncertainty. This confirms the assumption of hyperscaling with correlation length exponent ν=1/4ν= 1/4ν=1/4 and dynamical exponent z=4z = 4z=4. In contrast the metal-insulator transition in the generic band insulators in all dimensions as well as in the one-dimensional Hubbard model satisfy the hyperscaling assumption with exponents ν=1/2ν= 1/2ν=1/2 and z=2z = 2z=2.
We consider relativistic U(1) gauge theories in 2+12+12+1 dimensions, with Nb{N}_{b}Nb​ species of complex bosons and Nf{N}_{f}Nf​ species of Dirac fermions at finite temperature. The quantum phase transition between the Higgs and Coulomb phases is described by a conformal field theory (CFT). At large Nb{N}_{b}Nb​ and Nf{N}_{f}Nf​, but for arbitrary values of the ratio Nb∕Nf{N}_{b}∕{N}_{f}Nb​∕Nf​, we present computations of various critical exponents and universal amplitudes for these CFTs. We make contact with the different spin liquids, charge liquids, and deconfined critical points of quantum magnets that these field theories describe. We compute physical observables that may be measured in experiments or numerical simulations of insulating and doped quantum magnets.
We use a high-precision Monte Carlo simulation to determine the universal specific-heat amplitude ratio A + / A − in the three-dimensional Ising model via the impact angle ϕ of complex temperature zeros. We also measure the correlation-length critical exponent ν from finite-size scaling and the specific-heat exponent α through hyperscaling. Extrapolations to the thermodynamic limit yield ϕ = 59.2(1.0)°, A + / A − = 0.56(3), ν = 0.630 48(32) and α = 0.1086(10). These results are compatible with some previous estimates from a variety of sources and rule out recently conjectured exact values.
We use the two-step density-matrix renormalization group method to elucidate the long-standing issue of the universality class of the Mott transition in the Hubbard model in two dimensions. We studied a spatially anisotropic two-dimensional Hubbard model with a nonperfectly nested Fermi surface at half-filling. We find that unlike the pure one-dimensional case where there is no metallic phase, the quasi-one-dimensional model displays a genuine metal-insulator transition at a finite value of the interaction. The critical exponent of the correlation length is found to be $\ensuremath{\nu}\ensuremath{\approx}1.0$. This implies that the fermionic Mott transition belongs to the universality class of the 2D Ising model.
The phase transitions in the ground state of (1+1)-dimensional Ising models in a transverse field are studied for spin S=1/2, 1, 3/2, and 2 using the finite-size scaling method. The critical fields, thermal and correlation exponents are calculated as a function of S. The critical fields are monotonic functions of S but the exponents are S-independent and take their values for S=1/2. A novel calculational method based on universality of ratios of finite-size correlation length amplitudes is proposed and tested. It results in a much improved accuracy of extrapolation procedures. Such an universality appears to be a consequence of recently postulated conformal invariance of correlation functions of a class of two-dimensional classical models.