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Physical review. B, Condensed matter

Hyperuniversality in quantum critical phenomena

Kihong Kim, Peter B. Weichman

Published Jun 1, 1991• 34 citations
Publisher

Abstract

We apply generalizations of two-scale-factor universality, or hyperuniversality, to quantum phase transitions at zero temperature. We find new universal amplitude combinations involving the superfluid density in two- and three-dimensional Bose systems, as well as confirm and extend previous proposals for universal transport coefficients in two-dimensional superconducting films and magnetic-field-induced metal-insulator transitions. We also give a general proof for the existence of universal jumps in two-dimensional superfluids at finite temperature using finite-size hyperuniversality.

DOI
10.1103/physrevb.43.13583

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We apply generalizations of two-scale-factor universality, or hyperuniversality, to quantum phase transitions at zero temperature. We find new universal amplitude combinations involving the superfluid density in two- and three-dimensional Bose systems, as well as confirm and extend previous proposals for universal transport coefficients in two-dimensional superconducting films and magnetic-field-induced metal-insulator transitions. We also give a general proof for the existence of universal jumps in two-dimensional superfluids at finite temperature using finite-size hyperuniversality.
We discuss some problems related to quantum critical phenomena in superfluids and superconductors. In Ch. 1, we apply generalizations of hyperuniversality to quantum phase transitions at zero temperature. We find new universal amplitude combinations involving the superfluid density in Bose systems, as well as confirm and extend previous proposals for universal transport coefficients in two-dimensional superconducting films and magnetic-field-induced metal-insulator transitions. In Ch. 2, we apply the double-dimensionality expansion of Dorogovstev to derive renormalization-group recursion relations for a Bose fluid in a random external potential. We find a nontrivial fixed point. The onset of mean-field behavior for dimensions d > d[subscript]c = 4 is unconventional, yielding discontinuous exponents. Including positive temperatures, we give a clear picture of various crossover regimes. In Ch. 3, We give a detailed derivation of the critical thermodynamics of O(n) spin models, correct to O([epsilon] = 4 - d), using a generalization of the renormalization-group trajectory integral and noncritical matching technique introduced by Rudnick and Nelson. We especially emphasize the coexistence-curve behavior for spins with a continuous symmetry (n [...] 2), deriving detailed expressions for the renormalized spin-wave stiffness and longitudinal susceptibility.
We study the relativistic O (N) model, a quantum generalization of the φ⁴ theory used in statistical physics to study some phase transitions. This model describes quantum phase transitions such as the Mott insulator-superluid transition in boson gases trapped in optical latices or the paramagnetic-antiferromagnetic transition in magnets. In two space dimensions, these systems exhibit strong correlations near the transition. We study them using the nonperturbative renormalization group, an implementation of Wilson’s renormalization group. We focus on the universal properties in the vicinity of the zero-temperature quantum phase transition. We determine the universal scaling functions which define the thermodynamics and we show that these functions are related to those describing the critical Casimir forces in a three-dimensional system. Then, we study the excitation spectrum in the zero-temperature ordered phase. For N = 2 and 3, we establish the existence of an amplitude mode, also called “Higgs mode” by analogy with the Higgs mechanism in high-energy physics. For N = 1, we show the existence of a bound state at dimensions close to three. Finally, we compute the frequency-dependent conductivity at zero temperature and confirm its universal character, in particular at the transition. We prove that one of the components of the conductivity tensor in the ordered phase is a “superuniversal” quantity depending neither on the distance to the critical point nor on N.
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.
In contrast to the case of free fermions which have all states localized in two dimensions, dirty, repulsively-interacting bosons exhibit a zero-temperature localization phase transition between superconducting and insulating states as disorder is increased. The conductivity σ* = g (2 e ) 2 / h at the quantum critical point is a universal critical amplitude analogous to the universal jump in superfluid density at the classical Kosterlitz-Thouless transition. Using numerical Monte Carlo methods we compute the value of g for disorder-tuned transitions with and without long-range Coulomb interactions. We argue that this model is relevant to experiments on superconducting thin films and Josephson junction arrays.
We develop a scheme for analytic continuation of the strong-coupling perturbation series of the pure Bose-Hubbard model beyond the Mott insulator-to-superfluid transition at zero temperature, based on hypergeometric functions and their generalizations. We then apply this scheme for computing the critical exponent of the order parameter of this quantum phase transition for the two-dimensional case, which falls into the universality class of the three-dimensional XYXYXY model. This leads to anontrivial test of the universality hypothesis.
Topological phase transitions constitute a new class of quantum critical phenomena. They cannot be described within the usual framework of the Landau theory since, in general, the different phases cannot be distinguished by an order parameter, neither can they be related to different symmetries. In most cases, however, one can identify a diverging length at these topological transitions. This allows us to describe them using a scaling approach and to introduce a set of critical exponents that characterize their universality class. Here we consider some relevant models of quantum topological transitions associated with well-defined critical exponents that are related by a quantum hyperscaling relation. We extend to these models a finite-size scaling approach based on techniques for calculating the Casimir force in electromagnetism. This procedure allows us to obtain universal Casimir amplitudes at their quantum critical points. Our results verify the validity of finite-size scaling in these systems and confirm the values of the critical exponents obtained previously.