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Physical review. B, Solid state

Renormalization-group approach to a Bose system

K. K. Singh

Published Oct 1, 1975• 44 citations
Publisher

Abstract

The renormalization-group approach is extended to a Bose system within the framework of perturbation theory. Recursion relations are obtained for the parameters of the Hamiltonian and the correlation length index $\ensuremath{\nu}$ is evaluated. The values of $\ensuremath{\nu}$ and of the critical exponent $\ensuremath{\eta}$ (evaluated in an earlier paper) agree with those derived for a classical system characterized by a two-component (real) order parameter. The agreement suggests that the universality hypothesis holds irrespective of the quantum or classical nature of a phase transition.

DOI
10.1103/physrevb.12.2819

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The renormalization-group approach is extended to a Bose system within the framework of perturbation theory. Recursion relations are obtained for the parameters of the Hamiltonian and the correlation length index \ensuremathν\ensuremath{\nu}\ensuremathν is evaluated. The values of \ensuremathν\ensuremath{\nu}\ensuremathν and of the critical exponent \ensuremathη\ensuremath{\eta}\ensuremathη (evaluated in an earlier paper) agree with those derived for a classical system characterized by a two-component (real) order parameter. The agreement suggests that the universality hypothesis holds irrespective of the quantum or classical nature of a phase transition.
The critical behavior in the condensed phase of a Bose system is studied by the renormalization-group method using Bogolubov's well-known prescription for eliminating the condensate operators. Scaling relations and an equation of state in the Griffiths form are derived with explicit expressions for the critical exponents and the scaling function up to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$. The results strengthen the conclusion reached in an earlier paper, namely, that critical behavior in a given symmetry class is independent of the classical or quantum nature of a phase transition.
It is shown that the renormalization group approach can be extended to a Bose system without imposing an ad hoc upper momentum cut-off. The strategy adopted is to eliminate momenta between infinity and a small but arbitrary momentum p 0 by performing a preliminary partial trace on the partition function. Recursion relations for the parameters of the resulting Hamiltonian are derived in a form which enables scaling functions for the susceptibility and the order parameter to be calculated. The functions exhibit a crossover from dilute gas critical behaviour to that of a fully interacting system. The results are in agreement with those derived by Weichman et al. (1986) who relied upon a mapping of the Bose system onto a classical, two-component spin model. It is contended that a rigorous justification for the mapping emerges from the authors' approach.
The critical behavior of dense and dilute Bose systems is considered within the same framework in terms of an additional parameter y introduced in the renormalization-group transformation. The dense case is obtained when y=0, and the formalism becomes identical to the one previously devised by Singh which gives a two-thirds correlation-length critical exponent. The dilute case corresponds to the condition y=d>2, where the renormalization procedure leads to Bose-Einstein condensation in the vicinity of the critical point. As a consequence, the appearance of a crossover exponent results in a condensate correlation-length critical exponent equal to unity. The critical exponents in both cases are in agreement with an experiment done recently by Crooker et al., where a crossover from a two-thirds to a linear power law is observed in the temperature dependence of the superfluid mass with a systematic reduction in density. The critical masses of the condensate and noncondensate are found to be zero and infinite, respectively.
We present a nonperturbative renormalization group (RG) approach to the Bose-Hubbard model. By taking as initial condition of the RG flow the (local) limit of decoupled sites, we take into account both local and long-distance fluctuations in a nontrivial way. This approach yields a phase diagram in very good quantitative agreement with the quantum Monte Carlo results and reproduces the two universality classes of the superfluid--Mott-insulator transition with a good estimate of the critical exponents. Furthermore, it reveals the crucial role of the ``Ginzburg length'' as a crossover length between a weakly and a strongly correlated superfluid phase.
Abstract Chapter 6 describes how the perturbative renormalization group (RG) of quantum field theory has made it possible to derive universal properties of continuous macroscopic phase transitions. The RG initially emerged as a consequence of the necessity to cancel infinities that appear in the perturbative expansion (the renormalization procedure) and the possibility of defining the parameters of the renormalized theory at different momentum scales. Although the field theory RG is now understood to be an asymptotic form, it has made it possible to confirm the Wilson–Fisher fixed point and led to an understanding of universality for a large class of critical phenomena. In the framework of dimensional continuation, zeros of RG beta functions, which correspond to Wilson–Fisher’s fixed points, have made it possible to recover Wilson–Fisher’s epsilon expansion, proving scaling relations and calculating critical exponents. Series summation methods have then been used to generate precise values of exponents.
This book provides an elementary introduction to the notions of continuum limit and universality in statistical systems with a large number of degrees of freedom. The existence of a continuum limit requires the appearance of correlations at large distance, a situation that is encountered in second order phase transitions, near the critical temperature. In this context, the book emphasizes the role of gaussian distributions and their relations with the mean field approximation and Landau′s theory of critical phenomena. The book shows that quasi-gaussian or mean-field approximations cannot describe correctly phase transitions in three space dimensions. The book assigns this difficulty to the coupling of very different physical length scales, even though the systems we will consider have only local, that is, short range, interactions. To analyze the unusual situation, a new concept is required: the renormalization group, whose fixed points allow understanding the universality of physical properties at large distance, beyond mean-field theory. In the continuum limit, critical phenomena can be described by quantum field theories. In this framework, the renormalization group is directly related to the renormalization process; that is, the necessity to cancel the infinities that arise in straightforward formulations of the theory. The book discusses the renormalization group in the context of various relevant field theories. This leads to proofs of universality and to efficient tools for calculating universal quantities in a perturbative framework. Finally, the book constructs a general functional renormalization group, which can be used when perturbative methods are inadequate.