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Physical review. B, Condensed matter

Nonlinear quantum and classical renormalization-group trajectories for the interacting Bose fluid

Dimitri D. Vvedensky, R. J. Creswick

Published Dec 1, 1986• 2 citations
Publisher

Abstract

A nonlinear analysis of differential renormalization-group (RG) equations for the interacting Bose fluid is carried out to O(\ensuremath{\epsilon}) for T=0 and in the limit n\ensuremath{\rightarrow}\ensuremath{\infty} for finite T. The 1/n expansion is shown to provide a more convenient framework than the \ensuremath{\epsilon} expansion for deriving global RG solutions that match onto the classical (T\ensuremath{\rightarrow}\ensuremath{\infty}) and quantum (T\ensuremath{\rightarrow}0) limits and which incorporate the interplay between classical and quantum fluctuations.

DOI
10.1103/physrevb.34.7760

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A nonlinear analysis of differential renormalization-group (RG) equations for the interacting Bose fluid is carried out to O(\ensuremath{\epsilon}) for T=0 and in the limit n\ensuremath{\rightarrow}\ensuremath{\infty} for finite T. The 1/n expansion is shown to provide a more convenient framework than the \ensuremath{\epsilon} expansion for deriving global RG solutions that match onto the classical (T\ensuremath{\rightarrow}\ensuremath{\infty}) and quantum (T\ensuremath{\rightarrow}0) limits and which incorporate the interplay between classical and quantum fluctuations.
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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 derive an approximate closed form for the infinitesimal generator of the renormalization group for the interacting Bose fluid. The Bose-condensed phase is treated by the method of Bogoliubov, and a simple scaling law is found for the condensate density. It is shown that the quantum-mechanical features are irrelevant close to the $\ensuremath{\lambda}$ line. This is demonstrated by calculating the critical behavior in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions to order $\ensuremath{\epsilon}$.
We study renormalization group (RG) flows in far-from-equilibrium states. The study is made tractable by focusing on states that are spatially homogeneous, time independent, and scale invariant. Such states, in which mode k has occupation numbers n k ∼ k − γ , are well known in nonlinear physics. RG flow in such states is qualitatively different from that in the vacuum—a positive γ decreases the dimension of an operator, turning marginal interactions into relevant interactions. We compute one-loop beta functions. Depending on the sign of the beta function, backreaction may either cause a minor shift of the state in the IR or completely change the nature of the state. Focusing on nearly marginal interactions, we construct an analog of the epsilon expansion and IR fixed points, with epsilon now set by the scaling of the interaction rather than the spacetime dimension. In the language of RG flow, critical balance scaling—which has applications in fields as varied as astrophysics and ocean waves—corresponds to the state dynamically adjusting itself along the RG flow until the interaction becomes marginal.
We apply perturbative renormalization-group theory to the symmetric phase of a dilute interacting Bose gas which is trapped in a three-dimensional harmonic potential. Using Wilson's energy-shell renormalization and the $\ensuremath{\epsilon}$ expansion, we derive the flow equations for the system. We relate these equations to the flow for the homogeneous Bose gas. In the thermodynamic limit, we apply our approach to examine the transition temperature of the harmonically trapped Bose gas as a function of the scattering length. The results are compared to previous studies of the problem.