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The European Physical Journal B

Renormalization group equations with several length scales and crossover scaling functions for axial anisotropy

Heinz Horner

Published Jun 1, 1976• 16 citations
Publisher
DOI
10.1007/bf01352714

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Abstract The phase transition to superfluidity and the BCS‐BEC crossover for an ultracold gas of fermionic atoms is discussed within a functional renormalization group approach. Non‐perturbative flow equations, based on an exact renormalization group equation, describe the scale dependence of the flowing or average action. They interpolate continuously from the microphysics at atomic or molecular distance scales to the macroscopic physics at much larger length scales, as given by the interparticle distance, the correlation length, or the size of the experimental probe. We discuss the phase diagram as a function of the scattering length and the temperature and compute the gap, the correlation length and the scattering length for molecules. Close to the critical temperature, we find the expected universal behavior. Our approach allows for a description of the few‐body physics (scattering and molecular binding) and the many‐body physics within the same formalism.
Abstract We derive a theory of crossover scaling based on a scaling variable gξg, where g is the anisotropy parameter inducing the crossover and ξg is the correlation length in the presence of g. Our considerations are field theoretic and based on a renormalization group with a g dependent differential generator that interpolates between qualitatively different degrees of freedom. ξg is a nonlinear scaling field for this renormalization group and interpolates between (T – Tc(g))–v0 and (T – Tc(g))–v∞ (v0 and v∞ being the isotropic and anisotropic exponents respectively). By expanding about a ‘floating’ fixed point we can make corrections to scaling small throughout the crossover. In this formulation effective scaling exponents obey standard scaling laws, e. g. γeff = veff(2 – ɳeff). We discuss its advantages giving for various crossovers explicit supporting perturbative calculations of the susceptibility, which is found to conform to the general form derived from the g dependent renormalization group.
A new qualitative view of vortex length–scale dependence has been found in recent electrical transport measurements performed on strongly anisotropic Bi2Sr2CaCu2O8 single crystals in zero magnetic field. This indicates the need for a better description of the 3D/2D crossover in vortex dimensionality. The vortex-dominated properties of high transition temperature superconductors with extremely high anisotropy (layered systems) are reasonably well described in the framework of the layered XY model which can be mapped onto the layered sine-Gordon model. For the latter we derive an exact renormalization group (RG) equation using Wegner's and Houghton's approach in the local potential approximation. The agreement of the UV scaling laws found by us by linearizing the RG equations with those obtained previously in the literature in the dilute gas approximation makes apparent the improvement which can be achieved by solving our RG equations numerically.