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arXiv:cond-mat/0202209

Dimensional Crossover in the Non-Linear Sigma Model

Denjoe O’Connor, Christopher Rhodes Stephens, José Antonio Santiago

Published Feb 12, 2002• 1 citation
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Abstract

We consider dimensional crossover for an O(N) model on a d-dimensional layered geometry of thickness L, in the sigma-model limit, using ``environmentally friendly'' renormalization. We show how to derive critical temperature shifts, giving explicit results to one loop. We also obtain expressions for the effective critical exponents delta and beta effective that interpolate between their characteristic fixed point values assocliated with a d and (d-1)-dimensional system in the limits T -> T_c(L), with L(T-T_c(L))^ν->infty, and T->T_c(L) for L fixed respectively, where T_c(L) is the L-dependent critical temperature of the system.

DOI
10.48550/arxiv.cond-mat/0202209

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We consider dimensional crossover for an O(N) model on a d-dimensional layered geometry of thickness L, in the sigma-model limit, using ``environmentally friendly'' renormalization. We show how to derive critical temperature shifts, giving explicit results to one loop. We also obtain expressions for the effective critical exponents delta and beta effective that interpolate between their characteristic fixed point values assocliated with a d and (d-1)-dimensional system in the limits T -> T_c(L), with L(T-T_c(L))^ν->infty, and T->T_c(L) for L fixed respectively, where T_c(L) is the L-dependent critical temperature of the system.
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Series for the Wilson functions of an ``environmentally friendly'' renormalization group are computed to two loops, for an O(N) vector model, in terms of the ``floating coupling,'' and resummed by the Pad\'e method to yield crossover exponents for finite size and quantum systems. The resulting effective exponents obey all scaling laws, including hyperscaling in terms of an effective dimensionality, deff{\mathit{d}}_{\mathrm{eff}}deff​=4-${\ensuremath{\gamma}}_{\ensuremath{\lambda}}$, which represents the crossover in the leading irrelevant operator, and are in excellent agreement with known results.
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A classical n-vector model is considered which is infinitely extended in d-1 dimensions and periodic of period b in the dth dimension. Asymptotically for large values of b the critical temperature is expected to obey a scaling law T c (b)-T c ( infinity ) approximately Ab - lambda . The authors have calculated the shift exponent lambda to first order in a 1/n expansion for dimensions between three and four. To this order the result is consistent with the assertion lambda = nu d -1 where nu d is the correlation length exponent of the infinitely extended d-dimensional system. From the equivalence between this model and (d-1)-dimensional quantum mechanical systems the behaviour of critical lines near the displacive limit (T c =0) can be derived.
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.