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arXiv:hep-th/9310198 · International Journal of Modern Physics A

“ENVIRONMENTALLY FRIENDLY” RENORMALIZATION

DENJOE O’CONNOR, Christopher Rhodes Stephens

Published Jun 30, 1994• 45 citations
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Abstract

We analyze the renormalization of systems whose effective degrees of freedom are described in terms of fluctuations which are “environment”-dependent. Relevant environmental parameters considered are: temperature, system size, boundary conditions and external fields. The points in the space of “coupling constants” at which such systems exhibit scale invariance coincide only with the fixed points of a global renormalization group which is necessarily environment-dependent. Using such a renormalization group we give formal expressions to two loops for effective critical exponents for a generic crossover induced by a relevant mass scale g. These effective exponents are seen to obey scaling laws across the entire crossover, including hyperscaling, but in terms of an effective dimensionality, deff=4−γλ, which represents the effects of the leading irrelevant operator. We analyze the crossover of an O(N) model on a d-dimensional layered geometry with periodic, antiperiodic and Dirichlet boundary conditions. Explicit results to two loops for effective exponents are obtained using a [2/1] Padé-resummed coupling, for: the “Gaussian model” (N=−2), spherical model (N=∞), Ising model (N=1), polymers (N=0), XY model (N=2) and Heisenberg (N=3) models in four dimensions. We also give two-loop Padé resummed results for a three-dimensional Ising ferromagnet in a transverse magnetic field and corresponding one-loop results for the two-dimensional model. One-loop results are also presented for a three-dimensional layered Ising model with periodic, Dirichlet and antiperiodic boundary conditions. Asymptotically the effective exponents are in excellent agreement with known results.

DOI
10.1142/s0217751x94001151

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We analyze the renormalization of systems whose effective degrees of freedom are described in terms of fluctuations which are “environment”-dependent. Relevant environmental parameters considered are: temperature, system size, boundary conditions and external fields. The points in the space of “coupling constants” at which such systems exhibit scale invariance coincide only with the fixed points of a global renormalization group which is necessarily environment-dependent. Using such a renormalization group we give formal expressions to two loops for effective critical exponents for a generic crossover induced by a relevant mass scale g. These effective exponents are seen to obey scaling laws across the entire crossover, including hyperscaling, but in terms of an effective dimensionality, deff=4−γλ, which represents the effects of the leading irrelevant operator. We analyze the crossover of an O(N) model on a d-dimensional layered geometry with periodic, antiperiodic and Dirichlet boundary conditions. Explicit results to two loops for effective exponents are obtained using a [2/1] Padé-resummed coupling, for: the “Gaussian model” (N=−2), spherical model (N=∞), Ising model (N=1), polymers (N=0), XY model (N=2) and Heisenberg (N=3) models in four dimensions. We also give two-loop Padé resummed results for a three-dimensional Ising ferromagnet in a transverse magnetic field and corresponding one-loop results for the two-dimensional model. One-loop results are also presented for a three-dimensional layered Ising model with periodic, Dirichlet and antiperiodic boundary conditions. Asymptotically the effective exponents are in excellent agreement with known results.
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.
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.
A semi-microscopic scaling-field theory is developed for crossover phenomena near critical and tricritical points. The theory is based on a renormalization-group description of a model with two competing fixed points (such as a critical and a tricritical fixed point) in terms of scaling fields. The coupled nonlinear differential equations for scaling fields are truncated such as to preserve the physics essential for crossover phenomena. The approach allows the explicit calculation of thermodynamic functions for (i) tricritical systems and (ii) critical systems with an irrelevant scaling field. We obtain, for example, an explicit expression for the scaling function of the susceptibility, which describes the crossover from the tricritical to the critical region. The idea of "flow diagrams" in the scaling-field space is used to characterize crossover phenomena globally in the whole critical region. The concept of asymptotic critical exponents is generalized and effective critical exponents are introduced as logarithmic derivatives of thermodynamic quantities with respect to experimental fields and scaling fields, respectively. By using the method of effective exponents the size of the crossover region between regions of different asymptotic critical behavior is estimated. For the susceptibility, the width of the crossover region in decades of the effective temperature variable is roughly equal to the inverse of the crossover exponent. In the case of a critical system with a slow transient the asymptotic critical exponent is only reached extremely close to the critical point (unless the amplitude of the transient vanishes). It might then be impossible to determine the asymptotic exponent experimentally or by conventional series-expansion techniques, and an analysis of the data in terms of effective exponents is the alternative. The scaling-field approach is applied to three systems with crossover phenomena: (i) the model for tricritical systems with molecular-field tricritical exponents, (ii) the Ashkin-Teller model in three dimensions, and (iii) a model for phase transitions with Fisher exponent renormalization due to a constraint.
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.
Abstract Chapter 8 introduces the key ideas of the renormalization group, including how they provide a theoretical scheme and a proper language to face critical phenomena. It covers the scaling transformations of a system and their implementations in the space of the coupling constants and reducing the degrees of freedom. From this analysis, the reader is led to the important notion of relevant, irrelevant and marginal operators and then to the universality of the critical phenomena. Furthermore, the chapter also covers (as regards the RG) transformation laws, effective Hamiltonians, the Gaussian model, the Ising model, operators of quantum field theory, universal ratios, critical exponents and β-functions.
Monte Carlo simulations of the critical behaviour of disordered Ising systems have led to concentration-dependent critical exponents which seem to violate universality. We apply the renormalization group to investigate the crossover effects which cause the observed behaviour. We improve the three-loop expansion of the Callan-Symanzik and Wilson functions by a Pade-Borel approximation and solve the flow equations for various initial points. The exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point.