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arXiv:hep-th/9310086 · Physical Review Letters

Effective critical exponents for dimensional crossover and quantum systems from an environmentally friendly renormalization group

Denjoe O’Connor, Christopher Rhodes Stephens

Published Jan 24, 1994• 41 citations
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Abstract

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, ${\mathit{d}}_{\mathrm{eff}}$=4-${\ensuremath{\gamma}}_{\ensuremath{\lambda}}$, which represents the crossover in the leading irrelevant operator, and are in excellent agreement with known results.

DOI
10.1103/physrevlett.72.506

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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-\ensuremathγ\ensuremathλ{\ensuremath{\gamma}}_{\ensuremath{\lambda}}\ensuremathγ\ensuremathλ​, which represents the crossover in the leading irrelevant operator, and are in excellent agreement with known results.
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.
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.
The RG functions of the 2D nnn-vector ϕ4ϕ^4ϕ4 model are calculated in the five-loop approximation. Perturbative series for the βββ function and critical exponents are resummed by the Pade-Borel and Pade-Borel-Leroy techniques, resummation procedures are optimized and an accuracy of the numerical results is estimated. In the Ising case n=1n = 1n=1 as well as in the others (n=0n = 0n=0, n=−1n = -1n=−1, n=2,3,...32n = 2, 3,...32n=2,3,...32) an account for the five-loop term is found to shift the Wilson fixed point location only briefly, leaving it outside the segment formed by the results of the corresponding lattice calculations; even error bars of the RG and lattice estimates do not overlap in the most cases studied. This is argued to reflect the influence of the singular (non-analytical) contribution to the βββ function that can not be found perturbatively. The evaluation of the critical exponents for n=1n = 1n=1, n=0n = 0n=0 and n=−1n = -1n=−1 in the five-loop approximation and comparison of the numbers obtained with their known exact counterparts confirm the conclusion that non-analytical contributions are visible in two dimensions.
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.
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.
We present a field-theoretical treatment of the critical behavior of a three-dimensional weakly diluted quenched Ising model. To this end we analyze in the replica limit $\stackrel{\ensuremath{\rightarrow}}{n}0$ the five-loop renormalization-group functions of the ${\ensuremath{\varphi}}^{4}$ theory with O(n)O(n)O(n)-symmetric and cubic interactions [H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B 342, 284 (1995)]. The minimal subtraction scheme allows one to develop either the $\sqrt{\ensuremath{\varepsilon}}$-expansion series or to proceed within the three-dimensional approach, performing expansions in terms of renormalized couplings. Doing so, we compare both perturbation approaches and discuss their convergence and possible Borel summability. To study the crossover effect we calculate the effective critical exponents. We report resummed numerical values for the effective and asymptotic critical exponents. The results obtained within the three-dimensional approach agree pretty well with recent Monte Carlo simulations. $\sqrt{\ensuremath{\varepsilon}}$ expansion does not allow reliable estimates for d=3.d=3.d=3.