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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Finite-size scaling in space-time

Michael F. Zimmer

Published May 1, 1998• 0 citations
Publisher

Abstract

Finite-size scaling relations are obtained for the time-dependent Ginzburg-Landau equation in a finite space-time volume. Universal ratios above the upper critical dimension (UCD) are demonstrated to hold true under a mild restriction on its aspect ratio. A perturbation expansion is carried out to order one loop below the UCD, demonstrating the validity of the scaling relations. These results are of practical use in determining the dynamic critical exponent via space-time simulations.

DOI
10.1103/physreve.57.5190

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Finite-size scaling relations are obtained for the time-dependent Ginzburg-Landau equation in a finite space-time volume. Universal ratios above the upper critical dimension (UCD) are demonstrated to hold true under a mild restriction on its aspect ratio. A perturbation expansion is carried out to order one loop below the UCD, demonstrating the validity of the scaling relations. These results are of practical use in determining the dynamic critical exponent via space-time simulations.
In the 1960's, four famous scaling relations were developed which relate the six standard critical exponents describing continuous phase transitions in the thermodynamic limit of statistical physics models. They are well understood at a fundamental level through the renormalization group. They have been verified in multitudes of theoretical, computational and experimental studies and are firmly established and profoundly important for our understanding of critical phenomena. One of the scaling relations, hyperscaling, fails above the upper critical dimension. There, critical phenomena are governed by Gaussian fixed points in the renormalization-group formalism. Dangerous irrelevant variables are required to deliver the mean-field and Landau values of the critical exponents, which are deemed valid by the Ginzburg criterion. Also above the upper critical dimension, the standard picture is that, unlike for low-dimensional systems, finite-size scaling is non-universal. Here we report on new developments which indicate that the current paradigm is flawed and incomplete. In particular, the introduction of a new exponent characterising the finite-size correlation length allows one to extend hyperscaling beyond the upper critical dimension. Moreover, finite-size scaling is shown to be universal provided the correct scaling window is chosen. These recent developments also lead to the introduction of a new scaling relation analogous to one introduced by Fisher 50 years ago.
We develop a critical finite-force-range scaling theory for D-dimensional scalar ${\mathrm{\ensuremath{\varphi}}}^{\mathit{n}}$ field theories that is based on a scaling ansatz equivalent to a Ginzburg criterion. To investigate its relationship to other scaling theories we derive equivalent results from renormalization groups and from finite-size crossover scaling for systems with weak long-range forces. By comparing our finite-range scaling relations with finite-size scaling relations for hypercylindrical systems above the upper critical dimension Dc{\mathit{D}}_{\mathit{c}}Dc​, we arrive at a criterion of critical equivalence that provides an asymptotic mapping between the two kinds of systems. We apply our scaling relations to a ${\mathrm{\ensuremath{\varphi}}}^{4}$ Ginzburg-Landau Hamiltonian, to the one-dimensional Kac model with exponentially decaying interactions, and to the N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} quasi-one-dimensional Ising (Q1DI) model, in which each spin interacts with O(N) others. Near the Gaussian mean-field critical point the Ginzburg-Landau Hamiltonians for all three models become identical, but for the Q1DI model this requires a length rescaling. For the Kac model the resulting scaling relations are those of a D=1 quartic field theory, and for the Q1DI model they are those of a cylindrical Ising system above Dc{\mathit{D}}_{\mathit{c}}Dc​. Results of specialized numerical scaling techniques applied to transfer-matrix calculations for the Q1DI model with N\ensuremath{\le}1024 strongly support our theoretically obtained scaling relations.
Scaling, hyperscaling and finite-size scaling were long considered problematic in theories of critical phenomena in high dimensions. The scaling relations themselves form a model-independent structure that any model-specific theory must adhere to, and they are accounted for by the simple principle of homogeneity. Finite-size scaling is similarly founded on the fundamental idea that only two length scales enter the game — namely system length and correlation length. While all scaling relations are quite satisfactory for multitudes of physical systems in low dimensions, one fails in high dimensions. The aberrant scaling relation is called hyperscaling and involves dimensionality itself. Finite-size scaling also appears to fail in high dimensions. Developed in the 1930s, Landau mean-field theory is valid in such high-dimensional systems. However, it too does not accord with hyperscaling and finite-size scaling there. The advent of renormalization-group theory in the 1970s brought deeper fundamental insights into critical phenomena, allowing systems to be viewed at different scales. Above a critical dimensionality, higher-order Renormalization Group (RG) eigenvalues become irrelevant and scaling is governed by the Gaussian fixed point. Although obeying all scaling relations including hyperscaling, and although it appears to successfully explain scaling in the correlation sector, the Gaussian fixed point fails to capture the free energy and derivatives, even in infinite volume. In the 1980s, to fix this for the magnetisation, specific heat and susceptibility, Fisher introduced the notion of dangerous irrelevant variables. Since the correlation sector did not appear to be broken, no attempt was made to repair it and Fisher’s modified RG formalism worked quite well in the thermodynamic limit of infinite volume. However, finite-size scaling fails. Also in the 1980’s, Binder, Nauenberg, Privman and Young extended Fisher’s concept to the free energy itself and to finite-size systems. While putting Fisher’s ideas on a more fundamental footing, the failure of finite-size scaling there still presented a problem. This appeared to be resolved by the introduction of “thermodynamic length” to replace correlation length as the length scale that controls Finite-Size Scaling (FSS). Thus hyperscaling and FSS were both sacrificed in favour of the RG patched together by ad hoc solutions. In the 1990’s, Luijten and Blöte went a long way to resolving the dilemma by adding corrections to scaling to the above considerations. It was clear that both of these played a role. However, as with previous authors, and adhering to the principle of not fixing that appears not to be broken, they did not address correlation length directly. Here we report on developments over the past decade which went a long way to addressing these long-standing problems The key to unlocking these, and extending their validity to the high-dimensional regime, was to relax assumptions that the correlation length has to be bounded by the physical length of bounded systems. This allowed and necessitated the extension of Fisher’s concept of dangerous irrelevant variables to the correlation sector.
The validity of modified finite-size scaling above the upper critical dimension is demonstrated for the quantum phase transition whose dynamical critical exponent is z=2. We consider the N -component Bose-Hubbard model, which is exactly solvable and exhibits mean-field type critical phenomena in the large- N limit. The modified finite-size scaling holds exactly in that limit. However, the usual procedure, taking the large system-size limit with fixed temperature, does not lead to the expected (and correct) mean-field critical behavior because of the limited range of applicability of the finite-size scaling form. By quantum Monte Carlo simulation, it is shown that the modified finite-size scaling holds in the case of N=1.