© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:cond-mat/0604226 · Physical Review E

Mean crossover functions for uniaxial three-dimensional Ising-like systems

Yves Garrabos, Claude Bervillier

Published Aug 11, 2006• 15 citations

Abstract

We give simple expressions for the mean of the max and min bounds of the critical-to-classical crossover functions, previously calculated [Bagnuls and Bervillier, Phys. Rev. E 65, 066132 (2002)] within the massive renormalization scheme of the Phi(d)4(n) model in three dimensions (d = 3) and scalar order parameter (n = 1) of the Ising-like universality class. The mean functions are determined relying on the properties of the theoretical functions in the two limiting three-dimensional (3D) Ising-like and mean-field-like descriptions close to the Wilson-Fisher fixed point and to the Gaussian fixed point, respectively. Such descriptions correspond to the preasymptotic domains near each fixed point where a Wegner expansion restricted to two terms (leading and first confluent terms) is valid. The Ising-like preasymptotic domain description includes the correlations between parameters due to the error-bar determination of the exponents and amplitude combinations very close to the Wilson-Fisher fixed point. Adding the equivalent description of the mean field preasymptotic domain close to the Gaussian fixed point leads to define each mean crossover function with three calculated parameters. Fixing a unique value of one parameter whatever the selected mean crossover function, we use this parameter as a relative sensor to estimate the dominant nature, either (Ising-like) critical, or (mean-field-like) classical, of the crossover behavior. Finally, we obtain an explicit criterion to measure the extension of the Ising-like preasymptotic domain which can then permit to coherently account for measurements performed in systems where the asymptotical approach to the critical point remains finite, using a well-controlled number of system-dependent parameters (like in the subclass of one-component fluids).

DOI: 10.1103/physreve.74.021113

Related papers

Classical-to-critical crossovers from field theory

C. Bagnuls, Claude Bervillier

We extend our previous determinations of nonasymptotic critical behavior of Phys. Rev. B 32, 7209 (1985) and 35, 3585 (1987) to accurate expressions of the complete classical-to-critical crossover (in three-dimensional field theory) in terms of the temperaturelike scaling field (i.e., along the critical isochore) for (1) the correlation length, the susceptibility, and the specific heat in the homogeneous phase for the n-vector model (n=1 to 3) and (2) the spontaneous magnetization (coexistence curve), the susceptibility, and the specific heat in the inhomogeneous phase for the Ising model (n=1). The present calculations include the seventh-loop order of Murray and Nickel and closely account for the up-to-date estimates of universal asymptotic critical quantities (exponents and amplitude combinations) provided by Guida and Zinn-Justin [J. Phys. A 31, 8103 (1998)].

arXiv:hep-th/0112209•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jun 28, 2002•43 citations

Construction of an effective Hamiltonian for a three-dimensional Ising universality class

A. Brognara, Alberto Parola, L. Reatto

The asymptotic and preasymptotic critical behavior in fluids, mixtures, and uniaxial magnets is believed to be described by an effective straight phi(4) scalar field theory with suitable, nonuniversal, coupling constants. The critical parameters as well as the extent of crossovers and corrections to the leading critical behavior in physical systems, crucially depends on the choice of these couplings. Here we propose a new method for deriving the effective scalar field theory appropriate to a microscopic model in this universality class. Use is made of the hierarchical reference theory, which implements the basic ideas of Wilson momentum space renormalization group to microscopic Hamiltonians. The effective low-energy field theory is then analyzed by the minimal subtraction scheme of Schloms and Dohm. We discuss the application of this method to the three-dimensional Ising model and to the liquid-vapor phase transition. We make comparison with high-temperature expansion results and with experimental data for rare gas.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jul 25, 2001•7 citations

RENORMALIZATION GROUP DOMAINS OF THE SCALAR HAMILTONIAN

Bagnuls, Bervillier

Using the local potential approximation of the exact renormalization group (RG) equation, we show various domains of values of the parameters of the O(1) -symmetric scalar Hamiltonian.In three dimensions, in addition to the usual critical surface S c (attraction domain of the Wilson-Fisher fixed point), we explicitly show the existence of a first-order phase transition domain S f separated from S c by the tricritical surface S t (attraction domain of the Gaussian fixed point).S f and S c are two distinct domains of repulsion for the Gaussian fixed point, but S f is not the basin of attraction of a fixed point.S f is characterized by an endless renormalized trajectory lying entirely in the domain of negative values of the ϕ 4 -coupling.This renormalized trajectory also exists in four dimensions making the Gaussian fixed point ultra-violet stable (and the ϕ 4 4 renormalized field theory asymptotically free but with a wrong sign of the perfect action).We also show that a very retarded classical-to-Ising crossover may exist in three dimensions (in fact below four dimensions).This could be an explanation of the unexpected classical critical behaviour observed in some ionic systems.

Condensed Matter Physics•Jan 1, 2000•17 citations

Renormalization group domains of the scalar Hamiltonian

C. Bagnuls, Claude Bervillier

Using the local potential approximation of the exact renormalization group (RG) equation, we show the various domains of values of the parameters of the O(1)-symmetric scalar Hamiltonian. In three dimensions, in addition to the usual critical surface $S_{c}$ (attraction domain of the Wilson-Fisher fixed point), we explicitly show the existence of a first-order phase transition domain $S_{f}$ separated from $S_{c}$ by the tricritical surface $S_{t}$ (attraction domain of the Gaussian fixed point). $S_{f}$ and $S_{c}$ are two distinct domains of repulsion for the Gaussian fixed point, but $S_{f}$ is not the basin of attraction of a fixed point. $S_{f}$ is characterized by an endless renormalized trajectory lying entirely in the domain of negative values of the $\phi ^{4}$-coupling. This renormalized trajectory exists also in four dimensions making the Gaussian fixed point ultra-violet stable (and the $\phi_{4}^{4}$ renormalized field theory asymptotically free but with a wrong sign of the perfect action). We also show that very retarded classical-to-Ising crossover may exist in three dimensions (in fact below four dimensions). This could be an explanation of the unexpected classical critical behavior observed in some ionic systems.

arXiv:hep-th/0002254•Feb 29, 2000•10 citations

Universality and quantum effects in one-component critical fluids

Yves Garrabos

Nonuniversal scale transformations of the physical fields are extended to pure quantum fluids and used to calculate the susceptibility, the specific heat, and the order parameter density along the critical isochore of near its liquid-vapor critical point. Within the so-called preasymptotic domain, where the Wegner expansion restricted to the first term of confluent corrections to scaling is expected to be valid, the results are in agreement with the experimental measurements and recent predictions, either based on the minimal-substraction renormalization and the massive renormalization schemes within phi (4)(d = 3) (n = 1)the model, or based on the crossover parametric equation of state for Ising-like systems.

arXiv:cond-mat/0512408•Physical Review E•May 12, 2006•7 citations

Critical properties of the three-dimensional equivalent-neighbor model and crossover scaling in finite systems

Erik Luijten

Accurate numerical results are presented for the three-dimensional equivalent-neighbor model on a cubic lattice, for 12 different interaction ranges (coordination number between 18 and 250). These results allow the determination of the range dependences of the critical temperature and various critical amplitudes, which are compared to renormalization-group predictions. In addition, the analysis yields an estimate for the interaction range at which the leading corrections to scaling vanish for the spin-1 / 2 model, and confirms earlier conclusions that the leading Wegner correction must be negative for the three-dimensional (nearest-neighbor) Ising model. By complementing these results with Monte Carlo data for systems with coordination numbers as large as 52 514, the full finite-size crossover curves between classical and Ising-like behavior are obtained as a function of a generalized Ginzburg parameter. Also, the crossover function for the effective magnetic exponent is determined.

arXiv:cond-mat/9811332•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•May 1, 1999•26 citations

We give simple expressions for the mean of the max and min bounds of the critical-to-classical crossover functions, previously calculated [Bagnuls and Bervillier, Phys. Rev. E 65, 066132 (2002)] within the massive renormalization scheme of the Phi(d)4(n) model in three dimensions (d = 3) and scalar order parameter (n = 1) of the Ising-like universality class. The mean functions are determined relying on the properties of the theoretical functions in the two limiting three-dimensional (3D) Ising-like and mean-field-like descriptions close to the Wilson-Fisher fixed point and to the Gaussian fixed point, respectively. Such descriptions correspond to the preasymptotic domains near each fixed point where a Wegner expansion restricted to two terms (leading and first confluent terms) is valid. The Ising-like preasymptotic domain description includes the correlations between parameters due to the error-bar determination of the exponents and amplitude combinations very close to the Wilson-Fisher fixed point. Adding the equivalent description of the mean field preasymptotic domain close to the Gaussian fixed point leads to define each mean crossover function with three calculated parameters. Fixing a unique value of one parameter whatever the selected mean crossover function, we use this parameter as a relative sensor to estimate the dominant nature, either (Ising-like) critical, or (mean-field-like) classical, of the crossover behavior. Finally, we obtain an explicit criterion to measure the extension of the Ising-like preasymptotic domain which can then permit to coherently account for measurements performed in systems where the asymptotical approach to the critical point remains finite, using a well-controlled number of system-dependent parameters (like in the subclass of one-component fluids).

We extend our previous determinations of nonasymptotic critical behavior of Phys. Rev. B 32, 7209 (1985) and 35, 3585 (1987) to accurate expressions of the complete classical-to-critical crossover (in three-dimensional field theory) in terms of the temperaturelike scaling field (i.e., along the critical isochore) for (1) the correlation length, the susceptibility, and the specific heat in the homogeneous phase for the n-vector model (n=1 to 3) and (2) the spontaneous magnetization (coexistence curve), the susceptibility, and the specific heat in the inhomogeneous phase for the Ising model (n=1). The present calculations include the seventh-loop order of Murray and Nickel and closely account for the up-to-date estimates of universal asymptotic critical quantities (exponents and amplitude combinations) provided by Guida and Zinn-Justin [J. Phys. A 31, 8103 (1998)].

The asymptotic and preasymptotic critical behavior in fluids, mixtures, and uniaxial magnets is believed to be described by an effective straight phi(4) scalar field theory with suitable, nonuniversal, coupling constants. The critical parameters as well as the extent of crossovers and corrections to the leading critical behavior in physical systems, crucially depends on the choice of these couplings. Here we propose a new method for deriving the effective scalar field theory appropriate to a microscopic model in this universality class. Use is made of the hierarchical reference theory, which implements the basic ideas of Wilson momentum space renormalization group to microscopic Hamiltonians. The effective low-energy field theory is then analyzed by the minimal subtraction scheme of Schloms and Dohm. We discuss the application of this method to the three-dimensional Ising model and to the liquid-vapor phase transition. We make comparison with high-temperature expansion results and with experimental data for rare gas.

Using the local potential approximation of the exact renormalization group (RG) equation, we show various domains of values of the parameters of the O(1) -symmetric scalar Hamiltonian.In three dimensions, in addition to the usual critical surface S c (attraction domain of the Wilson-Fisher fixed point), we explicitly show the existence of a first-order phase transition domain S f separated from S c by the tricritical surface S t (attraction domain of the Gaussian fixed point).S f and S c are two distinct domains of repulsion for the Gaussian fixed point, but S f is not the basin of attraction of a fixed point.S f is characterized by an endless renormalized trajectory lying entirely in the domain of negative values of the ϕ 4 -coupling.This renormalized trajectory also exists in four dimensions making the Gaussian fixed point ultra-violet stable (and the ϕ 4 4 renormalized field theory asymptotically free but with a wrong sign of the perfect action).We also show that a very retarded classical-to-Ising crossover may exist in three dimensions (in fact below four dimensions).This could be an explanation of the unexpected classical critical behaviour observed in some ionic systems.

Using the local potential approximation of the exact renormalization group (RG) equation, we show the various domains of values of the parameters of the O(1)-symmetric scalar Hamiltonian. In three dimensions, in addition to the usual critical surface

S_{c}

(attraction domain of the Wilson-Fisher fixed point), we explicitly show the existence of a first-order phase transition domain

S_{f}

separated from

S_{c}

by the tricritical surface

S_{t}

(attraction domain of the Gaussian fixed point).

S_{f}

and

S_{c}

are two distinct domains of repulsion for the Gaussian fixed point, but

S_{f}

is not the basin of attraction of a fixed point.

S_{f}

is characterized by an endless renormalized trajectory lying entirely in the domain of negative values of the

\phi ^{4}

-coupling. This renormalized trajectory exists also in four dimensions making the Gaussian fixed point ultra-violet stable (and the

\phi_{4}^{4}

renormalized field theory asymptotically free but with a wrong sign of the perfect action). We also show that very retarded classical-to-Ising crossover may exist in three dimensions (in fact below four dimensions). This could be an explanation of the unexpected classical critical behavior observed in some ionic systems.

Nonuniversal scale transformations of the physical fields are extended to pure quantum fluids and used to calculate the susceptibility, the specific heat, and the order parameter density along the critical isochore of near its liquid-vapor critical point. Within the so-called preasymptotic domain, where the Wegner expansion restricted to the first term of confluent corrections to scaling is expected to be valid, the results are in agreement with the experimental measurements and recent predictions, either based on the minimal-substraction renormalization and the massive renormalization schemes within phi (4)(d = 3) (n = 1)the model, or based on the crossover parametric equation of state for Ising-like systems.

Accurate numerical results are presented for the three-dimensional equivalent-neighbor model on a cubic lattice, for 12 different interaction ranges (coordination number between 18 and 250). These results allow the determination of the range dependences of the critical temperature and various critical amplitudes, which are compared to renormalization-group predictions. In addition, the analysis yields an estimate for the interaction range at which the leading corrections to scaling vanish for the spin-1 / 2 model, and confirms earlier conclusions that the leading Wegner correction must be negative for the three-dimensional (nearest-neighbor) Ising model. By complementing these results with Monte Carlo data for systems with coordination numbers as large as 52 514, the full finite-size crossover curves between classical and Ising-like behavior are obtained as a function of a generalized Ginzburg parameter. Also, the crossover function for the effective magnetic exponent is determined.