Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Finite-size scaling close to the critical point: renormalisation group and epsilon expansion

A. M. Nemirovsky, Karl F. Freed

Published Mar 11, 1986• 13 citations
Publisher

Abstract

The authors present an effective free-energy functional formalism which allows systematic epsilon -expansion methods to be used deep inside the critical region. Applications are provided to a d-dimensional layered geometry with periodic boundary conditions, and calculations are presented for the shift of the critical temperature and its associated exponent lambda . The theory shows how the (d-1)-dimensional physics emerges with power law dependences on the interlayer separation.

DOI
10.1088/0305-4470/19/4/020

Related papers

Renormalisation group treatment of finite size scaling with ε expansion

A. M. Nemirovsky, Karl F. Freed

The use of renormalisation group techniques away from the critical point is shown to enable the calculation of finite size scaling corrections to scaling functions with epsilon -expansion methods. The authors consider the N-vector phi 4 theory for a layered geometry with periodic boundary conditions and evaluate the correlation function, susceptibility, correlation lengths and shift in the critical temperature to order epsilon .

Journal of Physics A Mathematical and General•Apr 21, 1985•23 citations

Surface transition and epsilon expansion

A. M. Nemirovsky, Karl F. Freed

The authors provide the first description of the surface transition that goes beyond mean-field theory and that develops a systematic epsilon -expansion method for this transition. The theory exhibits the emergence of (d-1)-dimensional physics with a complicated dependence on c (the surface interaction parameter). A calculation is provided for the lowest correction to the shift in the critical temperature (from its mean-field value).

Journal of Physics A Mathematical and General•Nov 11, 1985•7 citations

Finite-size scaling above the upper critical dimension

Matthew Wittmann, A. P. Young

We present a unified view of finite-size scaling (FSS) in dimension d above the upper critical dimension, for both free and periodic boundary conditions. We find that the modified FSS proposed some time ago to allow for violation of hyperscaling due to a dangerous irrelevant variable applies only to k=0 fluctuations, and "standard" FSS applies to k≠0 fluctuations. Hence the exponent η describing power-law decay of correlations at criticality is unambiguously η=0. With free boundary conditions, the finite-size "shift" is greater than the rounding. Nonetheless, using T-T(L), where T(L) is the finite-size pseudocritical temperature, rather than T-T(c), as the scaling variable, the data do collapse onto a scaling form that includes the behavior both at T(L), where the susceptibility χ diverges like L(d/2), and at the bulk T(c), where it diverges like L(2). These claims are supported by large-scale simulations on the five-dimensional Ising model.

arXiv:1410.5296•Physical Review E•Dec 29, 2014•56 citations

Critical temperature shifts for layered periodic systems in 1/n expansion

Paul R. Gerber

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.

Journal of Physics A Mathematical and General•Mar 1, 1978•4 citations

Critical temperature shifts for finite slabs in the ε-expansion

Alan J. Bray, M A Moore

The critical temperature T c (b) of a d-dimensional system with finite thickness b in one of its dimensions approaches its bulk limit as b to infinity according to T c ( infinity )-T c (b) varies as b - lambda . To first order in epsilon =4-d the shift exponent lambda is found to be given by lambda =2-((n+2)/(n+8)) epsilon +O( epsilon 2 ), and thus satisfies the scaling relation lambda = nu d -1 , where nu d is the correlation length exponent of the bulk system and n is the order parameter dimensionality.

Journal of Physics A Mathematical and General•Apr 1, 1978•22 citations

Renormalization-Group Theory of Critical Phenomena in Confined Systems: Order-Parameter Distribution Function

X.S. Chen, V. Dohm

We present a renormalization-group study of the order-parameter distribution function near the critical point of O(n) symmetric three-dimensional (3D) systems in a finite geometry. The distribution function is calculated within the φ4 field theory for a 3D cube with periodic boundary conditions by means of a novel approach that appropriately deals with the Goldstone modes below T c . Results are given for both vanishing and finite external field h. The results describe finite-size effects near the critical point in the h– T-plane including the first-order transition at the coexistence line at h = 0 below T c . Quantitative theoretical predictions of the finite-size scaling function are presented for the Ising (n=1), XY(n=2) and Heisenberg (n=3) models. Good agreement is found with recent Monte Carlo data.

International Journal of Modern Physics B•May 30, 1998•7 citations
The authors present an effective free-energy functional formalism which allows systematic epsilon -expansion methods to be used deep inside the critical region. Applications are provided to a d-dimensional layered geometry with periodic boundary conditions, and calculations are presented for the shift of the critical temperature and its associated exponent lambda . The theory shows how the (d-1)-dimensional physics emerges with power law dependences on the interlayer separation.
We present a unified view of finite-size scaling (FSS) in dimension d above the upper critical dimension, for both free and periodic boundary conditions. We find that the modified FSS proposed some time ago to allow for violation of hyperscaling due to a dangerous irrelevant variable applies only to k=0 fluctuations, and "standard" FSS applies to k≠0 fluctuations. Hence the exponent η describing power-law decay of correlations at criticality is unambiguously η=0. With free boundary conditions, the finite-size "shift" is greater than the rounding. Nonetheless, using T-T(L), where T(L) is the finite-size pseudocritical temperature, rather than T-T(c), as the scaling variable, the data do collapse onto a scaling form that includes the behavior both at T(L), where the susceptibility χ diverges like L(d/2), and at the bulk T(c), where it diverges like L(2). These claims are supported by large-scale simulations on the five-dimensional Ising model.
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.
The critical temperature T c (b) of a d-dimensional system with finite thickness b in one of its dimensions approaches its bulk limit as b to infinity according to T c ( infinity )-T c (b) varies as b - lambda . To first order in epsilon =4-d the shift exponent lambda is found to be given by lambda =2-((n+2)/(n+8)) epsilon +O( epsilon 2 ), and thus satisfies the scaling relation lambda = nu d -1 , where nu d is the correlation length exponent of the bulk system and n is the order parameter dimensionality.
We present a renormalization-group study of the order-parameter distribution function near the critical point of O(n) symmetric three-dimensional (3D) systems in a finite geometry. The distribution function is calculated within the φ4 field theory for a 3D cube with periodic boundary conditions by means of a novel approach that appropriately deals with the Goldstone modes below T c . Results are given for both vanishing and finite external field h. The results describe finite-size effects near the critical point in the h– T-plane including the first-order transition at the coexistence line at h = 0 below T c . Quantitative theoretical predictions of the finite-size scaling function are presented for the Ising (n=1), XY(n=2) and Heisenberg (n=3) models. Good agreement is found with recent Monte Carlo data.