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

Physical review. B, Solid state

Universal relations among critical amplitude. Calculations up to order ε2 for systems with continuous symmetry

Claude Bervillier

Published Dec 1, 1976• 151 citations
Publisher

Abstract

A general derivation of the two-scale-factor universality is given using the renormalized ${\ensuremath{\varphi}}^{4}$ theory. As a consequence ten universal relations among critical amplitudes are obtained, any other universal relations being a combination of them. The situation is shown to be the same as for the scaling laws for critical exponents. The calculations up to order ${\ensuremath{\epsilon}}^{2}$ for systems with continuous symmetry are performed for some of these relations such as the specific-heat amplitudes ratio and ${R}_{\ensuremath{\xi}}^{+}$, which relates the amplitudes of the correlation length and of the specific heat, etc. Comparison with experiments and series results and, in particular, the discussion of the superfluid helium and RbMn${\mathrm{F}}_{3}$ cases, are repeated with our improved numerical results.

DOI
10.1103/physrevb.14.4964

Related papers

Two-scale-factor universality and the renormalization group

Pierre C. Hohenberg, Amnon Aharony A, Bertrand I. Halperin +1 more

The hypothesis of two-scale-factor universality, originally proposed by Stauffer, Ferer, and Wortis, is shown to follow from the renormalization-group approach, for systems close to their critical point. Values of the universal ratios involving correlation length and specific-heat amplitudes are obtained from the $\ensuremath{\epsilon}$ expansion, for Ising, $X\ensuremath{-}Y$, and Heisenberg models. In the latter two cases the correlation function has a power-law behavior at large distances below ${T}_{c}$, and the (transverse) correlation length is defined in terms of the stiffness constant ${\ensuremath{\rho}}_{s}$. Experimental values of the correlation lengths and amplitude ratios are determined for superfluid $^{4}\mathrm{He}$, which is $X\ensuremath{-}Y$-like, and for the Heisenberg antiferromagnet ${\mathrm{RbMnF}}_{3}$. Comparisons are made between the values of the amplitude ratios coming from $\ensuremath{\epsilon}$ expansions, series, and experiments.

Physical review. B, Solid state•Apr 1, 1976•278 citations

Universal relations among critical amplitudes of surface quantities

Hans Werner Diehl, Gerhard Gompper, W. Speth

Universal relations among critical amplitudes of surface quantities are discussed for d-dimensional semi-infinite systems near their bulk critical temperature ${T}_{c}^{b}$. Based on the field-theoretic renormalization-group approach, a general derivation of the expected multi-scale-factor universality is given and various universal ratios involving critical amplitudes of the ordinary and special transitions both above and below ${T}_{c}^{b}$ are calculated to first order in \ensuremath{\epsilon}=4-d. In addition, an error in the published one-loop result for the universal order-parameter profile at the ordinary transition of the semi-infinite Ising model is corrected.

Physical review. B, Condensed matter•May 1, 1985•20 citations

Critical confluent corrections: Universality and estimates of amplitude ratios from field theory at d=3

C. Bagnuls, Claude Bervillier

We estimate, at $d=3$, from ${\ensuremath{\varphi}}^{4}$ field theory the universal ratios of confluent critical correction amplitudes relative to the correlation length, the susceptibility, and the specific heat above ${T}_{c}$. Our central values are comparable to these available from analysis of high-temperature series and the $\ensuremath{\epsilon}$ expansion. Furthermore, experiments in progress on a binary mixture are in good agreement with one of our results, the ratio of the confluent corrections for the specific heat and susceptibility which have not been previously estimated. We also present, in detail, a derivation of the universality of these quantities in the framework of field theory.

Physical review. B, Condensed matter•Aug 1, 1981•67 citations

Universal combination of critical amplitudes from field theory

Claude Bervillier, Claude Godrèche

Near a continuous phase transition, there exist universal combinations of critical amplitudes. We present an estimation of such a universal quantity, in the framework of the ${\ensuremath{\varphi}}^{4}$ field theory in three dimensions, using information recently available in the literature, and already used to determine precise values for critical exponents. This leads to an accuracy only obtained, hitherto, in the framework of field theory, for critical exponents. The universal quantity calculated, often denoted by $R_{\ensuremath{\xi}}^{}{}_{}{}^{+}$, combines the amplitude of the correlation length and that of the singular part of the specific heat in the vicinity of the critical point.

Physical review. B, Condensed matter•Jun 1, 1980•108 citations

Universality of correlation-length amplitudes and their relation with critical exponents for quantum systems

K. A. Penson, M. Kolb

Recently it has been found that for several two-dimensional classical models the amplitude of the correlation length of a finite system at a critical point is universal and related to the correlation-function exponent $\ensuremath{\eta}$. Here we propose that for quantum systems the ratios of two such amplitudes are universal and are related to the ratios of the corresponding exponents. This proposition is confirmed numerically for the transverse Ising model for spin $S=\frac{1}{2},1,\frac{3}{2}, \mathrm{and} 2$. It is used to calculate $\ensuremath{\eta}$ in a simplified way.

Physical review. B, Condensed matter•Mar 1, 1984•24 citations

Universal ratios among corrections-to-scaling amplitudes in the large- n limit

Haim Sompolinsky, Amnon Aharony A

The thermodynamic functions of an $n$-component $d$-dimensional ${\ensuremath{\sigma}}^{4}$ theory are written in the form ${f}_{i}={A}_{i}{|t|}^{\ensuremath{-}{\ensuremath{\lambda}}_{i}}(1+{a}_{i}{|t|}^{\ensuremath{\Delta}})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$. The correction-to-scaling amplitudes ${a}_{i}$ are calculated to leading order in $\frac{1}{n}$ for $2<d<4$, and the universality of their ratios is explicitly shown. The effective exponents ${\ensuremath{\lambda}}_{i,\mathrm{eff}}={\ensuremath{\lambda}}_{i}\ensuremath{-}{a}_{i}\ensuremath{\Delta}{|t|}^{\ensuremath{\Delta}}$ violate the thermodynamic scaling laws already at this leading order. The amplitudes ${a}_{i}$ have singularities (as functions of $d$) at the dimensionalities ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{l}=2+\frac{4}{l},l=3,4,5,\dots{}$. Above ${T}_{c}$, the specific heat and the susceptibility correction amplitudes ${a}_{c}^{+}$ and ${a}_{\ensuremath{\chi}}^{+}$ are discontinuous at ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{l}$ and at ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{2l}(l\ensuremath{\ge}2)$, respectively. Below ${T}_{c}$, the magnetization and the stiffness correction amplitudes are zero, ${a}_{M}={a}_{{\ensuremath{\rho}}_{s}}=0$, while ${a}_{c}^{\ensuremath{-}}$ diverges at ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{2l+1}(l\ensuremath{\ge}1)$ but is continuous at ${d}_{2l}$ (in particular at $d=3$). Results are also given for universal ratios among corrections-to-scaling amplitudes for the dependence of thermo-dynamic functions on the magnetization at $T={T}_{c}$ and at a nonzero field.

Physical review. B, Condensed matter•Apr 1, 1981•2 citations
A general derivation of the two-scale-factor universality is given using the renormalized \ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4 theory. As a consequence ten universal relations among critical amplitudes are obtained, any other universal relations being a combination of them. The situation is shown to be the same as for the scaling laws for critical exponents. The calculations up to order \ensuremathϵ2{\ensuremath{\epsilon}}^{2}\ensuremathϵ2 for systems with continuous symmetry are performed for some of these relations such as the specific-heat amplitudes ratio and R\ensuremathξ+{R}_{\ensuremath{\xi}}^{+}R\ensuremathξ+​, which relates the amplitudes of the correlation length and of the specific heat, etc. Comparison with experiments and series results and, in particular, the discussion of the superfluid helium and RbMnF3{\mathrm{F}}_{3}F3​ cases, are repeated with our improved numerical results.
The hypothesis of two-scale-factor universality, originally proposed by Stauffer, Ferer, and Wortis, is shown to follow from the renormalization-group approach, for systems close to their critical point. Values of the universal ratios involving correlation length and specific-heat amplitudes are obtained from the $\ensuremath{\epsilon}$ expansion, for Ising, $X\ensuremath{-}Y$, and Heisenberg models. In the latter two cases the correlation function has a power-law behavior at large distances below Tc{T}_{c}Tc​, and the (transverse) correlation length is defined in terms of the stiffness constant ${\ensuremath{\rho}}_{s}$. Experimental values of the correlation lengths and amplitude ratios are determined for superfluid 4He^{4}\mathrm{He}4He, which is $X\ensuremath{-}Y$-like, and for the Heisenberg antiferromagnet RbMnF3{\mathrm{RbMnF}}_{3}RbMnF3​. Comparisons are made between the values of the amplitude ratios coming from $\ensuremath{\epsilon}$ expansions, series, and experiments.
Universal relations among critical amplitudes of surface quantities are discussed for d-dimensional semi-infinite systems near their bulk critical temperature Tcb{T}_{c}^{b}Tcb​. Based on the field-theoretic renormalization-group approach, a general derivation of the expected multi-scale-factor universality is given and various universal ratios involving critical amplitudes of the ordinary and special transitions both above and below Tcb{T}_{c}^{b}Tcb​ are calculated to first order in \ensuremath{\epsilon}=4-d. In addition, an error in the published one-loop result for the universal order-parameter profile at the ordinary transition of the semi-infinite Ising model is corrected.
We estimate, at d=3d=3d=3, from ${\ensuremath{\varphi}}^{4}$ field theory the universal ratios of confluent critical correction amplitudes relative to the correlation length, the susceptibility, and the specific heat above Tc{T}_{c}Tc​. Our central values are comparable to these available from analysis of high-temperature series and the $\ensuremath{\epsilon}$ expansion. Furthermore, experiments in progress on a binary mixture are in good agreement with one of our results, the ratio of the confluent corrections for the specific heat and susceptibility which have not been previously estimated. We also present, in detail, a derivation of the universality of these quantities in the framework of field theory.
Near a continuous phase transition, there exist universal combinations of critical amplitudes. We present an estimation of such a universal quantity, in the framework of the ${\ensuremath{\varphi}}^{4}$ field theory in three dimensions, using information recently available in the literature, and already used to determine precise values for critical exponents. This leads to an accuracy only obtained, hitherto, in the framework of field theory, for critical exponents. The universal quantity calculated, often denoted by $R_{\ensuremath{\xi}}^{}{}_{}{}^{+}$, combines the amplitude of the correlation length and that of the singular part of the specific heat in the vicinity of the critical point.
Recently it has been found that for several two-dimensional classical models the amplitude of the correlation length of a finite system at a critical point is universal and related to the correlation-function exponent $\ensuremath{\eta}$. Here we propose that for quantum systems the ratios of two such amplitudes are universal and are related to the ratios of the corresponding exponents. This proposition is confirmed numerically for the transverse Ising model for spin S=12,1,32,and2S=\frac{1}{2},1,\frac{3}{2}, \mathrm{and} 2S=21​,1,23​,and2. It is used to calculate $\ensuremath{\eta}$ in a simplified way.
The thermodynamic functions of an nnn-component ddd-dimensional ${\ensuremath{\sigma}}^{4}$ theory are written in the form ${f}_{i}={A}_{i}{|t|}^{\ensuremath{-}{\ensuremath{\lambda}}_{i}}(1+{a}_{i}{|t|}^{\ensuremath{\Delta}})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$. The correction-to-scaling amplitudes ai{a}_{i}ai​ are calculated to leading order in 1n\frac{1}{n}n1​ for 2<d<42<d<42<d<4, and the universality of their ratios is explicitly shown. The effective exponents ${\ensuremath{\lambda}}_{i,\mathrm{eff}}={\ensuremath{\lambda}}_{i}\ensuremath{-}{a}_{i}\ensuremath{\Delta}{|t|}^{\ensuremath{\Delta}}$ violate the thermodynamic scaling laws already at this leading order. The amplitudes ai{a}_{i}ai​ have singularities (as functions of ddd) at the dimensionalities ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{l}=2+\frac{4}{l},l=3,4,5,\dots{}$. Above Tc{T}_{c}Tc​, the specific heat and the susceptibility correction amplitudes ac+{a}_{c}^{+}ac+​ and ${a}_{\ensuremath{\chi}}^{+}$ are discontinuous at ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{l}$ and at ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{2l}(l\ensuremath{\ge}2)$, respectively. Below Tc{T}_{c}Tc​, the magnetization and the stiffness correction amplitudes are zero, ${a}_{M}={a}_{{\ensuremath{\rho}}_{s}}=0$, while ${a}_{c}^{\ensuremath{-}}$ diverges at ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{2l+1}(l\ensuremath{\ge}1)$ but is continuous at d2l{d}_{2l}d2l​ (in particular at d=3d=3d=3). Results are also given for universal ratios among corrections-to-scaling amplitudes for the dependence of thermo-dynamic functions on the magnetization at T=TcT={T}_{c}T=Tc​ and at a nonzero field.