Universal amplitude ratios for constrained critical systems

The current theory of critical phenomena in constrained systems is re-examined. It is found that renormalization of critical exponents is possible even if alpha <0 in the unconstrained system. The renormalized behaviour occurs at a tricritical point which is of the same type as found recently in magnetothermomechanics. A soluble model illustrates this behaviour.

Journal of Physics C Solid State PhysicsApr 21, 197416 citations

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

Claude Bervillier

Physical review. B, Solid stateDec 1, 1976151 citations

Universal amplitude ratios for constrained critical systems

Izmailian, N., Ralph Kenna B

Published Jan 1, 2014• 5 citations

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Abstract

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Universal amplitude ratios for constrained critical systems

N Sh Izmailian, R Kenna

The critical properties of systems under constraint differ from their ideal counterparts through Fisher renormalization. The mathematical properties of Fisher renormalization applied to critical exponents are well known: the renormalized indices obey the same scaling relations as the ideal ones and the transformations are involutions in the sense that re-renormalizing the critical exponents of the constrained system delivers their original, ideal counterparts. Here we examine Fisher renormalization of critical amplitudes and show that, unlike for critical exponents, the associated transformations are not involutions. However, for ratios and combinations of amplitudes which are universal, Fisher renormalization is involutory.The critical properties of systems under constraint differ from their ideal counterparts through Fisher renormalization. The mathematical properties of Fisher renormalization applied to critical exponents are well known: the renormalized indices obey the same scaling relations as the ideal ones and the transformations are involutions in the sense that re-renormalizing the critical exponents of the constrained system delivers their original, ideal counterparts. Here we examine Fisher renormalization of critical amplitudes and show that, unlike for critical exponents, the associated transformations are not involutions. However, for ratios and combinations of amplitudes which are universal, Fisher renormalization is involutory.

arXiv:1402.4673Journal of Statistical Mechanics Theory and ExperimentJul 11, 20145 citations

Critical Phenomena for Systems under Constraint

N. Sh. Izmailian, Ralph Kenna B

It is well known that the imposition of a constraint can transform the properties of critical systems. Early work on this phenomenon by Essam and Garelick, Fisher, and others, focused on the effects of constraints on the leading critical exponents describing phase transitions. Recent work extended these considerations to critical amplitudes and to exponents governing logarithmic corrections in certain marginal scenarios. Here these old and new results are gathered and summarised. The involutory nature of the transformations between the critical parameters describing ideal and constrained systems are also discussed, paying particular attention to matters relating to universality.It is well known that the imposition of a constraint can transform the properties of critical systems. Early work on this phenomenon by Essam and Garelick, Fisher, and others, focused on the effects of constraints on the leading critical exponents describing phase transitions. Recent work extended these considerations to critical amplitudes and to exponents governing logarithmic corrections in certain marginal scenarios. Here these old and new results are gathered and summarised. The involutory nature of the transformations between the critical parameters describing ideal and constrained systems are also discussed, paying particular attention to matters relating to universality.

arXiv:1406.3865Jun 15, 20140 citations

Critical phenomena for systems under constraint

Izmailian, Ralph Kenna B

It is well known that the imposition of a constraint can transform the properties of critical systems.Early work on this phemomenon by Essam and Garelick, Fisher, and others, focused on the effects of constraints on the leading critical exponents describing phase transitions.Recent work extended these considerations to critical amplitudes and to exponents governing logarithmic corrections in certain marginal scenarios.Here these old and new results are gathered and summarised.The involutory nature of transformations between the critical parameters describing ideal and constrained systems are also discussed, paying particular attention to matters relating to universality.It is well known that the imposition of a constraint can transform the properties of critical systems.Early work on this phemomenon by Essam and Garelick, Fisher, and others, focused on the effects of constraints on the leading critical exponents describing phase transitions.Recent work extended these considerations to critical amplitudes and to exponents governing logarithmic corrections in certain marginal scenarios.Here these old and new results are gathered and summarised.The involutory nature of transformations between the critical parameters describing ideal and constrained systems are also discussed, paying particular attention to matters relating to universality.

Condensed Matter PhysicsSep 1, 20141 citation

Fisher renormalization for logarithmic corrections

Ralph Kenna B, Hsiao‐Ping Hsu, Christian von Ferber

For continuous phase transitions characterized by power-law divergences, Fisher renormalization prescribes how to obtain the critical exponents for a system under constraint from their ideal counterparts. In statistical mechanics, such ideal behaviour at phase transitions is frequently modified by multiplicative logarithmic corrections. Here, Fisher renormalization for the exponents of these logarithms is developed in a general manner. As for the leading exponents, Fisher renormalization at the logarithmic level is seen to be involutory and the renormalized exponents obey the same scaling relations as their ideal analogues. The scheme is tested in lattice animals and the Yang–Lee problem at their upper critical dimensions, where predictions for logarithmic corrections are made.For continuous phase transitions characterized by power-law divergences, Fisher renormalization prescribes how to obtain the critical exponents for a system under constraint from their ideal counterparts. In statistical mechanics, such ideal behaviour at phase transitions is frequently modified by multiplicative logarithmic corrections. Here, Fisher renormalization for the exponents of these logarithms is developed in a general manner. As for the leading exponents, Fisher renormalization at the logarithmic level is seen to be involutory and the renormalized exponents obey the same scaling relations as their ideal analogues. The scheme is tested in lattice animals and the Yang–Lee problem at their upper critical dimensions, where predictions for logarithmic corrections are made.

arXiv:0810.2719Journal of Statistical Mechanics Theory and ExperimentOct 31, 20087 citations

Fisher renormalization in systems with finite specific heat

V. Dohm

Journal of Physics C Solid State PhysicsApr 21, 197416 citations

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

Claude Bervillier

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.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.

Physical review. B, Solid stateDec 1, 1976151 citations

{\mathrm{F}}_{3}

cases, are repeated with our improved numerical results.