3D Ising model: the scaling equation of state

Guida, Riccardo; Zinn-Justin, Jean

doi:10.1016/s0550-3213(96)00704-3

Cited by 210 publications

(397 citation statements)

References 68 publications

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“…The summation methods used in [1] were based on a Borel transformation and conformal map. The same ideas have later been applied to the ε-expansion [15] when five loop series became available [16], and more recently to the equation of state by Guida-Zinn-Justin [3]. With time the method has been refined and the efficiency improved by various tricks but the basic principles have not changed.…”

Section: Critical Exponents and Series Summationmentioning

confidence: 99%

“…A rather powerful method, motivated by the results obtained within the ε-expansion scheme, is based on the parametric representation [3].…”

Section: The Problem Of the Low Temperature Phasementioning

confidence: 99%

“…Table ?.? contains the results of [3] together with other published estimates of the coefficients of the small z expansion of F (z) coming from high temperature series [50,52,53], simulations [54] and derivative expansion of the Exact RG [55,56]. …”

Section: The Small Field Expansionmentioning

confidence: 99%

“…We therefore review the methods, based on renormalized φ 4 3 quantum field theory and renormalization group, which have led to a precise determination of critical exponents of the N -vector model [1,2] and of the equation of state of the 3D Ising model [3]. These results are among the most precise available probing field theory in a nonperturbative regime.…”

Section: Introductionmentioning

confidence: 99%

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Precise determination of critical exponents and equation of state by field theory methods

Zinn-Justin¹

2001

Physics Reports

139

166

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Renormalization group, and in particular its Quantum Field Theory implementation has provided us with essential tools for the description of the phase transitions and critical phenomena beyond mean field theory. We therefore review the methods, based on renormalized φ 4 3 quantum field theory and renormalization group, which have led to a precise determination of critical exponents of the N -vector model [1,2] and of the equation of state of the 3D Ising model [3]. These results are among the most precise available probing field theory in a nonperturbative regime.Precise calculations first require enough terms of the perturbative expansion. However perturbation series are known to be divergent. The divergence has been characterized by relating it to instanton contributions. The information about large order behaviour of perturbation series has then allowed to develop efficient "summation" techniques, based on Borel transformation and conformal mapping [4].We first discuss exponents and describe our recent results [2]. Compared to exponents, the determination of the scaling equation of state of the 3D Ising model involves a few additional (non-trivial) technical steps, like the use of the parametric representation, and the order dependent mapping method. From the knowledge of the equation of state a number of ratio of critical amplitudes can also be derived.Finally we emphasize that few physical quantities which are predicted by renormalization group to be universal have been determined precisely, and much work remains to be done. Considering the steady increase in the available computer Talk given at the Conference "Renormalization Group 2000, Taxco (Mexico), 11-15 Jan. 1999 * Laboratoire de la Direction des Sciences de la Matière du Commissariatà l'Energie Atomique 2 resources, many new calculations will become feasible. In addition to the infinite volume quantities, finite size universal quantities would also be of interest, to provide a more direct contact with numerical simulations. Let us also mention dynamical observables, a largely unexplored territory.

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Section: Critical Exponents and Series Summationmentioning

confidence: 99%

“…A rather powerful method, motivated by the results obtained within the ε-expansion scheme, is based on the parametric representation [3].…”

Section: The Problem Of the Low Temperature Phasementioning

confidence: 99%

Section: The Small Field Expansionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Precise determination of critical exponents and equation of state by field theory methods

Zinn-Justin¹

2001

Physics Reports

139

166

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“…The similar estimations can be generalized to the models with an n-component order parameter [21] and MC simulations [22]. The scaling function of the susceptibility (the solid line is the CV method result, the dashed line is the result obtained in [21,22]).…”

Section: Discussionmentioning

confidence: 72%

Free energy and equation of state of Ising-like magnet near the critical point

Kozlovskii¹,

Pylyuk²,

Prytula³

2006

Nuclear Physics B

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The description of a three-dimensional Ising-like magnet in the presence of an external field in the vicinity of the critical point by the collective variables method is proposed. Using the renormalization group transformations, the scaling region size is defined as a function of temperature and field. The obtained expressions for the free energy, equation of state and susceptibility allow one to analyse their dependence on microscopic parameters of the system. The critical exponents of the correlation length and order parameter are calculated as well. The results agree qualitatively with ones obtained within the framework of the parametric representation of the equation of state and Monte-Carlo simulations. The calculations do not involve any parametrization, phenomenological assumptions and adjustable parameters. The approach can be extended to models with a multicomponent order parameter.

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New Developments in the Continuous Renormalization Group

Morris

NATO Science Series: B:

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3D Ising model: the scaling equation of state

Cited by 210 publications

References 68 publications

Precise determination of critical exponents and equation of state by field theory methods

Precise determination of critical exponents and equation of state by field theory methods

Free energy and equation of state of Ising-like magnet near the critical point

New Developments in the Continuous Renormalization Group

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