Critical exponents of the<b><i>N</i></b>-vector model

Guida, Riccardo; Zinn-Justin, Jean

doi:10.1088/0305-4470/31/40/006

Cited by 831 publications

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“…The equation of state is the relation between magnetic field H, magnetization M = φ (the "bare" field expectation value) and the temperature which is Table 2 Critical exponents of the O(N ) models from ε-expansion [2]. …”

Section: The Scaling Equation Of Statementioning

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

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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: The Scaling Equation Of Statementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Zinn-Justin¹

2001

Physics Reports

Self Cite

139

166

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“…It is slightly lower than the result of numerical estimate of this exponent for the Ising model, ν c = 0.630 (see, e.g., [6,15]). The difference in the values of ν and ν c is considered to be rather our limitation of the ρ 4 -model than the approximation during its calculation.…”

Section: Solution To the Recurrence Relation Near Critical Pointmentioning

confidence: 60%

“…Indeed, the Ising model may be really described by the model ρ 2m with m → ∞ [2]. To achieve a real correspondence between ν and ν c we have to use at least the model ρ 6 .…”

Section: Solution To the Recurrence Relation Near Critical Pointmentioning

confidence: 99%

“…One of such approximations in studying the critical behavior of the statistical systems consists in using the Gaussian distribution for the order parameter. In such a way, for the scalar ϕ 4 theory in d = 3 [6], the values of the critical exponents have been obtained by means of resummation of the series of Gaussian perturbation theory. At present these values are considered to be the most reliable.…”

Section: Introductionmentioning

confidence: 99%

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Recurrence relations for the three-dimensional Ising-like model in the external field

Kozlovskii¹

2005

Condens. Matter Phys.

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The method for calculation of the partition function of lattice model for the magnet in the external field near critical point (CP) is proposed. The recurrence relations and their explicit solution near the critical point are founded. It is shown that dependence on temperature of thermodynamic functions near CP, when the field value comes down to zero, is in good agreement with the previous results obtained using the collective variable method. The phase transition temperature (when h = 0) is calculated and the dependence on parameters of interaction potential is found.

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