Critical behavior of the three-dimensional Ising model: A high-resolution Monte Carlo study

Ferrenberg, Alan M.; Landau, D. P.

doi:10.1103/physrevb.44.5081

Cited by 775 publications

(684 citation statements)

References 38 publications

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“…We will show therefore in the following results for lateral lattice size L = 60 for the 2D case, and L = 24 for thin films with thicknesses L z = 4 and 6. In order to check the first-order nature of a weak firstorder transition, the histogram technique is very efficient [23,24]. But in our case as will be seen below, the reorientation is a very strong first-order transition.…”

Section: Model and Methodsmentioning

confidence: 99%

“…We did not use the theory of finite-size scaling [22][23][24] because the calculation of critical exponents is not the purpose of the present work. However, in order to appreciate finite-size effects, we carried out simulations in the 2D case for sizes from L × L = 24 × 24 to 60 × 60 and in the case of thin films from L × L × L z = 12 × 12 × 4 to 48 × 48 × 6.…”

Section: Model and Methodsmentioning

confidence: 99%

“…They serve as testing means for new theoretical developments. Over the years, the standard MC method [21] has been improved by the finite-size scaling theory [22] and by other high-performance techniques such as histogram techniques [23,24], cluster updating algorithms [25][26][27] and Wang-Landau flat-histogram method [28]. We have now at hand these efficient techniques to deal with complex systems.…”

Section: Introductionmentioning

confidence: 99%

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Re-orientation transition in molecular thin films: Potts model with dipolar interaction

Hoang¹,

Kasperski²,

Puszkarski³

et al. 2013

J. Phys.: Condens. Matter

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We study the low-temperature behavior and the phase transition of a thin film by Monte Carlo simulation. The thin film has a simple cubic lattice structure where each site is occupied by a Potts parameter which indicates the molecular orientation of the site. We take only three molecular orientations in this paper which correspond to the 3-state Potts model. The Hamiltonian of the system includes: (i) the exchange interaction Jij between nearest-neighbor sites i and j (ii) the long-range dipolar interaction of amplitude D truncated at a cutoff distance rc (iii) a single-ion perpendicular anisotropy of amplitude A. We allow Jij = Js between surface spins, and Jij = J otherwise. We show that the ground state depends on the the ratio D/A and rc. For a single layer, for a given A, there is a critical value Dc below (above) which the ground-state (GS) configuration of molecular axes is perpendicular (parallel) to the film surface. When the temperature T is increased, a re-orientation transition occurs near Dc: the low-T in-plane ordering undergoes a transition to the perpendicular ordering at a finite T , below the transition to the paramagnetic phase. The same phenomenon is observed in the case of a film with a thickness. We show that the surface phase transition can occur below or above the bulk transition depending on the ratio Js/J. Surface and bulk order parameters as well as other physical quantities are shown and discussed.

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Section: Model and Methodsmentioning

confidence: 99%

Section: Model and Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Re-orientation transition in molecular thin films: Potts model with dipolar interaction

Hoang¹,

Kasperski²,

Puszkarski³

et al. 2013

J. Phys.: Condens. Matter

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show abstract

“…Only recently have consistent, but significantly more precise, experimental results been reported in low gravity superfluid experiments [41]. Also the precision of results coming from high temperature expansions [18][19][20][21][22][23][24][25][26][27][28][29] and various numerical simulations [30][31][32][33][34][35][36][37][38][39][40] on the lattice has kept steadily improving.…”

Section: Exponentsmentioning

confidence: 99%

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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“…Our result is valid only in the strong coupling limit, and in order to make it general a different approach seems to be necessary. The use [14] 0.518 (7) 1.970(11) 0.6289 (8) of effective theories for SU(2) may help to solve the problem [15].…”

Section: Discussionmentioning

confidence: 99%

Percolation and deconfinement in SU(2) gauge theory

Fortunato¹,

Satz²

2001

Nuclear Physics A

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The deconfinement transition in SU(2) gauge theory and the magnetization transition in the Ising model belong to the same universality class. The critical behaviour of the Ising model can be characterized either as spontaneous breaking of the Z 2 symmetry of spin states or as percolation of appropriately defined spin clusters. We show that deconfinement in SU(2) gauge theory can be specified as percolation of Polyakov loop clusters with Fortuin-Kasteleyn bond weights, leading to the same critical exponents as the conventional order-disorder description based on the Polykov loop expectation value.

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Critical behavior of the three-dimensional Ising model: A high-resolution Monte Carlo study

Cited by 775 publications

References 38 publications

Re-orientation transition in molecular thin films: Potts model with dipolar interaction

Re-orientation transition in molecular thin films: Potts model with dipolar interaction

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

Percolation and deconfinement in SU(2) gauge theory

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