Ground States and Thermal States of the Random Field Ising Model

Wu, Yong; Machta, Jonathan

doi:10.1103/physrevlett.95.137208

Cited by 47 publications

(57 citation statements)

References 28 publications

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“…Hence the critical behavior is the same everywhere along the phase boundary and we can predict it simply by staying at T = 0 and crossing the phase boundary at the critical field point. This is a convenient approach because we can determine the ground states of the system exactly using efficient optimization algorithms [53][54][55][56][57][58][59][60][61][62][63][64][65][66]79,[88][89][90][91][92][93] through an existing mapping of the ground state to the maximum-flow optimization problem [94][95][96]. A clear advantage of this approach is the ability to simulate large system sizes and disorder ensembles in rather moderate computational times.…”

Section: Zero-temperature Algorithmmentioning

confidence: 99%

“…Although sophisticated improvements have appeared [48][49][50][51][52], these simulations produced low-accuracy data because they were limited to linear sizes of the order of L max 32. Larger sizes can be achieved at T = 0, through the wellknown mapping of the ground state to the maximum-flow optimization problem [53][54][55][56][57][58][59][60][61][62][63][64][65][66]. Yet, T = 0 simulations lack many tools, standard at T > 0.…”

Section: Introductionmentioning

confidence: 99%

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Efficient numerical methods for the random-field Ising model: Finite-size scaling, reweighting extrapolation, and computation of response functions

Fytas

Martı́n-Mayor²

2016

Phys. Rev. E

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It was recently shown [Phys. Rev. Lett. 110, 227201 (2013)] that the critical behavior of the random-field Ising model in three dimensions is ruled by a single universality class. This conclusion was reached only after a proper taming of the large scaling corrections of the model by applying a combined approach of various techniques, coming from the zero-and positive-temperature toolboxes of statistical physics. In the present contribution we provide a detailed description of this combined scheme, explaining in detail the zero-temperature numerical scheme and developing the generalized fluctuation-dissipation formula that allowed us to compute connected and disconnected correlation functions of the model. We discuss the error evolution of our method and we illustrate the infinite limit-size extrapolation of several observables within phenomenological renormalization. We present an extension of the quotients method that allows us to obtain estimates of the critical exponent α of the specific heat of the model via the scaling of the bond energy and we discuss the self-averaging properties of the system and the algorithmic aspects of the maximum-flow algorithm used.

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Section: Zero-temperature Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Efficient numerical methods for the random-field Ising model: Finite-size scaling, reweighting extrapolation, and computation of response functions

Fytas

Martı́n-Mayor²

2016

Phys. Rev. E

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“…This also shows that for each disorder configuration the staggered magnetization changes discontinuously at W c . Such discontinous jumps for bond energy were also reported for 3d RFIM [18,22,34]. The domains of the metastable state in the COP and the domains in ground state of disordered phase are plotted in figure 4 for L=48.…”

Section: Resultsmentioning

confidence: 77%

First order transition in two dimensional coulomb glass

Bhandari

Malik

2017

J. Phys.: Conf. Ser.

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Abstract.We have studied the ground states of two dimensional lattice model of Coulomb Glass via Monte Carlo annealing. Our results show a possibility of existence of a critical disorder (W c ) below which the system is in the charge ordered phase and above it the system is in the disordered phase. We have used finite size scaling to calculate W c = 0.2413, the critical exponent of magnetization β = 0 indicating discontinuity in magnetization and the critical exponent of correlation length ν = 1.0. The distribution of staggered magnetization for different disorder strengths shows a three peak structure. We thus predict that two dimensional Coulomb Glass shows a first order transition at T=0. IntroductionSince the introduction of Random field Ising Model, a lot of discussion has been done on the possibility of existence of ferromagnetic ordering below a critical disorder strength (W c ) in two dimensional (2D) RFIM. From Imry-Ma arguments [1], one finds that the lower critical dimension (d c ) below which the ferromagnetic ordered phase gets destroyed is d c ≤ 2, where d = 2 was considered as a limiting case. By doing field theoretical calculations [2,3], later it was suggested that d c = 3. In 1983, Binder [4] gave an argument that the roughening of domain walls would lead to the destruction of ferromagnetic ordering in 2d RFIM. Then an exact result was given by Bricmont and Kupiainen [5] in 1987 where they showed that there exist a ferromagnetic ordering in three dimensional (3d) RFIM. A rigorous theoretical proof was then given by Aizenman and Wehr [6] in 1989 that no long range ordering is possible in 2d RFIM case. Scaling theory for 3d RFIM assuming second order transition at T=0 leads to modified hyperscaling laws [7]. However, in contradiction to all the above arguments Frontera and Vives [8] in 1999 showed numerical signs of transition in 2d RFIM at zero temperature below a critical random field strength. In 2012, Aizenmann [9] too claims existence of phase transition in 2d RFIM. Spasojevic et al [10] have also given some numerical evidences which proves the existence of phase transition in 2d nonequilibrium RFIM at T = 0. Recently, Suman and Mandal [11] have studied some aspects of nonequilibrium 2d RFIM at low temperature. They have commented that 2d RFIM exhibits a phase transition in the disorder parameter even at T > 0. These numerical work suggests a possibility of presence of long range ordering in 2d RFIM at low disorder strengths. If we assume that the above arguments which are claiming that at T = 0 there is a finite disorder strength below which long range ordering in 2d RFIM exist, then the order of transition is another question which is not answered very clearly. This question remains unclear even for 3d RFIM where there is no controversy over the presence of phase transition at zero temperature. Some earlier work suggested a first order transition [12][13][14][15][16][17][18] but there are arguments [19][20][21][22] which favour second order

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“…the sim-ulation results in Refs. [58,59,60]) so that one indeed expects to observe cusp behavior in the field dependence of the 2-replica correlation functions. However, important questions are then whether such a phenomenon survives at long distances, e.g.…”

Section: B Renormalization Groupmentioning

confidence: 99%

Spontaneous versus explicit replica symmetry breaking in the theory of disordered systems

Mouhanna

Tarjus²

2010

Phys. Rev. E

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We investigate the relation between spontaneous and explicit replica symmetry breaking in the theory of disordered systems. On general ground, we prove the equivalence between the replicon operator associated with the stability of the replica symmetric solution in the standard replica scheme and the operator signaling a breakdown of the solution with analytic field dependence in a scheme in which replica symmetry is explicitly broken by applied sources. This opens the possibility to study, via the recently developed functional renormalization group, unresolved questions related to spontaneous replica symmetry breaking and spin-glass behavior in finite-dimensional disordered systems.

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Ground States and Thermal States of the Random Field Ising Model

Cited by 47 publications

References 28 publications

Efficient numerical methods for the random-field Ising model: Finite-size scaling, reweighting extrapolation, and computation of response functions

Efficient numerical methods for the random-field Ising model: Finite-size scaling, reweighting extrapolation, and computation of response functions

First order transition in two dimensional coulomb glass

Spontaneous versus explicit replica symmetry breaking in the theory of disordered systems

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