High-temperature expansion study of the Nishimori multicritical point in two and four dimensions

Singh, Rajiv R. P.; Adler, Joan

doi:10.1103/physrevb.54.364

Cited by 54 publications

(37 citation statements)

References 29 publications

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“…This coincides with numerical estimates of p at the multicritical point with high precision: 0.8905(5), 1) 0.886(3), 2) 0.8872(8), 3) 0.8906 (2) 4) and 0.8907(2).…”

supporting

confidence: 61%

“…[1][2][3][4][5][6][7][8][9][10][11] In this short note we develop a duality argument to predict the location of the multicritical point and the shape of the phase boundary in models of spin glasses on the square lattice.The first system we treat is a random Z q model with gauge symmetry which includes the AEJ Ising model and the Potts gauge glass. Following the notation of Wu and Wang, 12) the partition function is…”

mentioning

confidence: 99%

“…The prime in the sum indicates that the bond variables ij and ij are constrained such that their sums over a closed loop vanish. 12) Thus the duality transformation consists in replacing the exponential in (2) with that in (3):…”

mentioning

confidence: 99%

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Duality and Multicritical Point of Two-Dimensional Spin Glasses

2002

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Determination of the precise location of the multicritical point and phase boundary is a target of active current research in the theory of spin glasses. [1][2][3][4][5][6][7][8][9][10][11] In this short note we develop a duality argument to predict the location of the multicritical point and the shape of the phase boundary in models of spin glasses on the square lattice.The first system we treat is a random Z q model with gauge symmetry which includes the AEJ Ising model and the Potts gauge glass. Following the notation of Wu and Wang, 12) the partition function isis the quenched randomness, and VðÁÞ is a periodic function with period q. The sum in the exponent runs over neighbouring sites on the square lattice. It is straightforward to generalize the duality transformation in the Wu-Wang formalism 12) to the random model, and the result iswhere ij denotes i À j and ij is the bond variable on the dual lattice corresponding to ij and eṼ V is the Fourier transform of e V . The prime in the sum indicates that the bond variables ij and ij are constrained such that their sums over a closed loop vanish.12) Thus the duality transformation consists in replacing the exponential in (2) with that in (3):Average over quenched randomness is dealt with by the replica method. A generalization of the existing argument 13) suggests that it is convenient to consider the following quantity

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“…This coincides with numerical estimates of p at the multicritical point with high precision: 0.8905(5), 1) 0.886(3), 2) 0.8872(8), 3) 0.8906 (2) 4) and 0.8907(2).…”

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confidence: 61%

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confidence: 99%

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Duality and Multicritical Point of Two-Dimensional Spin Glasses

2002

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“…It is interesting to note that the correlation length exponent ν at the zero-temperature phase transition from the paramagnetic phase to the ferromagnetic phase in two dimension was found 219,31 to be very close to (or perhaps exactly equal to) the one for two-dimensional percolation. This fact, together with the geometric mechanism of the transition, makes it very attractive to speculate that the transition is due to a percolation of something that is defined geometrically.…”

Section: The Results Showed That [(∆E)mentioning

confidence: 99%

Recent Progress in Spin Glasses

Kawashima

Rieger

2013

Frustrated Spin Systems

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We review recent findings on spin glass models. Both the equilibrium properties and the dynamic properties are covered. We focus on progress in theoretical, in particular numerical, studies, while its relationship to real magnetic materials is also mentioned.The motivation that pulls researchers toward spin glasses is possibly not the potential future use of spin glass materials in "practical" applications. It is rather the expectation that there must be something very fundamental in systems with randomness and frustration. It seems that this expectation has been acquiring a firmer ground as the difficulty of the problem is appreciated more clearly. For example, a close relationship has been established between spin glass problems and a class of optimization problems known to be N P -hard. It is now widely accepted that a good (i.e. polynomial) computational algorithm for solving any one of N P -hard problems most probably does not exist. Therefore, it is quite natural to expect that the Ising spin glass problem, as one of the problem in the class, would be very hard to solve, which indeed turns out to be the case in a vast number of numerical studies. None the less, it is often the case that only numerical studies can decide whether a certain hypothetical picture applies to a given instance. Consequently, the major part of what we present below is inevitably a de-

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“…16,17,18,19,20) It was suggested by the ǫ-expansion method and a symmetry argument 17,18) that the multicritical point must be located on the line. This was confirmed by MC simulations 15,21) and high-temperature series expansions.…”

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confidence: 99%

Random Fixed Point of Three-Dimensional Random-Bond Ising Models

Hukushima

2000

J. Phys. Soc. Jpn.

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The fixed-point structure of three-dimensional bond-disordered Ising models is investigated using the numerical domain-wall renormalization-group method. It is found that, in the ±J Ising model, there exists a non-trivial fixed point along the phase boundary between the paramagnetic and ferromagnetic phases. The fixed-point Hamiltonian of the ±J model numerically coincides with that of the unfrustrated random Ising models, strongly suggesting that both belong to the same universality class. Another fixed point corresponding to the multicritical point is also found in the ±J model. Critical properties associated with the fixed point are qualitatively consistent with theoretical predictions.KEYWORDS: domain wall renormalization group, fixed point, random system, spin glass, MC simulationThe influence of quenched disorder on model systems has attracted considerable interest in the field of statistical physics. The first remarkable criterion was given by Harris, 1) who claimed that if the specific-heat exponent α pure of the pure system is positive, disorder becomes relevant, implying that a new random fixed point governs critical phenomena of the random system. One of the simplest models belonging to such a class is the three-dimensional (3D) Ising model. Critical exponents associated with the random fixed point have been investigated for dilution-type disorder by experimental, 2, 3, 4) theoretical 5, 6) and numerical approaches. 7,8) Recently, extensive Monte Carlo (MC) studies 8) have clarified that the critical exponents of the 3D site-diluted Ising model are independent of the concentration of the site dilution p, suggesting the existence of a random fixed point. These results were obtained by carefully taking into account correction for finite-size scaling, unless the exponents clearly depended on p.8) Experimentally, the critical exponents of randomly diluted antiferromagnetic Ising compounds, Fe x Zn 1−x F 2 3, 4) and Mnare distinct from those of the pure 3D Ising model. While the existence of the random fixed point has been established for the 3D site-diluted Ising model, the idea of the universality class for random systems, namely classification by fixed points, has not been explored yet as compared with various pure systems. In particular, the question as to whether the random fixed point is universal irrespective of the type of disorder or not is a non-trivial problem. In the present work, we study critical phenomena associated with the ferromagnetic phase transition in 3D site-and bond-diluted and ±J Ising models. The main purpose is to determine the fixed-point structure of 3D random-bond Ising models by making use of a numerical renormalization-group (RG) analysis. Our strategy is based on the domain-wall RG (DWRG) method proposed by McMillan.9, 10) This method has been applied to a 2D frustrated randombond Ising model 9, 10) where there is no random fixed point, and recently to a 2D ± J frustrated random-bond three-state Potts model 11) which displays a non-trivial random fixed point. In this paper,...

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High-temperature expansion study of the Nishimori multicritical point in two and four dimensions

Cited by 54 publications

References 29 publications

Duality and Multicritical Point of Two-Dimensional Spin Glasses

Duality and Multicritical Point of Two-Dimensional Spin Glasses

Recent Progress in Spin Glasses

Random Fixed Point of Three-Dimensional Random-Bond Ising Models

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