92 Years of the Ising Model: A High Resolution Monte Carlo Study

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

doi:10.1088/1742-6596/1012/1/012002

Cited by 10 publications

(12 citation statements)

References 33 publications

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“…These phases are separated by the usual 3d Ising transition; on the cubic lattice this at β = β c ≈ 0.221 (see e.g. [44] for a recent high-precision determination of this critical coupling). The Ising CFT is strongly coupled with no obvious small parameters; however a great deal is known about it, and the current most precise determination of the critical exponents arises from the conformal bootstrap [45] (see [46] for a review).…”

Section: Lattice Construction 21 Usual 3d Ising Modelmentioning

confidence: 97%

Toward a 3d Ising model with a weakly-coupled string theory dual

Iqbal

McGreevy

2020

SciPost Phys.

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It has long been expected that the 3d Ising model can be thought of as a string theory, where one interprets the domain walls that separate up spins from down spins as two-dimensional string worldsheets. The usual Ising Hamiltonian measures the area of these domain walls. This theory has string coupling of unit magnitude. We add new local terms to the Ising Hamiltonian that further weight each spin configuration by a factor depending on the genus of the corresponding domain wall, resulting in a new 3d Ising model that has a tunable bare string coupling g_sgs. We use a combination of analytical and numerical methods to analyze the phase structure of this model as g_sgs is varied. We study statistical properties of the topology of worldsheets and discuss the prospects of using this new deformation at weak string coupling to find a worldsheet description of the 3d Ising transition.

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Section: Lattice Construction 21 Usual 3d Ising Modelmentioning

confidence: 97%

Toward a 3d Ising model with a weakly-coupled string theory dual

Iqbal

McGreevy

2020

SciPost Phys.

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“…For the susceptibility, we include the leading confluent correction in our discussion by extending the scaling ansatz Equation (7) by an irrelevant scaling variable u

χ_{ℓ} false(T false) ≃ χ_{0} ε^{- γ} Z false(ℓ / ξ, u ℓ^{- ω} false), ℓ, ξ ≫ σ,

with the 3D‐Ising correction exponent [ 50,55,56,61 ]

ω \approx 0.83

; here,

y = ℓ / ξ

means

y = false(ℓ / ξ_{0} false) ε^{ν}

. For finite systems, the function

χ_{ℓ}

is analytic in T , but also in u ; in particular, the scaling function

scriptZ

is analytic in its second argument and obeys

Z (y, 0) = Z (y)

.…”

Section: Scaling Analysis Of Sub‐system Fluctuationsmentioning

confidence: 99%

“…[50,54] These variables encode microscopic details of the system that fade out upon coarse-graining by the RG flow; yet, the confluent corrections are associated with universal critical exponents. This type of corrections was analysed in simulation studies of, for example, the 3D Ising model, [55,56] the 3D Heisenberg model, [57] the statistics of percolation clusters, [58,59] and critical transport on such structures. [59,60] For the susceptibility, we include the leading confluent correction in our discussion by extending the scaling ansatz Equation (7) by an irrelevant scaling variable u…”

Section: Confluent Corrections To Scalingmentioning

confidence: 99%

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Continuous Demixing Transition of Binary Liquids: Finite‐Size Scaling from the Analysis of Sub‐Systems

Pathania

Chakraborty

Höfling

2021

Advcd Theory and Sims

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A binary liquid near its consolute point exhibits critical fluctuations of local composition and a diverging correlation length. The method of choice to calculate critical points in the phase diagram is a finite‐size scaling analysis, based on a sequence of simulations with widely different system sizes. Modern, massively parallel hardware facilitates that instead cubic sub‐systems of one large simulation are used. Here, this alternative is applied to a symmetric binary liquid at critical composition and different routes to the critical temperature are compared: 1) fitting critical divergences of the composition structure factor, 2) scaling of fluctuations in sub‐volumes, and 3) applying the cumulant intersection criterion to sub‐systems. For the last route, two difficulties arise: sub‐volumes are open systems, for which no precise estimate of the critical Binder cumulant Uc is available. Second, the boundaries of the simulation box interfere with the sub‐volumes, which is resolved here by a two‐parameter finite‐size scaling. The implied modification to the data analysis restores the common intersection point, yielding Uc=0.201±0.001, universal for cubic Ising‐like systems with free boundaries. Confluent corrections to scaling, which arise for small sub‐system sizes, are quantified and the data are compatible with the universal correction exponent ω≈0.83.

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“…Pα3 Pα4 0.4407 0 1.4134(3) 0.5135 (3) 1.7963(5) 1.4146 (7) 0.5134 (7) 1.799 (2) Ising model. This model does not have an analytical solution, but is known to experience a continuous transition at K (0) nn = 0.22165... [9]. To compute the CMTS at this nearest neighbor critical point, we use n = 3, L = 64, and the b = 2 marjority rule with a random pick on tie.…”

Section: B 3d Istropic Ising Modelmentioning

confidence: 99%

Determination of the critical manifold tangent space and curvature with Monte Carlo renormalization group

Car

2019

Phys. Rev. E

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We show that the critical manifold of a statistical mechanical system around a critical point is locally accessible through correlation functions at that point. A practical numerical method is presented to determine the tangent space and the curvature to the critical manifold with Variational Monte Carlo Renormalization Group. Because of the use of a variational bias potential of the coarsegrained variables, critical slowing down is greatly alleviated in the Monte Carlo simulation. In addition, this method is free of truncation error. We study the isotropic and anisotropic Ising model on a two-dimensional square lattice, and the isotropic Ising model on a cubic lattice to illustrate the method.

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92 Years of the Ising Model: A High Resolution Monte Carlo Study

Cited by 10 publications

References 33 publications

Toward a 3d Ising model with a weakly-coupled string theory dual

Toward a 3d Ising model with a weakly-coupled string theory dual

Continuous Demixing Transition of Binary Liquids: Finite‐Size Scaling from the Analysis of Sub‐Systems

Determination of the critical manifold tangent space and curvature with Monte Carlo renormalization group

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