Critical Behavior of the Two-Dimensional Ising Susceptibility

Orrick, William P.; Nickel, B. G.; Guttmann, A J; Perk, Jacques H. H.

doi:10.1103/physrevlett.86.4120

Cited by 50 publications

(54 citation statements)

References 19 publications

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“…Actually the latter is "Painlevé like" since its series expansion can be obtained from ¶ For the leading amplitude,χ (2) andχ (4) give 1/12π + I − 4 ≃ 1.0009593 · · · /12π which is very close to 1.0009603 · · · /12π for the fullχ [6]. a program of polynomial growth which uses exclusively a quadratic finite difference double recursion generalizing the Painlevé equations [15,16]. The difficulty to link holonomic versus non-linear descriptions of physical problems is typically the kind of problems one faces with the Feynman diagram approach of particle physics, but the susceptibility of the Ising model is, obviously, the simplest non trivial example to address such an important issue.…”

Section: Singular Behavior Ofχ (4)mentioning

confidence: 97%

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Square lattice Ising model susceptibility: connection matrices and singular behaviour of χ⁽³⁾and χ⁽⁴⁾

Zenine¹,

Boukraa²,

Hassani³

et al. 2005

J. Phys. A: Math. Gen.

197

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Abstract. We present a simple, but efficient, way to calculate connection matrices between sets of independent local solutions, defined at two neighboring singular points, of Fuchsian differential equations of quite large orders, such as those found for the third and fourth contribution (χ (3) and χ (4) ) to the magnetic susceptibility of the square lattice Ising model. We deduce all the critical behavior of the solutions χ (3) and χ (4) , as well as the asymptotic behavior of the coefficients in the corresponding series expansions. We confirm that the newly found quadratic singularities of the Fuchsian ODE associated with χ (3) are not singularities of the particular solution χ (3) itself. We use the previous connection matrices to get the exact expressions of all the monodromy matrices of the Fuchsian differential equation for χ (3) (and χ (4) ) expressed in the same basis of solutions. These monodromy matrices are the generators of the differential Galois group of the Fuchsian differential equations for χ (3) (and χ (4) ), whose analysis is just sketched here. As far as the physics implications of the solutions are concerned, we find challenging qualitative differences when comparing the corrections to scaling for the full susceptibillity χ at high temperature (resp. low temperature) and the first two terms χ (1) and χ (3) (resp. χ (2) and χ (4) ) .

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Section: Singular Behavior Ofχ (4)mentioning

confidence: 97%

“…The results agree with previous results of B. Nickel, but the correction terms are new ‡, in particular the term 3 ln(2)/32/π 2 in (26). In terms of the τ = (1/s − s)/2 variable introduced in [6,15,16], the singular part (26) reads:χ…”

Section: Singular Behavior Ofχmentioning

confidence: 99%

Square lattice Ising model susceptibility: connection matrices and singular behaviour of χ⁽³⁾and χ⁽⁴⁾

Zenine¹,

Boukraa²,

Hassani³

et al. 2005

J. Phys. A: Math. Gen.

197

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“…and also the susceptibility χ is basically known to arbitrary precision from very long series expansions, e.g., Orrick et al [14] give the formula …”

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confidence: 99%

Monte Carlo study of the evaporation/condensation transition of Ising droplets

Nussbaumer¹,

Bittner²,

Neuhaus³

et al. 2006

Europhys. Lett.

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Abstract. -In recent analytical work, Biskup et al. (Europhys. Lett., 60 (2002) 21) studied the behaviour of d-dimensional finite-volume liquid-vapour systems at a fixed excess δN of particles above the ambient gas density. By identifying a dimensionless parameter Δ(δN ) and a universal constant Δc(d), they showed in the limit of large system sizes that for Δ < Δc the excess is absorbed in the background ("evaporated" system), while for Δ > Δc a droplet of the dense phase occurs ("condensed" system). Also the fraction λΔ of excess particles forming the droplet is given explicitly. Furthermore, they argue that the same is true for solid-gas systems. By making use of the well-known equivalence of the lattice-gas picture with the spin-(1/2) Ising model, we performed Monte Carlo simulations of the Ising model with nearest-neighbour couplings on a square lattice with periodic boundary conditions at fixed magnetisation, corresponding to a fixed particles excess. To test the applicability of the analytical results to much smaller, practically accessible system sizes, we measured the largest minority droplet, corresponding to the solid phase, at various system sizes (L = 40, . . . , 640). Using analytic values for the spontaneous magnetisation m0, the susceptibility χ and the Wulff interfacial free energy density τW for the infinite system, we were able to determine λΔ numerically in very good agreement with the theoretical prediction.Introduction. -The formation and dissolution of equilibrium droplets at a first-order phase transition is one of the longstanding problems in statistical mechanics [1]. Quantities of particular interest are the size and free energy of a "critical droplet" that needs to be formed before the decay of the metastable state via homogeneous nucleation can start. For large but finite systems, this is signalised by a cusp in the probability density of the order parameter φ towards the phase-coexistence region as depicted in figs. 1 and 2 for the example of the twodimensional (2D) Ising model, where φ = m is the magnetisation. This "transition point" separates an "evaporated" phase with many very small bubbles of the "wrong" phase around the peak at φ 0 from the "condensed phase" phase, in which a large droplet has formed; for configuration snapshots see fig. 3. The droplet eventually grows further towards φ = 0 until it

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“…It turns out that there exist only integer corrections to scaling (for a recent and detailed discussions we refer readers to Refs. [11,12,13]). Values of the critical exponents and of some amplitude ratios are presented in Table 1.…”

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confidence: 99%

Universal ratios of critical amplitudes in the Potts model universality class

Berche¹,

Butera

Janke

et al. 2009

Computer Physics Communications

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Monte Carlo (MC) simulations and series expansions (SE) data for the energy, specific heat, magnetization, and susceptibility of the three-state and four-state Potts model and Baxter-Wu model on the square lattice are analyzed in the vicinity of the critical point in order to estimate universal combinations of critical amplitudes. We also form effective ratios of the observables close to the critical point and analyze how they approach the universal critical-amplitude ratios. In particular, using the duality relation, we show analytically that for the Potts model with a number of states q ≤ 4, the effective ratio of the energy critical amplitudes always approaches unity linearly with respect to the reduced temperature. This fact leads to the prediction of relations among the amplitudes of correction-to-scaling terms of the specific heat in the low-and high-temperature phases. It is a common belief that the four-state Potts and the Baxter-Wu model belong to the same universality class. At the same time, the critical behavior of the four-state Potts model is modified by logarithmic corrections while that of the Baxter-Wu model is not. Numerical analysis shows that critical amplitude ratios are very close for both models and, therefore, gives support to the hypothesis that the critical behavior of both systems is described by the same renormalization group fixed point.

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Critical Behavior of the Two-Dimensional Ising Susceptibility

Cited by 50 publications

References 19 publications

Square lattice Ising model susceptibility: connection matrices and singular behaviour of χ⁽³⁾and χ⁽⁴⁾

Square lattice Ising model susceptibility: connection matrices and singular behaviour of χ⁽³⁾and χ⁽⁴⁾

Monte Carlo study of the evaporation/condensation transition of Ising droplets

Universal ratios of critical amplitudes in the Potts model universality class

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Critical Behavior of the Two-Dimensional Ising Susceptibility

Cited by 50 publications

References 19 publications

Square lattice Ising model susceptibility: connection matrices and singular behaviour of χ(3)and χ(4)

Square lattice Ising model susceptibility: connection matrices and singular behaviour of χ(3)and χ(4)

Monte Carlo study of the evaporation/condensation transition of Ising droplets

Universal ratios of critical amplitudes in the Potts model universality class

Contact Info

Product

Resources

About

Square lattice Ising model susceptibility: connection matrices and singular behaviour of χ⁽³⁾and χ⁽⁴⁾

Square lattice Ising model susceptibility: connection matrices and singular behaviour of χ⁽³⁾and χ⁽⁴⁾