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AIP conference proceedings

Correlation length of the simple cubic layer Potts model

Shingo Ohya

Published Jan 1, 2001• 0 citations
Publisher

Abstract

The correlation length of the simple cubic semi-infinite lattice (l×m×∞) Potts model is numerically studied by using the eigenvalues of the transfer matrix. From the intersection of a correlation length curve family, the critical coupling parameter Kc=J/kTc for the simple cubic layer lattice (l×∞×∞) is calculated.
DOI
10.1063/1.1386830

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An energy curve family for a semi-infinite square lattice (l × ∞) of the q-state Potts model with periodic boundary conditions has an l-independent common fixed point, which gives the exact critical values of the coupling parameter K = J/kJ and the internal energy E(K), as a natural consequence of the self-dual property of the model. An energy curve intersection procedure (ECIP) is proposed to find the critical properties of the Potts model through numerical investigation of the transfer matrix. The ECIP shows that the lattices (l × ∞) for two different small values of l can reproduce the transition temperature and the critical internal energy of the bulk lattice (∞ × ∞) without use of the duality argument. As an applicable example of the ECIP, a semi-infinite simple cubic lattice (l × m × ∞) of the Potts model is preliminarily studied to obtain approximate critical quantities of the layer lattice (l × ∞ × ∞). A brief comment on the specific heat exponent μ estimated using the ECIP is added.
The internal energy density of the Potts model on a semi-infinite strip with a width L has been conjectured to have no finite-size corrections at the critical point K = K c . By factorizing the transfer matrix for the kagome lattice with larger widths, we have found that this conjecture is not correct in that the internal energy density varies slightly with L at the critical point. From this size dependence of the internal energy density, we obtain an upper bound as K c < 1.0565615, which is close to a recent estimate K JS = 1.0565600(7) by Jacobsen and Scullard [J. Phys. A 45, 494003 (2012)]. We also obtain a lower bound as K c > 1.0560 by calculating the correlation length along the strips.
Using the transfer matrices constructed by Kramers and Wannier's successive screw method for the 2-d (l × 蜴) and 3-d (l ×m ×蜴) q state Potts model (2 蠆q 蠆7), we examine the extent to which the presently available supercomputers can provide us with finite size effects and pseudocritical properties of the systems. We estimate the bulk coupling parameters Kc from the specific heat anomaly in two and three dimensions and also from the correlation length anomaly in two dimensions. In two dimensions, our results Kc are in good agreement with those given by the Potts duality formula. In three dimensions, however, we need extended precision computations to obtain the numerical results with sufficient accuracy. We regard the behavior of the exponent µ, which characterizes the relation between the specific heat peak Cv0(l) and the lattice size l, as an indicator to discriminate the order of the phase transition by investigating Cv0(l) for several small values of l.
For paper II, see ibid, vol. 27, p.1503 (1994). The finite-lattice method of series expansion has been used to extend low-temperature series for the partition function, order parameter and susceptibility of the 3-state Potts model on the simple cubic lattice to order z 43 and the high-temperature expansion of the partition function to order v 21 . We use the numerical data to show that the transition is first-order, and estimate the latent heat, the discontinuity in the magnetization, and a number of other critical parameters.
Within a real space renormalisation group framework, the authors discuss the criticality of the full anisotropic (arbitrary J x , J y and J z )q-state Potts ferromagnet in the simple cubic lattice. Several previously known exact results for the d=1 and d=2 particular cases are recovered. Furthermore they obtain: (i) the q-dependence of the d=3 correlation length critical exponent nu 3 (in particular, if q to 0, nu 3 (q) approximately nu 3 (0)+ nu 3 '(0)q where the present approximate values are nu 3 (0) approximately=1.105 and nu 3 '(0) approximately=-0.66); (ii) the q-dependence of the d=2 to or from d=3 crossover critical exponent phi 23 (in particular, phi 23 varies as 1/ square root q if q to 0); (iii) through a convenient numerical extrapolation, a quite accurate proposal for the critical temperatures corresponding to arbitrary ratios, J y /J x and J z /J x and values of q.
We calculate the large-q expansion of the second moment correlation length at the first order phase transition point of the q-state Potts model in two dimensions both in the ordered and disordered phases to order 21 in 1/q1/\sqrt{q}1/q​. They coincide with each other to the third term of the series but differ a little in higher orders. Numerically the ratio of the second moment correlation length in the two phases is not far from unity in all region of q>4. The ratio of the second moment correlation length to the standard correlation length in the disordered phase is far from unity, which suggests that the second largest and smaller eigenvalues of the transfer matrix form a continuum spectrum not only in the large-q region but also in all the region of q>4.