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arXiv:hep-lat/0001010

Large-q expansion for the second moment correlation length in the two-dimensional q-state Potts model

H. Arisue

Published Jan 12, 2000• 0 citations
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Abstract

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 $1/\sqrt{q}$. 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.

DOI
10.48550/arxiv.hep-lat/0001010

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We have studied spin-spin correlation functions in the ordered phase of the two-dimensional q -state Potts model with q = 10, 15, and 20 at the first-order transition point β t . Through extensive Monte Carlo simulations we obtain strong numerical evidence that the correlation length in the ordered phase agrees with the exactly known and recently numerically confirmed correlation length in the disordered phase: ξ o (β t ) = ξ d (β t ). As a by-product we find the energy moments in the ordered phase at β t in very good agreement with a recent large q -expansion.
We have calculated the large-q expansion for the energy cumulants at the phase transition point in the two-dimensional q-state Potts model to the 23rd order in 1/q1/\sqrt{q}1/q​ using the finite lattice method. The obtained series allow us to give very precise estimates of the cumulants for q>4q>4q>4 on the first order transition point. The result confirm us the correctness of the conjecture by Bhattacharya et al. on the asymptotic behavior of the cumulants for q→4+q \to 4_+q→4+​.
The surface and bulk properties of the two-dimensional Q>4Q>4Q>4 state Potts model in the vicinity of the first-order bulk transition point have been studied by exact calculations and by density-matrix renormalization-group techniques. For the surface transition the complete analytical solution of the problem is presented in the $\stackrel{\ensuremath{\rightarrow}}{Q}\ensuremath{\infty}$ limit, including the critical and tricritical exponents, magnetization profiles, and scaling functions. We have shown in an accurate numerical study that the universality class of the surface transition is independent of the value of Q>4.Q>4.Q>4. For the bulk transition we have numerically calculated the latent heat and the magnetization discontinuity and we have shown that the correlation lengths in the ordered and disordered phases are identical at the transition point.
We performed Monte Carlo simulations of the two-dimensional q-state Potts model with q=10, 15, and 20 to study the energy and magnetization cumulants in the ordered and disordered phase at the first-order transition point βtβ_tβt​. By using very large systems of size 300 x 300, 120 x 120, and 80 x 80 for q=10, 15, and 20, respectively, our numerical estimates provide practically (up to unavoidable, but very small statistical errors) exact results which can serve as a useful test of recent resummed large-q expansions for the energy cumulants by Bhattacharya `et al.' [J. Phys. I (France) 7 (1997) 81]. Up to the third order cumulant and down to q=10 we obtain very good agreement, and also the higher-order estimates are found to be compatible.
We performed Monte Carlo simulations of the two-dimensional q-state Potts model with q=10, 15, and 20 to study the energy and magnetization cumulants in the ordered and disordered phase at the first-order transition point betat\\beta_tbetat​. By using very large systems of size 300 x 300, 120 x 120, and 80 x 80 for q=10, 15, and 20, respectively, our numerical estimates provide practically (up to unavoidable, but very small statistical errors) exact results which can serve as a useful test of recent resummed large-q expansions for the energy cumulants by Bhattacharya `et al.' [J. Phys. I (France) 7 (1997) 81]. Up to the third order cumulant and down to q=10 we obtain very good agreement, and also the higher-order estimates are found to be compatible.