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arXiv:hep-lat/9609047 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Monte Carlo study of cluster-diameter distribution: An observable to estimate correlation lengths

Wolfhard Janke E, Stefan Kappler

Published Aug 1, 1997• 2 citations
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Abstract

We report numerical simulations of two-dimensional $q$-state Potts models with emphasis on a new quantity for the computation of spatial correlation lengths. This quantity is the cluster-diameter distribution function ${G}^{\mathrm{diam}}(x)$, which measures the distribution of the diameter of stochastically defined cluster. Theoretically it is predicted to fall off exponentially for large diameter $x$, ${G}^{\mathrm{diam}}\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}x/\ensuremath{\xi})$, where $\ensuremath{\xi}$ is the correlation length as usually defined through the large-distance behavior of two-point correlation functions. The results of our extensive Monte Carlo study in the disordered phase of the models with $q=10$, 15, and $20$ on large square lattices of size $300\ifmmode\times\else\texttimes\fi{}300$, $120\ifmmode\times\else\texttimes\fi{}120$, and $80\ifmmode\times\else\texttimes\fi{}80$, respectively, clearly confirm the theoretically predicted behavior. Moreover, using this observable we are able to verify an exact formula for the correlation length ${\ensuremath{\xi}}_{d}({\ensuremath{\beta}}_{t})$ in the disordered phase at the first-order transition point ${\ensuremath{\beta}}_{t}$ with an accuracy of about $1%$--$2%$ for all considered values of $q$. This is a considerable improvement over estimates derived from the large-distance behavior of standard (projected) two-point correlation functions, which are also discussed for comparison.

DOI
10.1103/physreve.56.1414

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We report numerical simulations of two-dimensional qqq-state Potts models with emphasis on a new quantity for the computation of spatial correlation lengths. This quantity is the cluster-diameter distribution function Gdiam(x){G}^{\mathrm{diam}}(x)Gdiam(x), which measures the distribution of the diameter of stochastically defined cluster. Theoretically it is predicted to fall off exponentially for large diameter xxx, Gdiam\ensuremath∝exp(\ensuremath−x/\ensuremathξ){G}^{\mathrm{diam}}\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}x/\ensuremath{\xi})Gdiam\ensuremath∝exp(\ensuremath−x/\ensuremathξ), where \ensuremathξ\ensuremath{\xi}\ensuremathξ is the correlation length as usually defined through the large-distance behavior of two-point correlation functions. The results of our extensive Monte Carlo study in the disordered phase of the models with q=10q=10q=10, 15, and 202020 on large square lattices of size 300\ifmmode×\else\texttimes\fi300300\ifmmode\times\else\texttimes\fi{}300300\ifmmode×\else\texttimes\fi300, 120\ifmmode×\else\texttimes\fi120120\ifmmode\times\else\texttimes\fi{}120120\ifmmode×\else\texttimes\fi120, and 80\ifmmode×\else\texttimes\fi8080\ifmmode\times\else\texttimes\fi{}8080\ifmmode×\else\texttimes\fi80, respectively, clearly confirm the theoretically predicted behavior. Moreover, using this observable we are able to verify an exact formula for the correlation length \ensuremathξd(\ensuremathβt){\ensuremath{\xi}}_{d}({\ensuremath{\beta}}_{t})\ensuremathξd​(\ensuremathβt​) in the disordered phase at the first-order transition point \ensuremathβt{\ensuremath{\beta}}_{t}\ensuremathβt​ with an accuracy of about 11%1--22%2 for all considered values of qqq. This is a considerable improvement over estimates derived from the large-distance behavior of standard (projected) two-point correlation functions, which are also discussed for comparison.
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 present results of Monte Carlo simulations of random bond Potts models in two dimensions, for different numbers of Potts states q. We introduce a simple scheme which yields continuous self-dual distributions of the interactions. As expected, we find multifractal behavior of the correlation functions at the critical point and obtain estimates of the exponent ${\ensuremath{\eta}}_{n}$ for several moments n of the correlation functions, including typical $(\stackrel{\ensuremath{\rightarrow}}{n}0)$, average (n=1)(n=1)(n=1), and others. In addition, for q=8q=8q=8, we find that there is only a single correlation length exponent $\ensuremath{\nu}$ describing the correlation length away from criticality. This is numerically very close to the pure Ising value $\ensuremath{\nu}=1$.
We have calculated the large-qqq series of the energy cumulants, the magnetization cumulants and the correlation length at the first order phase transition point both in the ordered and disordered phases for the qqq-state Potts model in two dimensions. The series enables us to estimate the numerical values of the quantities more precisely by a factor of 102−10410^2 - 10^4102−104 than the Monte Carlo simulations. From the large-qqq series of the eigenvalues of the transfer matrix, we also find that the excited states form a continuum spectrum and there is no particle state at the first order phase 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.
A new kind of finite-size phenomenon in the five-state Potts model in two dimensions, a ``false'' ground state, was recently reported by Katznelson and Lauwers based on Monte Carlo studies of large lattices. We reexamine the behavior of this weakly first-order system by performing extensive Monte Carlo simulations, and demonstrate that the ``novel'' states previously observed in internal-energy-distribution histograms tend to disappear when runs of sufficient length are used. We present evidence that very strong critical slowing down and a pseudodivergent correlation length are responsible for the observed effects. Our additional analysis of finite-size effects in the ten-state Potts model on small lattices (L\ensuremath{\le}10) shows a crossover of scaling from ${L}^{\mathrm{\ensuremath{-}}d}$ behavior (observed earlier on larger lattices) to a nonanalytic form. This result is used to discuss the observed nonanalytic scaling of thermodynamic quantities in the five-state Potts model.
Motivated by recent experiments on phase behavior of systems confined in porous media, we have studied the effect of randomness on the nature of the phase transition in the two-dimensional Potts model. To model the effects of the porous matrix we introduce a random distribution of couplings P(J(ij))=pdelta(J(ij)-J1)+(1-p)delta(J(ij)-J2) in the q state Potts Hamiltonian. An extensive Monte Carlo study is made on this system for q=5. We studied two different cases of disorder (a) J(1)/J(2)-->infinity and p=0.8 and (b) J(1)/J(2)=10 and p=0.5. We observed, in both cases, that the weak first order transition that appears in the pure case, changes to a second-order transition. A finite size scaling analysis shows that the correlation length exponent nu is close to 1 and the best fit to the dependence of the specific heat on system size is logarithmic. This suggests that both cases belong to the universality class of the Ising model. In contrast, the magnetic exponents point to a different universality class.