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International Journal of Modern Physics C

CRITICAL SURFACE EXPONENTS IN 2-D 10-STATE POTTS MODEL

Leo Kärkkäinen, Kari Rummukainen

Published Oct 1, 1992• 0 citations
Publisher

Abstract

At low temperature, the Potts models can have interfaces between domains ordered to different directions. We use the difference of periodic and twisted boundary conditions to study planar order-order interfaces. We show that the interface tension, energy and width display critical properties as one approaches the critical point. We measure the critical exponents connected to their power like singular behaviour at the critical point.

DOI
10.1142/s0129183192000750

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Boundary critical behaviour of two-dimensional random Potts \n\tmodels

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\n \nUsing extensive Monte Carlo simulations, transfer matrix techniques \nand\nconformal invariance, ferromagnetic random\n$q-$state Potts models for $3\\le q\\le 8$ are studied in the vicinity \nof the \ncritical temperature.\nIn particular the surface and bulk magnetization exponents $\\beta_1$ \nand \nβ are found\nmonotonically increasing with q. At the critical temperature, \ndifferent \nmoments (n)\nof the magnetization profiles are calculated which are all found \nto accurately follow predictions of \nconformal invariance. The critical correlation functions \nshow multifractal behaviour, the decay exponents of the different \nmoments\nboth in the volume and at the surface, are $n-$dependent.\n \n\n

•Oct 19, 2012•4 citations
At low temperature, the Potts models can have interfaces between domains ordered to different directions. We use the difference of periodic and twisted boundary conditions to study planar order-order interfaces. We show that the interface tension, energy and width display critical properties as one approaches the critical point. We measure the critical exponents connected to their power like singular behaviour at the critical point.
A convenient way to study phase transitions is by considering finite spin systems of linear size LLL with boundary conditions that produce a system-size interface. In this paper, we study the statistical properties of such an interface in a disordered Potts ferromagnet in dimension d=2d=2d=2 within Migdal--Kadanoff real-space renormalization. We first focus on the interface free-energy and energy to measure the singularities of the average and random contributions, as well as the corresponding histograms, both in the low-temperature phase and at criticality. We then consider the critical behavior of the interfacial adsorption of nonboundary states. Our main conclusion is that all singularities involve the correlation length ${\ensuremath{\xi}}_{\mathit{av}}(T)\ensuremath{\sim}{({T}_{c}\ensuremath{-}T)}^{\ensuremath{-}\ensuremath{\nu}}$ appearing in the average free-energy $\overline{F}\ensuremath{\sim}{[L∕{\ensuremath{\xi}}_{\mathit{av}}(T)]}^{{d}_{s}}$ of the interface of dimension ${d}_{s}=d\ensuremath{-}1$, except for the free-energy width $\ensuremath{\Delta}F\ensuremath{\sim}{[L∕{\ensuremath{\xi}}_{\mathit{var}}(T)]}^{\ensuremath{\theta}}$ that involves the droplet exponent $\ensuremath{\theta}$ and another correlation length ${\ensuremath{\xi}}_{\mathit{var}}(T)$ that diverges more rapidly than ${\ensuremath{\xi}}_{\mathit{av}}(T)$. We compare with the spin-glass transition in d=3d=3d=3, where ${\ensuremath{\xi}}_{\mathit{var}}(T)$ is the ``true'' correlation length and where the interface energy presents unconventional scaling with a chaos critical exponent ${\ensuremath{\zeta}}_{c}>1∕\ensuremath{\nu}$ [M. Nifle and H. J. Hilhorst, Phys. Rev. Lett. 68, 2992 (1992)]. The common feature is that in both cases, the characteristic length scale Lch(T){L}_{\mathit{ch}}(T)Lch​(T) associated with the chaotic nature of the low-temperature phase diverges more slowly than the correlation length.
A review is given of recent work on the ordinary surface critical behaviour of systems in two dimensions. Several models of interest in statistical mechanics are considered: Potts model, percolation, Ising clusters, ZN-model, O(n) model and polymers. Numerical results for surface exponents, obtained by suitable finite size scaling extrapolations, are discussed in the light of recent advances based on the conformal invariance approach. Surface exponents are often seen as important tests of conformal invariance predictions. In other cases these exponents provide important information for a location of the problem within the classification schemes offered by the conformal approach, and a determination of its universality class. A relevant example of the first aspect is the study of the q-state Potts model with q near 4, for which an analytical study of logarithmic scaling corrections is needed to achieve a successful test. The latter point of view applies, e.g., to the more controversial cases of polymers at the theta point and critical Ising clusters. Emphasis is put on the importance of an integrated study of both bulk and surface properties. Relevant issues, like the possible existence of analytical expressions for the indices in particular model families, or of general relationships between bulk and surface exponents, are critically discussed. The new problem of critical behaviour at fractal boundaries is also considered for random (RW) and self-avoiding walks (SAW). From the numerical analysis of this problem remarkable universalities of the surface exponents seem to emerge, which, in the case of SAW’s, are still far from being understood.
We develop an interface approach to bulk phase transitions and ordered states by using Monte Garlo simulations, and apply it to antiferromagnetic q -state Potts models with q =3∼6 on the simple cubic lattice. A stiffness exponent a defined by Δ F ∼ L a , where Δ F is the interface free energy for a system of size L , is introduced as a measure of the stiffness of the ordered phase against an external stress. Applying finite-size scaling to Δ F and to the squared order parameter enable one to determine each of T c , ν, β and γ in order where T c is the critical temperature and the others are critical exponents. This approach also provides a means to study properties of the ordered state through a and interface profiles. In the q =3 and 4 models the estimated exponents ν, β, γ and a indicate new universality classes. The q =5 model shows a phase transition, while the q =6 model does not. The q =3∼5 models have non-integer values of a at low temperatures. The phase transitions in these models are confirmed to be due to entropy gains.
In this paper we briefly review our finding about the effects of boundary conditions and lattice shapes on scaling functions in critical phenomena. We present new examples to show that different boundary conditions and lattice shapes may give quite different scaling functions for lattice phase transition models, but they give consistent critical point, critical exponents, and thermodynamic order parameter from renormalization group calculations. We also review our recent results about the universal scaling func-tions for bond and site percolations on planar lattices with different boundary conditions and lattice shapes. PACS. 05.5O.+q- Lattice theory and statistics; Ising problems. PACS. 64.60.Fr- Equilibrium properties near critical points, critical exponents. PACS. 75.10.Hk- Classical spin models. I.
\n \nUsing extensive Monte Carlo simulations, transfer matrix techniques \nand\nconformal invariance, ferromagnetic random\nq−q-q−state Potts models for 3leqle83\\le q\\le 83leqle8 are studied in the vicinity \nof the \ncritical temperature.\nIn particular the surface and bulk magnetization exponents beta1\\beta_1beta1​ \nand \nβ are found\nmonotonically increasing with q. At the critical temperature, \ndifferent \nmoments (n)\nof the magnetization profiles are calculated which are all found \nto accurately follow predictions of \nconformal invariance. The critical correlation functions \nshow multifractal behaviour, the decay exponents of the different \nmoments\nboth in the volume and at the surface, are n−n-n−dependent.\n \n\n