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International Journal of Modern Physics B

SURFACE CRITICAL BEHAVIOUR OF SOME TWO-DIMENSIONAL LATTICE MODELS

Attilio L. Stella, Carlo Vanderzande

Published Jul 1, 1990• 4 citations
Publisher

Abstract

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.

DOI
10.1142/s0217979290000711

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•Dec 12, 2022•0 citations
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.
The analysis of renormalisation group equations for the q-state Potts model in two dimensions is generalised to include the presence of a surface field H s besides the bulk one, H B . Combined with conformal invariance results, this allows prediction of a surface susceptibility behaviour delta 2/f delta H s delta H B approximately L -1/8 (1-C'lnL) 15/16 for a q=4 system of size L at the critical temperature. This prediction is checked by a Monte Carlo based finite-size scaling analysis, which also nicely reproduces the exact and conjectured magnetic surface exponents for q=2 and 3, respectively. For the Baxter-Wu model similar methods give results consistent with the q=4 behaviour without logarithmic corrections (c'=0).
Using suitable Monte Carlo methods and finite-size scaling, we investigate critical and tricritical surface phenomena of two-dimensional Potts models. For the critical two- and three-state models, we determine a surface scaling dimension describing percolation properties of the so-called Potts clusters near the edges. On this basis, we propose an exact expression describing this exponent for the whole critical branch. For tricritical Potts models we find that varying the surface coupling constant or the surface magnetic field can induce a continuous phase transition. At bulk tricriticality and sufficiently strong surface couplings, spontaneous one-dimensional order occurs on the edges. We determine several critical exponents describing these edge transitions. On the basis of these results and conformal field theory, we conjecture exact expressions for these exponents.
The self-dual random-bond eight-state Potts model is studied numerically through large-scale Monte Carlo simulations using the Swendsen-Wang cluster flipping algorithm. We compute bulk and surface order parameters and susceptibilities and deduce the corresponding critical exponents at the random fixed point using standard finite-size scaling techniques. The scaling laws are suitably satisfied. We can conclusively rule out that the model belongs to the 2D Ising model universality class. The dimensions of the relevant bulk and surface scaling fields are found to be yh=1.849{y}_{h}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.849yh​=1.849, yt=0.977{y}_{t}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.977yt​=0.977, and yhs=0.54{y}_{{h}_{s}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.54yhs​​=0.54, in contrast to their Ising values 15/815/815/8, 111, and 1/21/21/2, respectively.
We use the density matrix renormalization group (DMRG) to perform a detailed study of the critical properties of the two-dimensional QQQ state Potts model, including the magnetization and energy-density profiles, bulk and surface critical exponents, and the Casimir amplitudes. We apply symmetry breaking boundary conditions to a $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ strip and diagonalize the corresponding transfer matrix for a series of moderately large systems (L< 64)(L<~64)(L< 64) by the DMRG method. The numerically very accurate finite lattice results are then extrapolated by efficient sequence extrapolation techniques. The density profiles and the Casimir amplitudes at the critical point are found to follow precisely the conformal predictions for Q=2Q=2Q=2 and 333. Similarly, the bulk and surface critical exponents of the models are in very good agreement with the conformal and exact values: their accuracy has reached or even exceeded the accuracy of other available numerical methods. For the Q=4Q=4Q=4 model both the profiles and the critical exponents show strong logarithmic corrections, which are also studied.
Determining whether an additional local symmetry affects the universality class of a sta-tistical model is an important issue in the theory of critical phenomena. A basic example is provided by the RP N −1 model, in which N -component spin variables at each lattice site interact through an Hamiltonian invariant under global O(N ) rotations and local spin re-versals. The local symmetry makes the difference with the usual O(N ) model and amounts to the head-tail symmetry characteristic of liquid crystals. In three dimensions, the weak first order transition observed in numerical simulations of the ferromagnetic model is con-sistent with the mean field scenario. On the other hand, in the two-dimensional case –the one we focus on in this thesis – fluctuations are stronger and minimize the reliability of mean field predictions, as illustrated by the phase transition of the three-state Potts model, which becomes continuous on planar lattices. For the RP N −1 model, the absence of spontaneous breaking of continuous symmetry in two dimensions generically suggests that criticality is limited to zero temperature, and Monte Carlo studies for T → 0 showed a fast growth of the correlation length which made particularly hard to reach the asymp-totic limit and draw conclusions. On the other hand, the possibility of finite temperature topological transitions similar to the Berezinskii-Kosterlitz-Thouless (BKT) one – which should definitely occur for RP 1 ∼ O(2) – and mediated by ”disclination” defects has also been debated in numerical studies. While two-dimensional criticality has allowed for an impressive amount of exact solutions thanks to lattice integrability and conformal field theory, models with local symmetries traditionally remained outside the range of application of these methods. In this thesis, however, we show how the renormalization group fixed points of the RP N −1 model and of its complex generalization, the CP N −1 model, can be accessed in an exact way. This is achieved in the scale invariant scattering framework which, as we review in the introductory part of the thesis, implements in the basis of particle excitations the infinite-dimensional conformal symmetry characteristic of critical points in two dimensions and has provided in the last few years new results for pure and disordered systems. In the last part of the thesis, we will exploit the generality of the scale invariant scattering method to progress with another long standing problem of two-dimensional criticality, namely that of spin clusters in Potts correlated percolation. It has been known for long time that the problem can be addressed considering two coupled Potts models, but the need to consider the number of states of one of these models as a continuous variable has severely limited the analytical or numerical study of the critical points. Also here, the ability of the scattering method to enforce conformal invariance for the internal symmetry characteristic of the universality class will allow us to obtain exact equations for the critical points and to determine their solutions in the relevant limits.