Antiferromagnetic Potts models on the square lattice

Abstract

We study the antiferromagnetic q-state Potts model on the square lattice for q = 3 and q = 4, using the Wang-Swendsen-Koteck y Monte Carlo algorithm and a new nite-size-scaling extrapolation method. For q = 3 we obtain good control up to correlation length 5000; the data are consistent with ( ) =

Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms

“…Typically one approaches the critical point with L ~ c~, where c ~ 2 -4, and then uses finite-size scaling [22,23] to extrapolate to the infinite-volume limit. Note Added 1996: Recently, radical advances have been made in applying finite-size scaling to Monte Carlo simulations (see [97,98] and especially [99,100,101,102,103]); the preceding two sentences can now be seen to be far too pessimistic. For reliable extrapolation to the infinite-volume limit, L and ~ must both be » I, but the ratio L/ ~ can in some cases be as small as 10-3 or even smaller (depending on the model and on the quality of the data).…”

“…The idea ofthe embedding algorithms is to find Ising-like variables underlying a general spin variable, and then to update the resulting Ising model using the ordinary SW algorithm 'Note Added 1996: In addition to the generalizations discussed below, see [120,112] for SW-type algorithms for the Ashkin-Teller model; see [121] for a clever SW-type algorithm for the fully frustrated Ising model; and see [122,123,124,100] for a very interesting SW-type algorithm for antiferromagnetic Potts models, based on the "embedding" idea to be described below. tNote Added 1996: For at least some versions of the multi-grid SW algorithm, it can be proven [125] that the bound ZMosw ~ a/ v holds.…”

Complex-temperature phase diagram of Potts and RSOS models

“…Now, the triangular-lattice model does in fact exhibit c = 1 behavior elsewhere (for x = x − ), but the compactification radius is different from that of the square-lattice theory and accordingly the critical exponents differ. Finally, (Q, v) = (4, −1) is a critical c = 2 theory on the triangular lattice [21,22], but is non-critical on the square lattice [23].…”

Extrapolating Monte Carlo Simulations to Infinite Volume: Finite-Size Scaling atξ/L≫1

“…We expect this relative variance-time product for errors (i)+(ii) only] to scale as 4]. We ran on lattices L = 32; 64; 128; 256; 512; 1024; 1536 at 153 di erent pairs ( ; L) in the range 5 < 1 < 20000.…”

“…9) points di er from the exact value more than one standard deviation, and none by more than two. Details on all of these models will be reported separately 4,6]. The method is easily generalized to a model controlled by an RG xed point having k relevant operators.…”