Triangular SOS models and cubic-crystal shapes

We study a percolation problem on a substrate formed by two-dimensional XY spin configurations using Monte Carlo methods. For a given spin configuration, we construct percolation clusters by randomly choosing a direction x in the spin vector space, and then placing a percolation bond between nearest-neighbor sites i and j with probability p ij = max(0,1 − e −2Ks x i s x j ), where K > 0 governs the percolation process. A line of percolation thresholds K c (J ) is found in the low-temperature range J J c , where J > 0 is the XY coupling strength. Analysis of the correlation function g p (r), defined as the probability that two sites separated by a distance r belong to the same percolation cluster, yields algebraic decay for K K c (J ), and the associated critical exponent depends on J and K. Along the threshold line K c (J ), the scaling dimension for g p is, within numerical uncertainties, equal to 1/8. On this basis, we conjecture that the percolation transition along the K c (J ) line is of the Berezinskii-Kosterlitz-Thouless type.

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“…3) were fitted by Eq. (8). This fit yields J c = 1.120 (9), which is consistent with the result from χ .…”

Section: A Square Latticesupporting

confidence: 86%

“…We find that the data for 16 L 1024 and 1.100 J 1.125 are well described by Eq. (8). The fit yields J c = 1.124 (3).…”

Section: A Square Latticementioning

confidence: 96%

Berezinskii-Kosterlitz-Thouless-like percolation transitions in the two-dimensionalXYmodel

Deng

Blöte

2011

Phys. Rev. E

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“…For this model we know [16][17][18] the exact transition temperature for the antiferromagnetic model v c,AF = −1 = v c,BK . The partition function is given by a formula similar to that of the square lattice, and the dimensionality of T j (2, L) is the same as for the square lattice.…”

Section: Ising Model (P = 4)mentioning

confidence: 99%

Complex-temperature phase diagram of Potts and RSOS models

2006

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We study the phase diagram of Q-state Potts models, for Q = 4 cos 2 (π/p) a Beraha number (p > 2 integer), in the complex-temperature plane. The models are defined on L × N strips of the square or triangular lattice, with boundary conditions on the Potts spins that are periodic in the longitudinal (N ) direction and free or fixed in the transverse (L) direction. The relevant partition functions can then be computed as sums over partition functions of an A p−1 type RSOS model, thus making contact with the theory of quantum groups. We compute the accumulation sets, as N → ∞, of partition function zeros for p = 4, 5, 6, ∞ and L = 2, 3, 4 and study selected features for p > 6 and/or L > 4. This information enables us to formulate several conjectures about the thermodynamic limit, L → ∞, of these accumulation sets. The resulting phase diagrams are quite different from those of the generic case (irrational p). For free transverse boundary conditions, the partition function zeros are found to be dense in large parts of the complex plane, even for the Ising model (p = 4). We show how this feature is modified by taking fixed transverse boundary conditions.

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“…This can be seen from Fig. 1: If one places a dimer across every frustrated bond, one obtains the corresponding dimer covering of the dual hexagonal lattice, where each lattice site in the hexagonal lattice is touched by one and only one dimer 22 . If the two smallest couplings are equal, J 1 = J 3 < J 2 , then dimers can occupy bonds in directions 1 and 3 only and there is still a huge ground state degeneracy.…”

Section: A From Spins To Dimersmentioning

confidence: 99%

Ordering of geometrically frustrated classical and quantum triangular Ising magnets

Jiang

Emig

2006

Phys. Rev. B

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A systematic study of both classical and quantum geometric frustrated Ising models with a competing ordering mechanism is reported in this paper. The ordering comes in the classical case from a coupling of 2D layers and in the quantum model from the quantum dynamics induced by a transverse field. By mapping the Ising models on a triangular lattice to elastic lattices of non-crossing strings, we derive an exact relation between the spin variables and the displacement field of the strings. Using this map both for the classical (2+1)D stacked model and the quantum frustrated 2D system, we obtain a microscopic derivation of an effective Hamiltonian which was proposed before on phenomenological grounds within a Landau-Ginzburg-Wilson approach. In contrast to the latter approach, our derivation provides the coupling constants and hence the entire transverse field-versus-temperature phase diagram can be deduced, including the universality classes of both the quantum and the finite-temperature transitions. The structure of the ordered phase is obtained from a detailed entropy argument. We compare our predictions to recent simulations of the quantum system and find good agreement. We also analyze the connections to a dimer model on the hexagonal lattice and its height profile representation, providing a simple derivation of the continuum free energy and a physical explanation for the universality of the stiffness of the height profile for anisotropic couplings.

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Triangular SOS models and cubic-crystal shapes

Cited by 165 publications

References 16 publications

Berezinskii-Kosterlitz-Thouless-like percolation transitions in the two-dimensionalXYmodel

Berezinskii-Kosterlitz-Thouless-like percolation transitions in the two-dimensionalXYmodel

Complex-temperature phase diagram of Potts and RSOS models

Ordering of geometrically frustrated classical and quantum triangular Ising magnets

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