Abstract
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.
Highlights
The Q-state Potts model [1,2] can be defined for general Q by using the Fortuin–Kasteleyn (FK) representation [3, 4]
Χ1,3 for x ∈ (−∞, xc,1) ∪ For L = 6 the amount of memory needed for the computation of the phase diagram on the real x-axis is very large, so we have focused on trying t√o obtain the largest real phasetransition point
First we present an argument why the limiting curves corresponding to just the sector χ1,1 almost coincide with those for fully free boundary conditions
Summary
The Q-state Potts model [1,2] can be defined for general Q by using the Fortuin–Kasteleyn (FK) representation [3, 4]. The zero-temperature triangular-lattice Ising antiferromagnet, (Q, v) = (2, −1), is critical and becomes in the scaling limit a free Gaussian field with central charge c = 1 [16,17,18], whereas the corresponding square-lattice model is noncritical, its partition function being trivially Z = 2. While this observation does not in itself imply non-universality, since the critical temperature is expected to be lattice dependent (as is the value of xFM(Q)), the point to be noticed is that for no value of v does the Q = 2 square-lattice model exhibit c = 1 critical behavior. An appendix gives some technical details on the dimensions of the transfer matrices used
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