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Register a new userExact Partition Functions for the $q$-State Potts Model with a Generalized Magnetic Field on Lattice Strip Graphs
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We calculate the partition function of the $q$-state Potts model on arbitrary-length cyclic ladder graphs of the square and triangular lattices, with a generalized external magnetic field that favors or disfavors a subset of spin values ${1,...,s}$ with $s le q$. For the case of antiferromagnet spin-spin coupling, these provide exactly solved models that exhibit an onset of frustration and competing interactions in the context of a novel type of tensor-product $S_s otimes S_{q-s}$ global symmetry, where $S_s$ is the permutation group on $s$ objects.
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We determine the general structure of the partition function of the $q$-state Potts model in an external magnetic field, $Z(G,q,v,w)$ for arbitrary $q$, temperature variable $v$, and magnetic field variable $w$, on cyclic, Mobius, and free strip graphs $G$ of the square (sq), triangular (tri), and honeycomb (hc) lattices with width $L_y$ and arbitrarily great length $L_x$. For the cyclic case we prove that the partition function has the form $Z(Lambda,L_y times L_x,q,v,w)=sum_{d=0}^{L_y} tilde c^{(d)} Tr[(T_{Z,Lambda,L_y,d})^m]$, where $Lambda$ denotes the lattice type, $tilde c^{(d)}$ are specified polynomials of degree $d$ in $q$, $T_{Z,Lambda,L_y,d}$ is the corresponding transfer matrix, and $m=L_x$ ($L_x/2$) for $Lambda=sq, tri (hc)$, respectively. An analogous formula is given for Mobius strips, while only $T_{Z,Lambda,L_y,d=0}$ appears for free strips. We exhibit a method for calculating $T_{Z,Lambda,L_y,d}$ for arbitrary $L_y$ and give illustrative examples. Explicit results for arbitrary $L_y$ are presented for $T_{Z,Lambda,L_y,d}$ with $d=L_y$ and $d=L_y-1$. We find very simple formulas for the determinant $det(T_{Z,Lambda,L_y,d})$. We also give results for self-dual cyclic strips of the square lattice.
We calculate zeros of the $q$-state Potts model partition function on $m$th-iterate Sierpinski graphs, $S_m$, in the variable $q$ and in a temperature-like variable, $y$. We infer some asymptotic properties of the loci of zeros in the limit $m to infty$ and relate these to thermodynamic properties of the $q$-state Potts ferromagnet and antiferromagnet on the Sierpinski gasket fractal, $S_infty$.
We study the metastable equilibrium properties of the Potts model with heat-bath transition rates using a novel expansion. The method is especially powerful for large number of state spin variables and it is notably accurate in a rather wide range of temperatures around the phase transition.
We demonstrate that the occurrence of symmetry breaking phase transitions together with the emergence of a local order parameter in classical statistical physics is a consequence of the geometrical structure of probability space. To this end we investigate convex sets generated by expectation values of certain observables with respect to all possible probability distributions of classical q-state spins on a two-dimensional lattice, for several values of q. The extreme points of these sets are then given by thermal Gibbs states of the classical q-state Potts model. As symmetry breaking phase transitions and the emergence of associated order parameters are signaled by the appearance ruled surfaces on these sets, this implies that symmetry breaking is ultimately a consequence of the geometrical structure of probability space. In particular we identify the different features arising for continuous and first order phase transitions and show how to obtain critical exponents and susceptibilities from the geometrical shape of the surface set. Such convex sets thus also constitute a novel and very intuitive way of constructing phase diagrams for many body systems, as all thermodynamically relevant quantities can be very naturally read off from these sets.
Fortuin-Kastelyn clusters in the critical $Q$-state Potts model are conformally invariant fractals. We obtain simulation results for the fractal dimension of the complete and external (accessible) hulls for Q=1, 2, 3, and 4, on clusters that wrap around a cylindrical system. We find excellent agreement between these results and theoretical predictions. We also obtain the probability distributions of the hull lengths and maximal heights of the clusters in this geometry and provide a conjecture for their form.
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