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Register a new userZeros of the Potts Model Partition Function on Sierpinski Graphs
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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, $S_infty$.
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The equation of state of a system at equilibrium may be derived from the canonical or the grand canonical partition function. The former is a function of temperature T, while the latter also depends on the chemical potential mu for diffusive equilibrium. In the literature, often the variables beta=(k_BT)^{-1} and fugacity z=exp(beta mu) are used instead. For real beta and z, the partition functions are always positive, being sums of positive terms. Following Lee, Yang and Fisher, we point out that valuable information about the system may be gleaned by examining the zeros of the grand partition function in the complex z plane (real beta), or of the canonical partition function in the complex beta plane. In case there is a phase transition, these zeros close in on the real axis in the thermodynamic limit. Examples are given from the van der Waal gas, and from the ideal Bose gas, where we show that even for a finite system with a small number of particles, the method is useful.
Phase transition of the two- and three-state quantum Potts models on the Sierpinski pyramid are studied by means of a tensor network framework, the higher-order tensor renormalization group method. Critical values of the transverse magnetic field and the magnetic exponent $beta$ are evaluated. Despite the fact that the Hausdorff dimension of the Sierpinski pyramid is exactly two $( = log_2^{~} 4)$, the obtained critical properties show that the effective dimension is lower than two.
We present an analytic study of the Potts model partition function on two different types of self-similar lattices of triangular shape with non integer Hausdorff dimension. Both types of lattices analyzed here are interesting examples of non-trivial thermodynamics in less than two dimensions. First, the Sierpinski gasket is considered. It is shown that, by introducing suitable geometric coefficients, it is possible to reduce the computation of the partition function to a dynamical system, whose variables are directly connected to (the arising of) frustration on macroscopic scales, and to determine the possible phases of the system. The same method is then used to analyse the Hanoi graph. Again, dynamical system theory provides a very elegant way to determine the phase diagram of the system. Then, exploiting the analysis of the basins of attractions of the corresponding dynamical systems, we construct various examples of self-similar lattices with more than one critical temperature. These multiple critical temperatures correspond to crossing phases with different degrees of frustration.
A phenomenological approach to the ferromagnetic two dimensional Potts model on square lattice is proposed. Our goal is to present a simple functional form that obeys the known properties possessed by the free energy of the q-state Potts model. The duality symmetry of the 2D Potts model together with the known results on its critical exponent {alpha} allow to fix consistently the details of the proposed expression for the free energy. The agreement of the analytic ansatz with numerical data in the q=3 case is very good at high and low temperatures as well as at the critical point. It is shown that the q>4 cases naturally fit into the same scheme and that one should also expect a good agreement with numerical data. The limiting q=4 case is shortly discussed.
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.
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