Partition function zeros at first-order phase transitions: Pirogov-Sinai theory (original) (raw)
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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. ...
2002
The zeros of the partition function of the ferromagnetic q-state Potts model with long-range interactions in the complex-q plane are studied in the mean-field case, while preliminary numerical results are reported for the finite 1d chains with powerlaw decaying interactions. In both cases, at any fixed temperature, the zeros lie on the arc-shaped contours, which cross the positive real axis at the value for which the given temperature is transition temperature. For finite number of spins the positive real axis is free of zeros, which approach to it in the thermodynamic limit. The convergence exponent of the zero closest to the positive real-q axis is found to have the same value as the temperature critical exponent 1/ν. PACS: 05.50.+q, 64.60.Cn
The partition function zeros in the one-dimensional q-state Potts model
Journal of Physics A: Mathematical and General, 1994
The Zeros of the partition function in the one-dimensional y-state Potts model with a h i t r w and continuous q > 0 have been studied using a tnnsfer matrix. The location of zeros and the Yang-Lee edge singularity hove been analysed, and two different regimes. corresponding to q > 1 and q c I, have been observed. A duality relation has also been derived. which relates the zeros in the complex held plane to those in the complex tempemure plane.
Complex-q zeros of the partition function of the Potts model with long-range interactions
Physica A: Statistical Mechanics and its Applications
The zeros of the partition function of the ferromagnetic q-state Potts model with long-range interactions in the complex-q plane are studied in the mean-field case, while preliminary numerical results are reported for the finite 1d chains with powerlaw decaying interactions. In both cases, at any fixed temperature, the zeros lie on the arc-shaped contours, which cross the positive real axis at the value for which the given temperature is transition temperature. For finite number of spins the positive real axis is free of zeros, which approach to it in the thermodynamic limit. The convergence exponent of the zero closest to the positive real-q axis is found to have the same value as the temperature critical exponent 1/ν.