Numerical Study of Partition Function Zeros on Recursive Lattices Ruben Ghulghazaryan and Nerses Ananikian Yerevan Physics Institute

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SINGULARITES AT A DENSE SET OF TEMPERATURE IN HUSIMI TREE Paper by N.S. Ananikian, A.E. Alahverdian, and S.K. Dallakian, Phys. Rev. E57, 24519982A, 1998. Hamiltonian for the model where i takes ±1. The partition function The recurrence relation

Fisher Zeros by Direct Calculations Zeros of partition function searched from equation

Cayley-Type Recursive Lattice Multi-site interaction Ising model on a Cayley- type lattice is considered The Hamiltonian of the model is Si takes values ±1 The first sum goes over all p-polygons Π is the product of all spins on a p-polygon The second sum goes over all sites on the lattice

Generalized Recurrence Relation The partition function has the form The generalized recurrence relation has the form The values of external parameters at which the phase transition occurs defined by neutral periodical points the mapping f(x) Mandelbrot-like set is defined as a set of parameters of rational function for which f(x) has neutral periodic cycles.

Basin of Attraction of a Fixed Point If x* is an attracting fixed point of f, the basin of attraction of x denoted by A(x*) consists of all x such that fn(x*) is defined for all n 1 and fn(x)x*. The connected component of A(x*) containing x* is called the immediate basin of attraction of x* and is denoted by A*(x*). If  is an attracting cycle of period n, then each of the fixed points of fn(x*) has its own basin A(x*) and A() is the union of these basins. Critical points of the mapping are defined as solutions of f(c)=0.

Theorems

The Numerical Algorithm Find all the critical points of the mapping: Investigates the convergence of all the orbits of critical points (critical orbits) to any attracting periodic cycle. If all critical orbits converge, for example, after n iterations, one says that the system is in a stable equilibrium state, Otherwise the system undergoes a phase transition. Note that most of the critical orbits of Calyley0type mapping after the first iteration are intersected It is enough to consider only orbits of points 0, z, 1 and -1 for p>2, and 1/z , z for p=2 (Bethe lattice) For Husimi lattice p=3

Fisher zeros for Husimi Lattice Fisher zeros for Husimi lattice case 1.

Fisher zeros for Husimi Lattice Fisher zeros for Husimi lattice case 2.

Fisher zeros and “Yang-Lee” zeros

Q-State Potts Model on Bethe Lattice The dynamics of metastability regions of the Q- state Potts model on the Bethe lattice was studied in R. G. Ghulghazaryan, N. S. Ananikian, and P. M. A. Sloot, PHYSICAL REVIEW E 66, 046110, 2002. The Hamiltonian has the form The recurrence relation has the form where and

Yang-Lee Zeros for Recurrence Lattices The Yang-Lee zeros may be found from the condition |f ’(x1)|=|f ’(x2)|  1 where x1,2 are two attractive fixed points of the mapping. The boundary of Metastability regions may be found from the condition f (x)=x and |f ’(x)|=1. The critical temperature of the ferromagnetic Potts model is derived from the conditions f (x)=x, f ’(x)=1, f ’’(x)=0

Metastability Regions and Yang-Lee zeros Q-state Potts model on the Bethe lattice with coordination number =3 and Q=2 (Ising model). Plots of Bethe-Potts function indicating the existence of neutral fixed points for Different temperatures and magnetic field for =3 and Q=2 T<Tc (z=6) & exp(h/kT)=- T=Tc (z=3) h=0 T<Tc (z=6) and exp(h/kT)=+

Dynamics of Metastability regions of Q-State Potts Model with Q

Julia sets for Metastability Region Points

Boundary of Mandelbrot-like set and Fractal Dimensionality of the Julia Set The boundary of the Mandelbrot-like set of the Potts-Bethe mapping for z=11, Q=9 and =3. At the points k (=-4) and l (=1.0+2.0 I) outside of the metastability region the Potts-Bethe mapping has an attracting fixed point and, also, an attractive periodical point of period 2 and 4 respectively. Values of fractal dimensions of Julia sets of the Potts-Bethe mapping near the first order phase transition point (a Yang-Lee zero) c =0.74

Properties of Yang-Lee Zeros Location of Yang-Lee zeros is related to the phase coexistence lines in the complex magnetic field plane. The Yang-Lee zeros correspond to first order phase transition on the complex plane. For any temperature there exists a metastability region on the complex magnetic field plane where there are two attractive fixed points. The Yang-Lee zeros always located inside the metastability region. The fractal dimension of a Julia set for a value of magnetic field corresponding to a Yang-Lee zero is a local minimum as a function of magnetic field.

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