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The XXII International Workshop High Energy Physics and Quantum Field Theory June 24 ­ July 1, 2015 INFLUENCE OF LONG-RANGE INTERACTIONS ON THE PHASE TRANSITION TEMPERATURE OF THE ISING MODEL
A. Biryukov, Ya. Degtyarova Samara State Univercity, Russia


ISING MODEL AND ITS RESEARCHING
Ising model has been studied since 1924 for the description of phase transitions in magnetic materials.
E. Ising, Beitrag zur Theorie des Ferro- und Paramagnetimus, Hamburg, 1924 L. Onsager, Crystalstatistics. A two-dimensional transitions. // PhysRev, 1944; model with order-disorder

B.M. McCoy, Tai Tsun Wu, The two-dimensional Ising model. // Cambridge, MA: Harvard Univ.Press, 1973

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ISING MODEL AND ITS RESEARCHING
In recent the papers devoted to the Ising models with interaction between nearest and second nearest neighbors have appeared.
A.J. Ramirez-Pastor, F. Nieto, E.E. Vogel, Ising lattices with ±J second-nearestneighbor interactions. // PhysRev, 1997, v. 55, 21, p. 14323-14329; Rosana A. dos Anjos, J. Roberto Viana, J. Ricardo de Sousa, J. A. Plascak. Threedimensional Ising model with nearest- and next-nearest-neighbor interactions // PhysRev E. Vol. 76(022103), 2007; M. Picco, Critical behavior of the Ising model with long-range interactions. // arXiv:1207.1018v1, 2012; M. Ch. Angelini, G. Parisi, F. Ricci-Tersenghi. Relations between short-range and longrange Ising models // PhysRev E. Vol. 89(062120), 2014.

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ISING MODEL DESCRIPTION

Ising model may be described by the lattice with spins in the nodes. The lattice supposed with periodic boundary conditions. Lattice sides directed along the coordinate axes parallel to them. Coordinates of the spin are integer values (i, j).

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ISING MODEL
Ising model hamiltonian has a structure:

with J ­ interaction value between spins Si and Sj.

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ISING MODEL WITH LONG-RANGE INTERACTIONS
Interaction value between spins defined as

Is the nearest spin interaction value Is dimensional of the lattice Is parameter Is the distance between spins

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TWO-DIMENSIONAL ISING MODEL WITH LONG-RANGE INTERACTIONS

is the distance between spins for all is the interaction constant between spins Magnetization value: Normalization value: is the Boltzmann constant is the temperature is the linear size of the lattice

and

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The summing is over points which are situated in the circle of the radius R and the center (i, j):

Periodic boundary conditions are taken into account for the summing 8


PHASE TRANSITION TEMPERATURE OF THE TWO-DIMENSIONAL ISING MODEL

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THREE-DIMENSIONAL ISING MODEL WITH LONG-RANGE INTERACTIONS

is the distance between spins for all is the interaction constant between spins Magnetization value: Normalization value:

and


PHASE TRANSITION TEMPERATURE OF THE THREE-DIMENSIONAL ISING MODEL


For

, performed in K. Binder's work for the nearest-

interaction three-dimensional model.

K. Binder, E. Luijten. Monte Carlo test of renormalization group predictions for critical phenomena in Ising models. // Phys Rep, 344, 2001. pp.179 -- 253.


PHASE TRANSITION TEMPERATURE

for two-dimensional model for three-dimensional model The relative intensity interaction J (R) between the spins located at a distance R, and spins at a distance R = 1 represented by the expression then


PHASE TRANSITION TEMPERATURE DEPENDENCE FROM THE INTERACTION AREA RADIUS
We'll presume into account up to a distance the intensity of the interaction between the spins is taken i.e.


PHASE TRANSITION TEMPERATURE DEPENDENCE FROM THE INTERACTION AREA RADIUS

Two-dimensional model

Three-dimensional model


CONCLUSIONS


Phase transition temperature essentially depends on the interaction radius area between spins.



This dependence expressed analytically.


THANKS FOR YOUR ATTENTION!