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Phase diagram of d = 4 Ising Model with two couplings

José L. Alonso, J. M. Carmona, Jesús Clemente-Gallardo, L. A. Fernández, D. Íñiguez, Alfonso Tarancón, C.L. Ullod

Published May 23, 1996• 6 citations
OpenAlex

Abstract

We study the phase diagram of the four dimensional Ising model with first and second neighbour couplings, specially in the antiferromagnetic region, by using Mean Field and Monte Carlo methods. From the later, all the transition lines seem to be first order except that between ferromagnetic and disordered phases in a region including the first-neighbour Ising transition point. 1 Antiferromagnetism has been considered in a great variety of models in order to find properties not present in the purely ferromagnetic (FM) system. Mostly they have been models in two and three dimensions. For instance, in several works on High Temperature Superconductivity the transition from paramagnetic (PM) to non-pure FM ordered phases has been studied [1, 2, 3, 4]. On the other hand, in relation to the finite temperature phase transition in pure gauge SU(3), the d = 3 three state Potts model with negative second neighbour couplings was unsuccessfully considered to find a new critical behaviour [5, 6]. In four dimensions, in diluted systems recently new critical exponents have been

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We study the phase diagram of the four dimensional Ising model with first and second neighbour couplings, specially in the antiferromagnetic region, by using Mean Field and Monte Carlo methods. From the later, all the transition lines seem to be first order except that between ferromagnetic and disordered phases in a region including the first-neighbour Ising transition point. 1 Antiferromagnetism has been considered in a great variety of models in order to find properties not present in the purely ferromagnetic (FM) system. Mostly they have been models in two and three dimensions. For instance, in several works on High Temperature Superconductivity the transition from paramagnetic (PM) to non-pure FM ordered phases has been studied [1, 2, 3, 4]. On the other hand, in relation to the finite temperature phase transition in pure gauge SU(3), the d = 3 three state Potts model with negative second neighbour couplings was unsuccessfully considered to find a new critical behaviour [5, 6]. In four dimensions, in diluted systems recently new critical exponents have been
We performed a Monte Carlo simulation on a four-dimensional hypercube for the three-state Potts model with ferromagnetic nearest-neighbours interactions and antiferromagnetic next-nearest-neighbours interactions. We find that on the ferromagnetic branch of the critical line the phase transition is always of first order. This result, which is in agreement with Landau and mean-field theories, precludes the possibility of defining a continuum limit of the model and confirms the difficulties met in the construction of a four-dimensional nontrivial scalar field theory.
Abstract We derive the phase diagram of the one-dimensional three-state Potts model with an additional mean-field interaction in the canonical ensemble. The free energy is obtained by mapping the model onto the spin-1 Blume–Emery–Griffiths model and solving it by using a Hubbard–Stratonovich transformation combined with the transfer matrix method. A complex structure with lines of first-order transitions, two triple points and a critical point appears at finite temperature. The phase diagram is two-dimensional, since there are two adjustable parameters, the nearest-neighbour coupling K and the temperature T . We show that the phase diagram does not present second-order phase transition lines, due to the fact that the order parameter is not a symmetry-breaking one. Quite remarkably, we are able to determine analytically one of the first-order phase transition lines. We also prove that, when the nearest-neighbour coupling K is large and negative, the first-order transition temperature becomes asymptotically independent of K .
Studies the four-state antiferromagnetic Potts model with next-nearest-neighbour ferromagnetic interactions on the simple cubic lattice by the Monte Carlo method. The ratio J 2 /J 1 of the strength of the next-nearest-neighbour interactions to that of the nearest-neighbour interactions is to be 10. The author analyses the finite-size effects of the energy, the specific heat, the fourth-order cumulant, the magnetization, the susceptibility and the effective transition temperatures. With the help of the finite-size scaling theory the author concludes that the transition of the model is assigned to be of first order.
Positions of critical temperatures of antiferromagnetic Ising model with ferromagnetic next nearest neighbour interactions on the triangular lattice are investigated by making use of their properties of the Kosterlitz-Thouless type of phase transition. In particular, finite-size scaling method and Monte Carlo Renormalization Group methods (MCRG) with flow diagrams of the anomalous dimension η show the fixed line structure where the correlation length diverges. Another MCRG with flow diagram of the stiffness constant also shows a critical property clearly. The temperature dependence of the long range order is also discussed in order to locate the lower critical point.
The one-dimensional spin-½ Ising model with very long-range ferromagnetic interaction and first and second neighbor antiferromagnetic interactions is solved exactly in a direct magnetic field. A study of the finite temperature behavior confirms the existence of up to four first-order phase transitions ending in classical critical points. We find a confluence of three critical lines at a ``tricritical'' point with exponents β=14 , δ=5, γ=γ′=1,α=0, and α′=12. We find also a confluence of four critical lines at a ``tetracritical'' point with exponents β=14, δ=7, γ=γ′=1, α=0, and α′=23. We show the existence of triple lines and ``quadruple'' lines, along which three or four phases are simultaneously in contact with each other, respectively. For some special values of the interaction strengths, five phases are in contact with each other at a ``quintuple'' point. We remark that our ``multiple'' lines and ``multicritical'' points are not ``true'' multiple lines and multicritical points but that they would be so in a somewhat extended model. This model satisfies a generalized form of the ordinary phase rule of Gibbs. In distinction to various models considered recently, our model does not exhibit a singular phase-boundary diameter.
The Ising model with nearest and next nearest neighbor antiferromagnetic interactions on the triangular lattice displays, for Jnnn/Jnn = 0.1, three phase transitions in different universality classes as the magnetic field is increased. We have studied this model using Monte Carlo and renormalization group techniques. The transition from the paramagnetic to the 2 × 1 phase (universality class of the Heisenberg model with cubic anisotropy) is found to be first order; the transition from the paramagnetic phase to the [Formula: see text] phase (universality class of the three state Potts model) is continuous; and the transition from the paramagnetic to the 2 × 2 phase (universality class of the four state Potts model) is found to change from first order to continuous as the field is increased. We have mapped out the phase diagram and determined the critical exponents for the continuous transitions. A novel technique, using a Landau-like free energy functional determined from Monte Carlo calculations, to distinguish between first order and continuous transitions, is described.