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Physical Review Letters

Existence of a Phase Transition on a Classical Lattice Spin System with Three-Spin Interactions

P. Surànyi

Published Sep 20, 1976• 5 citations
Publisher

Abstract

It is shown that an appropriately defined magnetization has a zero thermal average above a critical temperature and a nonzero average below it for a specific lattice Hamiltonian with three-spin interactions in two or more dimensions. This work implies the existence of a phase transition in Reggeon field theory.
DOI
10.1103/physrevlett.37.725

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In this work, Markov Chain-Monte Carlo technique was used to study the phase transition in two and three dimensional Ising Model (IM) in a square and cubic lattice. The study of temperature dependence of average magnetization and specific heat in different magnetic fields has been carried out in the 3x3 and 3x3x3 lattice with periodic boundary. Critical temperature point kBTc / J for 2D and 3D Ising Model has been observed at around 2.2 and 4.3 respectively at zero field. Our work satisfies Onsager’s critical value in 2D IM. The simulation suggests bifurcation in average magnetization below critical temperature Tc. Temperature plays the role of increasing randomness of spins. We found that Ising Model in small lattice size still retains interesting features like spontaneous magnetization and symmetry breaking below Tc at B = 0. At a non-zero field, the likelihood of spins to prefer certain alignment depends on the direction of the external field and magnitude of magnetization depends on magnitude of field ±B. Specific heat Cv, which gives us fluctuation of energy at particular temperature, has been found to attain maximum value at critical regions.
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Using the Ising Model, the phase transition between ferromagnetic and paramagnetic phases of a 2-D square spin lattice are explored. A phase transition into the ferromagnetic phase occurs when the temperature of the system falls below some critical value. The ferromagnetic phase is characterized by a non-zero magnetization on the macroscopic level. Using the Mote Carlo method to obtain system's configurations (i.e., sets of microstates) in thermal equilibrium, we will explore system's macroscopic properties such as magnetization, magnetic susceptibility, and specific heat capacity as functions of temperature. We will evaluate the transition temperature, and discuss how the lattice size affects the results.
The three-spin interaction Hamiltonian ${\mathcal{H}}^{(3)}\ensuremath{\equiv}\ensuremath{-}{J}_{3}\ensuremath{\Sigma}{〈\mathrm{ijk}〉}^{}{s}_{i}{\ensuremath{\sigma}}_{j}{\ensuremath{\sigma}}_{k}\ensuremath{-}H\ensuremath{\Sigma}{i}^{}{s}_{i}\ensuremath{-}{H}^{\ensuremath{'}}\ensuremath{\Sigma}{j}^{}{\ensuremath{\sigma}}_{j}$ (the sss and $\ensuremath{\sigma}$ spins belong to different sublattices) is solved for some two-dimensional lattices by a generalized star-triangle transformation. The ${H}^{\ensuremath{'}}=0$ internal energy, specific heat, and magnetization are explicitly calculated, and the singularity structures of other functions (e.g.xxx are studied. Although the actual critical-point exponents remain the same as for systems with two-spin interaction, the three-spin interaction gives rise to different amplitude and background terms. In particular, the ${H}^{\ensuremath{'}}=0$ magnetization is nonzero for all finite TTT, because of the lack of spin-reversal symmetry; the diameter is seen explicitly to have an energy singlularity, with critical exponent $1\ensuremath{-}\ensuremath{\alpha}$.
The location of the phase transition in the two dimensional Ising model were determined using Monte Carlo simulation with importance sampling. The magnetization persite ,energy per site , magnetic susceptibility , specific heat of a Ferromagnetic materials were calculated as a function of temperature for spin lattice interaction in zero and nonzero magnetic field . There is thus a phase transition defined by the Curie temperature. The Monte Carlo method was used to check the results and to confirm the phase transition . The internal interaction results were found to be consistent with what was expected. As a magnetic field is applied, the spins tend to align with it for ˃ and its effect is not significant at a very high temperature because of the thermal agitation. For ˂ , the alignment of the spins is possible only if the amplitude of the field is big enough.