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Journal of the Physical Society of Japan

Phase Diagram of a Short-Range ±JHeisenberg Modelin Three Dimensions

Fumitaka Matsubara, Taketoshi Iyota, Sakari Inawashiro

Published Dec 15, 1991• 14 citations
Publisher

Abstract

The short-range ± J Heisenberg model with the concentration of ferromagnetic bonds p on a simple cubic lattice is studied using a hybrid Monte-Carlo spin-dynamics (HMCSD) method. Using a finite-size scaling analysis, we find the critical concentration of p c =0.79±0.01 and the ferromagnetic phase transition for p > p c which belongs to the same universality class as that of the pure ferromagnet. We also find that no random spin freezing occurs in our model.

DOI
10.1143/jpsj.60.4022

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The short-range ± J Heisenberg model with the concentration of ferromagnetic bonds p on a simple cubic lattice is studied using a hybrid Monte-Carlo spin-dynamics (HMCSD) method. Using a finite-size scaling analysis, we find the critical concentration of p c =0.79±0.01 and the ferromagnetic phase transition for p > p c which belongs to the same universality class as that of the pure ferromagnet. We also find that no random spin freezing occurs in our model.
The ferromagnetic Ising spin-(1/2 model in a finite simple-cubic lattice \ensuremath{\Lambda} is studied by Monte Carlo methods when two subsets of the lattice sites in \ensuremath{\Lambda}, say ${\ensuremath{\Omega}}^{+}$ and ${\ensuremath{\Omega}}^{\mathrm{\ensuremath{-}}}$, contain (the same number of) spins fixed at \ifmmode\pm\else\textpm\fi{}1, respectively, the global defect concentration being x\ensuremath{\le}0.25. We study the thermodynamic properties of the model for different choices of \ensuremath{\Omega}=${\ensuremath{\Omega}}^{+}$\ensuremath{\cup}${\ensuremath{\Omega}}^{\mathrm{\ensuremath{-}}}$. A finite-size-scaling analysis reveals that the transition remains second order with pure critical exponents for regularly spaced defects, the critical temperature varying with the symmetry of \ensuremath{\Omega}. Any small randomness in \ensuremath{\Omega}, however, makes the transition weakly first order; the transition becomes more abrupt for defects located fully at random, and the long-range order is suppressed when the numbers of defects in ${\ensuremath{\Omega}}^{+}$ and ${\ensuremath{\Omega}}^{\mathrm{\ensuremath{-}}}$ differ from each other. We also discuss our findings in relation to the random-field and frustration problems.
A short-range \ifmmode\pm\else\textpm\fi{}J Heisenberg spin glass with and without anisotropy on a simple cubic lattice is studied by using a hybrid Monte Carlo spin-dynamics method. The anisotropy leads to a cusplike peak of the susceptibility \ensuremath{\chi} and a very slow relaxation of the spin configuration at low temperatures. From a finite-size-scaling analysis of the spin-glass susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{sg}}$, we find, for the first time, that the anisotropy induces a phase transition.
We have performed large-scale numerical simulations for the ± J Ising model on the simple cubic and the square lattices with asymmetric weight of ferromagnetic and antiferromagnetic bonds. The Monte Carlo renormalization group method is used to estimate T c , γ/ν, β/ν and ν at various points along the phase boundary between ferromagnetic and paramagnetic phases including the tricritical point. The obtained values of T c and the resulting phase diagrams of the simple cubic and the square lattices are much more accurate than those estimated so far. In the simple cubic lattice, we find evidence for the possibility that universality of the ferromagnetic critical exponents holds along the critical line except at the tricritical point, while weak universality holds even at the tricritical point. Weak universality seems to hold also in the square lattice.
The three-dimensional quenched random bond diluted $({J}_{1}\ensuremath{-}{J}_{2})$ quantum Heisenberg antiferromagnet is studied on a simple-cubic lattice. Using extensive stochastic series expansion quantum Monte Carlo simulations, we perform very long runs for an $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ lattice up to L=48L=48L=48. By employing a standard finite-size scaling method, the numerical values of the N\'eel temperature are determined with high precision as a function of the coupling ratio r=J2/J1r={J}_{2}/{J}_{1}r=J2​/J1​. Based on the estimated critical exponents, we find that the critical behavior of the considered model belongs to the pure classical three-dimensional O(3)O(3)O(3) Heisenberg universality class.
We determine the ferromagnetic phase boundary for the short range ±J bond frustrated Heisenberg model in three dimensions using a very efficient Monte Carlo algorithm which eliminates the critical slowing down usually experienced at a second order phase transition. The phase boundary is identified by measuring the correlation length directly, a method which we show to be superior to more conventional methods such as the crossing of the Binder cumulant. The critical concentration of antiferromagnetic bonds beyond which ferromagnetism is lost is xc=0.208(2).
We have performed Monte Carlo simulations of the ferromagnetic order-disorder transition in an off-lattice version of the Heisenberg model. Varying the temperature at a fixed density, we located the transition for three different densities by means of a finite-size scaling analysis. The obtained critical exponents differ from those for the lattice Heisenberg model. Although the limited size of our systems warrants caution, the results question whether the lattice and the off-lattice model are in the same universality class.