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arXiv:0912.3990 · Physical Review B

Critical entropy of quantum Heisenberg magnets on simple-cubic lattices

Stefan Weßel

Published Feb 10, 2010• 32 citations
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Abstract

We analyze the temperature dependence of the entropy of the spin-1/2 Heisenberg model on the three-dimensional simple-cubic lattice, for both the case of antiferromagnetic and ferromagnetic nearest-neighbor exchange interactions. Using optimized extended ensemble quantum Monte Carlo simulations, we extract the entropy at the critical temperature for magnetic order from a finite-size scaling analysis. For the antiferromagnetic case, the critical entropy density equals $0.341(5){k}_{B}$, whereas for the ferromagnet, a larger value of $0.401(5){k}_{B}$ is obtained. We compare our simulation results to estimates put forward recently in studies assessing means of realizing the antiferromagnetic N\'eel state in ultracold fermion gases in optical lattices.

DOI
10.1103/physrevb.81.052405

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We analyze the temperature dependence of the entropy of the spin-1/2 Heisenberg model on the three-dimensional simple-cubic lattice, for both the case of antiferromagnetic and ferromagnetic nearest-neighbor exchange interactions. Using optimized extended ensemble quantum Monte Carlo simulations, we extract the entropy at the critical temperature for magnetic order from a finite-size scaling analysis. For the antiferromagnetic case, the critical entropy density equals 0.341(5)kB0.341(5){k}_{B}0.341(5)kB​, whereas for the ferromagnet, a larger value of 0.401(5)kB0.401(5){k}_{B}0.401(5)kB​ is obtained. We compare our simulation results to estimates put forward recently in studies assessing means of realizing the antiferromagnetic N'eel state in ultracold fermion gases in optical lattices.
The three-dimensional spin-1/2 Heisenberg model on a simple cubic lattice is studied for ferromagnetic and antiferromagnetic interactions using the Decoupled Cell Method for quantum Monte Carlo. Results for the relaxation time τL are determined for both ferromagnetic and antiferromagnetic systems and found to be similar to those found for the classical (s → ∞) Heisenberg model. The scaling of τL with system size is used to extract the dynamical critical exponent z for the two systems. The values of z = 1.98 ± 0.12 for the ferromagnet and z = 1.94 ± 0.09 for the antiferromagnet are in good agreement with theoretical predictions and previous Monte Carlo studies of the classical Heisenberg model.
Abstract The Monte Carlo replica technique is used to study phase transitions and the thermodynamic and critical properties of the three-dimensional Heisenberg antiferromagnetic model on the body-centered cubic lattice with the inclusion of the interaction of the nearest and the next-nearest neighbors. The studies are performed for the proportions of the values of the exchange interactions of the nearest and the next-nearest neighbors in the range of values of k [0.0, 0.6]. The phase transition character is analyzed based on the histogram method. The overall set of static critical indices is calculated in terms of the theory of the finite-dimensional scaling. It is shown that the class of the critical behavior universality of this model is conserved in the above range of values of k.
Phase transitions and critical properties in the antiferromagnetic Heisenberg model on a layered cubic lattice with allowance for intralayer next nearest neighbor interactions have been studied using the replica Monte Carlo algorithm. The character of phase transitions has been analyzed using the histogram method and the Binder cumulant method. It has been found that a transition from the collinear to paramagnetic phase in the model under study occurs as a second order phase transition. The statistical critical exponents of the specific heat α, susceptibility γ, order parameter β, and correlation radius ν, as well as the Fisher index η, have been calculated using the finite-size scaling theory. It has been shown that the three-dimensional Heisenberg model on the layered cubic lattice with allowance for the next nearest neighbor interaction belongs to the same universality class of the critical behavior as the antiferromagnetic Heisenberg model on a layered triangular lattice.
Using extensive Monte Carlo simulations, we study the equilibrium properties of the simple-cubic, classical Heisenberg ferromagnet. We employ very long runs for L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L lattices to obtain high-precision data for the magnetization probability distribution. Using finite-size scaling for L\ensuremath{\le}24 and an optimized multiple-histogram data analysis, we obtain an accurate value of the inverse critical temperature J/kB{\mathit{k}}_{\mathit{B}}kB​Tc{\mathit{T}}_{\mathit{c}}Tc​=0.6929\ifmmode\pm\else\textpm\fi{}0.0001, which is higher than previously accepted estimates. Calculated values of various static exponents are in excellent agreement with renormalization-group and \ensuremath{\epsilon}-expansion predictions.
Using both recently developed cluster-algorithm and histogram methods, we have carried out a high-resolution Monte Carlo study of static critical properties of classical ferromagnetic Heisenberg models. Extensive Monte Carlo simulations were performed at several temperatures in the critical region, using an improved cluster-updating scheme, on L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L simple-cubic and body-centered-cubic systems with L\ensuremath{\le}40. Thermodynamic quantities as a function of temperature in the vicinity of the critical point were obtained by an optimized multiple-histogram method, and the critical temperature and static critical exponents were extracted using finite-size scaling. Our best estimates for the inverse critical temperatures are 0.693 035(37) for the simple-cubic system and 0.486 798(12) for the body-centered-cubic system. Estimated static critical exponents for both systems agree with each other within their respective error bars, and the mean estimates \ensuremath{\nu}=0.7048(30) and \ensuremath{\gamma}=1.3873(85) are also in excellent agreement with field-theoretic predictions 0.705(3) and 1.386(4).
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.