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arXiv:cond-mat/9804234 · Physical Review Letters

Critical Temperature and the Transition from Quantum to Classical Order Parameter Fluctuations in the Three-Dimensional Heisenberg Antiferromagnet

Anders W. Sandvik

Published Jun 8, 1998• 108 citations
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Abstract

We present results of extensive quantum Monte Carlo simulations of the three-dimensional $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ Heisenberg antiferromagnet. Finite-size scaling of the spin stiffness and the sublattice magnetization gives the critical temperature ${T}_{\mathrm{c}}/J\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.946\ifmmode\pm\else\textpm\fi{}0.001$. The critical behavior is consistent with the classical 3D Heisenberg universality class, as expected. We discuss the general nature of the transition from quantum mechanical to classical (thermal) order parameter fluctuations at a continuous ${T}_{\mathrm{c}}>0$ phase transition.

DOI
10.1103/physrevlett.80.5196

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We present results of extensive quantum Monte Carlo simulations of the three-dimensional S=1/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2S=1/2 Heisenberg antiferromagnet. Finite-size scaling of the spin stiffness and the sublattice magnetization gives the critical temperature Tc/J=0.946\ifmmode±\else\textpm\fi0.001{T}_{\mathrm{c}}/J\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.946\ifmmode\pm\else\textpm\fi{}0.001Tc​/J=0.946\ifmmode±\else\textpm\fi0.001. The critical behavior is consistent with the classical 3D Heisenberg universality class, as expected. We discuss the general nature of the transition from quantum mechanical to classical (thermal) order parameter fluctuations at a continuous Tc>0{T}_{\mathrm{c}}>0Tc​>0 phase transition.
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.
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.
Spin-glass (SG) and chiral-glass (CG) orderings in three-dimensional (3D) Heisenberg spin glass with and without magnetic anisotropy are studied by using large-scale off-equilibrium Monte Carlo simulations. A characteristic time of relaxation, which diverges at a transition temperature in the thermodynamic limit, is obtained as a function of the temperature and the system size. Based on the finite-size-scaling analysis for the relaxation time, it is found that in the isotropic Heisenberg spin glass, the CG phase transition occurs at a finite temperature, while the SG transition occurs at a lower temperature, which is compatible with zero. Our results of the anisotropic case support the chirality scenario for the phase transitions in the 3D Heisenberg spin glasses.
We study the quantum phase transition out of the N\'eel state in SU(3) and SU(4) generalizations of the Heisenberg antiferromagnet with a sign-problem-free four-spin coupling (so-called JQJQJQ model), by extensive quantum Monte Carlo simulations. We present evidence that the SU(3) and SU(4) order parameters and the SU(3) and SU(4) stiffness go to zero continuously without any evidence of a first-order transition. However, we find considerable deviations from simple scaling laws for the stiffness even in the largest system sizes studied. We interpret these as arising from multiplicative scaling terms in these quantities that affect the leading behavior, i.e., they will persist in the thermodynamic limit, unlike the conventional additive corrections from irrelevant operators. We conjecture that these multiplicative terms arise from dangerously irrelevant operators whose contributions to the quantities of interest are nonanalytic.
We have carried out extensive Monte Carlo simulations of the dynamic critical behavior of L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L simple-cubic classical Heisenberg models with periodic boundary conditions. The equilibrium relaxation time \ensuremath{\tau} is determined from the long-time exponential decay of the time-displaced correlation function of the magnetization. Using finite-size scaling and corrections to scaling we find that at the critical point Tc{\mathit{T}}_{\mathit{c}}Tc​=(0.6929${)}^{\mathrm{\ensuremath{-}}1}$, \ensuremath{\tau} grows like \ensuremath{\tau}\ensuremath{\simeq}Lz{\mathit{L}}^{\mathit{z}}Lz with z=1.96(6). In the paramagnetic region we show that the relaxation time is represented by a universal scaling function \ensuremath{\tau}${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{z}}$ vs (T/Tc{\mathit{T}}_{\mathit{c}}Tc​-1)${\mathit{L}}^{1/\ensuremath{\nu}}$ with z=1.96(6). This estimate is consistent with theoretical predictions and results of experimental studies concerning the dynamical critical behavior of Heisenberg magnets perturbed by order-parameter nonconserving dipolar interactions. We also describe how a simple over-relaxation algorithm of Creutz and of Brown and Woch can be incorporated into the Metropolis sampling method for the classical Heisenberg model, thereby significantly reducing the critical slowing down; the resultant correlation time varies as \ensuremath{\tau}\ensuremath{\sim}(T/Tc{\mathit{T}}_{\mathit{c}}Tc​-1${)}^{\mathrm{\ensuremath{-}}0.77}$, which implies z\ensuremath{\simeq}1.10.
We investigate the quantum phase transition in an S=1/2S=1/2S=1/2 dimerized Heisenberg antiferromagnet in three spatial dimensions. By performing large-scale quantum Monte Carlo simulations and detailed finite-size scaling analyses, we obtain high-precision results for the quantum critical properties at the transition from the magnetically disordered dimer-singlet phase to the antiferromagnetically ordered N\'eel phase. This transition breaks O(N)\text{O}(N)O(N) symmetry with N=3N=3N=3 in D=3+1D=3+1D=3+1 dimensions. This is the upper critical dimension, where multiplicative logarithmic corrections to the leading mean-field critical properties are expected; we extract these corrections, establishing their precise forms for both the zero-temperature staggered magnetization ms{m}_{s}ms​ and the N\'eel temperature TN{T}_{N}TN​. We present a scaling ansatz for TN{T}_{N}TN​, including logarithmic corrections, which agrees with our data and indicates exact linearity with ms{m}_{s}ms​, implying a complete decoupling of quantum and thermal fluctuation effects even arbitrarily close to the quantum critical point. We also demonstrate the predicted NNN-independent leading and subleading logarithmic corrections in the size dependence of the staggered magnetic susceptibility. These logarithmic scaling forms have not previously been identified or verified by unbiased numerical methods, and we discuss their relevance to experimental studies of dimerized quantum antiferromagnets such as TlCuCl3{\mathrm{TlCuCl}}_{3}TlCuCl3​.