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Physical review. B, Condensed matter

Low-temperature behavior of the correlation length and the susceptibility of a quantum Heisenberg ferromagnet in two dimensions

Peter Kopietz, Sudip Chakravarty

Published Sep 1, 1989• 63 citations
Publisher

Abstract

The low-temperature behavior of quantum Heisenberg models can be obtained by combining a renormalization-group analysis of the quantum models with a Monte Carlo simulation of the corresponding classical model, where the ``spins'' are represented by unit vectors defined on a lattice. In this approach renormalization-group equations for the quantum models lead to an effective classical O(3) nonlinear \ensuremath{\sigma} model in appropriate regimes of the parameter space. In the present work we apply this technique, which was recently used by Chakravarty, Halperin, and Nelson to discuss the low-temperature behavior of quantum antiferromagnets, to ferromagnets and calculate the correlation length and the susceptibility of a two-dimensional quantum ferromagnet at low temperatures. The results obtained with this method are then compared with those obtained from a direct simulation of the spin S=1/2 quantum ferromagnet. Excellent agreement between the two approaches leads us to believe that the analytic expressions obtained in the renormalization-group method are accurate at low temperatures. This strengthens the validity of the procedure adopted for antiferromagnets as well, where no such simulation results are yet known. The present paper also further substantiates recent work on the properties of solid $^{3}\mathrm{He}$ in two dimensions.

DOI
10.1103/physrevb.40.4858

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The low-temperature behavior of quantum Heisenberg models can be obtained by combining a renormalization-group analysis of the quantum models with a Monte Carlo simulation of the corresponding classical model, where the ``spins'' are represented by unit vectors defined on a lattice. In this approach renormalization-group equations for the quantum models lead to an effective classical O(3) nonlinear \ensuremath{\sigma} model in appropriate regimes of the parameter space. In the present work we apply this technique, which was recently used by Chakravarty, Halperin, and Nelson to discuss the low-temperature behavior of quantum antiferromagnets, to ferromagnets and calculate the correlation length and the susceptibility of a two-dimensional quantum ferromagnet at low temperatures. The results obtained with this method are then compared with those obtained from a direct simulation of the spin S=1/2 quantum ferromagnet. Excellent agreement between the two approaches leads us to believe that the analytic expressions obtained in the renormalization-group method are accurate at low temperatures. This strengthens the validity of the procedure adopted for antiferromagnets as well, where no such simulation results are yet known. The present paper also further substantiates recent work on the properties of solid 3He^{3}\mathrm{He}3He in two dimensions.
The renormalization-group recursion relations for the spin-S nearest-neighbor quantum Heisenberg ferromagnet recently derived by Chakravarty and the present author are analyzed in one dimension. It is found that the susecptibility \ensuremath{\chi} is given at low temperatures T by T\ensuremath{\chi}=${C}_{\ensuremath{\chi}}^{(0)}$(JS2{\mathrm{JS}}^{2}JS2/T) [1+${C}_{\ensuremath{\chi}}^{(1)}$\ensuremath{\eta}+O(${\ensuremath{\eta}}^{2}$)], where J is the exchange coupling and the dimensionless expansion parameter \ensuremath{\eta} is given by \ensuremath{\eta}=${\ensuremath{\pi}}^{\mathrm{\ensuremath{-}}1}$(T/JS3{\mathrm{JS}}^{3}JS3)1/2{)}^{1/2})1/2. For the correlation length \ensuremath{\xi} we find (\ensuremath{\xi}/a)=${C}_{\ensuremath{\xi}}^{(0)}$(JS2{\mathrm{JS}}^{2}JS2/T) [1+${C}_{\ensuremath{\xi}}^{(1)}$\ensuremath{\eta}+O(${\ensuremath{\eta}}^{2}$)], where a is the lattice spacing. The values of the coefficients ${C}_{\ensuremath{\chi}}^{(j)}$ and ${C}_{\ensuremath{\xi}}^{(j)}$ (j=0,1), have been determined by fitting the above expressions with Monte Carlo data obtained for the S=1/2 Heisenberg chain at temperatures as low as T/J=0.01.
Monte Carlo renormalization-group methods were applied to the classical three-component Heisenberg model on a two-dimensional lattice. Expectation values of local correlations of spins and various sized block spins were computed using traditional Monte Carlo methods. By matching quantities at different length scales generated by different Monte Carlo Hamiltonians we directly determined the renormalization of the nearest-neighbor coupling. Using these data and the results of high-temperature expansions and low-temperature renormalization-group calculations we have determined that this model does not have a phase transition. We have also obtained the amplitude for the low-temperature divergence of the susceptibility and the correlation length.
The scaling function for the spin correlation length \ensuremath{\xi} of the two-dimensional (2D) quantum nonlinear \ensuremath{\sigma} model is obtained by Monte Carlo simulations. This is used to extract the temperature dependence of the spin correlation length, which sheds light on the nature of the ground state of low-dimensional quantum antiferromagnets. A crossover from quantum critical to renormalized classical behavior of the correlation length is found. Comparison is made with numerical results on the 2D square lattice Heisenberg model, as well as with neutron-scattering and nuclear magnetic relaxation experiments.
The renormalization-group method is reformulated in a way that allows the analysis of two-dimensional Heisenberg ferromagnets with small spin. A set of recursion equations is derived in a one-loop approximation. The low-temperature asymptotics of the correlation length and the uniform susceptibility are obtained. The comparison of the results with the results obtained within the momentum shell recursion technique shows that even at one-loop approximation there is a sensitive renormalization due to small spin.
Spin-1/2 nearest neighbor Heisenberg antiferromagnet and XY model on a square lattice are studied via large scale quantum Monte Carlo simulations using a fast and efficient multispin coding algorithm on the Caltech/JPL MarkIIIfp parallel supercomputer, based on the Suzuki-Trotter transformation. We performed simulations with very good statistics on lattices as large as 128x128 spins, in the temperature range from 0.1 to 2.5 in units of the effective exchange coupling J. We calculated energy, specific heat, magnetic susceptibilities and also spin correlation functions from which we deduce the correlation lengths. For the Heisenberg model, at temperatures higher than J the results are in excellent agreement with high-temperature series expansion. At low temperatures the long wavelength behavior is essentially classical. Our data show that the correlation length and staggered susceptibility are quantitatively well described by the renormalized classical picture at the 2-loop level of approximation. From the divergence of the correlation length, we deduce the value of the quantum renormalized spin stiffness, [rho][subscript s]/J = 0.199(2). We give evidence that the correlation function is of the Ornstein-Zernicke type. By comparing the largest measured correlation lengths with neutron scattering experiments on La2CuO4, we deduce the value of effective exchange coupling J = 1450±30 K. By measuring the imaginary time-dependent correlation functions, we show that the dynamics of the model can be well understood within a Bose liquid-type picture. The spin waves are rather sharp throughout most of the Brillouin zone and the damping is weakly dependent on the wave vector. In the case of the XY model, convincing numerical evidence is obtained on square lattices as large as 96x96 that the spin-1/2 XY model undergoes a Kosterlitz-Thouless (KT) phase transition at kTc/J = 0.350(4). The correlation length and in-plane susceptibility diverge at Tc precisely according to the form predicted by Kosterlitz and Thouless for the classical XY model. The specific heat increases very rapidly on heating near Tc and exhibits a peak around kT/J = 0.45. We also measure the spin stiffness and the correlation function exponent below the transition temperature. Within the statistical accuracy of the measurements, the results are well described by the square root singularity (with a nonuniversal amplitude) below Tc, and they have the universal values in agreement with KT theory at Tc.
The critical behavior of 2D magnetic easy-plane systems has mainly been studied by the classical XY model. However, the z components of the spins have to be considered in order to describe real systems, and their fluctuations cannot positively be neglected when quantum effects are to be included, quantum spins being intrinsically three-component objects. Therefore, Monte Carlo simulations are performed for the Heisenberg ferromagnet with easy-plane anisotropy (XXZ model) on a two-dimensional square lattice with a twofold aim: first, to obtain accurate quantitative results about the critical behavior of the classical model, showing the relevant role played by the out-of-plane fluctuations; second, to open the way for approaching the quantum thermodynamics by means of the effective Hamiltonian method that reduces the quantum thermodynamics of the XXZ ferromagnet to the investigation of an effective classical model with temperature-dependent renormalized interaction parameters. Specific heat, magnetic susceptibility, and correlation length are calculated in the critical region for lattice sizes up to 128×128. These quantities preserve the Kosterlitz–Thouless behavior of the XY model. For the transition temperature of the classical XX0 model we obtain the estimate kBTc/(JS̃2)=0.69±0.01.