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arXiv:hep-th/0611264 · Journal of Physics A Mathematical and Theoretical

Quasi-long-range ordering in a finite-size 2D classical Heisenberg model

O. Kapikranian, Bertrand Berche, Yurij V. Holovatch

Published Mar 20, 2007• 20 citations
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Abstract

We analyse the low-temperature behaviour of the classical isotropic ferromagnetic Heisenberg model on a two-dimensional square lattice of finite size. Presence of a residual magnetization in a finite-size system enables us to use a low-temperature approximation, which is however more restricting than the usual spin-wave approximation known to give reliable results for the XY model at low temperatures T . For the system considered, we find that the spin–spin correlation function decays as 1/ r η( T ) for large separations r bringing about the presence of a quasi-long-range ordering. We give analytic estimates for the exponent η( T ) in different regimes and support our findings by Monte Carlo simulations of the model on lattices of different sizes at different temperatures.

DOI
10.1088/1751-8113/40/14/001

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We analyse the low-temperature behaviour of the classical isotropic ferromagnetic Heisenberg model on a two-dimensional square lattice of finite size. Presence of a residual magnetization in a finite-size system enables us to use a low-temperature approximation, which is however more restricting than the usual spin-wave approximation known to give reliable results for the XY model at low temperatures T . For the system considered, we find that the spin–spin correlation function decays as 1/ r η( T ) for large separations r bringing about the presence of a quasi-long-range ordering. We give analytic estimates for the exponent η( T ) in different regimes and support our findings by Monte Carlo simulations of the model on lattices of different sizes at different temperatures.
We study the square-lattice XY model in the presence of random phase shifts. We consider two different disorder distributions with zero average shift and investigate the low-temperature quasi-long-range order phase which occurs for sufficiently low disorder. By means of Monte Carlo simulations we determine several universal quantities which are then compared with the analytic predictions of the random spin-wave theory. We observe a very good agreement which indicates that the universal long-distance behavior in the whole low-disorder low-temperature phase is fully described by the random spin-wave theory.
The thermodynamic properties and critical behavior of a Heisenberg ferromagnet in two dimensions are studied using the Handscomb quantum Monte Carlo method. The spins (with quantum number S=12) are arranged on a square lattice and have a long-range interaction with a power-law decrease, J(r)∝r−p, with distance r. The parameter p is taken to be in the range 2<p⩽6. The system has an ordering transition at a nonzero temperature Tc when 2<p<4. The dependence of Tc on p and the critical behavior of the susceptibility are investigated. Properties of spin correlation functions in different regimes (of p and T) are discussed. The results are compared with previous approximate analytic quantum theories.
We have considered a classical spin system, consisting of 3-component unit vectors, associated with a two-dimensional lattice {uk, k ∈ Z2}, and interacting via a translationally and rotationally invariant pair potential, of the long-range ferromagnetic form [Formula: see text] Here ∊ is a positive constant setting energy and temperature scales (i.e. T* = k B T/∊), and xj denotes dimensionless coordinates of the lattice sites. This potential model is known rigorously to possess an ordering transition at finite temperature, and has been characterized quantitatively by Monte Carlo simulation, whose results suggest a continuous transition taking place at [Formula: see text]; comparison is also reported with other theoretical treatments, such as Spherical Model, Molecular Field and Two Site Cluster approximations.
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.
The thermodynamic properties of a classical spin-SSS Heisenberg ferromagnetic chain with long-range interactions in the presence of an external magnetic field are investigated by means of the spectral density method within the framework of the two-time Green's functions in classical statistical mechanics. For exchange interactions decaying as ${r}^{\ensuremath{-}\ensuremath{\alpha}},$ it is shown that long-range order exists at finite temperature for $1<\ensuremath{\alpha}<2$ and that no transition to a ferromagnetic phase at any finite temperature occurs for $\ensuremath{\alpha}>~2,$ consistently with a recent extension of the Mermin-Wagner theorem to spin models with long-range interactions. Besides, by means of a suitable modification of the lowest-order moment equations for the spectral density, the critical temperature and the main critical exponents for $1<\ensuremath{\alpha}<2$ and the low-temperature behavior of the paramagnetic susceptibility for $\ensuremath{\alpha}>~2$ are also obtained. Our results appear in a reasonable agreement with the available exact ones and Monte Carlo calculations.