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Monte Carlo studies of two dimensional quantum spin systems

M. S. Makivić

Published Jul 20, 2007• 0 citations
Publisher

Abstract

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.

DOI
10.7907/tgsp-gc60

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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 two-dimensional spin-1/2 XY model is investigated via an extensive quantum Monte Carlo simulation on square lattices as large as 128\ifmmode\times\else\texttimes\fi{}128. The transverse susceptibility and correlation length show precise thermodynamic scaling behaviors of a phase transition of the Kosterlitz-Thouless type at kTc{\mathit{kT}}_{\mathit{c}}kTc​/J=0.353(3). The correlation functions near the transition point exhibit a universal scaling behavior. Various critical exponents are determined and our results demonstrate that the universality class of the two-dimensional quantum system with continuous symmetry is the same as its classical analog. The specific heat exhibits a finite peak around kT/J=0.45, decreases very rapidly near Tc{\mathit{T}}_{\mathit{c}}Tc​, and falls off as T2{\mathit{T}}^{2}T2 near T=0, validating the spin-wave treatment. The ground-state energy is -0.5543(1)J per spin.
Convincing numerical evidence is obtained via extensive quantum Monte Carlo simulations on square lattices as large as 96\ifmmode\times\else\texttimes\fi{}96 that the spin-1/2 XY model undergoes a Kosterlitz-Thouless phase transition at kTc{\mathit{kT}}_{\mathit{c}}kTc​/J=0.350(4). Correlation length and in-plane susceptibility diverge at Tc{\mathit{T}}_{\mathit{c}}Tc​ precisely according to the form predicted by Kosterlitz and Thouless for the classical XY model. The specific heat increases very rapidly near Tc{\mathit{T}}_{\mathit{c}}Tc​ and exhibits a peak around kT/J=0.45.
Results of Monte Carlo simulations of the ferromagnetic classical XXZ model on a square lattice are reported for interactions of the out of plane spin components \ensuremath{\lambda} varying between 0 and 0.99. For all the value of \ensuremath{\lambda} the signature of a Berezinskii-Kosterlitz-Thouless (BKT) phase transition was observed, and the corresponding critical temperature was located both by BKT fits and finite-size scaling methods. Data for in-plane and out of plane susceptibility, correlation length, specific heat, and internal energy are given in a broad range of temperature and for lattice sizes ranging from 32\ifmmode\times\else\texttimes\fi{}32 to 256\ifmmode\times\else\texttimes\fi{}256.
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.
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.
In this work, we benchmark the well-controlled and numerically accurate exponential thermal tensor renormalization group (XTRG) in the simulation of interacting spin models in two dimensions. Finite temperature introduces a finite thermal correlation length $\ensuremath{\xi}$, such that for system sizes $L\ensuremath{\gg}\ensuremath{\xi}$ finite-size calculations actually simulate the thermodynamic limit. In this paper, we focus on the square lattice Heisenberg antiferromagnet (SLH) and quantum Ising models (QIM) on open and cylindrical geometries up to width W=10W=10W=10. We explore various one-dimensional mapping paths in the matrix product operator (MPO) representation, whose performance is clearly shown to be geometry dependent. We benchmark against quantum Monte Carlo (QMC) data, yet also the series-expansion thermal tensor network results. Thermal properties including the internal energy, specific heat, and spin structure factors, etc. are computed with high precision, obtaining excellent agreement with QMC results. XTRG also allows us to reach remarkably low temperatures. For SLH, we obtain an energy per site ${u}_{g}^{*}\ensuremath{\simeq}\ensuremath{-}0.6694(4)$ and a spontaneous magnetization ${m}_{S}^{*}\ensuremath{\simeq}0.30(1)$ already consistent with the ground-state properties, which is obtained from extrapolated low-TTT thermal data on $W\ensuremath{\le}8$ cylinders and $W\ensuremath{\le}10$ open strips, respectively. We extract an exponential divergence versus TTT of the structure factor S(M)S(M)S(M), as well as the correlation length $\ensuremath{\xi}$, at the ordering wave vector $M=(\ensuremath{\pi},\ensuremath{\pi})$, which represents the renormalized classical behavior and can be observed over a narrow but appreciable temperature window, by analyzing the finite-size data by XTRG simulations. For the QIM with a finite-temperature phase transition, we employ several thermal quantities, including the specific heat, Binder ratio, as well as the MPO entanglement to determine the critical temperature Tc{T}_{c}Tc​.