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Physical review. B, Condensed matter

Monte Carlo studies of one-dimensional quantum Heisenberg and XY models

John J. Cullen, David Paul Landau

Published Jan 1, 1983• 118 citations
Publisher

Abstract

The Suzuki-Trotter transformation has been used to transform $N$-spin $S=\frac{1}{2}$ Heisenberg and $\mathrm{XY}$ chains into two-dimensional ($N\ifmmode\times\else\texttimes\fi{}2m$) classical Ising models with complicated interactions. A Monte Carlo method using multispin flipping techniques is then used to study these models for $\frac{\mathrm{kT}}{|J|}\ensuremath{\ge}0.0125$ and $m\ensuremath{\le}10$. Finite-size scaling is used to analyze the low-temperature susceptibility of the ferromagnetic Heisenberg chain, and we find a power-law divergence $\ensuremath{\chi}={T}^{\ensuremath{-}\ensuremath{\gamma}}$ with $\ensuremath{\gamma}=1.32\ifmmode\pm\else\textpm\fi{}0.13$. Difficulties in the method, particularly due to singular bonding, are discussed.

DOI
10.1103/physrevb.27.297

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The Suzuki-Trotter transformation has been used to transform NNN-spin S=12S=\frac{1}{2}S=21​ Heisenberg and XY\mathrm{XY}XY chains into two-dimensional (N\ifmmode×\else\texttimes\fi2mN\ifmmode\times\else\texttimes\fi{}2mN\ifmmode×\else\texttimes\fi2m) classical Ising models with complicated interactions. A Monte Carlo method using multispin flipping techniques is then used to study these models for kT∣J∣\ensuremath≥0.0125\frac{\mathrm{kT}}{|J|}\ensuremath{\ge}0.0125∣J∣kT​\ensuremath≥0.0125 and m\ensuremath≤10m\ensuremath{\le}10m\ensuremath≤10. Finite-size scaling is used to analyze the low-temperature susceptibility of the ferromagnetic Heisenberg chain, and we find a power-law divergence \ensuremathχ=T\ensuremath−\ensuremathγ\ensuremath{\chi}={T}^{\ensuremath{-}\ensuremath{\gamma}}\ensuremathχ=T\ensuremath−\ensuremathγ with \ensuremathγ=1.32\ifmmode±\else\textpm\fi0.13\ensuremath{\gamma}=1.32\ifmmode\pm\else\textpm\fi{}0.13\ensuremathγ=1.32\ifmmode±\else\textpm\fi0.13. Difficulties in the method, particularly due to singular bonding, are discussed.
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.
We propose a new efficient scheme for the quantum Monte Carlo study of quantum critical phenomena in quantum spin systems. Rieger and Young's Trotter-number-dependent finite-size scaling in quantum spin systems and Ito et al 's evaluation of the transition point with the non-equilibrium relaxation in classical spin systems are combined and generalized. That is, only one Trotter number and one inverse temperature proportional to system sizes are taken for each system size, and the transition point of the transformed classical spin model is estimated as the point at which the order parameter shows power-law decay. The present scheme is confirmed by the determination of the critical phenomenon of the one-dimensional asymmetric XY model in the ground state. The estimates of the transition point and the critical exponents , and are in good agreement with the exact solutions. The dynamical critical exponent is also estimated as , which is consistent with that of the two-dimensional Ising model.
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.