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 in two dimensions.
We study the continuum limit of the quantum nonlinear \ensuremath{\sigma} model in 2+1 dimensions and at finite temperature T using both Monte Carlo simulation on large-size lattices (
1002\ifmmode\times\else\texttimes\fi{}8 is our largest-size lattice) and saddle-point approximation. At zero temperature, we find the critical point
gc that separates the quantum disordered phase from the phase with spontaneous symmetry breaking (nonzero staggered magnetization). We calculate the model's renormalization group \ensuremath{\beta} function close to the critical point. Using the \ensuremath{\beta} function, we rescale the correlation lengths calculated at various values of the coupling constant (spin stiffness) and temperature and find that they all collapse on the same curve \ensuremath{\xi}/${a}_{\ensuremath{\sigma}}$=f(T/${T}_{\ensuremath{\sigma}}$). Even though the lattice spacing vanishes, a finite unit of length ${a}_{\ensuremath{\sigma}}$ and a temperature scale ${T}_{\ensuremath{\sigma}}$ is generated via dimensional transmutation. Assuming that the nonlinear \ensuremath{\sigma} model and the spin-(1/2 antiferromagnetic (AF) Heisenberg model are equivalent at low temperature, we relate the units ${a}_{\ensuremath{\sigma}}$ and ${T}_{\ensuremath{\sigma}}$ to the lattice spacing
aH and the AF coupling J of the Heisenberg model so that the correlation lengths obtained from the simulation of the two models agree. In order to achieve this agreement we find that (a) the spin-(1/2 AF Heisenberg model should order at T=0 and (b) the relationship between the scales ${a}_{\ensuremath{\sigma}}$,${T}_{\ensuremath{\sigma}}$ and
aH, J is obtained, and f(T/${T}_{\ensuremath{\sigma}}$) can be accurately approximated by an exponential of ${T}_{\ensuremath{\sigma}}$/T below
gc. We obtain a reasonable fit to the neutron scattering data of the insulator
La2CuO4 by taking J=1270 K, a value close to that reported by Raman scattering experiments.
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.
Analytic expressions for the temperature dependences of the correlation length are given for antiferromagnetic spin-
21 Heisenberg ladders using a finite-size nonlinear \ensuremath{\sigma} model approach. These calculations rely on identifying three successive crossover regimes as a function of temperature. In each of these regimes, precise and controlled approximations are formulated. The analytical results are found to be in excellent agreement with Monte Carlo simulations for the Heisenberg Hamiltonian.
In this thesis the quantum Monte Carlo loop algorithm is applied to study the properties of correlation functions in low-dimensional quantum antiferromagnets. Two different types of lattice models are investigated. The first system is the quantum critical XXZ chain. The second class of models emerges when coupling the square lattice Heisenberg model to phonons. In case of the quantum critical XXZ chain, the loop algorithm is used to compute the two-point spin and dimer correlation functions at low temperatures. Starting from the well-known asymptotic expressions from conformal field theory for the decay of the correlations, the crossover from the finite-temperature to the ground state behavior is investigated. In addtion, the Monte Carlo data are used to extract numerical estimates for the correlation amplitudes. The deviations from asymptotic scaling at low finite temperatures and intermediate distances are examined. The results are also used to study the algebraic and logarithmic corrections in the ground state. The method is further applied to compute the low-temperature spin correlations of the square lattice Heisenberg model coupled to dispersionless Einstein phonons. Also, statically dimerized Heisenberg models in two dimensions are investigated. Here the quantum Monte Carlo technique is used to determine the specific pattern of dimerized bonds which has the largest gain in the magnetic energy. A direct comparison of ground state energies is only possible for a small number of dimerized systems. For this reason an alternative approach is developed, which allows to study a significantly larger class of models. Starting from a Landau expansion of the free energy, this amounts to a computation of specific structure factors of the dimer correlations of the uniform Heisenberg model. As a consequence, a Landau theory for the two-dimensional counterpart of the spin-Peierls transition is established. This adiabatic approach also applies to the Heisenberg model coupled to dispersive phonons. In this case the data for the dimer correlations are used to study the impact of the spin-phonon coupling on the phonon dispersion relations. In particular, the softening of phonon modes is investigated, which is relevant for experimental studies on substances with a quasi two-dimensional magnetic structure.
We discuss the renormalization of the quantum antiferromagnet in two dimensions. With an analysis of the spin-wave Hamiltonian of the quantum Heisenberg model or the quantum nonlinear \ensuremath{\sigma} model (NL\ensuremath{\sigma}M), we consider O(n) invariant observable 〈
Sl\ensuremath{\cdot}
Sj〉 and evaluate \ensuremath{\beta} functions in the dimensional regularization method up to two-loop order. We show that quantum and classical NL \ensuremath{\sigma} M are connected by a simple crossover function. It follows from this crossover phenomenon that the critical temperature
Tc for d>2 is reduced due to the quantum effects and that the thermal fluctuations reduce the critical coupling constant
gc. The disappearance of
gc indicates thermal restoration of the asymptotic freedom. Finally we present an exact expression of the correlation length.