Coarse-grained loop algorithms for Monte Carlo simulation of quantum spin systems

Recently, Syljuasen and Sandvik proposed a new framework for constructing algorithms of quantum Monte Carlo simulation. While it includes new classes of powerful algorithms, it is not straightforward to find an efficient algorithm for a given model. Based on their framework, we propose an algorithm that is a natural extension of the conventional loop algorithm with the split-spin representation. A complete table of the vertex density and the worm-scattering probability is presented for the general XXZ model of an arbitrary S with a uniform magnetic field.Comment: 12 pages, 7 figures, insert a word "squared" in the first line of the caption of Fig.7 and correct the label of vertical axis of Fig.

Accessing the dynamics of large many-particle systems using Stochastic Series Expansion

The Stochastic Series Expansion method (SSE) is a Quantum Monte Carlo (QMC) technique working directly in the imaginary time continuum and thus avoiding "Trotter discretization" errors. Using a non-local "operator-loop update" it allows treating large quantum mechanical systems of many thousand sites. In this paper we first give a comprehensive review on SSE and present benchmark calculations of SSE's scaling behavior with system size and inverse temperature, and compare it to the loop algorithm, whose scaling is known to be one of the best of all QMC methods. Finally we introduce a new and efficient algorithm to measure Green's functions and thus dynamical properties within SSE.Comment: 11 RevTeX pages including 7 figures and 5 table

Stochastic series expansion method with operator-loop update

A cluster update (the ``operator-loop'') is developed within the framework of a numerically exact quantum Monte Carlo method based on the power series expansion of exp(-BH) (stochastic series expansion). The method is generally applicable to a wide class of lattice Hamiltonians for which the expansion is positive definite. For some important models the operator-loop algorithm is more efficient than loop updates previously developed for ``worldline'' simulations. The method is here tested on a two-dimensional anisotropic Heisenberg antiferromagnet in a magnetic field.Comment: 5 pages, 4 figure

Quantum Monte Carlo with Directed Loops

We introduce the concept of directed loops in stochastic series expansion and path integral quantum Monte Carlo methods. Using the detailed balance rules for directed loops, we show that it is possible to smoothly connect generally applicable simulation schemes (in which it is necessary to include back-tracking processes in the loop construction) to more restricted loop algorithms that can be constructed only for a limited range of Hamiltonians (where back-tracking can be avoided). The "algorithmic discontinuities" between general and special points (or regions) in parameter space can hence be eliminated. As a specific example, we consider the anisotropic S=1/2 Heisenberg antiferromagnet in an external magnetic field. We show that directed loop simulations are very efficient for the full range of magnetic fields (zero to the saturation point) and anisotropies. In particular for weak fields and anisotropies, the autocorrelations are significantly reduced relative to those of previous approaches. The back-tracking probability vanishes continuously as the isotropic Heisenberg point is approached. For the XY-model, we show that back-tracking can be avoided for all fields extending up to the saturation field. The method is hence particularly efficient in this case. We use directed loop simulations to study the magnetization process in the 2D Heisenberg model at very low temperatures. For LxL lattices with L up to 64, we utilize the step-structure in the magnetization curve to extract gaps between different spin sectors. Finite-size scaling of the gaps gives an accurate estimate of the transverse susceptibility in the thermodynamic limit: chi_perp = 0.0659 +- 0.0002.Comment: v2: Revised and expanded discussion of detailed balance, error in algorithmic phase diagram corrected, to appear in Phys. Rev.

Critical exponents of the quantum phase transition in a planar antiferromagnet

We have performed a large scale quantum Monte Carlo study of the quantum phase transition in a planar spin-1/2 Heisenberg antiferromagnet with CaV4O9 structure. We obtain a dynamical exponent z=1.018+/-0.02. The critical exponents beta, nu and eta agree within our errors with the classical 3D O(3) exponents, expected from a mapping to the nonlinear sigma model. This confirms the conjecture of Chubukov, Sachdev and Ye [Phys. Rev. B 49, 11919 (1994)] that the Berry phase terms in the planar Heisenberg antiferromagnet are dangerously irrelevant.Comment: 5 pages including 4 figures; revised version: some minor changes and added reference

The role of winding numbers in quantum Monte Carlo simulations

We discuss the effects of fixing the winding number in quantum Monte Carlo simulations. We present a simple geometrical argument as well as strong numerical evidence that one can obtain exact ground state results for periodic boundary conditions without changing the winding number. However, for very small systems the temperature has to be considerably lower than in simulations with fluctuating winding numbers. The relative deviation of a calculated observable from the exact ground state result typically scales as $T^{\gamma}$, where the exponent $\gamma$ is model and observable dependent and the prefactor decreases with increasing system size. Analytic results for a quantum rotor model further support our claim.Comment: 5 pages, 5 figure

Finite-Size Scaling of the Ground State Parameters of the Two-Dimensional Heisenberg Model

The ground state parameters of the two-dimensional S=1/2 antiferromagnetic Heisenberg model are calculated using the Stochastic Series Expansion quantum Monte Carlo method for L*L lattices with L up to 16. The finite-size results for the energy E, the sublattice magnetization M, the long-wavelength susceptibility chi_perp(q=2*pi/L), and the spin stiffness rho_s, are extrapolated to the thermodynamic limit using fits to polynomials in 1/L, constrained by scaling forms previously obtained from renormalization group calculations for the nonlinear sigma model and chiral perturbation theory. The results are fully consistent with the predicted leading finite-size corrections and are of sufficient accuracy for extracting also subleading terms. The subleading energy correction (proportional to 1/L^4) agrees with chiral perturbation theory to within a statistical error of a few percent, thus providing the first numerical confirmation of the finite-size scaling forms to this order. The extrapolated ground state energy per spin, E=-0.669437(5), is the most accurate estimate reported to date. The most accurate Green's function Monte Carlo (GFMC) result is slightly higher than this value, most likely due to a small systematic error originating from ``population control'' bias in GFMC. The other extrapolated parameters are M=0.3070(3), rho_s = 0.175(2), chi_perp = 0.0625(9), and the spinwave velocity c=1.673(7). The statistical errors are comparable with those of the best previous estimates, obtained by fitting loop algorithm quantum Monte Carlo data to finite-temperature scaling forms. Both M and rho_s obtained from the finite-T data are, however, a few error bars higher than the present estimates. It is argued that the T=0 extrapolations performed here are less sensitive to effects of neglectedComment: 16 pages, RevTex, 9 PostScript figure

Ground state parameters, finite-size scaling, and low-temperature properties of the two-dimensional S=1/2 XY model

We present high-precision quantum Monte Carlo results for the S=1/2 XY model on a two-dimensional square lattice, in the ground state as well as at finite temperature. The energy, the spin stiffness, the magnetization, and the susceptibility are calculated and extrapolated to the thermodynamic limit. For the ground state, we test a variety of finite-size scaling predictions of effective Lagrangian theory and find good agreement and consistency between the finite-size corrections for different quantities. The low-temperature behavior of the susceptibility and the internal energy is also in good agreement with theoretical predictions.Comment: 6 pages, 8 figure

Pseudo-gap behavior in dynamical properties of high-Tc cuprates

Dynamical properties of 2D antiferromagnets with hole doping are investigated to see the effects of short range local magnetic order on the temperature dependence of the dynamical magnetic susceptibility. We show the pseudo-gap like behavior of the temperature dependence of the NMR relaxation rate. We also discuss implications of the results in relations to the observed spin gap like behavior of low-doped copper oxide high-$T_c$ superconductors.Comment: 3 pages, Revtex, with 2 eps figures, to appear in J.Phys.Soc.Jpn. Vol.67 No.

Quantum Monte Carlo simulation for the conductance of one-dimensional quantum spin systems

Recently, the stochastic series expansion (SSE) has been proposed as a powerful MC-method, which allows simulations at low $T$ for quantum-spin systems. We show that the SSE allows to compute the magnetic conductance for various one-dimensional spin systems without further approximations. We consider various modifications of the anisotropic Heisenberg chain. We recover the Kane-Fisher scaling for one impurity in a Luttinger-liquid and study the influence of non-interacting leads for the conductance of an interacting system.Comment: 8 pages, 9 figure