This study delves into the pivotal role of equilibration Monte Carlo steps (š steps ) in the Stochastic Series Expansion method-a cutting-edge quantum Monte Carlo technique known for its precision in simulating low-dimensional quantum systems.The research focuses on how varying equilibration steps impact the accuracy and stability of ground state energy, sublattice magnetization, specific heat, and susceptibility results, by analyzing the effect of equilibration steps on these physical properties across different temperatures and lattice sizes.Our findings demonstrate that while ground state energy and sublattice magnetization achieve remarkable stability, susceptibility and specific heat present variabilities, especially at low temperatures.Ground state energy stabilizes after approximately 50 equilibration steps.However, excessive equilibration steps lead to unnecessary computational costs without further accuracy improvements.Furthermore, increasing the measurement steps significantly reduces error bars, enhancing reliability, as demonstrated by a 30% reduction in standard deviation with higher counts.Low temperatures exhibit greater variability in specific heat and susceptibility due to complex quantum effects.
We investigate the Stochastic Series Expansion (SSE) technique, guided by Sandvik's algorithm of the quantum Monte Carlo simulation for spin-1/2 antiferromagnetic Heisenberg linear chain. The impact of the Monte Carlo steps, lattice sizes, and various temperatures on the variability and fidelity of simulation outcomes are studied. The findings show that the Monte Carlo steps where measurements are made (Msteps), and lattice size are crucial for stabilizing the results. Our results show that for larger lattice sizes, such as L=1024, achieving reliable outcomes necessitates extensive Monte Carlo simulations, specifically. The convergence in the properties is achieved with a high number of Monte Carlo steps starting from 7Ć10^4. Strong fluctuations are calculated between the results for each magnetic property and the existence of negative and positive in the specific heat at low temperatures.
Background: Antiferromagnetic systems exhibit complex quantum behaviours that require accurate numerical methods to analyse. The Stochastic Series Expansion (SSE) is a quantum Monte Carlo technique that simulates these systems. This study examines the effect of the number of bins on the accuracy of physical property measurements using SSE. Materials and Methods: The SSE simulations were performed using Fortran 90 on a Ryzen 7 processor. The system was initialised with varying lattice sizes (64, 128, 256, and 1024) and dimensionless temperatures (1/32, 1/16, 1/2, and 4) to analyse different configurations. Nbins and Monte Carlo steps were adjusted systematically to investigate their impact on the results. Results were visualised in Origin Lab. Results: Increasing the number of bins (Nbins) reduced fluctuations, leading to reliable results, especially at low temperatures. Lower temperatures lead to higher fluctuations in susceptibility and specific heat of the system. The Neel temperatures were observed around dimensionless temperature. Indicating phase transitions. Conclusions: The study shows that Nbins are crucial for accurate results in SSE simulations. Although the method is effective for most properties, specific heat calculations require higher computational costs and present limitations in precision.
Abstract Motivated by the need for high-precision benchmark results for basic quantum many-body models, several ground state properties of the square lattice S = 1 / 2 Heisenberg antiferromagnet are computed (the energy, order parameter, spin stiffness, spinwave velocity, long-wavelength susceptibility and staggered susceptibility) using extensive quantum Monte Carlo simulations with the stochastic series expansion method. Moderately sized lattices are studied at temperatures T sufficiently low to realize the T ā 0 limit, with significantly reduced statistical errors relative to previous works. Results for periodic L Ć L lattices with L ā [ 6 , 96 ] are tabulated versus L and extrapolations to infinite system size are carried out. The relative statistical errors of finite-size energies are below 10 ā7 . The extrapolated ground state energy density is e 0 = ā 0.669 441 857 ( 7 ) , where the number within parentheses is the statistical error (one standard deviation) of the preceding digit. This value represents an improvement in precision of three orders of magnitude over the previous best result. The leading and subleading finite-size corrections to e 0 are in full quantitative agreement with predictions from chiral perturbation theory, thus further supporting the soundness of both the extrapolations and the theory. The extrapolated sublattice magnetization is m s = 0.307 447 ( 2 ) , which agrees well with previous estimates but with a much smaller statistical error. The coefficient of the linear in L ā1 correction to m s 2 agrees with the value from chiral perturbation theory and the presence of a factor ln Ī³ ā” ( L ) in the second-order correction is also confirmed, with the previously unknown value of the exponent being Ī³ = 0.82 ( 4 ) . The finite-size corrections to the staggered susceptibility point to logarithmic corrections in this quantity as well. To facilitate benchmarking of methods for which periodic boundary conditions are challenging, results for systems with open and cylindrical boundaries are also listed and their spatially inhomogeneous order parameters are analyzed.
The Antiferromagnetic Heisenberg Model with spin 1/2 on a square lattice was simulated by using the Stochastic Series Expansion(SSE) Quantum Monte Carlo Method.The temperature dependence of the internal energy,specific heat and uniform susceptibility was observed.The results showed that specific heat has a maximum at kT/J=0.6 and uniform susceptibility saturates at kT/J=1 for the isotropy case.And the size of lattice also affects the temperature dependence of these thermodynamic quantities.For the anisotropy case,energy decreases with anisotropic parameter g increasing.Susceptibility changes slowly with temperature for g1 and exponentially decay for g1 in the low temperature region and its behaviors approach consistent for different g in the high temperature region.
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 $L\ifmmode\times\else\texttimes\fi{}L$ lattices with
L up to
16. The finite-size results for the energy
E, the sublattice magnetization
M, the long-wavelength susceptibility ${\ensuremath{\chi}}_{\ensuremath{\perp}}(q=2\ensuremath{\pi}/L)$, and the spin stiffness ${\ensuremath{\rho}}_{s},$ 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 $\ensuremath{\sigma}$ 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 $(\ensuremath{\sim}{1/L}^{4})$ agrees with chiral perturbation theory to within a statistical error of a few percent, thus providing numerical confirmation of the finite-size scaling forms to this order. The extrapolated ground- state energy per spin is $E=\ensuremath{-}0.669437(5)$. The result from previous Green's function Monte Carlo (GFMC) calculations 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), ${\ensuremath{\rho}}_{s}=0.175(2)$, ${\ensuremath{\chi}}_{\ensuremath{\perp}}=0.0625(9)$, and the spin-wave velocity
c=1.673(7). The statistical errors are comparable with those of previous estimates obtained by fitting loop algorithm quantum Monte Carlo data to finite-temperature scaling forms. Both
M and ${\ensuremath{\rho}}_{s}$ 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 neglected higher-order corrections, and therefore should be more reliable.
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.