Monte Carlo simulation of quantum statistical lattice models

In this article we review recent developments in computational methods for quantum statistical lattice problems. We begin by giving the necessary mathematical basis, the generalized Trotter formula, and discuss the computational tools, exact summations and Monte Carlo simulation, that will be used to examine explicit examples. To illustrate the general strategy, the method is applied to an analytically solvable, non-trivial, model: the one-dimensional Ising model in a transverse field. Next it is shown how to generalized Trotter formula most naturally leads to different path-integral representations of the partition function by considering one-dimensional fermion lattice models. We show how to analyze the different representations and discuss Monte Carlo simulation results for one-dimensional fermions. Then Monte Carlo work on one- and two-dimensional spin-$\text{1}\text{2}$ models based upon the Trotter formula approach is reviewed and the more dedicated Handscomb Monte Carlo method is discussed. We consider electron-phonon models and discuss Monte Carlo simulation data on the Molecular Crystal Model in one, two and three dimensions and related one-dimensional polaron models. Exact numerical results are presented for free fermions and free bosons in the canonical ensemble. We address the main problem of Monte Carlo simulations of fermions in more than one dimension: the cancellation of large contributions. Free bosons on a lattice are compared with bosons in a box and the effects of finite size on Bose-Einstein condensation are discussed.     

Quantum statistical monte carlo methods and applications to spin systems

A short review is given concerning the quantum statistical Monte Carlo method based on the equivalence theorem⁽¹⁾ thatd-dimensional quantum systems are mapped onto (d+1)-dimensional classical systems. The convergence property of this approximate tansformation is discussed in detail. Some applications of this geneal appoach to quantum spin systems are reviewed. A new Monte Carlo method, “thermo field Monte Carlo method,” is presented, which is an extension of the projection Monte Carlo method at zero temperature to that at finite temperatures. Invited talk presented at “Frontiers of Quantum Monte Carlo,” Los Alamos National Laboratory, September 3–6, 1985.     

Simulation of carrier capture in quantum well lasers due to strong inelastic scattering

The effects of strong inelastic scattering on carrier transport over and capture into the quantum wells of quantum well lasers are simulated. In contrast to most semiconductor devices, strong scattering is beneficial to the operation of quantum well lasers. However, such strong inelastic scattering in nanostructures can be expected to produce intermediate degrees of phase coherence, limiting the applicability of both classical models, such as Bethe thermionic emission theory, and commonly used quantum mechanical treatments, such as Fermi's Golden Rule. Two computational approaches are demonstrated for simulating such transport with intermediate degrees of phase coherence. First, absorbing potentials are used within Schrödinger's equation to represent inelastic scattering. This simple approach both exhibits much of the essential physics of such transport and is computationally efficient. Then a more rigorous approach, Schrödinger equation (based) Monte Carlo (SEMC), is demonstrated. While SEMC is rigorously quantum mechanical, the numerical algorithm has more in common with semiclassical Monte Carlo methods than path integral-based quantum Monte Carlo methods. Both of these methods demonstrate nonlinear variations in carrier capture with variations in scattering, and the destruction of quantum resonances for transmission over the quantum well.     

A new Monte Carlo power method for the eigenvalue problem of transfer matrices

We propose a new Monte Carlo method for calculating eigenvalues of transfer matrices leading to free energies and to correlation lengths of classical and quantum many-body systems. Generally, this method can be applied to the calculation of the maximum eigenvalue of a nonnegative matrix  such that all the matrix elements of Â^k are strictly positive for an integerk. This method is based on a new representation of the maximum eigenvalue of the matrix  as the thermal average of a certain observable of a many-body system. Therefore one can easily calculate the maximum eigenvalue of a transfer matrix leading to the free energy in the standard Monte Carlo simulations, such as the Metropolis algorithm. As test cases, we calculate the free energies of the square-lattice Ising model and of the spin-1/2XY Heisenberg chain. We also prove two useful theorems on the ergodicity in quantum Monte Carlo algorithms, or more generally, on the ergodicity of Monte Carlo algorithms using our new representation of the maximum eigenvalue of the matrixÂ.     

Comparison between semiclassical and full quantum transport analysis of THz quantum cascade lasers

We implement and compare two theoretical models for stationary electron transport in quantum cascade lasers and Stark ladders. The first one, the nonequilibrium Green's function method is a very general scheme to include coherent quantum mechanics and incoherent scattering with phonons and device imperfections self-consistently. However, it is numerically very demanding and cannot be used for systematic device parameter scans. For this reason, we also implement the approximate, numerically efficient ensemble Monte Carlo method and assess its applicability on the above mentioned transport problems. We identify a transport regime in which results of both methods quantitatively agree. In this regime, the ensemble Monte Carlo method is well suited to propose design improvements.     

QWalk: A quantum Monte Carlo program for electronic structure

We describe QWalk, a new computational package capable of performing quantum Monte Carlo electronic structure calculations for molecules and solids with many electrons. We describe the structure of the program and its implementation of quantum Monte Carlo methods. It is open-source, licensed under the GPL, and available at the web site http://www.qwalk.org.     

二维纳米点阵列的Monte-Carlo模拟

介绍了一种利用标准单电子隧穿理论与Monte-Carlo 方法模拟二维量子点阵列的程序.数值计算结果表明,二维量子点阵在低温下有库仑充电行为,其量子功能作用使它显示出可喜的研究价值和应用前景. 关键词： Monte-Carlo 模拟 单电子 量子点阵     

Diagonalizations on a correlated basis

Unsymmetrical quantum-dot systems are generally difficult to study using wave-function techniques, like quantum Monte Carlo (QMC) or exact diagonalization (ED) methods. The initial trial wave function for Monte Carlo methods is difficult to find, and the exact diagonalization method can only handle very few particles.In this article a two-dimensional semiconductor quantum dot containing a non-centered impurity ion is studied, using a new exact wave-function method. Results are analyzed and compared to density-functional-theory calculations. The computational method allows one to relax the commonly used lowest-Landau level (LLL) approximation, and it's effects are studied, e.g., on the charge and current density profiles.The method, which is a combination of QMC and ED methods, is described. It combines the scalability of Monte Carlo methods with the benefits of exact diagonalization, and allows one to accurately obtain the wave function for unsymmetrical quantum dots up to more than ten electrons. Also, excited states are accessible and are analyzed in this article.     

(3 1)-Dimensional Quantum Mechanics from Monte Carlo Hamiltonian: Harmonic Oscillator

In Lagrangian formulation, it is extremely difficult to compute the excited spectrum and wavefunctions ora quantum theory via Monte Carlo methods. Recently, we developed a Monte Carlo Hamiltonian method for investigating this hard problem and tested the algorithm in quantum-mechanical systems in 1 1 and 2t1 dimensions. In this paper we apply it to the study of thelow-energy quantum physics of the (3 1)-dimensional harmonic oscillator.``     

Real-time path integral approach to nonequilibrium many-body quantum systems

A real-time path-integral Monte Carlo approach is developed to study the dynamics in a many-body quantum system coupled to a phonon background until reaching a nonequilibrium stationary state. The approach is based on augmenting an exact reduced equation for the evolution of the system in the interaction picture which is amenable to an efficient path integral (worldline) Monte Carlo approach. Results obtained for a model of inelastic tunneling spectroscopy reveal the applicability of the approach to a wide range of physically important regimes, including high (classical) and low (quantum) temperatures, and weak (perturbative) and strong electron-phonon couplings.     

A quantum Monte Carlo study of the localizable entanglement in anisotropic ferromagnetic Heisenberg chains

We study the localizable entanglement in large one-dimensional anisotropic XYZ ferromagnetic Heisenberg chains interacting with a uniform magnetic field. With the use of quantum Monte Carlo simulations we calculate the bounds of localizable entanglement by means of the correlation functions. The present quantum Monte Carlo method has the advantage over existing methods that it can be readily applied to fully anisotropic magnetic chains.     

Solving quantum rotor model with different Monte Carlo techniques

We systematically test the performance of several Monte Carlo update schemes for the (2+1)d XY phase transition of quantum rotor model. By comparing the local Metropolis (LM), LM plus over-relaxation (OR), Wolff-cluster (WC), hybrid Monte Carlo (HM), hybrid Monte Carlo with Fourier acceleration (FA) schemes, it is clear that among the five different update schemes, at the quantum critical point, the WC and FA schemes acquire the smallest autocorrelation time and cost the least amount of CPU hours in achieving the same level of relative error, and FA enjoys a further advantage of easily implementable for more complicated interactions such as the long-range ones. These results bestow one with the necessary knowledge of extending the quantum rotor model, which plays the role of ferromagnetic/antiferromagnetic critical bosons or Z₂ topological order, to more realistic and yet challenging models such as Fermi surface Yukawa-coupled to quantum rotor models.     

Nonlinear phenomenology from quantum mechanics: soliton in a lattice

We study a soliton in an optical lattice holding bosonic atoms quantum mechanically using both an exact numerical solution and quantum Monte Carlo simulations. The computation of the state is combined with an explicit account of the measurements of the numbers of the atoms at the lattice sites. In particular, importance sampling in the quantum Monte Carlo method arguably produces faithful simulations of individual experiments. Even though the quantum state is invariant under lattice translations, an experiment may show a noisy version of the localized classical soliton.     

巨正则量子Monte Carlo方法和二维Hubbard模型的研究

用巨正则量子Monte Carlo方法,计算了二维单带Hubbard模型的局域磁矩、磁化率、交错磁化率和内能等物理量.结果表明,用巨正则量子Monte Carlo方法能够统一地研究Hubbard模型的关联强度从弱至强区域的性质,它是处理强关联多体系统的有效方法.     

Autocorrelations and acceptances with noisy Monte Carlo methods for quantum chromodynamics

Using a noisy Monte Carlo algorithm we investigate the possible advantages of a large step size Monte Carlo method against a small step size Langevin method for quantum chromodynamics (QCD). It is shown that for a suitably defined cost function, we have no strong dependence on the step size, and that for small step sizes the system progresses through phase space at a rate compatible with a random walk.     

Quantum mechanical Monte Carlo approach to electron transport at heterointerface

With the progress of LSI technology, the electronic device size is scaled down to the sub 0.1μ m region. In such an ultrasmall device, it is indispensable to take quantum mechanical effects into account in device modeling. In this paper, we present a newly developed quantum Monte Carlo device simulation applicable to ultrasmall semiconductor devices. In this model, the quantum effects are represented in terms of quantum mechanically corrected potential in the classical Boltzmann equation. It is demonstrated that the quantum transport effects such as tunneling and energy quantization in ultrasmall semiconductor devices are obtained for the first time by using the standard Monte Carlo techniques.     

沉积速率和生长停顿对InAs/GaAs量子点超晶格生长影响的综合分析

采用动力学蒙特卡罗模型模拟了沉积速率和生长停顿对GaAs衬底中垂直耦合InAs 量子点超晶格生长早期阶段的影响.通过对生长表面形态、岛平均尺寸、岛尺寸分布及其标准差等方面的研究,发现综合控制沉积速率和生长停顿时间能够得到大小均匀、排列有序的岛阵列.这对后续量子点超晶格生长过程中量子点的定位有重要影响. 关键词： 动力学蒙特卡罗模拟 量子点超晶格 外延生长     

Monte Carlo techniques for real-time quantum dynamics

The stochastic-gauge representation is a method of mapping the equation of motion for the quantum mechanical density operator onto a set of equivalent stochastic differential equations. One of the stochastic variables is termed the “weight”, and its magnitude is related to the importance of the stochastic trajectory. We investigate the use of Monte Carlo algorithms to improve the sampling of the weighted trajectories and thus reduce sampling error in a simulation of quantum dynamics. The method can be applied to calculations in real time, as well as imaginary time for which Monte Carlo algorithms are more-commonly used. The Monte-Carlo algorithms are applicable when the weight is guaranteed to be real, and we demonstrate how to ensure this is the case. Examples are given for the anharmonic oscillator, where large improvements over stochastic sampling are observed.     

Equilibrium properties of hard non-sphere fluids

Derivation of the thermodynamic properties of fluids of hard non-spherical molecules of arbitrary symmetry is based on the decoupling approximation. Theoretical expressions are given and calculations made for the equation of state and virial coefficients for hard ellipsoids. These results are compared with Monte Carlo values and show fair agreement in all cases. The theoretical predictions for the equation of state for binary mixtures are compared with the Monte Carlo results for hard spheres and hard prolate spherocylinders. Theoretical expressions for the first order quantum correction to the free energy, pressure and virial coefficients are also given. The quantum effects increase with increase of density and with increase of anisotropy parameter.