An algorithm is developed for determining the exact ground state properties of quantum many-body systems which is equally applicable to bosons and fermions. The Schroedinger eigenvalue equation for the ground state energy is recast into the form of a many-dimensional integral through the use of the Hubbard-Stratonovitch representation of the imaginary time many- body evolution operator. The resulting functional integral is then evaluated stochastically. The algorithm is tested for an exactly soluble boson system and is then extended to include fermions and repulsive potentials. Importance sampling is crucial to the success of the method, particularly for more complex systems. Improved computational efficiency is attained by performing the calculations in momentum space.
An algorithm is developed for determining the exact ground state properties of quantum many-body systems which is equally applicable to bosons and fermions. The Schroedinger eigenvalue equation for the ground state energy is recast into the form of a many-dimensional integral through the use of the Hubbard-Stratonovitch representation of the imaginary time many- body evolution operator. The resulting functional integral is then evaluated stochastically. The algorithm is tested for an exactly soluble boson system and is then extended to include fermions and repulsive potentials. Importance sampling is crucial to the success of the method, particularly for more complex systems. Improved computational efficiency is attained by performing the calculations in momentum space.
Abstract This paper describes a different Q11C method for solving the Schrödinger equation, one unlike variational, diffusion, and Green’s function methods. It can be applied to many-body systems to produce solutions for ground states. For systems of fermions, antisymmetrization can be imposed to overcome the problem of node location, but the instability of the method is a serious shortcoming, so that the method as described is limited to treating relatively simple systems. Nevertheless, this auxiliary field quantum 1,fonte Carlo (AFMC) method was demonstrated to be successful for test cases reported in this paper, and in time has been improved to allow treatment of realistic systemsa. The method makes use of the imaginary-time propagation of an initial wavefunction toward the ground-state function. It requires use of a trial wavefunction which is the product of single-particle orbitals for bosons or an antisymmetrized product for fermions. A Hubbard-Stratonovich transformation of the propagator changes the many-body problem to a problem of a collection of single particles interacting with a fluctuating (auxiliary) field which replaces the many-body interactions. The authors developed an integral expression for the energy which, after discretization, could be evaluated by a Metropolis Monte Carlo integration forward in time. The algorithm was tested for one-dimensional systems of particles: 6 to 20 bosons with varied interactions, 2 to 6 neutron-proton pairs, others with repulsive potentials. Importance sampling was found to be critical and computational requirements to be large. For spin systems on a lattice the AFMC method was considered feasible, but for general multidimensional systems the method appeared to be limited by its computation requirements.
We formulate a quantum Monte Carlo (QMC) method for calculating the ground state of many-boson systems. The method is based on a field-theoretical approach, and is closely related to existing fermion auxiliary-field QMC methods which are applied in several fields of physics. The ground-state projection is implemented as a branching random walk in the space of permanents consisting of identical single-particle orbitals. Any single-particle basis can be used, and the method is in principle exact. We illustrate this method with a trapped atomic boson gas, where the atoms interact via an attractive or repulsive contact two-body potential. We choose as the single-particle basis a real-space grid. We compare with exact results in small systems and arbitrarily sized systems of untrapped bosons with attractive interactions in one dimension, where analytical solutions exist. We also compare with the corresponding Gross-Pitaevskii (GP) mean-field calculations for trapped atoms, and discuss the close formal relation between our method and the GP approach. Our method provides a way to systematically improve upon GP while using the same framework, capturing interaction and correlation effects with a stochastic, coherent ensemble of noninteracting solutions. We discuss various algorithmic issues, including importance sampling and the back-propagation technique for computing observables, and illustrate them with numerical studies. We show results for systems with up to N approximately 400 bosons.
Abstract. For many-body lattice quantum systems the existence of a Feynman-Kac–type formula based on Poisson stochastic processes has recently allowed to derive an exact Monte Carlo algorithm for the evolution operator at real or imaginary times. We extend this algorithm to the study of exact time-dependent correlation functions. The techniques generally employed in Monte Carlo simulations to control fluctuations, namely reconfigurations and importance sampling, are adapted to the present algorithm and their validity is rigorously proved. We complete the analysis by several examples for the hard-core boson Hubbard model and the Heisenberg model. PACS numbers: 02.50.-r, 02.70.SsExact Monte Carlo time dynamics in many-body lattice quantum systems 2 1.
We formulate a quantum Monte Carlo (QMC) method for calculating the ground state of many-boson systems. The method is based on a field-theoretical approach, and is closely related to existing fermion auxiliary-field QMC methods which are applied in several fields of physics. The ground-state projection is implemented as a branching random walk in the space of permanents consisting of identical single-particle orbitals. Any single-particle basis can be used, and the method is in principle exact. We apply this method to an atomic Bose gas, where the atoms interact via an attractive or repulsive contact two-body potential parametrized by the s-wave scattering length. We choose as the single-particle basis a real-space grid. We compare with exact results in small systems, and arbitrarily-sized systems of untrapped bosons with attractive interactions in one dimension, where analytical solutions exist. Our method provides a way to systematically improve upon the mean-field Gross-Pitaevskii (GP) method while using the same framework, capturing interaction and correlation effects with a stochastic, coherent ensemble of non-interacting solutions. to study the role of many-body correlations in the ground state, we examine the properties of the gas, such as the energetics, condensate fraction, and the density and momentum distributions as a function of the number of particles and the scattering length, both in the homogenous and trapped gases. Results are presented for systems with up to 1000 bosons. Comparing our results to the mean-field GP results, we find significant departure from mean field at large positive scattering lengths. The many-body correlations tend to increase the kinetic energy and reduce the interaction energy compared to GP. In the trapped gases, this results in a qualitatively different behavior as a function of the scattering length. Possible experimental observation is discussed.
On the basis of a Feynman–Kac-type formula involving Poisson stochastic processes, a Monte Carlo algorithm has recently been introduced, which describes exactly the real- or imaginary-time evolution of many-body lattice quantum systems. We extend this algorithm to the exact simulation of time-dependent correlation functions. The techniques generally employed in Monte Carlo simulations to control fluctuations, namely reconfigurations and importance sampling, are adapted to the present algorithm and their validity is rigorously proved. We complete the analysis by several examples for the hard-core boson Hubbard model and for the Heisenberg model.
Quantum Monte Carlo simulations provide a powerful tool to compute static properties of quantum many-particles models and realistic systems. A vast array of ground-state energies and properties defined by equal-time correlation functions can be computed with high accuracy. On the other hand, the extensions needed to study time-displaced correlation functions, which provide access to excited states, have been challenging, especially for fermionic strongly correlated systems. The authors present a methodology that allows a high-accuracy calculation of the dynamical Green functions of many-fermions systems in the framework of the auxiliary-field quantum Monte Carlo method. They extract the the charge gap of the repulsive two-dimensional Hubbard model at half-filling as a function of the interaction strength and propose strategies for applying their methodology to more realistic systems.