Properties Of Quantum Many-body Systems Research Articles

Worm algorithm and diagrammatic Monte Carlo: A new approach to continuous-space path integral Monte Carlo simulations

A detailed description is provided of a new worm algorithm, enabling the accurate computation of thermodynamic properties of quantum many-body systems in continuous space, at finite temperature. The algorithm is formulated within the general path integral Monte Carlo (PIMC) scheme, but also allows one to perform quantum simulations in the grand canonical ensemble, as well as to compute off-diagonal imaginary-time correlation functions, such as the Matsubara Green function, simultaneously with diagonal observables. Another important innovation consists of the expansion of the attractive part of the pairwise potential energy into elementary (diagrammatic) contributions, which are then statistically sampled. This affords a complete microscopic account of the long-range part of the potential energy, while keeping the computational complexity of all updates independent of the size of the simulated system. The computational scheme allows for efficient calculations of the superfluid fraction and off-diagonal correlations in space-time, for system sizes which are orders of magnitude larger than those accessible to conventional PIMC. We present illustrative results for the superfluid transition in bulk liquid 4He in two and three dimensions, as well as the calculation of the chemical potential of hcp 4He.

Microscopic functional theory of dielectrics

Density functional theory (DFR) rests upon the proofs by Hohenberg and Kohn, which show that all properties of quantum many-body systems are functionals of the ground-state density, and the Kohn–Sham construction, in which the exchange–correlation energy is a functional only of the density. DFT has been widely assumed to apply directly to the intrinsic bulk properties of extended systems such as crystals, e.g., the static dielectric properties of insulators. However, in 1995, Godby, Ghosez, and Godby pointed out that the assumptions of DFT do not strictly apply to the case of a crystal in a finite electric field, since there is no ground state, and they argued that the description of intrinsic bulk dielectric phenomena in a crystal requires a functional of both the bulk density and the polarization. Here, we discuss the results of recent work which show that in the limit of long wavelengths both the bulk (periodic) density and the bulk average macroscopic polarization are necessary to fully describe the dielectric properties of insulating materials. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 567–572, 1998

Critical Properties of Quantum Many-Body Systems with 1/r2 Interaction

We review recent results obtained for a class of one-dimensional quantum models with $1/r^2$ long-range interaction. Based on the asymptotic Bethe-ansatz solution and conformal field theory, we study critical properties of the continuum boson model, the SU($\nu$) spin chain, the OSp($\nu$,1) supersymmetric {\it t-J} model, and a new hierarchy of models related to the fractional quantum Hall effect. We further investigate the class of $1/r^2$ models with harmonic confinement by means of a newly proposed method of the renormalized-harmonic oscillator solution.

Critical Properties of Quantum Many-Body Systems with 1/r2 Interaction

We review recent results obtained for a class of one-dimensional quantum models with 1/r2 long-range interaction. Based on the asymptotic Bethe-ansatz solution and conformal field theory, we study critical properties of the continuum boson model, the SU(ν) spin chain, the OSp(ν, 1) supersymmetric t-J model and a new hierarchy of models related to the fractional quantum Hall effect. We further investigate the class of 1/r2 models with harmonic confinement by means of a newly proposed method of the renormalized-harmonic oscillator solution.

Frequency moment sum rules, recurrence relations and continued fractions in nonequilibrium statistical mechanics

Dynamical properties of quantum many-body systems, e.g., the electron gas, lattice spins are usually studied via frequency moment sum rules. For example, the dynamic structure factor of the density-density response function in an interacting electron gas can be given a continued fraction expansion in terms of frequency moments. Since only a few moments are known, one is faced with the problem of extracting best information from a continued fraction representation. It is desirable to obtain an approximate form which is sufficiently accurate for comparison with experimental data. The method of recurrence relations provides a convergent procedure for obtaining such physical quantities using approximate higher order frequency moments. This idea seems to work very well for a strongly interacting electron gas at metallic densities. The basic ideas behind this procedure are explained.

Thermodynamical properties of quantum many-body systems.

We present a method to compute exactly the partition function of quantum many-body systems. This approach involves the evaluation of a set of coefficients for the continued-fraction expansion of a resolventlike operator. The main advantage of this method is that this set of coefficients is unique and allows one to calculate the partition function at any temperature. As an illustration, the method is applied to a simple one-dimensional spinless model.

Dynamical properties of quantum many-body systems at zero temperature.

We present a method to compute dynamical correlation functions in quantum many-body systems at zero temperature. The ground state of a finite system is evaluated exactly by a modified Lanczos method and from it the real-frequency correlation functions are obtained by a projective technique for the memory-function formalism. We apply the method to the density-density correlation function of a one-dimensional spinless fermion system. Exact results are recovered in the noninteracting limit; in the strong-coupling limit our results are compared with Monte Carlo simulation and analytical approximations.

Auxiliary field Monte-Carlo for quantum many-body ground states

We develop an algorithm 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 as a many-dimensional integral using the Hubbard-Stratonovitch representation of the imaginary-time many-body evolution operator. The integral is then evaluated stochastically. We test the algorithm for an exactly soluble boson system with an attractive potential and then extend it to fermions and repulsive potentials. Importance sampling is crucial to the success of the method, particularly for more complex systems. Computational efficiency is improved by performing the calculations in Fourier space.

High-temperature thermodynamics of the Hubbard model: An exact numerical solution

A new first-principles method for numerically calculating finite-temperature properties of quantum many-body systems is described. It is used to compute the thermodynamic properties and magnetic correlations of the Hubbard model, with negligible error at temperatures near the bandwidth or above. Results for the half-filled band and a simple cubic or one-dimensional lattice confirm the existence of a high-temperature peak in the specific heat which has been associated with a smooth but rapid change in conductivity. Correlations calculated for the non-half-filled band simple cubic Hubbard model suggest condensation, an entirely unexpected phenomenon whose existence is shown (by unrelated methods) in Paper II.