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arXiv:nucl-th/0009005 · International Journal of Modern Physics B

QUANTUM MONTE CARLO METHODS FOR NUCLEI AT FINITE TEMPERATURE

Y. Alhassid

Published May 10, 2001• 29 citations
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Abstract

We discuss finite temperature quantum Monte Carlo methods in the framework of the interacting nuclear shell model. The methods are based on a representation of the imaginary-time many-body propagator as a superposition of one-body propagators describing non-interacting fermions moving in fluctuating auxiliary fields. Fermionic Monte Carlo calculations have been limited by a "sign" problem. A practical solution in the nuclear case enables realistic calculations in much larger configuration spaces than can be solved by conventional methods. Good-sign interactions can be constructed for realistic estimates of certain nuclear properties. We present various applications of the methods for calculating collective properties and level densities.

DOI
10.1142/s0217979201005945

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International Journal of Modern Physics B•May 10, 2001•0 citations
We discuss finite temperature quantum Monte Carlo methods in the framework of the interacting nuclear shell model. The methods are based on a representation of the imaginary-time many-body propagator as a superposition of one-body propagators describing non-interacting fermions moving in fluctuating auxiliary fields. Fermionic Monte Carlo calculations have been limited by a "sign" problem. A practical solution in the nuclear case enables realistic calculations in much larger configuration spaces than can be solved by conventional methods. Good-sign interactions can be constructed for realistic estimates of certain nuclear properties. We present various applications of the methods for calculating collective properties and level densities.
We discuss finite temperature quantum Monte Carlo methods in the framework of the interacting nuclear shell model. The methods are based on a representation of the imaginary-time many-body propagator as a superposition of one-body propagators describing non-interacting fermions moving in fluctuating auxiliary fields. Fermionic Monte Carlo calculations have been limited by a ``sign'' problem. A practical solution in the nuclear case enables realistic calculations in much larger configuration spaces than can be solved by conventional methods. Good-sign interactions can be constructed for realistic estimates of certain nuclear properties. We present various applications of the methods for calculating collective properties and level densities.
The shell model provides a powerful framework for nuclear structure calculations. The nucleons beyond an inert magic core are confined in a valence shell and interact through an effective two-body potential generally determined from the G-matrix method. However, the applicability of the shell model remains limited by the exponential growth of the many-body space with the number of valence nucleons and with the size of the one-body space. The quantum Monte Carlo (QMC) methods may thus be considered as a potentially attractive alternative to the direct diagonalization of the Hamiltonian matrix. The many-body problem is indeed reduced to a set of stochastic one-body problems describing independent particles in a fluctuating external field. Up to date, the QMC approaches of the shell model lead to the ground-state and the thermodynamic properties, but a detailed spectroscopy of nuclei is not reachable. Furthermore, they suffer from a common pathology of fermionic QMC methods, namely the sign/phase problem, which strongly contaminates the convergence of the calculation. In this context, we will present a new QMC method, providing the "yrast spectroscopy" of nuclei, and in which the sign/phase problem is managed through the constrained path approximation that is widely used in the ab initio nuclear calculations and in condensed matter physics.
The formalism for the auxiliary-field Monte Carlo approach to the nuclear shell model is presented. The method is based on a linearization of the two-body part of the Hamiltonian in an imaginary-time propagator using the Hubbard-Stratonovich transformation. The foundation of the method, as applied to the nuclear many-body problem, is discussed. Topics presented in detail include: (1) the density-density formulation of the method, (2) computation of the overlaps, (3) the sign of the Monte Carlo weight function, (4) techniques for performing Monte Carlo sampling, and (5) the reconstruction of response functions from an imaginary-time auto-correlation function using MaxEnt techniques. Results obtained using schematic interactions, which have no sign problem, are presented to demonstrate the feasibility of the method, while an extrapolation method for realistic Hamiltonians is presented. In addition, applications at finite temperature are outlined.
The formalism for the auxiliary-field Monte Carlo approach to the nuclear shell model is presented.The method is based on a linearization of the two-body part of the Hamiltonian in an imaginary-time propagator using the Hubbard-Stratonovich transformation.The foundation of the method, as applied to the nuclear many-body problem, is discussed.Topics presented in detail include: (1) the density-density formulation of the method, (2) computation of the overlaps, (3) the sign of the Monte Carlo weight function, (4) techniques for performing Monte Carlo sampling, and ( 5) the reconstruction of response functions from an imaginary-time auto-correlation function using MaxEnt techniques.Results obtained using schematic interactions, which have no sign problem, are presented to demonstrate the feasibility of the method, while an extrapolation method for realistic Hamiltonians is presented.In addition, applications at finite temperature are outlined.
We present novel Monte Carlo methods for treating the interacting shell model that allow exact calculations much larger than those heretofore possible. The two-body interaction is linearized by an auxiliary field; Monte Carlo evaluation of the resulting functional integral gives ground-state or thermal expectation values of few-body operators. The ``sign problem'' generic to quantum Monte Carlo calculations is absent in a number of cases. We discuss the favorable scaling of these methods with nucleon number and basis size and their suitability to parallel computation.
The formulation and recent applications of the Quantum Monte Carlo diagonalization (QMCD) method are reported. The QMCD has been proposed for solving the quantum many-body inteacting systems, providing us with energy eigenvalues, transition matrix elements and wave functions. Its application to the nuclear shell model is referred to as the Monte Carlo Shell Model. By the Monte Carlo Shell Model calculations, the level structure of low-lying states can be studied with realistic interactions, providing a useful tool for nuclear spectroscopy. The Monte Carlo Shell Model has been applied to the study of a variety of nuclei, and can be characterized as the importance truncation scheme to the full diagonalization which is infeasible in many cases due to extremely large dimensions.