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Journal of Physics G Nuclear and Particle Physics

Hybrid schemes for the calculation of low-energy levels of shell model Hamiltonians

G. Puddu

Published Jan 19, 2006• 22 citations
Publisher

Abstract

We investigate the use of hybrid schemes for the calculation of low-energy eigenstates of shell model Hamiltonians. These schemes mix the basic ideas behind the quantum Monte Carlo diagonalization (QMCD) method and the VAMPIR method. In applications to some fp nuclei using the GXPF1 effective Hamiltonian, we find most practical and time saving the use gradient methods (typical of the VAMPIR approach) with QMCD trial wavefunctions, which are Slater determinants for fp nuclei. We find that the number of many-body states needed to reach reasonable values of the ground-state energies for a given angular momentum is small. We also explore the possibility of using these hybrid methods for model Hamiltonians with a strong repulsive core.

DOI
10.1088/0954-3899/32/3/007

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We investigate the use of hybrid schemes for the calculation of low-energy eigenstates of shell model Hamiltonians. These schemes mix the basic ideas behind the quantum Monte Carlo diagonalization (QMCD) method and the VAMPIR method. In applications to some fp nuclei using the GXPF1 effective Hamiltonian, we find most practical and time saving the use gradient methods (typical of the VAMPIR approach) with QMCD trial wavefunctions, which are Slater determinants for fp nuclei. We find that the number of many-body states needed to reach reasonable values of the ground-state energies for a given angular momentum is small. We also explore the possibility of using these hybrid methods for model Hamiltonians with a strong repulsive core.
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.
We propose a quantum Monte Carlo diagonalization method (QMCD) for solving the quantum many-body interacting systems. Not only the ground state but also low-lying excited states are obtained with their wavefunctions. Consequently the level structure of low-lying states can be studied with realistic interactions. The developments in the formulation of the QMCD are described with an illustrative and intuitive example. The QMCD is finally characterized as a `importance truncation' scheme to the shell model. After testing this method for Cr, we present first results for energy levels and E2 properties of Ge, indicating its large and -soft deformation. The doubly closed shell probability of Ni is shown to be only 49% in a full pf shell calculation, in contrast to the corresponding probability of Ca which reaches 86%. The prolate deformed excited band is obtained as a result of the QMCD calculation for non-yrast states, in a good agreement to recent experimental data. This band seems to have an SU (3) (-like) structure which was originally suggested by Elliott for sd-shell nuclei.
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.
We present a hybrid quantum-classical Green's function Monte Carlo (GFMC) algorithm for estimating the excited states of the nuclear shell model. The conventional GFMC method, widely used to find the ground state of a quantum many-body system, is plagued by the sign problem, which leads to an exponentially increasing variance with the growth of system size and evolution time. This issue is typically mitigated by applying classical constraints but at the cost of introducing bias. Our approach uses quantum subspace diagonalization (QSD) on a quantum computer to prepare a quantum trial state, replacing the classical trial state in the GFMC process. We also incorporated a modified classical shadow technique in the implementation of QSD to optimize quantum resource utilization. Besides, we extend our hybrid GFMC algorithm to find the excited states of a given quantum system. Numerical results suggest our method largely enhances accuracy in determining excited state energies, offering an improvement over the conventional method.
We present a hybrid quantum-classical Green's function Monte Carlo (GFMC) algorithm for estimating the excited states of the nuclear shell model. The conventional GFMC method, widely used to find the ground state of a quantum many-body system, is plagued by the sign problem, which leads to an exponentially increasing variance with the growth of system size and evolution time. This issue is typically mitigated by applying classical constraints but at the cost of introducing bias. Our approach uses quantum subspace diagonalization (QSD) on a quantum computer to prepare a quantum trial state, replacing the classical trial state in the GFMC process. We also incorporated a modified classical shadow technique in the implementation of QSD to optimize quantum resource utilization. Besides, we extend our hybrid GFMC algorithm to find the excited states of a given quantum system. Numerical results suggest our method largely enhances accuracy in determining excited-state energies, offering an improvement over the conventional method.
Conventional diagonalization methods to calculate nuclear energy levels in the framework of the configuration-interaction (CI) shell model approach are prohibited in very large model spaces. The shell model Monte Carlo (SMMC) is a powerful technique for calculating thermal and ground-state observables of nuclei in very large model spaces, but it is challenging to extract nuclear spectra in this approach. We present a novel method to extract low-lying energy levels for given values of a set of good quantum numbers such as spin and parity. The method is based on imaginary-time one-body density correlation matrices that satisfy asymptotically a generalized eigenvalue problem. We validate the method in a light nucleus that allows comparison with exact diagonalization results of the CI shell model Hamiltonian. The method is applicable to other finite-size quantum many-body systems that can be described within a CI shell model approach.