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International Journal of Modern Physics E

NUCLEAR PAIRING AS RANDOM WALK OF COOPER PAIRS

J. Dudek, Jean Baptiste Faes

Published Feb 1, 2004• 0 citations
Publisher

Abstract

We develop a stochastic approach to obtain the energies and wave functions of the full spectrum of the nuclear pairing Hamiltonian (i.e. not only the ground-state but also the excited states of the nuclear many-body systems). We assume that nuclear Cooper pairs may spontaneously jump from one energy configuration to another, the mechanism resembling that of the random walk on a mesh of non-equidistant points. A probability distribution associated with such a random walk is modelled and the resulting solutions tested using an exact quantum algorithm. We use the Hamiltonian parameters characteristic for the nuclear scale: at present an agreement between the quantum and stochastic treatment is of the order of a few permille in terms of eigen-energies.

DOI
10.1142/s0218301304001953

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We develop a stochastic approach to obtain the energies and wave functions of the full spectrum of the nuclear pairing Hamiltonian (i.e. not only the ground-state but also the excited states of the nuclear many-body systems). We assume that nuclear Cooper pairs may spontaneously jump from one energy configuration to another, the mechanism resembling that of the random walk on a mesh of non-equidistant points. A probability distribution associated with such a random walk is modelled and the resulting solutions tested using an exact quantum algorithm. We use the Hamiltonian parameters characteristic for the nuclear scale: at present an agreement between the quantum and stochastic treatment is of the order of a few permille in terms of eigen-energies.
We present a variational Monte Carlo method that solves the nuclear many-body problem in the occupation number formalism exploiting an artificial neural network representation of the ground-state wave function. A memory-efficient version of the stochastic reconfiguration algorithm is developed to train the network by minimizing the expectation value of the Hamiltonian. We benchmark this approach against widely used nuclear many-body methods by solving a model used to describe pairing in nuclei for different types of interaction and different values of the interaction strength. Despite its polynomial computational cost, our method outperforms coupled-cluster and provides energies that are in excellent agreement with the numerically exact full configuration-interaction values.
Ground-state properties in a model quantum field theory are calculated by stochastic evaluation of a path integral representation of the many-body propagator. The model consists of nonrelativistic nucleons coupled to vector and scalar mesons in one spatial dimension. Binding energies and density distributions are calculated for bound states of up to twenty nucleons. The binding energy as a function of density, the nucleon-nucleon correlation function, and the meson-meson correlation function are evaluated in nuclear matter. Exact ground-state solutions to the full field theory are shown to differ relatively little from those of the potential theory corresponding to the static limit. The exact solutions differ substantially from those of the mean-field (Hartree) approximation, but are quite similar to those obtained in the Hartree-Fock approximation.NUCLEAR STRUCTURE Meson-nucleon field theory. Monte Carlo solution. Nuclear matter.
Pairing correlations in nuclei play a decisive role in determining nuclear drip-lines, binding energies, and many collective properties. In this work a new Configuration-Space Monte-Carlo (CSMC) method for treating nuclear pairing correlations is developed, implemented, and demonstrated. In CSMC the Hamiltonian matrix is stochastically generated in Krylov subspace, resulting in the Monte-Carlo version of Lanczos-like diagonalization. The advantages of this approach over other techniques are discussed; the absence of the fermionic sign problem, probabilistic interpretation of quantum-mechanical amplitudes, and ability to handle truly large-scale problems with defined precision and error control, are noteworthy merits of CSMC. The features of our CSMC approach are shown using models and realistic examples. Special attention is given to difficult limits: situations with non-constant pairing strengths, cases with nearly degenerate excited states, limits when pairing correlations in finite systems are weak, and problems when the relevant configuration space is large.
Background: The nuclear many-body system is a strongly correlated quantum system, posing serious challenges for perturbative approaches starting from uncorrelated reference states. The last decade has witnessed considerable progress in the accurate treatment of pairing correlations, one of the major components in medium-sized nuclei, reaching accuracies below the 1% level of the correlation energy. Purpose: Development of a quantum many-body method for pairing correlations that is (a) competitive in the 1% error range, and (b) can be systematically improved with a fast (exponential) convergence rate. Method: The present paper capitalizes upon ideas from Richardson-Gaudin integrability. The proposed method is a two-step approach. The first step consists of the optimization of a Richardson-Gaudin ground state as variational trial state. At the second step, the complete set of excited states on top of this Richardson-Gaudin ground state is used as an optimal basis for a Configuration Interaction method in an increasingly large effective Hilbert space. Results: The performance of the variational Richardson-Gaudin (varRG) and Richardson-Gaudin Configuration Interaction (RGCI) method is benchmarked against exact results using an effective GGG-matrix interaction for the Sn region. The varRG already reaches accuracies around the 1% level of the correlation energies, and the RGCI step sees an additional improvement scaling exponentially with the size of the effective Hilbert space. Conclusions: The Richardson-Gaudin models of integrability provide an optimized complete basis set for pairing correlations.
The problem of one pair of identical nucleons sitting in N{\cal N}N single particle levels of a potential well and interacting through the pairing force is treated introducing, in the Hamiltonian formalism, even Grassmann variables. The eigenvectors are analytically expressed solely in terms of these with coefficients fixed by the eigenvalues and the single particle energies. For these a specific model is needed: in the case of the harmonic oscillator well, for any strength of the pairing interaction, an accurate expression is derived for both the collective eigenvalue and for those trapped in between the single particle levels. Notably the latter are labelled through an index upon which they depend parabolically.