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arXiv:2308.14698

Circumventing the odd particle-number sign problem in the shell model Monte Carlo

Y. Alhassid, P. Fanto, C. Özen

Published Aug 28, 2023• 0 citations
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Abstract

The shell model Monte Carlo (SMMC) method is a powerful method for calculating exactly (up to statistical errors) thermal observables and statistical properties of atomic nuclei. However, its application has been limited by a sign problem at low temperatures that arises from the projection onto odd particle number even for good-sign interactions. Here, we develop a technique - the partition function extrapolation method (PFEM) - to extract the ground-state energy of an odd-mass nucleus from the excitation partition function calculated at temperatures at which this sign problem is moderate. We validate the PFEM in heavy even-mass nuclei and systematically calculate ground-state energies for isotopic chains of heavy odd-mass nuclei. The PFEM can be extended to other finite-size quantum many-body systems.

DOI
10.48550/arxiv.2308.14698

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The shell model Monte Carlo (SMMC) method is a powerful method for calculating exactly (up to statistical errors) thermal observables and statistical properties of atomic nuclei. However, its application has been limited by a sign problem at low temperatures that arises from the projection onto odd particle number even for good-sign interactions. Here, we develop a technique - the partition function extrapolation method (PFEM) - to extract the ground-state energy of an odd-mass nucleus from the excitation partition function calculated at temperatures at which this sign problem is moderate. We validate the PFEM in heavy even-mass nuclei and systematically calculate ground-state energies for isotopic chains of heavy odd-mass nuclei. The PFEM can be extended to other finite-size quantum many-body systems.
The shell model Monte Carlo method is a powerful method for calculating exactly (up to statistical errors) thermal observables and statistical properties of atomic nuclei. However, its application has been limited by a sign problem at low temperatures that arises from the projection onto odd number of particles even for good-sign interactions. Here, we develop a technique---the partition function extrapolation method (PFEM)---to extract the ground-state energy of an odd-mass nucleus from the excitation partition function calculated at temperatures at which this sign problem is moderate. We validate the PFEM in heavy even-mass nuclei and systematically calculate ground-state energies for isotopic chains of heavy odd-mass nuclei. The PFEM can be extended to other finite-size quantum many-body systems.
The shell model Monte Carlo method is a powerful technique to calculate thermal and ground-state properties of strongly correlated finite-size systems. However, its application to odd-particle-number systems has been hampered by the sign problem that originates from the projection on an odd number of particles. We circumvent this sign problem for the ground-state energy by extracting the ground-state energy of the odd-particle-number system from the asymptotic behavior of the imaginary-time single-particle Green's function of the even-particle-number system. We apply this method to calculate pairing gaps of nuclei in the iron region. Our results are in good agreement with experimental pairing gaps.
The shell model Monte Carlo (SMMC) method enables calculations in model spaces that are many orders of magnitude larger than those that can be treated by conventional methods, and is particularly suitable for the calculation of level densities in the presence of correlations. We review recent advances and applications of SMMC for the microscopic calculation of level densities. Recent developments include (i) a method to calculate accurately the ground-state energy of an odd-mass nucleus, circumventing a sign problem that originates in the projection on an odd number of particles, and (ii) a method to calculate directly level densities, which, unlike state densities, do not include the spin degeneracy of the levels. We calculated the level densities of a family of nickel isotopes 59−64Ni and of a heavy deformed rare-earth nucleus 162Dy and found them to be in close agreement with various experimental data sets.
The auxiliary-field Monte Carlo (AFMC) method, also known as the shell model Monte Carlo (SMMC), enables us to calculate microscopically statistical and collective properties of nuclei in the presence of strong correlations. These calculations can be carried out in shell model spaces that are many orders of magnitude larger than spaces that can be treated with conventional diagonalization methods. A recent major development has been the extension of the AFMC approach to heavy nuclei. Such applications to heavy nuclei have been a major challenge. On the conceptual level, a crucial question is whether a truncated spherical shell model Hamiltonian can describe the proper collectivity observed in heavy nuclei and, in particular, the rotational character of strongly deformed nuclei. On the technical level, the low excitation energies make it necessary to perform calculations down to much lower temperatures. At such low temperatures the propagator becomes ill-conditioned and requires the introduction of stabilization methods.
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.
Nuclear energy levels are usually calculated using conventional diagonalization methods in the framework of the configuration-interaction (CI) shell model but these methods are prohibited in very large model spaces. The shell model Monte Carlo (SMMC) method 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 within SMMC 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 correlation matrices (ITCMs) of one-body densities 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 broadly applicable to quantum many-body systems in other disciplines.