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The nuclear many-body problem for large, many-shell nuclei: An exact solution using tensor networks

Roman Rausch, Cassian Plorin, Matthias Peschke, Christoph Karrasch

Published May 14, 2021• 0 citations
OpenAlex

Abstract

We introduce an exact numerical technique to solve the nuclear pairing Hamiltonian and to determine properties such as the even-odd mass differences or spectral functions for any element within the periodic table for any number of nuclear shells. In particular, we show that the nucleus is a system with small entanglement and can thus be described efficiently using a one-dimensional tensor network (matrix-product state) despite the presence of long-range interactions. Our approach is numerically cheap and accurate to essentially machine precision, even for large nuclei. We apply this framework to compute the even-odd mass differences of all known lead isotopes from $^{178}$Pb to $^{220}$Pb in the very large configuration space of 13 shells between the neutron magic numbers 82 and 184 (i.e., two major shells) and find good agreement with the experiment. To go beyond the ground state, we calculate the two-neutron removal spectral function of $^{210}$Pb which relates to a two-neutron pickup experiment that probes neutron-pair excitations across the gap of $^{208}$Pb. Finally, we discuss the capabilities of our method to treat pairing with non-zero angular momentum. This is numerically more demanding, but one can still determine the lowest excited states in the full configuration space of one major shell with modest effort, which we demonstrate for the $N=126$, $Z\geq 82$ isotones.

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Matrix‐Product State Approach to the Generalized Nuclear Pairing Hamiltonian

Roman Rausch, Cassian Plorin, Matthias Peschke +1 more

Abstract It is shown that from the point of view of the generalized pairing Hamiltonian, the atomic nucleus is a system with small entanglement and can thus be described efficiently using a 1D tensor network (matrix‐product state) despite the presence of long‐range interactions. The ground state can be obtained using the density‐matrix renormalization group (DMRG) algorithm, which is accurate up to machine precision even for large nuclei, is numerically as cheap as the widely used Bardeen‐Cooper‐Schrieffer (BCS) approach, and does not suffer from any mean‐field artifacts. This framework is applied to compute the even‐odd mass differences of all known lead isotopes from to in a very large configuration space of 13 shells between the neutron magic numbers 82 and 184 (i.e., two major shells) and find good agreement with the experiment. Pairing with non‐zero angular momentum is also considered and the lowest excited states in the full configuration space of one major shell is determined, which is demonstrated for the , isotones. To demonstrate the capabilities of the method beyond low‐lying excitations, the first 100 excited states of with singlet pairing and the two‐neutron removal spectral function of are calculated, which relate to a two‐neutron pickup experiment.

arXiv:2105.07012•Annalen der Physik•Mar 4, 2024•0 citations

Tensor-decomposition techniques for ab initio nuclear structure calculations: From chiral nuclear potentials to ground-state energies

A. Tichai, Roman Schutski, Gustavo E. Scuseria +1 more

Background: The computational resources needed to generate the ab initio solution of the nuclear many-body problem for increasing mass number and/or accuracy necessitates innovative developments to improve upon (i) the storage of many-body operators and (ii) the scaling of many-body methods used to evaluate nuclear observables. The storing and efficient handling of many-body operators with high particle ranks is currently one of the major bottlenecks limiting the applicability range of ab initio studies with respect to mass number and accuracy. Recently, the application of tensor decomposition techniques to many-body tensors has proven highly beneficial to reduce the computational cost of ab initio calculations in quantum chemistry and solid-state physics.Purpose: The impact of applying state-of-the-art tensor factorization techniques to modern nuclear Hamiltonians derived from chiral effective field theory is investigated. Subsequently, the error induced by the tensor decomposition of the input Hamiltonian on ground-state energies of closed-shell nuclei calculated via second-order many-body perturbation theory is benchmarked.Methods: The first proof-of-principles application of tensor-decomposition techniques to the nuclear Hamiltonian is performed. Two different tensor formats are investigated by systematically benchmarking the approximation error on matrix elements stored in various bases of interest. The analysis is achieved while including normal-ordered three-nucleon interactions that are nowadays used as input to the most advanced ab initio calculations in medium-mass nuclei. With the aid of the factorized Hamiltonian, the second-order perturbative correction to ground-state energies is decomposed and the scaling properties of the underlying tensor network are discussed.Results: The employed tensor formats are found to lead to an efficient data compression of two-body matrix elements of the nuclear Hamiltonian. In particular, the sophisticated tensor hypercontraction scheme yields low tensor ranks with respect to both harmonic-oscillator and Hartree-Fock single-particle bases. It is found that the tensor rank depends on the two-body total angular momentum $J$ for which one performs the decomposition, which is itself directly related to the sparsity of the corresponding tensor. Furthermore, including normal-ordered two-body contributions originating from three-body interactions does not compromise the efficient data compression. Ultimately, the use of factorized matrix elements authorizes controlled approximations of the exact second-order ground-state energy corrections. In particular, a small enough error is obtained from low-rank factorizations in $^{4}\mathrm{He},^{16}\mathrm{O}$, and $^{40}\mathrm{Ca}$.Conclusions: It is presently demonstrated that tensor-decomposition techniques can be efficiently applied to systematically approximate the nuclear many-body Hamiltonian in terms of lower-rank tensors. Beyond the input Hamiltonian, tensor-decomposition techniques can be envisioned to scale down the cost of state-of-the-art nonperturbative many-body methods in order to extend ab initio studies to (i) higher precisions, (ii) larger masses, and (iii) nuclei of doubly open-shell character.

arXiv:1810.08419•Physical Review C•Mar 18, 2019•22 citations

Many-Body Perturbation Theory for Ab Initio Nuclear Structure

A. Tichai

The solution of the quantum many-body problem for medium-mass nuclei using realistic nuclear interactions poses a superbe challenge for nuclear structure research. Because an exact solution can only be provided for the lightest nuclei, one has to rely on approximate solutions when proceeding to heavier systems. Over the past years, tremendous progress has been made in the development and application of systematically improvable expansion methods and an accurate description of nuclear observables has become viable up to mass number $A \approx 100$. While closed-shell systems are consistently described via a plethora of different many-body methods, the extension to genuine open-shell systems still remains a major challenge and up to now there is no \textit{ab initio} many-body method which applies equally well to systems with even and odd mass numbers. The goal of this thesis is the development and implementation of innovative perturbative approaches with genuine open-shell capabilities. This requires the extension of well-known single-reference approaches to more general vacua. In this work we choose two complementary routes for the usage of generalized reference states. First, we derive a new \emph{ab initio} approach based on multi-configurational reference states that are conveniently derived from a prior no-core shell model calculation. Perturbative corrections are derived via second-order many-body perturbation theory, thus, merging configuration interaction and many-body perturbation theory. The generality of this ansatz enables for a treatment of medium-mass systems with arbitrary mass number, as well as the extension to low-lying excited states such that ground and excited states are treated on an equal footing. In a complementary approach, we use reference states that break a symmetry of the underlying Hamiltonian. In the simplest case this corresponds to the expansion around a particle-number-broken Hartree-Fock-Bogoliubov vacuum which is obtained from a mean-field calculation. Pairing correlations are already absorbed in the reference state. The mild scaling allows for investigations up to tin isotopic chains. All benchmarks were performed using state-of-the-art chiral two- plus three-nucleon interactions thus allowing for a universal description of nuclear observables in the medium-mass regime.

•Jan 1, 2017•1 citation

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

Y. Alhassid, P. Fanto, C. Özen

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.

Physical Review C•Dec 23, 2024•1 citation

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

Y. Alhassid, P. Fanto, C. Özen

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.

arXiv:2308.14698•Aug 28, 2023•0 citations

Pairing of fermions in atomic traps and nuclei

H. Heiselberg

Pairing gaps for fermionic atoms in harmonic-oscillator traps are calculated for a wide range of interaction strengths and particle numbers, and compared to pairing in nuclei. Especially systems where the pairing gap exceeds the single-level spacing but is smaller than the shell splitting $\ensuremath{\Elzxh}\ensuremath{\omega}$ are studied, which applies to most trapped Fermi atomic systems as well as to finite nuclei. When solving the gap equation for a large trap with such multilevel pairing, one finds that the matrix elements between nearby harmonic-oscillator levels and the quasi particle energies lead to a double logarithm of the gap, and a pronounced shell structure at magic numbers. It is argued that neutron and proton pairing in nuclei belongs to the class of multilevel pairing, that their shell structure follows naturally, and that the gaps scale as $\ensuremath{\sim}{A}^{\ensuremath{-}1/3}$---all in qualitative agreement with odd-even staggering of nuclear binding energies. Pairing in large systems is related to that in the bulk limit. For large nuclei the neutron and proton superfluid gaps approach the asymptotic value in infinite nuclear matter: $\ensuremath{\Delta}\ensuremath{\simeq}1.1\mathrm{MeV}.$

arXiv:physics/0304005•Physical Review A•Nov 24, 2003•36 citations
We introduce an exact numerical technique to solve the nuclear pairing Hamiltonian and to determine properties such as the even-odd mass differences or spectral functions for any element within the periodic table for any number of nuclear shells. In particular, we show that the nucleus is a system with small entanglement and can thus be described efficiently using a one-dimensional tensor network (matrix-product state) despite the presence of long-range interactions. Our approach is numerically cheap and accurate to essentially machine precision, even for large nuclei.
We apply this framework to compute the even-odd mass differences of all known lead isotopes from 178^{178}178Pb to 220^{220}220Pb in the very large configuration space of 13 shells between the neutron magic numbers 82 and 184 (i.e., two major shells) and find good agreement with the experiment. To go beyond the ground state, we calculate the two-neutron removal spectral function of 210^{210}210Pb which relates to a two-neutron pickup experiment that probes neutron-pair excitations across the gap of 208^{208}208Pb. Finally, we discuss the capabilities of our method to treat pairing with non-zero angular momentum. This is numerically more demanding, but one can still determine the lowest excited states in the full configuration space of one major shell with modest effort, which we demonstrate for the N=126N=126N=126, Z≥82Z\geq 82Z≥82 isotones.
Abstract It is shown that from the point of view of the generalized pairing Hamiltonian, the atomic nucleus is a system with small entanglement and can thus be described efficiently using a 1D tensor network (matrix‐product state) despite the presence of long‐range interactions. The ground state can be obtained using the density‐matrix renormalization group (DMRG) algorithm, which is accurate up to machine precision even for large nuclei, is numerically as cheap as the widely used Bardeen‐Cooper‐Schrieffer (BCS) approach, and does not suffer from any mean‐field artifacts. This framework is applied to compute the even‐odd mass differences of all known lead isotopes from to in a very large configuration space of 13 shells between the neutron magic numbers 82 and 184 (i.e., two major shells) and find good agreement with the experiment. Pairing with non‐zero angular momentum is also considered and the lowest excited states in the full configuration space of one major shell is determined, which is demonstrated for the , isotones. To demonstrate the capabilities of the method beyond low‐lying excitations, the first 100 excited states of with singlet pairing and the two‐neutron removal spectral function of are calculated, which relate to a two‐neutron pickup experiment.
Background: The computational resources needed to generate the ab initio solution of the nuclear many-body problem for increasing mass number and/or accuracy necessitates innovative developments to improve upon (i) the storage of many-body operators and (ii) the scaling of many-body methods used to evaluate nuclear observables. The storing and efficient handling of many-body operators with high particle ranks is currently one of the major bottlenecks limiting the applicability range of ab initio studies with respect to mass number and accuracy. Recently, the application of tensor decomposition techniques to many-body tensors has proven highly beneficial to reduce the computational cost of ab initio calculations in quantum chemistry and solid-state physics.Purpose: The impact of applying state-of-the-art tensor factorization techniques to modern nuclear Hamiltonians derived from chiral effective field theory is investigated. Subsequently, the error induced by the tensor decomposition of the input Hamiltonian on ground-state energies of closed-shell nuclei calculated via second-order many-body perturbation theory is benchmarked.Methods: The first proof-of-principles application of tensor-decomposition techniques to the nuclear Hamiltonian is performed. Two different tensor formats are investigated by systematically benchmarking the approximation error on matrix elements stored in various bases of interest. The analysis is achieved while including normal-ordered three-nucleon interactions that are nowadays used as input to the most advanced ab initio calculations in medium-mass nuclei. With the aid of the factorized Hamiltonian, the second-order perturbative correction to ground-state energies is decomposed and the scaling properties of the underlying tensor network are discussed.Results: The employed tensor formats are found to lead to an efficient data compression of two-body matrix elements of the nuclear Hamiltonian. In particular, the sophisticated tensor hypercontraction scheme yields low tensor ranks with respect to both harmonic-oscillator and Hartree-Fock single-particle bases. It is found that the tensor rank depends on the two-body total angular momentum JJJ for which one performs the decomposition, which is itself directly related to the sparsity of the corresponding tensor. Furthermore, including normal-ordered two-body contributions originating from three-body interactions does not compromise the efficient data compression. Ultimately, the use of factorized matrix elements authorizes controlled approximations of the exact second-order ground-state energy corrections. In particular, a small enough error is obtained from low-rank factorizations in 4He,16O^{4}\mathrm{He},^{16}\mathrm{O}4He,16O, and 40Ca^{40}\mathrm{Ca}40Ca.Conclusions: It is presently demonstrated that tensor-decomposition techniques can be efficiently applied to systematically approximate the nuclear many-body Hamiltonian in terms of lower-rank tensors. Beyond the input Hamiltonian, tensor-decomposition techniques can be envisioned to scale down the cost of state-of-the-art nonperturbative many-body methods in order to extend ab initio studies to (i) higher precisions, (ii) larger masses, and (iii) nuclei of doubly open-shell character.
The solution of the quantum many-body problem for medium-mass nuclei using realistic nuclear interactions poses a superbe challenge for nuclear structure research. Because an exact solution can only be provided for the lightest nuclei, one has to rely on approximate solutions when proceeding to heavier systems. Over the past years, tremendous progress has been made in the development and application of systematically improvable expansion methods and an accurate description of nuclear observables has become viable up to mass number A≈100A \approx 100A≈100. While closed-shell systems are consistently described via a plethora of different many-body methods, the extension to genuine open-shell systems still remains a major challenge and up to now there is no \textit{ab initio} many-body method which applies equally well to systems with even and odd mass numbers. The goal of this thesis is the development and implementation of innovative perturbative approaches with genuine open-shell capabilities. This requires the extension of well-known single-reference approaches to more general vacua. In this work we choose two complementary routes for the usage of generalized reference states. First, we derive a new \emph{ab initio} approach based on multi-configurational reference states that are conveniently derived from a prior no-core shell model calculation. Perturbative corrections are derived via second-order many-body perturbation theory, thus, merging configuration interaction and many-body perturbation theory. The generality of this ansatz enables for a treatment of medium-mass systems with arbitrary mass number, as well as the extension to low-lying excited states such that ground and excited states are treated on an equal footing. In a complementary approach, we use reference states that break a symmetry of the underlying Hamiltonian. In the simplest case this corresponds to the expansion around a particle-number-broken Hartree-Fock-Bogoliubov vacuum which is obtained from a mean-field calculation. Pairing correlations are already absorbed in the reference state. The mild scaling allows for investigations up to tin isotopic chains. All benchmarks were performed using state-of-the-art chiral two- plus three-nucleon interactions thus allowing for a universal description of nuclear observables in the medium-mass regime.
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 (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.
Pairing gaps for fermionic atoms in harmonic-oscillator traps are calculated for a wide range of interaction strengths and particle numbers, and compared to pairing in nuclei. Especially systems where the pairing gap exceeds the single-level spacing but is smaller than the shell splitting $\ensuremath{\Elzxh}\ensuremath{\omega}$ are studied, which applies to most trapped Fermi atomic systems as well as to finite nuclei. When solving the gap equation for a large trap with such multilevel pairing, one finds that the matrix elements between nearby harmonic-oscillator levels and the quasi particle energies lead to a double logarithm of the gap, and a pronounced shell structure at magic numbers. It is argued that neutron and proton pairing in nuclei belongs to the class of multilevel pairing, that their shell structure follows naturally, and that the gaps scale as $\ensuremath{\sim}{A}^{\ensuremath{-}1/3}$---all in qualitative agreement with odd-even staggering of nuclear binding energies. Pairing in large systems is related to that in the bulk limit. For large nuclei the neutron and proton superfluid gaps approach the asymptotic value in infinite nuclear matter: $\ensuremath{\Delta}\ensuremath{\simeq}1.1\mathrm{MeV}.$