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Physical Review

"Seniority-Zero" Excited States of the Pairing Hamiltonian

Richard R. Chasman

Published Apr 26, 1965• 20 citations
Publisher

Abstract

The seniority-zero excited-state spectrum of the Hamiltonian $H=\ensuremath{\Sigma}\stackrel{}{k>0}{\ensuremath{\epsilon}}_{k}({{a}_{k}}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{k}+{{a}_{\ensuremath{-}k}}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{\ensuremath{-}k})\ensuremath{-}G \ensuremath{\Sigma}\stackrel{}{k,l>0}{{a}_{k}}^{\ifmmode\dagger\else\textdagger\fi{}}{{a}_{\ensuremath{-}k}}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{\ensuremath{-}l}{a}_{l}$ is studied in detail. Approximate excited-state wave functions of the form ${\ensuremath{\psi}}^{\ensuremath{\lambda}}=\ensuremath{\Sigma}\stackrel{}{\ensuremath{\alpha}}F(\ensuremath{\lambda},\ensuremath{\alpha})\ensuremath{\Sigma}\stackrel{}{i}D(\ensuremath{\alpha},i){{a}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}{{a}_{\ensuremath{-}i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Sigma}\stackrel{}{j>i}D(\ensuremath{\alpha},j){{a}_{j}}^{\ifmmode\dagger\else\textdagger\fi{}}{{a}_{\ensuremath{-}j}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\cdots}|0〉$ are developed. These solutions are compared with exact solutions of a small system and the agreement is quite good. A somewhat larger system is studied in order to see how the excited-state spectrum changes as the number of separable functions is increased. The method is applied to heavy nuclei and is in good agreement with observed ${0}^{+}$ excited states for nuclei having 144 to 150 neutrons. A theoretical ${0}^{+}$ spectrum is displayed for each even neutron system from 144 to 152 neutrons.

DOI
10.1103/physrev.138.b326

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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.
A new technique is developed in this paper for solving Hamiltonians of the form $H=\ensuremath{\Sigma}\stackrel{}{k>0}{\ensuremath{\epsilon}}_{k}({{a}_{k}}^{+}{a}_{k}+{{a}_{\ensuremath{-}k}}^{+}{a}_{\ensuremath{-}k})\ensuremath{-}\ensuremath{\Sigma}\stackrel{}{k,l>0}G(k,l){{a}_{k}}^{+}{{a}_{\ensuremath{-}k}}^{+}{a}_{\ensuremath{-}l}{a}_{l},$ where G(k,l)G(k, l)G(k,l) may have the forms: G(k,l)=GG(k, l)=GG(k,l)=G (independent of kkk and lll), G(k,l)=GkGlG(k, l)={G}_{k}{G}_{l}G(k,l)=Gk​Gl​, or G(k,l)=G(k,l)G(k, l)=G(k, l)G(k,l)=G(k,l) (nonseparable in kkk and lll). The eigenvalues obtained with this technique are compared with the exact solutions of Kerman, Lawson, and Macfarlane. The agreement with the exact solutions is fairly good.
Pair vibrations are studied for a Hamiltonian with neutron–neutron, proton–proton and neutron–proton pairing. The spectrum is found to be rich in strongly correlated, low-lying excited states. Changing the ratio of diagonal to off-diagonal pairing matrix elements is found to have a large impact on the excited-state spectrum. The variational configuration interaction (VCI) method, used to calculate the excitation spectrum, is found to be in very good agreement with exact solutions for systems with large degeneracies having equal T=0 and T=1 pairing strengths.
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.