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arXiv:2405.18312 · Physical Review Letters

Construction of Continuous Collective Energy Landscapes for Large Amplitude Nuclear Many-Body Problems

Paul Carpentier, N. Pillet, Denis Lacroix A, N. Dubray, David Regnier

Published Oct 8, 2024• 10 citations
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Abstract

Several protocols are proposed to build continuous energy surfaces of many-body quantum systems, regarding both energy and states. The standard variational principle is augmented with constraints on state overlap, ensuring arbitrary precision on continuity. As an illustration, the lowest energy and excited state paths relevant for the ^{240}Pu asymmetric fission are studied. The scission is clearly signed, with a neutron excess in the neck, the ultimate glue before its rupture. Our approach can potentially connect any couple of Hilbert space states, which opens up new horizons for various applications.

DOI
10.1103/physrevlett.133.152501

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Several protocols are proposed to build continuous energy surfaces of many-body quantum systems, regarding both energy and states. The standard variational principle is augmented with constraints on state overlap, ensuring arbitrary precision on continuity. As an illustration, the lowest energy and excited state paths relevant for the ^{240}Pu asymmetric fission are studied. The scission is clearly signed, with a neutron excess in the neck, the ultimate glue before its rupture. Our approach can potentially connect any couple of Hilbert space states, which opens up new horizons for various applications.
At present there are two vastly different ab initio approaches to the description of the many-body dynamics: the density functional theory (DFT) and the functional integral (path integral) approaches. On one hand, if implemented exactly, the DFT approach can allow in principle the exact evaluation of arbitrary one-body densities. However, when applied to large amplitude collective motion (LACM), this approach needs to be extended in order to accommodate the phenomenon of surface-hopping, when adiabaticity is strongly violated and the description of a system using a single (generalized) Slater determinant is not valid anymore. The functional integral approach on the other hand does not appear to have such restrictions, but its implementation does not appear to be a straightforward endeavor. However, within a functional integral approach one seems to be able to evaluate in principle any kind of observable, such as the fragment mass and energy distributions in nuclear fission. These two radically different approaches can likely be brought together by formulating a stochastic time-dependent DFT approach to many-body dynamics.
Nuclear fission presents a unique example of quantum entanglement in strongly interacting many-body systems. A heavy nucleus can split into hundreds of combinations of two complementary fragments in the fission process. The entanglement of fragment wave functions is persistent even after separation and impacts the partition of particles and energies between fragments. Based on microscopic dynamical calculations of the fission of 240 Pu, this work finds that dynamical quantum entanglement is indispensable in the appearance of sawtooth distributions of average excitation energies of fragments and thus neutron multiplicities, but not in average neutron excess of fragments. Both sawtooth slopes from particle-number projections are found to be steep – a feature which can be alleviated by random fluctuations. The persistent entanglement is mainly due to non-adiabatic dynamics since the final splitting is so fast that the non-localization of wave functions is kept during the separation. These findings may impact the understanding of quantum entanglement more broadly in mesoscopic systems.
Determining quantum excited states is crucial across physics and chemistry but presents significant challenges for variational methods, primarily due to the need to enforce orthogonality to lower-energy states, often requiring state-specific optimization, penalty terms, or specialized ansatz constructions. We introduce a novel variational principle that overcomes these limitations, enabling the \textit{simultaneous} determination of multiple low-energy excited states. The principle is based on minimizing the trace of the inverse overlap matrix multiplied by the Hamiltonian matrix, Tr(S−1H)\mathrm{Tr}(\mathbf{S}^{-1}\mathbf{H})Tr(S−1H), constructed from a set of \textit{non-orthogonal} variational states {∣ψi⟩}\{|ψ_i\rangle\}{∣ψi​⟩}. Here, Hij=⟨ψi∣H∣ψj⟩\mathbf{H}_{ij} = \langleψ_i | H | ψ_j\rangleHij​=⟨ψi​∣H∣ψj​⟩ and Sij=⟨ψi∣ψj⟩\mathbf{S}_{ij} = \langleψ_i | ψ_j\rangleSij​=⟨ψi​∣ψj​⟩ are the elements of the Hamiltonian and overlap matrices, respectively. This approach variationally optimizes the entire low-energy subspace spanned by {∣ψi⟩}\{|ψ_i\rangle\}{∣ψi​⟩} without explicit orthogonality constraints or penalty functions. We demonstrate the power and generality of this method across diverse physical systems and variational ansatzes: calculating the low-energy spectrum of 1D Heisenberg spin chains using matrix product states, finding vibrational spectrum of Morse potential using quantics tensor trains for real-space wavefunctions, and determining excited states for 2D fermionic Hubbard model with variational quantum circuits. In all applications, the method accurately and simultaneously obtains multiple lowest-lying energy levels and their corresponding states, showcasing its potential as a unified and flexible framework for calculating excited states on both classical and quantum computational platforms.
In most nuclear many-body methods, observables are calculated using many-body wave functions explicitly. The variational two-particle reduced density matrix method is one of the few exceptions to the rule. Ground-state energies of both closed-shell and open-shell nuclear systems can indeed be evaluated by minimizing a constrained linear functional of the two-particle reduced density matrix. However, it has virtually never been used in nuclear theory, because nuclear ground states were found to be well overbound, contrary to those of atoms and molecules. Consequently, we introduced new constraints in the nuclear variational two-particle reduced density matrix method, developed recently for atomic and molecular systems. Our calculations then show that this approach can provide a proper description of nuclear systems where only valence neutrons are included. For the nuclear systems where both neutrons and protons are active, however, the energies obtained with the variational two-particle reduced density matrix method are still overbound. The possible reasons for the noticed discrepancies and solutions to this problem will be discussed.
We investigate an approach for studying the ground state of a quantum many-body Hamiltonian that is based on treating the correlation functions as variational parameters. In this approach, the challenge set by the exponentially-large Hilbert space is circumvented by approximating the positivity of the density matrix, order-by-order, in a way that keeps track of a limited set of correlation functions. In particular, the density-matrix description is replaced by a correlation matrix whose dimension is kept linear in system size, to all orders of the approximation. Unlike the conventional variational principle which provides an upper bound on the ground-state energy, in this approach one obtains a lower bound instead. By treating several one-dimensional spin 1/21/21/2 Hamiltonians, we demonstrate the ability of this approach to produce long-range correlations, and a ground-state energy that converges to the exact result. Possible extensions, including to higher-excited states are discussed.
A variational principle of maximal decoupling between the collective and noncollective subspaces has recently been proposed to pin down the collective degrees of freedom of the nuclear many-body problem. This maximal-decoupling variational principle is further utilized here to determine in an optimal way the auxiliary Hamiltonians to be used in severely truncated boson-space calculations for nuclear collective motions. In the present method one treats the spurious-state problem in truncated boson-space calculations by using appropriate auxiliary boson Hamiltonians. The viability of the method is demonstrated through its application to a simplified shell model containing only the monopole pairing interactions.