Stochastic TDHF in an exactly solvable model

doi:10.1016/j.aop.2016.07.008

Annals of Physics

Volume 373, October 2016, Pages 216-229

https://doi.org/10.1016/j.aop.2016.07.008 Get rights and content

Abstract

We apply in a schematic model a theory beyond mean-field, namely Stochastic Time-Dependent Hartree–Fock (STDHF), which includes dynamical electron–electron collisions on top of an incoherent ensemble of mean-field states by occasional 2-particle–2-hole ( $2 p 2 h$ ) jumps. The model considered here is inspired by a Lipkin–Meshkov–Glick model of $Ω$ particles distributed into two bands of energy and coupled by a two-body interaction. Such a model can be exactly solved (numerically though) for small $Ω$ . It therefore allows a direct comparison of STDHF and the exact propagation. The systematic impact of the model parameters as the density of states, the excitation energy and the bandwidth is presented and discussed. The time evolution of the STDHF compares fairly well with the exact entropy, as soon as the excitation energy is sufficiently large to allow $2 p 2 h$ transitions. Limitations concerning low energy excitations and memory effects are also discussed.

Introduction

Time-dependent mean-field methods are widely used tools to describe the dynamics of many-fermion systems, for example in the framework of time-dependent density functional theory in electronic systems [1], [2], [3] or of time-dependent Hartree–Fock (TDHF) in nuclei [4], [5], [6]. However, at high excitations and/or over long simulation times, dynamical correlations, neglected in mean-field propagation, become increasingly important. These have been studied extensively in homogeneous systems as quantum liquids [7], [8]. Dynamical correlations for finite systems are much more demanding and have been treated mostly in semi-classical approximation by the Vlasov–Uehling–Uhlenbeck (VUU) approach which has found wide spread application, e.g., in nuclear physics [9], [10], in laser excitation of metal clusters [11], [12], [3], or in electron transport in wires [13]. A fully quantum-mechanical description of dynamical correlations in finite fermion systems is much more demanding. One promising line of development is Stochastic TDHF (STDHF) where correlations are handled in terms of an ensemble of mean-field states generated by stochastic jumps into 2-particle–2-hole states [14], [15]. Recently, first practical tests came up in one-dimensional many-electron systems [16], [17].

The aim of this paper is to continue testing of STDHF by comparison with an exact solution. To this end, we employ a sufficiently simple schematic model. Starting point is the Lipkin–Meshkov–Glick (LMG) model [18], [19], [20]. It reduces the dynamics in many-body systems to one degenerated band of occupied levels and another degenerated band of unoccupied levels modeling the typical energy separation of the HOMO–LUMO gap in closed shell systems. A two-body interaction is added which generates one prominent coherent resonance excitation. Depending on the interaction strength, one can simulate a variety of many-body effects as, e.g., spontaneous symmetry breaking or large-amplitude collective motion and it has been used for this purpose particularly in nuclear physics [21]. The LMG model is closely related to models of coupled spin-1/2 systems as used in quantum optics [22]. The difference lies mainly in the shaping of interaction which ranges all over the system in the LMG model while nearest neighbor coupling is often used in other realizations. The LMG model has also been used as a test model in a nuclear context [23], [24], [25], [26]. It can be modified to allow for a description of dissipation by allowing a certain spread of excitation energies over the levels [27]. The energies of the levels are distributed stochastically and that is why we called this extension a Stochastic Two-Level Model (STLM). The simplicity of the model allows an exact solution and so we use STLM here for testing STDHF.

The paper is organized as follows. In Section 2, we first present the STLM, then the TDHF, the STDHF and the exact propagation thereof. We also detail the initial excitation used and the observables studied in this work. And we finally end the theory section on numerical technicalities. In Section 3, we discuss the corresponding results: we start with a typical test case and then study the impact of varying the model parameters and the initial excitation. We finally give some conclusions and perspectives in Section 4.

Section snippets

A stochastic two-level model

The STLM is sketched in Fig. 1. The model consists in two bands of single-particle (s.p.) levels, the lower band denoted by the principle quantum number $s = - 1$ and the upper one by $s = + 1$ . Each band contains an even number of $Ω$ levels denoted by the secondary quantum number $m$ running from $- j$ to $+ j$ in steps of 1 such that $Ω = 2 j + 1$ (and $j$ is then half integer). In the example displayed in Fig. 1, the $j = 9 / 2$ yields a sub-shell with 10 different $m$ values from $- 9 / 2$ to $+ 9 / 2$ . S.p. states are thus represented

A first test case

We start with the analysis of a typical test case from the perspective of the difference of s.p. density matrices, the entropy, and the expectation value of the dipole. The time evolution of these observables is displayed in Fig. 3. In the upper panel, we compare the density matrix $ρ$ obtained in HF and that in STDHF with respect to the exact density matrix $ρ_{ex}$ in terms of norm of the difference $δ_{ρ}$ , defined in Eq. (16), of TDHF or STDHF with respect to the exact solution. STDHF provides a much

Conclusion and perspectives

We have investigated the Stochastic Time-Dependent Hartree–Fock (STDHF) approach in a schematic model which allows a comparison with the exact solution. The model consists of $N$ fermions in two bands of single-particle (s.p.) levels, the lower one fully occupied and the upper one empty. This is augmented by a two-body interaction, which is motivated first from a typical pairing interaction and modified by gradually mixing more and more couplings between different s.p. levels. The model has much

Acknowledgments

This work was supported by the CNRS and the Midi-Pyrénées region (doctoral allocation number 13050239), and the Institut Universitaire de France. It was granted access to the HPC resources of IDRIS under the allocation 2014-095115 made by GENCI (Grand Equipement National de Calcul Intensif), and of CalMiP (Calcul en Midi-Pyrénées) under the allocation P1238.

References (34)

J.A. Maruhn et al.
Comput. Phys. Comm.
(2014)
P.-G. Reinhard et al.
Ann. Phys.
(1992)
N. Slama et al.
Ann. Phys. (NY)
(2015)
H.J. Lipkin et al.
Nuclear Phys.
(1965)
N. Meshkov et al.
Nuclear Phys.
(1965)
A.J. Glick et al.
Nuclear Phys.
(1965)
G. Holzwarth
Nuclear Phys. A
(1973)
Y. Abe et al.
Phys. Rep. C
(1996)
E.K.U. Gross et al.
Top. Curr. Chem.
(1996)
M.A.L. Marques et al.
Annu. Rev. Phys. Chem.
(2004)

T. Fennel et al.

Rev. Modern Phys.

(2010)

K.T.R. Davies et al.

C. Simenel

Eur. Phys. J. A

(2012)

L.P. Kadanoff et al.

Quantum Statistical Mechanics

(1962)

R. Balescu

Equilibrium and Non Equilibrium Statistical Mechanics

(1975)

G.F. Bertsch et al.

Phys. Rep.

(1988)

D. Durand et al.

Nuclear Dynamics in the Nucleonic Regime

(2000)

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Stochastic TDHF in an exactly solvable model

Abstract

Introduction

Section snippets

A stochastic two-level model

A first test case

Conclusion and perspectives

Acknowledgments

Comput. Phys. Comm.

Ann. Phys.

Ann. Phys. (NY)

Nuclear Phys.

Nuclear Phys.

Nuclear Phys.

Nuclear Phys. A

Phys. Rep. C

Top. Curr. Chem.

Annu. Rev. Phys. Chem.

Rev. Modern Phys.

Eur. Phys. J. A

Quantum Statistical Mechanics

Equilibrium and Non Equilibrium Statistical Mechanics

Phys. Rep.

Nuclear Dynamics in the Nucleonic Regime