Stochastic Time-dependent Hartree-Fock Research Articles

An average stochastic approach to two-body dissipation in finite fermion systems

We present average stochastic time-dependent Hartree–Fock (ASTDHF) as a theory for efficient handling of incoherent dynamical correlations in finite fermion systems at fully quantum-mechanical level. The basic time evolution is given by time-dependent mean-field theory. The two-body collisions beyond mean-field propagation are evaluated stochastically in an incoherent ensemble of mean-field states as was done in the previously developed STDHF. The new, and time saving, feature in ASTDHF is that it employs one common mean field throughout the whole ensemble. A detailed presentation of ASTDHF is given and extensive comparison with STDHF and other approaches to dissipation are discussed. ASTDHF is found to be a very efficient approach for well bound systems where mean-field fluctuations remain small.

A collisional extension of time-dependent Hartree–Fock

We propose a collisional extension of time-dependent mean-field theories on the basis of a recently proposed stochastic extension of mean-field dynamics (stochastic time-dependent Hartree–Fock, STDHF). The latter theory is unfortunately too involved to envision practical applications in realistic systems in the near future and is thus bound to model systems. It is also hard to explore moderate to low energies with STDHF, because of vanishing transition probabilities that are impossible to sample properly. For such moderately excited situations covering small fluctuations, we compactify sampling by employing the same average mean field for all STDHF trajectories. The new approach, coined average STDHF (ASTDHF), ignores the fluctuations of the mean field but still accounts correctly for the collisional correlations responsible for dissipative features on top of mean-field dynamics. We detail the main features of the new approach in relation to existing equations, in particular quantum kinetic theories. The new theory is directly connected to STDHF, both formally and practically. We thus discuss in detail how the two approaches are related to each other. We apply the new scheme to illustrative examples taking as benchmark STDHF dynamics in 1D. ASTDHF provides results that are in remarkable agreement with the more elaborate STDHF. It makes it a promising approach to deal with dissipative dynamics in finite quantum systems, because of its moderate cost allowing applications in realistic systems and the possibility of exploring any excitation energy range where collisional correlations are expected to play a role.

Stochastic TDHF in an exactly solvable model

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 (2p2h) 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 2p2h transitions. Limitations concerning low energy excitations and memory effects are also discussed.

Stochastic time-dependent Hartree-Fock methods for fermions: Quasiprobability distributions, master equations, and convergence towards exact quantum dynamics

The time-dependent fermionic Hartree-Fock equations can be stochastically extended in such a way as to become the exact representation of quantum dynamics. This fact was first observed in the work of Juillet and Chomaz [Phys. Rev. Lett. 88, 142503 (2002)]. During the past decade, this observation has led to the emergence of a whole family of stochastic wave-function methods for fermions. The common feature of all these methods is that they are based on the expansion of the density operator over the dyadic product of the two fermionic Slater determinant states. In this work, we develop a unified and rigorous foundation for this family of methods. We find a general form of stochastic equations and describe the sufficient conditions under which these methods converge towards exact quantum dynamics. To achieve these goals, we employ the representation of quantum dynamics in generalized phase space. In particular, we consider the quasiprobability distributions which emerge in these stochastic methods and their master equations. It is shown that the convergence towards exact quantum dynamics is controlled by the problem of boundary terms. We provide an example of stochastic Hartree-Fock method which is well-defined and free from this problem.

Stochastic quantum dynamics beyond mean field

Mean-field approaches where a complex fermionic many-body problem is replaced by an ensemble of independent particles in a self-consistent mean field can describe many static and dynamical aspects. It generally provides a rather good approximation for the average properties of one-body degrees of freedom. However, the mean-field approximation generally fails to produce quantum fluctuations of collective motion. To overcome this difficulty, noise can be added to the mean-field theory leading to a stochastic description of the many-body problem. In the present work, we summarize recent progress in this field and discuss approaches where fluctuations have been added either to the initial time, like in the stochastic mean-field theory or continuously in time as in the stochastic time-dependent Hartree-Fock. In some cases, the initial problem can even be reformulated exactly by introducing Quantum Monte Carlo methods in real time. The possibility to describe superfluid systems is also invoked. Successes and shortcomings of the different beyond mean-field theories are discussed and illustrated.

Exact norm-conserving stochastic time-dependent Hartree–Fock

We derive an exact single-body decomposition of the time-dependent Schroedinger equation for N pairwise-interacting fermions. Each fermion obeys a stochastic time-dependent norm-preserving wave equation. As a first test of the method we calculate the low energy spectrum of Helium. An extension of the method to bosons is outlined.

Stochastic TDHF and the Boltzman-Langevin equation

Outgoing from a time-dependent theory of correlations, we present a stochastic differential equation for the propagation of ensemble of Slater determinants, called stochastic time-dependent Hartree-Fock (stochastic TDHF). These ensembles are allowed large fluctuations to develop in the Hartree-Fock mean fields. An alternative stochastic differential equation, the Boltzmann-Langevin equation, can be derived from stochastic TDHF by averaging over subensembles with small fluctuations.