The semiclassical propagator in fermionic Fock space

Abstract

We present a rigorous derivation of a semiclassical propagator for anticommuting (fermionic) degrees of freedom, starting from an exact representation in terms of Grassmann variables. As a key feature of our approach the anticommuting variables are integrated out exactly, and an exact path integral representation of the fermionic propagator in terms of commuting variables is constructed. Since our approach is not based on auxiliary (Hubbard-Stratonovich) fields, it surpasses the calculation of fermionic determinants yielding a standard form D[ψ, ψ * ]e iR[ψ,ψ * ] with real actions for the propagator. These two features allow us to provide a rigorous definition of the classical limit of interacting fermionic fields and therefore to achieve the long-standing goal of a theoretically sound construction of a semiclassical van Vleck-Gutzwiller propagator in fermionic Fock space.As an application, we use our propagator to investigate how the different universality classes (orthogonal, unitary and symplectic) affect generic many-body interference effects in the transition probabilities between Fock states of interacting fermionic systems.

Spectral statistics of chaotic many-body systems

“…In particular keeping the sum over different solutions of the Gross-Pitaevski equation allows to account for crucial interference effects between such solutions. An approach where the semiclassical propagator of bosonic many-particle systems was used to study these effects was pioneered in [10] for coherent backscattering, see also [11] for applications to fermionic systems.…”

Post-Ehrenfest many-body quantum interferences in ultracold atoms far out of equilibrium

“…Finally, we mention that with the help of a semiclassical propagator in fermionic Fock space (such as proposed in Ref. [32]), the present method can be extended to the case of fermionic atoms. This opens various perspectives for studying the interplay of scrambling and localization phenomena in the context of ultracold (bosonic or fermionic) gases.…”

“…Going beyond TWA without resorting to rather involved numerical "quantum" methods based, e.g., on the time-dependent density matrix renormalization group (t-DMRG) [28][29][30] or on matrix product states (MPS) [31] (which would fail to reach the mesoscopic regime) requires the implementation of a truly semiclassical technique. It would account for the phases that are associated with the mean-field (MF) trajectories of the classical TWA sampling [32][33][34][35]. Exploiting the formal similarity between the N → ∞ limit of the bosonic manybody systems and the → 0 limit of a one-body prob- lem, a proper theory can be constructed for the manybody case by generalizing the time-dependent semiclassical techniques developed in the one-body context [36].…”

The semiclassical propagator in Fock space: dynamical echo and many-body interference

“…This prediction, successfully confirmed in numerical calculations presented in [5], shows the power of the semiclassical thinking in many-body systems, a road that has been shown to be useful for other set-ups like the 1-site Bose-Hubbard system [44], the WKB approach for the 2-site case of [45][46][47][48][49], bosonic transport in optical lattices [50], connecting soliton-like solutions of discrete nonlinear equations with properties of the quantum spectra [51], investigating spectral statistics of fully chaotic systems [52][53][54][55] and going beyond the classical truncated Wigner method to describe dynamical processes in many-body systems [56]. While the extension of our methods to non-relativistic Fermionic fields on the lattice (partially initiated in [57]) is the subject of work in progress, the ultimate goal of addressing fully fledged quantum fields in the continuum, or equivalently an infinite number of possible single-particle orbitals, will face fundamental aspects of such systems in the presence of interactions 2 like renormalization issues, well beyond the present tools of semiclassical analysis. However, non-relativistic bounded one-dimensional systems such as interacting bosons with delta interactions in a ring (the Lieb-Liniger model) are of high interest because of their experimental realization in cold atom systems; exploring the formal construction of the semiclassical approximation there is a realistic goal.…”