Gaussian phase-space representations for fermions

Abstract

We introduce a positive phase-space representation for fermions, using the most general possible multimode Gaussian operator basis. The representation generalizes previous bosonic quantum phase-space methods to Fermi systems. We derive equivalences between quantum and stochastic moments, as well as operator correspondences that map quantum operator evolution onto stochastic processes in phase space. The representation thus enables first-principles quantum dynamical or equilibrium calculations in many-body Fermi systems. Potential applications are to strongly interacting and correlated Fermi gases, including coherent behavior in open systems and nanostructures described by master equations. Examples of an ideal gas and the Hubbard model are given, as well as a generic open system, in order to illustrate these ideas.

Exact classical stochastic representations of the many-body quantum dynamics

“…Moreover, the overcompleteness of the basis leads to the existence of equivalent transformations of quasiprobability master equations, which allows us to perform their stochastic unraveling. Such a structure is already known to occur in the representations of quantum mechanics in generalized phase spaces [23][24][25][26]. We also demonstrate that the stochastic wave-function methods [8,[12][13][14][15][16][17][18][19][20]36] also fall into this category of stochastic representations.…”

“…The space L is usually called the generalized phase space [23][24][25][26], due to its intrinsic analogy to the phase space in the deformation quantization [27]. We suppose that the stochastic representation provides us with a methodology for how to assign a positive quasiprobability distribution P (λ) to anyρ ∈ P + .…”

“…(5)] together with the trace formula (6) are the most important components of the stochastic representation: they completely define the mathematical structure of our stochastic representation, which is given by Eqs. (19), (21), (22) and (25).…”

“…First, these operators α allow us to transform the representation H [Eq. (25)] of the quantum Liouville superoperator into the form which admits the stochastic interpretation; second, the operators α allow us to adjust the quasiprobability master equation [Eq. (26)] so that it is valid and that it possesses no systematic errors with respect to the exact quantum dynamics [Eq.…”

Superconductivity in a Hubbard-Fröhlich model and in cuprates

“…Further support was provided by dynamical mean-field theory ͑DMFT͒ and dynamical cluster approximation ͑DCA͒ results ͑for a review, see Ref. 11͒. However, recent studies by Aimi and Imada, 12 using a sign-problem-free Gaussian-basis Monte Carlo ͑GBMC͒ algorithm 13 showed that the Hubbard model does not account for high-temperature superconductivity either. This remarkable result is in line with earlier numerical studies using the auxiliary-field quantum Monte Carlo ͑Ref.…”