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arXiv:2209.13223 · Journal of Physics A Mathematical and Theoretical

Fermion quadrature bases for Wigner functionals

Filippus Stefanus Roux

Published May 7, 2024• 1 citation
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Abstract

Abstract A Grassmann functional phase space is formulated for the definition of fermionic Wigner functionals by identifying suitable fermionic operators that are analogues to boson quadrature operators. Instead of the Majorana operators, we use operators that are defined with relative spin transformations between the ladder operators. The eigenstates of these operators are shown to provide orthogonal bases, provided that the dual space is defined with the incorporation of a spin transformation. These bases then serve as quadrature bases in terms of which Wigner functionals are defined in a way equivalent to the bosonic case. As an application, we consider a two-level fermion system.

DOI
10.1088/1751-8121/ad483b

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Abstract A Grassmann functional phase space is formulated for the definition of fermionic Wigner functionals by identifying suitable fermionic operators that are analogues to boson quadrature operators. Instead of the Majorana operators, we use operators that are defined with relative spin transformations between the ladder operators. The eigenstates of these operators are shown to provide orthogonal bases, provided that the dual space is defined with the incorporation of a spin transformation. These bases then serve as quadrature bases in terms of which Wigner functionals are defined in a way equivalent to the bosonic case. As an application, we consider a two-level fermion system.
We discuss a scheme for performing Jordan-Wigner transformation for various lattice fermion systems in two and three dimensions which keeps internal and spatial symmetries manifest. The correspondence between fermionic and bosonic operators is established with the help of auxiliary Majorana fermions. The current construction is applicable to general lattices with even coordination numbers and an arbitrary number of fermion flavors. The approach is demonstrated on the single-orbital square, triangular, and cubic lattices for spin-1/21/21/2 fermions. We also discuss the relation to some quantum spin liquid models.
Abstract This chapter formally presents the representation of the quantum density operator by phase space distribution functionals of phase space fields, leading to expressions for quantum correlation functions in terms of phase space functional integrals, where the field functions—complex fields for bosons, Grassmann fields for fermions—replace field annihilation and creation operators. The emphasis is on double-phase-space normally ordered (positive P type) normalised distribution functionals, with distinct fields for annihilation and creation field operators; however, symmetrically ordered (Wigner type) distribution functionals are also considered, along with unnormalised B distribution functionals, which lead to phase space functional integrals for Fock position state populations and coherences. Characteristic functionals are first defined and then shown to be related to distribution functionals via phase space functional integrals. The existence of the distribution functionals is demonstrated from previous results for equivalent distribution functions by using mode expansions.
Using the quadrature bases that incorporate the spatiotemporal degrees of freedom, we develop a Wigner functional theory for quantum optics, as an extension of the Moyal formalism. Since the spatiotemporal quadrature bases span the complete Hilbert space of all quantum optical states, it does not require factorization as a tensor product of discrete Hilbert spaces. The Wigner functions associated with such a space become functionals and operations are expressed by functional integrals -- the functional version of the star product. The resulting formalism enables tractable calculations for scenarios where both spatiotemporal degrees of freedom and particle-number degrees of freedom are relevant. To demonstrate the approach, we compute examples of Wigner functionals for a few well-known states and operators.
Abstract Fermionic phase space representations are a promising method for studying correlated fermion systems. The fermionic Q-function and P-function have been defined using Gaussian operators of fermion annihilation and creation operators. The resulting phase-space of covariance matrices belongs to the symmetry class D, one of the non-standard symmetry classes. This was originally proposed to study mesoscopic normal-metal-superconducting hybrid structures, which is the type of structure that has led to recent experimental observations of Majorana fermions. Under a unitary transformation, it is possible to express these Gaussian operators using real anti-symmetric matrices and Majorana operators, which are much simpler mathematical objects. We derive differential identities involving Majorana fermion operators and an antisymmetric matrix which are relevant to the derivation of the corresponding Fokker–Planck equations on symmetric space. These enable stochastic simulations either in real or imaginary time. This formalism has direct relevance to the study of fermionic systems in which there are Majorana type excitations, and is an alternative to using expansions involving conventional Fermi operators. The approach is illustrated by showing how a linear coupled Hamiltonian as used to study topological excitations can be transformed to Fokker–Planck and stochastic equation form, including dissipation through particle losses.
The basic ideas of second quantization and Fock space are extended to density operator states, used in treatments of open many‐body systems. This can be done for fermions and bosons. While the former only requires the use of a non‐orthogonal basis, the latter requires the introduction of a dual set of spaces. In both cases an operator algebra closely resembling the canonical one is developed and used to define the dual sets of bases. We here concentrated on the bosonic case where the unboundedness of the operators requires the definitions of dual spaces to support the pair of bases. Some applications, mainly to non‐equilibrium steady states, will be mentioned.