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arXiv:2208.04460

Density Matrix of the Fermionic Harmonic Oscillator

Batool A. Abu Saleh

Published Aug 8, 2022• 0 citations
arXivHTMLPDF

Abstract

The path integral technique is used to derive a possible expression for the density operator of the fermionic harmonic oscillator. In terms of the Grassmann variables, the fermionic density operator can be written as: $ρ_F (β)=c^* (β)c(β) \pm c^*(β)c(β)e^{-βω}$, where +(-) means that the sum over all antiperiodic (periodic) orbits. Our density operator is then used to obtain the usual fermionic partition function which describes the fermionic oscillator in thermal equilibrium. Also, according to the periodic orbit $c(β)=c(0)$, the graded fermionic partition function is obtained.

DOI
10.48550/arxiv.2208.04460

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The path integral technique is used to derive a possible expression for the density operator of the fermionic harmonic oscillator. In terms of the Grassmann variables, the fermionic density operator can be written as: ρF(β)=c∗(β)c(β)±c∗(β)c(β)e−βωρ_F (β)=c^* (β)c(β) \pm c^*(β)c(β)e^{-βω}ρF​(β)=c∗(β)c(β)±c∗(β)c(β)e−βω, where +(-) means that the sum over all antiperiodic (periodic) orbits. Our density operator is then used to obtain the usual fermionic partition function which describes the fermionic oscillator in thermal equilibrium. Also, according to the periodic orbit c(β)=c(0)c(β)=c(0)c(β)=c(0), the graded fermionic partition function is obtained.
Abstract The functional integral representation of the density matrix at thermal equilibrium in non-relativistic quantum mechanics (QM) with many degrees of freedom, in the grand canonical formulation is introduced. In QM, Hamiltonians H(p,q) can be also expressed in terms of creation and annihilation operators, a method adapted to the study of perturbed harmonic oscillators. In the holomorphic formalism, quantum operators act by multiplication and differentiation on a vector space of analytic functions. Alternatively, they can also be represented by kernels, functions of complex variables that correspond in the classical limit to a complex parametrization of phase space. The formalism is adapted to the description of many-body boson systems. To this formalism corresponds a path integral representation of the density matrix at thermal equilibrium, where paths belong to complex spaces, instead of the more usual position–momentum phase space. A parallel formalism can be set up to describe systems with many fermion degrees of freedom, with Grassmann variables replacing complex variables. Both formalisms can be generalized to quantum gases of Bose and Fermi particles in the grand canonical formulation. Field integral representations of the corresponding quantum partition functions are derived.
An operator whose position representation is a Gaussian function is referred to as a Gaussian operator. In common with the harmonic oscillator Hamiltonian operator, the eigenvectors of the Gaussian operator are a product of Hermite polynomial functions and a Gaussian weighting factor. It is demonstrated that the thermal equilibrium density operator of a one-dimensional harmonic oscillator can be expressed in Gaussian operator form, including its eigenvalues. These can be obtained without first solving for the eigenfunctions of the harmonic oscillator Hamiltonian.
The manner in which probability amplitudes of paths sum up to form wave functions of a harmonic oscillator, as well as other, simple 1-dimensional problems, is described. Using known, closed-form, path-based propagators for each problem, an integral expression is written that describes the wave function. This expression conventionally takes the form of an integral over initial locations of a particle, but it is re-expressed here in terms of a characteristic momentum associated with motion between the endpoints of a path. In this manner, the resulting expression can be analyzed using a generalization of stationary-phase analysis, leading to distributions of paths that exactly describe each eigenstate. These distributions are valid for all travel times, but when evaluated for long times they turn out to be real, non-negative functions of the characteristic momentum. For the harmonic oscillator in particular, a somewhat broad distribution is found, peaked at value of momentum that corresponds to a classical energy which in turn equals the energy eigenvalue for the state being described.
We present a path integral formalism for expressing matrix elements of the density matrix of a quantum many-body system between any two coherent states in terms of standard Matsubara action with periodic (antiperiodic) boundary conditions on bosonic (fermionic) fields. We show that this enables us to express several entanglement measures for bosonic and fermionic many-body systems described by a Gaussian action in terms of the Matsubara Green function. We apply this formalism to compute various entanglement measures for the two-dimensional Bose-Hubbard model in the strong-coupling regime, both in the presence and absence of Abelian and non-Abelian synthetic gauge fields, within a strong-coupling mean-field theory. In addition, our method provides an alternative formalism for addressing time evolution of quantum many-body systems, with Gaussian actions, driven out of equilibrium without the use of Keldysh technique. We demonstrate this by deriving analytical expressions of the return probability and the counting statistics of several operators for a class of integrable models represented by free Dirac fermions subjected to a periodic drive in terms of the elements of their Floquet Hamiltonians. We provide a detailed comparison of our method with the earlier, related, techniques used for similar computations, discuss the significance of our results, and chart out other systems where our formalism can be used.
In this paper, we consider Dirac fermion confined in harmonic potential and submitted to a constant magnetic field. The corresponding solutions of the energy spectrum are obtained by using the path integral techniques. For this, we begin by establishing a symmetric global projection, which provides a symmetric form for the Green function. Based on this, we show that it is possible to end up with the propagator of the harmonic oscillator for one charged particle. After some transformations, we derive the normalized wave functions and the eigenvalues in terms of different physical parameters and quantum numbers. By interchanging quantum numbers, we show that our solutions have interesting properties. The density of current and the nonrelativistic limit are analyzed where different conclusions are obtained. Finally, the completeness of the Dirac oscillator eigenfunctions is proved by using the standard properties of the generalized Laguerre polynomials.