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The Journal of Chemical Physics

Quantum path integrals and reduced fermion density matrices: One-dimensional noninteracting systems

J. C. Light, Ji-Bing Yuan

Published Jan 15, 1973• 32 citations
Publisher

Abstract

A new approach has been used to determine directly the reduced single particle density matrix for noninteracting fermions in a one-dimensional potential. A time evolution operator closely related to the classical limit of the Feynman path integral propagator is derived which has superior behavior in turning point and nonclassical regions. This is used to determine the reduced density matrix in terms of the phase and time of oscillation at the Fermi energy. The density so obtained has the correct quantum oscillations and nonclassical tails and is a distinct improvement over the Thomas-Fermi density. Two simple examples are presented and implications for electronic structure calculations are discussed.

DOI
10.1063/1.1679253

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A new approach has been used to determine directly the reduced single particle density matrix for noninteracting fermions in a one-dimensional potential. A time evolution operator closely related to the classical limit of the Feynman path integral propagator is derived which has superior behavior in turning point and nonclassical regions. This is used to determine the reduced density matrix in terms of the phase and time of oscillation at the Fermi energy. The density so obtained has the correct quantum oscillations and nonclassical tails and is a distinct improvement over the Thomas-Fermi density. Two simple examples are presented and implications for electronic structure calculations are discussed.
Using an integral expression which is based on the use of discretized path integrals, the one-particle density matrix is evaluated for individual fermions in a one- and three-dimensional harmonic well. In this regard, the present paper improves and extends work previously done by Handler. By deriving a closed form expression for the path integral with finite discretization, the density is expressed as a one-dimensional integral that can be evaluated numerically. In addition, the trace of this density matrix, which corresponds to the number of particles present as a function of the Fermi energy, is evaluated in closed form. For finite discretization, the function is continuous in the Fermi energy, but approaches the correct discontinuous limit as the discretization becomes large. Oscillations are observed at the discontinuities analogous to the Gibbs phenomenon of Fourier analysis. Calculations of the density show a marked improvement in the classically allowed region when the discretization is only one level beyond the simple Thomas–Fermi theory. The density also penetrates into the classically forbidden region as the discretization is increased, but the presence of negative oscillations requires a higher level of discretization for a proper description in this region.
The single particle reduced density matrices for fermions in a linear potential in one and three dimensions are derived. An idempotent reduced density matrix for a nonuniform field with a single turning point is obtained. The density expression contains the contributions from the direct path and the path with one turning point. When applied to a potential well the corrections from the long-time paths are taken from the theory of Light and Yuan. Only classical quantities of momentum, time, and action enter in the final expression. The density distribution for the nonclassical region is more accurate than that given by the formula of Light and Yuan.
For common condensed phase problems described by a low-dimensional system coupled to a harmonic bath, Feynman’s path integral formulation of time-dependent quantum mechanics leads to expressions for the reduced density matrix of the system where the effects of the harmonic environment enter through an influence functional that is nonlocal in time. In a recent Letter [Chem. Phys. Lett. 221, 482 (1994)], we demonstrated that the range of the nonlocal interactions is finite even at zero temperature, such that the nonlocal kernel extends over only a few time steps if the path integral is expressed in terms of accurate quasiadiabatic propagators. This feature arises from disruption of phase coherence in macroscopic environments and leads to Markovian dynamics for an augmented reduced density tensor, permitting iterative time evolution schemes. In the present paper we analyze the structure and properties of the relevant tensor propagator. Specifically, we show that the tensor multiplication scheme rigorously conserves the trace of the reduced density matrix, and that in cases of short-range nonlocality it leads to Redfield-type equations which are correct to all orders in perturbation theory and which take into account memory effects. We also argue that a simple eigenvector analysis reveals (without actual iteration) the nature of the dynamics and of the equilibrium state, and directly yields quantum reaction or relaxation rates.
The quantizer–dequantizer method is employed. Using the construction of probability distributions describing density operators of a quantum system states, the connection between the Feynman path integral and the time evolution of the density operator (Landau density matrix) as well as the wave function of the stateconsidered. For single–mode systems with continuous variables, a tomographic propagator is introduced in the probability representation of quantum mechanics. An explicit expression for the probability in terms of the Green function of the Schrödinger equation is obtained. Equations for the Green functions defined by arbitrary integrals of motion are derived. Examples of probability distributions describing the evolution of state of a free particle, as well as states of systems with integrals of motion that depend on time (oscillator type) are discussed.
One-particle properties of noninteracting fermions in a one-dimensional harmonic trap and at zero temperature are studied. Exact expressions and asymptotic results for a large fermion number N are given for the particle density distribution n0(z,N).{n}_{0}(z,N).n0​(z,N). For large N and near the classical boundary at the Fermi energy, the density displays increasing fluctuations. A simple scaling of these tails of the density distribution with respect to N is established. The Fourier transform of the density distribution is calculated exactly. It displays a small but characteristic hump near 2kF,{2k}_{F},2kF​, with kF{k}_{F}kF​ being a properly defined Fermi wave number. This is due to Friedel oscillations, which are identified and discussed. These quantum effects are missing in the semiclassical approximation. Momentum distributions are also evaluated and discussed. As an example of a time-dependent one-particle problem, we calculate exactly the evolution of the particle density when the trap is suddenly switched off, and find a simple scaling behavior, in agreement with recent general results.
We introduce a new approach for calculating quantum time-correlation functions and time-dependent expectation values in many-body thermal systems; both electronically adiabatic and non-adiabatic cases can be treated. Our approach uses a path integral simulation to sample an initial thermal density matrix; subsequent evolution of this density matrix is equivalent to solution of the time-dependent Schrödinger equation, which we perform using a linear expansion of Gaussian wavepacket basis functions which evolve according to simple classical-like trajectories. Overall, this methodology represents a formally exact approach for calculating time-dependent quantum properties; by introducing approximations into both the imaginary-time and real-time propagations, this approach can be adapted for complex many-particle systems interacting through arbitrary potentials. We demonstrate this method for the spin Boson model, where we find good agreement with numerically exact calculations. We also discuss future directions of improvement for our approach with a view to improving accuracy and efficiency.