Computation of many-particle quantum trajectories with exchange interaction: application to the simulation of nanoelectronic devices

Abstract

Abstract. Following Ref. [Oriols X 2007 Phys. Rev. Lett., 98 066803], an algorithm to deal with the exchange interaction in non-separable quantum systems is presented. The algorithm can be applied to fermions or bosons and, by construction, it exactly ensures that any observable is totally independent from the interchange of particles. It is based on the use of conditional Bohmian wave functions which are solutions of singleparticle pseudo-Schrödinger equations. The exchange symmetry is directly defined by demanding symmetry properties of the quantum trajectories in the configuration space with a universal algorithm, rather than through a particular exchange-correlation functional introduced into the single-particle pseudo-Schrödinger equation. It requires the computation of N 2 conditional wave functions to deal with N identical particles. For separable Hamiltonians, the algorithm reduces to the standard Slater determinant for fermions, or permanent for bosons. A numerical test for a two-particle system, where exact solutions for non-separable Hamiltonians are computationally accessible, is presented. The numerical viability of the algorithm for quantum electron transport (in a far-from equilibrium time-dependent open system) is demonstrated by computing the current and fluctuations in a nano-resistor, with exchange and Coulomb interactions among electrons.Computation of many-particle quantum trajectories with exchange interaction 2

Applied Bohmian mechanics

“…See Ref. [227] for details. Once the exact 2D wave function Φ(x 1 , x 2 , t) is known, we can compute the exact 2D Bohmian trajectories straightforwardly.…”

“…However, their numerical values are in principle unknown and need some educated guesses [9,227]. On the other hand, U a (r a , r α b (t), t) is the part of the total potential energy that appears in many-body Schrödinger equation with an explicit dependence on r a .…”

“…(31) for each particle. From a practical point of view, all quantum trajectories r α (t) have to be computed simultaneously [9,124,227].…”

“…However, in order to exactly compute the terms G a and J a we must know the total wave function, which is in principle unknown. There are however a few ways to introduce the symmetry of the wave function without dealing directly with these two coupling terms [35,142,227].…”

Can the wave function in configuration space be replaced by single-particle wave functions in physical space?

“…(42). This type of computation has already been used to study quantum transport in nano electronic devices (Albareda et al 2013;Oriols & Mompart 2012;Albareda et al 2009Albareda et al , 2010Traversa et al 2011;Alarcón et al 2013). A commercial software named BITLLES 11 has been developed following these ideas.…”