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Physical Review B

Effect of symmetry in the many-particle Wigner function

E. Cancellieri, Paolo Bordone, Carlo Jacoboni

Published Dec 4, 2007• 12 citations
Publisher

Abstract

An analysis of the Wigner function for identical particles is presented. Four situations have been considered. (i) The first is scattering process between two indistinguishable particles described by a minimum uncertainty wave packets showing the exchange and correlation effects in Wigner phase space. (ii) An equilibrium ensemble of $N$ particles in a one-dimensional box and in a one-dimensional harmonic potential is considered second, showing that the reduced one-particle Wigner function, as a function of the energy defined in the Wigner phase space, tends to the Fermi-Dirac or to the Bose-Einstein distribution function, depending on the considered statistics. (iii) The third situation is reduced one-particle transport equation for the Wigner function, in the case of interacting particles, showing the need for the two-particle reduced Wigner function within the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy scheme. (iv) Finally, the electron-phonon interaction in the two-particle case is considered, showing coparticipation of two electrons in the interaction with the phonon bath.

DOI
10.1103/physrevb.76.214301

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An analysis of the Wigner function for identical particles is presented. Four situations have been considered. (i) The first is scattering process between two indistinguishable particles described by a minimum uncertainty wave packets showing the exchange and correlation effects in Wigner phase space. (ii) An equilibrium ensemble of NNN particles in a one-dimensional box and in a one-dimensional harmonic potential is considered second, showing that the reduced one-particle Wigner function, as a function of the energy defined in the Wigner phase space, tends to the Fermi-Dirac or to the Bose-Einstein distribution function, depending on the considered statistics. (iii) The third situation is reduced one-particle transport equation for the Wigner function, in the case of interacting particles, showing the need for the two-particle reduced Wigner function within the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy scheme. (iv) Finally, the electron-phonon interaction in the two-particle case is considered, showing coparticipation of two electrons in the interaction with the phonon bath.
An analysis of the Wigner function for identical particles is presented. Four situations have been considered. i) A scattering process between two indistinguishable electrons described by a minimum uncertainty wave packets showing the exchange and correlation hole in Wigner phase space. ii) An equilibrium ensemble of N electrons in a one-dimensional box and in a one-dimensional harmonic potential showing that the reduced single particle Wigner function as a function of the energy defined in the Wigner phase-space tends to a Fermi distribution. iii) The reduced one-particle transport-equation for the Wigner function in the case of interacting electrons showing the need for the two-particle reduced Wigner function within the BBGKY hierarchy scheme. iv) The electron-phonon interaction in the two-particle case showing co-participation of two electrons in the interaction with the phonon bath.
Scattering of non-interacting, identical bosons or fermions by a one-dimensional Dirac delta function barrier potential underlines the importance of the role of statistics (that is, whether the particles obey Fermi–Dirac or Bose–Einstein statistics) in the scattering. We consider an initial wave function for the system that corresponds to one particle incident from the left and one from the right of the potential barrier. For bosons, both particles are scattered either to the left or to the right if the intensity reflection coefficient is 1/2, provided the left and right propagating wave packets fully overlap in the scattering region. For fermions, the particles “pass through” one another, provided the left and right propagating wave packets fully overlap in the scattering region, with zero probability that both particles are scattered to the left or right, consistent with the Pauli exclusion principle.
A numerical method, suitable for the simulation of the time evolution of quantum spin models of arbitrary lattice dimension, is presented. The method combines sampling of the Wigner function with evolution equations obtained from the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Quantum correlations are treated through both the Wigner function sampling and the BBGKY evolution, bringing about highly accurate estimates of correlation functions. The method is particularly suitable for long-range interacting systems, and we demonstrate its power by comparing with exact results as well as other numerical methods. As an application we compute spin squeezing in a frustrated two-dimensional lattice with antiferromagnetic power-law interactions and a transverse field, making predictions pertinent to future ion trap experiments.
We consider the dynamics of |NNN| bosons in three dimensions. We assume that the pair interaction is given by |N3β−1V(Nβ⋅)N^{3\beta -1}V(N^{\beta }\cdot )N3β−1V(Nβ⋅)|⁠. By studying an associated many-body wave operator, we introduce a Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy which takes into account all of the interparticle singular correlation structures developed by the many-body evolution from the beginning. Assuming energy conditions on the |NNN|-body wave function, for |$\beta \in ( 0,1] ∣⁠,wederivetheGross–Pitaevskiihierarchywith2−bodyinteraction.Inparticular,weestablishthat,inthe∣|⁠, we derive the Gross–Pitaevskii hierarchy with 2-body interaction. In particular, we establish that, in the |∣⁠,wederivetheGross–Pitaevskiihierarchywith2−bodyinteraction.Inparticular,weestablishthat,inthe∣N\rightarrow \infty ∣limit,all∣| limit, all |∣limit,all∣k∣−bodyscatteringprocessesvanishif∣|-body scattering processes vanish if |∣−bodyscatteringprocessesvanishif∣k\geqslant 3∣andthusprovideadirectanswertoaquestionraisedbyErdo¨setal.[30].Moreover,thisnewBBGKYhierarchysharesthelimitpointswiththeordinaryBBGKYhierarchystronglyfor∣| and thus provide a direct answer to a question raised by Erdös et al. [30]. Moreover, this new BBGKY hierarchy shares the limit points with the ordinary BBGKY hierarchy strongly for |∣andthusprovideadirectanswertoaquestionraisedbyErdo¨setal.[30].Moreover,thisnewBBGKYhierarchysharesthelimitpointswiththeordinaryBBGKYhierarchystronglyfor∣\beta \in ( 0,1) ∣andweaklyfor∣| and weakly for |∣andweaklyfor∣\beta =1∣⁠.SincethisnewBBGKYhierarchyconvertstheproblemfromatwo−bodyestimatetoaweakerthree−bodyestimateforwhichwehavetheestimatestoachieve∣|⁠. Since this new BBGKY hierarchy converts the problem from a two-body estimate to a weaker three-body estimate for which we have the estimates to achieve |∣⁠.SincethisnewBBGKYhierarchyconvertstheproblemfromatwo−bodyestimatetoaweakerthree−bodyestimateforwhichwehavetheestimatestoachieve∣\beta <1∣⁠,itthenallowsustoprovethatalllimitpointsoftheordinaryBBGKYhierarchysatisfythespace–timeboundconjecturedbyKlainermanandMachedon[48]for∣|⁠, it then allows us to prove that all limit points of the ordinary BBGKY hierarchy satisfy the space–time bound conjectured by Klainerman and Machedon [48] for |∣⁠,itthenallowsustoprovethatalllimitpointsoftheordinaryBBGKYhierarchysatisfythespace–timeboundconjecturedbyKlainermanandMachedon[48]for∣\beta \in (0,1)$|⁠.