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Physical Review B

Antisymmetric wave functions for mixed fermion states and energy convexity

Antonios Gonis, X.-G. Zhang, D. M. C. Nicholson, G. Malcolm Stocks

Published Jul 15, 2011• 7 citations
Publisher

Abstract

We show how ensembles or mixed states can be described in terms of pure states that for Fermions lead to wave functions that are antisymmetric with respect to interchange of particle coordinates (and spin). The pure states are constructed in an augmented Hilbert space spanned by products of ensemble states projected onto mutually nonoverlapping coordinates that prevents the appearance of interference terms under antisymmetrization. As a demonstration of this new formalism, and under the assumptions of a positive interparticle interaction and a corresponding energy that is extensive in the number of particle pairs (pair extensive), we prove the convexity relation, ${E}_{v}[N\ensuremath{-}1]+{E}_{v}[N+1]\ensuremath{\geqslant}2{E}_{v}[N]$, where ${E}_{v}[N]$ denotes the total ground-state energy of $N$ Fermions (electrons) under an external potential $v(\mathbf{r})$.

DOI
10.1103/physrevb.84.045121

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We show how ensembles or mixed states can be described in terms of pure states that for Fermions lead to wave functions that are antisymmetric with respect to interchange of particle coordinates (and spin). The pure states are constructed in an augmented Hilbert space spanned by products of ensemble states projected onto mutually nonoverlapping coordinates that prevents the appearance of interference terms under antisymmetrization. As a demonstration of this new formalism, and under the assumptions of a positive interparticle interaction and a corresponding energy that is extensive in the number of particle pairs (pair extensive), we prove the convexity relation, Ev[N\ensuremath−1]+Ev[N+1]\ensuremath⩾2Ev[N]{E}_{v}[N\ensuremath{-}1]+{E}_{v}[N+1]\ensuremath{\geqslant}2{E}_{v}[N]Ev​[N\ensuremath−1]+Ev​[N+1]\ensuremath⩾2Ev​[N], where Ev[N]{E}_{v}[N]Ev​[N] denotes the total ground-state energy of NNN Fermions (electrons) under an external potential v(r)v(\mathbf{r})v(r).
Consider a pair of particles that may be either two fermions or two bosons. e Schrödinger equation for the system is written in terms of the Hamiltonian operator for the system as H( , ) ( , ) ( , ),1 2 1 2 1 2c c= E (12.1) where 1 represents the coordinates of particle 1 and 2 the coordinates of particle 2. e Hamiltonian (1, 2) must be symmetric under particle interchange because of the indistinguishability of the particles, giving H H( , ) ( , ).1 2 2 1= (12.2) e pair wave function χ(1, 2) may be either symmetric or antisymmetric. is can be seen by introducing the permutation operator P12, which permutes the coordinates of the particles.
The symmetric properties of wave functions of bosons and fermons were proposed by the principles of quantum mechanics.The calculation of the expectation value of the square of the separation distance between two particles confirms that bosons with symmetric wave functions under exchange tend to be attractive,while fermions with antisymmetric wave functions under exchange have some repulsive properties.Based on this point of view,the electron wave function is product of position wave function and spin wave function,the position wave functions of oppositely aligned electrons must be symmetric and attraction appears,resulting in electron pairs.
A method for the calculation of translationally invariant wave functions for systems of identical fermions with arbitrary potential of pair interaction is developed.It is based on the well-known result that the essential dynamic part of Hamiltonian for the system of identical particles is the reduced Hamiltonian operator describing a relative movement of two particles inside the system.The eigenfunctions of this operator take into account all correlations caused by interaction.These eigenfunctions are basic elements for building the components (i.e., the functions with a lower degree of antisymmetry) used to construct the total antisymmetric wave function of the system.The main problem of this approach appears to be the antisymmetrization of the components.The developed universal algorithm for antisymmetrization gives a possibility to carry out this operation in a simple way and to keep numerical approximations under control.
We analyze the problem of quantifying entanglement in pure and mixed states of fermionic systems with a fixed number parity yet not necessarily a fixed particle number. The mode entanglement between one single-particle level and its orthogonal complement is first considered, and an entanglement entropy for such a partition of a particular basis of the single-particle Hilbert space H\mathcal{H}H is defined. The sum over all single-particle modes of this entropy is introduced as a measure of the total entanglement of the system with respect to the chosen basis and it is shown that its minimum over all bases of H\mathcal{H}H is a function of the one-body density matrix. Furthermore, we show that if minimization is extended to all bases related through a Bogoliubov transformation, then the entanglement entropy is a function of the generalized one-body density matrix. These results are then used to quantify entanglement in fermion systems with four single-particle levels. For general pure states of such a system a closed expression for the fermionic concurrence is derived, which generalizes the Slater correlation measure defined by J. Schliemann et al. [Phys. Rev. A 64, 022303 (2001)], implying that particle entanglement may be seen as minimum mode entanglement. It is also shown that the entanglement entropy defined before is related to this concurrence by an expression analogous to that in the two-qubit case. For mixed states of this system the convex roof extension of the previous concurrence and entanglement entropy is evaluated analytically, extending the results in previous reference to general states.
Enforcing the Pauli principle in many-body systems of fermions to ensure an antisymmetric wave function is typically a numerically expensive task. Numerical methods, such as coupled-cluster, correlated basis function, Monte Carlo, and traditional configuration interaction, scale with powers of NNN that make the determination of the energy spectrum of these systems a challenge. In a recent paper, we successfully obtained energies for large systems of fermions in the unitary regime in which we applied the Pauli principle ``on paper'' without obtaining an explicit wave function. This approach, an extension of the symmetry invariant perturbation method to fermions, applies the Pauli principle by imposing restrictions on the quantum numbers used to obtain the energy. As a followup to this study, we perform an explicit test of the validity of this application of the Pauli principle by comparing our energy results against the energies corresponding to explicitly antisymmetrized wave functions for an exactly solvable system of harmonically confined, harmonically interacting fermions. Our results show that our simple method of applying the Pauli principle selects from a spectrum of possible many-body energies only those energies that correspond to explicitly antisymmetrized wave functions. Our results were tested for values of NNN through $\ensuremath{\approx}10 000$ and for weak and strong interactions both repulsive and attractive.
Extending the one-particle Schrodinger equation from one to three spatial dimensions enabled us to describe electron states in the hydrogen atom and in other hydrogen-like systems. This chapter generalizes the Schrodinger equation from one to two and then to many particles bound as a single system. For all particles of half-integral spin, the total wave function, as for electrons, is always antisymmetric with respect to interchange, whereas with all particles of integral spin the total wave function is always symmetric. The chemical and spectroscopic characteristics of an element depend on the details of the quantum state of its outermost electrons. The chemical valence depends upon the extent to which the total number of electrons in an atom represents an excess or a deficit with respect to the more stable configuration of a completed shell. The Schrödinger equation for helium contains no explicit mention of the spin states of the electrons.