A simple position operator for periodic systems
Emilia Aragao Valença, Diego Moreno, Stefano Battaglia, Gian Luigi Bendazzoli, Stefano Evangelisti, Thierry Leininger, Nicolas Suaud, J.A. Berger
aa r X i v : . [ c ond - m a t . o t h e r] N ov A simple position operator for periodic systems
Emilia Aragao Valen¸ca, Diego Moreno, Stefano Battaglia,
1, 2
Gian Luigi Bendazzoli, Stefano Evangelisti, ∗ Thierry Leininger, Nicolas Suaud, and J. A. Berger
1, 4, † Laboratoire de Chimie et Physique Quantiques, IRSAMC, CNRS, Universit´e de Toulouse, UPS, France Dipartimento di Chimica, Biologia e Biotecnologie, Universit`adegli Studi di Perugia, Via Elce di Sotto 8, 06123 Perugia, Italy Universit`a di Bologna, Bologna, Italy European Theoretical Spectroscopy Facility (ETSF) (Dated: November 14, 2018)We present a position operator that is compatible with periodic boundary conditions (PBC). It is aone-body operator that can be applied in calculations of correlated materials by simply replacing thetraditional position vector by the new definition. We show that it satisfies important fundamental aswell as practical constraints. To illustrate the usefulness of the PBC position operator we apply it tothe localization tensor, a key quantity that is able to differentiate metallic from insulating states. Inparticular, we show that the localization tensor given in terms of the PBC position operator yieldsthe correct expression in the thermodynamic limit. Moreover, we show that it correctly distinguishesbetween finite precursors of metals and insulators.
Expectation values that involve the position oper-ator ˆ r play a prominent role in both molecular andcondensed-matter physics. Many important quanti-ties are expressed in term of ˆ r , e.g. , the multipole mo-ments and the localization tensor. The latter quantitywas introduced by Resta and co-workers [1–3] follow-ing an idea of Kohn [4] that information about elec-tron localization should be obtained from the ground-state wave function (see also Ref. [5]). The localiza-tion tensor is able to distinguish between conductorsand insulators: when the number of electrons tendsto infinity it diverges in the case of a conductor, whileit remains finite in the case of an insulator. It hasbeen applied to study the metallic behavior of clus-ters [6–14] and has recently also been used to investi-gate Wigner localization [15].In its standard definition the position operator ˆ r issimply defined as the multiplication with the positionvector r . However, this definition is not compatiblewith periodic boundary conditions (PBC), since r isnot a periodic function. This is a problem, since manyquantities of interest are related to the solid statewhich are conveniently described using PBC. There-fore, it is of great interest to search for a positionoperator that is compatible with PBC while reducingto the position vector r in the appropriate limit. Wewill provide such a definition of the position opera-tor in this work. For notational convenience we willconsider only one dimension in the remainder of thiswork. All our findings can be generalized to higher di-mensions. We use Hartree atomic units ( ~ = 1, e = 1, m e = 1, 4 πǫ = 1).We study a system of length L whose electronicmany-body wave function Ψ( t ) satisfies PBC, i.e. , foreach x i the following condition holds,Ψ( x , · · · , x i , · · · , x N ) = Ψ( x , · · · , x i + L, · · · , x N ) , (1)where N is the number of electrons. We are lookingfor a position operator that is compatible with PBC.We denote such an operator as ˆ q . Let us summarize important criteria that ˆ q shouldsatisfy: 1) ˆ q should be invariant with respect to atranslation L ; 2) ˆ q should reduce to the standardposition operator ˆ x = x for finite systems describedwithin PBC, i.e. , in a supercell approach ( L → ∞ forfixed N ) [16] one should obtain results that coincidewith those obtained within open-boundary conditions(OBC). 3) The distance defined in terms of ˆ q shouldbe gauge-invariant, i.e. , it should be independent ofthe choice of the origin. This criterium is importantsince the main purpose of a position operator is toyield the correct distance between two spatial coor-dinates. Finally, we add a fourth criterium: 4) Fora system of many particles, ˆ q should be a one-bodyoperator, as is ˆ x . Although the last criterium is not afundamental one, it is crucial if we want to apply thenew operator to realistic systems.In a seminal work Resta proposed a definition forthe expectation value of the total position operatorˆ X = P Ni =1 x i that is compatible with PBC [17]. Fol-lowing a similar strategy Resta and coworkers alsoproposed an expression for the localization tensor thatis compatible with PBC [1, 2]. Despite the importantprogress made in these works there are also severalshortcomings to this approach: 1) they provide defi-nitions for expectation values but not a definition forthe position operator itself ; 2) the operators are N -body which make them unpractical for the calculationof expectation values of real correlated systems withmany electrons. Finally, we note that their localisa-tion tensor is ”formally infinite (even at finite N ) inthe metallic case” [1]. Therefore, their approach isnot applicable to finite precursors of metals.Instead, in this work we propose a definition forthe position operator itself . We will demonstrate thatit satisfies the four criteria mentioned above. More-over, we will explicitly show that it correctly yieldsthe macroscopic polarization in the thermodynamiclimit as well as a useful expression for the localizationtensor. The latter gives finite values at finite N and L while yielding the correct values in the thermody-namic limit.In order to treat PBC systems, we associate to theelectron position x the complex position q L ( x ), definedas q L ( x ) = L π i (cid:20) exp (cid:18) π i L x (cid:19) − (cid:21) , (2)with i the imaginary unit. The complex position q L ( x )is a continuous and infinitely derivable function of x .In complete analogy with the quantum treatment ofthe position operator ˆ x , we define the action of thecomplex position operator ˆ q as the multiplication with q L ( x ).Let us review the criteria mentioned above with re-spect to q L ( x ). 1) q L ( x ) is trivially invariant undera translation, i.e. , q L ( x + L ) = q L ( x ). Therefore thecomplex position q L ( x ), unlike the ordinary position x , satisfies the PBC constraint. 2) q L ( x ) reduces tothe standard position operator x in the limit L → ∞ ,in the sense of a supercell approach mentioned above.This can be shown by expanding the exponential func-tion: exp (cid:0) π i L x (cid:1) = 1 + π i L x + O (1 /L ) [18]. 3)the distance defined in terms of q L ( x ) is gauge in-dependent. By defining the difference q L,x ( x , x ) = q L ( x − x ) − q L ( x − x ), where x is the (arbitrary)origin, it can be verified that q L,x ( x , x ) = L π i e − π i L x h e π i L x − e π i L x i . (3)Therefore, the distance q q † L,x ( x , x ) q L,x ( x , x ) isindependent of x , as it should. We note that a propo-sition made in a short comment by Zak [19], in whichthe position operator is defined in terms of a sine func-tion, does not satisfy this important constraint and,therefore, the corresponding distance depends on thechoice of the origin. 4) q L ( x ) is a one-body operator.Let us now demonstrate how the complex positionin Eq. (2) yields a useful expression for the localizationtensor within PBC using an approach that is com-pletely analogous to the OBC case. The localizationtensor λ is defined as the total position spread (TPS)per electron where the TPS is a one-body operatorthat is defined as the second cumulant moment of thetotal position operator ˆ X = P Ni =1 x i [20]: λ ( N ) = 1 N h h Ψ | ˆ X | Ψ i − h Ψ | ˆ X | Ψ i i . (4)The second term in the square brackets ensures gaugeinvariance with respect to the choice of the originof the coordinate system. The localization tensor istranslationally invariant.In a complete analogy with the OBC definition,within PBC we replace the position of a particle x i by its complex position q L ( x i ). In such a way, thecomplex total position operator is still a one-body op- erator, defined as ˆ Q L = N X i =1 ˆ q L ( x i ) . (5)The localisation tensor within PBC λ L is also a realquantity, like λ . It is defined as the second cumulantmoment of the complex total position operator perelectron: λ L ( N ) = 1 N h h Ψ | ˆ Q † L ˆ Q L | Ψ i − h Ψ | ˆ Q † L | Ψ ih Ψ | ˆ Q L | Ψ i i (6)in complete analogy with the OBC definition inEq. (4). Equation (6), together with Eq. (2), is themain result of this work. The simple expression iscompletely general, it can be applied to correlatedmany-body wave functions and is valid for both fi-nite systems and infinite systems, e.g. , by taking thethermodynamic limit. In the special case of a single-determinant wave function, it can be shown that theexpression in Eq. (6) coincides with the result ob-tained in Ref. [2] in the thermodynamic limit.We will now demonstrate that the PBC localisationtensor given by Eq. (6) can differentiate between met-als and insulators and that it can be applied to bothfinite and infinite systems. To do so we will apply itto model systems. First, we will treat the simple caseof N non-interacting electrons in a one-dimensionalbox of length L . This model system can be seen as aprototype of a conductor. For this reason, it is par-ticularly important that the formalism we propose inthis work can be applied to such a system. In analogyto the OBC case [3], one expects that the localizationtensor λ L ( N ), diverges in the thermodynamic limit.We consider N = 2 m + 1 non-interacting electronswhere m is a non-negative integer. For the sake of sim-plicity, we assume that the particles are spinless. Inthe case of particles with spin, the final result can betrivially obtained by multiplying the spinless-particleresult by the spin multiplicity of a single particle. Theeigenfunctions of the Hamiltonian of this system areperiodic orbitals given by φ n ( x ) = 1 √ L exp (cid:18) i 2 πnL x (cid:19) , (7)where n is an integer. Since the particles do notinteract, the ground-state wave function is a singleSlater determinant of the occupied orbitals, given by | Φ i = | φ − m · · · φ m i . The corresponding localisationtensor reads λ L ( N ) = L π N (cid:20) h Φ | N X i =1 e − π i L x i N X j =1 e π i L x j | Φ i− h Φ | N X i =1 e − π i L x i | Φ ih Φ | N X j =1 e π i L x j | Φ i (cid:21) . (8)Inserting a complete set of states in the first term inthe square brackets yields λ L ( N ) = L π N X I =0 N X i,j =1 h Φ | e − i πL x i | Φ I ih Φ I | e i πL x j | Φ i (9)= L π N X | p |≤ m X | l | >m h φ p | e − i πL x | φ l ih φ l | e i πL x | φ p i , (10)where in the last step we used the Slater-Condon rulesfor one-electron operators [21]. Inserting Eq. (7) intothe above expression leads to the following expressionfor the matrix element h φ p | e − i πL x | φ l i ,1 L Z L exp (cid:18) i 2 π ( l − p − L x (cid:19) dx = δ l − p − . (11)Therefore, there is only one nonzero contribution inthe double summation over p and l in Eq. (10), namelywhen p = m and l = m + 1. We can write the finalresult as λ L ( N ) = L N N π , (12)from which we can deduce the behavior of the local-isation tensor in the thermodynamic limit. Since, inthis limit, N/L remains constant, the localisation ten-sor diverges linearly with N , as one would expect fora measure of conductivity applied to a perfect con-ductor.Finally, we consider a dimerized chain containing4 n + 2 atoms at half filling, i.e. , N = 4 n + 2, in atight-binding model. The Hamiltonian is given byˆ H = N X i =1 − t i ( a † i a i +1 + a † i +1 a i ) (13)where a † i ( a i ) is a creation (annihilation) operator andthe hopping parameter t i = 1 − ( − i δ with 0 ≤ δ ≤ δ = 1 while there is no dimerization when δ = 0.The latter system can be interpreted as a precursorof a metal since in the thermodynamic limit this sys-tem becomes metallic. It is convenient to express theHamiltonian in the site basis. Upon diagonalizationwe thus obtain the eigenfunctions which, when in-serted in Eqs. (4) and (6), yield the OBC and PBClocalisation tensors of the dimerized chain.In Fig. 1 we report λ ( N ), as defined in Eq. (4) forOBC, as well as λ L ( N ), as defined in Eq. (6) for PBC,as a function of the number of electrons for variousvalues of δ . The length of the unit cell has been setto unity and we have set L = N such that the density N/L = 1. First of all, we see that the PBC local- N λ ( L ) ( N ) δ=0δ=0.04δ=0.1δ=0.5δ=1 ∞ ∞ FIG. 1. The OBC localisation tensor λ ( N ) (solid lines)and the PBC localisation tensor λ L ( N ) (dashed lines) as afunction of the number of electrons N for various values ofthe dimerization parameter δ in a tight-binding model (seeEq. (13)). The numbers next to each curve are the valuesof the PBC localisation tensor in the thermodynamic limit.They were obtained from Eq. (18) with d = 1. Inset: λ ( L ) ( N ) for δ = 0 .