Quantum continuum mechanics in a strong magnetic field
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Quantum continuum mechanics in a strong magnetic field
S. Pittalis and G. Vignale
Department of Physics, University of Missouri-Columbia, Columbia, Missouri 65211
I. V. Tokatly
ETSF Scientific Development Centre, Departamento de F´ısica de Materiales,Universidad del Pa´ıs Vasco UPV/EHU, Av. Tolosa 72, E-20018 San Sebasti´an, Spain andIKERBASQUE, Basque Foundation for Science, E-48011, Bilbao, Spain (Dated: December 13, 2018)We extend a recent formulation of quantum continuum mechanics [J. Tao et. al, Phys. Rev.Lett. , 086401 (2009)] to many-body systems subjected to a magnetic field. To accomplishthis, we propose a modified Lagrangian approach, in which motion of infinitesimal volume elementsof the system is referred to the “quantum convective motion” that the magnetic field producesalready in the ground-state of the system. In the linear approximation, this approach results ina redefinition of the elastic displacement field u , such that the particle current j contains both anelectric displacement and a magnetization contribution: j = j + n ∂ t u + ∇× ( j × u ), where n and j are the particle density and the current density of the ground-state and ∂ t is the partial derivativewith respect to time. In terms of this displacement, we formulate an “elastic approximation”analogous to the one proposed in the absence of magnetic field. The resulting equation of motionfor u is expressed in terms of ground-state properties – the one-particle density matrix and thetwo-particle pair correlation function – and in this form it neatly generalizes the equation obtainedfor vanishing magnetic field. I. INTRODUCTION
Consideration of an hydrodynamical formulation of theelectron dynamics goes back to the early days of quantummechanics . In the modern language of time-dependent(current-)density-functional theory (TD(C)DFT), thisnaturally leads to appealing time-dependent orbital-freemethods . In this spirit, in two recent papers ,the problem of calculating the linear response of ageneric quantum many-body systems to an external time-dependent potential has been reformulated in the lan-guage of quantum continuum mechanics (QCM). In thisapproach, the non-equilibrium state of the system is de-scribed in terms of an elastic “displacement field” u ( r , t )(a function of cartesian coordinates r and time t ), suchthat an infinitesimal volume element of the system that islocated at point r in the equilibrium state will be locatedat r + u ( r , t ) in the non-equilibrium state. The particlecurrent density is connected to the displacement by therelation j ( r , t ) = n ( r ) ∂ t u ( r , t ) , (1)where ∂ t represents a partial derivative with respect totime and n ( r ) is the ground-state density. An “elasticapproximation” was then introduced, based on the ideathat the time evolution of the wave function can be de-scribed as a geometric deformation of the ground-statewave function, the deformation being defined by the dis-placement field u . More precisely, the wave function ofthe deformed state | ψ [ u ] i was expressed in terms of theground-state wave function | ψ i in the following manner: | ψ [ u ] i = exp (cid:20) − i Z d r ˆ j ( r ) · u ( r ) (cid:21) | ψ i , (2) where ˆ j ( r ) is the canonical current density operator act-ing as the generator of differential translations – a dif-ferent translation at each point in space. (Here and inthe following, we set the mass of the particles m = 1).Starting from Eq. (2), a closed equation of motion for u could be derived, which is demonstrably exact for (i)one-particle systems, and (ii) many-particle systems sub-jected to high-frequency fields. This equation has theform n ( r ) ∂ t u ( r , t ) = − n ( r ) u · ∇ ( ∇ V ) − δT [ u ] δ u ( r , t ) − δW [ u ] δ u ( r , t ) − n ( r ) ∇ V ( r , t ) , (3)where V ( r ) is a static potential that defines the na-ture of the many-body system, V ( r , t ) is the small time-dependent potential to which the many-body system re-sponds, T [ u ] is the kinetic energy density of the de-formed state (2) expanded to second-order in u , and W [ u ] is the electron-electron interaction energy densityof the deformed state, also expanded to second-order in u . The functional derivatives that appear on the righthand side of Eq. (3) are actually linear integro-differentialoperators acting on u . The explicit form of these op-erators was derived in Refs. 6,7. The final expressionsinvolve only the following ground-state properties: theone-particle density matrix and the two-particle correla-tion function, which are quantities that may be computedby means of quantum Monte Carlo methods .Quantum continuum mechanics holds great promise asa tool for simplifying and streamlining the calculationof excitation spectra, particularly in situations in whichthe ground-state correlations are well understood and thespectrum is dominated by collective excitations. Anotherinteresting possibility is to use quantum continuum me-chanics as a tool for efficiently approximating the density-density response function of the non-interacting Kohn-Sham system. This possibility has been recently pur-sued by Gould and Dobson. The Kohn-Sham responsefunction is then used by these authors, in combinationwith the random phase approximation, to generate moreaccurate exchange-correlation energy functionals, whichcapture dispersion forces between metals and insulators.Moreover, the use of quantum continuum mechanics hasallowed to better understand and simplify the deriva-tion of the expression of the high-frequency limit of the exact exchange-correlation kernel of TD(C)DFT . Weshould also mention that, another interesting approachthat holds promise to overcome present limitations of theavailable approximate KS methods is the semiclassicaldensity-matrix time dependent propagation of Refs. .The theory of Refs. 6,7 was based on the assumptionthat the ground-state of the many-body system is time-reversal invariant and has no spin-orbit coupling, so thatthe ground-state current-density j ( r ) = 0. This paperis concerned with the extension of that theory to many-electron systems in the presence of a static magnetic field B ( r ) = ∇ × A ( r ), where A ( r ) is a static vector poten-tial. Thus, we consider a system of N identical particlesdescribed by the time-dependent Hamiltonianˆ H ( t ) = N X j =1 (cid:20) ( − i ∇ j + A ( r j )) V ( r j ) + V ( r j , t ) (cid:21) + 12 X j = k W ( | r j − r k | ) (4)where W ( | r − r ′ | ) is the electron-electron interaction po-tential (we set e = ~ = c = 1). The time-dependentmany-body wave function Ψ( r , . . . , r N , t ) is the solutionof the Schr¨odinger equation i∂ t Ψ( r , . . . , r N , t ) = H Ψ( r , . . . , r N , t ) (5)with initial conditionΨ( r , . . . , r N ,
0) = Ψ ( r , . . . , r N ) . (6)There are compelling reasons for working out the gen-eralization of QCM to systems subjected to magneticfields. As discussed in Refs. 6,7, the elastic approxi-mation is, in essence, a collective approximation for in-homogeneous systems. It condenses the excitation spec-trum of the many-electron system into a simpler spec-trum of collective excitations, which still carry the exactfull spectral strength (this is a consequence of the exact-ness of the theory in the high-frequency limit). Naturally,such an approximation becomes more trustworthy whenthe excitations under study are truly collective, as op-posed to incoherent single-particle excitations. But, itis well known that a strong magnetic field, by quench-ing the kinetic energy of an electronic system, suppressessingle-particle behavior and promotes collective behav-ior. Indeed, one of the first successful theories of the ex-citation spectrum of the homogeneous two-dimensional electron gas (2DEG) at high magnetic field was basedon a single-mode approximation very similar to our elas-tic approximation . Nanopatterned electronic systemsat high magnetic field (e.g., quantum dots) also exhibitstrongly collective behavior. One can, for example, fab-ricate a lattice of quantum dots (nanopillars) by chemi-cal etching on a 2DEG and observe the collective ex-citations of the resulting electronic system by Ramanscattering . By doing this experiment, new collectivemodes have been recently discovered , which have nocounterpart in the homogeneous 2DEG. We believe thatour continuum mechanics will be useful precisely for amicroscopic study of these collective modes.The generalization of Eq. (3) to systems described bythe Hamiltonian (4) is not as straightforward as onemight initially think. Of course, the presence of the vec-tor potential modifies the form of the kinetic energy andintroduces a Lorentz force term, but this is not the maindifficulty. The main difficulty arises from the fact thatthe ground-state current no longer vanishes: it has a fi-nite expectation value, j ( r ). This introduces an ambi-guity in the definition of the displacement field.The first possibility we explored is to define u in closeanalogy to Eq. (1) such that j ( r , t ) = j ( r ) + n ( r ) ∂ t u ( r , t ) . (7)With this definition, it is possible to derive within stan-dard linear response theory, a closed equation of motionfor u . This will be done in Section II. While this equationis formally elegant and gives some insight into the gen-eral properties of the solutions, it is very difficult to putit in an explicit and therefore useful form. The reason isthat one needs to calculate the ground-state expectationvalue of complicated commutators: the amount of alge-bra involved is formidable. By contrast, in the treatmentof the zero field case we could rely on a direct calculationof the energy of the deformed state (2) – a comparativelysimpler task that did not require the evaluation of com-plicated commutators.Eq. (7) assumes that the excess current j − j is en-tirely due to the time derivative of the displacement field,but, in the presence of a magnetic field, even a time-independent displacement can produce an excess current(see below). After all, the ground-state is perfectly sta-tionary, and yet it does carry a current j . However, atvariance with a time-dependent current, the current as-sociated with a static deformation must necessarily havevanishing divergence in order to satisfy the continuityequation. Thus, for example, one must have ∇ · j = 0.Taking into account this condition, we are free to add tothe right hand side of Eq. (7) the curl of a “magnetiza-tion field”. In other words, we expect the most generalform of the relation between current and displacement tohave the form j ( r , t ) = j ( r ) + n ( r ) ∂ t u ( r , t ) + ∇ × M ( r , t ) , (8)where M ( r , t ) is a functional of u . The divergence of thelast term on the right hand side is guaranteed to be zero.This is completely analogous to the material current inelectrodynamics, which is customarily written as the sumof the time derivative of the electric polarization and thecurl of the magnetization. To determine the form of M ( r , t ) up to the linear or-der in u , we assume that (i) M ( r , t ) is a local func-tional of the displacement, i.e., it depends on u ( r , t ) atthe same point of space and time, (ii) A uniform displace-ment u ( r ) = u must cause the ground-state current tobe rigidly displaced by j ( r ) → j ( r − u ), i.e. we musthave ∇ × M ( r , t ) = − ( u · ∇ ) j ( r ) for uniform u . Thiscondition implies that M ( r ) = u × j ( r ) and fixes therelation between current and displacement in the form j ( r , t ) = j ( r ) + n ( r ) ∂ t u ( r , t ) + ∇ × [ u ( r , t ) × j ( r )] . (9)Remarkably, this relation emerges naturally from theLagrangian formulation of the problem, which we presentin Section III. We remind the reader that in Refs. 6,7 theelastic approximation was derived via a transformationto a local non-inertial reference frame – the so-called co-moving frame – in which the density is constant and thecurrent density is zero. To achieve these conditions, thedisplacement field u , which defines the transformationto the co-moving frame, was related to the current viaEq. (1). In the present situation, the physically “nat-ural” requirement for the co-moving frame is that thedensity remain constant and the current density remainequal to the ground-state value j . In other words, anobserver “riding” on a volume element should not detectany change in the density or the current density. Wefound that this requirement determines the relation be-tween the displacement field and the current density inthe form of Eq. (9) just as we found from the heuristicargument given above.Adopting the mentioned special co-moving frame is thecrucial insight that allows us to arrive at an explicit equa-tion of motion for u in the presence of a magnetic field.Throughout the paper, we will consider a general situ-ation of non-collinear and non-uniform magnetic fields(neglecting spin-degrees of freedom). The equation ofmotion in the practically most common case of a uniform magnetic field is obtained from the equation of motion(3) by the following simple replacements:1. Replace the time derivative ∂ t by the “convectivederivative” D t = ∂ t + v · ∇ (10)where v = j /n .2. Include on the right hand side a “Lorentz forceterm” D t u × B . (11)3. Calculate T [ u ] and W [ u ] respectively from the ex-pectation values of the kinetic energy and electron-electron interaction energy operators evaluated in the deformed ground-state wave function defined,just as in Eq. (2). The electron-electron interac-tion energy term remains formally unaffected, whilethe kinetic energy term is modified by the inclu-sion of an effective vector potential accounting forthe external static vector potential and the cor-responding contribution of the convective motionin the ground-state (remarkably, this modificationvanishes for one-electron systems).The paper is organized as follows:In Section II we derive an approximate equation ofmotion for the conventional displacement field from thehigh-frequency expansion of the exact current-current re-sponse function. We show that this approximation is ex-act for one-particle systems and discuss the difficultiesarising when one attempts to explicitly evaluate the for-mal expressions appearing in this equation.In Section III we address the difficulties discussed inSection II by resorting to a non-standard Lagrangian for-mulation appropriate for systems in magnetic field. Tothis end, we formulate the quantum many-body dynam-ics in a special co-moving frame, such that both the den-sity and the current density retain their initial (ground-state) values at all times. We arrive at an exact, but stillnot explicit equation of motion for the displacement fieldin terms of the Hamiltonian in the stress tensor of theco-moving frame.In Section IV we introduce the elastic approximationand obtain a closed, fully nonlinear equation of motionfor the displacement field in this approximation.Finally, in Section V, we linearize the elastic equa-tion of motion and obtain an explicit, linear equationof motion for the displacement field. It is shown thatthe displacement field obtained in this manner is not theconventional one, but is related to the current density byEq. (8).Section VI concludes the paper with a summary of themain results. II. DERIVATION FROM LINEAR RESPONSETHEORY
A formally exact equation of motion for the cur-rent density response of a many-particle system inthe linear response regime can be easily derived fromstandard linear response theory.
To this end, itis convenient to replace the external potential V ( r , t )by a time-dependent vector potential A ( r , t ) = − R t −∞ ∇ V ( r , t ′ ) dt ′ , which is physically equivalent sinceit gives rise to the same time-dependent electric field andno time-dependent magnetic field. Assuming further thatthe time-dependence of the external field is periodic withangular frequency ω and switched on adiabatically at t = −∞ with the system initially in the ground state | ψ i , we obtain the standard result for the Fourier com-ponent of the current density at frequency ω : j ,µ ( r , ω ) = Z d r ′ χ µν ( r , r ′ , ω ) ∂ ν V ( r ′ ) iω , (12)where χ µν ( r , r ′ , ω ) is the current-current response func-tion (Einstein summation convention is used throughoutthe paper) χ µν ( r , r ′ , ω ) = n ( r ) δ ( r − r ′ ) − i Z ∞ dte iωt h ψ | [ˆ j ( r , t ) , ˆ j ( r ′ , | ψ i (13)and ∂ ν V ( r ′ ) iω is the Fourier component of A ( r , t ) at fre-quency ω . In the above equation (13) ˆ j ( r , t ) is thecurrent-density operator defined asˆ j ( r ) = 12 N X j =1 n [ − i ˆ ∇ j + A (ˆ r j )] , δ ( r − ˆ r j ) o , (14)where { ˆ A, ˆ B } = ˆ A ˆ B + ˆ B ˆ A is the anticommutator. Thisoperator evolves in time under the unperturbed Hamil-tonian, i.e., the Hamiltonian (4) without the V term.The first term on the right hand side of Eq. (12), isthe so-called diamagnetic response, which involves onlythe ground-state density n ( r ). A formal inversion ofEq. (12) yields immediately an equation of motion forthe current density: iω Z d r ′ [ χ − ] µν ( r , r ′ , ω ) j ,ν ( r ′ , ω ) = ∂ µ V ( r , ω ) . (15)Obviously, this result is purely formal, since we haveno way to exactly calculate the current-current responsefunction of a many-body system, let alone invert it. How-ever Eq. (15) can serve as a convenient starting point forconstructing approximate theories. Following the ideasproposed in Refs. we consider the high frequency limitof Eq. (15) and then interpret it as an approximate equa-tion of motion for the induced current density.We start by observing that, at high frequency, thecurrent-current response function has the well-known ex-pansion χ µν ( r , r ′ , ω ) = n ( r ) δ ( r − r ′ ) δ µν − i B µν ( r , r ′ ) ω + M µν ( r , r ′ ) ω + O (cid:18) ω (cid:19) , (16)where B µν ( r , r ′ ) = i h ψ | [ˆ j µ ( r ) , ˆ j ν ( r ′ )] | ψ i , (17)and M µν ( r , r ′ ) = −h ψ | [[ ˆ H , ˆ j µ ( r )] , ˆ j ν ( r ′ )] | ψ i . (18)A few words of caution should be added at this point.The expansion written for χ in Eq. (16) is done under the assumption that the real-space current-current responsefunction has a regular Taylor expansion in inverse pow-ers of 1 /ω at high-frequency. However, this is not alwaysthe case. If the initial (ground) state is not sufficientlysmooth, the unboundedness of the kinetic energy opera-tor may cause χ µν ( ω ) to develop a non-analytic behav-ior (e. g. fractional powers of ω ) at high frequency .Therefore, strictly speaking, the expansion of Eq. (16)assumes that proper smoothness conditions are imposedon the initial state of the system. Fortunately, after in-verting Eq. (16) to get the high frequency expansion ofthe operator χ − µν entering the left hand side of Eq. (15),all smoothness restriction can be relaxed. It turns outthat such obtained high frequency form of the inverseresponse function is generally valid. We prove this inSecs. IV and V by rederiving the equation of motion forthe current density via the Lagrangian frame formalism,which can be viewed as a direct construction of a highfrequency/short time limit of χ − µν . In Appendix B we ex-plicitly demonstrate that for the one-particle system theabove formal inversion procedure indeed yields the exactform of the inverse current response function. A possiblenonanalytic behavior in ω of the response is correctly re-covered in our theory because at the level of Eq. (15) itis encoded in the space part of the operator χ − µν actingon the current density.Thus, we invert Eq. (16) and plug the result intoEq. (15). This yields ω j ,µ + iω Z d r ′ B µν ( r , r ′ ) j ,ν ( r ′ ) n ( r ′ ) − Z d r ′ K µν ( r , r ′ ) j ,ν ( r ′ ) n ( r ′ ) = ∂ µ V (19)where K µν ( r , r ′ ) = M µν ( r , r ′ )+ Z d r ′′ B µγ ( r , r ′′ ) 1 n ( r ′′ ) B γν ( r ′′ , r ′ ) . (20)Evaluation of the current-current commutator inEq. (17) is relatively straightforward and yields B µν ( r , r ′ ) = j ,µ ( r ′ ) ∂δ ( r − r ′ ) ∂r ν + j ,ν ( r ) ∂δ ( r ′ − r ) ∂r µ + ǫ µνγ B ,γ ( r ) n ( r ) δ ( r − r ′ ) , (21)where j ( r ) is the ground-state current. Notice that B µν ( r , r ′ ) is imaginary and antisymmetric.The evaluation of the double-commutator K µν ( r , r ′ ),while in principle equally straightforward, is in practicean extremely cumbersome task, leading to very compli-cated expression to which we are not able to attach anytransparent physical meaning. We do not undertake thistask here, since in the next section we will develop an al-ternative approach, which leads more directly to a phys-ically meaningful equation of motion.Even without knowing the explicit form of K we canlearn something from the form of Eq. (19). To beginwith, we may express the response of the current in termsof the displacement defined in the conventional way, i.e.according to Eq. (7). By doing so, we obtain ω n u + iω B · u − K · u = n ∇ V , (22)where we have adopted a compact notation in which O · u ≡ Z d r ′ O µν ( r , r ′ ) u ν ( r ′ ) . (23)It is easy to prove that M must be a positive-definiteoperator, since, by definition, it is related to the second-order term in the expansion of the energy of the distortedground-state (2) in powers of u .Setting the right-hand side to zero in Eq. (22), we rec-ognize a generalized eigenvalue problem of the form (formore details see Appendix A) ω n u + iω B · u − K · u = 0 . (24)Under the stronger assumption that not only M , butalso K is positive definite, one can show (see AppendixA) that the generalized eigenvalue problem in Eq. (24)has real solutions ω whose modes satisfy a generalizedorthogonality relation h u B , i B · u A i + ( ω A + ω B ) h u B , n u A i = 0 , ω A = ω B (25)with the scalar product defined as follows h u B , u A i ≡ Z d r u ∗ B,µ ( r ) u A,µ ( r ) . (26)The eigenvalues are naturally interpreted as (approx-imate) excitation energies and the corresponding eigen-vectors are matrix elements of the current density oper-ator between the ground-state and the excited state inquestion (see Ref. 7 and Appendix B in the present pa-per). Thus, we see that the assumption of positivity of K is, in practice, equivalent to the expectation that ourapproximation does not lead to spurious instabilities (i.e.complex excitation energies).A natural question at this point is: how reliable ourapproximate equation of motion (22) will be in situationsother than the very high frequency limit? The answer tothis question has already been discussed extensively inRef. 7, so we only summarize the main points addingthe required modifications where needed.First of all, the fact that the proposed current-currentresponse function has the exact high-frequency behaviorup to order 1 /ω implies that the first moment of thespectral function R ∞ dω ω ℑ mχ µν ( r , r ′ , ω ) ≃ M µν ( r , r ′ ) isexactly reproduced. Thus, while the energy of individualexcitations may be misrepresented, the first moment ofthe current-density fluctuation excitation spectrum (or,equivalently, the third moment of the density fluctuationexcitation spectrum) is correctly reproduced.Second, our equation assumes an exact and explicitform for one-particle systems. This is rigorously proved in Appendix B and Appendix C. In particular, in Ap-pendix B we show that χ − ( ω ) · j = A + iB/ω + C/ω for any frequency ω , where A , B , and C are coefficients ex-pressible in terms of the ground-state density and currentdensity. The coefficient A is related to the diamagneticresponse, while B and C may be related to the coeffi-cients of ω − and ω − in the high-frequency expansionof χ .The fact that the form derived by the above proce-dure becomes exact in one-particle systems has a deeperphysical significance. Recall that in Refs. 6,7 the equa-tion for the displacement was derived by describing thedynamics in a co-moving reference frame, defined as annon-inertial frame in which the density remains constantand the current density is always zero. So the high-frequency approximation could be restated in terms ofan “anti-adiabatic approximation” or “elastic approxi-mation” introduced as the assumption that not only thedensity and the current, but the wave function itself re-mains unchanged in the co-moving reference frame. Thisassumption is correct for one-particle systems, because insuch systems the density and the current density uniquelydetermine the wave function, up to a trivial phase factor.We will show that this approach – transformation toa co-moving frame and the requirement of a stationarywave function in that frame – also works in the presenceof a magnetic field, but with an important difference: thecurrent density in the co-moving frame will not be zero,but will be equal to the current density in the ground-state. Because of this redefinition the relation betweencurrent response and displacement field will be drasti-cally changed: Eq. (1) will be replaced by Eq. (8). How-ever, as a result of this modification, we will be able toobtain an explicit equation of motion for the new u , by-passing the need to calculate cumbersome commutators.The final form of the equation of motion will be a naturalextension of the one derived in Refs. 6,7 in the absenceof a magnetic field. III. LAGRANGIAN FORMULATION
The key physical idea of the Lagrangian formulationin application to many-body dynamics is to separate theconvective motion of the electron fluid from the motion ofelectrons relative to the convective flow. The former typeof motion is characterized by the trajectories of infinites-imal volume elements, while the latter is described by amany-body Schr¨odinger equation in a local non-inertialreference frame attached to those elements .In the standard Lagrangian formalism, the convec-tive dynamics of the system is described by a set oftrajectories r ( ξ , t ), which represent the motion of an in-finitesimal volume elements initially located at ξ . Thesetrajectories satisfy the first-order differential equation ∂ t r ( ξ , t ) = j ( r ( ξ , t ) , t ) n ( r ( ξ , t ) , t ) (27)with initial condition r ( ξ ,
0) = ξ . Notice that we are noteven making the linear response approximation at thisstage: the quantity n ( r ( ξ , t ) , t ) is the actual particle den-sity along the trajectory and reduces to the ground-statedensity n ( ξ ) only at the initial time. The displacementfield u ( ξ , t ) is defined via the relation r ( ξ , t ) = ξ + u ( ξ , t ).This standard definition is, however, inconvenient ifthe initial stationary/ground state already carries a non-zero current-density j µ = n v µ , which is the case ina magnetized electronic system. Indeed, even in theabsence of the external driving field the ground stateflow drags the volume elements and produces a time-dependent displacement in a system that, physically,should be considered as stationary and undeformed. It isnatural to try and exclude the equilibrium convective mo-tion from the trajectory function, i.e., to consider a vol-ume element as moving only when it moves relatively tothe ground-state flow. To accomplish this in the presenceof the magnetic field we adopt a modified Lagrangian de-scription, where the motion of the volume elements is de-scribed relatively to the ground state flow. Thus, whenthe system is in the ground-state the displacement van-ishes and the volume elements are regarded as stationary.An observer “riding on” these volume elements does notmove at all (relatively to the laboratory) and sees thecurrent density j . Now, when the system is out of equi-librium a finite displacement appears and the volume el-ements begin to move. However, to an observer attachedto the material elements the current density still appearsto be j . Just as in the standard Lagrangian descriptionan observer riding on the volume element sees stationarydensity n and zero current density, in the present de-scription an observer riding on the volume element seesstationary density n and current density j . The task at hand is now to implement the transforma-tion that makes the density and the current density equalto their ground-state values at all times. This is done intwo steps. First we learn how to describe the quantummany-body dynamics in a generic non-inertial referenceframe. Then we fix the reference frame from the condi-tion that density and current density retain their ground-state values at all times. We carry out the above stepswithout assuming that the displacement is small, i.e., ina fully non linear way. The linearization of the equationsof motion will be carried out only at the very end of thenext Section.
A. Transformation to a non-inertial referenceframe
Let us consider a local reference frame moving along aprescribed trajectory r ( ξ , t ). The form of the quantummany-body dynamics in such a frame is worked out infull detail in Refs. 25–27. Here we present only the keyresults that are needed for our purposes.First of all, the transformed many-body wave function has the form e Ψ( ξ , . . . , ξ N , t ) = N Y j =1 g ( ξ j , t ) e − iS cl ( ξ j ,t ) × Ψ( r ( ξ , t ) , . . . , r ( ξ N , t ) , t ) , (28)where g µν ( ξ , t ) = ∂r α ( ξ , t ) ∂ξ µ ∂r α ( ξ , t ) ∂ξ ν (29)is the metric tensor induced by the (non-singular) trans-formation r → ξ , √ g ≡ p det g µν is the Jacobian of thesame transformation and S cl ( ξ , t ) = Z t dt (cid:20)
12 (˙ r ( ξ , t )) − ˙ r ( ξ , t ) A ( r ( ξ , t ) , t ) − V ( r ( ξ , t ) , t )] . (30)is the classical action of a particle moving along the as-signed trajectory. The factor Q Nj =1 g ( ξ j , t ) in Eq. (28)preserves the standard normalization of the wave func-tion h e Ψ | e Ψ i = 1 after a non-volume-preserving transfor-mation of coordinates. Equation (28) is a generaliza-tion of the transformation to a homogeneously acceler-ated frame, which is used, for example, in the proofs ofa harmonic potential theorem .The transformed wave function e Ψ( ξ , . . . , ξ N , t ) satis-fies the Schr¨odinger equation i∂ t e Ψ( ξ , . . . , ξ N , t ) = e H [ g µν , A ] e Ψ( ξ , . . . , ξ N , t ) (31)with the transformed Hamiltonian e H [ g µν , A ] = N X j =1 g − j ˆ K j,µ √ g j g µνj K j,ν g − j + 12 X k = j W ( l ξ k ξ j )(32)where ˆ K j,µ = − i∂ ξ µj + A µ ( ξ j , t ), A µ ( ξ , t ) = ∂r ν ∂ξ µ A ,ν ( r ( ξ , t )) − ∂r ν ∂ξ µ ˙ r ν ( r ( ξ , t ) , t )+ ∂ ξ µ S cl ( ξ , t ) , (33)and l ξ k ξ j is the distance between j th and k th particlesin the non-inertial frame (i.e., the length of geodesic con-necting points ξ j and ξ k in the space with metric g µν ).The metric tensor g µ,ν and “effective” vector potential A in Eq. (33) describe the combined action of externalforces and inertial forces in the local non-inertial frame.The transformed densities are obtained from the re-duced one-particle density matrix e γ ( ξ , ξ ′ , t ) = N R N Y j =2 d ξ j × e Ψ ∗ ( ξ , . . . , ξ N , t ) e Ψ( ξ ′ , . . . , ξ N , t )(34)through the formulas: e n ( ξ , t ) = e γ ( ξ , ξ , t ) , (35) e j µ ( ξ , t ) = g µν ( ξ , t ) he j p,ν ( ξ , t ) + e n ( ξ , t ) A ν ( ξ , t ) i , (36)where e j p,µ ( ξ , t ) = i ξ µ ′ → ξ µ ( ∂ ξ µ − ∂ ξ µ ′ ) e γ ( ξ , ξ ′ , t ) , (37)is the (covariant) paramagnetic current. It is extremelyimportant to carefully keep track of covariant (lower in-dices) and contravariant (upper indices) components ofvectors and tensors. The two types of components areconnected by the metric tensor via the standard formu-las V µ = g µν V ν V µ = g µν V ν , (38)where g µν is the inverse of g µν . Armed with these defi-nitions, one can readily verify that the expressions of thedensities in the non-inertial frame are related to the onesin the laboratory frame by the relations˜ n ( ξ , t ) = √ gn ( r ( ξ , t ) , t ) , (39)˜ j ν ( ξ , t ) = √ g ∂ξ ν ∂x µ [ j µ ( r ( ξ , t ) , t ) − n ( r ( ξ , t ) , t ) ˙ r µ ( r ( ξ , t ) , t )] . (40)The local conservation laws for the transformed many-particle problem read as follows ∂ t e n + ∂ ξ µ e j µ = 0 (41)and ∂ t e j µ − e j ν ( ∂ ξ ν A µ − ∂ ξ µ A ν ) − e n ∂ t A µ + √ g D ν e P νµ = 0 , (42)where e P νµ is a mixed component of the stress tensor,which, in Ref. 25 was proved to have the form e P µν ( ξ , t ) = − √ g * e Ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ e H [ g αβ , A ] δg µν ( ξ , t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e Ψ + , (43)where the functional derivative with respect to g µν is tobe taken fixed A .The quantity D ν e P νµ in Eq. (43) stands for the covariantdivergence of the stress tensor and is explicitly given by D ν e P νµ = 1 √ g ∂ ν √ g e P νµ − e P αβ ∂ µ g αβ . (44)A rather lengthy calculation shows that √ gD ν e P νµ = ∂r ν ∂ξ µ * e Ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ e Hδr ν !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e Ψ + , (45)where the functional derivative with respect to r ν is tobe taken at constant A . It is important to notice thatin the above equation, the transformed hamiltonian ˜ H – a functional of g µν and A [see Eq. (32)] – is actuallytreated as a functional of the trajectory r µ and A . Thisis permissible because g µν itself is a functional of r µ – seeEq. (29).We will make heavy use of this identity in the nextsection. Now let us learn how to fix our reference frameso that the density and current density become constantsof the motion. B. Fixing the reference frame
As discussed in the introduction to this section, ourgoal is to separate the convective motion of the electronfluid from the motion of electrons relative to the con-vective flow. This is achieved by following the volumeelements along their trajectories, so that the density andthe current density becomes stationary, with values equalto the initial one. Mathematically, this translates to thesingle condition e j µ ( ξ , t ) = e j µ ( ξ ,
0) = j µ ( ξ ) . (46)It is important to notice that our condition must beimposed on the contravariant components e j µ of the cur-rent density, rather than on the covariant components e j µ .The two choices are, at first sight, equally plausible, butnot equivalent, since the metric tensor g µν , which con-nects covariant and contravariant components of vectors,is time-dependent. What forces our choice is the factthat the contravariant component of the current densityappears naturally in the continuity equation (41). Thus,with this choice, the continuity equation (41) guaranteesthat we get e n ( ξ , t ) = e n ( ξ , t = 0) = n ( ξ ) (47)as soon as Eq. (46) is satisfied.Given Eq. (46) and Eq. (47), the effective potentialacting on the system is determined through Eq. (36) asfollows A µ ( ξ , t ) = g µν ( ξ , t ) v ν ( ξ ) − e j p,µ ( ξ , t ) n ( ξ ) , (48)where v µ ( ξ ) = j µ ( ξ ) /n ( ξ ) . Eq. (48) is a gauge-fixingcondition which determines the effective vector potential A and, thus, a particular non-inertial frame. C. Equation of motion for the Lagrangiantrajectory
After fixing the reference frame the system is com-pletely characterized by two dynamical variables – thetrajectory of volume elements r µ ( ξ , t ), and the trans-formed many-body wave function e Ψ( t ) that describes mi-croscopic motion relative to the convective flow. Thewave function e Ψ( t ) satisfies the Schr¨odinger equation(31). In order to obtain a physicaly trasparent form of theequation of motion for the Lagrangian trajectory r µ ( ξ , t )we differentiate Eq. (33) with respect to time and we get¨ r µ + ∂ µ V ( r ) + [˙ r × B ( r )] µ + ∂ξ ν ∂r µ ∂ t A ν = − ∂ µ V ( r , t ) . (49)In this equation we insert the value of ∂ t A ν determinedthrough the local momentum balance equation [Eq. (42)],i.e., n ∂ t A µ = ∂ t ( g µν j ν ) − j ν ( ∂ ξ ν A µ − ∂ ξ µ A ν )+ √ gD ν e P νµ , (50)where we have made use of the conditions (46) and (47)and lowered the index of the contravariant current e j µ = g µν e j ν through the action of the metric tensor g µν . Andnow, on the right hand side of Eq. (50) we plug in the ex-pression (33) for A µ in the combination ( ∂ ξ ν A µ − ∂ ξ µ A ν )(notice that the contribution of the classical action termvanishes, since the curl of a gradient is zero), and makeuse of the identity (45) for √ gD ν e P νµ . The result of thesemanipulations is¨ r µ + 2 v ν ∂ ν ˙ r µ + [( D t r ) × B ] µ + ∂ µ V ( r )+ 1 n * e Ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ e Hδr ν !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e Ψ + = − ∂ µ V ( r , t ) , (51)where D t = ∂ t + v ν ∂ ν (52)is a “convective time derivative”, which takes into ac-count the ground-state flow.In view of the last result, it is desirable to change all the time derivatives in Eq. (51) to convective ones. Tothis end, we note that¨ r µ + 2 v ν ∂ ν ˙ r µ = (cid:0) ∂ t + 2 v ν ∂ ν ∂ t (cid:1) r µ (53)and “complete the square” to get¨ r µ + 2 v ν ∂ ν ˙ r µ = D t r µ − v α ∂ α v β ∂ β r µ = D t r µ + 1 n δW [ g µν ] δr µ , (54)where W [ g µν ] ≡ Z d ξ n ( ξ ) g µν ( ξ , t ) v µ ( ξ ) v ν ( ξ ) . (55)Using Eq. (54) in Eq. (51), we obtain D t r µ + [( D t r ) × B ] µ + ∂ µ V ( r )+ 1 n (* e Ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ e Hδr µ !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e Ψ + + δW δr µ (cid:27) = − ∂ µ V ( r , t ) . (56)We stress that the latter equation, Eq. (56), is an exactnonlinear equation for the trajectory of current-carryingmaterial elements in the quantum many-body system,provided e Ψ( t ) entering the stress force is the solution tothe Schr¨odinger equation (31). IV. ELASTIC APPROXIMATION
We now introduce our elastic approximation by setting e Ψ( ξ , t ) ≡ Ψ ( ξ ) , (57)It is clear that this approximation is consistent with therequirement of stationary transformed densities, but can-not be exact in general, since the correct form of e Ψ isdetermined by Eq. (31).The approximation in Eq. (57) applies to short-time dynamics, or to fast-driving fields such that there is notime for the wave function to adjust to minimize the en-ergy in the presence of fast-varying external conditions.Here, we may recognize the high-frequency approximation discussed in Section II. The same approximation may becharacterized as an anti-adiabatic approximation in thefollowing sense. The wave function in the Eulerian (iner-tial) frame is obtained as an instantaneous deformationof the initial wave function. While the initial wave func-tion minimizes the energy (i.e., it is the ground-state), itis clear that the deformed wave function does not mini-mize, at any given instant, the energy of the system. Thedeformation is similar to the change in the shape of anelastic body. Hence, the term elastic approximation .Let us now replace Eq. (57) into the trajectory equa-tion (56). Because the ground-state wave function Ψ does not depend on the Lagrangian trajectories, we canpull the functional derivative with respect to r µ out ofthe quantum average. In other words, we can write * e Ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ e Hδr µ !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e Ψ + + δW δr µ = δδr µ h Ψ | e H + ˆ W | Ψ i (cid:12)(cid:12)(cid:12)(cid:12) A (58)where ˆ W is the operatorˆ W [ g µν ] ≡ Z d ξ ˆ n ( ξ ) g µν ( ξ , t ) v µ ( ξ ) v ν ( ξ ) , (59)such that h ψ | ˆ W | ψ i = W .The functional derivative with respect to r µ at con-stant A can be written as an unrestricted functionalderivative minus counter-terms, which cancel the un-wanted contributions from the variation of A . In for-mulas, we have δδr µ ( ξ ) h Ψ | e H + ˆ W | Ψ i (cid:12)(cid:12)(cid:12)(cid:12) A = δδr µ ( ξ ) h Ψ | e H + ˆ W | Ψ i− Z d ξ ′ δ A ν ( ξ ′ ) δr µ ( ξ ) (cid:18) δδ A ν ( ξ ′ ) h Ψ | e H | Ψ i (cid:12)(cid:12)(cid:12)(cid:12) r (cid:19) , (60)where we have used the fact that ˆ W does not depend on A . Now we recall the identityˆ e j µ ( ξ ′ ) = δ ˜ Hδ A µ ( ξ ′ ) (61)and the fact that ˆ e j µ as well as A ν [see Eq. (48)] mustbe here evaluated at Ψ . In particular, we notice that,according to Eq. (48) δ A ν ( ξ ′ ) δr µ ( ξ ) = δg να ( ξ ′ , t ) v α ( ξ ′ ) δr µ ( ξ ) . (62)Therefore, Eq. (60) can be compactly and suggestivelyrestated as δδr µ ( ξ ) h Ψ | e H + ˆ W | Ψ i (cid:12)(cid:12)(cid:12)(cid:12) A = δE el [ g µν ] δr µ ( ξ ) ≡ −F el ,µ ( ξ ) . (63)Here E el [ g µν ] is an elastic energy defined as E el [ g µν ] = h ψ | ˜ H [ g µν , A = − v p ] | ψ i , (64)where v p = j p /n is the paramagnetic velocity in theground-state (in a proper gauge, this quantity alwaysvanishes in a one-particle system). We shall refer to F el ,as the elastic force acting on the systems: this is givenby the functional derivative of the elastic energy withrespect to the Lagrangian trajectory.Plugging the above definitions and formulas into thetrajectory equation (56), we finally obtain D t r µ + [( D t r ) × B ] µ + ∂ µ V − n F el ,µ = − ∂ µ V . (65)The situation is actually analogous to what happens inthe case of a classical spring: displace it from the equi-librium, compute the gain in energy, get the elastic forcefrom the derivative of the energy with respect to dis-placement. Classical elasticity theory generalizes this tocontinuum media, for which the “stress field” is the func-tional derivative of the energy with respect to the strainfield. Here, in a quantum many-body system, the cor-responding quantity is the functional derivative of the(transformed) quantum energy with respect to the tra-jectories of the infinitesimal volume elements. Due tothe presence of an external static magnetic field, thosematerial elements carry a non-vanishing current.The quantum elastic energy can be naturally split intoa kinetic component and an electron-electron interactioncomponent. In detail, we find E el [ g µν ] = T el [ g µν ] + W el [ g µν ] , (66)where T el [ g µν ] = Z d ξ (cid:20) √ gg µν (cid:18) ∂ µ r n g / (cid:19) (cid:18) ∂ ν r n g / (cid:19) + 12 ∆ T (0) µν g µν (cid:21) , (67)∆ T (0) µν = 12 lim ξ ′ → ξ (ˆ π ∗ µ ˆ π ′ ν + ˆ π ∗ ν ˆ π ′ µ ) γ ( ξ , ξ ′ ) − ( ∂ µ √ n )( ∂ ν √ n ) , (68) γ ( r , r ′ ) is the ground-state one-particle density matrix γ ( ξ , ξ ′ ) = N R N Y j =2 d r j × Ψ ∗ ( ξ , . . . , ξ N )Ψ ( ξ ′ , . . . , ξ N ) , (69)ˆ π is the operator of the kinematic momentum for therelative motion ˆ π = − i ∇ − v p , (70) W el [ g µν ] = 12 Z d ξ d ξ ′ W ( r ( ξ ) − r ( ξ ′ ))Γ ( ξ , ξ ′ ) . (71)and Γ ( r , r ′ ) = N ( N − Z N Y j =3 d r j × Ψ ∗ ( ξ , ξ ′ , . . . , ξ N )Ψ ( ξ , ξ ′ , . . . , ξ N ) (72)is the ground-state pair distribution function.The identification of the elastic energy defined byEqs. (66-72) is the key result of this section. In com-parison with the non-magnetic case, the essential dif-ference is the redefinition of the kinetic contribution [seeEq. (67), Eq. (68), and Eq. (70)]. V. LINEARIZED EQUATION OF MOTION
The equation of motion for the Lagrangian trajectoryderived in the previous section are fully nonlinear. Herewe present the linearized form of those equations, whichare expected to be useful for systems performing smalloscillations about the ground-state.First of all, we insert Eqs. (46) and (47) on the lefthand sides of Eqs. (35) and (36). This gives n ( ξ ) = √ gn ( r ( ξ , t ) , t ) , (73) j ν ( ξ ) = √ g ∂ξ ν ∂x µ [ j µ ( r ( ξ , t ) , t ) − n ( r ( ξ , t ) , t ) v µ ( r ( ξ , t ) , t )] , (74)where r ( ξ , t ) = ξ + u ( ξ , t ) . (75)Expanding to first order in u we get n ( ξ , t ) = n ( ξ ) − ∇ · [ n ( ξ ) u ( ξ )] (76)and j ( ξ , t ) = j ( ξ ) + n ( ξ ) u ( ξ ) + ∇ × [ u ( ξ ) × j ( ξ )] . (77)0Eq. (77) expresses the response of the current density, j − j , as the sum of two terms: the first accounts for thepolarization and the second for the orbital magnetizationof the quantum medium. We have thus succeeded inderiving the relation (8) between the linearized responseof the current and the displacement field that was putforward in the introduction on a heuristic basis.We notice that, up to the linear order, there is no dif-ference between the Lagrangian and the Eulerian descrip-tion of the displacement field, thus u ( ξ ) = u ( r ). In thisspirit, we replace ξ by r is all the expressions below. Itis also worth emphasizing that Eqs. (76) and (77) aregeneral and hold true independently of the elastic or anyother approximation.The linearized equation of motion of the displacementis readily obtained D t u ( r , t ) + [ D t u ( r , t )] × B ( r ) + ( u · ∇ ) ∂ µ V ( r )+ v × ( u · ∇ ) B ( r ) − n ( r ) F el ( r , t )= −∇ V ( r , t ) . (78)where F el ( r , t ) is the (linearized) elastic force. The termson the second line result from the expansion of the clas-sical forces to first order in u . The linearized elastic forceis given by F el ,µ ( r , t ) = − Z d r ′ δ E el [ u ] δu µ ( r ) δu ν ( r ′ ) (cid:12)(cid:12)(cid:12)(cid:12) u =0 u ν ( r ′ , t ) , (79)and can be naturally separated into kinetic and electron-electron interaction contributions: F el ,µ ( r ) = F kinel ,µ ( r ) + F intel ,µ ( r ) . (80)Explicit expressions for the two components are ob-tained by closely following the steps outlined in Ref. 6.The final expressions are F kinel ,µ ( r ) = ∂ α [2 T mag νµ, ( r ) u να ( r ) + T mag να, ( r ) ∂ µ u ν ( r )] − ∂ ν ∂ µ [ n ( r ) ∂ ν ∇ · u ( r )]+ 14 ∂ ν (cid:8) (cid:2) ∇ n ( r ) (cid:3) u νµ ( r ) + [ ∂ ν n ( r )] ∂ µ ∇ · u ( r )+ [ ∂ µ n ( r )] ∂ ν ∇ · u ( r ) − ∂ µ [( ∂ α n ( r )) u να ( r )] } , (81)( u νµ ( r ) ≡ ( ∂ ν u µ ( r )+ ∂ µ u ν ( r )) / F intel ,µ ( r ) = Z d r ′ K µν ( r , r ′ )[ u ν ( r ) − u ν ( r ′ )] . (82)In Eq. (81), we have defined T mag µν, ( r ) = 12 (cid:0) ∂ µ ∂ ′ ν + ∂ ν ∂ ′ µ (cid:1) γ ( r , r ′ ) | r = r ′ − ∇ n ( r ) δ µν + 3 n ( r ) v p ,µ ( r ) v p ,ν ( r ) , (83)where γ ( r , r ′ ) is the ground-state one-particle densitymatrix [see Eq. (69)]. In Eq. (82) K µν ( r , r ′ ) = Γ ( r , r ′ ) ∂ µ ∂ ′ ν W ( | r − r ′ | ) , (84) where W ( | r − r ′ | ) is the interaction potential andΓ ( r , r ′ ) is the ground-state pair distribution function[see Eq. (72)].Analysis of the one-particle case within the Lagrangianformulation is reported in the Appendix C, thus this alsocompletes the analysis started in the Appendix B withinthe Euler approach. VI. CONCLUSIONS
In summary, we recapitulate the essential changes thatmust be made in order to go from the quantum contin-uum mechanics in the absence of magnetic field to the onein the presence of magnetic field, within the frameworkof the “elastic approximation”:(i) A Lorentz-force term must be added to the equa-tion of motion for the displacement field, and ap-propriately linearized, taking into account the pres-ence of a non-vanishing velocity field v ( r ) in theground-state.(ii) Everywhere, the time derivative ∂ t must be re-placed by the convective derivative D t = ∂ t +( v · ∇ ). The replacement must also be done withinthe Lorentz force term, compatibly with the re-quirements of linearization.(iii) The kinetic contribution to the elastic energy mustbe calculated taking into account the replacementof the canonical momentum operator − i ∇ by thekinematic momentum ˆ π = − i ∇ − v p .(iv) The relation between linearized current densityand displacement field is changed from Eq. (1) toEq. (8).The Fourier transform of Eq. (78) yields the followinggeneralized eigenvalue problem ω u + iω [2 ( v · ∇ ) u + ( u × B )] − ( v · ∇ ) u − [( v · ∇ u ) × B ] − ( u · ∇ ) ∂ µ V ( r ) − v × ( u · ∇ ) B ( r ) + 1 n F el = 0 . (85)Finally, we have an explicit form of all the terms: this isa major step forward.In conclusion, we have presented results that open thepossibility to obtain the response of the current, andthus the excitation energies, of systems in strong mag-netic fields avoiding the solution of the time-dependentSchr¨odinger equation and making use only of ground-state properties. Given the required ground-state prop-erties, the complexity of the problem to be solved doesnot increase with the number of the particles in the sys-tem. The presented framework is expected to be usefulin dealing with large systems and with current-carryingstates exhibiting an elastic behavior.1 Acknowledgments
S.P. and G.V were supported by DOE grant DE-FG02-05ER46203. I.V.T. was supported by the Span-ish MICINN, Grant No. FIS2010-21282-C02-01, and“Grupos Consolidados UPV/EHU del Gobierno Vasco”,Project No. IT-319-07. S.P. and G.V. acknowledge inter-esting discussions with Zeng-hui Yang about non-analytictime behavior of quantum states.
Appendix A: Generalized eigenvalue problem
Eq. (24) can be represented in the form ω ˜ u + iω e B · ˜ u − e K · ˜ u = 0 , (A1)where we introduced the following notations˜ u µ ( r ) = p n ( r ) u µ ( r ) , (A2)˜ B µ,ν ( r , r ′ ) = i p n ( r ) B µ,ν ( r , r ′ ) 1 p n ( r ′ ) , (A3)˜ K µν ( r , r ′ ) = 1 p n ( r ) K µ,ν ( r , r ′ ) 1 p n ( r ′ ) . (A4)Moreover, Eq. (85) shares the same structure although(as we have explained) it is for a different dispalcement.The operators acting on e u have properties˜ B µ,ν ( r , r ′ ) = ˜ B ∗ µ,ν ( r , r ′ ) , ˜ B µ,ν ( r , r ′ ) = − ˜ B ν,µ ( r ′ , r ) (A5)and˜ K µ,ν ( r , r ′ ) = ˜ K ∗ µ,ν ( r , r ′ ) , ˜ K µ,ν ( r , r ′ ) = ˜ K ν,µ ( r ′ , r ) . (A6)Eq. (A1) has the form of a non-standard eigenvalue prob-lem which may be found for example also in the analysisof the modes of rotating stars and in the analysis of themagneto hydrodynamic of hot plasma .In terms of the scalar product [the difference withEq. (26) is related to the rescaling of the displacementfield, Eq. (A2)] h ˜ u B , ˜ u A i ≡ Z d r ˜ u ∗ B,µ ( r )˜ u A,µ ( r ) , (A7)the orthogonality relation get modified as follows h ˜ u B , i e B · ˜ u A i + ( ω A + ω B ) h ˜ u B , ˜ u A i = 0 , (A8)where ω A and ω B are assumed to be real and differentfrom each. In fact, the operator defining the problemin Eq. (A1) is Hermitian for real ω but it is also ω -dependent; thus, solutions of Eq. (A2) correspondingto different ω do not need to be orthogonal in the usualsense (as for e B ≡ ω . For this, Eq. (A1)can be brought the quadratic form ω h ˜ u , ˜ u i + i ω h ˜ u , e B · ˜ u i − h ˜ u , e K · ˜ u i = 0 . (A9)Real-valued ω are obtained for positive definitive discrim-inant of Eq. (A9). Since h ˜ u , ˜ B · ˜ u i is a purely imaginaryquantity, the stability condition may be stated in termsof the stronger requirement h ˜ u , e K · ˜ u i > . (A10)Eq. (A10) is not manifestly satisfied for the equationsunder considerations. Nevertheless, for Eq. (22) andEq. (85), we expected to find stable solutions becausethose same equations are valid for small displacementsand short-time intervals: within these conditions, thesystem must stay “close” to the the initial minimum (theground-state) of the unperturbed Hamiltonian. Appendix B: One-particle case in the Eulerdescription with the standard displacement
Here, we show that the inversion of the current re-sponse for one particle system can be easily worked outfrom the linearized Schroedinger equation or, equiva-lently, from the linearized Euler equations for the den-sities. In this way, we are able to show that the approx-imation put forward in the high-frequency limit providethe exact excitation energies for one particle systems.We start by observing that, the wave function may bewritten as follows Ψ = √ ne iϕ , (B1)thus, j = n ∇ ϕ + n A , (B2)and ∇ × v = B . (B3)As a result, the local balance equation for the linear mo-mentum reads as follows ∂ t j µ + ∂ µ [ V B + 12 j n + V ] = ∂ t A µ (B4)where V B = − ∇ √ n √ n (B5)is the well-known Bohm potential . Let V = V , and A = A + A , (B6)2the linearization of Eq. (B4) yields ∂ t j n + ( ∇ · j ) n v + ∇ (cid:26) V ,B + v · j n − v n n (cid:27) = ∂ t A (B7)where V ,B = − " ∇ n √ n √ n − ∇ √ n √ n (cid:18) n n (cid:19) , (B8)moreover, we may also remind that ∂ t n = −∇ · j . (B9)From Eq. (B7), we obtain an important relation forthe inverse of the current-current response function: χ − ( ω ) · j = j n + iω (cid:20) ( ∇ · j ) v n + ∇ (cid:18) v · j n (cid:19)(cid:21) + 1 ω ∇ (cid:20) √ n (cid:18) − ∇ ∇ √ n √ n − v (cid:1) ∇ · j √ n (cid:21) . (B10)This expression tells us that the frequency dependencyincludes only 1 /ω and 1 /ω terms: therefore, it is ap-parent that the inversion in high-frequency limit of thecurrent response function is exact for one particle sys-tems.Let us consider purely longitudinal perturbation, fromEq. (B7) we get the equation for modes described interms of j = − iωn u ω u + iω (cid:20) ( ∇ · n u ) v n + ∇ ( v · u ) (cid:21) + ∇ (cid:20) √ n (cid:18) − ∇ ∇ √ n √ n − v (cid:19) ∇ · ( n u ) √ n (cid:21) = ∇ V , (B11)Now, we find the exact displacement for the one-particlesystem. For this purpose, we use the expression of thelinearized Schr¨odinger equation. Without loosing gener-ality, one may shift to zero the energy of ground-stateenergy and choose the ground-state wave function inEq. (B1) to be a real-valued function (i.e.; ϕ = 0). Inthe given gauge, the relation A = j n = v (B12)holds true. By substitution in Eq. (B11), we can verifythat the solutions have form u = [ j ] n n , ω = ω n (B13)where [ j ] n = − i ∇ (cid:18) Ψ n Ψ (cid:19) + Ψ Ψ n A . (B14) are the matrix element of the current operator evaluatedfor the eigenstates, Ψ n , of ˆ H and ω n are the correspond-ing excitation energies. In verifying the above results, itmay be useful to remind the continuity relations ∇ · [ j ] n = iω n Ψ Ψ n . (B15) Appendix C: One-particle case in the Lagrangiandescription with the new displacement
Let us consider the one-particle case but within theelastic approximation as obtained within the Lagrangiandescription. In this case, we readily arrive at E N =1el [ g µν ] = Z d ξ √ gg µν × (cid:18) ∂ µ r n g / (cid:19) (cid:18) ∂ ν r n g / (cid:19) . (C1)It is remarkable that, Eq. (C1) has the same form asin the limit of vanishing magnetic field. Moreover, it ispossible to directly verify that the Lagrangian with theelastic energy as in Eq. (C1): L = Z d ξ n ( ξ ) (cid:20)
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