Sum Rules and Asymptotic Behaviors for Optical Conductivity of Nonequilibrium Many-Electron Systems
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Sum Rules and Asymptotic Behaviors for Optical Conductivityof Nonequilibrium Many-Electron Systems
Akira
Shimizu ∗ and Tatsuro Yuge † Department of Basic Science, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8902, Japan IIAIR, Tohoku University, Aoba-ku, Sendai, Miyagi 980-8578, Japan
For many-electron systems, we consider a nonequilibrium state (NES) that is driven by apump field(s), which is either an optical field or a longitudinal electric field. For the differentialoptical conductivity describing the differential response of the NES to a probe optical field,we derive exact sum rules and asymptotic behaviors, which open wide possibilities for experi-ments. In deriving these results, we have also derived universal properties of general differentialresponse functions of time-dependent NESs of general systems.
KEYWORDS: sum rule, nonlinear nonequilibrium, optical properties, pump-probe
Introduction –
The optical conductivity tensor σ eq αβ ( ω )describes the response of an equilibrium state to a probeoptical field. It gives much information on electronicproperties of condensed matter. In particular, ithas been shown that the integrals of Re σ eq αβ ( ω ) and ω Im σ eq αβ ( ω ) over the frequency ω are directly relatedto basic properties of the system such as the single-particle distribution and band dispersion. Such rela-tions, called sum rules , are therefore useful for exploringelectron systems, and have been successfully utilizedfor analyzing a large variety of electron systems.
However, since an equilibrium state (of each system) isuniquely determined by a small number of parameters(such as temperature), the number of controllable param-eters that affect the sum (integral) values is very small.This fact has severely limited the usage of sum rules.This limitation can be removed by considering the op-tical conductivity of a nonequilibrium state (NES). ANES can be created and driven by a pump field A , whichis assumed to be an optical field and/or a longitudinalelectric field (generated by, say, a battery). The responseof the NES to a probe optical field is characterized bythe differential optical conductivity tensor σ Aαβ [definedby eqs. (1)-(3)].
Unlike equilibrium states and σ eq αβ ,the NES and σ Aαβ depend strongly on the magnitude andfunctional form of A ( t ). Therefore, by tuning A ( t ) as anew controllable parameter, one will be able to make thesum rules for σ Aαβ much more informative than those for σ eq αβ . However, the problem was that, until now, the sumrules for σ Aαβ were unknown.Note that two different configurations are possible inexperiments on σ Aαβ : (i) A ( t ) is turned off before a ( t ) isapplied and (ii) A ( t ) is present when a ( t ) is applied. Wehere call both configurations pump-probe experiments . Inconfiguration (i), the NES (created by A ( t ) beforehand)might sometimes be approximated as a quasi-equilibriumstate (QES), and the sum rules of σ eq αβ are often substi-tuted for those of σ Aαβ . However, in general, the tran- ∗ E-mail address: [email protected] † E-mail address: [email protected] sient NES is not well approximated as a QES, and thissubstitution has not been justified. In configuration (ii),such substitution is obviously wrong because the NESdriven by A ( t ) is far from quasi-equilibrium because, forexample, strong mixing phenomena such as frequencymixing take place. Therefore, until now, reliable sumrules for σ Aαβ were unknown in either configuration.In this paper, we derive sum rules for σ Aαβ ( ω ) [eqs. (23)and (24)], and its asymptotic behaviors [eqs. (25) and(26)], for a general class of models for many-electron sys-tems. They hold rigorously in both configurations (i) and(ii), even when many-body interactions are strong. Differential optical conductivity of NESs –
Supposethat an optical field, described by a vector potential A ( t )(in the Coulomb gauge), and/or a longitudinal electricfield, described by a scalar potential φ ( r , t ), is applied toan electron system. Since A and φ induce optical excita-tion and electrical conduction, respectively, the systembecomes a NES, whose density operator is denoted byˆ ρ A ( t ). We therefore call A ≡ ( A , φ ) the pump field . Itcan be strong such that perturbation expansion in powersof A breaks down . Furthermore, we do not assumeany specific functional form (such as periodicity) for thetime dependence of A .One can study properties of a NES created by A bymeasuring the response to another optical field a ( t ),which we call a probe field . It brings the system into an-other NES, ˆ ρ A + a ( t ). We are interested in the change,induced by a ( t ), in the current density j ,∆ j ( t ) ≡ h ˆ j i A + a t − h ˆ j i At , (1)where h·i A + a t ≡ Tr[ˆ ρ A + a ( t ) · ] and h·i At ≡ Tr[ˆ ρ A ( t ) · ].When a ( t ) is weak, ∆ j ( t ) is well described in terms ofthe differential optical conductivity tensor σ Aαβ as∆ j α ( t ) = X β Z t −∞ σ Aαβ ( t − t ′ ; t ) f β ( t ′ ) dt ′ + o ( f ) . (2)Here, f ( t ) = − ˙ a ( t ) is the probe electric field, and α, β = x, y, z . Since the NES varies as a function of time,so does σ Aαβ . That is, σ Aαβ depends not only on the time
Letter A. Shimizu and T. Yuge delay τ ≡ t − t ′ but also on t . Furthermore, as eq. (15)shows, σ Aαβ is generally a nonlinear functional of A [whileit is independent of a ]. Throughout this paper, the su-perscript A , such as those in σ Aαβ and h·i At , denotes sucha functional dependence. Equations (1) and (2) and thecausality, σ Aαβ ( τ ; t ) = 0 for τ < , (3)define the differential optical conductivity tensor of theNES driven by A . It contains much more informationthan that of equilibrium states, σ eq αβ ( ω ), as we will discusslater.Experimentally, a ( t ) is usually taken as monochro-matic, and thus f ( t ) = f e − iωt + c.c. Then, eq. (2) reads∆ j α ( t ) = X β σ Aαβ ( ω ; t ) f β e − iωt + c.c. + o ( f ) , (4)where σ Aαβ ( ω ; t ) ≡ R ∞−∞ σ Aαβ ( τ ; t ) e iωτ dτ is the Fouriertransform (FT) with respect to the time delay τ . Onecan measure σ Aαβ ( ω ; t ) directly by such experiments us-ing eq. (27). Since σ Aαβ ( τ ; t ) is real, Re σ Aαβ ( ω ; t ) andIm σ Aαβ ( ω ; t ) are even and odd functions of ω , respec-tively. We study sum rules for them. For example, weconsider W Aαβ ( t ) ≡ Z ∞−∞ Re σ Aαβ ( ω ; t ) dω, (5)which is called the optical spectral weight . This quan-tity is of central interest in many theories and experi-ments. Model and definitions –
We consider a many-electronsystem in the presence of electron-electron and electron-phonon interactions as well as random potentials. Theelectrons move on a regular lattice, whose dimensionalityand symmetries are arbitrary .We assume that the system is described, in the energyscale of interest, by the general Hamiltonian;ˆ H = ˆ H e + ˆ H ei + ˆ H ee + ˆ H ep + ˆ H p . (6)Here, ˆ H e is the kinetic-energy term of electrons; ˆ H e ≡ P k ,σ ε ( k )ˆ n k σ , where ε ( k ) denotes the energy disper-sion of the band of interest, and ˆ n k σ ≡ ˆ c † k σ ˆ c k σ . Here,ˆ c k σ ≡ P l e i k · l ˆ c l σ / √ N , where ˆ c l σ annihilates an electronon site l with spin σ , and N is the number of unit cells.ˆ H ei ≡ P l ,σ u l ˆ n l σ is a random potential (with a randomon-site energy u l and ˆ n l σ ≡ ˆ c † l σ ˆ c l σ ), which may be pro-duced, for example, by impurities. Furthermore, ˆ H ee isthe sum of electron-electron interactions. We assume thatˆ H ee is a function of ˆ n l σ ’s. ˆ H ep is the electron-phononinteraction, and ˆ H p denotes the Hamiltonian of freephonons. This general model includes many models suchas the Hubbard model (for which ˆ H ee = U P l ˆ n l ↑ ˆ n l ↓ ,ˆ H ei = ˆ H ep = ˆ H p = 0). Our results hold irrespective ofthe details and magnitudes of ˆ H ee , ˆ H ei and ˆ H ep .For later use, we define the velocity vector and inversemass tensor as v α ( k ) ≡ ~ ∂∂k α ε ( k ) , m − αβ ( k ) ≡ ~ ∂ ∂k α ∂k β ε ( k ) . To consider interactions with A and a , we assume thatthe spatial variations of A and a can be neglected. Thisapproximation is good in most experimental configura-tions. The directions of A , ∇ φ and a are arbitrary. Un-der these conditions, we may incorporate the interactionswith A and a by the Peierls substitution, and the interac-tion with φ by the Coulomb interaction with the chargeof electrons. Then, the Hamiltonian in the presence of A , φ and a is given byˆ H A + a = X k ,σ ε ( k − ( e/ ~ ) A ( t ) − ( e/ ~ ) a ( t ))ˆ n k σ + ˆ H ei + e X l (cid:16) X σ ˆ n l σ − n bg l (cid:17) φ ( l , t ) + ˆ H ee + ˆ H ep + ˆ H p . (7)Here, e is the electron charge, and − en bg l is a backgroundcharge on site l . By differentiating ˆ H A + a with A + a , weobtain the current density asˆ j α = eV X k ,σ v α ( k − ( e/ ~ ) A ( t ) − ( e/ ~ ) a ( t ))ˆ n k σ (8)= ˆ j vα + ˆ j mα + o ( a ) , (9)whereˆ j vα ≡ eV X k ,σ v α ( k − ( e/ ~ ) A ( t ))ˆ n k σ , (10)ˆ j mα ≡ − e V X k ,σ,β m − αβ ( k − ( e/ ~ ) A ( t ))ˆ n k σ a β ( t ) . (11)When A = 0, ˆ j mα represents the diamagnetic currentinduced by a . When A = 0, the diamagnetic currentis induced by both A and a , and thus is included in bothˆ j vα and ˆ j mα .Since ˆ j mα is O ( a ), ∆ j ( t ) defined by eq. (1) is given by∆ j ( t ) = ∆ j v ( t ) + j m ( t ) + o ( a ) . (12)Here, ∆ j vα ( t ) ≡ h ˆ j vα i A + a t − h ˆ j vα i At and j mα ( t ) ≡− P β d Aαβ ( t ) a β ( t ), where d Aαβ ( t ) ≡ e V X k ,σ m − αβ ( k − ( e/ ~ ) A ( t )) h ˆ n k σ i At . (13)For a simple cubic lattice, for example, P αβ d Aαβ ( t ) isproportional to the expectation value of the kinetic en-ergy.While j m ( t ) responds to a ( t ) instantaneously, ∆ j v ( t )responds with a finite delay as∆ j vα ( t ) = X β Z t −∞ Φ Aαβ ( t − t ′ ; t ) a β ( t ′ ) dt ′ + o ( a ) . (14)Here, Φ Aαβ ( τ ; t ) is the response function describing thedifferential response of ∆ j v ( t ) to a ( t ). We denote itsFT with respect to the time delay τ by Ξ Aαβ ( ω ; t ). Since f ( t ) = − ˙ a ( t ), eqs. (2) and (12)-(14) yield the differentialoptical conductivity tensor as σ Aαβ ( ω ; t ) = − iω + i h Ξ Aαβ ( ω ; t ) − d Aαβ ( t ) i . (15)Both Ξ Aαβ and d Aαβ are nonlinear functionals of A , and so . Phys. Soc. Jpn. Letter A. Shimizu and T. Yuge 3 is σ Aαβ . Universal properties of response functions of time-dependent NESs –
To derive sum rules for σ Aαβ , we notethat Ξ
Aαβ in eq. (15) should satisfy all the universal prop-erties that were found in ref. 13 for general response func-tions of general systems. Since ref. 13 assumed steady
NESs driven by a static pump field, we here generalize itstheory to time-dependent NESs, which are realized, forexample, by the application of a time-dependent pumpfield. For this general discussion, we omit vector and ten-sor indices.We denote the pump and probe fields by A ( t ) and a ( t ), respectively. In nonequilibrium statistical mechan-ics (e.g., in the Kubo formula and in refs. 13–15), it isusually assumed (implicitly) that an observable of inter-est is independent of a ( t ). However, we here consider thegeneral case where an observable of interest, denoted byˆ Q a ( t ) , is a function of a ( t ), because this is the case for ˆ j α given by eq. (8). Then, by expanding ˆ Q a ( t ) in powers of a ( t ), we obtainˆ Q a ( t ) = ˆ Q + ˆ Q a ( t ) + o ( a ) , (16)where ˆ Q and ˆ Q are operators independent of a ( t ). Wehave obtained such an expansion in eq. (9), where ˆ Q =ˆ j vα and ˆ Q a ( t ) = ˆ j mα . The response to a ( t ), ∆ Q a ( t ) ≡h ˆ Q a ( t ) i A + at − h ˆ Q a ( t ) i At , is therefore given by∆ Q a ( t ) = ∆ Q ( t ) + h ˆ Q i At a ( t ) + o ( a ) , (17)where ∆ Q ( t ) ≡ h Q i A + at − h Q i At . Since the response func-tion of the second term on the right-hand side is sim-ply given by h ˆ Q i At , let us consider the non-trivial term∆ Q ( t ). Unlike h ˆ Q i At , ∆ Q ( t ) depends on ˆ ρ A + a ( t ) (theNES in the presence of both A and a ). We therefore haveto use the theory of ref. 13 to evaluate ∆ Q ( t ).When a ( t ) is sufficiently weak, ∆ Q ( t ) responds to a ( t )linearly as∆ Q ( t ) = Z t −∞ Φ A ( t − t ′ ; t ) a ( t ′ ) dt ′ + o ( a ) . (18)This and the causality condition, Φ A ( τ ; t ) = 0 for τ < A ( τ ; t ) of theNES. Its FT with respect to the time delay τ is denotedby Ξ A ( ω ; t ). It is straightforward to generalize the theoryof ref. 13 to the case where A and the NES are time-dependent. We then obtain the following results.The dispersion relations, such asRe Ξ A ( ω ; t ) = Z ∞−∞ P ω ′ − ω Im Ξ A ( ω ′ ; t ) dω ′ π , (19)are satisfied. Furthermore, the sum rules Z ∞−∞ Re Ξ A ( ω ; t ) dωπ = h ˆ C i At , (20) Z ∞−∞ n ω Im Ξ A ( ω ; t ) − h ˆ C i At o dωπ = h ˆ D i At (21)hold. Here, ˆ C ≡ [ ˆ R, ˆ Q ] /i ~ and ˆ D ≡ − [ ˆ Q, [ ˆ R, ˆ H A +ˆ H ′ ]] / ~ , where ˆ R denotes the operator that couples to a ( t ) via the interaction term − ˆ Ra ( t ), ˆ H A is the Hamil- tonian of the target system in the presence of A [suchas eq. (7) with a = 0], and ˆ H ′ is the interaction be-tween the target system and other systems such as heatreservoirs and electric leads. In general, these opera-tors (such as ˆ Q and ˆ R ) are additive operators or theirdensities.
13, 15)
Equation (21) also gives the asymptoticbehavior for large ω as ω Im Ξ A ( ω ; t ) → h ˆ C i At . (22)In deriving these results following ref. 13, we have usedthe von Neumann equation for the density operator of ahuge system, which includes not only the target system ofinterest but also environments and a source of the pumpfield, as well as all interactions among them. [Althoughsuch a huge system is analyzed, we have successfully de-rived, as in ref. 13, the relations among quantities of onlythe target system.] Therefore, these results are rigorousand apply to all physical systems , as long as the linearrelation given by eq. (18) holds.
13, 14)
Main results –
Let us apply the above results to σ Aαβ of the system described by eq. (7). By expanding ˆ H A + a in powers of a ( t ), we find that ˆ R = V ˆ j vβ for a β ( t ). ForΞ Aαβ , which is the FT of Φ
Aαβ of eq. (14), ˆ Q = ˆ j vα . Thesum rules for σ Aαβ are obtained from the properties ofΞ
Aαβ through eq. (15).For the optical spectral weight [defined by eq. (5)],eq. (19) for ω = 0 yields W Aαβ ( t ) = πd Aαβ ( t ) . (23)Note that this result relies only on eqs. (15) and (19).That is, this sum rule is derived only from the causality[eq. (3)] and the specific form of the current [eq. (9)]: Noother relations are necessary for deriving this sum rule.For ω Im σ Aαβ , on the other hand, eqs. (15) and (20) yieldthe following sum rule: Z ∞−∞ (cid:8) ω Im σ Aαβ ( ω ; t ) − d Aαβ ( t ) (cid:9) dω = 0 . (24)This and eq. (22), respectively, give the asymptotic be-haviors for large ω as ω Im σ Aαβ ( ω ; t ) → d Aαβ ( t ) , (25) ω Re σ Aαβ ( ω ; t ) → . (26)Equations (23)-(26) are our main results. They arerigorous (to the same degree as the Kubo formula is)within the general model defined by eq. (7), even when A ( t ) , φ ( t ) , ˆ H ee , ˆ H ep and ˆ H ei are strong. For example, ourresults hold for any possible phases of the system that isdescribed by eq. (7). That is, our results are completelyvalid as long as the target system is well described bythe Hamiltonian of eq. (7). Conversely, if experimentalresults disagree with our results, it means that the sys-tem is not described by eq. (7) (because, say, transitionto another band takes place). Such rigor seems impor-tant for the application of the sum rules and asymptoticbehaviors.Note that the effects of ˆ H ee , ˆ H ep , ˆ H ei and φ on the sumand asymptotic values appear only through the distribu-tion function h ˆ n k σ i At . In contrast, the effects of A on J. Phys. Soc. Jpn.
Letter A. Shimizu and T. Yuge the sum and asymptotic values appear not only through h ˆ n k σ i At but also through m − αβ ( k − ( e/ ~ ) A ). In either case,the decoherence of electrons affects the sum and asymp-totic values only through the broadening of h ˆ n k σ i At . Possible applications –
For σ eq αβ , the sum rule for W eq αβ ≡ R ∞−∞ Re σ eq αβ ( ω ) dω reads W eq αβ /π = d eq αβ ≡ ( e /V ) P k ,σ m − αβ ( k ) h ˆ n k σ i eq . For each system, d eq αβ depends only on the temperature T and doping density n d . In pump-probe experiments, in contrast, W Aαβ /π = d Aαβ ( t ) can be studied as a function of T, n d and A . Thisopens wide possibilities for studying many-electron sys-tems. For example, suppose that an ordered phase is re-alized as an equilibrium state. By measuring σ eq αβ , oneobtains the value of d eq αβ for the ordered phase. Then, astatic A = (0 , φ ) is applied to induce a DC electric cur-rent while keeping T equal to that for A = 0 (by, forexample, using a good heat sink). By measuring σ (0 ,φ ) αβ by applying a ( t ), one now obtains, from eq. (23) or (25),the value of d (0 ,φ ) αβ = ( e /V ) P k ,σ m − αβ ( k ) h ˆ n k σ i (0 ,φ ) fora non -ordered phase, because the order would be de-stroyed by the electric current if |∇ φ | was larger thana certain value. One thus obtains the values of d αβ withand without the order at the same T and n d . Alterna-tively, suppose that no order is present in an equilibriumstate. Then, a coherent optical field A = ( A ( t ) ,
0) is ap-plied. This would induce an electron-hole ( eh ) correla-tion. Hence, by measuring σ Aαβ , one obtains the value of d Aαβ for the state with the eh correlation. Method of measuring σ Aαβ ( ω ; t ) — σ Aαβ ( ω ; t ) can bemeasured, for example, by the following process.Step 1: Prepare the system in some initial state atan initial time t = 0. Apply a pump field A ( t ) only,and measure the current density j ( t ) continuously for asufficiently long time. Then, turn off A ( t ), and at anotherinitial time prepare the system in the same initial state asthat at t = 0. Redefine the origin of time ( t = 0) as thisnew initial time. Apply the same pump field A ( t ) again,and measure the current density j ( t ) continuously. Byrepeating these procedures sufficiently many times, oneobtains many independent records of j ( t ). The averageof these records gives h ˆ j i At .Step 2: Perform the same sequence of experiments us-ing the pump and probe fields instead of the pump field.Here, the pump field A ( t ) is taken to be the same as thatof Step 1. One then obtains h ˆ j i A + a t . From this and theresult of Step 1, one obtains ∆ j ( t ) = h ˆ j i A + a t − h ˆ j i At .If one takes the probe field as a monochromatic one, f ( t ) = f e − iωt + c.c. , and if one takes f parallel to the β -axis (i.e., f α = f δ αβ ), then eq. (4) yields ∆ j α ( t ) = σ Aαβ ( ω ; t ) f e − iωt + c.c. + o ( f ).Step 3: Perform the same sequence of experiments us-ing the same pump field and another (phase shifted)probe field f ′ ( t ) = f ′ e − iωt + c.c. , where f ′ = i f . Onethen obtains h ˆ j i A + a ′ t . From this and the result of Step1, one obtains ∆ j ′ ( t ) ≡ h ˆ j i A + a ′ t − h ˆ j i At . According toeq. (4), it is expressed as ∆ j ′ α ( t ) = σ Aαβ ( ω ; t ) if e − iωt + c.c. + o ( f ).From these experimental results, one can evaluate σ Aαβ ( ω ; t ) using σ Aαβ ( ω ; t ) = lim f → ∆ j α ( t ) − i ∆ j ′ α ( t )2 f e − iωt . (27) Concluding remarks —
Our results hold in bothconfigurations (i) and (ii), which were discussed inthe introduction. In configuration (i), eq. (23) reads W Aαβ ( t ) = ( πe /V ) P k ,σ m − αβ ( k ) h ˆ n k σ i At . Comparing thiswith the corresponding result for σ eq αβ , W eq αβ =( πe /V ) P k ,σ m − αβ ( k ) h ˆ n k σ i eq , we find that the resultfor W Aαβ ( t ) is obtained simply by replacing the equi-librium electron distribution h ˆ n k σ i eq with the nonequi-librium one h ˆ n k σ i At . Hence, the analysis of the pump-probe experiments in ref. 11, which substituted the sumrule of W eq αβ for that of W Aαβ ( t ), is now justified. Inconfiguration (ii), on the other hand, eq. (23) reads W Aαβ ( t ) = ( πe /V ) P k ,σ m − αβ ( k − ( e/ ~ ) A ( t )) h ˆ n k σ i At . Since the pump field enters the inverse mass tensor, thesimple replacement of h ˆ n k σ i eq with h ˆ n k σ i At in the sumrule of W eq αβ does not yield the correct result.Finally, we point out that the present results can begeneralized. Suppose that the current density takes ageneral form;ˆ j α = ˆ J Aα − X β ˆ D Aαβ a β ( t ) + o ( a ) . Here, ˆ J Aα and ˆ D Aαβ are arbitrary vector and tensor oper-ators, respectively, which may be functions of A . [Equa-tion (9) takes this form.] Then the sum rules eqs. (23)and (24) are respectively generalized as Z ∞−∞ Re σ Aαβ ( ω ; t ) dω = π h ˆ D Aαβ i At , (28) Z ∞−∞ n ω Im σ Aαβ ( ω ; t ) − h ˆ D Aαβ i At o dω = 0 . (29)Furthermore, generalizations to the case where the probefield is a longitudinal AC electric field and to higher-orderresponses [following ref. 15] are straightforward.We thank T. Oka and N. Tsuji for directing our at-tention to this problem and for helpful discussions. Thiswork was supported by KAKENHI Nos. 22540407 and23104707, and by a Grant-in-Aid for the GCOE Pro-gram “Weaving Science Web beyond Particle-Matter Hi-erarchy”.
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