Optical response of highly excited particles in a strongly correlated system
OOptical response of highly excited particles in a strongly correlated system
Zala Lenarˇciˇc , Denis Goleˇz , Janez Bonˇca , and Peter Prelovˇsek , J. Stefan Institute, SI-1000 Ljubljana, Slovenia and Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia
We present a linear-response formalism for a system of correlated electrons out of equilibrium, as relevant forthe probe optical absorption in pump-probe experiments. We consider the time dependent optical conductivity σ ( ω, t ) and its nonequilibrium properties. As an application we numerically study a single highly excitedcharged particle in the spin background, as described within the two-dimensional t - J model. Our results showthat the optical sum rule approaches the equilibrium-like one very fast, however, the time evolution and the finalasymptotic behavior of the absorption spectra in the finite systems considered still reveal dependence on thetype of initial pump perturbation. This is observed in the evolution of its main features: the mid-infrared peakand the Drude weight. PACS numbers: 71.27.+a, 78.47.-p
I. INTRODUCTION
Time-resolved pump-probe optical spectroscopy representsa new powerful tool to study materials with strongly cor-related electrons and offers a direct approach to far-from-equilibrium phenomena, probing in particular the relaxationand thermalization processes . It appears that strongly cor-related systems are in general characterized very fast relax-ation processes, emerging from the inherent strong interac-tions. Theoretical studies of nonequilibrium dynamics andtransient phenomena in correlated models confirm this, evenin particular states as the Mott-Hubbard insulator .In theory, probing the transient nonequilibrium state witha weak electromagnetic pulse naturally leads to the linear-response approach. The optical conductivity σ ( ω, t ) is thetime-dependent dynamical quantity directly relevant to pump-probe optical spectroscopy and represents the response to theprobing electric field in a nonequilibrium situation. Some careis needed to define and properly extract the sum rules and apossible dissipationless component – Drude weight (chargestiffness). Recently, such a formalism has been proposed forcontinuous correlated systems . Adapted for the applicationof the dynamical mean-field theory (DMFT), it has been usedfor the analysis of the Hubbard model or with the em-phasis on the description of time-dependent photoemissionspectroscopy . Another definition has been used to examinethe dynamics of the Hubbard-Holstein model . A station-ary response, as a characteristic of a nonequilibrium quenchedstate, has also been considered recently and studied explic-itly for a hard-core boson model and in connection with thefluctuation-dissipation relation . There are also studies inwhich the effect of the probe field after the intensive pulse ex-citation is directly followed by introduction of the (classical)driving time-dependent electric field .The aim of this paper is to introduce the differential opticalconductivity σ ( ω, t ) representing the causal linear response ofthe lattice current to an arbitrary electric-field pulse E ( t (cid:48) > t ) ,acting on a general nonequilibrium many-body wave function | ψ ( t ) (cid:105) within a tight-binding model of correlated electrons.Such a formulation allows the definition and the considera-tion of the optical sum rule at any time, as well as the possible existence of the Drude weight D ( t ) as the dissipationless re-sponse.As a nontrivial example we test the formalism by numer-ical investigation of a single highly excited charged particle(hole) within Mott-Hubbard insulator, as represented by thetwo-dimensional (2D) t - J model. It seems plausible that inthe long-time and thermodynamic limit, anomalous σ ( ω, t ) should approach the ground state (g.s.) response σ ( ω ) , sincethe particle is expected to relax to the g.s. by emitting the extraenergy via relevant bosonic excitations, i.e., via magnons. Onthe other hand, in a closed finite system one would expect theresponse to approach a thermal equilibrium response σ th ( ω ) ,characterized by T > . Our numerical solutions indicatethat these presumptions are only partly realized for concreteexamples and that for the long-time response also the initialstate plays a role, clearly visible at least within the limitationof our finite systems and finite evolution times.The paper is organized as follows. In Sec. II we present thelinear response formalism for the optical conductivity σ ( ω, t ) within the tight-binding model for the general nonequilibriummany-body wave function and related density matrix. Sincethe calculation of the time-dependent σ ( ω, t ) in principle in-volves a two-time evolution, implementation with a single-time evolution is developed in order to reduce the numericalcomplexity as described in Sec. III. Section IV is devoted tothe numerical study of a nontrivial test case, representing theoptical response of the excited particle in the strongly corre-lated background, as given by the t - J model with a singlehole. Conclusions and open questions are discussed in Sec. V. II. NONEQUILIBRIUM OPTICAL LINEAR RESPONSE
In a general single-band tight-binding model of correlatedelectrons (with charge e ) , assuming the system with periodicboundary conditions (PBCs), the action of the electromagneticfield can be introduced via the vector potential A ( t ) throughthe usual gauge (Peierls) construction. The latter neglects theinter-band tunneling in multi-orbital models, and the field-induced distortions of the orbitals, but remains appropriate forthe single-orbital case and weak fields. We consider the tight- a r X i v : . [ c ond - m a t . s t r- e l ] A p r binding Hamiltonian (using (cid:126) = 1 ) up to O ( A ) , H ( A ( t )) = − (cid:88) i,j,s t ij exp( ie A ( t ) · R ij ) c † js c is + H int ≈ H − e A ( t ) · j + e A ( t ) · τ A ( t ) , (1)written with the particle current j and the stress tensor τ oper-ators, j = i (cid:88) i,j,s t ij R ij c † js c is , τ = (cid:88) i,j,s t ij R ij ⊗ R ij c † js c is , (2)where R ij = R j − R i . The electrical current j e ( t ) = − ∂H/∂ A ( t ) = e j − e τ A ( t ) (3)is a sum of the particle current and the diamagnetic contri-bution. We treat the case in which the unperturbed system isdescribed by a pure (nonequilibrium) many-body wave func-tion (wf.) | ψ ( t ) (cid:105) , with the time evolution operator U ( t (cid:48) , t ) =exp[ − iH ( t (cid:48) − t )] corresponding to the time-independent H .We use the standard formalism to evaluate the linear re-sponse of (cid:104) j (cid:105) t (cid:48) to a general A ( t (cid:48)(cid:48) ) applied at t , whereas thediamagnetic part is already linear in A ( t (cid:48) ) . Introducing thenotation for expectation values (cid:104) O (cid:105) t = (cid:104) ψ ( t ) | O | ψ ( t ) (cid:105) and theinteraction representation B I ( t (cid:48) ) = U † ( t (cid:48) , t ) BU ( t (cid:48) , t ) , (cid:104) j e (cid:105) t (cid:48) = e (cid:104) j (cid:105) t (cid:48) − e A ( t (cid:48) ) (cid:104) τ (cid:105) t (cid:48) + e (cid:90) t (cid:48) t dt (cid:48)(cid:48) χ ( t (cid:48) , t (cid:48)(cid:48) ) A ( t (cid:48)(cid:48) ) ,χ ( t (cid:48) , t (cid:48)(cid:48) ) = iθ ( t (cid:48) − t ”) (cid:104) [ j I ( t (cid:48) ) , j I ( t (cid:48)(cid:48) )] (cid:105) t . (4)Differential conductivity σ ( t (cid:48) , t ) is defined through the re-sponse to electric field E (¯ t ) , δ (cid:104) j e (cid:105) t (cid:48) = V (cid:90) t (cid:48) t d ¯ t σ ( t (cid:48) , ¯ t ) E (¯ t ) , (5) V being the volume of the system. Taking into account A ( t (cid:48)(cid:48) ) = − (cid:82) t (cid:48)(cid:48) t E (¯ t ) d ¯ t and Eqs. (4),(5) we get σ ( t (cid:48) , t ) = e V [ (cid:104) τ (cid:105) t (cid:48) − (cid:90) t (cid:48) t dt (cid:48)(cid:48) χ ( t (cid:48) , t (cid:48)(cid:48) )] . (6)For a nonstationary state there is no unique definition ofthe frequency-dependent σ ( ω, t ) . We choose plausiblerelation reflecting the causality and switching-on of the fieldat time t , i.e., E (¯ t < t ) = 0 , σ ( ω, t ) = (cid:90) t m ds σ ( t + s, t ) e iωs (7)where t m is the width of window in which we do the Fouriertransformation, so that formally t m → ∞ but is in practicethe maximum time of probe duration. With such a defini-tion, Eq. (7), we avoid the ambiguity of including times priorto the pump pulse. The sum rule for so defined σ (cid:48) ( ω, t ) =Re σ ( ω, t ) then follows directly from Eq. (7), (cid:90) ∞−∞ dω σ (cid:48) ( ω, t ) = πσ ( t, t ) = πe V (cid:104) τ (cid:105) t . (8) It is evident that the sum rule, Eq. (8), is a time-dependentquantity, i.e., (cid:104) τ (cid:105) evaluated at the time t when the probe fieldis applied. Moreover, independent of the precise form of theFourier transform it remains proportional to the (cid:104) τ (cid:105) at the timeheld fixed in the transformation.One can define also the Drude weight D ( t ) as the dissipa-tionless component, σ (cid:48) ( ω, t ) = 2 πe D ( t ) δ ( ω ) + σ (cid:48) reg ( ω, t ) D ( t ) = 12 V t m (cid:90) t m ds [ (cid:104) τ (cid:105) t + s − (cid:90) s ds (cid:48) χ ( t + s, t + s (cid:48) )] , (9)again for t m → ∞ . Equation (9) is a generalization of theequilibrium expression, D = (1 / V )( (cid:104) τ (cid:105) − χ (cid:48) ( ω = 0)) .In contrast to the sum rule, D ( t ) following from Eq. (9) isexpected to be dominated by t (cid:48) , t (cid:48)(cid:48) >> t in Eq. (6). Its depen-dence on t is revealed if written in the basis of eigenstates | φ m (cid:105) of H . In the standard notation for matrix elements (cid:104) φ m | O | φ n (cid:105) = O mn and amplitudes (cid:104) φ m | ψ (0) (cid:105) = a m ( t = 0 chosen arbitrarily, e.g., when the nonequilibrium state is pre-pared), we can express D ( t ) in the eigenbasis, assuming thatthere are no degeneracies, D ( t ) = 1 V (cid:88) m | a m | (cid:2) τ mm − (cid:88) n (cid:54) = m | j mn | ( (cid:15) n − (cid:15) m ) (cid:3) ++ 12 V (cid:88) m,n (cid:54) = m a ∗ m a n j mn ( j mm − j nn )( (cid:15) m − (cid:15) n ) e i ( (cid:15) m − (cid:15) n ) t . (10)Obviously, the last term provides dependence on t if nonzero.However, it is expected to vanish if averaged over t , yieldingstationary D ( t ) = D which is dependent only on the (initial)nonequilibrium state | ψ (0) (cid:105) through a m . Moreover, the firsttwo terms in Eq. (10) resemble the equilibrium expression ,with thermal weights substituted by projection weights | a m | .However, the latter derivation is feasible only for the time-independent H . One can express limiting D also for thecase with degeneracies, where it is of more general form, D = 1 V (cid:88) (cid:15) m = (cid:15) n a ∗ m a n (cid:2) τ mn − (cid:88) (cid:15) l (cid:54) = (cid:15) m j ml j ln ( (cid:15) l − (cid:15) m ) (cid:3) , (11)still being independent of the choice of the basis within de-generate sector.We have so far considered the response of a pure state | ψ ( t ) (cid:105) and a t -independent unperturbed H . The formal-ism can be extended also to more general density-matrix aswell as the time dependent H ( t ) , e.g., representing the pres-ence of the pump. In this case, the response to the perturba-tion V ( t ) = f ( t ) V for an ensemble of pure states or time-dependent H ( t ) is derived by expanding the density matrixup to the first order in f ( t ) , ρ ( t ) = ρ ( t )+ ρ ( t )+ O ( f ) , andusing the von-Neumann equation ∂ρ ( t ) /∂t = − i [ H ( t ) + f ( t ) V, ρ ( t )] . The linear response of general operator B at time t (cid:48) to perturbation applied at t is then δ (cid:104) B (cid:105) t (cid:48) = T r [ ρ ( t (cid:48) ) B ] (12) = − i (cid:90) t (cid:48) t dt (cid:48)(cid:48) f ( t (cid:48)(cid:48) ) T r [ ρ ( t )[ B I ( t (cid:48) ) , V I ( t (cid:48)(cid:48) )]] , (13) ρ ( t ) = U ( t, ρ (0) U † ( t, ,ρ ( t (cid:48) ) = i (cid:90) t (cid:48) t dt (cid:48)(cid:48) f ( t (cid:48)(cid:48) ) U ( t (cid:48) , t (cid:48)(cid:48) )[ ρ ( t (cid:48)(cid:48) ) , V ] U † ( t (cid:48) , t (cid:48)(cid:48) ) , where the time-evolution operator is U ( t (cid:48) , t ) =ˆ T [exp( − i (cid:82) t (cid:48) t H ( t (cid:48)(cid:48) ) dt (cid:48)(cid:48) )] . In such a formulation thelinear response of the particle current to the field appliedat t , written with density matrix for a single pure state ρ ( t ) = | ψ ( t ) (cid:105)(cid:104) ψ ( t ) | is δ (cid:104) j (cid:105) t (cid:48) = − i (cid:90) t (cid:48) t dt (cid:48)(cid:48) ( − e A ( t (cid:48)(cid:48) )) T r [ ρ ( t )[ j I ( t (cid:48) ) , j I ( t (cid:48)(cid:48) )] . (14)Optical conductivity, possibly generalized also to an ensembleof pure states, is then σ ( t (cid:48) , t ) = e V (cid:32) (cid:104) τ (cid:105) t (cid:48) − i (cid:90) t (cid:48) t dt (cid:48)(cid:48) T r [ ρ ( t )[ j I ( t (cid:48) ) , j I ( t (cid:48)(cid:48) )]] (cid:33) , (15)where now (cid:104) τ (cid:105) t = T r [ ρ ( t ) τ ] . III. NUMERICAL IMPLEMENTATION
Let us discuss here the numerical implementation for σ ( ω, t ) only for the case of single wf. | ψ ( t ) (cid:105) and time-independent H . The apparent disadvantage of the definitionwith Eq. (6) is that for a fixed t the evaluation of χ ( t (cid:48) , t (cid:48)(cid:48) ) viaEq. (4) requires the propagation U ( t (cid:48) , t (cid:48)(cid:48) ) for each t (cid:48)(cid:48) , and fi-nally for the numerical calculation at chosen t m , O ( t m / ∆ t ) operations are needed with ∆ t being the integration time step.Instead it is more efficient to calculate the integral insidethe matrix element, Eq.(4), performing discrete steps, (cid:90) t (cid:48) t dt (cid:48)(cid:48) U ( t (cid:48) , t (cid:48)(cid:48) ) j | ψ ( t (cid:48)(cid:48) ) (cid:105) ≈ ∆ t n m (cid:88) n =0 U n m − n +1∆ j | ψ ( t + n ∆ t ) (cid:105) , (16)where n m = ( t (cid:48) − t ) / ∆ t − and U ∆ = U ( t (cid:48)(cid:48) + ∆ t, t (cid:48)(cid:48) ) =˜ U (∆ t ) propagation for ∆ t . The sum (16) can be then evalu-ated recursively, | S (cid:105) = j | ψ ( t ) (cid:105) , | S n (cid:105) = j | ψ ( t + n ∆ t ) (cid:105) + U ∆ | S n − (cid:105) (17)so that finally σ ( t (cid:48) , t ) = ( e /V )( (cid:104) τ (cid:105) t (cid:48) + 2∆ t Im (cid:104) ψ ( t (cid:48) ) | j ˜ U (∆ t ) | S n m (cid:105) ) . (18)Such a procedure reduces the number of operations to O ( t m / ∆ t ) . We note that quite an analogous procedure canbe applied for other transient correlation functions or for thetime-dependent H ( t ) . IV. SINGLE EXCITED PARTICLE
In order to test the feasibility of the above formalism andcontribute to the discussion of transient optical response ofnonequilibrium strongly correlated systems, we investigate inthe following the case of a single excited charge carrier (hole) N h = 1 , doped in an antiferromagnetic Mott-Hubbard insula-tor. We consider the standard single-band t - J model, H = − t h (cid:88) (cid:104) i,j (cid:105) ,s (˜ c † i,s ˜ c j,s + H.c. ) + J (cid:88) (cid:104) i,j (cid:105) ( S i · S j − n i n j ) , (19)where ˜ c i,s = c i,s (1 − n i, − s ) are fermion operators, projectedonto the space with no double occupancy, describing hoppingbetween the nearest neighbor sites only. We consider in theconcrete example the 2D square lattice, relevant for cuprates,with J/t h = 0 . . Further on we use t h = 1 as the unit ofenergy, as well of time t = (cid:126) /t h = 1 . Lattice spacing is set a = 1 so that V = N , as well e = 1 .The intention is to consider the situation relevant for thepump-probe experiments on cuprates . One can imagine twodifferent situations:a) First is the photodoping of the Mott-Hubbard insulator,where a low concentration of highly excited charge carriers(holons and doublons) is created within an otherwise insulat-ing AFM system. Both types of carriers would exhibit, at leastin the transient stage, an independent response to the probepulse. To simulate this situation we consider as the initial wf.a single hole localized on one site, | ψ (0) (cid:105) being the eigenstateof the model, Eq. (19), where effective t h is put to zero.b) Another setting is a weakly doped AFM insulator, repre-sented by the g.s. of a single hole within the t - J model. Theeffect of a strong pump pulse applied to it can be simulated byintroduction of a phase shift in the hopping term of Eq. (19),changing t h → t ijh = t h exp( iθ ij ) , where we perform themaximum shift in the chosen direction x , i.e. θ x = π . Bothscenarios correspond to the same change in the total kineticenergy E kin , calculated from the expectation value of the t h term in Eq. (19). As obvious for the initially localized holewith t h ( t = 0) = 0 , the excited state has E kin ( t = 0) = 0 .For the choice θ x = π the effect of particular phase shift isto change the sign of hopping in the x direction. Due to rota-tional invariance this again yields zero total kinetic energy.Results for a single hole within the t - J model are obtainedvia two numerical methods. One is the exact diagonalization(ED) of small systems employing Lanczos method, where westudy the 2D square lattices with N ≤ sites and p.b.c.First we find the g.s. | ψ (0) (cid:105) . By solving time-dependentSchr¨odinger equation, time evolution of | ψ ( t (cid:48) ) (cid:105) and evalua-tion of recursive relations, Eq. (17), is obtained by employ-ing the Lanczos basis . Since the available square lattices( N = 18 , , ) are in general not rotationally invariant weperform the averaging of σ (cid:48) = ( σ (cid:48) xx + σ (cid:48) yy ) / for the case b)together with corresponding pulses θ α = θ x , θ y .Another method to evaluate σ ( ω, t ) is the diagonalizationwithin the limited functional space (EDLFS) The advan-tage of the EDLFS method in the equilibrium regime followsfrom a systematical construction of states with distinct config-urations of local spin excitations in the proximity of the hole.In this way (in contrast to the ED on small system) the methodin principle deals with an infinite system. In practice the effec-tive size of the system is larger than in ED, but still limited bythe number of basis states taken into account. The EDLFS re-mains efficient even when applied to nonequilibrium systems,as long as the spin disturbance caused by the local quench re-mains within the confines of generated spin excitations .In the considered case spin excitations extend up to L = 16 lattice sites away from the hole.First we analyze the time variation of (cid:104) τ αα (cid:105) , representingthe sum rule Eq. (8), which is for the tight-binding model(19) related to the kinetic energy, (cid:104) τ αα (cid:105) t = − E kin,α ( t ) ,where only hopping in α direction is taken into account.In Fig. 1 we present ED and EDLFS results for (cid:104) τ (cid:105) calcu-lated directly from kinetic energy for | ψ (0) (cid:105) correspondingto localised and π -pulse excitations, respectively. Averaging (cid:104) τ (cid:105) = ( (cid:104) τ xx (cid:105) + (cid:104) τ yy (cid:105) ) / is employed in ED calculations,whereas for EDLFS (cid:104) τ (cid:105) = (cid:104) τ xx (cid:105) . For comparison we showalso the g.s. E kin,x = E kin / ∼ − . . From Fig. 1 it fol-lows that for both types of initial excited states the decay of (cid:104) τ (cid:105) is very fast. The corresponding short time t d can be re-lated to the formation of the spin polaron and can be explainedwith the generation of string states . It is expected to scale as t d ∝ ( J/t h ) − / .We confirm in Fig. 1 that both methods, the ED and theEDLFS, give quite consistent result for (cid:104) τ (cid:105) t for short times t < t d ∼ . (for chosen J = 0 . ). For intermediate times t d < t < t i ∼ the decay of E kin,x evaluated with EDis somewhat slower, which could indicate the influence offinite-size and p.b.c. effects. Namely, in small systems con-sidered with the ED spin excitations populate the lattice andconsequently influence further hole relaxation. The spread ofspin excitation is expected to saturate in t i ∼ √ N /J , yield-ing approximately stationary response afterwards. Within theEDLFS with in principle an infinite lattice, such effects are notpresent or appear only at later times due to the restricted basis.More than an artifact, ED results on small lattices should berelevant for the optical sum rules of systems with finite densityof carriers, while those from EDLFS corresponds to systemswith vanishing density.Although for both types of excitations E kin ( t = 0) ∼ ,the distinction in sum rules is apparent for times shorter than t < t d , featuring the fact that the state of initially localizedhole has the rotational symmetry, whereas with π − pulse thissymmetry is broken, with E kin,x ( t = 0) = − E kin,y ( t = 0) .It is evident that for long times t > t i kinetic energy ap-proaches or oscillates around a quasi-stationary value, de-noted by E kin ( t → ∞ ) . Within the ED latter still somewhatdepends on the system size N . The larger is the system underconsideration, closer is this value to the g.s. E kin , as markedin Fig. 2.Since the ED simulates a fixed-size system, and the excitedstate is quite far from the g.s., a plausible interpretation couldbe investigated within the concept of thermalization, i.e. theapproach to the equilibrium state with a finite effective tem-perature T eff > .One could argue that different E kin ( t → ∞ ) originate in EDLFS, locED E kin, x (cid:45) (cid:45)(cid:60) Τ (cid:62) a (cid:76) EDLFS, Θ x (cid:61)Π ED E kin, x (cid:45) t (cid:45)(cid:60) Τ (cid:62) b (cid:76) Figure 1. (Color online) Sum rule (cid:104) τ (cid:105) vs. time t for J = 0 . asobtained using the ED on N = 26 sites as well as the EDLFS. Re-sults are presented for initial states of: a) the localized hole and b)the π -pulse. The g.s. E kin,x value is also displayed. different T eff , depending on the type of quench and size ofthe system. In a finite system T eff is set by the excitationenergy so that the canonical expectation value of the energyequals the total initial energy (cid:104) ψ (0) | H | ψ (0) (cid:105) = E tot . At N = 26 the effective temperature is approximately T eff =0 . , . for the initially localized hole and π -pulsed hole, re-spectively, and increases as N decreases.Our observation is that E kin ( T eff ) is still lower than E kin ( t → ∞ ) for all finite systems considered, as seen inFig. 2 This suggests that system cannot completely thermal-ize, possibly due to the discreteness of spin excitations in fi-nite systems. In this connection we notice that T > calcu-lation of the same model, Eq. (19), with a single hole usingthe finite temperature Lanczos method surprisingly revealthat E kin ( T ∼ J ) does not essentially differ from g.s. E kin ,Fig. 2.In Fig. 3 we finally present results for the time-dependentoptical spectra per hole ˜ σ (cid:48) ( ω, t ) = N σ (cid:48) ( ω, t ) as obtained byED and EDLFS, respectively, following the described proce- (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:45) (cid:45) (cid:45) T E k i n (cid:144) locN (cid:61) Θ x (cid:61)Π locN (cid:61) Θ x (cid:61)Π g.s. Figure 2. Equilibrium kinetic energy E kin / vs. temperature T forsingle hole on a system with N = 26 sites. Dots mark E kin ( t →∞ ) / for both types of excitations in N = 20 , systems and g.s.. dure. The cases of initially localized and π -pulse excited holesare compared. With respect to Fig. 1 chosen times representdifferent evolution stages: a) t ∼ response of the initial ex-cited state | ψ (0) (cid:105) , b) response at approximately characteristicdecay time t = 1 . ∼ t d , and c) t = 10 ∼ t i already relaxedbut not yet fully stationary response. To mimic the station-ary response for finite systems ˜ σ (cid:48) ( ω, ¯ t ) , obtained by averageover responses in interval t ∈ [20 , (cid:29) t d , is presented inFig. 4 and compared with the g.s. ˜ σ ( ω ) and with the thermal-equilibrium result ˜ σ th ( ω ) at effective T eff (cid:38) J , all obtainedwithin the ED.In the initial stage the response is very incoherent and forthe π − pulse case even predominantly negative ˜ σ (cid:48) ( ω, t ) < ,which is compatible with the sum rule (cid:104) τ (cid:105) t ∼ < in Fig. 1.The latter indicates on highly nonequilibrium state | ψ (0) (cid:105) cor-responding to an inverse particle population .With some quantitative difference between both methods,within the g.s. ˜ σ ( ω ) two features are well visible and of t (cid:61) (cid:61) (cid:61) (cid:45) Σ (cid:142) ' (cid:72) Ω , t (cid:76) ED, loca (cid:76) (cid:45) Σ (cid:142) ' (cid:72) Ω , t (cid:76) EDLFS, locb (cid:76) (cid:45) Σ (cid:142) ' (cid:72) Ω , t (cid:76) ED, Θ x (cid:61)Π c (cid:76) (cid:45) Ω Σ (cid:142) ' (cid:72) Ω , t (cid:76) EDLFS, Θ x (cid:61)Π d (cid:76) Figure 3. (Color online) Time-dependent optical conductivity ˜ σ (cid:48) ( ω, t ) (per hole) vs. ω as calculated at different times t =0 , . , for: a) the initially localized hole with the ED on N = 26 sites, b) localized hole with with the EDLFS, c) the π -pulse excitedhole with the ED, and d) the π -pulse excited hole within EDLFS.For comparison the g.s. ˜ σ ( ω ) is shown. Broadening of spectra δω = 0 . is used. particular interest. One is the mid-infrared (MIR) peak at ω ∼ . J , which is the signature of the string-like excitedparticle states within the 2D AFM background. We see inFigs. 3a,b that the MIR-like peak appears also in ˜ σ ( ω, t ) andstabilizes very fast at t ∼ t d for the localized hole, indepen-dent of the numerical method employed. On the other hand,after the π -pulse (Fig. 3c,d) the MIR peak is less pronounced.Especially within the ED, ˜ σ ( ω, t > t d ) is closer to the ther-mal ˜ σ th ( ω ) at appropriate T eff = 0 . . In the equilibriumat such high T > J the MIR peak (Figs. 3c, 4b) is alreadysmeared out due to thermally disordered AFM spin back-ground. Within the EDLFS results (Fig. 3d) the MIR peakrecovers better, which could be attributed to a lower densityof spin excitations in effectively bigger systems.The second feature, pronounced within the g.s. is the Drudeweight D , i.e. peak at ω = 0 which accounts for ∼ / of theweight in the sum rule at T = 0 . It should acquire a finitewidth in the equilibrium at T > due to scattering processes(i.e. in the strict sense D = 0 is expected for a nonintegrablesystem). Although within both methods used (Eq. 10 wouldrequire full ED method) we cannot strictly establish and deter-mine the value of D ( t ) as defined in Eqs. (9,10), it is evidentthat with the respect of low- ω response studied, excited par-ticles reveal quite different behavior. Within the ED π -pulseexcited hole shows essentially no Drude peak, i.e., no cor-responding low- ω remainder at t > . On the other hand,initially localized hole displays a substantial low- ω peak andpresumably D ( t ) > (note that we use broadening δω = 0 . )at all t > , although the weight is smaller than in ˜ σ ( ω ) .Using the EDLFS we notice a weak remainder of the low- ω contribution even for π -pulse, yet much smaller than for thelocalized hole. This findings indirectly support Eq. (10), thatalso the initial wf. (and not just its total energy) determines g.s.T (cid:61) Σ (cid:142) ' (cid:72) Ω , t (cid:76) a (cid:76) ED, loc g.s.T (cid:61) Ω Σ (cid:142) ' (cid:72) Ω , t (cid:76) b (cid:76) ED, Θ x (cid:61)Π Figure 4. (Color online) Long-time response ˜ σ (cid:48) ( ω, ¯ t ) vs. ω ascalculated with ED on N = 26 , obtained by average over re-sponses in interval t ∈ [20 , (cid:29) t d for a) initially localized hole,and b) the π -pulse excited hole. Results are compared to the g.s. ˜ σ ( ω ) and the thermal ˜ σ th ( ω ) at corresponding effective tempera-tures T = 0 . , . . Spectra are broadened with δω = 0 . . the limiting Drude weight value D . V. CONCLUSIONS
We have presented a formalism for the linear optical con-ductivity response σ ( ω, t ) of a nonequilibrium (excited) stateof a strongly correlated system, where probe pulse is taken asa perturbation in the linear order. In the absence of an uniqueapproach we have chosen the definition reflecting the onset ofthe probe electric field pulse at time t , which is in contrast tosome other studies. Such an approach allows the discussionof the optical sum rule as well as the dissipationless Drudeweight at any time t > . On the other hand, the definition in-troduces a complication due to an additional time integrationwhich we circumvent by a particular numerical implementa-tion.The presented test case of a single highly excited chargecarrier within the t - J model already shows several featuresand opens questions relevant for the theoretical analysis ofthe pump-probe spectroscopy results. Independently of theinitial state we observe a fast relaxation of several observ-ables, e.g. the kinetic energy and the optical sum rule, to-wards the respective g.s. value. Still, more specific featuresof the transient and long-time optical response, as the MIRpeak and the Drude component D ( t ) , appear non-universal.Our results reveal that they do not depend merely on the ex-citation energy, as e.g. expected from the canonical thermal-ization, but as well on the character and wf. of the initial excited state. The persistance of this feature in the thermo-dynamic limit remains an open question. One one hand, itappears plausible that in an infinite system the local state ofthe quasiparticle will correspond to the ground state. Nev-ertheless, this statement is, e.g., not evident for a quantumsystem with gapped bosonic excitations. To this end our find-ings show lack of canonical thermalization, observed beforein theoretical as well as experimental studies . Since weaddress excited systems, the concepts of thermalization andrelaxation to the g.s. response are only partly applicable, es-pecially for dynamical quantities, and remain the challengealso for further studies. We should note as well that in theapplication of our formalism and results to the pump-probeexperiments some care is needed when energy absorption ofparticular probe pulses is measures, which cannot be directlycompared to time-dependent σ (cid:48) ( ω, t ) calculated in the presentstudy.Z. L. and D. G. acknowledge discussions with M. Ecksteinand A. Silva. J. B. acknowledges stimulating discussion withS. A. Trugman and the financial support by the Center for In-tegrated Nanotechnologies, an Office of Science User Facilityoperated for the U.S. This work has been supported by theProgram P1-0044 and the project J1-4244 of the SlovenianResearch Agency (ARRS). VI. REFERENCES K. Matsuda, I. Hirabayashi, K. Kawamoto, T. Nabatame, T. Tok-izaki and A. Nakamura, Phys. Rev. B , 4097 (1994). H. Okamoto, T. Miyagoe, K. Kobayashi, H. Uemura, H. Nishioka,H. Matsuzaki, A. Sawa and Y. Tokura, Phys. Rev. B , 060513(2010); Phys. Rev. B , 125102 (2011). S. Wall et al, Nature Phys. , 114 (2011). F. Novelli, D. Fausti, J. Reul, F. Cilento, P. H. M. van Loosdrecht,A. A. Nugroho, T. T. M. Palstra, M. Gr¨uninger and F. Parmigiani,Phys. Rev. B , 165135 (2012). S. Dal Conte et al. , Science , 6076 (2012). C. Giannetti et al. , Nature Communications , 353 (2011). M. Eckstein and P. Werner, Phys. Rev. B , 035122 (2011). M. Eckstein and P. Werner, Phys. Rev. Lett. , 126401 (2013). Z. Lenarˇciˇc and P. Prelovˇsek, Phys. Rev. Lett. , 016401 (2013). A. Shimizu and T. Yuge, J. Phys. Soc. Jpn. , 093706 (2011). M. Eckstein and M. Kollar, Phys. Rev. B , 205119 (2008). M. Eckstein, M. Kollar, P. Werner, Phys. Rev. B , 115131(2010). J. K. Freericks, H. R. Krishnamurthy, T. Pruschke, Phys. Rev. Lett. , 136401 (2009). G. De Filippis, V. Cataudella, E. A. Nowadnick, T. P. Devereaux,A. S. Mishchenko and N. Nagaosa, Phys. Rev. Lett. , 176402(2012). D. Rossini, R. Fazio, V. Giovannetti, and A. Silva, arXiv1310.4757. L. Foini, L. F. Cugliandolo and A. Gambassi, Phys. Rev. B ,212404 (2011). L. Foini, L. F. Cugliandolo and A. Gambassi, J. Stat. Mech. ,P09011 (2012). B. Moritz, A. F. Kemper, M. Sentef, T. P. Devereaux, and J. K.Freericks, Phys. Rev. Lett. , 077401 (2013). P. Nozi`eres,
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