Phase-space approach to polaron response: Kadanoff and FHIP re-examined
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Phase-space approach to polaron response: Kadanoff andFeynman-Hellwarth-Iddings-Platzmann re-examined
Dries Sels ∗ and Fons Brosens † Physics Department, University of Antwerp, Universiteitsplein 1, 2060 Antwerpen, Belgium
A method is presented to obtain the linear response coefficients of a system coupled to a bath. Themethod is based on a systematic truncation of the Liouville equation for the reduced distributionfunction. The first order truncation results are expected to be accurate in the low temperature andweak coupling regime. Explicit expressions for the conductivity of the Fr¨ohlich polaron are obtained,and the discrepancy between the Kadanoff and the Feynman-Hellwarth-Iddings-Platzmann mobilityis elucidated.
I. INTRODUCTION
Since its inception, the mobility of the Fr¨ohlich polaron [1–3] has been the subject of many theoretical studies.For an excellent in-depth overview and discussion we refer to a textbook by Alexandrov and Devreese [4] and tolecture notes by Devreese [5]. A prominent approach was proposed by Feynman et al. [6] (hereafter referred toas FHIP), based on the path-integral formalism. This method is nonperturbative in the sense that no expansionin the coupling constant is assumed, but it is limited to first order in the applied electric field. However, in theasymptotic limit of weak electron-phonon coupling and low temperature, the FHIP polaron mobility differs by afactor of 3 / (2 ~ βω LO ) –with ω LO the dispersionless longitudinal optical phonon frequency, and β = 1 / ( k B T ) where k B is Boltzmann’s constant and T is the temperature– from the mobility which Kadanoff [7] found later on from theBoltzmann equation within the relaxation time approximation. As already pointed out in FHIP, the same factor of3 / (2 ~ βω LO ) appears in comparison with earlier results [8–10]. It has been argued in [11, 12] that this discrepancymight be due to interchanging two limits (with both the frequency of the applied electric field and the electron-phononcoupling strength tending to zero). But this mathematical argument implicitly assumes the Kadanoff result to bevalid, which we dispute.In the present paper, we propose an alternative approach, based on the dynamics of the Wigner distributionfunction [13]. The methodology is basically inspired by the Feynman-Vernon influence functionals [14], rather thanon Feynman’s variational path integral treatment of the ground state energy of the polaron [15]. However, instead ofconsidering the path integral for the wave function of a system, we contributed in [16] to a path integral descriptionof the Wigner distribution function. Concentrating on a particle that linearly interacts with a set of independentharmonic oscillators, the influence functional for the Wigner distribution function could be reduced to a double pathintegral in the path variables of the particle, if the oscillators are initially in thermodynamical equilibrium. In asubsequent paper [17] we derived a perturbation series for the propagator of the reduced Wigner function (i.e., theWigner function for the particle of interest). By exactly resumming this series, we found a Dyson integral equationfor the reduced propagator, from which the equation of motion for the reduced Wigner function could be derived.For general temperature and interaction strength, the resulting equation with a dressed propagator is still underinvestigation. We here concentrate on linear response at weak coupling and low temperature, in order to elucidatethe discrepancy between the FHIP and the Kadanoff mobility.The paper is organized as follows. In section II we extract the assumptions and results from the papers [16, 17] whichare relevant for our present purpose. In section III we present an approximate, however systematically improvable,truncation method to derive the linear response coefficients from the equation of motion for the reduced Wignerfunction. We present a detailed discussion on the conductivity of the Fr¨olich polaron in section IV, after which weconclude in V. Supplementary information on the used truncation scheme is provided in appendix A. Additionalcalculations on the relaxation time approximation and on FHIP are found in appendix B and C respectively. ∗ Corresponding author: [email protected] † [email protected] II. REDUCED WIGNER FUNCTION FOR A GENERIC POLARON SYSTEM
Consider the following generic polaron Hamiltonian H = p m − e E ( t ) · x + X k ~ ω k (cid:18) b † k b k + 12 (cid:19) + X k (cid:16) γ ( k ) exp ( − i k · x ) b † k + γ ∗ ( k ) exp ( i k · x ) b k (cid:17) , (II.1)where ( x , p ) represent the electron coordinate and momentum operator. It is coupled to some bosonic field b k in aisotropic translational invariant way, i.e. γ ( k ) = γ ( k ) = γ ( | k | ). Also the phonon frequency ω k = ω | k | is isotropic. Theelectron is subject to a time dependent but homogeneous electric field E ( t ) . Because the system is translational invariant in the absence of the field E ( t ) , we suppose that the electron distri-bution is homogeneous, and that the phonon bath is initially in thermal equilibrium: f ( r , p , { x k , p k } , t = −∞ ) = f ( p , t = −∞ ) Y k tanh β ~ ω k π ~ exp − tanh β ~ ω k ~ ω k (cid:18) p k m + mω k x k (cid:19)! . (II.2)Knowledge of the (reduced) Wigner distribution function f ( p , t ) would allow to calculate the current density, andhence the conductivity σ J ( t ) = em Z p f ( p , t ) d p (II.3)= Z t −∞ σ ( t − t ′ ) E ( t ′ )d t ′ . (II.4)In general, σ is a tensor but, due to the cylindrical symmetry of (II.1), it becomes diagonal. The Wigner-Liouvilleequation for the case of a phonon bath which initially is in thermal equilibrium, and for a general potential V ( x ,t ) , was derived in a recent paper [17]. For the electronic Hamiltonian p m − e E ( t ) · x under consideration here, the relevantequations (I.2–I.4) of Ref. [17] simplify into (cid:18) ∂∂t + e E ( t ) · dd p (cid:19) f ( p , t ) = X k | γ ( k ) | ~ × Z Z Z
Θ ( t ′ ≤ t ) (cid:18) ( n B ( ω k ) + 1) cos ( k · ( x − x ′ ) + ω k ( t − t ′ ))+ n B ( ω k ) cos ( k · ( x − x ′ ) − ω k ( t − t ′ )) (cid:19) × (cid:0) K (cid:0) x , p − ~ k , t | x ′ , p ′ + ~ k , t ′ (cid:1) − K (cid:0) x , p + ~ k , t | x ′ , p ′ + ~ k , t ′ (cid:1)(cid:1) f ( p ′ , t ′ ) d t ′ d x ′ d p ′ , (II.5)with K ( x , p , t | x ′ , p ′ , t ′ ) = δ (cid:18) p − p ′ − Z tt ′ e E ( σ ) d σ (cid:19) δ (cid:18) x − x ′ − p ′ m ( t − t ′ ) − Z tt ′ e E ( σ ) m ( t − σ ) d σ (cid:19) , (II.6) n B ( ω k ) = 1 / (cid:0) e β ~ ω k − (cid:1) . (II.7)Note that we have dropped the position dependence of the distribution function because both the initial state (II.2)and the electric field are homogeneous. In the absence of the electric field, the time evolution of the Wigner distributionbecomes ∂f E =0 ( p , t ) ∂t = X k | γ ( k ) | ~ Z ∞ ( n B ( ω k ) + 1) cos (cid:16)(cid:16) ( p + ~ k ) − p m ~ − ω k (cid:17) s (cid:17) + n B ( ω k ) cos (cid:16)(cid:16) ( p + ~ k ) − p m ~ + ω k (cid:17) s (cid:17) f E =0 ( p + ~ k , t − s ) − ( n B ( ω k ) + 1) cos (cid:16)(cid:16) ( p + ~ k ) − p m ~ + ω k (cid:17) s (cid:17) + n B ( ω k ) cos (cid:16)(cid:16) ( p + ~ k ) − p m ~ − ω k (cid:17) s (cid:17) f E =0 ( p , t − s ) d s. (II.8)It seems unlikely that this integro-differential equation can be solved in closed form. Even a stationary solution f stat E =0 ( p ) in the absence of an electric field obeys a non-trivial integral equation. Using R ∞ cos ( as ) d s = πδ ( a ), someelementary algebra reveals that, within the continuum limit, it satisfies the balance equation Z Π ( p + ~ k → p ) f stat E =0 ( ~ k + p ) d k = f stat E =0 ( p ) Π ( p ) , (II.9)where we adopt an analogous notation as introduced by Devreese and Evrard [24], and defineΠ ( p + ~ k → p ) = V | γ ( k ) | (2 π ) ~ ( n B ( ω k ) + 1) δ (cid:16) ( p + ~ k ) − p m − ~ ω k (cid:17) + n B ( ω k ) δ (cid:16) ( p + ~ k ) − p m + ~ ω k (cid:17) , (II.10)Π ( p ) = Z Π ( p → p + ~ k ) d k . (II.11)Even this equation is hard to solve in its generality. One can however check by straightforward algebra that f stat E =0 ( p ) ∝ exp (cid:0) − β p / m (cid:1) satisfies Eq.(II.9). In order to elucidate the discrepancy between the mobility results of FHIP andKadanoff, we limit the further discussion to linear response at weak coupling and low temperature. III. LINEAR RESPONSE AT WEAK COUPLING AND LOW TEMPERATURE
Limiting the discussion to first order in the electric field and to first order in | γ ( k ) | , the dependence on E of thereduced Wigner propagator (II.6) can be neglected, and the Wigner-Liouville equation (II.5) simplifies into (cid:18) ∂∂t + e E ( t ) · dd p (cid:19) f ( p , t ) = X k | γ ( k ) | ~ × Z t −∞ f ( p + ~ k , s ) ( n B ( ω k ) + 1) cos (cid:16) ( t − s ) (cid:16) k · p + ~ k m − ω k (cid:17)(cid:17) + n B ( ω k ) cos (cid:16) ( t − s ) (cid:16) k · p + ~ k m + ω k (cid:17)(cid:17) − f ( p , s ) ( n B ( ω k ) + 1) cos (cid:16) ( t − s ) (cid:16) k · p + ~ k m + ω k (cid:17)(cid:17) + n B ( ω k ) cos (cid:16) ( t − s ) (cid:16) k · p + ~ k m − ω k (cid:17)(cid:17) d s. (III.1)It seems impossible to solve this highly non-Markovian initial value problem exactly.Here we propose an approach which is inspired by the truncated Wigner approximation as, e.g., extensively dis-cussed by Polkovnikov [18]. Its application to general coupling strength and arbitrary temperature is under currentinvestigation. However, for sufficiently small electron-phonon coupling strength γ ( k ) and sufficiently low temperature,the truncation after the first moment is justified, as argued in detail in Appendix A. It results in the following equationof motion (A.4) for the current density: d J ( t ) dt + Z t −∞ J ( s ) χ ( t − s )d s = e m E ( t ) , (III.2)where the memory function χ of the system is given by χ ( t ) = t X k | γ ( k ) | ~ k m ( n B ( ω k ) + 1) sin (cid:16) t (cid:16) ~ k m + ω k (cid:17)(cid:17) + n B ( ω k ) sin (cid:16) t (cid:16) ~ k m − ω k (cid:17)(cid:17) . (III.3)The definition (II.4) of the conductivity thus yields the following relation between the Laplace transform L ( σ, Ω) ofthe conductivity and the Laplace transform L ( χ, Ω) of the memory function: L ( σ, Ω) = e m
1Ω + L ( χ, Ω) , (III.4)from which one can, for example, immediately extract the (long wavelength) optical absorption coefficient [19]Γ( ω ) = Z n Re [ L ( σ, iω )] , (III.5)where n is the crystals refractive index and Z = ( ǫ c ) − is the impedance of free space. Further results of coursedepend on the specifics of the system at hand. Here we apply the proposed model to the Fr¨ohlich polaron. IV. FR ¨OHLICH POLARON
For the optical Fr¨ohlich polaron one considers ω k = ω LO to be constant. The coupling | γ ( k ) | = ~ ω LO k παV r ~ mω LO (IV.1)scales with the dimensionless coupling constant α. Then, in the continuum limit, the remaining integral in Eq. (III.3)is Gaussian and results in χ ( t ) = 2 αω LO √ π (cid:20) (2 n B ( ω LO ) + 1) cos ( ω LO t ) √ ω LO t − sin ( ω LO t ) √ ω LO t (cid:21) (IV.2)= αω LO (cid:2) (2 n B ( ω LO ) + 1) J − / ( ω LO t ) − J / ( ω LO t ) (cid:3) , where J ± / denotes the Bessel function of the first kind of order ± / . The Laplace transform [20] of χ is given by L ( χ, Ω) = αω LO r(cid:16) Ω ω LO (cid:17) + 1 (2 n B ( ω LO ) + 1) vuuts(cid:18) Ω ω LO (cid:19) + 1 + Ω ω LO − vuuts(cid:18) Ω ω LO (cid:19) + 1 − Ω ω LO . (IV.3)Consequently the low temperature DC-conductivity (III.4) is σ DC = lim Ω → L ( σ, Ω) = 3 e αmω LO n B ( ω LO ) ≈ e αmω LO e β ~ ω LO . (IV.4)It should immediately be noted that this result differs by a factor of 3 from that of Kadanoff [7] and by a factor of(2 ~ βω LO ) from that of FHIP [6], i.e., σ DC = 3 σ DC Kadanoff = 2 ~ βω LO σ DC FHIP . (IV.5)The result is however in agreement with a prediction made by Los’ [21], based on a Green’s superoperator calculationof Kubo’s formula. It was already argued by FHIP, that in the Ω → / (2 ~ βω LO ) discrepancy was caused by aninterchange of the Ω → α → p → . One might guess that interchanging the limits by first taking the limit of Ω → , hence t → ∞ , and then the limit of p → E exactly subtracts 2 / αω LO ) from the inverse scattering rate resulting in amobility which is three times higher than the one calculated within the relaxation time approximation. It is clearthat the present approach does not violate particle number conservation, neither do FHIP and Los’.The additional 2 ~ βω LO difference with FHIP however remains to be explained. In appendix C we reexamine theFHIP approximation in the language of the distribution function rather than path integrals for the reduced densitymatrix. This illuminates the main problem in the FHIP approximation. First and foremost, unlike what is arguedby FHIP, it is detrimental to assume an initial product state between the bath and the system for the evolution ofthe model. Although the true system will quickly thermalize to the temperature of the bath, the model system ofFHIP does not thermalize, because it is completely harmonic and consequently fully integrable. In order to obtaina physical trial distribution one must assume that the complete model system was in thermal equilibrium instead ofin a product state of the system with a thermal bath. Apart from this small change the analysis in appendix C iscompletely in line with FHIP. The final low temperature DC conductivity however reads σ DC = 3 e αm ∗ ω LO e β ~ ω LO . where the effective mass m ∗ /m = v /w is defined in terms of Feynman’s variational parameters. Since w ≈ v andthus m ∗ ≈ m for sufficiently small α, we recover the same result (IV.4) as derived by our linearized equation of motion.It is clear that the present FHIP reanalysis does not have the spurious 2 ~ βω LO terms. V. CONCLUSION
In conclusion we have presented a method to obtain the conductivity of a generic polaron. In the low temperatureand weak coupling regime a truncation after the first moment is justified and the conductivity is completely determinedby a single memory function χ. The method is used to study the conductivity of the Fr¨ohlich polaron. It is found thatthe present approach results in a conductivity which is three times higher than the one predicted by Kadanoff anddiffers from that of FHIP by a factor 2 ~ βω LO . Consequently we recover the result of Los’ [21]. In order to elucidate thedifference, we have reanalyzed the Boltzmann equation used by Kadanoff and the approach used by FHIP. Whereasthe relaxation time approximation used by Kadanoff explicitly violates particle number conservation, the methoddeveloped by FHIP does not. The FHIP approximation however relies on an unphyiscal initial state for Feynman’spolaron model. We find that a slightly modified version of both, which amends these two problems, accounts for theirdiscrepancy.
Appendix A: Truncated equation of motion
Multiplying the Liouville equation (III.1) with e p /m and integrating out the momentum yields Z e p m (cid:18) ∂∂t + e E ( t ) · dd p (cid:19) f ( p , t ) d p = X k | γ ( k ) | ~ × Z t −∞ R e p m f ( p − ~ k , s ) ( n B ( ω k ) + 1) cos (cid:16) ( t − s ) (cid:16) k · p − ~ k m + ω k (cid:17)(cid:17) + n B ( ω k ) cos (cid:16) ( t − s ) (cid:16) k · p − ~ k m − ω k (cid:17)(cid:17) d p − R e p m f ( p , s ) ( n B ( ω k ) + 1) cos (cid:16) ( t − s ) (cid:16) k · p + ~ k m + ω k (cid:17)(cid:17) + n B ( ω k ) cos (cid:16) ( t − s ) (cid:16) k · p + ~ k m − ω k (cid:17)(cid:17) d p d s. (A.1)Taking the expression (II.3) for the current density into account, the left hand side can directly be calculated. Afterthe substitution p − ~ k → p in the first term on the right hand side, one is left with d J ( t ) dt − e m E ( t ) = X k | γ ( k ) | ~ k em Z t −∞ Z f ( p , s ) ( n B ( ω k ) + 1) cos (cid:16) ( t − s ) (cid:16) k · p m + ~ k m + ω k (cid:17)(cid:17) + n B ( ω k ) cos (cid:16) ( t − s ) (cid:16) k · p m + ~ k m − ω k (cid:17)(cid:17) d p d s. (A.2)Using the k ↔ − k symmetry results in d J ( t ) dt − e m E ( t ) = − X k | γ ( k ) | ~ k Z t −∞ n B ( ω k ) sin (cid:16)(cid:16) ~ k m − ω k (cid:17) ( t − s ) (cid:17) + ( n B ( ω k ) + 1) sin (cid:16)(cid:16) ~ k m + ω k (cid:17) ( t − s ) (cid:17) × em Z f ( p , s ) sin (cid:18) k · p m ( t − s ) (cid:19) d p d s. (A.3)Since the current density (II.3) is of order E , the dominant contribution in the last line of this equation is providedby the small momenta. It thus seems reasonable to expand the sine function: em Z f ( p , s ) sin (cid:18) k · p m ( t − s ) (cid:19) d p = em Z f ( p , s ) k · p m ( t − s ) − (cid:18) k · p m (cid:19) ( t − s ) · · · ! d p = ( t − s ) k · J ( s ) m − ( t − s ) em Z f ( p , s ) (cid:18) k · p m (cid:19) d p + · · · . For general coupling strength γ ( k ) and temperature, this expansion seems not very useful. Indeed, the Wignerfunction broadens with increasing temperature. Furthermore, for strong coupling the initial phonon states are betterdescribed by a displaced and broadened Gaussian wave functions, as shown in the derivation of the optical absorptionof polarons in [22, 23]. The change in the initial phonon state will effect the influence phase [16] and consequently theself energy [17]. A dressed propagator will replace the free particle propagator (II.6). The extension of the presentresult to strong coupling will be a topic of forthcoming work.However, in the present paper we were mainly concerned with the discrepancy between the FHIP result and theKadanoff result for small electron-phonon coupling and low temperature. In that case, neither the electron-phononcoupling nor the temperature are able to broaden the distribution function substantially. Therefore, for γ ( k ) and T sufficiently small, one might truncate the expansion to the first moment, which results in d J ( t ) dt − e m E ( t ) ≈ − X k | γ ( k ) | ~ k Z t −∞ ( n B ( ω k ) + 1) sin (cid:16) ( t − s ) (cid:16) ~ k m + ω k (cid:17)(cid:17) + n B ( ω k ) sin (cid:16) ( t − s ) (cid:16) ~ k m − ω k (cid:17)(cid:17) ( t − s ) k · J ( s ) m d s. (A.4)Note that one can systematically improve the result [18] by the equations of motion for the higher moments. Appendix B: Relaxation time approximation
The purpose of this Appendix is to explain the discrepancy in (IV.5) by a factor of 3 between the DC conductivityof the Fr¨ohlich polaron which we derived in (IV.4), as compared to the Kadanoff result [7]. We thus consider thelinearized Liouville equation (III.1) for the reduced Wigner function. Using R t −∞ cos (( t − s ) a ) d s = πδ ( a ) one easilyderives that its stationary version is a Boltzmann equation e E · df ( p ) d p = − Π ( p ) f ( p ) + Z Π ( p + ~ k → p ) f ( p + ~ k ) d k , (B.1)with Π ( p + ~ k → p ) and Π ( p ) defined in (II.10) and (II.11).Because the unperturbed reduced Wigner distribution function at sufficiently low temperature peaks around p = 0 , one might argue that the dominant term in the right hand side is given by − f ( p ) lim p → Π ( p ) , which gives rise toa relaxation time approximation (RTA): e E · df ( p ) d p ≈ − f ( p ) τ with τ = 1lim p → Π ( p ) . The first moment of this equation with respect to p , taking (II.3) into account, then immediately leads to J = lim p → e /m Π ( p ) E hence σ DC RTA = lim p → e /m Π ( p ) . For the Fr¨ohlich polaron, with the constant frequency ω k = ω LO and the electron-phonon coupling (IV.1), thecorresponding function Π Fr¨ohlich ( p ) can easily be calculated in closed form:Π Fr¨ohlich ( p ) = 2 αω LO √ m ~ ω LO p ( n B ( ω LO ) + 1) Θ (cid:16) ~ ω LO < p m (cid:17) arccosh p √ m ~ ω LO + n B ( ω LO ) arcsinh p √ m ~ ω LO ! , This simple relaxation time approximation thus immediately gives the Kadanoff conductivity for the Fr¨ohlich polaron: σ DC Kadanoff = lim p → e /m Π Fr¨ohlich ( p ) ≈ e mαω LO e β ~ ω LO . However, the neglect of the integral term in (B.1) is an unwarranted approximation, essentially because it violatesthe particle number conservation. Indeed, consider the first moment of (B.1) with respect to p : e E = Z p Π ( p ) f ( p ) d p − Z Z p Π ( p + ~ k → p ) f ( p + ~ k ) d kdp . By the substitution p + ~ k → p in the last term, interchanging k ↔ − k and using the definition (III.3), the terms inΠ ( p ) cancel against each other, and one is left with eE = − E · Z Z ~ k Π ( p → p + ~ k ) f ( p ) d kdp , which shows that the in-scattering rate can not be neglected.At sufficiently low temperature, the distribution function peaks at ¯p = m J /e which is indeed near p = since ¯p ∝ E → . Replacing f ( p ) by δ ( p − m J /e ) then gives eE = − E · Z ~ k Π (cid:18) m J e → m J e + ~ k (cid:19) d k . (B.2)For the Fr¨ohlich polaron (IV.1), the evaluation of this integral is elementary and results in: eE = mω LO α √ r ~ ω LO m e mJ × ( n B ( ω LO ) + 1) Θ (cid:16) ~ ω LO < mJ e (cid:17) (cid:18) √ mJ √ e q mJ e − ~ ω LO + ~ ω LO arccosh (cid:16) Je √ m √ ~ ω LO (cid:17)(cid:19) + n B ( ω LO ) (cid:18) √ mJ √ e q mJ e + ~ ω LO − ~ ω LO arcsinh (cid:16) Je √ m √ ~ ω LO (cid:17)(cid:19) . (B.3)Keeping linear response in mind, it is obvious that this expression is only needed to first order in J = O ( E ) , suchthat the emission term does not contribute at sufficiently low temperature. The result is eE = 23 mω LO αn B ( ω LO ) Je + O (cid:0) J (cid:1) , (B.4)which is fully consistent with the conductivity (IV.4) derived above. Appendix C: FHIP with distribution function
In this section we present a calculation in the spirit of the FHIP approximation but using our phase space approach.It was shown in [17] how the path integral for the reduced Wigner function leads to the Liouville equation (II.5). Thepath integral for the reduced Wigner function is just the Weyl transform of the path integral for the density matrixused by FHIP. The basic approach in FHIP is to expand the action around Feynman’s linear polaron model, ratherthan around the free particle. In terms of the distribution function this means that f ( p ,t ) = f ( p ,t ) + f ( p ,t ) , (C.1)where f is a variational time dependent Wigner function which can be found by propagating the initial distributionalong a certain, so far free to choose, linear model. Similar as for the linear response at weak coupling (i.e., to firstorder in the deviation from the free particle), we now consider linear response to first order in the deviation from theFeynman polaron model, which means that (cid:18) ∂∂t + e E ·∇ p (cid:19) f ( p ,t ) = g ( p , t ) , (C.2)where g ( p , t ) , apart from the time evolution of f , is the right hand side of (III.1) with f replaced by f : g ( p , t ) = − (cid:18) ∂∂t + e E ( t ) · ∇ (cid:19) f ( p , t )+ X k | γ ( k ) | ~ Z t −∞ f ( p + ~ k , s ) ( n B ( ω k ) + 1) cos (cid:16) ( t − s ) (cid:16) k · p + ~ k m − ω k (cid:17)(cid:17) + n B ( ω k ) cos (cid:16) ( t − s ) (cid:16) k · p + ~ k m + ω k (cid:17)(cid:17) − f ( p , s ) ( n B ( ω k ) + 1) cos (cid:16) ( t − s ) (cid:16) k · p + ~ k m + ω k (cid:17)(cid:17) + n B ( ω k ) cos (cid:16) ( t − s ) (cid:16) k · p + ~ k m − ω k (cid:17)(cid:17) d s. (C.3)The time dependence of the distribution function f follows the classical equation of motion, and consequently f ( p ,t ) = Z t −∞ g (cid:18) p − Z tt ′ e E ( s )d s, t ′ (cid:19) d t ′ . Because of the particle number conservation of the trial distribution, and due to the linearity of the classical equationof motion, the expected current density of the perturbation around the model becomes J ( t ) = em Z p f ( p ,t )d p = em Z t −∞ Z p g ( p , t ′ )d p d t ′ . (C.4)The total current density is consequently given by J ( t ) = J ( t ) + J ( t ) , where J ( t ) is the current density of the model distribution function. In terms of Feynman’s variational parameters w and v, Feynman’s model distribution function reads f ( p ,t ) = (cid:18) β mπ (cid:19) / exp − β m (cid:18) p − w v Z t −∞ e E ( s )d s − v − w v Z t −∞ e E ( s ) cos v ( t − s )d s (cid:19) ! , provided we assume the model to be initially in canonical equilibrium at an effective temperature equal to the realtemperature β − . At this point the present discussion differs from that of FHIP, where the initial state of the modelis assumed to be a product state of the oscillators with the particle. It is argued by FHIP that the product stateansatz is admissible because ”... In the past only the oscillators were in thermal equilibrium at β − . As a result ofthe coupling the system will come very quickly to thermal equilibrium at the same temperature. [6] ” Although thismight be true for the real system, it does not apply to the model. Because of the linearity of the model it will neverthermalize. Consequently, the reduced model distribution function will endlessly oscillate even in the absence of anelectric field. In contrast, the present model distribution is the exact stationary distribution of the reduced Liouvilleequation in the absence of an electric field [17]. It should however also be noted that, as a consequence of the samelinearity, the expected model current density J ( t ) = w v Z t −∞ e E ( s ) m d s + v − w v Z t −∞ e E ( s ) m cos v ( t − s )d s, is not affected by the change in initial state, in contrast to the correction J ( t ). From the definition (II.4) ofthe conductivity, we furthermore find the following expression for the Laplace transform L ( σ , Ω) of the modelconductivity L ( σ , Ω) = e m (cid:18) w v
1Ω + v − w v Ω v + Ω (cid:19) . The first order correction J ( t ) consists of two parts, one that scales with the coupling constant and one that doesnot. The latter one is given by J , ( t ) = − Z t −∞ Z e p (cid:18) ∂∂t + e E ( t ) ·∇ p (cid:19) f ( p ,t ′ )d p d t ′ . = v − w v (cid:20)Z t −∞ e E ( s ) m d s − Z t −∞ e E ( s ) m cos v ( t − s )d s (cid:21) . The coupling dependent part leads to J , ( t ) = em Z t −∞ d t ′ Z t ′ −∞ d s X k | γ ( k ) | ~ k Z f ( p , s ) ( n B ( ω k ) + 1) cos (cid:16) ( t ′ − s ) (cid:16) k · p m + ~ k m + ω k (cid:17)(cid:17) + n B ( ω k ) cos (cid:16) ( t ′ − s ) (cid:16) k · p m + ~ k m − ω k (cid:17)(cid:17) d p , which within linear response, hence up to O ( E ) , simplifies to J , ( t ) = − Z t −∞ d t ′ Z t ′ −∞ d sχ β ( t ′ − s ) J ( s ) , with χ β ( t ) = t X k | γ ( k ) | ~ k m ( n B ( ω k ) + 1) sin (cid:16) t (cid:16) ~ k m + ω k (cid:17)(cid:17) + n B ( ω k ) sin (cid:16) t (cid:16) ~ k m − ω k (cid:17)(cid:17) exp (cid:18) − k mβ t (cid:19) . Note that lim β →∞ χ β ( t ) = χ ( t ) , where χ ( t ) is the memory function obtained by truncating the equation of motion forthe current density, as explained in appendix A. Consequently the low temperature, linear response, current densityup to first order around the Feynman polaron model is J ( t ) = Z t −∞ e E ( s ) m d s − Z t −∞ d t ′ Z t ′ −∞ d sχ ( t ′ − s ) J ( s ) . Hence the Laplace transform L ( σ, Ω) of the conductivity reads L ( σ, Ω) = L ( σ , Ω) + L ( σ , Ω) , where the correction to the model conductivity σ is given by L ( σ , Ω) = e m v − w Ω ( v + Ω ) − L ( σ , Ω) L ( χ, Ω)Ω . A more accurate conductivity can be found using the standard resummation argument L ( σ, Ω) = L ( σ , Ω) (cid:18) L ( σ , Ω) L ( σ , Ω) (cid:19) ≈ L ( σ , Ω)1 − L ( σ , Ω) L ( σ , Ω) , σ DC = lim Ω → L ( σ, Ω) = w v e αmω LO n B ( ω LO ) . Moreover, since v /w = m ∗ /m [6] we have σ DC = 3 e αm ∗ ω LO n B ( ω LO ) ≈ e αm ∗ ω LO e β ~ ω LO , consistent with our result (IV.4). ACKNOWLEDGMENTS
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