Physics 219 Summary of linear response theory

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1 1 Physics 219 Suary of liear respose theory I. INTRODUCTION We apply a sall perturbatio of stregth f(t) which is switched o gradually ( adiabatically ) fro t =, i.e. the aplitude of the perturbatio grows as e ɛt for ɛ 0 +. We assue that the perturbatio couples to soe operator of the syste B, so the additioal Hailtoia due to the perturbatio is H = e ɛt f(t)b. (1) For exaple, f(t) ight be a tie depedet agetic field ad B the total agetizatio of the syste. We wish to copute the respose of soe operator of the syste, A say, to the perturbatio. (I ay cases A will be the sae as B but this is ot always the case.) We shall just cosider the liear respose i.e. the respose to first order i the perturbatio. We will show i class, see also Refs. [1 3], that where A(t) = t χ AB (t t )e ɛt f(t ) dt, (2) i χ AB (t t ) = h [A(t), B(t )] (t > 0), 0 (t < 0). Sice we defie χ AB (t) = 0 for t < 0 we ca replace the upper liit i Eq. (2) by. The fact that there is o cotributio i Eq. (2) for t > t reflects causality, i.e. the perturbatio at tie t caot effect the syste at a earlier tie t. We deote (equilibriu) averages i the absece of the pertubatio by agular brackets, ad averages i the presece of the perturbatio by a overbar. It will also be coveiet to defie a fuctio φ AB (t) to be give by φ AB (t) = ī h [A(t), B(t )], (4) for all tie. Frequetly the perturbatio is at a frequecy ω, i.e. f(t) = f ω e iωt, i which case the syste respods at the sae frequecy (at least to first order i the perturbatio) ad we have A ω = χ AB (ω + iɛ)f ω, (5) where the coplex Fourier trasfor is defied by χ AB (z) = 0 (3) χ AB (t)e izt dt, (I(z) > 0), (6)

2 2 i which the coditio o I(z) is eeded to esure covergece of the itegral at. We defie real ad iagiary parts of the physical (i.e. z = ω + iɛ) respose fuctio by χ (ω) ad χ (ω) respectively, i.e. χ AB (ω + iɛ) = χ AB(ω) + iχ AB(ω). (7) If A ad B are Heritia (which they will be) the oe ca show that χ AB (t) (ad hece also φ AB (t)) is real. Furtherore, if A ad B have the sae sigature uder tie reversal (discussed i class), which will usually be the case, the φ AB (t) is a odd fuctio of tie. Fro ow o we will assue that A ad B are Heritia ad have the sae sigature uder tie reversal. Except where ecessary, we will also oit the subscript AB o the liear respose fuctios. It is coveiet to defie a real frequecy Fourier trasfor of φ(t) i the usual way, φ(ω) = φ(t)e iωt dt. (8) Coparig Eq. (8) with Eq. (6), ad otig that φ(t) ad χ(t) are the sae for t > 0 while χ( t) = 0, φ( t) = φ(t) for t < 0 we have φ(ω) = 2iχ (ω). (9) Fourier trasforig back to tie, ad usig Eq. (4) we see that χ (t) = 1 2 h [A(t), B(t )]. (10) II. ANALYTIC PROPERTIES OF THE LINEAR RESPONSE FUNCTION It is coveiet to also defie a liear respose fuctio χ(z) also for z i the lower half plae. It the turs out that χ(z) is a aalytic fuctio of z everywhere i the coplex plae, except alog the real axis. This ca be see fro the spectral represetatio χ(z) = 1 2π J(ω ) dω ω z where J(ω) is called the spectral fuctio ad is usually real. Notig that, (11) 1 ω ω iɛ = P 1 ω ω + iπδ(ω ω), (12) where P deotes the pricipal part, we see fro Eq. (11) that usually J(ω) = 2χ (ω), (= iφ(ω)), (13) so the spectral fuctio is just (twice) the iagiary part of the respose fuctio. The last equality i Eq. (13) coes fro Eq. (9). The sae arguet shows that usually χ(ω iɛ) = χ (ω) iχ (ω), (14) so χ(z) has a brachcut alog the real axis everywhere where χ (ω) is o-zero, ad the size of the discotiuity is 2iχ (ω).

3 3 C ω FIG. 1: Cotour i the coplex frequecy (z) plae used to derive the Kraers-Kroig relatios. The radius of the outer seicircle teds to ifiity. The radius of the sall seicircle about z = ω teds to zero. III. KRAMERS-KRÖNIG RELATIONS I the previous sectio we showed that the liear respose fuctio χ(z) is a aalytic fuctio of coplex frequecy z i the upper-half plae because of causality. We ow use this property to derive iportat relatioships betwee the real ad iagiary parts of χ(ω + iη) = χ (ω) + iχ (ω), due to Kraers ad Kroig. We evaluate χ(z) dz, (15) C z ω over the cotour show i Fig. (1). Because χ(z) is aalytic iside the regio of itegratio ad the pole at z = ω is also excluded, the itegral is 0. The cotributio fro the seicircle at ifiity vaishes if χ(z) 0 for z.[4] The itegral alog the real axis is a pricipal value itegral, ad the cotributio fro the seicircle is iπ ties the residue at z = ω, i.e. iπχ(ω + iη) = iπ[χ (ω) + iχ (ω)]. Hece we have P χ (ω ) + iχ (ω ) ω ω dω iπ [χ (ω) + iχ (ω)] = 0. (16) Equatio real ad iagiary parts gives the desired Kraers-Kroig relatios: χ (ω) = 1 π P χ (ω) = 1 π P χ (ω ) ω ω dω, (17) χ (ω ) ω ω dω. (18) IV. FLUCTUATION DISSIPATION THEOREM Let us deote the correlatio fuctio betwee A(t) ad B(t ) by S AB (t t ), i.e. S AB (t t ) = A(t)B(t ). (19)

4 4 O physical grouds it is clear that S AB oly depeds o the tie differece t t ad it is easy to show this usig the cyclic ivariace of the trace. Fro Eq. (4) we see that the liear respose fuctios are related to the correlatio fuctios by Fourier trasforig ad usig Eq. (9) gives φ AB (t) = ī h [S AB(t) S BA ( t)]. (20) χ AB(ω) = 1 2 h [S AB(ω) S BA (ω)]. (21) The two correlatio fuctios o the RHS of Eq. (21) are actually closely related. To see this let us write our the expressio for S AB (t) i ters of eigestates of the syste, so Siilarly S AB (t) = 1 S AB (ω) = 2π h e βe A B exp[i(e E )t/ h], (22) S BA ( ω) = 2π h Iterchagig with i the last expressio gives S BA ( ω) = 2π h e βe A B δ(e E + hω). (23) e βe B A δ(e E hω). (24) e β(e E) e βe B A δ(e E + hω) (25), Substitutig ito Eq. (21) gives = e β hω S AB (ω). (26) χ AB(ω) = 1 ( ) 1 e β hω S AB (ω) (FDT), (27) 2 h which is kow as the fluctuatio-dissipatio theore. (The word dissipatio is used because χ is the dissaptive part of the respose fuctio, i.e. the rate of eergy dissipatio is proportioal to χ.) Note that the classical liit correspods to β hω 1, i which case the fluctuatio dissipatio theore becoes φ AB (ω) = iβωs AB (ω), (28) where we used Eq. (9). Fourier trasforig back to tie, ad reeberig that φ(t) = χ(t) for t > 0 we have k B T χ AB (t) = S AB(t), (classical FDT), (29) t

5 5 where we used the fact that the Fourier trasfor of S(t)/ t is iωs(ω). If we itegrate Eq. (29) fro t = 0 to we get k B T χ AB (ω = 0) = S AB (t = 0) S AB (t = ). (30) For t. the syste will have lost the eory of its iitial state ad so we expect li t A(t)B(0) = A B. Hece Eq. (30) becoes k B T χ AB (ω = 0) = AB A B, (31) which is a special case of the classical FDT (et already i class) which is easily derived fro equilibriu classical statistical echaics. V. CONCLUSION I ay body theory, the approach take is to calculate the liear respose fuctio (ofte called a Gree s fuctio ad defied with a overall ius sig) ad the obtai the correlatio fuctios fro the fluctuatio-dissipatio theore. [1] M. Plischke ad B. Bergese Equilibriu Statistical Mechaics [2] W. Marshall ad S. W. Lovesey Theory of Theral Neutro Scatterig, Appedix B, Oxford Uiversity Press. [3] ubarev, Sov. Phys. Uspekhi, 3, 320 (1960) [4] If χ(z) cost. for z the χ(z) is replaced by χ(z) χ( ) i the itegrad i Eq. (15) ad i the subsequet results.