Landau-Silin Kinetic Equation in the Theory. of Normal Fermi Liquid

Transcription

1 Applied Mathematical Scieces, Vol. 8, 014, o. 107, HIKARI Ltd, Ladau-Sili Kietic Equatio i the Theory of Normal Fermi Liquid V.A. Daileko Departmet of Nuclear Physics, St. Petersburg State Uiversity, Neva River Embakmet 7/9, St. Petersburg , Russia K.A. Gridev Departmet of Nuclear Physics, St. Petersburg State Uiversity, Neva River Embakmet 7/9, St. Petersburg , Russia A.S. Kodratyev Departmet of Physics, Herze State Pedagogical Uiversity of Russia, Moika River Embakmet 48, St. Petersburg , Russia Copyright 014 V.A.Daileko, K.A. Gridev ad A.S. Kodratyev. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract I the frame of the Kadaoff-Baym versio of the Gree s fuctio method i quatum statistical mechaics it is proved that the Ladau-Sili kietic equatio for a ormal quatum Fermi liquid is valid i the case whe the widths of oeparticle eergy levels are take ito accout exactly. Mathematics Subject Classificatio: 8D0 Keywords: quatum liquid, kietic equatio, quatum Gree s fuctio, spectral fuctio, quasiparticle

2 5338 V.A.Daileko, K.A. Gridev ad A.S. Kodratyev 1 Itroductio The problem of derivig quatum kietic equatios durig decades determied oe of the mai directios of may-body system theory s developmet. Special attetio was attracted to trasport pheomea i quatum liquids both i the frame of pheomeological ad microscopic theories. Ladau-Sili (LS) kietic equatio i the theory of ormal Fermi liquid proved to be a effective tool for describig o-equilibrium properties of eutral ad charged Fermi liquids [1]. Ladau offered a kietic equatio for a eutral quasiparticle distributio fuctio i Wiger s semiclassical approach which may be applied to a macroscopic perturbatio, the scale of which is large compared to the atomic scale. Sili has first show that the equatio may be also applied to the systems of charged particles provided the difficulties associated with the log rage of the Coulomb iteractio betwee the particles are removed if we allow for the dyamic screeig of the particle motio i self-cosistet fashio. The experimetal discovery of the pheomea, theoretically predicted o the basis of the LS theory (zero soud i He-3 ad spi waves i o-ferromagetic metals) ad the excellet agreemet of the experimetal data with the theoretical predictios made this pheomeological theory a subject of ivestigatio o the basis of strict fudametal may-body theory. The most coveiet ad effective approach to the problem is based o the realtime Gree s fuctio formalism of Marti ad Schwiger, further developed by Kadaoff ad Baym (KB) [6]. It s likely to state that the KB approach developed for equilibrium ad o-equilibrium problems ad for zero ad fiite temperatures is capable of describig both the statistics ad dyamics of the may-body systems i a comprehesive way. A brief review of umerous various developmets of the KB method ad its applicatios to differet may-body systems may be foud i []. I this paper we will cocetrate o the problem of rigorous derivatio of the LS kietic equatio. The first microscopic derivatio of the LS equatio was preseted i [6]. The keystoe of the derivatio was a smalless of the widths of oe-particle eergy states ear the Fermi level. This approximatio ivolves a assumptio about the cotiuity of the self-eergy fuctio at the Fermi level. The cotiuity ca be proved i all orders of perturbatio theory, but it is ot ecessarily true for situatios, such as the superfluid ad supercoductig states, i which perturbatio theory is ot valid. Therefore, the preseted i [6] derivatio of the

3 Ladau-Sili kietic equatio 5339 LS kietic equatio applies oly to the so-called ormal fermio systems at absolute zero temperature. Ufortuately, the assumptios are ot valid for real fermio systems. We will show that these assumptios are oly sufficiet, but ot ecessary coditios for the validity of the LS equatio, which ca be proved for much more soft coditios. Experimetal discovery of a superfluidity of He-3 at low, but fiite temperature, iitiated a wave of attempts to derive a kietic or trasport equatios for a ormal Fermi liquid that would take ito accout fiite widths of oeparticle eergy levels ad would be valid at fiite temperature [,7-10,14]. Mai difficulty i derivig the LS kietic equatio was associated with the ecessity of a mathematically lawful elimiatio of secod Poisso bracket i the right side of the KB geeralized quatum kietic equatio (see below). The failure of the attempts was determied by the usage of wrog approximatios for the spectral fuctios which did t satisfy the KB equatio for the spectral fuctio i the case of slightly o-equilibrium systems. The approximatio for the spectral fuctio, which satisfies the KB equatio, was offered i [] o the basis of certai relatios i the theory of the Fourier trasform, but the problem of the covergece of the appearig expasio was ot cosidered. Thus, the validity of the LS kietic equatio was proved i [] oly with the precisio up to the liear terms i the widths of oe-particle eergy levels. The covergece of the expasio of the spectral fuctio was cosidered i [5], what made possible the rigorous derivatio of the LS equatio, exactly takig ito accout the widths of eergy levels. We will show that the LS kietic equatio i the case of slowly varyig i space ad time disturbaces takes ito accout the widths of eergy levels exactly. The followig sectio cotais a short presetatio of the ecessary formulas of KB theory which will be used for the aalysis of the expasio for the spectral fuctio ad for the derivatio of the LS equatio. I Sec. 3 a discussio ad summary are preseted. Spectral fuctio ad kietic equatio i Kadaoff-Baym formalism The KB formalism leads to the followig geeral expressio for the oe-particle spectral fuctio a ( p, ) of a system i equilibrium [6]:

4 5340 V.A.Daileko, K.A. Gridev ad A.S. Kodratyev a ( ) ), (1) ( e( )) ( ) / 4 ( where HF e( ) E ( p) Re ( ), () c ad E HF ( p) is a oe-particle eergy i the Hartree-Fock approximatio. Real ad imagiary ( ( p, ) ) parts of the correlatio self-eergy fuctio ( ) are related to each other through the Hilbert trasform: Re c ( ) P d ( p, ) Here P refers to a pricipal value itegratio. I the case of slowly varyig i space ad time disturbaces, after the trasitio to the Wiger coordiates c (3) R ( r t t t 1 r1 ) /, r r1 r1, T ( t1 t1) /, 1 1 (4) ad the performace of the Fourier trasform with respect to r ad t, all the quatities eterig the theory are cosidered to be the fuctios of p,, R, T. For example, a a( p ;. The fuctios HF E ad Re c iclude the iteractio with the exteral field. If we take ito accout oly the first derivatives with respect to slowly varyig quatities R, ad T i the KB equatios for the correlatio fuctios, we come to the followig equatio for the spectral fuctio a( p ; : e( p;, a( p; Reg( p;, ( p; 0, (5) ad to the geeralized KB kietic equatio for the correlatio fuctio g ( p ;

5 Ladau-Sili kietic equatio 5341 e( p;, g ( p; Reg( p;, ( p; g g ( p; (6) Here A, B is the geeralized Poisso bracket defied by the expressio: A B A B B A A, B A B T T p R R p, (7) The exact solutio of Eq. (5) is give by the expressio [6]: g( p z; HF z E ( p; Re ( p z; 1 c (8) I fact, the solutio (8) leads to almost the same evaluatio of the spectral fuctio a( p ; as i the equilibrium state: ( p; a( p; (9) ( e( p; ) ( p; / 4 Eq. (6) provides a exact descriptio of the respose to slowly varyig disturbace. All the quatities appearig i this equatio may be expressed i terms of correlatio ad self-eergy fuctios. The result (9) meas that i the case of slowly varyig disturbaces the approximatios for the o-equilibrium spectral fuctio may be writte i the same form as i the equilibrium case. I order to simplify the otatio, we will omit space ad time variables. I the KB formalism the quasiparticle eergy E E( p) is defied as a solutio of the equatio [6]: E ( p) e( ) E( p ) (10) Now we expad e( p, ) as a fuctio of i Taylor series ear the value E( p) ad save oly liear terms: e( ) e( ) E( p) E( p ) ( E( p)) (11)

6 534 V.A.Daileko, K.A. Gridev ad A.S. Kodratyev We substitute (11) ito Eq. (1) ad get the formula for a QP called the spectral fuctio of the quasiparticle state: a QP Z ( E( p)) ) (1) ( E( p)) Z ( E( p)) / 4 ( where Z Z( p) is a reormalizig factor defied by the expressio Z 1 e( ) 1 E( p ) It is easy to prove that Z 1 for all values of p whe correlatio eergy c is take ito accout. The quatity (13) Z ca be cosidered the width of the quasiparticle eergy level, ad Eq. (1) for the spectral fuctio may be rewritte as a QP ) Z ( E( p)) ( / 4 (14) The spectral fuctio (1) or (9) may be preseted i the form of the expasio i power series of the width of eergy levels with a help of the followig relatio []: dt exp( t ) exp( itx), 0 (15) x / 4 The expasio of the first expoet i the left side of (15) i Taylor series i powers of with the subsequet term-by-term itegratio with a help of the formulas for the Fourier trasforms of powers of t allows to geeralize the results of [6] ad to get the expasio of the spectral fuctio a ( p, ) i powers of. These formulas may be preseted i the form [4]: t exp( itx) () dt ( i) ( x), 0, 1,,..., (16)

7 Ladau-Sili kietic equatio 5343 t 1 1 exp( itx) dt si x 1 1!, 0,1,,... (17) The terms of the expasio cotaiig the eve powers of the eergy level s width ca be cosidered o the basis of the formula [1,13] / x ( x) ( x) (18) It is easy to prove by mathematical iductio that ( ) ( 1)! ( x) ( x) (19) x I correspodece with (16) ad (19) the terms with eve powers of geerate a geometric sequece, ad the sum of the sequece is equal to [5] ( e( )) ) ( (, )) e p (0) ( e( )) ( ) / 4 a eve ( Due to the presece of the Dirac delta fuctio i the umerator it is clear that the cotributio of the sum (0) to all expressios cotaiig itegratio with respect to frequecy variable equals zero. The terms with odd powers geerate a geometric sequece with the same commo ratio, ad its sum equals to the complete expressio (1) or (9) for the spectral fuctio. Takig ito accout the remark after (0), it is coveiet to write it i the form of the expasio a ) ( e) ( e) 3 4( e) 5 16( e) ( (1) The expasio for the real part of the Gree fuctio Re g with a help of (8) ca be preseted i the form 4 Re e 1 g ( ) 3 ( e) / 4 e 4( e) 16( e) 5... ()

8 5344 V.A.Daileko, K.A. Gridev ad A.S. Kodratyev These statemets geeralize the results obtaied i [] ad ope a way for the rigorous derivig the LS equatio. Mai problems here are associated with the ecessity of the elimiatio of the secod Poisso bracket i the left side of Eq. (6) i a mathematically lawful way. The reaso of the failure of the attempts of such elimiatio, for example i [14], was the usage of a improper approximatio to the spectral fuctio. A special ame puzzlig term for the secod Poisso brackets i Eqs. (5) ad (6) was offered i [14] after such useless attempts. The proof of the validity of the LS kietic equatio may be produced i the followig way. The first term i the expasio (1) for the spectral fuctio after the substitutio to the first Poisso bracket i Eq. (7) leads to the LS kietic equatio after itegratio with respect to frequecy [6]. This equatio may be writte i the form: T E E I collisio, (3) p R R p where is the quasiparticle distributio fuctio defied by the expressio: ( p; RT ) f ( p ; (4) E( p; RT ) Due to the property (0) which elimiates the extra terms with eve powers of i the expasio for the spectral fuctio, we see that there appear the couples of the terms from the expasios (1) ad () which, beig substituted to the first ad secod Poisso brackets correspodigly i the left sides of Eqs. (5) ad (6), cacel each other. Ideed, the correspodig couples of the terms are: 4 ( e) 1 ( 1) (5) i the expasio (1), ad 4 ( e) ( 1) 1 (6) i the expasio (). These terms make the followig cotributios to the metioed Poisso brackets:

9 Ladau-Sili kietic equatio 5345 ad 1 (7) 4 e, a ( ) ( 1) e, 1 4 Reg, ( ) ( 1) e, (8) Thus, we see that oly the first term i the expasio (1) for the spectral fuctio makes the cotributio to the left side of the kietic equatio (3). 3 Coclusios Ituitive cosideratios demad that i case of slowly varyig i space ad time disturbaces the kietic equatio for the distributio fuctio should have the form (3), whatever complex the eergies E(R,T) could be. But the questio arises, if a strog iteractio betwee the particles of a system may be take ito accout i terms of such oe-particle eergies. We have proved that i the cases, whe dyamic treatmet of a system uder cosideratio is possible, the kietic equatio for the quasiparticle distributio fuctio always has a form of LS equatio provided there is oe-to-oe correspodece betwee the eergies of quasiparticles ad oe-particle eergy states i the system. The widths of the oeparticle eergy levels do ot ifluece the form of the left side of kietic equatio, that describes the kietic effects of the potetial, i.e., how the potetial chages the eergy-mometum relatio from that of free particles to the more complex spectrum. The right side of the kietic equatio cotais collisio itegrals that describe the dyamical effect of collisios. The form of these itegrals may be affected by the widths of the eergy levels. Thus, the geeral picture of the behavior of a arbitrary fermio system ca be the followig. At zero or very close to zero temperature a system ca reveal superfluid or supercoductig behavior. I this situatio there is o oe-to-oe correspodece betwee oe-particle eergy levels ad complex quasiparticles (for example, Cooper pairs) ad the LS equatio ca t be used for the descriptio of trasport pheomea. As the temperature arises, the complex quasiparticle

10 5346 V.A.Daileko, K.A. Gridev ad A.S. Kodratyev states are destroyed, ad oe-particle eergy spectrum is restored. I this situatio the usage of the LS kietic equatio becomes lawful. The LS kietic equatio allows to aalyze the collective excitatio spectrum of a solid ad of a uclear matter, cosiderig that it is, i a good approximatio, a collectio of strogly iteractig electros ad ucleos correspodigly waderig solo i all directios. Whe discussig uclear collective motio, oe is usually makig aalogies to molecules ad their collective modes [3,11].. The situatio i case of molecules is govered by the adiabatic approximatio by Bor ad Oppeheimer which is justified due to differet time scales of uclear ad electroic motios. But i uclei there is o essetial differece betwee the masses of protos ad eutros, ad the choice of proper uclear collective coordiates is a log-stadig problem. The same aalogy ca be traced betwee the collective modes i uclear matter ad i solids with the similar situatio about the masses of their compoets. The existece of differet time scales i the adiabatic approximatio is a special case of the geeral Bogolubov s priciple of time scale hierarchy, which ca be determied ot oly be the iitial large differece of masses which permits separatig the system of coordiates ito slow (relevat) ad fast (irrelevat) oes. The time scale hierarchy may be caused by a strog iteractio betwee particles which reveals by the appearace of large effective masses. The questio that appears here, is whether this strog iteractio would t destroy the validity of kietic equatio used for the descriptio of the collective motio. The obtaied result for the LS equatio for the quasiparticle distributio fuctio makes lawful the followig cosideratios. The time T of the quasiparticle eergy formatio ca be estimated as ħ /EF, where EF is a Fermi eergy [1]. For a uclear system this time is of a order 10 - sec. ad it is the time separatio betwee sigle-particle ad collective motio. Thus, it opes the possibility for a rigorous explaatio of the validity of the adiabatic approximatio for the descriptio of the collective excitatio spectrum i uclear matter as it is accepted i literature [11]. This problem will be aalyzed i a separate paper. Refereces [1] G.B. Arfke ad H.J. Weber, Mathematical Methods for Physisists, Academic Press, Elsevier, 005.

11 Ladau-Sili kietic equatio 5347 [] M. Arshad, A.S. Kodratyev ad I. Siddique, Spectral fuctio ad kietic equatio for a ormal Fermi liquid, Physical Review B 76 (007), [3] A. Bohr ad B.R. Mottelso, Nuclear Structure, Bejami, New York, 1975, Vol. II. [4] Yu. A. Brychkov ad A.P. Prudikov, Itegral Trasform of Geeralized Fuctios, Gordo ad Breach, New York, [5] V.A. Daileko, K.A. Gridev ad A.S. Kodratyev, Spectral fuctios ad properties of uclear matter, Iteratioal Joural of Statistical Mechaics, Volume 013 (013), ID , [6] L.P. Kadaoff ad G. Baym, Quatum Statistical Mechaics, Bejami, New York, 196; Perseus Books, Cambridge, MA, [7] H.S. Kohler, Beyod the quasi-particle picture i uclear matter calculatios usig Gree s fuctio techiques, Joural of Physics: Coferece Series 35 (006), [8] H.S. Kohler ad K. Morewetz, Correlatios i may-body systems with two-time Gree s fuctios, Physical Review C 64 (001), [9] P. Lipavsky, V. Spicka ad B. Velicky, Geeralized Kadaoff-Baym asatz for derivig quatum trasport equatios, Physical Review B 34 (1986), [10] K. Morawetz ad G. Roepke, Memory effects ad virial correctios i oequilibrium dese systems, Physical Review E 51 (1995), [11] W. Nazarewicz, Diabaticity of uclear motio: problems ad perspectives, Nuclear Physics, A557, (1993), [1] P. Nozieres ad D. Pies, The Theory of Quatum Liquids, Perseus Books, Cambridge, MA, [13] L.I. Schiff, Quatum Mechaics, Mc Graw-Hill Book Compay, Ic. New York-Toroto-Lodo, [14] V. Spicka ad P. Lipavsky, Quasiparticle Boltzma equatio i semicoductors, Physical Review, B 5 (1995), Received: July 9, 014