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A semiclassical approach in the linear response theory
M. Combescure, D. Robert
To cite this version:
A semiclassical Approach in the Linear Response Theory
M. Combescure andD. Robert
Abstract. We consider a quantum system of non-interacting fermions at temperature T,in the frameworkof linear response theory. We show that semiclassicaltheory is an appropriate framework to describe some of their thermodynamicproperties,inparticularthroughasymptoticexpansionsin~ (Planckconstant)ofthedynamicalsusceptibilities. Weshowhowtheorbits oftheclassicalmotioninphasespacemanifestthemselvesintheseexpansions, intheregimewhereTisoftheorderof~.
Considerasystemofnon-interactingfermionscon nedbyanexternal po-tentialandin contactwithanexteriorreservoirat temperatureT. Assumethat a time-varying externalperturbationdrivesthesystemoutto,but nearof, its equi-libriumstate. Theresponseofthisquantumsystemtoanexternaltime-dependent perturbation is asubjectof high physical interest, which canbe investigated ex-perimentally, in particular the so-called \dynamical susceptibility". A complete rigorousanalysis ofthis problemis stilllacking, although recent progressis being madeintheunderstandingofnon-equilibriumstatisticalmechanism,andofitslink withtheunderlyingchaoticdynamics[11,12, 18].
A semi-empiricalroute whichhas beenproposed (seeclassicaltextbooks [14, 15])consists,forsmallperturbation,ofinvestigatingtheresponsefunction\to rst order oftheperturbation",i.e. theso-called\linear response theory". This semi-empirical routehas beengiven a rmer foundation (see thebook byBratelli and Robinson[4])wherealinkwiththeKMSconditionisestablished. (Seealsorecent progressin [18]).
In this paper we rederive, formally, the rst order response function for the quantum fermionic system under study, i.e. the so-called\generalized Kubo for-mula"(seealso[2]andinvestigatesemiclassicalexpansionsofit,assumingsuitable \chaoticity assumptions" on the one-body underlying classical dynamics. These semiclassicalexpansionsaredevelopedinasimilarspiritaspreviousstudiesonthe \semiclassical magnetic response for non-interacting electrons" [1, 5, 7, 10, 13, 16,17]i.e. weexhibitalowtemperatureregimewheretheclosedclassicalorbitsof one-particlemotion manifest themselvesasoscillating correctionsto the response function.
Weshallpresentherethephysicalobjectsunderstudy,andgivesome assump-tionsunderwhich amathematicaltreatmentcanbeapplied.
Consider a system of non-interacting fermions living in IR n
, each fermion being
c
subjecttoaone-bodyHamiltonian b
H whichistheWeylquantizationofaclassical HamiltonianH(q;p) oftheform
(0.1) H(q;p)= p 2 2m +V(q) withV 2C 1 (IR n )suchthat (0.2) V(q)c 0 (1+q 2 ) s=2 s;c 0 >0: Undertheseassumptions,
b
H isself-adjointinL 2
(IR n
)anditsspectrumispure point,andcontainedin[0;1). Assumeoursystemisincontactwithareservoirat temperatureT,andchemicalpotential, andde ne theFermi-Diracfunction, as usual: (0.3) f(x)= 1+e (x ) 1
where = 1=kT (k being the Boltzmann constant). Then the total nomber of fermionsin thesystemisrelatedtoT andby:
(0.4) N =f(
b H)
where thetraceistakenin theoperatorsense inH=L 2
(IR n
),andthetrace-one operator: (0.5) c eq =N 1 f( b H)
isthe"equilibriumstate"forthefermioncsystemunderconsideration.
Assume that this system, being in the far pastprepared in the state(0.5) is submittedadiabaticallytotheone-bodyperturbedHamiltonian
(0.6) b H(t)= b H+ b A F(t) where b Aisself-adjointinL 2 (IR n
)(beingtheWeylquantizationofasymbolA(q;p) thatweshallmakepreciselater),andF isanycontinuousfunctionthatequalsone onthepositiverealaxis,andisintegrableonthenegativerealaxis,forexample:
(0.7) F(t)=e t
t<0 F(t)=1 t0:
In the \linear response theory" we try to solve the following problem : nd a \density matrix" c
eq
(t) (i.e. atrace oneoperator in L 2
(IR n
)) which, to the rst orderin theperturbationsolves
(0.8) i~ @b @t = h b H(t);b i
with\initialcondition"att= 1being
(0.9) lim t! 1 c eq (t)=c eq :
LetV (t;t )betheunitaryevolutionoperatorinducedby b
(0.10) i~ dV (t;t 0 ) dt = b H(t)V (t;t 0 ) withV (t 0 ;t 0 )=1),andlet (0.11) U(t):=e it b H=~ :
Weknowusing[19]that V
(t;t 0
)solutionof(0.10)exists underthefollowing: Assumption 1: A(q)is amultiplicativefunction dominated byC(1+q
2
)in ab-solutevalue.
Wecancheckthat :
(0.12) (t;b t 0 ;)=V(t;t 0 )b eq V(t 0 ;t) solves(0.12)with(tb 0 )=b eq . Moreover(t;b t 0 ;)hasalimitast 0
! 1,inthetrace-classoperatornormsense, whichiscalledb
eq (t;)
ThenusingDuhamel'sformula,wecanwrite:
(0.13) b eq (t;) b eq = i~ Z t 1 dt 0 F(t 0 )[b eq ; b A t t 0 ]+o() b A t
beingbyde nition theHeisenbergobservableat timet (fortheevolution gov-ernedby b H) (0.14) b A t =U(t) b AU(t) :
Equation (0.13) is the linear response formula in this framework. It implies thatif
b
B issomeself-adjointoperatorthatwewanttomeasureinthe\stationary" stateb
eq
(t),the rstorder contributioninas!0totheresult:
(0.15) J(t)=Tr n b B(b eq (t;) b eq ) o isoftheform: (0.16) J L (t)= Z t 1 dt 0 F(t 0 )(t t 0 ) where (t) = 1 i~ Tr b B h b eq ; b A t i (0.17) = 1 i~ Tr b eq h b A; b B t i (0.18)
usingthecyclicityof thetrace.
(0.19) A;B (!)= Z +1 1 (t)e i!t dt
Ouraimistostudythesemiclassicalbehaviourof A;B indistributionalsense asT !0. Setting: (0.20) = ~ and (0.21) f (x)=(1+e x ) 1
andgivenanyrealg intheSchwartzclasswehaveformally:
(0.22) Z A;B (!)g(!)d!= 1 i~ Z +1 1 Tr f b H ~ ! h b A ; c B t i ! e g(t)dt
In order to avoid the singularity in x = 0 of f f (x), ( f f
being the Fourier Transformoff )wereplacef by: (0.23) f ; =f withe2C 1 0
(IR)andthereforestudyacorresponding"regularizedsusceptibility", orinotherwordsadistribution
A;B
(s;!)intwovariablessand!inD 0
(IR 2
)such that(0.22) isreplacedby:
Z Z A;B (s;!)e(s)g(!)dsd!= (0.24) 1 2i~ Z Z Tr e is( b H )=~ h b A; c B t i f f (s)e(s) eg(t)dsdt Let t
be the classical ow induced by Hamiltonian (0.1). Consider
the energysurfaceconservedbythe ow:
(0.25) = (q;p)2IR 2n :H(q;p)= We call d
the Liouville measure on
, sothat thecorrelation of classical ob-servablesAandB on isde nedby: (0.26) C A;B (t)= Z A:B t d where B t (z) = B[ t
(z)]. Moreover if is any periodic orbit on
, and the corresponding primitive orbit, with period T
, we introduce the correlation function (0.27) c (t)= Z T 0 A s (q;p)B s+t (q;p)ds (q;p)2
(0.28) c (t)= k =+1 X k = 1 c ;k e 2ik t=T
To each is associatedacorresponding"linearized Poincaremap" called P
, aclassicalactionalong called S
,and aMaslovindex
(see[6]). Letusassumethat
t on
satis esthe: Gutzwiller Assumption:
Assumption2Theperiodicorbits arenon-degenerate,i.ethePoincaremaps donothave1aseigenvalue(which impliesthat theyareisolated).
MoreoverB hasto obey:
Assumption3 j@ q @ p B(q;p)jC j j+j j2 j@ q VjC j j2 Ourresultisasfollows:
Theorem 0.1. Under Assumptions 1,2,3, we have, in distributional sense inD 0 (IR 2 ): A;B (s;!) i~ n Æ 0 (s) ] C 0 A;B (!)+ X j1 ~ j n j (s;!) + X :T 6=0 e i(S =~+ =2) ~sinh(T =)jdet(1 P )j 1=2 0 @ Æ T (s) X k c ;k Æ(! 2k T )+ X j1 ~ j j; (s;!) 1 A where j and j; are distributions in D 0 (IR 2
) such that Supp( j ) f0gIR , Supp( j; )fT gIR .
Proof: Thecomplete proofis givenin [8]. Theideaisto developthetraceas anintegralovercoherentstatesinH , andto obtaintheasymptoticexpansionsin ~bystationaryphasetheoremsasin[6].
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LPT B^
at. 210
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eParis-SudF-91405ORSAYFRANCE
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E-mailaddress: [email protected] 3.fr D epartementde Math ematiques Universit edeNantes, 2rue dela Houssini ere F-44322NANTESFRANCE
