metal – spin-density-wave transition in a quasi-one-dimensional...

PHYSICAL REVIEW B VOLUME 38, NUMBER 7 1 SEPTEMBER 1988

Metal —spin-density-wave transition in a quasi-one-dimensional conductor:Pressure and magnetic field effects

Gilles Montambaux'A T&T Bell Laboratories, 600 Mountain Avenue, Murray Hill, New Jersey 07974

(Received 4 March 1988)

This paper provides a mean-field description of the spin-density-wave (SDW) state of a quasi-

one-dimensional (quasi-1D) conductor with an open Fermi surface, as well as of its behavior in low

magnetic field. The variation of simple quantities with the degree of nesting is worked out: These

quantities are the specific-heat jump at the metal-SDW transition and its variation in a magnetic

field, and the increase of the transition temperature with the field. A quadratic behavior T, (H)

reproduces the experimental situation in the Bechgaard salt (TMTSF)2PF6 (TMTSF being

tetramethyltetraselenafulvalene) in which other experiments performed at different pressures should

provide a test for such a mean-field low-coupling description.

I. INTRODUCTION

The unusual features of the spin-density-wave (SDW)ordering in Bechgaard salts have been the subject of a lotof experimental and theoretical work. ' Variation ofexternal parameters such as pressure and magnetic fieldare known to act in a drastic way on the occurrence ofdifferent SDW orderings.

In (TMTSF)2C104 (tetramethyltetraselenafulvaniumperchlorate) where the ground state is superconductor inzero pressure, a magnetic field applied along the directionof lowest conductivity first destroys the superconductivi-ty around 0.3 T and then stabilizes a SDW phase above athreshold value of order 3 T. This phase in fact consistsin a series of sub phases. This has been proved by experi-ments on rnagnetoresistance, Hall effect, specific heat,and magnetization.

The origin of these subphases is the quantization of theorbital motion. ' Each subphase is labeled by a quantumnumber N which quantizes the orbital motion of itinerantelectrons. Classically speaking, while the electronic or-bits are open in the metallic phase, the electrons abovethe SDW gap form closed orbits and are quantized inLandau levels and energy is minimized when these Lan-dau levels are completely filled. This is possible, thanksto a variation of the nesting vector Q(H ) with the mag-netic field. In a fully quantum picture, there is tunnelingbetween closed orbits and the Landau levels broaden intoLandau bands separated by a series of small gaps, thelargest gap sitting at the Fermi level. " All these gaps,the amplitude of which vary with the field, induce a quiteunusual thermodynamics. ' Oscillations of the magneti-zation "' or the specific heat' with the field have beenshown to fit qualitatively experimental data. ' Thisquantized structure of the SDW phases occurs in the lim-it where the temperature T is small compared with thecharacteristic energy of the magnetic field co, orequivalently when the gaps energy scale 5 is smaller thanCO

On the other hand, only a few measurements have been

performed on compounds which already exhibit adensity-wave (DW) ground state at zero temperature,essentially NbSe3 and (TMTSF)2PF6." (TMTSF)zPF&exhibits a SDW in zero pressure and zero magnetic fieldwith an ordering temperature T, =12 K. An appliedpressure P' of order of 7 kbars destroys this state. Thesuperconducting state is stabilized between 7 and 8 kbarsand above this pressure a metallic behavior is recovered.At such a pressure the magnetic field has the same effectas in (TMTSF)2C104: it induces a cascade of SDW sub-

phases. ' ' The qualitative picture is essentially the samein both compounds although some aspects remain to beunderstood such as temperature dependence of the transi-tion fields, Hall effect, frequency of the transitions, etc.

In the present paper we are interested in the structureof this SDW observed in zero field, i.e., when P &P'. Itis well known that the metallic phase of quasi-1D com-pounds is very dependent on the geometry of the Fermisurface and very sensitive to the nesting properties. Webelieve that, at least for the TMTSF family, this phasecan be essentially described in a mean-field picture withonly one parameter which is the characteristic energy ofthe deviation from perfect nesting which we write tb

Such a picture has been proved to describe the essentialfeatures of the field induced SDW phases when P gP'.The goal of this paper is to give mean-field predictions forthe evolution of this zero field phase with tl', and to esti-mate an absolute value of this characteristic energy.

Another tool to explore this SDW phase is its studyunder magnetic field. In particular, it is worth knowing ifthe ambient pressure (or P & P*) SDW is linked with thehigh-pressure (P &P') field induced SDW phases. Toanswer this problem, experiments in a magnetic field havebeen performed by Kwak et al. ' ' around 6 kbars. TheSDW critical temperature decreases very rapidly withpressure around 6 kbars but could be maintained around2 K. This transition temperature is found to show anenhancement with magnetic field H. T, (P,H) has beenproved to be orbitally driven (only the component per-pendicular to the most conducting plane is relevant). The

38 4788 1988 The American Physical Society

38 METAL-SPIN-DENSITY-WAVE TRANSITION IN A QUASI-. . . 4789

temperature increases nearly quadratically with the field.According to their authors, these experiments suggestthat the high-pressure high-field phases and ambientSDW are manifestation of the same phase. If it is so, oneexpects the same model to describe these different limitsand their crossover namely this quadratic variation atlow field.

We first begin (Sec. II) to derive the features of the in-stability line and give universal variations of relevantquantities as a function of tb. Then, from the study of thesusceptibility in low magnetic field, we describe the ob-served variation T, (H). This variation has been de-scribed as a function of the interaction parameter in aslightly different model by Balseiro and Falicov. ' Weshall give here the evolution of T, (H) with the degree ofnesting and give quantitative estimate of this variationwhich can be directly compared to the experiments in(TMTSF)2PF6 In Sec. III we write down the thermo-dynamic equations of the ordered SDW phase and derivefrom a Landau expansion the variation of the specificheat jump at the transition with the degree of nesting aswell as with magnetic field. This variation comparedwith forthcoming experiments would be a very simpletest of this mean-field theory. These results are discussedand compared with experiments in Sec. IV.

Recent NMR experiments have shown that the SDWphase is separated in two by a transition line character-ized by a temperature T-3.5 K in ambient pressure anda decrease in pressure, at the rate 0.2 K/kbar. 20 On theother hand, different authors have pointed out that theSDW phase could be characterized by two slightlydifferent wave vectors, depending on pressure and tem-perature. ' Hasegawa and Fukuyama described thetransition between these two phases, only along the me-tallic phase. From a Ginzburg-Landau expansion, it ispossible to extend this study inside the ordered phase, toestablish if such a transition between two different wavevectors could explain the observed feature. This is donein Sec. V. The answer seems to be negative.

II. TRANSITION TEMPERATURE

E(k)=v(~ kl ~

kF)—+tJ(kJ b) .

This dispersion has been linearized around the Fermi lev-

el, along the direction of highest conductivity. U is theFermi velocity and t~(k~b } describes the periodic warp-

ing of the Fermi surface along the second direction ofhigh conductivity. The simplest choice '

t, (p) = —2tb cosp —2tb cos(2p) (2)

is expected to describe the main features of the geometryof the Fermi surface. With tb

——0, there is a perfect nest-

ing with vector Q& (2kF, m /——b) and deviations from per-fect nesting which destabilize the SDW phase are de-scribed by the unique parameter tb. It has been shownthat if nesting is good enough along the third direction,the corresponding dispersion can be forgotten so that weare dealing with a 2D electron gas (for temperatureshigher than a characteristic t,' which is believed to bevery small). The critical temperature T, (tb,H} is given by

Xo(Q, tb, T„H)= I/A, ,

A, being a phenomenological parameter describing thestrength of the interaction. In principleQ=( Q~~, Q, )=( 2k F+q~~, Q, }should be ~d~pt~d «optim-

ize Xo so that Q is a function of tb, T„and H. The sus-

ceptibility is written as

A. Susceptibility of the anisotropictwo-dimensional electron gas

Let us start with a SDW, stable below a critical tem-perature T, . We want to know the evolution of T, (tl'„H )

with nesting departure as well as with magnetic field, inthe limit of low field. This is done by studying the sus-ceptibility Xo(Q, tb, T,H) of the noninteracting electrongas. This gas is described by the following dispersion re-lation:

I

Xo(Q, tb, T,H)=TQ dx'6+(ice„,p, x', x)6 (icy„,p Q~b, x',x)e-2m.b

~n

(4)

where co~~ are the Matsubara frequencies and 6+ obeysthe following equation, within the gauge A= (O, Hx, O):

dice„+iv +vkF t~(p eHbx )—6+ (ice„,p—,x,x')

with

XO X XP(x,p }=q~~x — T p Qb — + T —p—

U Xp Xp

=5(x —x') .

Xo——2 g J'exp2T

bU 60 )p

—2'~ xRe(exp[i/(x, p )] )dx

The susceptibility can be written in the following form: Tj (p) = f~zt~(u)du ( ) is the average over the transversemomentum p.

xp is the characteristic length of the magnetic fieldxp=1/eHb. One can also define a characteristic energyco, of the magnetic field ~, =eHUb/2 which will be usefulin the following. When co, &&T or equivalently when xp

4790 GILLES MONTAMBAUX 38

is small compared to a typical thermal length

xr v——/2m T, there are many oscillations of the periodicfactor in the integral, leading to the oscillatory and quan-tized behavior at large fields. ' Here, we explore the op-posite limit co, «T I.n this case, the phase P can be ex-

panded as

. Ax A xexpiP(x, p) =exp(iAx } 1 i— +i

2xo 6x 0 +" 0.5-

(a)

(A') x + 0 ~ ~

8x0

where

A(p) =q~~+(1/U)[t, (p)+r J(p QJb—)]

and A' and A" are derivative with respect to p.

I

0.5

8. Instability line in zero magnetic Aeld

In this case, P= Ax. Summation in Eq. (5), leads to thewell-known result

go(Q, t&, T)=N o[l n(E o/T)+1'( —,')—Re(g( ,'+iB) ) ]—,

(10}

X~ 405

where B= A /4nTNo = .I/(2mvb is the density of statesat the Fermi level. Eo is a cutoff proportional to thebandwidth. f is the digamma junction. Asymptotic be-haviors, tI, gg t~" or tI, —t~", are given in Ref. 23.

As pointed out by various authors, ' two SDWphases can occur, characterized by slightly different wavevectors. At zero temperature, the best nesting vectorconnects the inflexion points of the Fermi surface. Itsvalue is given by

aA(p, Q~)= A(p, Q~)= ~ A(p, Q2)=0.

()p Qp2

(c}

0.5

At high temperature, when T becomes of order of tb, de-viation from perfect nesting becomes irrelevant and thebest nesting vector is Q, =(2kF, m/b). SDW ph. ases withwave vector Q, and Qz have been named SDW, , andSDW2 by Yamaji. ' Qz(T) varies with temperature andjumps to Q, at a critical temperature estimated byHasegawa and Fukuyama as T'=0 232t& (for t.&/tb

——20~2}. Such a SDWz phase occurring at low tem-perature will be studied at the end of this paper. For themoment we restrict our study to the SDW, phase withnonvarying nesting vector.

The critical line is given by the Stoner criterion (3).Given the interaction A., there is a critical line value tI, atwhich the SDW ground state disappears. The variationT, /t&* versus P=ts!t&' is a universal function ' re-called in Fig. 1(a). Useful quantities for further studiesare the dimensionless parameters which describe the evo-lution of go with temperature nesting or magnetic field,expressed in units of tb*, the only energy scale of theproblem. From Eq. (10), one has

0.5-

0.5

FIG. 1. {a)Universal variation T, /tb* as a function of the de-gree of nesting P=t&lt'* (Ref. 21). The experimental curveT, (P) in (TMFSF)2PF6, if well described by this theory, shouldgive the correspondence between the tb scale and the pressurescale, and tell in particular where the ambient pressure point in(TMTSF)2PF6 stands on this phase diagram (arrow). The smallportion in the low-temperature part of the phase diagram de-scribes a phase with temperature-dependent wave vector Q2(T}.(b) Variation of the dimensionless parameter t' BY&/Bt& vs thedegree of perfect nesting. {c)Variation of the dimensionless pa-rameter tI, 8+0/BT vs the degree of perfect nesting.


BXpt„'* = N—, [I+Im(B Q'( —,'+iB ) ) ],

C

tbt&* —— N—o Im(n. 'cos(2p )g'( ,'+—iB) ),

()tb T~

(12)One thus deduces

CO

T, (H}= T,(0}+f(P) (17)

where f' is the trigamrna function. These quantities areplotted in Figs. 1(b) and 1(c).

C. Finite magnetic field

We come to the variation of the susceptibility in lowfield, starting from the expansion of the phase in Eq. (8).The first term of this expansion leads to the zero-field sus-ceptibility. The second one gives zero contribution sinceA' is an odd function of p. Successive integrations byparts of the third and fourth terms yield

j (A'}Bra, bu xo (2'„iA )—

where P= tI', /t&' and the dimensionless parameter f(P) isdeduced from Eqs. (15) and (12) and is plotted in Fig. 3.As stressed by Balseiro and Falicov, ' the effect of themagnetic field is more important close to the marginallystable SDW.

Let us stress that this behavior is obtained when theexpansion of the phase [Eq. (8)] is licit. Roughly, this isthe case when co, /T& inf(1, ( T/tt', )

~ ], a conditionwhich is obeyed when tb is not to close to tb'. When tb

approaches tb', T, goes to zero and the domain of validi-

ty of Eq. (17) becomes vanishingly small.

III.THERMODYNAMICS OF THE ORDERED PHASE

or

t) Xp (ty ) (B')~ (4)(tl", ) = No —Re P' '( ,'+iB)—

/~2 T2 48~2

Starting from the equation for the Green functions inthe ordered phase (U =1):

[iso„—k t~(p)]G+—bF =1,(18)

P' ' being the hexagamma function. The susceptibilityincreases quadratically with the field. The above dimen-sionless parameter is shown in Fig. 2. From the Stonercriterion, one has

[i'„—k —qli

—t~(p Qtb )]F—+b, G =0,

and the self-consistency condition

t)Xp 1 t) Xpb, T+ , H'=0—.

2 "r)H(16)

one has

=Ty Jdk(F)~n

(19)

0.08

0.2—

0.06—

OlX0.04—3

0.02—

0'0.5 0

0 0.5

FIG. 2. This curve exhibits the variation of the suscetibilityunder low magnetic field, through the dimensionless parameter(4*}'8Xo/Bco, which is a universal function of il=r& jtl', *

FIG. 3. The curvature of the variation T,{H) is expressedthrough a universal function f of the degree of nesting P, de-

duced from the ratio of the quantities shown in Figs. 1{c)and 2.


(20)

This equation gives, in principle, access to the thermo-dynamics of the SDW phase in the whole plane (T, ts ).Here we want to focus on one simply measurable quanti-

ty, the specific heat jurnp at the transition. It is foundfrom a Landau expansion of Eq. (20) around T, . Onededuces the ratio r =(b,C/C)/(b, C/C)ncs normalized tothe BCS value:

where r is the jump in zero field, found in Eq. (21). Thefunction g(p) is shown in Fig. 5. As for T, (H), this be-havior breaks down very close to the critical tb'. At lowtemperature, quantization effects are expected with oscil-latory behavior of r(H) versus the magnetic field. '3

IV. DISCUSSION OF THE RESULTS

(21)

Its variation versus the degree of nesting p=tb/ts' is

shown on Fig. 4. It has the BCS value in the case of per-fect nesting and decreases with deviation tb.

Another interesting quantity to evaluate experimental-ly would be the evolution of this specific heat jump withthe magnetic geld. As for T, (H), we expect an increaseof this jump with the field because the field improves theeffective nesting of the Fermi surface. We have calculat-ed this increase in the limit co, & T, . This evaluation ismore complicated since it involves the calculation of thefourth order term of the Ginzburg-Landau expansion ofthe free energy under magnetic field. This is done in theAppendix. As a result the specific heat jump, normalizedto the BCS value, varies as

Nr(H) =r(0) 1+g(P)

(tb")

We have obtained quantitative predictions for the vari-ation of simply measurable quantities with the deviationfrom perfect nesting. It is believed that this deviationfrom perfect nesting increases with increasing pressure,leading to a progressive destruction of the SDW state.One can expect roughly

ts tb =a(p —p ) .

It would be useful to know the parameters tb* and u or,in other words, where does the ambient pressure pointstands on the phase diagram of Fig. 1(a). At the presenttime experimental points T, (P) are insufficient to giveunambiguously a set of parameters. For future investiga-tions we add the variation of the slope BT,/r)tb deducedfrom the model (Fig. 6). In principle, the location of theambient pressure point comes immediately from thespecific-heat jump at the transition. But just one point isnot a really good test of the theory. More interestingwould be the investigation of the variation of this jurnp

0.8—

0.6—

0.5

0.4—

0.2—

0.5

0.5

FIG. 4. Variation of the specific-heat jump at the metal-SDWtransition, normalized to the BCS value vs the degree of nestingP=t&lt&* of the Fermi surface.

FIG. 5. The curvature of the variation r(H) of the specific-heat jump at the metal-SDW transition is expressed through auniversal function g of the degree of nesting P.


tb=0

4-

3c

0.5

FIG. 6. Slope of the variation T, (tb) as a function ofP= tb ltb

with pressure and with magnetic field.The variation in low field T, (H) is also an interesting

probe of this SDW phase. The theory qualitatively ex-plains the observed variation and a correct order of mag-nitude can be obtained from the currently estimatedvalue of tb (5 K & tb & 15 K). However, contrary to thespecific-heat jump, the curvature of the variation T, (H)is not a universal function of p=tbltb' since ( I/tb')f (p)depends independently on tb*. Moreover, the ratioco, /H =cub/2 estimated as 0.86 KlT in this simple mod-el is not well known. At last, the experiment' ' wasdone close to the critical pressure P„ i.e., close to tb', in

a region where f(P) varies rapidly. From the observedcurvature, one has'

tb

Ig Igtb tb

-0.035 K ' at 6.3 kbars

which can be easily satisfied for reasonable values of tb

and tb*. As for the specific heat, a study in different pres-sures would be precious. From Fig. 3, one qualitativelyexpects the curvature of T, (H) to diminish at lower pres-sure. This is observed on the measurement of T, (H)on the same sample at two different pressures:f(p)/tb* ——0.035 K ' at 6.3 kbars and f(p)/tb ——0.023K ' at 6.1 kbars. A precise measurement of T, (H) atambient pressure would tell us how far is the Fermi sur-face from perfect nesting. The quadratic behavior de-rived analytically here is only valid as long as co, &tb.When co, » tb, the critical temperature T, (tb,H) is ex-pected to saturate at T, (0,0) so that the perfect nestingstate can be reached in principle. Since tb is probably oforder of 10 K, it is not sure at all that this variation canbe reached. At last, an inflexion point in the variationT, (H) should be observed at low pressure if tbltb' issmall enough. Figure 7 shows qualitatively the expected

FIG. 7. Qualitative variation of the critical temperature T,vs magnetic field for various values of t,'. The larger tb, thelower T, . At a given pressure the T, (H) curve saturates at largefield. Is this saturation visible at ambient pressure Po? At lowT„oscillations in T, (H) will occur for H & T, .

results. We a1so expect the same kind of behavior andpossible saturation in the field evolution of the specificheat jump r(H).

The behavior at low field derived in this paper is validas long as co, & T. In the range T, &co, & tb, one can ex-

pect the quantization effects become relevant and induceperiodic variations of T, (H) reminiscent of the quantumcascade of phase transitions observed at low temperature.

In conclusion we have given predictions for the varia-tions of some measurable quantities associated with themetal-SDW transition. Experiments in (TMTSF)2PF6could easily test the validity of this mean-field low-coupling theory. In reality, the dispersion relation ismore complicated than the one used here and the Fermisurface is not so simple. But we still expect the essentialof the physics be described by only one parameter whichis the degree of perfect nesting. For a more realistic Fer-mi surface, such a parameter tb can be also extracted.

We think that these quantitative predictions of thenesting model will allow us to test its validity to describeproperly the SDW ordering in (TMTSF)2PF6, especiallygiven the possibility of another mechanism (the so-calledantiferromagnetic interchain exchange mechanism ) pro-posed to describe the stabilization of the SDW at lowpressure (P «P ).

V. POSSIBILITY OF A DIFFERENT NESTING VECTOR

As stressed in the Introduction, a transition between aphase characterized by Q, =(2k, n. /b) and a phasecharacterized by a temperature-dependent wave vectorQ2(T) can take place. Hasegawa and Fukuyama (HF)have studied this change of regime, along the metal-SDWtransition line. It is worth predicting how the calculat-


ed transition point extend in the ordered phase, to knowif the observed transition inside the (TMTSF)zPF6 SDWphase could be described by such a wave-vector change.

The structure of this transition line around the HFpoint can be inferred from a Landau-Ginzburg expan-sion. It can be shown that

1 a 8(T,co, )8( T, co, ) =8(T,O)+—

2 ac02

2Sco =0

(A2)

The specific-heat jump is given by iI), C= —T(a F/aT ).We want to evaluate this jurnp at the critical temperatureT, (H} at which A(T, co, }=0. Thus

8m. T, Np asap aXpbF= dT+, dtbRe(@"(,'+—iB) ) aT

2I 3 '[ T, (H), cp, ] I

~

8[T,(H), co, ](A3)

where the different quantities aYplaT, aXplatb, and 8 de-

pend on the chosen nesting vector. From this expansion,the slopes dT, /dt„' of the metal-SDW, , metal-SDW2, andSDWi-SDWz lines can be deduced. Using the HF pa-ram«ers tb!t(', =20''2,

Uq(( 0 48——tb, . bqi=~ —0.02121,we have found

where A' is the derivative of Eq. (Al) with respect to T.As a result we get the expansion in co,

ac ac ' a'Xp/a'~, aT1+

C T(H) C T(H) ag IaT

BT —1 BT —1

dt„' where

1 aBCO

BBco(A4)

The slopes are quasivertical but the two lines cross with afinite angle. (Contrary to Fig. 1 of HF, we expect thecritical line of the metal-SDW2 transition to be vertical atvanishing temperature for a thermodynamic reason:since there is no entropy at T=O, the Clapeyron formulaleads dT/dP= ao. ) From evaluation of Re(g"( ,'+iB })—for the two vectors Q, and Q2, one finds the slope of theSDW&-SDW2 first-order transition

aT 1

atb 175

0ACC T (H)

is the expression (21) at the temperature T, (H). The re-

sult can be written in the form

hCT (H)

hC N1+g (f3)

r, (tb" ' (A5)

where g is a function of the only parameter 13=tbltb'.(b,c/C)

i r is the specific heat jump in zero magneticC

field [Eq. (21)]. In Eq. (A4), the first term in c0, can beobtained from derivation of Eq. (15). One gets:

which is also a quasivertical line and cannot account forthe observed transition in the SDW phase of (TMTSF)2PF,.

'a xp

acp~aT(A6)

N0Im B)B') |)"'(,'+iB) ), —

48~ T

APPENDIX: CALCULATION OF THESPECIFIC-HEAT JUMP h, c /CAT FINITE MAGNETIC FIELD

8 is defined in the text. f' ' is a derivative of the diagam-ma function 1i):

At a given temperature T and close to co, =0, the freeenergy is written as F= —A /2B with

(X)1q'"'(x) =( —I)"+'(n!) y

p (p+x)" +' (A7)

1 a'Xp(T, cp, )A ( T, cu, ) =Xp( T,0)+-

ON co =0C

co, —I /A, ,

(Al)

The second term in co, is more complicated. It resultsfrom the variation in magnetic field of the fourth-orderterm B of the Ginzburg-Landau expansion of the free en-ergy. At finite field this term has to be written in themixed representation:

8=T y (, f dy f dw f dzG+(p, x y)G (p Qib, y, z}G+(—p, z, w)G (p —Q)b, w, x)exp[ iQ(((x —y—+z —w)]) .

(A8)The Green function G+ is defined in Eq. (5). ( ) is an average over the transverse momentum p. As in Sec. II A, theoscillatory factor of the Green function can be expanded as soon as x «x0 which is the case if T))co,. One gets

8=4 g f exp

2TRe(exp[i t(X(),p)])d dyw dz (A9)


with

i A'X i A"X

exp[i/(X, p)]=exp iAX 1 — +2xp

(A )'X'8 2 (A10)

NpB(T,0) = — ( f"( ,'+iB—)), (A 1 1)

which enters in the calculation of the zero-field specific-

where we have written symbolically X"=x"—y"+z"—u". As expected, the result is independent of xwhich we fix to be 0. The integrals in Eq. (A9) are per-formed in the domains y &0, y &z, ur &z, and m &0. Atzero magnetic field, the first term of Eq. (A10) leads to

1 BB Np6

((A')'tj' '( —'+iB))2 t)to, 18(4n T)

(A12)

Numerical estimations of the expressions found in thisAppendix gives the variation g(P) shown in Fig. 5.

heat jump [Eq. (21)]. The second term of Eq. (A10) givesno contribution. After successive by parts integrations ofthe last two terms of Eq. (10), a lengthy by straightfor-ward calculation gives

Also at Laboratoire de Physique des Solides, CNRS, UniversiteParis-Sud, 91405 Orsay, France.

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