Aitchison, Ian Johnston Rhind

Ao, P

Thouless, David James

Zhu, X M

CERN Document Server

CERN-TH.7385/94EFFECTIVE THEORIES OF BCS SUPERCONDUCTORS AT T=0Ian J.R. Aitchison1;2;y, Ping Ao2, David J. Thouless2, and X.-M. Zhu21TH. Division, CERN, CH-1211, Geneva, 23, Switzerlandand2Department of Physics, FM-15University of Washington, Seattle, WA 98195, USAAbstractWe show that the low frequency, long wavelength dynamics for the phase of the pair eldfor an BCS-type s-wave superconductor at T=0 is equivalent to that of a time-dependentnon-linear Schrodinger Lagrangian (TDNLSL), when terms required by Galilean invarianceare included. If the modulus of the pair eld is also allowed to vary, the system is equivalentto two coupled TDNLSL's.PACS#s:74.20.-z, 67.50.FiCERN-TH.7385/94August 1994y Permanent address: Physics Department, Theoretical Physics1 Keble Road, Oxford OX1 3NP, UK1The classic Ginzburg-Landau (GL) theory [1] is very successful[2,3] in describing a largeclass of static superconducting phenomena near the critical temperature Tc, and its formwas established by Gorkov [4] shortly after the microscopic BCS theory [5]. Subsequently, anumber of attempts [2,3,6,7] were made to obtain a generalized GL theory for time-dependentphenomena, and for temperatures well below Tc, but a consensus has still not been reachedon the form of such a theory at T=0. In this letter we shall show that the eective theoryat T=0 is equivalent to a time-dependent non-linear Schrodinger Lagrangian (TDNLSL).At rst sight, this result might seem almost obvious: after all, the energy density in GLtheory looks formally like that of a non-linear Schrodinger theory so that it seems naturalto extend it to the corresponding time-dependent theory as, indeed, Feynman assumed [8]in his discussion of the dynamics of superconductors and of the Josephson eects. Yetneither the earlier discussions[2,3,6], nor recent work based on the eective action formalismof quantum eld theory [7,9], appears to lead to this conclusion. This is in contrast to thecase of a Bose superuid, such as4He, which is well described by a TDNLSL near T=0[10].Indeed, there is considerable current interest in probing the relationship and "crossover"between BCS and Bose superuidity[11]. Our result implies that both are fundamentallythe same, at least near T=0; in particular, the existence of the Magnus force for a vortexline in a superconductor follows naturally. The last point is pertinent to the discussion ofvortex dynamics in superconductors within the eective theory formulation [12].Three of the present authors have, in fact, recently shown [13] that the motion of thecondensate is described by a non-linear Schrodinger equation at T=0, using a density matrixapproach. But this left open the question how this could be reconciled with the earlierwork [2,3,6,7,9], which was generally based on Green's function (or related) techniques, andapparently led to a quite dierent result. The solution of this problem is contained in thepresent paper, and it is essentially very simple. We take the Goldstone mode Lagrangian2which has recently been derived from BCS theory [9], after being proposed on symmetrygrounds [14], and show that it is equivalent to a TDNLSL. This Lagrangian also correspondsprecisely to the early results of Kemoklidze and Pitaevskii [15], who started from Gorkov'sequations [5]. We also extend this to include variations in the modulus of the pair elds,and show that the dynamics is then that of two coupled TDNLSL's. The thread that unitesall these approaches is ultimately Galilean invariance. Since the microscopic starting pointis always Galilean invariant, one expects any eective theory to preserve this symmetry, apoint emphasized in Ref. 15, and the Schrodinger Lagrangian is the simplest such available.We begin by recalling briey the formalism and results of [9], the latter coinciding withthe proposal of [14]. The BCS Lagrangian, for s-wave pairing and in the absence of externalelds, isL =X (x) i@t+r22m+ ! (x) + g "(x) #(x) #(x) "(x) (1)where  describes electrons with spin  = ("; #),  = k2F=2m is the Fermi energy in thenormal state, and x = (x; t). Introducing the auxiliary(pair) elds (x) and (x), andintegrating out the electron elds, one obtains the eective actionS[;] =  iT r lnG 1 1gZd4xj(x)j2(2)where the Nambu Green function satises0BB@O1(x)(x) O21CCAG(x; x0) = (x  x0) (3)with O1= i@t+ r2=2m + , O2= i@t  r2=2m   , and Tr includes interval and space-time indices. To obtain from (2) an eective Lagrangian in terms of the degrees of freedomrepresented by , a possible procedure [7] is to set (x) = 0+ 0(x) where 0is theposition of the minimum of S for space-time independent , and where 0is assumed to beslowly varying in both space and time. One then expands Tr lnG 1in powers of derivatives3of 0. There are, however, two (related) objections to this. First, we are dealing with thespontaneous breaking of a local U(1) phase invariance, which implies that at a temperaturefar from the transition temperature, the most important degree of freedom is the phase of ,which is the relevant Goldstone eld. It is this eld, rather than the real and/or imaginaryparts of , which should carry the low frequency and long wavelength dynamics. Secondly,the ansatz (x) = 0+0(x) violates the Galilean invariance possessed by (1), which implies[15] that(r   vt; t) exp(2imv  r   imv2t) (4)should satisfy the same equation of motion as (r; t). We shall return to the question ofGalilean invariance below.We therefore set(x) = ei(x)j(x)j (5)and j(x)j = j0j+j(x)j, where we are interested in the low frequency and long wavelengthuctuations of (x) and j(x)j. However, although j(x)j=j0j is expected to be small,and a simple expansion of the sort mentioned above for Tr lnG 1could easily be set upin terms of derivatives of j(x)j if (x) were zero, it is crucial to recognize that (x) isnot small in general, so that the phase factor in (5) cannot be expanded, but must betreated as a whole. This prevents a straightforward expansion of Tr lnG 1when (5) issubstituted into (2). Fortunately, this dicult can be easily circumvented [9,16]. DeningU(x) = exp(i(x)3=2), we can writeTr lnG 1= Tr lnG 1UU 1= Tr lnU~G 1U 1= Tr ln~G 1(6)where~G 1= G 10(1  G0)4G 10=0BB@O1jjjj O21CCA(7)and = ir2=4m + ir  r=2m+ (_=2 + (r)2=4m)3  jj 1: (8)Minimizing (2) with  = jj = 0 yields the usual gap equation for j0j. The dynamics of and jj is contained inSeff[; jj] = iT r1Xn=11n(G0)n 1gZjj2d4x ; (9)where we note that  contains just j(x)j and derivatives of (x), in terms of which (as-sumed small) quantities a useful expansion can be conducted, following standard techniques[17].We now concentrate on (x), and set jj = 0 for the time being. The results of Ref.9(see also[14]) then giveLeff() = 0(_ + (r))2=4m) 12N(0)(_ + (r)2=4m)2(10)where 0= k3F=32is the electron density at T=0, N(0) is the density of states ( for onespin projection) at the Fermi surface and we have adopted a convenient normalization; notethat N(0) = 0=2mv2awhere va= vF=p3 is the velocity of propagation of the Bogoliubov-Anderson mode. We proceed to our main result, which is the demonstration that (10) isequivalent to a TDNLSL. The equation of motion which follows from (10) is@@t+r  j = 0 (11)where  0=  N(0)(_ + (r)2=4m)   (12)andj = r=2m: (13)5Equations (11)-(13) are, in fact, precisely those obtained (to this order in derivatives, andchanging an overall sign) by putting jj = 0 in Eqns.(21)-(23) of Ref.15. Consider now thenon-linear Schrodinger LagrangianL = i _  14mr  r   V (14)where the mass has been chosen to be 2m, and V will be assumed to be a function of j jonly. Our observation is that if we set =p exp(i) (15)where  and  are as dened above, then the equations of motion that follow from L are(up to the given order in derivatives) the same as (11)-(13). This is easy to verify: putting(15) into (14) and discarding a total derivative we obtainL =  _   (r)2=4m  (r)2=16m   V () (16)leading to the equation of motion (11) with j given by (13), anddVd=  (_ + (r)2=4m)   (r)2=16m2+r2=8m2: (17)We now chooseV = (   0)2=2N(0) (18)and solve (17) by expanding in derivatives. The lowest order solution is exactly (12), so thatall of (11)-(13) have been recovered. We have shown that the dynamics of the Goldstoneeld (x) is given (to the relevant order in derivatives) by the TDNLSL (14), where  isgiven by (15),  by (12) and V by (18).Before discussing the inclusion of the eld j(x)j we comment further on (10)-(13). Werst note that the Galilean invariance requirement (see Eqn.(4)) implies that0(r0; t0) = (r; t) +mv2t  2mv  r (19)6where r0= r   vt, t0= t. From (19) we easily nd0(r; t0) = (r; t); j0(r0; t0) = j(r; t)  v(r; t) (20)so that Leffis a Galilean scalar, as is the original L of (1), while  and j transformcovariantly. Indeed, a simple alternative route to (10) is to start from a Lagrangian whichdescribes just the Bogoliubov-Anderson mode, vizLa=12_2 12v2a(r)2: (21)Now (21) is clearly not a scalar under (19), but it can be made so by adding terms of theform _(r)2and ((r)2)2. The requirement that the resulting L be a scalar under (19)(up to constants and total derivatives) determines  and  uniquely to be 1=4m and 1=32m2respectively, and the Lagrangian is then proportional to Leff(). Further, simple linearresponse theory (assuming, as always, a derivative expansion) gives[18]   N(0)_; j  0r=2m: (22)The rst relation can be converted into (12) by requiring that  is a Galilean scalar, while thesecond has to be replaced by (13) to ensure that j transforms covariantly. The requirementthat the Galilean symmetry possessed by the original theory (1) should be respected by theeective theory is a powerful constraint.In view of its relative unfamiliarity, it maybe worth noting that Leff() (or equivalentlyL ) embodies the usual phenomenology of superuid dynamics at T=0 (see, for example[19,20]). We identify r=2m with the superuid velocity vs, and multiply  and j of (12),(13) by m to convert them to mass density and ux, mand jm. Eq. (11) is then the law ofmass conservation, following from the fact that Leffdoes not depend explicitly on . Eq.(12)is equivalent [15] to Bernoulli's equation, if we make use of   2N(0) and p  0.7Since Leffdoes not depend explicitly on t, we have the energy conservation relation@E@t+r Q = 0 (23)where using the canonical denitions (with suitable normalization), one ndsE 12mv2s; Q = jm12v2s+ ; (24)and we have dropped a quantity of order   in E. Finally, since Leffis translationinvariant we have the momentum conservation relation@jm@t+r  = 0 (25)where the momentum ux density tensor isij= mvsivsj+ p ij: (26)Eq.(25) is equivalent to Euler's equation. In Ref.14, the proportionality between the mo-mentum density and the number current j (dened by @L=@(r)), which is included in (25),was taken as a constraint on possible Lagrangians L. If L is a function solely of the Galileanscalar g =_ + (r)2=4m, then@L@_=@L@g;@L@(r)=@L@gr2m: (27)Since  is a phase variable, we can interpret @L=@_ and @L=@(r) as being proportionalto a conserved number density  and number current density j respectively, so that (27)becomes just (13). The momentum density is then automatically proportional to j. Onceagain, Galilean invariance is the essential principle.We now turn to the inclusion of the eld j(x)j. Leff(; jj) can be extracted from(9), up to a given order in derivatives, but calculations rapidly become laborious. For our8present purpose, we will simply use the result of Ref.15 which, using the normalization of(10), givesLeff(; ) = 0"_ +(r)24m+(r)24m# N(0)2"_ +(r)24m+(r)24m#2 N(0)2"_+r  r2m#2+ 6N(0)j0j22(28)where (x)  j(x)j=(p3j0j) and we have retained corresponding terms in  and .The quadratic terms in  yield the amplitude collective mode found in Ref.18 (and are alsoin agreement with the result of Ref.7); we have omitted higher powers of . The term in _ ismade Galilean invariant by the addition of r  r=2m, since (x) is a scalar.There are now clearly two independent degrees of freedom involved, and correspondinglywe nd that (28) is equivalent to two TDNLSL's. That is, the equations of motion for  and which follow from (28) are identical to those arising fromL 1; 2= i 1_ 1 14mr 1 r 1  (j 1j2  0=2)=N(0)+ i 2_ 2 14mr 2 r 2  (j 2j2  0=2)=N(0) +32N(0)j0j2[Im ln( 1= 2)]2(29)where 1=q(+ )=2 exp(i( + ));  2=q(  )=2 exp(i(   )): (30)For example, corresponding to (12), we have= 0 N(0) _ +(r)24m+(r)24m!=@Leff(; )@_(31)and=  N(0) _+r  r2m!=@Leff(; )@ _: (32)Eqn.(29) represents a system of two TDNLSL's coupled via the "mass" term in (28). Ex-pressions for all the conserved quantities can be found as before, and will include quantumcorrections to the semiclassical results of (23)-(26).9The inclusion of electromagnetism in the above formalism is straightforward. Considerthe formulation in terms of Leff(; ). Since  is the phase of a eld with charge  2e,gauge invariance implies that_ and r must appear in the combinations_   2eA0andr + 2eA (e > 0), where A0and A are the electromagnetic potentials. The eld , onthe other hand, is electromagnetically neutral. The leading order electromagnetic chargeand current densities are obtained by multiplying  and j in (22) by  e and making theabove replacements for_ and r. One then obtains the usual results [7]. In terms of theSchrodinger formulation, one simply makes the expected minimal coupling substitutions:i@t! i@t+ 2eA0and  ir !  ir+ 2eA in (14) or (29) (note from (30) that both  1and 2have charge  2e).When the above analysis is extended to higher order derivative terms, it is clear ondimensional grounds that some characteristic scale must enter. In fact, such higher termsenter in the form @t=j0j and vFr=j0j  r (see for example Eqn.(35) of [9]), where  isthe coherence length. The basis of the expansion is therefore the usual assumption [15] thatthe characteristic frequency !, and wavenumber k, of variations of (x) satisfy !  j0j,k   1. Indeed, (28) already yields a static solution for  which decays exponentially overa characteristic distance =6. Such a solution is of the type expected far from a vortex core.Inclusion of appropriate higher derivative terms should make possible some predictions aboutthe vortex core structure.The TDNLSL formulation provides, we believe, a simple and unifying framework for thediscussion of dynamical eects in BCS superconductors at T=0. Results which have beenknown for many years [2,3,6,15], as well as those obtained by quite dierent methods onlyrecently [9,14], are all seen to be in agreement with each other, and with the TDNLSLformulation.Acknowledgements: This work was supported in part by the US National Science10Foundation under grant No. DMR-9220733. The work of I.J.R.A, while at the University ofWashington, was also supported by a grant from the US Department of Energy to the nucleartheory group at UW, and by the physics Physics Department, UW. I.J.R.A. is grateful tothe members of the UW Physics Department for their hospitality and support.References1. V. I. Ginzburg and L. D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950).2. See, for example, N. R. Werthamer in Superconductivity, Vol.1, edited by R.D. Parks,M. Dekker, New York, 1969.3. M. Cyrot, Rep. Prog. Phys. 36, 103(1973) gives a critical review.4. L. P. Gorkov, Zh. Eksp. Teor. 36, 1918 (1959).5. J. Bardeen, L. N. Cooper and J. R. Schrieer, Phys. Rev. 108, 1175 (1957).6. M. Dreschler and W. Zwerger, Ann. Phys. 1, 15(1992).7. H. T. C. Stoof, Phys. Rev. 47, 7979 (1993).8. R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics,vol. III, Addison-Wesley, 1965. See also R. P. Feynman, Statistical Mechanics, Reading,Massachusetts, 1972.9. A. M. J. Schakel, Mod. Phys. Lett, B 14, 927 (1990); see also Ch. G. van Weert,in Thermal Field Theories, Proc. 2nd Workshop on Thermal Field Theories and TheirApplications, Tsukuba, Japan, July 1990, edited by H. Ezawa, T. Arimitsu and Y. Hashimoto(North Holland, 1991).10. E. P. Gross, Nuovo Cimento 20, 454 (1961); L. P. Pitaevskii, Sov. Phys. JETP 12,155 (1961).11. See, for example, C. A. R. Sa de Melo, Mohit Randeira and Jan R. Engelbrecht,Phys. Rev. Lett. 71, 3202 (1993).12. A. T. Dorsey, Phys. Rev. B46, 8376 (1992).1113. P. Ao, D. J. Thouless, and X.-M. Zhu, preprint.14. M. Greiter, F. Wilczek and E. Witten, Mod. Phys. Lett. B3, 903 (1989).15. M. P. Kemoklidze and L. P. Pitaevskii, Zh. Eksp. Teor. 50, 160(1966).16. U. Eckern, G. Schon and V. Ambegaokar, Phys. Rev. 30, 6419 (1984).17. C. M. Fraser, Z. Phys. C 28, 101 (1985); I. J. R. Aitchison and C. M. Fraser, Phys.Rev. D 31, 2605 (1985).18. See, for example, E. Abrahams and T. Tsuneto, Phys. Rev. 152, 416(1966).19. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press Inc, New York,1959, Section 130.20. M. J. Stephen, Phys. Rev. 139A, 197 (1965); O. Betbeder-Matibet and P. Nozieres,Ann. Phys. 51, 392 (1969); A. G. Aronov, Y. 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Effective theories of BCS superconductors at T=0

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