PHYSICAL REVIEW B

VOLUME 55, NUMBER 22

1 JUNE 1997-II

BCS to Bose crossover: Broken-symmetry state Jan R. Engelbrecht Los Alamos National Laboratory, CMS MS-K765, Los Alamos, New Mexico 87545 and Department of Physics, Boston College, Chestnut Hill, Massachusetts 02167

Mohit Randeria Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439 and Theoretical Physics Group, Tata Institute of Fundamental Research, Bombay 400005, India

C. A. R. Sa´ de Melo Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439 ~Received 26 December 1995! A functional integral formulation, used previously to calculate T c and describe normal state properties of the BCS-Bose crossover, is extended to T,T c . The saddle point approximation is shown to be qualitatively correct for T!T c for all couplings, in contrast to the situation above T c . Several features of the crossover are described. The difference between the T50 ‘‘pair size’’ and the ~prefactor of the T dependent! GinzburgLandau coherence length is pointed out: the two quantities are the same only in the BCS limit. The evolution of the collective modes from the BCS to the Bose regime is discussed together with the mixing of the amplitude and phase in the absence of a particle-hole symmetry. @S0163-1829~97!07317-7#

In this paper we consider the problem of the crossover from BCS theory to Bose condensation using the functional integral formulation. This is an old problem1,2 which has recently been revisited3–9 because of the interest in high-T c superconductors which have a coherence length comparable to the interparticle spacing. The authors3 recently used a functional integral approach to study the normal state, the evolution of T c and the timedependent Ginzburg-Landau ~TDGL! theory going from the weak coupling BCS regime to the formation and condensation of composite bosons. We extend that analysis to the broken symmetry state below T c here. We show that, even in the absence of an obvious small expansion parameter, this formulation permits us to make physically motivated approximations for all values of the coupling constant at low temperatures T!T c . In addition, our analysis provides an interpolation scheme between two physically rather different limits. Previous work on the BCS-Bose crossover used a variety of techniques in different regimes, e.g., variational wave functions,1,6 random-phase approximation7 ~RPA! at T50, diagrams without2 and with self-consistency,9 and numerical simulations8 above T c . One of the merits of the present approach is that, in addition to recovering previously known results within a single formalism, we are able to go beyond them and gain further insight into the BCS-Bose crossover. We will derive results valid for T,T c . In the broken symmetry phase, we argue that the effects of fluctuations are small even for strong coupling, provided T!T c . For T.T c ~normal state! and for intermediate and strong couplings, the emerging pair degrees of freedom can only be adequately described by retaining dynamic fluctuations about the trivial saddle point. In contrast, the nontrivial saddle point below T c already includes the nonperturbative effects of the bound state formation and condensation. 0163-1829/97/55~22!/15153~4!/$10.00

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Among the new results presented below are a discussion of the distinction between the T50 pair size and the GL coherence length and the sharpness of the crossover at T50. In fact, the asymptotic expressions for the order parameter and the chemical potential in both the BCS ~weak attraction! and dilute Bose gas ~strong attraction! limits approximate the full solutions very well. We also discuss the evolution of the collective excitations, and in particular the presence of a linear time derivative term in the T50 effective action, which is prohibited by particle-hole symmetry in the weak coupling limit. Furthermore, we emphasize that the amplitude and phase modes mix with increasing coupling, as particle-hole symmetry is lost. We study a three dimensional ~3D! continuum model of fermions of mass m, density n5k F 3 /3p 2 , described by the imaginary time action per unit volume S5 * b0 d t * dx$ ¯ c s (x) ] t c s (x)1H(x) % , where

c s ~ x ! Kc s ~ x ! 2g ¯ c ↑~ x !¯ c ↓~ x ! c ↓~ x ! c ↑~ x ! . H~ x !5¯ Here b is the inverse temperature, K52¹ 2 /2m2 m , m is the chemical potential, x5(x, t ), and \5k B 51. The cutoff associated with the attractive interaction g will be discussed below. We follow the standard approach of introducing D(x) which couples to ¯ c¯ c , and integrating out the fermions. The resulting effective action S eff@ D # 5

E tE H b

d

0

dx

u D~ x !u 2 1 2 Trlnb G21 @ D ~ x !# g b

J

~1!

is written in terms of the inverse Nambu propagator G21 52 ] t 2Ks z 1D ~ x ! s 1 1D * ~ x ! s 2 ,

~2!

where s 6 5( s x 6i s y )/2, and the s j ’s are Pauli matrices. 15 153

© 1997 The American Physical Society

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Consider a uniform static saddle point D(x)5D 0 which satisfies d S eff@ D 0 # / d D 0 50. Fourier transforming from x to k5(k,ik n ), where ik n 5i(2n11) p / b , and doing the frequency sum, the saddle point condition can be written as 1/g5 ( ktanh(bEk/2)/2E k , where E k5 Aj k2 1D 0 2 with j k5 e k2 m and e k5 u ku 2 /2m. We regulate the ultraviolet divergence in the gap equation by expressing our final results in terms of an s-wave scattering length10 a s which describes the two-body attraction g in a low density system (k F 21 ! range of attraction!. The ‘‘renormalized’’ gap equation in terms of a s is given by N0 5 k Fa s

(k

F

G

1 tanh~ b E k /2 ! 2 , 2ek 2E k

~3!

where N0 5mk F /4p . In weak coupling BCS theory m differs from its noninteracting value e F by a negligible amount @order (D 0 / e F ) 2 #. However, with increasing coupling the momentum distribution n k is increasingly broadened ~see below! and we need to compute the resulting change in m . Using n52 ] V/ ] m , the saddle point approximation for the thermodynamic potential V 0 5S eff@ D 0 # / b , leads to n52 b 21 ( k Tr@ G0 (k) ] G21 0 (k)/ ] m # , where G0 (k) is G@ D(x)5D 0 # transformed to k space. This can be further simplified to n5

1 L3

(k

F

12

S DG

jk bEk tanh . Ek 2

~4!

It is important to ask if the approximate number equation ~4! is adequate outside the weak coupling limit. Above, and at, T c we know from Ref. 3 that the saddle point number equation leads to qualitatively wrong answers with increasing g. The reason is that, as it stands, Eq. ~4! accounts only for the thermodynamics of broken and condensed pairs — it does not include the contributions from pair excitations ~‘‘collective modes’’!. We would thus expect that just below T c it would prove inadequate with increasing coupling. However, we argue that for T!T c , the corrections to Eq. ~4! are small for all couplings. To see this we anticipate some results to be derived below. For small q the collective mode spectrum is linear, v 5c u qu , and thus the expression for the thermodynamic potential has to be modified to V.V 0 @ D 0 # 1T ( qln@12exp(bcuqu ) # . In the absence of longrange interactions, such a propagating mode exists for all values of the coupling, and thus the corrections to the number equation are down by factors a (T/ e F ) 4 , where a is a bounded function of the coupling strength (k F a s ) 21 . Thus, these corrections are down by (T/ e F ) 4 in weak coupling, and even smaller, of order (T/ e F ) 4 Ak F a s , in the Bose limit.11 First we consider the ground state properties. At T50 the saddle point gap and number equations simplify to ( k( e k21 2E k21 )52N0 /(k F a s ) and ( k(12 j k /E k)5n, which are identical to the variational equations obtained by Leggett.1 The solutions are plotted in Fig. 1 together with the analytic results of the two asymptotic limits. In the weak coupling BCS limit @ 1/(k F a s )→2`# we obtain m (0)5 e F 5k F 2 /2m and D 0 (0)58e 22 e F exp@2p/(2kFuasu)#. In the strong coupling limit @ 1/(k F a s )→1`# we obtain m (0)52E b /212 e F (k F a s )/(3 p ), which goes to one-half the pair binding energy E b 51/ma 2s asymptotically. The or-

FIG. 1. Solutions of the gap and number equations at T50: D 0 and m as a function of the dimensionless coupling 1/(k F a s ). Analytical results for weak and strong coupling are shown as dashed lines.

der parameter is given by D 0 (0)5(16/p ) 1/2e F / Ak F a s . We note from Fig. 1 that the BCS-Bose crossover at T50 is very sharp and occurs in the range 21,1/(k F a s ),1. Also, the asymptotic expressions hold very well outside this interval. In particular the BCS weak coupling asymptotic result at T50 is found to work over the entire range of couplings where the system is highly degenerate. The momentum distribution n(k) for the fermions is shown in Fig. 2. With increasing coupling the pairs become more tightly bound ~see below! and thus n(k) becomes broader as fermions with larger k’s participate in the bound state formation. The pair size may be defined by j 2pair 2 52 ^ c ku ¹ ku c k& / ^ c ku c k& , where the zero-temperature ‘‘pair wave function’’ is given by c k5D 0 /2E k . As shown in Fig. 3, j pair is a monotonically decreasing function of the attraction, going from the exponentially large value k F /mD 0 in the

FIG. 2. The evolution of the momentum distribution n(k) as a function of e k for couplings 1/(k F a s ) of 22, 21, 20.5, 0, 0.5, and 1. The dotted line intersects n(k) for each coupling when e k5 m .

BCS TO BOSE CROSSOVER: BROKEN-SYMMETRY STATE

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T c provided D 0 (T)! v as discussed in Ref. 3, and ~2! at T50 where the thermal factors make the qp-qh term vanish.11 At T50, we take the analytic continuation iq m → v 1i0 1 , and write M 11(q, v )5M E11(q, v ) O 1M 11(q, v ) which are even (E) and odd (O) in v ; M 12(q, v ) is even in v . It is useful to define h (x)5 @ l(x)1i u (x) # / A2 where l(x) and u (x) are real, and may be identified with amplitude and phase fluctuations, respectively. The effective action is then of the form FIG. 3. The pair size j pair ~broken line! and the GL coherence length j 0 ~solid line! in units of k F 21 as functions of the coupling 1/(k F a s ).

BCS limit to a s , the size of the two-body bound state, in the Bose limit. This behavior of the pair size must be contrasted with the Ginzburg-Landau coherence length j 0 @where j GL(T). j 0 u (T2T c )/T c u 21/2#; see Fig. 3. From the analysis of Ref. 3 we see that while the two length scales are essentially the same in weak coupling, j 0 has a minimum value of order k F 21 and then increases at strong coupling because of the diluteness of the Bose gas: j 0 ;k F 21 / Ak F a s for g→`. ~In the absence of the diluteness condition, the Bose limit j 0 would be expected to saturate to the distance between particles independent of the size of composite bosons.! To investigate Gaussian fluctuations about the saddle point D 0 , we write D(x, t )5D 0 1 h (x, t ), and expand S eff to second order in h to obtain S Gauss5S eff@ D 0 # 1 21 ( q Trh † M h . Here h † 5 @ h * (q), h (2q) # and we have Fourier transformed to q5(q,iq m ) with iq m 5i2m p / b . The inverse fluctuation propagator M is a 232 matrix defined below.12 We simplify notation by denoting j k and E k by j and E, and j k1q and E k1q by j 8 and E 8 , respectively. Further, using u 2 5(11 j /E)/2 and v 2 5(12 j /E)/2, f 51/ @ exp(bE)11# and f 8 51/@ exp(bE8)11#, we can write the matrix elements of M as 1 M 11~ q ! 5 1 g 1

(k @ f 2 f 8 #

H

u 2v 82 v 2u 82 2 iq n 1E2E 8 iq n 2E1E 8

(k @ 12 f 2 f 8 #

H

J

J

u 2u 82 v 2v 82 2 iq n 2E2E 8 iq n 1E1E 8

~5! and M 12~ q ! 5

(k @ 12 f 2 f 8 # 1

(k @ f 2 f 8 #

H

H

u v u 8v 8 u v u 8v 8 2 iq n 2E2E 8 iq n 1E1E 8

J

J

u v u 8v 8 u v u 8v 8 2 , iq n 1E2E 8 iq n 2E1E 8 ~6!

with M 22(q)5M 11(2q) and M 21(q)5M 12(2q). We note in passing that the quasiparticle-quasihole ~qp-qh! terms, with the @ f (E)2 f (E 8 ) # prefactor, have the usual Landau theory singularity for q and v going to zero. A small q and v expansion is possible only in two cases:13 ~1! just below

~ l *u * !

S

M E111M 12

iM O 11

2iM O 11

M E112M 12

DS D l

u

.

~7!

Since M O 11(q,0)50, the amplitude and phase modes decouple at v 50. Further M E11(0,0)5M 12(0,0) ensures that the phase mode for q50 is gapless, i.e., the Goldstone mode. We now make a small q and v expansion ( v , u qu 2 /m!mink$ E k% ) of the effective action using M E111M 125A1C u qu 2 2D v 2 1•••, M O and 115B v 1•••, M E112M 125Q u qu 2 2R v 2 1•••. We note that the amplitude and phase are coupled via the linear time derivative term iB v in Eq. ~7!. The coefficients A,B,C, etc., for arbitrary couplings, are given by A5 ( kD 0 2 /2E 3 , B5 ( kj /4E 3 , C5 ( k$ (123X) j /m2 @ 1210X(12X) # Y % /8E 3 , D5 ( k(1 2X)/8E 3 , Q5 ( k$ j /m2(113X)Y % /8E 3 , and finally, R5 ( k1/8E 3 , with the notation X[D 0 2 /E 2 and Y [ u ku 2 cos2u/m2. The collective mode dispersion obtained by solving detM@ q, v (q) # 50 is v (q)5c u qu with the sound velocity c 2 5Q/ @ B 2 /A1R # . The corresponding eigenvector of M is (l, u )5(2ic u qu B/A,1), which is a pure phase mode for q50, but for nonzero u qu has an admixture of the amplitude controlled by B. For arbitrary coupling the integrals involved in A,B, etc., have to be performed numerically; here we shall only treat the analytically tractable limiting cases. The weak coupling limit is particularly simple with all integrals peaked near the Fermi surface and B50 due to particle-hole symmetry ~the integrand is odd under j k→2 j k). With N( e F ) the density of states at e F for one spin, we find Q54N( e F ) e F /3mD 0 2 and R52N( e F )/D 0 2 . This leads to the well-known14 result c5 v F / A3 corresponding to the phase mode. To study the amplitude mode at T50 we go back to the matrix M prior to making a small v expansion. In weak coupling the amplitude is completely decoupled from the phase mode since M O 11(q, v )50 by particle-hole symmetry. Next, it can be shown that M E11(0,2D 0 )1M 12(0,2D 0 )50. Thus the amplitude mode has a gap 2D 0 at q50. Further, in the weak coupling limit, Q/C5D/R53 and thus the stiffness for amplitude distortions is exactly one-third that for phase distortions. As the attraction increases, the particle-hole symmetry of the weak coupling limits is lost and the off-diagonal term in Eq. ~7! proportional to i v (T c / e F ) increases accordingly. This leads to a mixing of amplitude and phase for v ,qÞ0. In the strong coupling limit A5 p D 0 2 K/ 4u m u 3/2, B5 p K/8u m u 1/2, R5 p K/16u m u 3/2, and C5 p K/ 32m u m u 1/2, where K5N( e F )/ Ae F . Here the dominant time dependence arises from the linear time derivative in the offdiagonal matrix elements, in contrast to the weak coupling

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limit. To exhibit the resulting amplitude-phase mixing it is instructive to write the three propagators obtained from M21 :

^ l q* l q & 5 ^ u *q u q & 5

u qu 2 /2m B , B @ v B ~ q! 2 2 v 2 #

2m B c 2 1 u qu 2 /2m B , B @ v B ~ q! 2 2 v 2 #

^ l *q u q & 5

~8!

2i v . B @ v B ~ q! 2 2 v 2 #

Note that all the propagators have poles at v B (q) 2 5c 2 u qu 2 1( u qu 2 /2m B ) 2 , where m B 52m and c 2 5D 0 2 /8m u m u 5 v 2F (k F a s )/3p . However only the phase propagator has a nonzero residue in the limit as q and v go to zero. From the analysis above T c we had deduced3 that the strong coupling limit is a dilute Bose gas of pairs with density n B 5n/2, mass m B 52m and scattering length a B 52a s with n B a 3B !1. Here we find exactly the same result from an analysis of the collective modes at T50 since our spectrum is identical to the Bogoliubov spectrum with c 2 54 p n B a B /m 2B .

1

A. J. Leggett, in Modern Trends in the Theory of Condensed Matter, edited by A. Pekalski and R. Przystawa ~SpringerVerlag, Berlin, 1980!. 2 P. Nozieres and S. Schmitt-Rink, J. Low Temp. Phys. 59, 195 ~1985!. 3 C. A. R. Sa´ de Melo, M. Randeria, and J. Engelbrecht, Phys. Rev. Lett. 71, 3202 ~1993!. 4 For a recent review with references, see M. Randeria, in BoseEinstein Condensation, edited by A. Griffin, D. Snoke, and S. Stringari ~Cambridge University Press, Cambridge, England, 1994!. 5 M. Dreschler and W. Zwerger, Ann. Phys. ~Leipzig! 1, 15 ~1992!. 6 M. Randeria, J. Duan, and L. Shieh, Phys. Rev. B 41, 327 ~1990!. 7 L. Belkhir and M. Randeria, Phys. Rev. B 49, 6829 ~1994!; T. Kostryko and R. Micnas, ibid. 46, 11 025 ~1993!; R. Coˆte´ and A. Griffin, ibid. 48, 10 404 ~1993!. 8 M. Randeria, N. Trivedi, A. Moreo, and R. T. Scalettar, Phys. Rev. Lett. 69, 2001 ~1992!; N. Trivedi and M. Randeria, ibid. 75, 312 ~1995!; R. Micnas, M. H. Pedersen, S. Schafroth, T. Schneider, J. J. Rodrı´guez-Nu´˜ nez, and H. Beck, Phys. Rev. B 52, 16 223 ~1995!. 9 R. Haussmann, Z. Phys. B 91, 291 ~1993!; Phys. Rev. B 49, 12 975 ~1994!.

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It is remarkable that all these results which are known for Bose systems15 are derived here in the composite boson limit starting with the underlying fermion degrees of freedom. The ability of our formulation, which at first sight might have been expected to work only in weak coupling, to reproduce the strong coupling Bose limit gives us some confidence in the validity of the intermediate coupling results where no obvious small expansion parameter is known. In conclusion, we have investigated the evolution of the condensate, its low-energy excitations and the pair wave function from the BCS to the dilute Bose gas limits, within a single framework based on a functional integral formulation with a complex pairing field.16 The BCS-Bose crossover in the condensate occurs in the regime 21,1/(k F a s ),1 and the asymptotic forms hold outside this regime. The pair wave function is distinct from the Ginzburg coherence length outside the BCS limit. The low-energy collective excitations evolve from a phase-only mode ~weak coupling! towards a strongly mixed amplitude-phase mode ~strong coupling!. M.R. and C.SdM. were supported by the Division of Materials Sciences, office of BES-DOE Contract No. W-31109-ENG-38. J.R.E. was supported by the DOE.

10

With increasing attraction 1/a s increases monotonically from 1/a s →2` ~for g→0 1 ) to 1/a s →1` ~for g→`). 1/a s 50 is the threshold for a two-body bound state in vacuum, beyond which a s is the size of this bound state with binding energy E b 51/ma 2s ~using a reduced mass m/2). 11 Haussmann’s self-consistent generalization ~Ref. 9! of the Gaussian theory takes into account the feedback between the field D(x) and the qp-qh excitations. Since these excitations vanish as T→0 we argue that self-consistency does not play an important role in the low-temperature limit we consider. 12 Notice that there are some formal similarities in the BCS to Bose crossover problem between superconductors ~particle-particle channel! and excitonic systems ~particle-hole channel!. See, for instance, R. Coˆte´ and A. Griffin, Phys. Rev. B 37, 4539 ~1988!. 13 E. Abrahams and T. Tsuneto, Phys. Rev. 152, 416 ~1966!. 14 P. W. Anderson, Phys. Rev. 112, 1900 ~1958!. 15 V. N. Popov, Functional Integrals and Collective Excitations ~Cambridge University Press, Cambridge, England, 1987!. 16 In order to deal with the effects of a Coulomb repulsion on the collective mode, one needs to consider the coupling between the particle-particle channel ~treated here! and the particle-hole, density fluctuation channel ~Refs. 7 and 14!. Recently, this has been analyzed within a functional integral framework by C. Castellani, C. Di Castro, and S. Di Paulo ~private communication!.