- Email: [email protected]
Density functional theory of fermion condensation
.__ kl!li!d
CQs
30 November 1998
-!B
ELSEVIER
PHYSICS
LETTERS
A
Physics Letters A 249 (1998) 237-241
Density functional theory of fertnion condensation V.R. Shaginyan ’ Petersburg Nuclear Physics Institute, Russian Academy of Science, Gatchina I88350, Russia
Received 11 May 1998; accepted for publication 20 September 1998 Communicated by V.M. Agranovich
Abstract A density functional formulation of the fermion condensation is presented. It is demonstrated that the standard KohnSham scheme is not valid beyond the fetion condensation phase transition since the functional starts to depend on the quasiparticle density matrix. Our consideration also furnishes an opportunity to calculate the real single particle excitation spectra of superconducting systems within the density functional theory. @ 1998 Elsevier Science B.V. PACS: 71.27.+a; 74.20.-z Keywords: Strongly correlated systems; Superconducting state
Several years ago an extremely powerful method was developed for measuring the electronic structure close to the Fermi level [ 11. As the result, a plateau adjacent to the Fermi level has been observed in the electronic spectra of a number of strongly correlated metals. For instance, the spectra of SrzRuOa [2,3] or YBa&!uhOs [4] contain very smooth segments on the Fermi surface. It is a remarkable thing that these spectra have not been reproduced in theoretical calculations [ 5,2]. We submit that the above-described behavior of the electronic spectra can be understood within the framework of the theory of fermion condensation, which was predicted in Ref. [ 61 and associated with the rearrangement of the single-particle degrees of freedom in strongly correlated Fermi systems. The main feature of the fermion condensation (FC) is the appearance of a plateau in the single-particle excitation spectrum at the Fermi level [ 6-91. It has been demonstrated that in the vicinity of, but before, the
density-wave-instability point of a Fermi system the FC phase transition can occur, that is FC may take place if the effective coupling constant is sufficiently strong [ 91. This makes one think that the FC is a rather widespread phenomenon inherent in strongly correlated Fermi systems. For example, FC can arise in such an unusual system as fermions locked in vortex cores in a superfluid Fermi liquid [ lo]. On the other hand, it was demonstrated that the charge-density-wave instability takes place in three-dimensional [ 11,121 and two-dimensional electron liquid [ 131. Thus, the electronic systems of some strongly correlated metals are suited for searching for FC [ 14,151. Now let us outline the key points of the FC theory [ 8,161. FC is related to a new class of solutions of the Fermi-liquid-theory equation [ 171
a(F - PN) = E(p,T) MP, T)
-
p(T)
-
Tln
= 0, ’ E-mail: [email protected].
for the quasiparticle
0375-9601/98/$ - see front matter @ 1998 Elsevier Science B.V. All rights reserved. PII SO375-9601(98)00736-l
1 - n(p,T) MP? T)
(1) distribution
function
n (p, T) , de-
pending on the momentum p and temperature T. Here F is the free energy, p is the chemical potential, while s(p,T) = SEo/Gn(p,T) is the quasiparticle energy, being a functional of n(p,T) just like the energy Ea and the other thermodynamic functions. Eq. (1) is usually rewritten in the form of the Fermi-Dirac distribution n(p,T)
=
[l+exp (E’p7,-“)I-‘-
(2)
In homogeneous matter, the standard solution ar;(p,T = 0) = Bfpn - p), with PF being the Fermi momentum, is obtained assuming de@, T = 0) /dp to be positive and finite near the Fermi level, and T-dependent corrections to the effective mass M*, quasiparticle energy, and the other quantities start with T2-terms [ 181. New solutions of Eq. (2) possess at low T a spectrum ~(p, T) linear in T [ 14,151:
~P>T) - P(T) = T~o(P>>
f3)
within the interval pi < p < pf occupied by the fermion condensate. Inserting (3) into (2), one finds its distribution m(p) it T = 0 no(p)
= (1 +eVo@))--l,
pi
(4)
differs from the Fermi step. At T -+ being continuous within the FC interval, admits a finite limit for the logarithm in Eq. (l),yieldingatT=O [6] which drastically
0, the function
no(p),
When condensation is just starting, the momenta obey pi = &$ = pf. This fact means, as follows from Eq. (5), that A4* -+ 00. Then, at finite temperatures and when FC has taken place, the effective mass, as follows from Eq. (3), M* N 1/T. Within the FC interval the solution no(p) deviates from the Fermi step function nt.7(p) in such a way that the energy E(P) stays constant: e(p) = ,a, while outside of this region, no(p) coincides with f%F(p). We see that the occupation nnmbers m(p) become variational parameters: the solution no(p) emerges if the energy E;I is lowered by alteration of the standard occupation numbers. We recall that the idea of a multiconnected Fermi sphere, resulting in a lowering of the energy, was considered
in Ref. [ 191. FC can be considered as a generalization of this idea. The quasiparticle formalism is applicable to this problem if the damping of the condensate states is small compared to their energy. This condition holds for superfluid systems and for normal ones provided the ratio of the fermion condensate density pc to the density p of the system is small [ 15,161. As we have mentioned above, calculations within the density functional theory (DFT) failed to reproduce the smooth segments in the single-p~ticle excitation spectra, while these smooth segments are the “calling card” of FC. Thus, a density functional formulation of FC is of crucial importance, since it will allow one to understand the roots of these difficulties. As we shall see the local single-particle potential of DFT [20] beyond the FC phase transition becomes a nonlocal one. One of the remarkable peculiarities of the FC phase transition is related to spontaneous breaking of gauge symmetry: the superconductivity order parameter K(P) = (~~,~a_~,_~) has a nonzero value Q(P) = &o(p) [ 1 - PI@(P)] even if the gap d vanishes [ 161. On one hand, this cir~ums~n~e allows us to apply methods of the DFT of superconductivity ]2 1] to consider FC. On the other hand, this consideration also gains new insights into the density functional theory of superconductivity. As a result, one gets the possibility to calculate the single-particle excitation spectrum. In the DFT of superconductivity, there exists a unique functional F(T) of two densities, namely, the normal density of an electron system p and the anomalous density K,
17
dll,r2)
= (vh(f-1)#1(~2)).
(6)
P[p, K] is given by
In atomic units, the functional
F[p, K] = T,[P, K] - T&[P, K] + + -
J JPt’l) Ptrd [u&r)
lo -
I
xd’rt
rc*trt,r2)
- PIP(~) d3r
r2l
d3rl
d3r
2
V(ri,n,r3,r4'4)
d3r2d3rs d3r4 t F,,[p,
fctr3,+4) K]
VR. Shaginyan/Physics
= HP,
~1--T&b, ~1 +
J
[udr> -PIP(~) d3r. (7)
Here Ts[ p, K] and S, [p, K] stand for the kinetic energy and the entropy of a noninteracting system, while F,, [ p( r) , K( 1-1,r2) ] is the exchange-correlation freeenergy functional, V is a pairing interaction, and uext is an external potential. The fourth and fifth terms on the r.h.s. of (7) are the Hartree term due to the Coulomb forces and the pairing interaction, respectively. We suppose V to be sufficiently weak like the model BCSinteraction [ 221. The last equality in Eq. (7) can be considered as the definition of E. For the densities (6) one can employ a quite general form [ 231 dr1)
=
~11~~,,(~1)121~,,,12(1
-
.f-a,n>
u,n
(8)
~-,,-,(rl)lu~,,urr,n(l-2frr,n),
+$g,,(r2)
with the coefficients v,,, malization conditions
and u,,,,
obeying
1%,,12 + Iu,,,12 = 1.
(9) the nor-
pl(rl,r2),
=
CC&G,,(rl)
0-J + +Z,-,(rl) &,-dr2)
239
We demonstrate now that the functions &, are not defined by a local operator. For the sake of simplicity, we omit the spin variables and put T = 0. Minimization of F with respect to @,, leads to the eigenfunction problem
+
J
u&
@‘b, Kl aK(rl
r2)
b(rd
d3r2= kh(rl),
9
(12)
with A, being the Lagrangian multiplier, preserving the normalization of the eigenfunction $,,_ We see that if the anomalous density K were zero, then Iu,]* = 1 below the Fermi level, and Iu, I2 = 0 above the Fermi level, while functions &, would be given by a local operator. Now let us take F as a functional of pt, K. The functions 41 is subject to the orthonormality constraint
The constraints F with
( 13) may be introduced
by replacing
(10)
Here n denotes the quantum numbers such as the momentum p in the case of homogeneous matter or the crystal momentum and the band index in the solid state. We introduce a one-quasiparticle density matrix
m(r13r2)
Letters A 249 (1998) 237-241
4b(r2)
14d2(1 -fc,n>
where ain, are Lagrangian parameters. (14) with respect to 41, one obtains
IG
Minimizing
JPCmrdh(rdd3r2 +QQsf'(n,rd h(rdd3r2 (1%
I~,~l~f~,~l.
(11)
Considering Eqs. (8)) (9)) ( 11)) one can conclude that there exists a one-to-one correspondence between functions p, K and functions pt , K. Therefore the functional F can be treated as a functional of p, K or as a functional of pr , K. If one needs to calculate only the densities p, K then it is reasonable to employ the functional F [ p, K] . On the other hand, the functional F [ pl , K] may be used if it is necessary to gain a deeper insight into the problem, e.g., to calculate the singleparticle excitation spectra, the functions d,, etc.
with je and f’ given by JOtrl,r2)
=
f'trl,r2)
=
~FI[PI,K] Ql(rl,r2)'
@‘I[PIY Kl Mrl,
r2)
’
(16)
From Eq. (15) one can find that (17)
240
KR. Shaginyan/Physics
With this result, the functions @l turn out to be solutions of the eigenvalue equation
s
.f(r~,~,)
&(r2)
d3r2
= w&(rl),
(18)
while the operator f’ drops out of the problem in accordance with the results of Ref. [23]. Notice such a procedure to deal with a nonlocal operator was considered in Ref. [ 241. Eq. ( 18) can be rewritten as a single-particle equation
-;02 + Ue,t(Tl> +
s
,. vxc(r19r2)hdrd
I
+
with u,, being a nonlocal
h(rl,
r2> =
SF,, 6~1(rl r2> 9
d3r2= w$nt(rl),
(19)
potential
(20) *
(21) If V were zero, EI would represent the real singleparticle excitation spectra of the system. The energy el is perturbed by the BCS correlations, but, in fact, this perturbation is small. It is worth noting, as it will be seen below, one has to put V = 0 to derive equations for FC. The same consideration in the case of finite temperatures preserves Eqs. ( 18), ( 19). In order to satisfy the constraint (lo), it is convenient to take vl = cos 81, ul = sin 131.Then one finds
@[PI, ~1
energy El is given [ 261,
= (el - ,u)2 + A:.
(25)
It is helpful to verify Eqs. (22), (23) for homogeneous matter in the weak coupling limit V -+ 0 at T = 0 using the BCS model [22]. Recall that in this case V enters the functional F only through the Hartree term, while F,, is completely defined by the Coulomb interaction. One obtains (26)
where N( 0) is the correct density of the single-particle states at the Fermi level, given by Eq. (2 1) , while the functions 4k are the plane waves. If the potential uXc entering Eq. (19) were a local one, then vXc= const, and the energies EL would be defined by the singleparticle energies of a free electron gas. As a result, instead of the correct density of states in Eq. (26) one would have the density of states of a free electron gas [ 231. So, we are led to the conclusion that it is of crucial importance to take into account from the very beginning the nonlocality, or equivalently, velocitydependence of the single-particle potential (20). One can do that in the same way as it is commonly done in nuclear physics, where the density functional includes velocity-dependent components [ 271. We shall now give further proof of Eq. (5) deduced at T = 0. Consider once again Eq. (22) in the limit V = 0. In this case, Al = 0, and Eq. (22) can be written as (el - p) tan 281 = 0.
(27)
Eq. (27) requires that =
(El--p)tan201+Al=0.
W
(22)
The gap Al is given by (see, e.g., Refs. [23,25])
(23) We now have to minimize minimization yields
“fi=
Here the real excitation
A - exp[-l/N(O)V],
Eqs. ( 11)) ( 18)) one can also
Taking into account infer that
Letters A 249 (1998) 237-241
1
1 + exp(&/T)
.
F with respect to fl. The
&l - #u = 0,
if lull2 # 0, 1,
(28)
with el defined by Eq. (21). Therefore, the fermion condensation solution is a new solution of the old equations. On the other hand, it is seen from Eq. (28) that the standard Kohn-Sham scheme for the single particle equations [20] is no longer valid beyond the point of the FC phase transition, since one has to introduce the nonlocal single-particle potential (20)) while the quasiparticle occupation numbers Iv1I2 become variational parameters, minimizing the total energy. In the homogeneous limit Eq. (28) takes form of
W?. Shaginyan/Physics
Eq. (5), and the single-particle energy F(P) is given byEq. (5), whilen(p) = /uPI WithEbeingthefunctional of n(p). One can conclude from Eqs. (3)) (5), (28) that the FC is correlated to the unbounded growth of the density of states. As the result, FC serves as a source for new phase transitions which lift the degeneracy of the spectrum. We are going to analyze in brief the situation when the superconductivity wins the competition with the other phase transitions. Now let us switch on the interaction V. Then, as it follows from Eqs. (7) and (23), d N V, when V is sufficiently small [ 6,161, while in the BCS case d, given by Eq. (26), is exponentially small. Inserting the result d - V into Eq. (7)) one finds the pairing correction 6E,(T = 0) to the ground state energy at T=O [15], SE,(T
= 0) - $A.
(29)
This result is analogous to that in the strong coupling limit of the BCS theory [28] and differs drastically from the ordinary BCS result SE0 N A 2. In response to this, an essential increase of the critical magnetic field, destroying superconductivity, can be expected [ 151. The further investigation of this increase will be published elsewhere. Above the critical temperature the system under consideration is in its anomalous normal state, Eq. (3) is valid, and one can observe the smooth segments of the spectra at the Fermi level. In summary, we have developed the density functional formulation of the fermion condensation. As a result, it was demonstrated that the standard KohnSham scheme in not valid beyond the FC phase transition, since one has to introduce the occupation numbers, which serve as variational parameters, while the single-particle potential transforms into the nonlocal one. Our consideration yields a new set of the singleparticle equations for superconductors which allow one to calculate the normal and anomalous densities and the real single-p~ti~l~ excitation spectra as well. We have also shown that the superconducting state, superimposed on FC, and the ordinary BCS state diverge considerably. I thank J.W. Clark, R. Fisch, M. de Llano, I’. Schuck and G.E. Volovik for valuable discussions. I am also
Letters A 249 (1998) 237-241
241
indebted to VA. Khodel for many discussions. This research was supported in part by the Russian Foundation for Basic Research under Grant 98-02-16170.
References 111Z.-X. Shen, D.S.
Dessau, Phys. Ren. 253 (19951
1.
121T. Yokoya et al., Phys. Rev. -Len 76 (1996) 3069. 131 D.H. Lu et al., Phys. Rev. Lett. ‘76 ( 1996) 4845. [4l K. Gofron et al., Phys. Chem. Solids 54 (1993) 1193. 151 D.J. Singh, Phys. Rev. 54 (1995) 1358. [61 V.A. Khodel, V.R. Shaginyan, JETP Lett. 51 (1990) 553. [71 G.E. Volovik, JBTP Lett. 53 (1991) 208. 181 VA. Kbodel, V.R. Shaginyan, V.V. Kbodel, Phys. Rep. 249 (1994)
1.
E91 V.A. Khodel, V.R. Shaginyan,
M.V. Zverev, JBTP Lett. 65 (1997) 253. 1101 G.E. Volovik, JBTP Lett. 63 (1996) 763. 1111 H.B. Shore, E. Zaremba, J.H. Rose, L. Sander, Phys. Rev. B 18 (1978) 6506. fl2l M. Levy, J.l? Perdew, Phys. Rev. 48 ( 1993) 11638. [I31 L. Swierkowski, D. Neilson, J. Szymariski, Phys. Rev. Lett. 67 (1991) 240. Solid State I141 V.A. Khodel, J.W. Clark, V.R. Shaginyan, Commun. 96 (1995) 353. [I51 VA. Khodel, VR. Shaginyan, M.V. Zverev, M. Baldo, JBTP Lett. 65 (1997) 828. 1161 J. Dukelsky, VA. Khodel, l? Schuck, VR. Shaginyan, Z. Phys. 102 ( 1997) 245. 1171 L.D. Landau, JBTP 30 (1956) 1058. t1u E.M. Lifshitz, L.P. Pitaevskii, Statistical physics (Pergamon, Oxford, 1980). 119 M. de Llano, J.I? Vary, Phys. Rev. C 19 (1979) 1083; M. de Llano, A. Plastino, J.G. Zabolitsky, Phys. Rev. C 20 (1979) 2418. PO P. Hohenberg, W. Kohn, Phys. Rev. B 136 (1965) 864; W. Kohn, L.J. Sham, Phys. Rev. A 140 (1965) 1133; W. Kohn, P. Vashishta, Theory of the inhomogeneous electron gas (Plenum, New York, 1983). 1211 L.N. Oliveira, E.K.U. Gross, W. Kohn, Phys. Rev. Lett. 60 (1988) 2430. [22] J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 108 ( 1957) 1175. [23f E.K.U.Gross,S.K~,Int. J.Quant. Chem.25 (lQQ1) 289. 1241 T.L. Gilbert, Phys. Rev. A 12 (1975) 2117. 1251 D.R. Tilley, J. Tilley, Su~~uj~ty and su~~onductivi~ (Hilger, Bristol, 1980). 1261 E.M. Lifshitz, L.P. Pitaevskii, Statistical physics, Part II (Pergamon, Oxford, 1980). [27] P. Ring, P. Schuck, Nuclear many body problem (Springer, New York, 1980). [28] P. Nozieres, S. Schmitt-Rink, J. Low Temp. Phys. 59 (1985) 395.
CQs
30 November 1998
-!B
ELSEVIER
PHYSICS
LETTERS
A
Physics Letters A 249 (1998) 237-241
Density functional theory of fertnion condensation V.R. Shaginyan ’ Petersburg Nuclear Physics Institute, Russian Academy of Science, Gatchina I88350, Russia
Received 11 May 1998; accepted for publication 20 September 1998 Communicated by V.M. Agranovich
Abstract A density functional formulation of the fermion condensation is presented. It is demonstrated that the standard KohnSham scheme is not valid beyond the fetion condensation phase transition since the functional starts to depend on the quasiparticle density matrix. Our consideration also furnishes an opportunity to calculate the real single particle excitation spectra of superconducting systems within the density functional theory. @ 1998 Elsevier Science B.V. PACS: 71.27.+a; 74.20.-z Keywords: Strongly correlated systems; Superconducting state
Several years ago an extremely powerful method was developed for measuring the electronic structure close to the Fermi level [ 11. As the result, a plateau adjacent to the Fermi level has been observed in the electronic spectra of a number of strongly correlated metals. For instance, the spectra of SrzRuOa [2,3] or YBa&!uhOs [4] contain very smooth segments on the Fermi surface. It is a remarkable thing that these spectra have not been reproduced in theoretical calculations [ 5,2]. We submit that the above-described behavior of the electronic spectra can be understood within the framework of the theory of fermion condensation, which was predicted in Ref. [ 61 and associated with the rearrangement of the single-particle degrees of freedom in strongly correlated Fermi systems. The main feature of the fermion condensation (FC) is the appearance of a plateau in the single-particle excitation spectrum at the Fermi level [ 6-91. It has been demonstrated that in the vicinity of, but before, the
density-wave-instability point of a Fermi system the FC phase transition can occur, that is FC may take place if the effective coupling constant is sufficiently strong [ 91. This makes one think that the FC is a rather widespread phenomenon inherent in strongly correlated Fermi systems. For example, FC can arise in such an unusual system as fermions locked in vortex cores in a superfluid Fermi liquid [ lo]. On the other hand, it was demonstrated that the charge-density-wave instability takes place in three-dimensional [ 11,121 and two-dimensional electron liquid [ 131. Thus, the electronic systems of some strongly correlated metals are suited for searching for FC [ 14,151. Now let us outline the key points of the FC theory [ 8,161. FC is related to a new class of solutions of the Fermi-liquid-theory equation [ 171
a(F - PN) = E(p,T) MP, T)
-
p(T)
-
Tln
= 0, ’ E-mail: [email protected].
for the quasiparticle
0375-9601/98/$ - see front matter @ 1998 Elsevier Science B.V. All rights reserved. PII SO375-9601(98)00736-l
1 - n(p,T) MP? T)
(1) distribution
function
n (p, T) , de-
pending on the momentum p and temperature T. Here F is the free energy, p is the chemical potential, while s(p,T) = SEo/Gn(p,T) is the quasiparticle energy, being a functional of n(p,T) just like the energy Ea and the other thermodynamic functions. Eq. (1) is usually rewritten in the form of the Fermi-Dirac distribution n(p,T)
=
[l+exp (E’p7,-“)I-‘-
(2)
In homogeneous matter, the standard solution ar;(p,T = 0) = Bfpn - p), with PF being the Fermi momentum, is obtained assuming de@, T = 0) /dp to be positive and finite near the Fermi level, and T-dependent corrections to the effective mass M*, quasiparticle energy, and the other quantities start with T2-terms [ 181. New solutions of Eq. (2) possess at low T a spectrum ~(p, T) linear in T [ 14,151:
~P>T) - P(T) = T~o(P>>
f3)
within the interval pi < p < pf occupied by the fermion condensate. Inserting (3) into (2), one finds its distribution m(p) it T = 0 no(p)
= (1 +eVo@))--l,
pi
(4)
differs from the Fermi step. At T -+ being continuous within the FC interval, admits a finite limit for the logarithm in Eq. (l),yieldingatT=O [6] which drastically
0, the function
no(p),
When condensation is just starting, the momenta obey pi = &$ = pf. This fact means, as follows from Eq. (5), that A4* -+ 00. Then, at finite temperatures and when FC has taken place, the effective mass, as follows from Eq. (3), M* N 1/T. Within the FC interval the solution no(p) deviates from the Fermi step function nt.7(p) in such a way that the energy E(P) stays constant: e(p) = ,a, while outside of this region, no(p) coincides with f%F(p). We see that the occupation nnmbers m(p) become variational parameters: the solution no(p) emerges if the energy E;I is lowered by alteration of the standard occupation numbers. We recall that the idea of a multiconnected Fermi sphere, resulting in a lowering of the energy, was considered
in Ref. [ 191. FC can be considered as a generalization of this idea. The quasiparticle formalism is applicable to this problem if the damping of the condensate states is small compared to their energy. This condition holds for superfluid systems and for normal ones provided the ratio of the fermion condensate density pc to the density p of the system is small [ 15,161. As we have mentioned above, calculations within the density functional theory (DFT) failed to reproduce the smooth segments in the single-p~ticle excitation spectra, while these smooth segments are the “calling card” of FC. Thus, a density functional formulation of FC is of crucial importance, since it will allow one to understand the roots of these difficulties. As we shall see the local single-particle potential of DFT [20] beyond the FC phase transition becomes a nonlocal one. One of the remarkable peculiarities of the FC phase transition is related to spontaneous breaking of gauge symmetry: the superconductivity order parameter K(P) = (~~,~a_~,_~) has a nonzero value Q(P) = &o(p) [ 1 - PI@(P)] even if the gap d vanishes [ 161. On one hand, this cir~ums~n~e allows us to apply methods of the DFT of superconductivity ]2 1] to consider FC. On the other hand, this consideration also gains new insights into the density functional theory of superconductivity. As a result, one gets the possibility to calculate the single-particle excitation spectrum. In the DFT of superconductivity, there exists a unique functional F(T) of two densities, namely, the normal density of an electron system p and the anomalous density K,
17
dll,r2)
= (vh(f-1)#1(~2)).
(6)
P[p, K] is given by
In atomic units, the functional
F[p, K] = T,[P, K] - T&[P, K] + + -
J JPt’l) Ptrd [u&r)
lo -
I
xd’rt
rc*trt,r2)
- PIP(~) d3r
r2l
d3rl
d3r
2
V(ri,n,r3,r4'4)
d3r2d3rs d3r4 t F,,[p,
fctr3,+4) K]
VR. Shaginyan/Physics
= HP,
~1--T&b, ~1 +
J
[udr> -PIP(~) d3r. (7)
Here Ts[ p, K] and S, [p, K] stand for the kinetic energy and the entropy of a noninteracting system, while F,, [ p( r) , K( 1-1,r2) ] is the exchange-correlation freeenergy functional, V is a pairing interaction, and uext is an external potential. The fourth and fifth terms on the r.h.s. of (7) are the Hartree term due to the Coulomb forces and the pairing interaction, respectively. We suppose V to be sufficiently weak like the model BCSinteraction [ 221. The last equality in Eq. (7) can be considered as the definition of E. For the densities (6) one can employ a quite general form [ 231 dr1)
=
~11~~,,(~1)121~,,,12(1
-
.f-a,n>
u,n
(8)
~-,,-,(rl)lu~,,urr,n(l-2frr,n),
+$g,,(r2)
with the coefficients v,,, malization conditions
and u,,,,
obeying
1%,,12 + Iu,,,12 = 1.
(9) the nor-
pl(rl,r2),
=
CC&G,,(rl)
0-J + +Z,-,(rl) &,-dr2)
239
We demonstrate now that the functions &, are not defined by a local operator. For the sake of simplicity, we omit the spin variables and put T = 0. Minimization of F with respect to @,, leads to the eigenfunction problem
+
J
u&
@‘b, Kl aK(rl
r2)
b(rd
d3r2= kh(rl),
9
(12)
with A, being the Lagrangian multiplier, preserving the normalization of the eigenfunction $,,_ We see that if the anomalous density K were zero, then Iu,]* = 1 below the Fermi level, and Iu, I2 = 0 above the Fermi level, while functions &, would be given by a local operator. Now let us take F as a functional of pt, K. The functions 41 is subject to the orthonormality constraint
The constraints F with
( 13) may be introduced
by replacing
(10)
Here n denotes the quantum numbers such as the momentum p in the case of homogeneous matter or the crystal momentum and the band index in the solid state. We introduce a one-quasiparticle density matrix
m(r13r2)
Letters A 249 (1998) 237-241
4b(r2)
14d2(1 -fc,n>
where ain, are Lagrangian parameters. (14) with respect to 41, one obtains
IG
Minimizing
JPCmrdh(rdd3r2 +QQsf'(n,rd h(rdd3r2 (1%
I~,~l~f~,~l.
(11)
Considering Eqs. (8)) (9)) ( 11)) one can conclude that there exists a one-to-one correspondence between functions p, K and functions pt , K. Therefore the functional F can be treated as a functional of p, K or as a functional of pr , K. If one needs to calculate only the densities p, K then it is reasonable to employ the functional F [ p, K] . On the other hand, the functional F [ pl , K] may be used if it is necessary to gain a deeper insight into the problem, e.g., to calculate the singleparticle excitation spectra, the functions d,, etc.
with je and f’ given by JOtrl,r2)
=
f'trl,r2)
=
~FI[PI,K] Ql(rl,r2)'
@‘I[PIY Kl Mrl,
r2)
’
(16)
From Eq. (15) one can find that (17)
240
KR. Shaginyan/Physics
With this result, the functions @l turn out to be solutions of the eigenvalue equation
s
.f(r~,~,)
&(r2)
d3r2
= w&(rl),
(18)
while the operator f’ drops out of the problem in accordance with the results of Ref. [23]. Notice such a procedure to deal with a nonlocal operator was considered in Ref. [ 241. Eq. ( 18) can be rewritten as a single-particle equation
-;02 + Ue,t(Tl> +
s
,. vxc(r19r2)hdrd
I
+
with u,, being a nonlocal
h(rl,
r2> =
SF,, 6~1(rl r2> 9
d3r2= w$nt(rl),
(19)
potential
(20) *
(21) If V were zero, EI would represent the real singleparticle excitation spectra of the system. The energy el is perturbed by the BCS correlations, but, in fact, this perturbation is small. It is worth noting, as it will be seen below, one has to put V = 0 to derive equations for FC. The same consideration in the case of finite temperatures preserves Eqs. ( 18), ( 19). In order to satisfy the constraint (lo), it is convenient to take vl = cos 81, ul = sin 131.Then one finds
@[PI, ~1
energy El is given [ 261,
= (el - ,u)2 + A:.
(25)
It is helpful to verify Eqs. (22), (23) for homogeneous matter in the weak coupling limit V -+ 0 at T = 0 using the BCS model [22]. Recall that in this case V enters the functional F only through the Hartree term, while F,, is completely defined by the Coulomb interaction. One obtains (26)
where N( 0) is the correct density of the single-particle states at the Fermi level, given by Eq. (2 1) , while the functions 4k are the plane waves. If the potential uXc entering Eq. (19) were a local one, then vXc= const, and the energies EL would be defined by the singleparticle energies of a free electron gas. As a result, instead of the correct density of states in Eq. (26) one would have the density of states of a free electron gas [ 231. So, we are led to the conclusion that it is of crucial importance to take into account from the very beginning the nonlocality, or equivalently, velocitydependence of the single-particle potential (20). One can do that in the same way as it is commonly done in nuclear physics, where the density functional includes velocity-dependent components [ 271. We shall now give further proof of Eq. (5) deduced at T = 0. Consider once again Eq. (22) in the limit V = 0. In this case, Al = 0, and Eq. (22) can be written as (el - p) tan 281 = 0.
(27)
Eq. (27) requires that =
(El--p)tan201+Al=0.
W
(22)
The gap Al is given by (see, e.g., Refs. [23,25])
(23) We now have to minimize minimization yields
“fi=
Here the real excitation
A - exp[-l/N(O)V],
Eqs. ( 11)) ( 18)) one can also
Taking into account infer that
Letters A 249 (1998) 237-241
1
1 + exp(&/T)
.
F with respect to fl. The
&l - #u = 0,
if lull2 # 0, 1,
(28)
with el defined by Eq. (21). Therefore, the fermion condensation solution is a new solution of the old equations. On the other hand, it is seen from Eq. (28) that the standard Kohn-Sham scheme for the single particle equations [20] is no longer valid beyond the point of the FC phase transition, since one has to introduce the nonlocal single-particle potential (20)) while the quasiparticle occupation numbers Iv1I2 become variational parameters, minimizing the total energy. In the homogeneous limit Eq. (28) takes form of
W?. Shaginyan/Physics
Eq. (5), and the single-particle energy F(P) is given byEq. (5), whilen(p) = /uPI WithEbeingthefunctional of n(p). One can conclude from Eqs. (3)) (5), (28) that the FC is correlated to the unbounded growth of the density of states. As the result, FC serves as a source for new phase transitions which lift the degeneracy of the spectrum. We are going to analyze in brief the situation when the superconductivity wins the competition with the other phase transitions. Now let us switch on the interaction V. Then, as it follows from Eqs. (7) and (23), d N V, when V is sufficiently small [ 6,161, while in the BCS case d, given by Eq. (26), is exponentially small. Inserting the result d - V into Eq. (7)) one finds the pairing correction 6E,(T = 0) to the ground state energy at T=O [15], SE,(T
= 0) - $A.
(29)
This result is analogous to that in the strong coupling limit of the BCS theory [28] and differs drastically from the ordinary BCS result SE0 N A 2. In response to this, an essential increase of the critical magnetic field, destroying superconductivity, can be expected [ 151. The further investigation of this increase will be published elsewhere. Above the critical temperature the system under consideration is in its anomalous normal state, Eq. (3) is valid, and one can observe the smooth segments of the spectra at the Fermi level. In summary, we have developed the density functional formulation of the fermion condensation. As a result, it was demonstrated that the standard KohnSham scheme in not valid beyond the FC phase transition, since one has to introduce the occupation numbers, which serve as variational parameters, while the single-particle potential transforms into the nonlocal one. Our consideration yields a new set of the singleparticle equations for superconductors which allow one to calculate the normal and anomalous densities and the real single-p~ti~l~ excitation spectra as well. We have also shown that the superconducting state, superimposed on FC, and the ordinary BCS state diverge considerably. I thank J.W. Clark, R. Fisch, M. de Llano, I’. Schuck and G.E. Volovik for valuable discussions. I am also
Letters A 249 (1998) 237-241
241
indebted to VA. Khodel for many discussions. This research was supported in part by the Russian Foundation for Basic Research under Grant 98-02-16170.
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