close

Вход

Забыли?

вход по аккаунту

?

Dynamic Field-Theoretic Approach to a Disordered Model Superconductor.

код для вставкиСкачать
Annalen der Physik. 7. Folge, Band 46, Heft 1, 1989, S. 46-54
VEB J. A. Berth, Leipzig
Dynamic Field-Theoretic Approach
to a Disordered Model Superconductor
By E. KOLLEYand W. KOLLEY
Sektion Physik, Karl-Marx-Universitat Leipzig, DDR
Dedicated to Projessor Dr. G. Bojtu on the Occasion of his 60th Birthduy
A b s t r a c t . On the basis of the attractive Hubbard model with a Gaussian random potential we
describe superconductivity in the presence of quenched disorder. A dynamic path-integral formalism
with a n 8-component Keldysh-Nambn spinor is shown t o be convenient for solving the averaging
problem without the recourse t o the replica trick. The effective action is bosonized in terms of collective variables, namely local &-matricesand real specc pair fields. Saddle point values are calculated
in two simplified limits.
Dynamischer feldtheoretischcr Zugang zu einem ungeordneten
Modell-Supraleitcr
I n h a l t s u b e r s i c h t . Auf der Basis des attrsktiven Hubbard-Modells mit GauBschem Zufallspotential beschreiben wir Supraleitung bci ,,quenched” Unordnung. Gezeigt wird, daB ein dynamischer Wegintegral-Formalismus mit einem 8-komponentigen Keldysh-Nambu-Spinor geeignet ist
zur Losung des Mittelungsproblems ohne Replikatrick. Die effektive Wirkung wird mittels kollektiver
Variabler, namlich lokaler &-Matrizen und Paarfclder im direkten Raum, bosonisiert. Sattelpunktwerte werden in zwei vereinfachten Grenzfsllen berechnet.
1. Introduction
The closed time-path concept proposed by Schwinger [I] and Keldysh [2] (for reviews
see [3, 41) to construct real-time finite-temperature Green functions was recently shown
[5-91 to be a suitable tool for studying interacting electron systems in the presence of
quenched Gaussian disorder. The Green function generating functional represented
by a path integral on the Keldysh contour does not contain a normalization denominator
in contrast to Euclidean (imaginary time) versions of the quantum field theory. Therefore, the dynaniical approach permits to avoid the dangerous replica trick in averaging
over the quenched randomness.
Superconductivity can be treated by means of a combined Nambu-Keldysh technique
cf. [4] and references therein. On the other hand, functional-integral formalisms of
superconductivity are established mainly for clean systems, e.g. in [lo- 121, but also
in the presence of substitutional disorder [ L3, 141. The field-theoretic formulations
for dirty superconductors in [15] and supercontlucting glasses in [16] use the replica
method.
E. KOLLEY
and W. KOLLEY,
Disordered Model Superconductor
47
In this paper we extend our prevjoiis path-integral bosonization schenie [ 71 to a
supercontlucting phase with real-space pairing. Gaussian site-diagonal disorder and onsite attractive electron-electron interaction are described simultaneously. The attractive
Hubbard niodel was first considered by Anderson [17], cf. [I81 in the binary alloy
context. Local attraction was also taken into account in [ll,13, 161. In Section 2 we
derive the collective action and perform in Section 3 the saddle point approximation.
2. Path-Integral Bosonization Scheme
To model superconductivit,y in disordered tight-binding systems we choose the
Hamiltoriian
where ci, (ciu) denotes the creation (annihilation) operator of an electron a t lattice site 7;
M ith spin a. H consists of the free antl nonrandom contribution with the hopping integral
tz,, the raritlom part in ternis of the atomic potential E, subject to the Gaussian probability distribution
and the on-site attract,ive interaction [13, 16-18] with the parameter ;I> 0 (negative
Hubbartl interact'ion). The local coupling in (I) allows one t o describe real-space pairing
of electxons in the simplest way.
On the basis of (I)we ext'end in the following the functional-int,egral formalism from
[ 71 t o the supercontlucting case. Then the quenched-averagetl generating functional
f o r real-time finit'e-temperature Green functions can be represented by the path integral
Z(X, x) = J % g c expci(A,[Z, c ]
+ colsx + j&c)>
( 3)
over C:rassmann variables F,!,")(t) antl c$)(t) with the measure 9 Z 9 c =
n%;;)(t)
9c$)(t),
iun ,t
t being the t'ime. The branch index a = 1 (or 2) corresponds to t'he forward (backward)
path on the Keltlgsh contour. The external sources jij,")(t)and xi:)(t) are also anticomniuting fields. The doublet notation
2
to
C = (Z('), C@))
')
and c =
(z::)
-
is used,bo that,,e.g.,ca3x =
dt(Z$)(t),C'i2)(t)) (I
tvith the third Pauli matrix a3. The nornializa0 -1 (xi"(t))
.#(t)
tion reads Z[O, 01 = 1.
In (3) the fermionic action for the model (1) is given by
--31
W
A&, c] =
2 j-
dt dt'
c,,(t)(b;l(t- f))+icjo(L')
ijo - 00
+ il 2 a(*)
aa
dt $;)(I)
dn)= ( - l ) & + l
Fii)(t)~ $ ) ( t c&?(t),
)
-00
involving the disorder average over (2). The inverse unperturbed Green function GC1
(A means 2 x 2 matrix in the Kelclysh space) in (4) can be written in momentum k
48
Ann. Physik Leipzig 46 (1989) 1
antl frequency co representation as (cf. [a, 51)
(G:,k(W ))-I
A
= &'(w)(Gg;;(cu))-l
G-l(w),
+
with the E'ermi distribution f ( w ) = (8"' 1)-l a t the inverse temperature
inverses of the retarded ( r ) antl advanced ( a ) free propagators
enter
1
(s),including the band energy &k = 2 tiiexp(--ik(Ri
fl i + j
-
p. Here the
Ri)} ( N : nuniher of
lattice sites, Ri: position vector of site i ) and the chemical potential p.
I n view of (4) let us introduce the composite variables
(?; Q2- ) (the trace "Sp" goes over (i, c,a , t } )
Then in (3) the weighting factors exp -Sp
and exp (-d
2 d") \J(")
la)
arise from the disorder-induced quartic and the original
a
quartic interactions in (4), respectively. Botli ternis can be linearized by HubharclStratonovich transformations according to (A3) and (A4) in the Appendix A. Instead
of (3) one gets
Z[X,
1
x] = Tj.9;
.9c 9Q9 A * 9 A exp{i(AFn[Z, c, Q, A*, d] + Za3x + ?x,c)} ,
(9)
where A" = $9Q
expi-Sp Q2}, 9Qis defined in the Appendix A, and 9 A * 9 A =
n 9 ( R e d j a ) ( t ) / J S )9 ( I m A!")(t)/fG).The auxiliary fields Q and A are complex bosonic
In
t
(eornmuting) variables. The spatially local Q;$(t, t ' ) matrix is caused by the site-diagonal randomness, whereas A p ) ( t ) reflects the local pairing. The action containing now
both fermionic and bosonic fields becomes
stands for the 0 matrix (7).
Here f,Ec
I n order to integrate out the ferniionic variables in (9) we have t o construct a compact bilinear form C3-W which takes into account simultaneously the time-path doubling, the spin structure, and the particle-hole degrees of freedom. Therefore we choose
49
E. KOLLEY
and W. KOLLEY,
Disordered Model Superconductor
an 8-component formalism with the enlarged basis
-=
Q
(y,
pi
c1
i
--(2), ( 0 , (2), (l),
CJ.
CJ.
cJ. Ct
-(2),
q)),c=
This gives rise to rewrite (10) as
A p B [ C , C,
1 -
Q, A*, A ] = 2 CG-’[Q, A*, A ] C
+ i Sp + 2 dJ)Id(”’12,
Q2
a
(12)
expressed in terms of the inverse field-dependent one-particle propagator
1
1
G-l[Q, A*, A ] = - (to z,) 80,
@d&’- (to
- z,) 00,
@o(&;’)~
2
2
+
to the transposed matrix with respect to site and branch indices as well as time arguments; e.g., if drl
(6G1),.,“’(t,t’) then (6;’)’ 2 (b&i)7i”(t‘,
t ) . Indeed we are working
with three sets of Pauli matrices, acting in the Keldysh (designated by a ) , spin (a),
and Nambu (z) subspaces. Their combination spans an 8 x 8 space for the quasiparticle.
The symbol @ means the direct (tensorial) product. ao, a,,, and to are unit matrices.
The integration over the Grassmann variables in (9) on the basis of (11)ant1 (12),
where the original measure 9;9 c can be mapped onto 96 = fl 9c,(t) (since C is
L,t
only the regrouped
leads to
e
c),
Z[jj, x ]
=
1
7
J 9Q9 A * 9 A exp i AB[&,A*, A ]
((
1-
- - 2~
(14)
a[&,A*, A1 (T3@00@%) x)) >
( ~ 3 @ ~ 0 @ h )
with the collectit e bosonic action
i
AB[Q,d*,A] = --S~pln(-iG-~[Q,d*,A])
2
+ i S p & 2 + 2 dq)\L I ( ~ ) P .
ix
(15)
+ i d2y 4:;
I fF
8,1
i
C$
i1/2y
air'
I/hLi
idGQ;y
&I+
0
0
-
0
- (&1)T
fLi*
-i
- i fay
I/% tj;,
a;,
-
--i
fhd
(& o;,
- (6-1
0 )T - i V % Q ; f ,
3. Saddle Point Approximation
Up t o here the functional-integral formalism was developed without approximations.
Now we perform the saddle point approsinlation by applying the variational principle
t o the collective action (15) combined with (13) or (16). So one can establish a self-consistent mean-field theory.
I n particular, from (dA,[&, A * , A]/dQ$(t’, t ) ) = O one gets
(Gij(t‘, t )
-
Gl:(t, t ‘ ) ) ,
(17a)
Q?!
( t1 t ’ ) = 2f.t
- ]/F(G;?(t’, t ) - G:;(f, t ‘ ) ) ,
(17c)
@,“,(t, t’) = i
2
1/‘
-
(17d)
2
2
(G:’,2(t’,t ) - G:f(t, t ’ ) ) ,
where ‘‘ - ” indicates the saddle point value. The fields 0,8*,and 2 in the arguments
1
of the Green function G have been dropped. It is nient,ionedthat -(Gti(t’,t ) - Gn(t, t‘))
2
corresponds to the functional average - i ( c $ ? ) ( t ’ ) ~ $ ) ( t ) etc.
) ~ ~Altogether,
,
there are
1G equations of the type (17): the remaining ones belong to the other spin arrangements.
According to (6AB[Q,
A*, d ] / d A y ) ( t ) )= 0 and (6AB[Q,A*, d]/&ly)*(t)) = 0 the
saddle points for the anomalous fields obey
i
op)*(t)= 2 fh(Gf:(t, t ) - G;:(t,
-
i
t)),
-
A(i2)*(t)= - - 12 (Cf:(t, t ) - G::(t, t ) ) ,
2
-
i -
AF)(t)= - - 12 (Gi:(t, t ) - G:,(t, t ) ) ,
2
-
i
(18a)
-
A(%‘)(t)= -f A (Gi2,G(t,t ) - G::(t, t ) ) .
2
(18b)
(18c)
(18~1)
Let us now discuss tala limiting cases of the saddle-point equations: either A = 0 or
y = 0. To exploit (17) and (18)on the mean-field level we suppose the stationary, spatially homogeneous, and paramagnetic case.
E. KOLLEY
and W. KOLLEY,
Disordered Model Superconductor
51
3.1. Case h = 0
Such a disordered quantum system without superconductivity was treated earlier
in 173. Here we have only t o prove that for A = 0 our previous forniulation is reached.
P u t a$:i,(t,t ’ ) = &““’(t - t’) dOot.Then the spatial and temporal Fourier transform of (13)
is given by
-_
1
G,%J) = 7
(to
t3)@Go@(b<;(m)
i
S”(4)
LI
+
+
having dropped the &wgument in G;l(w). The inverse of (19) becomes
(22)
since, in view of (201, G~’(Lu)
= -(4E(w))ll= --&l(--w)
provided that &k = &-k due
to time-reversal symmetry. Altogether, one concludes from (17) and ( 2 0 ) the self-consistency condition for
-I
which fits into the formalism in 171.
One can solve ( 2 3 ) with ( 2 1 ) analytically [7] by assuming a semielliptic unperturbed
-
band. This results in theself-energy2?(w) = --i ( 2 y
$‘((u)
(z2w)&. md
=b l ( w )
?(m)
with I m P ( w ) = - n y ~ ( o )corresponding to the Born approximation. The density of
states
reflects the broadening of the band by disorder; w = 6t (for a S.C. lattice with nearestneighbour hopping t ) denotes the unperturbed half-bandwidth.
3.2. Case y = 0
This concerns superconductivity with local pairing but without disorder. On the
mean-field level we
restrict ourselves t o A($(t) = A(’). At y = 0 we abbreviate (13)
as G-1 = G;’ - A which can be inverted into
G = (I - GJ)-~G~ = G,
+ G,, OC, + ... ,
(25)
Ann. Physik Leipzig 46 (1989) 1
52
expressed in terms of the unperturbed propagator
and the anomalous self-energy part
-
A =
-
1
-i?1{(z1 + iz,) @03@a- (tl- iz,) @o,@i*).
2
(27)
Near the superconducting transition one needs from ( 2 5 ) only the term linear in 0.
According t o ( 2 6 ) and ( 2 7 ) one finds
- 1
:1
Go dGo= f l (T(tl.f iz2)@03@GodGF - 2 (tl- it,)@JU~@&'A~**~,).
1
(28)
Recalling the self-consistency conditions (18c) and (18d), and going over to the
Fourier transformed Green functions one obtains on the basis of (25) and (28) the
relations
-m
1
d(l)= -il- 2
(80,k(w) i G : k ( w ) ) l l
(29a)
-"
k
2
9
m
More explicitly, we have to solve the coupled equations
+ i ~ x ~J(l1) )+ i A x 102 J(2) = 0,
-ilxi1 J(1) + (1- iAxi2) ~ ( 2 =
) 0,
(1
which include the bare "susceptibilities"
m
As before, ek = & - k . By means of the 2 X 2 matrix G O , p ( ~ )being
,
available as the inverse
from (5),one shows immediately that xi2 = xi1.
A nontrivial solution of (30) a t J(')= jc2) implies the condition
+
2
iA(x;' - xi2) = 0 .
Using again the structure of (5) and (31) one evaluates
with the local unperturbed density of states po(w) =
-
(32)
1
TEN
Im
2
GE,k(cU) and K(y)=
k
{ dz(thz ) / ~The last step in (33) involves the standard BCS approximation, i. e. the
0
cutoff of the integral a t the Debye frequency w D due to a small w range of the attractive
iZ interhction. Hence, (32) leads with (33), if wD 3 T,., to the familiar weak-coupling
formula [la]
T, = 1.13 w D e
-- 1
k'O(O)
for the superconducting transition temperature T,.
(34)
E. KOLLEY
and W. KOLLEY,
Disordered Model Superconductor
63
4. Conclusion
We have studied a random attractive Hubbard model for a disordered superconductor. Our aim was to demonstrate the formal compatibility of the Keldysh and Nambu
descriptions on the basis of a path-integral framework involving thermal fermion propagators in 8 x 8 matrix form. From the saddle point approximation two cases were extracted as limits, viz. those without superconductivity and without disorder.
The resultant collective action could be subject to the loop expansion which amounts
to fluctuations around the saddle point. Moreover, Gaussian bond randomness around
a nonzero average hopping as well as nonlocal attractive electron-electron coupling
(e.g. via a contact term as in [14]) may be included in the present formalism. The outstanding problem in our scheme it to look for explicit solutions with respect to the interplay between disorder and electron pairing.
Appendix A. Hubbard-Stratonovich Transformations
Let us quote the Hubbard-Stratonovici transformations needed for decoupling
the quartic action terms in (3). I n contrast to Euclidean versions of the field theory we
are working here within a real-time finite-temperature formalism yielding a Feynmanlike weighting factor eiAF under the functional integral.
(i) Disorder-induced interaction
Starting point is the elementary Gaussian integral
-
J2y (ux+ i(wz* + w*z) + vy>>.
This formula can be translated into a 2 x 2 matrix notation provided that we restrict
ourselves to the Keldysh subspace. Having in mind (7) we identify u = C(’)c(l) = u*,
w = C(2)c(1), w* = C(1)c(2), and v = C(z)c(2)= v*. Combining these elements to the matrix
a (yw
=
iw*) , (AI) takes the form
V
where Q =
X
d Im
QZ1
(x z* ) with real diagonal elements and
Q12
= Q21* , and dQ =
-dQli dReQ2I
in
in12
dQ22
- . The trace “tr” runs over 2 x 2 matrices. The functional integral
JZ
generalization of (A2) becomes
Y
,+Y2a
-
=
8 9Qexp{-Sp
Q2
- I/3l/Sp
sQ)/S
9Qexp{-Sp
Q2},
where now the trace “Sp” goes over site, spin, time-path, and time arguments.
(A3)
Ann. Physik Leipzig 46 (1989) 1
54
(ii) Attractive Hubbard interaction
Concerning Fresnel-type integrals we use because of (8) the fiinctional identity
-
el-i.llAla
= J 9.A*9 .
exp{F
~ i
+ L~*LI)),
].A 12 - i JT@O*
written in short-hand notation with 9 A * 9 A =
n
(A41
$9
References
[13 SCHWINGER,
J.: J. Math. Phys. 0, (1961) 407.
[a] KELDYSH,
L. V.: Zh. Eksp. Tcor. Fiz. 47 (1964) 1616.
[3] CHOU,K.; Su, Z.; HAO,B.; Yu, L. : Phys. Reports 11s (1985) 1.
[4] RAMMER,
J.; SMITH,
H.: Rev. Mod. Phys. .5Y (1986) 323.
[5] BABICHENKO,
V. S.; KOZLOV,
A. N.: Solid State Commun. 59 (1986) 39.
[6] KOLLEY,
E.; KOLLEY,
W.: Phys. Lett. A124 (1987) 330.
[7] KOLLEY,
E.; KOLLEY,
W.: phys. stat. sol. (b) 145 (1988) 586.
[8] KOLLEY,
E.; KOLLEY,
W.: J. Phys. A 2 1 (1988) L333.
[9] WELLER,
W.; SOULEIMAN,
M.; KOLLEY,
W.: MECO 15, Abstract A-7.1, Karpacz 1988.
[lo] POPOV,
V. N.: Functional Integrals in Quantum Field Theory and Statistical Physics (in Russian). Moscow: Atomizdat 1976.
1111 BALDO,M.; GIANSIRACUSA,G.; LOMBARDO,
U.; Puccr, R.: Phys. Lett. 62 A (1977) 509.
[ld] KLEINERT,
H.: Fortschr. Wys. 26 (1978) 565.
[13] KOLLEY,
E.; KOLLEY,
W.: Commun. JINR, E17-80-714, Dubna 1980.
1141 KOLLEY,
E.; KOLLEY,
W.: phys. stat. sol. (b) 114 (1982) 135.
[ls] OPPERMANN,R.: Z. Phys. B 61 (1986) 89.
[l6] OPPERMANN,R.: Z. Phys. B 68 (1987) 389.
[17] ANDERSON,
P. W.: Phys. Rev. Lett. 34 (1975) 953.
[18] YOSHIOKA,
D.; FUKUYAMA,
H.: J. Phys. Soc. Jap. 54 (1985) 2996.
Bei der Redaktion eingegangen am 13. Juni 1988.
Anschr. d. Verf.: Prof. Dr. E. KOLLEY
Dr. W. KOLLEY
Sektion Physik
Karl-Marx-Universitgt Leipzig
LinnBstr. 5
Leipzig DDR-7010
Документ
Категория
Без категории
Просмотров
0
Размер файла
432 Кб
Теги
theoretical, mode, approach, disorder, field, superconductors, dynamics
1/--страниц
Пожаловаться на содержимое документа