PHYSICAL REVIEW B
VOLUME 60, NUMBER 10
1 SEPTEMBER 1999-II
Algorithm for obtaining the gradient expansion of the local density of states and the free energy
of a superconductor
Lorenz Bartosch and Peter Kopietz
Institut für Theoretische Physik, Universität Göttingen, Bunsenstrasse 9, D-37073 Göttingen, Germany
共Received 2 February 1999兲
We present an efficient algorithm for obtaining the gauge-invariant gradient expansion of the local density
of states and the free energy of a clean superconductor. Our method is based on a new mapping of the
semiclassical linearized Gorkov equations onto a pseudo-Schrödinger equation for a three-component wave
ជ (x), where one component is directly related to the local density of states. Because ␺ជ (x) satisfies a
function ␺
linear equation of motion, successive terms in the gradient expansion can be obtained by simple linear
iteration. Our method works equally well for real and complex order parameter, and in the presence of arbitrary
external fields. We confirm a recent calculation of the fourth order correction to the free energy by Kosztin,
Kos, Stone, and Leggett 关Phys. Rev. B 58, 9365 共1998兲兴, who obtained a discrepancy with an earlier result by
Tewordt 关Z. Phys. 180, 385 共1964兲兴. We also give the fourth order correction to the local density of states,
which has not been published before. 关S0163-1829共99兲10833-6兴
I. INTRODUCTION
The phenomenological Ginzburg-Landau theory has
proven to be a powerful method to study the physical properties of superconductors in spatially varying external fields.1
The form of the free energy F 兵 ⌬(r) 其 as a functional of the
complex superconducting order parameter ⌬(r) follows
from general symmetry arguments. Alternatively, for temperatures in the vicinity of the critical temperature and for
slowly varying external fields, F 兵 ⌬(r) 其 can be derived microscopically
from
the
Gorkov
equations
of
superconductivity.2 The microscopic approach can also be
used to obtain corrections to the Ginzburg-Landau free energy functional. Tewordt3 explicitly calculated F 兵 ⌬(r) 其 up
to fourth order in the gradients of ⌬(r). By comparing the
terms with four and two gradients, he could estimate the
range of validity of the usual approximation for the
Ginzburg-Landau free energy functional, where only terms
with two gradients are retained. Unfortunately, a direct expansion of the free energy of a superconductor in powers of
gradients of ⌬(r) is quite laborious, so that the result of
Tewordt is rather difficult to verify, and the calculation of
even higher orders seems almost impossible.
Recently Kosztin, Kos, Stone and Leggett,4 and Kos and
Stone5 共KKSL兲 developed new and more efficient algorithms
to obtain the gradient expansion of the free energy F 兵 ⌬(r) 其 .
Starting point of their calculations are the Bogoliubov–de
Gennes equations6 for the two-component wave functions
and eigenenergies of a superconductor in an external field.
Assuming that 兩 ⌬(r) 兩 is small compared with the Fermi energy, the low-energy physics can be obtained by linearizing
the energy dispersion in the Bogoliubov–de Gennes equations. Then one arrives at the so-called semiclassical Andreev equations,7 which involve only first order derivatives.
In the work4 KKSL then use the special properties of the
Andreev equations to show that for real ⌬(r) the gradient
expansion of F 兵 ⌬(r) 其 can be indirectly obtained from the
0163-1829/99/60共10兲/7452共6兲/$15.00
PRB 60
gradient expansion of the resolvent of a one-dimensional
Schrödinger operator, which is nothing but the square of the
Andreev-Hamiltonian. This resolvent satisfies a nonlinear
second order differential equation, the so-called GelfandDikii equation.4,5,8 KKSL proceed by solving the GelfandDikii equation iteratively, and then use the approximate solution to reconstruct the gradient expansion of F 兵 ⌬(r) 其 .
Although this strategy is more efficient than the direct approach used by Tewordt,3 this procedure still has the disadvantage that, in order to obtain all contributions to F 兵 ⌬(r) 其
with n gradients, one has to calculate terms of higher order
than n in the iterative solution of the Gelfand-Dikii equation.
In the case of complex ⌬(r) KKSL derive a 2⫻2 matrix
generalization of the Gelfand-Dikii equation, which is a form
of the semiclassical Eilenberger equation.9 In this case no
reshuffling of terms in the iterative procedure is necessary.
To take care of a zero-mode KKSL only have to consider the
next order in the gradient expansion. While in the work4
gradients of the magnitude and the phase of ⌬(r) are
handled separately, an expansion in terms of gradients of
⌬(r) itself is given in Ref. 5.
In this work we shall develop an alternative algorithm for
calculating the gradient expansion of the free energy of a
superconductor. We derive a pseudo-Schrödinger equation
which is equivalent to the Eilenberger equation and develop
a gradient expansion of the local density of states. In this
expansion there appears a zero mode which needs to be taken
care of. In contrast to KKSL we fix this problem by making
use of a nonlinear constraint which naturally appears in our
formalism. We do believe that this is an interesting and efficient alternative to the methods developed by KKSL. Our
method works equally well for real and complex ⌬(r) and
directly generates the gradient expansion of the local density
of states of the superconductor, from which the free energy
can be obtained by simple integration. As an application of
our method, we have explicitly calculated the local density
of states ␳ (r, ␻ ) and the free energy F 兵 ⌬(r) 其 up to fourth
order in the gradients. We confirm the result by KKSL4,5 for
7452
©1999 The American Physical Society
PRB 60
ALGORITHM FOR OBTAINING THE GRADIENT . . .
the fourth order correction to F 兵 ⌬(r) 其 , who obtained a discrepancy with the expression published by Tewordt.3
II. FROM THE GORKOV EQUATIONS TO A
PSEUDO-SCHRÖDINGER EQUATION
In this section we shall map the problem of calculating the
local density of states 共DOS兲 of a superconductor onto the
problem of calculating the time evolution of the wave function of an effective J⫽1 quantum spin in a time-dependent
complex magnetic field. For real ⌬ we have recently used a
similar mapping to study the effect of order parameter fluctuations in disordered Peierls chains.10 Here we generalize
this mapping to the case of complex ⌬.
A. Reduction to an effective one-dimensional problem
Starting point of our manipulations are the Gorkov equations for the matrix Green’s function of a clean superconductor in the magnetic field H(r)⫽“ r⫻A(r),
冉
␻ ⫺Hr
⌬ 共 r兲
⌬ * 共 r兲
␻ ⫺Hr*
冊
G共 r,r⬘ , ␻ 兲 ⫽ ␦ 共 r⫺r⬘ 兲 ␴ 0 ,
冋
e
⫺i“ r⫹ A共 r兲
c
Hr⫽
2m
册
⫺␮.
共2兲
␯3
具 G 共 r,r, ␻ 兲 典 n .
␯1 n
共3兲
Here ␯ d is the d-dimensional DOS of free fermions at the
Fermi energy 共including both spin directions兲, and 具 ••• 典 n
⬅ 兰 •••d⍀ n/4␲ denotes directional averaging over the directions of the three-dimensional unit vector n. The auxiliary
Green’s function Gn(r,r⬘ , ␻ ) satisfies
冉
␻ ⫺V 共 r兲 ⫹i v F n–ⵜr
⌬ 共 r兲
⌬ * 共 r兲
␻ ⫺V 共 r兲 ⫺i v F n–ⵜr
⫽ ␦ „n• 共 r⫺r⬘ 兲 …␴ 0 ,
冉
␻ ⫺V 共 x 兲 ⫹i ⳵ x
⌬共 x 兲
⌬ *共 x 兲
␻ ⫺V 共 x 兲 ⫺i ⳵ x
冊
G共 x,x ⬘ , ␻ 兲 ⫽ ␦ 共 x⫺x ⬘ 兲 ␴ 0 .
冊
Gn共 r,r⬘ , ␻ 兲
共4兲
where V(r)⫽(e/c)n–A(r) and v F is the Fermi velocity.
Note that Eq. 共4兲 is the Green’s function analog of the Andreev equations for the wave function.7 Introducing r⫽xn
共5兲
To eliminate the forward scattering potential V(x) in Eq. 共5兲
we set
G共 x,x ⬘ , ␻ 兲 ⫽e ⫺(i/2) ␪ (x) ␴ 3 G1 共 x,x ⬘ , ␻ 兲 e (i/2) ␪ (x ⬘ ) ␴ 3 ,
共6兲
where ␴ 3 is the usual Pauli matrix. Choosing the real function ␪ (x) such that
1
⳵ ␪ 共 x 兲 ⫽V 共 x 兲 ,
2 x
we obtain
冉
␻ ⫹i ⳵ x
˜ 共x兲
⌬
˜ *共 x 兲
⌬
␻ ⫺i ⳵ x
冊
G1 共 x,x ⬘ , ␻ 兲 ⫽ ␦ 共 x⫺x ⬘ 兲 ␴ 0 ,
共7兲
共8兲
with
˜ 共 x 兲 ⫽⌬ 共 x 兲 e i ␪ (x) .
⌬
2
We have set ប⫽1 and measure energies and frequencies
with respect to the chemical potential ␮ . Here c is the speed
of light and ⫺e is the charge of an electron. Note that
KKSL4,5 start from the Bogoliubov–de Gennes equations
共which are eigenvalue equations for the wave functions of the
superconductor兲 while our starting point are the Gorkov
equations 共which are differential equations for the Green’s
functions兲.
If 兩 ⌬(r) 兩 Ⰶ ␮ and if we are only interested in low-energy,
long wavelength properties of the superconductor, we may
linearize the energy dispersion. For calculating the local
DOS and the free energy, we only need to know the Green’s
function at coinciding points r⫽r⬘ . Following Waxman,11
we write
G共 r,r, ␻ 兲 ⬇
⫹r⬜ and ⳵ x ⬅n–ⵜr 共where n•r⬜ ⫽0), dropping the dependence on the parameters r⬜ and n, and setting for simplicity
v F ⫽1, we can write Eq. 共4兲 as a truly one-dimensional equation,
共1兲
where ␴ 0 is the 2⫻2 unit matrix, and the differential operator Hr is given by
7453
共9兲
B. Pseudo-Schrödinger equation for the local DOS
The gradient expansion of the free energy can be obtained
from the local DOS. Because G1 (x,x, ␻ ) has the same diagonal elements as G(x,x, ␻ ), the local DOS ␳ 1 (x, ␻ ) of our
effective one-dimensional model can be written as
␳ 1 共 x, ␻ 兲 ⫽⫺ ␲ ⫺1 Im Tr关 G1 共 x,x, ␻ ⫹i0 兲兴 ,
共10兲
where due to the trace we get a factor of 2 which takes both
spin directions into account. Note that in the matrix equation
共8兲 the derivative operator ⳵ x is proportional to the Pauli
matrix ␴ 3 . In the following it will be advantageous if the
derivative operator ⳵ x is proportional to the unit matrix. Using ␴ 23 ⫽ ␴ 0 , we obtain from Eq. 共8兲 a differential equation
with this property by setting10,5
G2 共 x,x ⬘ , ␻ 兲 ⫽ ␴ 3 G1 共 x,x ⬘ , ␻ 兲 ,
共11兲
so that Eq. 共8兲 implies
˜ 共 x 兲 ␴ ⫹ ⫹⌬
˜ * 共 x 兲 ␴ ⫺ 兴 G2 共 x,x ⬘ 兲
关 i ⳵ x ␴ 0 ⫹ ␻␴ 3 ⫺⌬
⫽ ␦ 共 x⫺x ⬘ 兲 ␴ 0 .
共12兲
For simplicity we have omitted the frequency label. The
three Pauli matrices are denoted by ␴ i , i⫽1,2,3, and ␴ ⫾
⫽ 12 关 ␴ 1 ⫾i ␴ 2 兴 . We now generalize the method described for
real ⌬ in Ref. 10 to the case of complex ⌬. We start by
making the non-Abelian Schwinger-ansatz10,12
G2 共 x,x ⬘ 兲 ⫽U 共 x 兲 G0 共 x,x ⬘ 兲 U ⫺1 共 x ⬘ 兲 ,
共13兲
where U(x) is an invertible 2⫻2 matrix. It is easy to see that
the solution of Eq. 共12兲 can indeed be written in this form
provided G0 and U satisfy
关 i ⳵ x ␴ 0 ⫹ ␻␴ 3 兴 G0 共 x,x ⬘ 兲 ⫽ ␦ 共 x⫺x ⬘ 兲 ␴ 0 ,
共14兲
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LORENZ BARTOSCH AND PETER KOPIETZ
PRB 60
˜ 共 x 兲␴⫹
i ⳵ x U 共 x 兲 ⫽ ␻ 关 U 共 x 兲 ␴ 3 ⫺ ␴ 3 U 共 x 兲兴 ⫹ 关 ⌬
˜ *共 x 兲 ␴ ⫺ 兴 U 共 x 兲 .
⫺⌬
共15兲
We parametrize U(x) as follows:10,13
U 共 x 兲 ⫽e i⌽ ⫹ (x) ␴ ⫺ e i⌽ ⫺ (x) ␴ ⫹ e i⌽ 3 (x) ␴ 3 .
2
˜ * ⫺⌬
˜ ⌽⫹
⳵ x ⌽ ⫹ ⫽⫺2i ␻ ⌽ ⫹ ⫹⌬
,
共17a兲
˜ 关 1⫺2⌽ ⫹ ⌽ ⫺ 兴 ,
⳵ x ⌽ ⫺ ⫽2i ␻ ⌽ ⫺ ⫺⌬
共17b兲
˜ ⌽ ⫹.
⳵ x ⌽ 3 ⫽⫺i⌬
共17c兲
˜ these equations reduce to the set of equations
For real ⌬
given in Ref. 10. Recently Schopohl13 used an analogous
procedure to map the semiclassical Eilenberger equations9
onto a similar set of nonlinear equations.
The one-dimensional local DOS defined in Eq. 共10兲 can
now be written as
␳ 1 共 x, ␻ 兲 ⫽ ␯ 1 ReR 共 x, ␻ ⫹i0 兲 ,
共18兲
where the complex variable R(x) is defined by 共suppressing
again the frequency label兲
R 共 x 兲 ⫽1⫺2⌽ ⫹ 共 x 兲 ⌽ ⫺ 共 x 兲 .
共19兲
In principle one could now try to solve Eqs. 共17a兲 and 共17b兲
˜ (x), and then
iteratively in powers of the derivatives of ⌬
obtain the gradient expansion of the local DOS from Eqs.
共18兲 and 共19兲. However, the structure of the iterative solution
becomes more transparent if we introduce the threecomponent complex vector
␺ជ 共 x 兲 ⫽
冉 冊冉
R共 x 兲
⫽
⫺ 冑2 关 1⫺⌽ ⫹ 共 x 兲 ⌽ ⫺ 共 x 兲兴 ⌽ ⫹ 共 x 兲
1⫺2⌽ ⫹ 共 x 兲 ⌽ ⫺ 共 x 兲
Z ⫺共 x 兲
冑2⌽ ⫺ 共 x 兲
冊
,
共20兲
such that the 2⫻2 matrix Green’s function at coinciding
space points,
1
g 共 x 兲 ⬅ 关 G2 共 x⫹0 ⫹ ,x 兲 ⫹G2 共 x,x⫹0 ⫹ 兲兴
2
共21兲
i
1
g共 x 兲⫽ R共 x 兲␴ 3⫹
关 Z 共 x 兲 ␴ ⫹ ⫹Z ⫹ 共 x 兲 ␴ ⫺ 兴 .
2
冑2 ⫺
共22兲
is equal to
Because the components of ␺ជ can be parametrized by only
two variables ⌽ ⫾ , they satisfy a constraint. Introducing the
row vector
˜␺ T 共 x 兲 ⫽ 关 ⫺Z ⫺ 共 x 兲 ,R 共 x 兲 ,⫺Z ⫹ 共 x 兲兴 ,
共23兲
the constraint can be simply written as
˜␺ T 共 x 兲 ␺ជ 共 x 兲 ⫽R 2 共 x 兲 ⫺2Z ⫹ 共 x 兲 Z ⫺ 共 x 兲 ⫽1,
which can also be expressed as
共24兲
共25兲
ជ (x) we find the equation of moDifferentiating the vector ␺
tion
共16兲
This leads to the following system of equations for the complex functions ⌽ ⫾ (x) and ⌽ 3 (x):
Z ⫹共 x 兲
1
g 2 共 x 兲 ⫽⫺ ␴ 0 .
4
ជ 共 x 兲 ⫽H 共 x 兲 ␺ជ 共 x 兲 ,
⫺ ⳵ x␺
共26兲
with the pseudo-Hamiltonian given by
H共 x 兲⫽
冉
2i ␻
˜ *共 x 兲
冑2⌬
0
˜ 共x兲
冑2⌬
0
˜ *共 x 兲
冑2⌬
0
˜ 共x兲
冑2⌬
⫺2i ␻
冊
.
共27兲
Note that H(x) can also be written as
˜ 共 x 兲 J ⫺ ⫹⌬
˜ *共 x 兲 J ⫹ ,
H 共 x 兲 ⫽2i ␻ J 3 ⫹⌬
共28兲
where the J i are spin J⫽1 operators in the representation
冉
1
0
0
J 3⫽ 0
0
0
0
0
⫺1
冊
,
冉 冊
0
1
0
J ⫹ ⫽ 冑2 0
0
0
†
1 ⫽J ⫺
.
0
0
共29兲
Using the alternative parametrization of ␺ជ (x) in terms of the
2⫻2 matix g(x) given in Eq. 共22兲, we find that our pseudoSchrödinger equation is equivalent to the Eilenberger equation,
˜ 共 x 兲 ␴ ⫹ ⫹i⌬
˜ * 共 x 兲 ␴ ⫺ ,g 共 x 兲兴 . 共30兲
⳵ x g 共 x 兲 ⫽ 关 i ␻␴ 3 ⫺i⌬
This can be seen by using the commutation relations for the
Pauli matrices and reducing Eq. 共30兲 to the equation of motion 共26兲. Thus we have found an unconventional derivation
of the Eilenberger equations which is based on the nonAbelian Schwinger ansatz.
Formally Eq. 共26兲 looks like the imaginary time Schrödinger equation for a J⫽1 quantum spin subject to an imaginary time-dependent magnetic field. Recall that the real part
ជ (x) can be identified
of the second component of the state ␺
with the local DOS. Because our pseudo-Schrödinger equation is linear, the gradient expansion of the local DOS can
now be generated by a straightforward iterative calculation
ជ (x) in powers of gradients. Let us emphasize
of the state ␺
that, apart from the semiclassical approximation 关Eq. 共3兲兴, so
far no approximation has been made. We have simply
mapped the original problem onto an effective pseudoSchrödinger equation, which is the most convenient starting
point for setting up the gradient expansion.
III. RECURSIVE ALGORITHM AND GRADIENT
EXPANSION
The gradient expansion of the local DOS is directly obtained from the second component of the gradient expansion
ជ (x). For convenience we develop the gradient expansion
of ␺
ជ (x) for imaginary frequencies ␻ ⫽iE, because then our
of ␺
pseudo-Hamiltonian 共27兲 is Hermitian and left and right
eigenvectors are identical. Suppose we expand the solution
of Eq. 共26兲 in the form
PRB 60
ALGORITHM FOR OBTAINING THE GRADIENT . . .
␺ជ 共 x 兲 ⫽
⬁
兺 ␺ជ n共 x 兲 ,
n⫽0
H 共 x 兲 ␺ជ 0 共 x 兲 ⫽0,
共32兲
ជ 0 (x) must be an eigenvector of H(x) with eigenvalue
i.e., ␺
zero. The existence of such an eigenvector follows trivially
from the fact that our pseudo-Hamiltonian 共28兲 can be interpreted as the Zeeman-Hamiltonian of a J⫽1 quantum spin in
an external magnetic field. Note that Eq. 共32兲 determines
␺ជ 0 (x) only up to an overall multiplicative factor, which is
ជ 0 (x) satisfy the
fixed by requiring that the components of ␺
constraint 共24兲. This yields 共with ␻ ⫽iE)
冉 冊
˜ *共 x 兲
⌬
1
␺ជ 0 共 x 兲 ⫽
2
冑E ⫹ 兩 ⌬˜ 共 x 兲 兩 2
冑2
E
⫺
.
共33兲
冑2
n⫽0,1, . . . .
共34兲
Because one of the eigenvalues of H(x) vanishes, the inverse
of H(x) does not exist, so that we cannot simply solve Eq.
共34兲 by multiplying both sides by H ⫺1 (x). As a conseជ n⫹1 (x) only up to a vector
quence, Eq. 共34兲 determines ␺
proportional to ␺ជ 0 (x),
␺ជ n⫹1 共 x 兲 ⫽⫺H⬜⫺1 共 x 兲 ⳵ x ␺ជ n 共 x 兲 ⫹c n⫹1 共 x 兲 ␺ជ 0 共 x 兲 ,
H共 x 兲
,
˜ 共 x 兲兩 2兴
4 关 E 2⫹兩⌬
c n⫹1 共 x 兲 ⫽⫺
1
2
n
兺 ˜␺ Ti 共 x 兲 ␺ជ n⫹1⫺i共 x 兲 ,
i⫽1
冋冕
1
f 共 x 兲 ⫽⫺ Re
␤
⫽⫺
n
兺 ˜␺ Ti ␺ជ n⫹1⫺i .
共38兲
i⫽1
⬁
⫺⬁
d ␻ 关 R 共 x, ␻ ⫹i0 兲 ⫺1 兴 ln共 1⫹e ⫺ ␤ ␻ 兲
d ␻ 关 N 共 x, ␻ ⫹i0 兲 ⫺ ␻ 兴
冋兺
2␲
Im
␤
␻ n ⬎0
册
1
e
␤␻
N 共 x,i ␻ n 兲 ⫺i ␻ n .
⫹1
册
册
共39兲
Here ␻ n ⫽(2n⫹1) ␲ / ␤ are fermionic Matsubara frequencies, and
N 共 x, ␻ 兲 ⫽
冕
␻
0
d ␻ ⬘ R 共 x, ␻ ⬘ 兲 .
共40兲
Note that ␯ 1 Re关 N(x, ␻ ⫹i0) 兴 is the integrated onedimensional local DOS. In Eq. 共39兲 we have used the fact
that R(x, ␻ ) is analytic in the upper half of the complex ␻
plane. The local DOS of the three-dimensional superconductor is then given by
␳ 共 r, ␻ 兲 ⫽ ␯ 3 具 Re R n共 r, ␻ ⫹i0 兲 典 n ,
共41兲
where we have now written R n(r, ␻ ) for R(x, ␻ ) to indicate
the dependence on all parameters. Similarly we write f n(r)
for f (x) and obtain for the difference between the free energies of the three dimensional system in the superconducting
and normal state,
共37兲
ជ i (x) by exchangwhere the vector ˜␺ i (x) is obtained from ␺
ing the first and third components and multiplying them by
⫺1; see Eq. 共23兲. For odd n we can show that c n (x)⫽0. We
thus obtain an explicit and very compact recursive algorithm
for calculating the gradient expansion of the local DOS. To
ជ (x) is given by Eq. 共33兲. This corzeroth order the vector ␺
responds to the adiabatic approximation of elementary quantum mechanics. The step n→n⫹1 is summarized as follows:
⬁
⫺⬁
冋冕
⫽⫺Re
共36兲
i.e., H⬜⫺1 (x) is proportional to H(x). To fix the constant
c n⫹1 (x) in Eq. 共35兲, we require that the components of
n⫹1 ជ
兺 i⫽0
␺ i (x) satisfy the constraint 共24兲. This implies
␺ជ 0
2
It is easy to implement this iterative algorithm on a symbolic
manipulation program 共such as MATHEMATICA兲. In this way
the lowest few terms in the gradient expansion can be obtained in a straightforward manner.
Given the gradient expansion of the local DOS 关which
can be directly obtained from the second component of
␺ជ (x)兴, we can calculate the free energy by simple integrations. Using the fact that in the normal state R⫽1, the difference between the free energy densities in the superconducting and normal state of our effective one-dimensional
model at inverse temperature ␤ is given by ␯ 1 f (x), where
共35兲
where H⬜⫺1 (x) is the inverse of H(x) in the subspace orជ 0 (x). Using the fact that the two nonvanishing
thogonal to ␺
˜ (x) 兩 2 兴 1/2, we
eigenvalues of H(x) are given by ⫾2 关 E 2 ⫹ 兩 ⌬
find
H⬜⫺1 共 x 兲 ⫽
␺ជ n⫹1 ⫽⫺H⬜⫺1 ⳵ x ␺ជ n ⫺
˜ 共x兲
⌬
For the higher order terms we obtain the simple recursion
relation
⳵ x ␺ជ n 共 x 兲 ⫽⫺H 共 x 兲 ␺ជ n⫹1 共 x 兲 ,
␺ជ i given for i⫽0, . . . ,n,
共31兲
ជ n (x) involves n derivatives with rewhere by definition ␺
spect to x. Obviously
7455
F 兵 ⌬ 共 r兲 其 ⫽
冕 冋
d 3 r ␯ 3 具 f n共 r兲 典 n⫹
册
兩 ⌬ 共 r兲 兩 2 关 H共 r兲 ⫺He 共 r兲兴 2
.
⫹
␭
8␲
共42兲
The second term is the field energy of the superconducting
pair potential 共where ␭ is the coupling constant of the
Gorkov electron-electron interaction兲, and the last term is the
magnetic field energy of the superconductor 关where He (r) is
the externally applied magnetic field9兴.
7456
LORENZ BARTOSCH AND PETER KOPIETZ
The above equations allow for a simple recursive calculation of the gradient expansion of the free energy. To compare our results with Tewordt3 and KKSL4,5 we use Eq. 共38兲
to calculate all terms in the gradient expansion up to fourth
PRB 60
order. Systematically adding total derivatives to the expressions for the free energy 共which do not change the bulk properties of the superconductor兲, we find the following expressions.
Zeroth order:
␳ (0) 共 r, ␻ 兲 ⫽ ␯ 3 ⌰ 共 ␻ 2 ⫺ 兩 ⌬ 兩 2 兲
F (0) 兵 ⌬ 共 r兲 其 ⫽
冕
冋
d 3r ⫺ ␯ 3
2␲
␤
Second order:
n
冓
Fourth order:
␳ (4) 共 r, ␻ 兲 ⫽ ␯ 3 兩 ␻ 兩 ⌰ 共 ␻ 2 ⫺ 兩 ⌬ 兩 2 兲
冓
冕
2␲
␤
d 3r
冑␻
兺 关 冑␻ 2n ⫹ 兩 ⌬ 兩 2 ⫺ ␻ n 兴 ⫹
␻ ⬎0
␳ (2) 共 r, ␻ 兲 ⫽ ␯ 3 兩 ␻ 兩 ⌰ 共 ␻ 2 ⫺ 兩 ⌬ 兩 2 兲 ⫺
F (2) 兵 ⌬ 共 r兲 其 ⫽ ␯ 3
兩␻兩
2
共43兲
,
册
兩 ⌬ 共 r兲 兩 2 关 H共 r兲 ⫺He 共 r兲兴 2
.
⫹
␭
8␲
5 关 n•ⵜ r兩 ⌬ 兩 2 兴 2 1 关共 n•ⵜ r兲 2 兩 ⌬ 兩 2 兴 ⫺3 兩 n•Dr⌬ 兩 2
⫺
32 共 ␻ 2 ⫺ 兩 ⌬ 兩 2 兲 7/2 8
共 ␻ 2 ⫺ 兩 ⌬ 兩 2 兲 5/2
兺
␻ ⬎0
n
⫺兩⌬兩2
冓
⫺
1 关 n•ⵜ r兩 ⌬ 兩 2 兴 2 1 兩 n•Dr⌬ 兩 2
⫹
32 共 ␻ 2n ⫹ 兩 ⌬ 兩 2 兲 5/2 8 共 ␻ 2n ⫹ 兩 ⌬ 兩 2 兲 3/2
冔
冔
共44兲
共45兲
,
n
共46兲
.
n
1155 关 n•ⵜ r兩 ⌬ 兩 2 兴 4
42 关 n•ⵜ r兩 ⌬ 兩 2 兴 2 „11关共 n•ⵜ r兲 2 兩 ⌬ 兩 2 兴 ⫺15兩 n•Dr⌬ 兩 2 …
⫹
2048 共 ␻ 2 ⫺ 兩 ⌬ 兩 2 兲 13/2 512
共 ␻ 2 ⫺ 兩 ⌬ 兩 2 兲 11/2
⫹
7 5 兩 n•Dr⌬ 兩 4 ⫹4 关 n•ⵜ r兩 ⌬ 兩 2 兴关共 n•ⵜ r兲 3 兩 ⌬ 兩 2 兴 ⫺10†n•ⵜ r共 兩 n•Dr⌬ 兩 2 关 n•ⵜ r兩 ⌬ 兩 2 兴 兲 ‡⫹3 关共 n•ⵜ r兲 2 兩 ⌬ 兩 2 兴 2
128
共 ␻ 2 ⫺ 兩 ⌬ 兩 2 兲 9/2
⫹
1 关共 n•ⵜ r兲 4 兩 ⌬ 兩 2 兴 ⫺5 关共 n•ⵜ r兲 2 兩 n•Dr⌬ 兩 2 兴 ⫹5 兩 共 n•Dr兲 2 ⌬ 兩 2
32
共 ␻ 2 ⫺ 兩 ⌬ 兩 2 兲 7/2
F (4) 兵 ⌬ 共 r兲 其 ⫽ ␯ 3
⫹
冕
2␲
␤
d 3r
兺
␻ n ⬎0
冓
⫺
冔
共47兲
,
n
35 关 n•ⵜ r兩 ⌬ 兩 2 兴 4
70 兩 n•Dr⌬ 兩 2 关 n•ⵜ r兩 ⌬ 兩 2 兴 2
⫹
2048 共 ␻ 2n ⫹ 兩 ⌬ 兩 2 兲 11/2 512
共 ␻ 2n ⫹ 兩 ⌬ 兩 2 兲 9/2
1 5 兩 n•Dr⌬ 兩 4 ⫺10兩 n•Dr⌬ 兩 2 关共 n•ⵜ r兲 2 兩 ⌬ 兩 2 兴 ⫹ 关共 n•ⵜ r兲 2 兩 ⌬ 兩 2 兴 2 1 兩 共 n•Dr兲 2 ⌬ 兩 2
⫺
128
32 共 ␻ 2n ⫹ 兩 ⌬ 兩 2 兲 5/2
共 ␻ 2n ⫹ 兩 ⌬ 兩 2 兲 7/2
e
Here ⌰(x) is the step function, and Dr⫽ⵜ r⫹2i A(r) is the
c
covariant gradient. In deriving the above expressions we
have used
具 共 n•A兲共 n•B兲共 n•C兲共 n•D兲 典 n⫽
冔
.
共48兲
n
1
关共 A–B兲共 C–D兲 ⫹ 共 A–C兲
15
⫻共 B–D兲 ⫹ 共 A–D兲共 B–C兲兴 .
共51兲
兩 共 n•ⵜ r兲 ⌬ 兩 ⫽ 兩 共 n•Dr兲 ⌬ 兩 ,
l˜
l
l⫽0,1,2, . . . .
共49兲
Note that all terms in our expansion are gauge invariant.
Since the terms involving an odd number of derivatives cancel after directional averaging we only presented the terms of
even order. The directional averaging is easily done using
1
3
具 共 n•A兲共 n•B兲 典 n⫽ A–B,
共50兲
Because after directional averaging the above expressions
look even more complicated, we do not give the explicit
results in this work. Comparing Eq. 共48兲 with the fourth
order correction to the free energy given by Tewordt3 and by
KKSL,4,5 we find that our result agrees with that of KKSL
关up to a total derivative which eliminates the first term on the
right-hand side of Eq. 共48兲兴. We have not been able to find in
the published literature an explicit expression for the fourth
order correction to the gauge invariant local DOS, so that we
cannot compare Eq. 共47兲 with the results of other authors.
PRB 60
ALGORITHM FOR OBTAINING THE GRADIENT . . .
IV. CONCLUSION
We have developed an efficient algorithm for calculating
the gradient expansion of the local DOS and the free energy
of a superconductor in an external magnetic field. The linearization of the energy dispersion allows to obtain the local
DOS directly from the second component of a three compoជ (x) satisfying a pseudo-Schrödinger
nent wave function ␺
equation, which is formally identical to the Schrödinger
equation of a J⫽1 quantum spin in a time-dependent complex magnetic field. The gradient-expansion of this pseudoSchrödinger equation turns out to be quite simple and
straightforward and can be easily implemented on a symbolic manipulation program such as MATHEMATICA. It seems
to us that our algorithm gives an interesting alternative to the
algorithms developed by KKSL.4,5 Our final result for the
1
See, for example, M. Tinkham, Introduction to Superconductivity,
2nd ed. 共McGraw-Hill, New York, 1996兲.
2
See, for example, A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical
Physics 共Dover, New York, 1963兲, Chap. 7.
3
L. Tewordt, Z. Phys. 180, 385 共1964兲.
4
I. Kosztin, S. Kos, M. Stone, and A. J. Leggett, Phys. Rev. B 58,
9365 共1998兲.
5
S. Kos and M. Stone, Phys. Rev. B 59, 9545 共1999兲.
6
P. G. de Gennes, Superconductivity of Metals and Alloys 共Benjamin, New York, 1966兲.
7457
contribution to the gauge-invariant free energy containing
four gradients agrees with KKSL5 and is in disagreement
with the expression given by Tewordt.3 We believe that our
algorithm could also prove advantageous in other problems
where fluctuation effects can be mapped onto a 2⫻2 matrix
equation for a Green’s function in external fields. One interesting example are two coupled Tomonaga-Luttinger models.
ACKNOWLEDGMENTS
We would like to thank Simon Kos for pointing out to us
that our pseudo-Schrödinger equation is in fact equivalent to
the Eilenberger equation. This work was financially supported by the DFG 共Grants No. Ko 1442/3-1 and Ko 1442/
4-1兲.
A. F. Andreev, Zh. Éksp. Teor. Fiz. 46, 1823 共1964兲 关Sov. Phys.
JETP 19, 1228 共1964兲兴.
8
I. M. Gelfand and L. A. Dikii, Russian Math. Surveys 30, 77
共1975兲.
9
G. Eilenberger, Z. Phys. 180, 385 共1964兲; 182, 427 共1965兲; 214,
195 共1968兲.
10
L. Bartosch and P. Kopietz, Phys. Rev. Lett. 82, 988 共1999兲.
11
D. Waxman, Ann. Phys. 共N.Y.兲 223, 129 共1993兲.
12
J. Schwinger, Phys. Rev. 128, 2425 共1962兲.
13
N. Schopohl, cond-mat/9804064 共unpublished兲.
7