8.514: Many-body phenomena in condensed matter and atomic physics Problem Set # 11 Due: 12/09/03
Field integral. Bosons and fermions.
Reading: M. Stone, The Physics of Quantum Fields
1. Bose gas
Consider a saddle point approximation for the functional integral theory of a weakly
nonideal Bose gas with a repulsive interaction,
Z
ZZ i
@t + c:c: ; H( ) d3xdt Z = d ]e h�i S S =
(1)
2
1
1
2
H( ) = ; 2 r ; + 2 ( )2 > 0
a) Show that the saddle point equation S= = 0 gives the Gross-Pitaevsky (GP) equation.
b) Perform a transformation to imaginary time, t !p;it, and consider uctuations
about the stationary solution of the GP equation, = nei , Evaluate the uctuation
determinant contribution to Z at nite temperature, and show that the answer agrees
with what is expected from Bogoliubov quasiparticles, treated as free bosons.
2. Ideal Fermi gas.
Consider functional integral representation of the ideal spinless Fermi gas,
Z
ZZ i
c0 d3 xdt Hc0 = ; 1 r2 ; (2)
@t + c:c: ; H
Z = d ] exp i
2
2
To nd the partition function at nite temperature, go to imaginary time, t ! ;it, and
integrate over the anticommuting variables and ,
c
(3)
Z = det @t ; H
0
Evaluate the determinant, taking into account that fermionic elds obey an antiperiodic
boundary condition, (t + ) P= ;(t). (In Fourier components, that is a sum over
Matsubara frequencies, (t) = n ei!n t n, !n = (2n + 1)T .) Show that the result agrees
with the partition function of an ideal Fermi gas.
3. BCS pairing. Functional integral approach.
Consider a Fermi gas with an attractive interaction,
0 ZZ
1
Z
X
i
c0 + # " " # d3 xdtA (4)
Z = d ] exp @i
@t + c:c: ; H
2
�"#
c0 = ; 1 r2 ; < 0
H
2
This problem describes BCS pairing. It is very convenient to construct the mean eld
theory using the functional integral representation.
a) Perform a transformation to imaginary time, t ! ;it, and apply the HabbardStratonovitch identity: decouple the quartic term with the help of integration over a
complex-valued eld (x t):
Z
RR
Z = d ]e; 21 j�(xt)j2dxdt Z� (5)
0 ZZ
1
X
i
Z� = d ] exp @;
@t + c:c: ; H0( ) ; " # ; # " d3xdtA
2
�"#
Z
Now one can formally integrate over and . Show that the result is
Z
RR j�(xt)j2 dxdt
1 2
@
(
x
t
)
t ; H0
; 21
Z = d ]e
det (
x t) @t + H0 H0 = ; 2 r ; (6)
b) Consider the saddle point approximation for the functional integral (6). Show that
the equation S= = 0, for a constant , has the form
X
= T
!n = (2n + 1)T p = p2 =2m ; (7)
2 + ! 2 + jj2
p!n p
n
Show that this equation is identical to the BCS gap equation.
c) The advantage of the functional integral approach is that it allows to consider
spatially varying order parameter without changing the formalism. In the vicinity of the
critical point, we expect to be small. Expand the eective action obtained above,
ZZ
(x t) t ; H0
S () = 21 j(x t)j2d3xdt ; ln det i@(
(8)
x t) ;i@t + H0
in powers of spatially inhomogeneous (assuming slow variation in space) and derive the
Ginzburg-Landau functional1
ZZ
1
2
4
2
S=
ajj + 2 bjj + cjrj d3xdt
(9)
Relate the coecients a, b, c to the parameters of the microscopic BCS hamiltonian. Show
that the coecient a of the quadratic term changes sign at the BCS transition point, while
b and c are positive.
4. Path integral for spin 1=2 (optional)
In this problem we rst introduce a fermionic representation of the spin 1=2 operators,
and then use it to derive the Wess-Zumino path integral for spin.
a) Consider two fermions, described by the canonical operators a12, a+12 and commutation relations ai aj + aj ai = 0, a+i aj + aj a+i = ij . Show that the operators
+ = a+2 a1 ; = a1+ a2 z = + ;] = a2+a2 ; a1+a1
(10)
obey Pauli commutation relations. Note that, since these operators commute with the
particle number operator nb = a+1 a1 + a+2 a2 , the 2 2 = 4-dimensional Hilbert space has
a canonical spin representation is realized.
a 2-dimensional subspace, nb = 1, in which
P
b) Consider the Hamiltonian Hn = ij a+i aj ij n , where n is a unit vector and ij are
Pauli matrices ( = x y z). Starting from the reprentation of the partition
R function for
R
i (ix�j @t xj ;Hn )dt
H as an integral over two anticommuting variables, Z = dx1dx2 dx1dx2 e
,
perform gaussian integration over xi and obtain an eective action for the dynanmcs of n.
Assuming slow time variation (for large ), expand in the powers of the time derivative
@t to the lowest order. the result is the path integral for a unit vector n(t) that describes
spin 1=2.
c) Add Zeeman coupling to external magnetic eld, B n, and obtain Bloch equations
as a saddle point approximation to this path integral.
The virtue of the GL equation is that it allows to study uctuations near the critical point, and,
more importantly, the complex behavior of superconductors in magnetic eld. The coupling to magnetic
2e
eld is via the long derivative ;ir ! ;ir ; hc
� A.
1