8.514: Many-body phenomena in condensed matter and atomic physics Problem Set # 9 Due: 11/20/03
(released late due to instructor il lness)
Path Integral
Reading: R. P. Feynman and A. P. Hibbs, Quantum Mechanics and Path Integrals
M. Stone, The Physics of Quantum Fields
1. Harmonic oscillator.
a) Starting from a path integral representation, obtain the Greens function of a harmonic
oscillator:
1=2
h 2
i
m!
im!
2
hx tjx 0i = 2i sin !t exp 2 sin !t (x + x ) cos !t ; 2xx
(1)
To nd the prefactor, evaluate the innite product of gaussian integrals using the formula
0
0
z = Y 1 ; z2
sin z n>0
2 n2
!
0
1
;
(2)
Note the periodicity in t and the singularities at !t = n, n 2 Z . Discuss their meaning
and check that in the limit ! ! 0 the Greens function of the Schrodinger equation for a
free particle is recovered.
b) Consider the density matrix (x x ) = hx je jxi of a harmonic oscillator at nite
temperature ( = 1=kBT ). It is convenient
(x x ) using the imaginary time
R to evaluate
S
path integral representation, (x x ) = dx( )]e , with the integral take over the paths
x( )0< < constrained by x(0) = x, x( ) = x . Find (x x ) by continuing the result of
part a) to an imaginary time t ! ;i .
R
Evaluate the partition function as Z = tr e =P (x x)dx and check that the result
agrees with the standard geometric series sum Z = n e E , En = h!
(n + 12 ).
0
0
;
H
0
0
;
0
;
0
H
;
n
2. Ising chain and the double well potential.
Consider a one-dimensional Ising problem with a model ferromagnetic interaction between the spins that falls exponentially:
X
Hsi] = ; 21 Jij sisj Jij = Je k i j si = 1
(3)
ij
;
j ; j
Using the Hubbard-Stratonovich transformation
Z Z Y
P A ss
N
P ;1
P hs
1
N=
2
e
= (2)
:::
dhie 2 A h h
1
2
i j
ij
;
;
i j
i j
ij
i
j;
i i
i
i=1
(4)
bring the Ising partition function to the form
Z
Z = :::
Z Y
N
i=1
dhie
;
1
2
P A;1 h h +P
i j
ij
i
j
i
ln(2 cosh hi )
Aij = J e
ki j
;
j ; j
(5)
Invert the matrix Aij (Hint: use Fourier representation) and show that
8C i = j
<
T
T coth k
Aij1 = : B i = j 1 B = J sinh
C
=
(6)
k
J
0 else
Use this result to rewrite the partition function as an imaginary time path integral
;
Z
Z
Z = ::: dh1 :::dhN exp ;
X B
!
2 (hi ; hi+1 ) + U (hi )
U (h) = (C ;B )h2 ;ln (2 cosh h)
i
(7)
Plot U (h) for dierent temperatures. Find the 'phase transition' temperature Tc below
which the potential U becomes a double well.
b) There is no ordering in the Ising model in 1D due to thermal uctuations. The
behavior at T < Tc can be understood by comparing with the imaginary time path integral
for a qunatum mechanical particle in a double well potential,
Z m
Z
2
dx( )] exp ; 2 x_ + U (x) d
(8)
(Argue that at the large radius of spin intercation, k 1, the discrete variable i can be
replaced by a continuous time variable.)
Discuss the meaning of varius statistical mechanical quantities in the quantum-mechanical
instanton tunneling landuage. Relate the correlation length of magnetic ordering in the
Ising chain with the tunnel even-odd level splitting. Use the instanton solution for the
double well potential to estimate the correltion length at T Tc.
c) Consider the above Ising model in the presence of an external magnetic eld,
X
X
Hsi ] = ; 12 Jij sisj ; Bsi
(9)
ij
i
2
Find the magnetization M = hsi as a function of B . (Hint: use the tunneling/instanton
picture and treat B as a double-well asymmetry parameter.)