LMU München
SoSe 2017
Lehrstuhl für Theoretische Nanophysik
PD Dr. F. Heidrich-Meisner
M.Sc. L. Stenzel
Many-Body Physics: Exercise set 1
Discussion of the solutions: Tuesday, May 9, 2017
Exercise 1.1: Spectral representations and Kramers-Kronig relations
The dynamical susceptibility is given by:
Z
Z
χret (q, ω) = −i dr dt χret (r, t)e −i(q·r−(ω+iη)t) ,
(1)
with
χret (r, t) = −iΘ(t)h[Â(t, r), Ô(0, 0)]i .
(2)
The eigenstates of the Hamiltonian Ĥ0 (with respect to which we compute expectation
values h·i) are Ĥ0 |ni = En |ni.
a. Show that χret (t) (we omit the dependence on r for simplicity) can be written as:
1 X i(En −Em )t
(3)
e
hn|Â|mihm|Ô|ni(e −βEn − e −βEm ) ,
χret (t) = −iΘ(t)
Z n,m
where Z =
−βEn
ne
P
is the partition function.
b. Compute the Fourier transformation χret (ω) of χret (t) using the result of (a) and show
that the imaginary part can be written as (for  = Ԇ )
X
π
Imχret (ω) = − (1 − e −βω )
|hn|Ô|mi|2 e −βEn δ(ω + En − Em ) .
(4)
Z
n,m
c. We next introduce a function χ(z) of an arbitrary complex frequency z via
χ(z) =
1 X
e −βEn − e −βEm
hn|Â|mihm|Ô|ni
.
Z n,m
z + En − Em
(5)
The retarded response function χret is obtained from χ(z) by analytic continuation
z → ω + iη. Assuming hn|Â|mihm|Ô|ni ∈ R, show that:
Z
−1
1
χ(z) =
dω 0
Imχret (ω) .
(6)
π
z − ω0
d. Use the last result to derive the Kramers-Kronig relation:
Z
−1
1
0
Reχret (ω) =
dω P
Imχret (ω 0 ) ,
π
ω − ω0
where P denotes the principal part.
e. Interpret your result.
(7)
Exercise 1.2: 2nd quantization, commutators and norm
a. Derive these commutation relations for bosons and fermions from the definitions of the
corresponding creation and annihilation operators in terms of their matrix elements
that were given in the lecture:
[ci , c†j ]+ = δij ,
[ai , a†j ] = δij .
(8)
b. Consider bosons, where a†µ creates a particle in a single-particle state |αi. Calculate
the norm of the state (a†1 )2 a†2 |0i. From this result, what is the norm of an arbitrary
state
(a†1 )n1 · · · (a†i )ni |0i .
(9)