Download Random Matrix Theory

21
To accomplish this we expand the exponential function to leave only the term that contains a complete set µ∗1 µ1 ...µ∗n µn
of Grassmann variables and apply the table of integration Eq.(142).
The corresponding integral over the usual complex variables would give the following result:
"
#
Z Y
X
Y π
∗
∗
.
(144)
dϕi dϕi exp −
ϕn an ϕn =
ai
n
i
i
We see a remarkable property: the product of the two integrals is equal to 1. This property remains true for any
Gaussian integrals of commuting and anti-commuting variables. In particular,
Z
Z −1 = Dµ∗ Dµ exp [iS± [µ]] .
(145)
Now the Green’s functions can be represented without the denominator:
Z
GR/A
=
∓i
Dψ ϕn ϕ∗m exp [iS± [ψ]] ,
nm
(146)
where
S± [ψ] = S± [ϕ] + S± [µ] = ±
X
ψi†[E± δij − Hij ] ψj .
(147)
i,j
Here we introduced the super-vectors ψ and ψ †:
†∗
∗
ψ = (ϕ , µ ),
ϕ
ψ=
µ
(148)
and the super-measure:
Dψ = Dϕ∗ Dϕ Dµ∗ Dµ.
(149)
The action Eq.(147) and the integration measure Eq.(149) are super-symmetric, i.e. the commuting and anticommuting variables enter in a fully symmetric way. The super-symmetry is however broken in the pre-exponent
in Eq.(146), as it depends only on the commuting variables.
Now when the problem of denominator is solved by the supersymmetry trick, the next step is to average over the
Gaussian ensemble of Hij . To this end we write:
X †∓i X †∓i
ψi ψj Hij =
ψi ψj Hij + ψj†ψi Hji
2 ij
ij
Averaging of the r.h.s. is done independently for each pair of i, j using the identity:
1
1
i
i
Aij †r.h.s. −
|Hij |2 = −
Hij ± Aij ψi†ψj
ψ ψj ψj†ψi .
Hji ± Aij ψj†ψi −
Aij
Aij
2
2
4 i
From now on for simplicity we will consider the case β = 2. Then
Z
Z
1
Aij ††∗
∗ exp − 1 |H˜ |2
|Hij |2 = dH˜ij dH˜ij
exp
−
ψ
dHij dHij
exp r.h.s. −
ψ
ψ
ψ
ij
j j i ,
Aij
Aij
4 i
where
i
H˜ij = Hij ± ψi†ψj .
2
The simplicity of the case β = 2 is that H˜ij belongs to the same manifold of complex numbers as Hij , so that one may
replace in the integral over the entire manifold H˜ij → Hij . Thus on the right hand side we obtain the normalization
integral for the random matrix ensemble averaging. So we obtain for the disorder average:

+


*
X †X
1
exp ∓i
ψi ψj Hij  = exp −
Aij ψi†ψj ψj†ψi  .
(150)
4
ij
ij