We want to generalize the setup of the previous sections by considering more than one complex system.
For this purpose we introduce the N–orbital model, where we consider an array of complex atoms with N orbitals each. Electrons can tunnel (or ”hop”) between neighbouring atoms. (This picture of electronic transport is commonly refered to as ”tight binding approximation”.)
For this model we want to use a random matrix approach. The calculations will be done for the GOE.
We consider the Hamiltonian
Hj,x;j′,x′ ≡ (Hxx′)jj′ , (34)
where j, j′ denote the orbitals and x, x′ the positions on the array (the ”lattice sites”). The matrix H is symmetric and its elements are Gaussian distributed. The distribution function reads
P ({H})Y
j≥j′
Y
(x,x′)∈B
dHj,x;j′,x′ =
= 1
N exp
−N 2
X
(x,x′)∈B
axx′Tr (Hxx′Hx′x)
Y
j≥j′
Y
(x,x′)∈B
dHj,x;j′,x′. (35)
Here B denotes the set of lattice bonds and axx′ is a positive, symmetric matrix.
7.1 Evaluation of the DOS for N ∼ ∞
First we recall that there is a representation of the δ function in terms of a Lorentzian, namely δ (Θ− Θl) =−1
πlim
ǫց0 Im
1
Θl− Θ + iǫ
| {z }
= −ǫ
(Θl−Θ)2+ǫ2
.
With this the averaged DOS can be written as hρ(Θ)iH=− lim
ǫց0
1
π|Λ|N Im Tr
(H− Θ + iǫ)−1
H , (36)
whereh. . .iH stands for an average with respect to P ({H}) and |Λ| denotes the number of lattice sites.
The Replica Trick
We introduce a real field φαjx with an integer parameter α — the so called replica index —, which runs from 1, . . . , n and which will be sent to 0 at the end of our calculations. But for now let us consider n replica of our model.
We start with two preliminary notes.
1. We note the following integral identity: 2. For notational convinience we define:
φβ, Aφβ ≡ XN
j,j′=1
X
x,x′∈Λ
φβj,xAjx;j′x′φβj′,x′ .
With this in mind we can write our resolvent as
(H− Θ + iǫ)−1jx;j′x′ = 2i
Interpretation: The functional integral describes a non-interacting many-body system, where the resolvent is that for a single particle. Many-body correlations can also be described by this functional integral.
The denominator can be evaluated yielding Z
Thus the resolvent is given as (H− Θ + iǫ)−1jx;j′x′ =
We see that the left–hand side of this equation does not depend on n. Therefore we can formally take the limit n → 0 on the right–hand side (replica trick). This frees us from evaluating the determinant, which would be a hopeless task.
Averaging with respect to the distribution of H we obtain D(H− Θ + iǫ)−1jx;j′x′
To go further we make two asumptions, namely:
1. limnց0andh. . .iH commute, 2. h. . .iH andR
. . . dφ commute.
With these the averaged resolvent reads D(H− Θ + iǫ)−1jx;j′x′
The averaging can now be performed yielding
The trace in the last line is performed over a product of symmetric n×n matrices with elements (Φx)ββ′ ≡ P
j′φβj′xφβj′′x.
Method of Composite Fields
To perform the integration (37) we introduce a new field Qx, which is conjugate to Φx. Denoting (axx′)−1 ≡ αxx′ we can express the exponential in (38) as an integral over this symmetric random matrix field Qααx ′. The averaged resolvent can now be written as
D
We have used a shorthand notation for the integration meassure here. The action S in the above expression is
For an appropriate choice of the integration paths of Qx and Φx the order of the integrations can be exchanged. For j = j′ and x = x′ one gets
= −2 limn
First, the φ–integration is performed.
Z
All that remains is the integration over the Q–field 1
Note that the dependence on the orbitals j = 1, . . . , N enters only through the prefactor N .
For large N (i. e. N ∼ ∞) we perform a saddle point integration. The condition for a saddle point is δQS′[Q] = 0 .
For our integral this gives 2X For simplicity we only consider a homogeneous replica–symmetric solution, i. e.
Qαβx = δαβQ0 . Denoting w≡P
x′ α−1
xx′ this gives a quadratic equation in Q0
2Q0w + 1
Θ− iǫ + Q0 = 0 Q0(Q0+ Θ− iǫ) + 1
2w = 0 . The solutions for this equation are
Q0=−(Θ − iǫ) ±
Therefore the density of states for our model is approximately ρ(Θ) = 1
πImQ0+ o(1 N)
and shows a semicircular behaviour like in the case of just one random matrix (see fig. 7.1).
Θ0
−R R
R
ρ
Figure 16: Density of states of the N orbital model. R =p 1/2w
7.2 Alternative Approach: Supersymmetric theory
The replica trick has some mathmatical difficulties, because in general it is not clear if the limit n→ 0 and the exchanges of the averaging with this limit and the integration are well defined. To avoid this one can instead introduce a two component field φβj with β = 1, 2. The first component (φ1j) is a complex number and the second (φ2j) is a so called Grassmann variable. A main property of the latter is that they are nilpotent i. e.
φ2jn
= 0 for n≥ 2 .
Therefore the expansion of an analytic function in a power series only has terms up to first order f (φ2j) = f (0) + f′(0)φ2j .
Futhermore, the integration in a Grassmann algebra is defined as a linear mapping to complex variables
as follows Z
With these definitions it can be shown that the Grassmann integral over a Gaussian gives the determinant of the kernel
whereas in the complex case the inverse of the determinant is obtained Z
Thus by integrating over the supersymmetric field the determinant cancels so that the replica trick is not needed..
Considering a random matrix H in the Gaussian unitary ensemble the averaged inverse resolvent reads
The procedure of the replica trick approach leads to a matrix field Q which has complex and Grassmann elements Qββ′ .
Q =
Q11 Q12 Q21 Q22
Like in the case discussed in the former section a saddle point approximation gives a semicircular law for the density of states.
PSfrag x–y plane
Θ
scattering medium
light source
observer
unscattered light scattered light
σ Γ
r1
r0 j0
Figure 17: Typical setup of an scattering experiment
7.3 DOS for N < ∞
For finite systems the density of states ρ(Θ) dΘ cannot be a semicircle, because the matrix elements are Gaussian distributed (i. e. can have values−∞ < Hij <∞).
A 1/N expansion does not give corrections outside the semicircle, which means that in this sense there are non perturbative corrections for N <∞.
Calculating these one finds the following scaling law:
ρ(Θ, N ) = N−ξρ N2ξ(|Θ| − Θ0)
+O N−2ξ
, (39)
with
ξ =
2
6−d : d < 4
1 : d > 4 . (40)
7.4 Level Repulsion
Another interesting feature of random matrix theory can be seen if one considers the distribution of the distance Sik =|Θi− Θk| of a pair of eigenvalues. Wigner surmised this distribution to be
P (S) = aβSβe−λβS2 , (41)
where
a1=π
2 λ1=π4 (GOE) , (42)
a2= 32
π λ2=π4 (GUE) , (43)
a4= 262144
729π3 λ4=9π64 (GSE) . (44)
Thus the probability that two eigenvalues have a distance S→ 0 is vanishing. This effect is called ”level repulsion”.