A. Zee, Quantum Field Theory in a Nutshell · Problem VI.7.2

Field Theory Redux

Physics — QFT

Problem

As another example from the literature on disorder, consider the following problem. Place

N

points randomly in a

D

-dimensional Euclidean space of volume

V

. Denote the locations of the points by

\vec{x}_i

(i = 1, \ldots, N)

. Let

f(\vec{x}) = (-)\int \frac{d^D k}{(2\pi)^D} \frac{e^{i \vec{k} \cdot \vec{x}}}{k^2 + m^2}

. Consider the

N

N

matrix

H_{ij} = f(\vec{x}_i - \vec{x}_j)

. Calculate

\rho(E)

, the density of eigenvalues of

H

, averaged over the ensemble of matrices, in the limit

N \to \infty

V \to \infty

, with the density of points

\rho_e = N/V

(not to be confused with

\rho(E)

of course) held fixed. [Hint: Use the replica method and arrive at the field theory action

S(\phi) = \int d^D x \left[ \sum_{a=1}^n (|\nabla \phi_a|^2 + m^2 |\phi_a|^2) - \rho_e e^{-(1/z) \sum_{a=1}^n |\phi_a|^2} \right]

. This problem is not entirely trivial; if you need help consult M. M\'ezard et al., Nucl. Phys. B559: 689, 2000, cond-mat/9906135.]