In this post I’d like to continue looking at the dynamics of disordered systems. In particular I want to discuss the dynamics of the basic discrete 1D Anderson within the approximation developed in the previous post. As we’ll see, this can be reduced to the solution of a partial integro-differential equation.

1. Dynamics of the discrete Anderson model

Recall that the hamiltonian for the discrete 1D Anderson model is given by

where addition is modulo . (We slightly deviate in notation in this post from the previous by numbering sites on the ring beginning with .)

We are interested in the evolution of the system averaged over the disorder:

This density operator contains all the observable information about the system.

It is convenient, for the approach adopted here, to assume that the system is initialised in an eigenstate of . We write these eigenstates as

We now develop an evolution equation for in the interaction picture via the approach described in the previous post (i.e. via introduction of a fictitious ancilla system per site):

where is the memory kernel. Carrying out the approximation described there yields

where

encodes the probability distribution for . If we now trace out the fictitious ancillas we arrive at

Writing out the commutators and carrying out the sum over gives:

where (which can be expressed in terms of a bessel function of the first kind, see the previous post). Notice a peculiarity of this evolution equation: if is a function solely of at then it remains so for all time. Thus, eg., if we can write

then we learn that for all we can always write

Although the expansion (8) is not completely general, it will allow us to study the dynamics of the system initialised in an eigenstate of as there exist functions such that

Let’s now go with the expansion (8) and try to derive an evolution equation for :

A change of variable gives us

Defining

allows us to write

where

and

Thus we arrive at the following evolution equation for

Putting back in the hermitian conjugate terms leaves us with

which can be rewritten as

Such an equation can be expressed in terms of the convolution operation (in the variable) as

This equation is amenable to solution via a fourier transform in and a laplace transform in . I’ll discuss it’s solution in another post.

This entry was posted on Tuesday, June 30th, 2009 at 11:11 am and is filed under disordered quantum systems. You can follow any responses to this entry through the RSS 2.0 feed.
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