Localization_length

The transfer matrix method makes it possible to solve the Schroedinger equation at a given energy E (not necessarily an energy level) assuming some boundary condition on the left side and propagating to the right side. The propagation equation is in 1D a simple recurrence relation between \psi_{i-1}, \psi_i and \psi{i+1}, so that the propagation is trivial. In dimension d, one has to propagate a structure of dimension d-1 representing the wavefunction in an hyperplane perpendicular to the direction of propagation. The numerical method is intrisincally unstable, so that it typically picks an exponentially growing solution, with a positive Lyapounov exponent. In 1D, the only Lyapounov exponent is the inverse of the localization length. For a certain system size (much longer than the localization length), the statistical distribution of the Lyapounov exponent is close to a Gaussian distribution whose width decays as the inverse square root of the system size. Thus, the longer the system, the better. The same uncertainty is obtained by using many small systems or a single long one, all what matters is the sum of all lengths. As the number of Flops is proportional to the system size, the optimum is to use a system size significantly longer than the localization length, but not much longer, for a better use of the caches.

This is only for non-interacting systems.

It works for 1d spin systems.

WARNING: Currently does not work for multidimensional systems