Stochastic integral representation for the dynamics of disordered systems

The dynamics of interacting quantum systems in the presence of disorder is studied and an exact representation for disorder-averaged quantities via Ito stochastic calculus is obtained. The stochastic integral representation affords many advantages, including amenability to analytic approximation, potential applicability to interacting systems, and compatibility with tensor network methods. The integral may be expanded to produce a series of approximations, the first of which already includes all diffusive corrections and, further, is manifestly completely positive. The addition of fluctuations leads to a convergent series of systematic corrections. As examples, expressions for the density of states and spectral form factor for the Anderson model are obtained.

Automated summary

The dynamics of interacting quantum systems in the presence of disorder is studied and an exact representation for disorder-averaged quantities via Ito stochastic calculus is obtained. The stochastic integral representation allows for analytic approximation, potential applicability to interacting systems, and compatibility with tensor network methods. The integral can be expanded to produce a series of approximations, the first of which includes all diffusive corrections and is manifestly completely positive. As examples, expressions for the density of states and spectral form factor for the Anderson model are obtained.