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 Itô 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.

Figure 1

The (dimensionless) quantity $X\left(t\right)\equiv \frac{1}{N}\mathbb{E}[tr\left(e^{itH}\right)]$ for the Anderson model on 30 sites (the $x$ axis is time in units where $\hslash =1$). (It turns out that the results for 30 sites are already indistinguishable from the $N=\infty$ limit.) The result of numerical sampling with 100 samples is shown in black. $X\left(t\right)=X_0\left(t\right)+X_2\left(t\right)$, the second-order result calculated via the stochastic Dyson series (47), is shown in blue (dashed).

Figure 2

The (dimensionless) spectral form factor $\frac{1}{N^2}\mathbb{E}\left[\right|tr\left(e^{itH}\right){|}^2]$ for the Anderson model on 30 sites (the $x$ axis is time in units where $\hslash =1$). Again, the results for 30 sites are indistinguishable from the $N=\infty$ limit. The result of numerical sampling with 100 samples is shown in black. The zeroth-order diffusion-corrected term from the stochastic Dyson series is shown in blue (dotted). For comparison, the zeroth-order result from ordinary time-dependent perturbation theory is shown in gray (dashed). Note that the zeroth-order stochastic Dyson series result already incorporates the dephasing decay resulting from the disorder average.