Full distribution of the ground-state energy of potentials with weak disorder

We study the full distribution P(E) of the ground-state energy of a single quantum particle in a potential V(x)=V_{0}(x)+sqrt[ε]v_{1}(x), where V_{0}(x) is a deterministic "background" trapping potential and v_{1}(x) is the disorder. We consider arbitrary trapping potentials V_{0}(x) and white-noise disorder v_{1}(x), in arbitrary spatial dimension d. In the weak-disorder limit ε→0, we find that P(E) scales as P(E)∼e^{-s(E)/ε}. The large-deviation function s(E) is obtained by calculating the most likely configuration of V(x) conditioned on a given ground-state energy E. For infinite systems, we obtain s(E) analytically in the limits E→±∞ and E≃E_{0} where E_{0} is the ground-state energy in the absence of disorder. We perform explicit calculations for the case of a harmonic trap V_{0}(x)∝x^{2} in dimensions d∈{1,2,3}. Next, we calculate s(E) exactly for a finite, periodic one-dimensional system with a homogeneous background V_{0}(x)=0. We find that, remarkably, the system exhibits a sudden change of behavior as E crosses a critical value E_{c}<0: At E>E_{c}, the most likely configuration of V(x) is homogeneous, whereas at E<E_{c} it is inhomogeneous, thus spontaneously breaking the translational symmetry of the problem. As a result, s(E) is nonanalytic: Its second derivative jumps at E=E_{c}. We interpret this singularity as a second-order dynamical phase transition.