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)=V0(x)+ϵv1(x), where V0(x) is a deterministic "background"trapping potential and v1(x) is the disorder. We consider arbitrary trapping potentials V0(x) and white-noise disorder v1(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≃E0 where E0 is the ground-state energy in the absence of disorder. We perform explicit calculations for the case of a harmonic trap V0(x)∝x2 in dimensions d∈{1,2,3}. Next, we calculate s(E) exactly for a finite, periodic one-dimensional system with a homogeneous background V0(x)=0. We find that, remarkably, the system exhibits a sudden change of behavior as E crosses a critical value Ec<0: At E>Ec, the most likely configuration of V(x) is homogeneous, whereas at E<Ec 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=Ec. We interpret this singularity as a second-order dynamical phase transition.