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 $V(\mathit{x})=V_0(\mathit{x})+\sqrt{\epsilon }{v}_1(\mathit{x})$, where $V_0(\mathit{x})$ is a deterministic ``background’’ trapping potential and $v_1(\mathit{x})$ is the disorder. We consider arbitrary trapping potentials $V_0(\mathit{x})$ and white-noise disorder $v_1(\mathit{x})$, in arbitrary spatial dimension $d$. In the weak-disorder limit $\epsilon \to 0$, we find that $P(E)$ scales as $P(E)\sim e^{-s(E)\mathrm{/}\epsilon }$. The large-deviation function $s(E)$ is obtained by calculating the most likely configuration of $V(\mathit{x})$ conditioned on a given ground-state energy $E$. For infinite systems, we obtain $s(E)$ analytically in the limits $E\to \pm \mathrm{\infty }$ and $E\simeq E_0$ where $E_0$ is the ground-state energy in the absence of disorder. We perform explicit calculations for the case of a harmonic trap $V_0(\mathit{x})\propto x^2$ in dimensions $d\in \{1,2,3\}$. 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.