Classical particle in a box with random potential: Exploiting rotational symmetry of replicated Hamiltonian

Abstract We provide a detailed discussion of the replica approach to thermodynamics of a single classical particle placed in a random Gaussian N ( ≫ 1 ) -dimensional potential inside a spherical box of a finite radius L = R N . Earlier solutions of R = ∞ version of this model were based on applying the Gaussian Variational Ansatz (GVA) to the replicated partition function, and revealed a possibility of glassy phases at low temperatures. For a general R , we show how to utilize instead the underlying rotational symmetry and to arrive to a compact expression for the free energy in the limit N → ∞ directly, without any need for intermediate variational approximations. This method reveals a striking similarity with the much-studied spherical model of spin glasses. Depending on the competition between the radius R and the curvature of the parabolic confining potential μ ⩾ 0 , as well as on the three types of disorder—short-ranged, long-ranged, and logarithmic—the phase diagram of the system in the ( μ , T ) plane undergoes considerable modifications. In the limit of infinite confinement radius our analysis confirms all previous results obtained by GVA. The paper has also a considerable pedagogical component by providing an extended presentation of technical details which are not always easy to find in the existing literature.