Bose-Einstein-Like condensation of deformed random matrix: A replica approach

Harukuni Ikeda

Published 2022-08-03Version 1

In this work, we investigate a symmetric deformed random matrix, which is obtained by perturbing the diagonal elements of the Wigner matrix by a constant $h_i$. The eigenvector $\vec{x}_{\rm min}$ of the minimal eigenvalue $\lambda_{\rm min}$ of the deformed random matrix tends to condensate at a site with the smallest $h_i$. In certain types of distribution of $h_i$ and in the limit of the large components, this condensation becomes a sharp phase transition, where the mechanism to cause the condensation can be identified with the Bose-Einstein condensation in a mathematical level. We study this Bose-Einstein like condensation phenomenon by means of the replica method. We first derive a formula to calculate the minimal eigenvalue and the overlap of $\vec{x}_{\rm min}$. Then, we apply the formula for two solvable cases: when the distribution $h_i$ has a double peak, and when it has a continuous peak. For the double peak, we find that at the transition point, the participation ratio changes discontinuously from a finite value to zero. On the contrary, in the case of the continuous distribution, the participation ratio continuously goes to zero.