Spontaneous symmetry breaking in U(N) invariant ensembles with a soft confinement potential
A solution is provided to the problem of finding the probability distribution of elements of a random matrix in terms of the distribution of eigenvalues and eigenvectors. It is then proved that completely isotropic eigenvectors can become localized when the eigenvalues increase exponentially. This general result confirms the prediction of a spontaneous breaking of the unitary transformation, U(N), invariance of random matrix ensembles, in the limit of extremely soft confinement. An algorithm is implemented to generate eigenvectors with broken symmetry. The theory is then verified numerically.
Pato
MP
Nuclear Theory and Elementary Particle Phenomenology Group, Instituto de Fisica, Universidade de Sao Paulo, Caixa Postal 66318, 05315-970 Sao Paulo, SP, Brazil.