Pseudo-Hermitian β-Hermite Ensemble with an Unbound Metric

It is shown that a pseudo-Hermitian matrix can be constructed which is iso-spectral to a Hermitian matrix of the β-Hermite ensemble. It is then verified that if the matrix, thus constructed, is pseudo-Hermitian with respect to an unbound metric, that is, a metric that diverges for arbitrary large matrix size, it is unstable to small perturbations. This means that it can undergo a spontaneous symmetry breaking no matter how small the perturbation is as the size of matrix increases such that at some point eigenvalues leave the real axis in conjugate pairs. It is shown that the eigenstates of the complex eigenvalues become localized. The phenomenon of pseudo-spectra is discussed and illustrated with a tight-binding model.