Structure of eigenstates and quench dynamics at an excited-state quantum phase transition

We study the structure of the eigenstates and the dynamics of a system that undergoes an excited-state quantum phase transition (ESQPT). The analysis is performed for two-level pairing models characterized by a U(n + 1) algebraic structure. They exhibit a second-order phase transition between two limiting dynamical symmetries represented by the U(n) and SO(n + 1) subalgebras. They are, or can be mapped onto, models of interacting bosons. We show that the eigenstates with energies very close to the ESQPT critical point, E-ESQPT, are highly localized in the U(n) basis. Consequently, the dynamics of a system initially prepared in a U(n)-basis vector with energy epsilon similar to E-ESQPT may be extremely slow. Signatures of an ESQPT can therefore be found in the structures of the eigenstates and in the speed of the system evolution after a sudden quench. Our findings can be tested experimentally with trapped ions.