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We study the structure of the eigenstates and the dynamics of a system thatundergoes an excited state quantum phase transition (ESQPT). The analysis isperformed for two-level pairing models characterized by a U(n+1) algebraicstructure. They exhibit a second order phase transition between two limitingdynamical 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 theeigenstates with energies very close to the ESQPT critical point, E_{ESQPT},are highly localized in the U(n)-basis. Consequently, the dynamics of a systeminitially prepared in a U(n)-basis vector with energy close to E_{ESQPT} may beextremely slow. Signatures of an ESQPT can therefore be found in the structuresof the eigenstates and in the speed of the system evolution after a suddenquench. Our findings can be tested experimentally with trapped ions.

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