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, EESQPT, are highly localized in the U(n)-basis. Consequently, the dynamics of a system initially prepared in a U(n)-basis vector with energy E EESQPT 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.L.F.S. thanks the ITAMP, where part of this work was done, for its hospitality. We thank Jose M. Arias, Jose E. Garcia-Ramos, Francesco Iachello, and Pedro Perez-Fernandez for useful discussions. L.F.S. was supported by the NSF Grant No. DMR-1147430. F.P.B. was funded by MINECO Grants No. FIS2011-28738-C02-02 and No. FIS2014-53448-C2-2-P and by the Spanish Consolider-Ingenio 2010 (CPANCSD2007-00042)