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

Abstract

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. Introduction.-Quantum phase transitions (QPTs) occur at zero temperature. They correspond to an abrupt change in the character of the ground state of a system when a control parameter passes a critical point [1]. The subject, which permeates condensed matter and nuclear physics, has become one of the highlights of experiments with cold gases, where transitions from a superfluid to a Mott insulator [2] and from a normal to a superradiant phase [3] have been observed. The investigations are not restricted to the properties of the ground state, but extend also to the dynamics of systems undergoing QPTs. In this context, one finds studies about the quantum analogue of the Kibble-Zurek mechanism [4], as well as the relaxation time [5][6][7][8], revivals [9,10], and temporal fluctuations [11,12] at critical points.Recently, the concept of ground state QPT has been generalized to encompass also QPTs occurring at excited states. These so-called ESQPT refer to a singularity in the energy spectrum caused by the clustering of excited levels at a critical energy [13][14][15][16]. This critical point can be reached either for a constant excited energy by varying the control parameter(s) or by fixing the latter and increasing the energy. ESQPTs have been investigated for a broad class of many-body quantum systems, such as the Lipkin-Meshkov-Glick (LMG) [17][18][19], the molecular vibron [15,20] In terms of dynamics, it has been shown that an ESQPT leads to random oscillations of the survival probability in isolated systems [21], to singularities in the evolution of observables [32], and to maximal decoherence in open systems [18,33]. Despite these works, studies of the effects of ESQPTs on systems' evolutions are still scarce.In this Rapid Communication, we provide insights into the

Excited-state quantum phase transitions in the interacting boson model: Spectral characteristics of0+states and effective order parameter

“…More recently, considerable attention has been given to the so-called excited-state quantum phase transition (ESQPT) [6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Unlike GSQPT, an ESQPT can occur not only with variation of the control parameters of a model Hamiltonian, but also with the increasing of the excitation energy.…”

Monodromy in Dicke superradiance

“…In the present case, the localization results in a sharp local decrease of the atom-field entanglement entropy [40] and has specific consequences for the dynamics of relaxation processes following a quantum quench [48,51]. In particular, a quench from the λ < λ c ground state to the λ γ > λ c critical region results in a slow decay of the initial state (due to its large overlap with the eigenstates in the critical region) [51], while a quench from the λ > λ c side leads, on contrary, to an immediate decay of the initial state and its weak re-occurrences [48]. A more detailed analysis of the quantum quench dynamics in the extended Dicke model is presently a subject of our study.…”

Excited-state quantum phase transitions studied from a non-Hermitian perspective

“…They have been linked with the bifurcation phenomenon [20] and with the exceedingly slow evolution of initial states with energy close to the ESQPT critical point [18][19][20]. Equivalently to what one encounters in QPTs, the nonanalycities associated with ESQPTs occur in the thermodynamic limit.…”

“…This divergence in the density states at the lowest energy moves to higher energies as the control parameter increases above the QPT critical point. The energy value where the density of states peaks marks the point of the ESQPT.ESQPTs have been analyzed in various theoretical models [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and have also been observed experimentally [22][23][24][25][26][27]. They have been linked with the bifurcation phenomenon [20] and with the exceedingly slow evolution of initial states with energy close to the ESQPT critical point [18][19][20].…”

“…The degeneracy is lifted for E > E c , where the eigenstates have structure closer to the U (1) symmetry. Quantities such as the participation ratio [18,20] and the fidelity [19] have been used to capture the abrupt changes in the structures of eigenstates caused by ESQPTs.…”