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arXiv:1506.06765 · Physical Review A

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

Léa Ferreira dos Santos, Francisco Pérez-Bernal

Published Nov 9, 2015• 80 citations
arXivHTMLPDF

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 $\mathrm{U}(n+1)$ algebraic structure. They exhibit a second-order phase transition between two limiting dynamical symmetries represented by the $\mathrm{U}(n)$ and $\mathrm{SO}(n+1)$ 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, ${E}_{\text{ESQPT}}$, are highly localized in the $\mathrm{U}(n)$ basis. Consequently, the dynamics of a system initially prepared in a $\mathrm{U}(n)$-basis vector with energy $\mathcal{E}\ensuremath{\sim}{E}_{\text{ESQPT}}$ 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.

DOI
10.1103/physreva.92.050101

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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)\mathrm{U}(n+1)U(n+1) algebraic structure. They exhibit a second-order phase transition between two limiting dynamical symmetries represented by the U(n)\mathrm{U}(n)U(n) and SO(n+1)\mathrm{SO}(n+1)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{E}_{\text{ESQPT}}EESQPT​, are highly localized in the U(n)\mathrm{U}(n)U(n) basis. Consequently, the dynamics of a system initially prepared in a U(n)\mathrm{U}(n)U(n)-basis vector with energy E\ensuremath∼EESQPT\mathcal{E}\ensuremath{\sim}{E}_{\text{ESQPT}}E\ensuremath∼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.
This work is concerned with the excited state quantum phase transitions (ESQPTs) defined in Ann. Phys. 323, 1106 (2008). In many‐body models that exhibit such transitions, the ground state quantum phase transition (QPT) occurs in parallel with a singularity in the energy spectrum that propagates to higher energies as the control parameter increases beyond the QPT critical point. The analysis of the spectrum has been a main tool for the detection of these ESQPTs. Studies of the effects of this transition on the system dynamics are more limited. Here, we extend our previous works and show that the evolution of an initial state with energy close to the ESQPT critical point may be extremely slow. This result is surprising, because it may take place in systems with long‐range interactions, where the dynamics is usually expected to be very fast. A timely example is the one‐dimensional spin‐1/2 model with infinite‐range Ising interaction studied in experiments with ion traps. Its Hamiltonian has a U(2) algebraic structure. More generally, the slow dynamics described here occurs in two‐level bosonic or fermionic models with pairing interactions and a Hamiltonian exhibiting a QPT between its limiting and dynamical symmetries. In this work, we compare the results for , and 3.
Excited-state quantum phase transitions (ESQPTs) are generalizations of quantum phase transitions to excited levels. They are associated with local divergences in the density of states. Here, we investigate how the presence of an ESQPT can be detected from the analysis of the structure of the Hamiltonian matrix, the level of localization of the eigenstates, the onset of bifurcation, and the speed of the system evolution. Our findings are illustrated for a Hamiltonian with infinite-range Ising interaction in a transverse field. This is a version of the Lipkin-Meshkov-Glick (LMG) model and the limiting case of the one-dimensional spin-12\frac{1}{2}21​ system with tunable interactions realized with ion traps. From our studies for the dynamics, we uncover similarities between the LMG and the noninteracting XX models.
Excited-state quantum phase transitions (ESQPTs) strongly influence the spectral properties of collective many-body quantum systems, changing degeneracy patterns in different quantum phases. Level degeneracies in turn affect the system's dynamics. We analyze the degeneracy dependence on the size of two-level boson models with a u(n+1)u(n+1)u(n+1) dynamical algebra, where nnn is the number of collective degrees of freedom. Below the ESQPT critical energy of these models, the energy gap between neighboring levels that belong to different symmetry sectors gets close to zero as the system size increases. We report and explain why this gap goes to zero exponentially for systems with one collective degree of freedom but algebraically in models with more than one degree of freedom. As a consequence, we show that the infinite-time average of out-of-time-order correlators is an ESQPT order parameter in finite systems with n=1n=1n=1, but in systems with n>1n>1n>1, this average only works as an order parameter in the mean-field limit.
Excited-state quantum phase transitions (ESQPTs) have been extensively studied in two-level models, but their characterization remains challenging in systems displaying mixed regular and chaotic dynamics. In this work, we investigate ESQPTs within the three-level Lipkin-Meshkov-Glick model, where an enlarged Hilbert space and multiple separatrices give rise to rich spectral structures strongly influenced by chaos. To investigate the different dynamical regions, we have calculated Poincaré sections and Peres lattices. In addition, by combining chaos-sensitive measures with standard ESQPT diagnostics, we provide a static analysis of ESQPT signatures in this model and establish a robust framework for future studies of its dynamical behavior. The degree of chaos and the Kullback-Leibler divergence are found to be very effective chaos-sensitive measures, which are complementary to ESQPT diagnostics such as the mean field limit and the participation ratio. Hence we provide a standard framework to work with ESQPTs in chaotic three-level systems.
Abstract We review the effects of excited-state quantum phase transitions (ESQPTs) in interacting many-body systems with finite numbers of collective degrees of freedom. We classify typical ESQPT signatures in the spectra of energy eigenstates with respect to the underlying classical dynamics and outline a variety of quantum systems in which they occur. We describe thermodynamic and dynamic consequences of ESQPTs, like those in microcanonical thermodynamics, quantum quench dynamics, and in the response to nearly adiabatic or periodic driving. We hint at some generalizations of the ESQPT concept in periodic lattices and in resonant tunneling systems.
We analyze excited-state quantum phase transitions (ESQPTs) in three schematic (integrable and nonintegrable) models describing a single-mode bosonic field coupled to a collection of atoms. It is shown that the presence of the ESQPT in these models affects the quantum relaxation processes following an abrupt quench in the control parameter. Clear-cut evidence of the ESQPT effects is presented in integrable models, while in a nonintegrable model the evidence is blurred due to chaotic behavior of the system in the region around the critical energy.