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arXiv:1604.06851 · Fortschritte der Physik

Effects of excited state quantum phase transitions on system dynamics

Francisco Pérez-Bernal, Léa Ferreira dos Santos

Published Jun 23, 2016• 36 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1002/prop.201600035

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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.
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 $\mathcal{E}\ensuremath{\sim}{E}_{\text{ESQPT}}$ 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.
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.
Dynamical signatures of quantum phase transitions for excited states Jakub Dolejší Abstract We study the impact of quantum phase transitions (QPTs) and excited- state quantum phase transitions (ESQPTs) on the validity of the adiabatic approximation for a slowly varying Hamiltonian. We compare two cases, when the initial state is the ground state of the initial Hamiltonian and when the initial state is a statistical mixture of excited states induced by a finite temperature. We use the Lipkin-Meshkov-Glick model of a spin lattice and obtain an abruptly decreasing scaling law of the ground-state population with a growing system size N. We comment on the justifiability of using the Landau-Zener formula to make a quantitative prediction in the case of a first-order and a second-order QPT. To achieve a truly adiabatic evolution in the thermodynamic limit, one would need to perform the Hamiltonian change during an impossibly long time period. It is possible, however, to obtain the same adiabatic final state in a given finite time period by inducing the quantum evolution with another Hamiltonian specifically devised for this purpose, thus employing the so called adiabatic shortcut. We verify the validity of adiabatic shortcuts in the presence of QPTs and ESQPTs and study the costs of performing such adiabatic...
The series of articles [Ann. Phys. 345, 73 (2014) and Ann. Phys. 356, 57 (2015)] devoted to excited-state quantum phase transitions (ESQPTs) in systems with f=2f=2f=2 degrees of freedom is continued by studying the interacting boson model of nuclear collective dynamics as an example of a truly many-body system. The intrinsic Hamiltonian formalism with angular momentum fixed to L=0L=0L=0 is used to produce a generic first-order ground-state quantum phase transition with an adjustable energy barrier between the competing equilibrium configurations. The associated ESQPTs are shown to result from various classical stationary points of the model Hamiltonian, whose analysis is more complex than in previous cases because of (i) a nontrivial decomposition to kinetic and potential energy terms and (ii) the boundedness of the associated classical phase space. Finite-size effects resulting from a partial separability of both degrees of freedom are analyzed. The features studied here are inherent in a great majority of interacting boson systems.