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arXiv:1901.09677 · AIP conference proceedings

Excited-state quantum phase transitions in bosonic lattice systems

Michal Macek

Published Jan 1, 2019• 1 citation
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Abstract

Concentrating on bosonic lattice systems, we ask whether and how Excited State Quantum Phase Transition (ESQPT) singularities occur in condensed matter systems with ground state QPTs. We study in particular the spectral singularities above the ground-state phase diagram of the boson Hubbard model. As a general prerequisite, we point out the analogy between ESQPTs and van Hove singularities (vHss).

DOI
10.1063/1.5124610

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Concentrating on bosonic lattice systems, we ask whether and how Excited State Quantum Phase Transition (ESQPT) singularities occur in condensed matter systems with ground state QPTs. We study in particular the spectral singularities above the ground-state phase diagram of the boson Hubbard model. As a general prerequisite, we point out the analogy between ESQPTs and van Hove singularities (vHss).
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) generalize the concept of quantum phase transitions to excited levels. Here we investigate how such ESQPTs can emerge in Kerr nonlinear oscillators, where a bosonic mode with a Kerr nonlinearity is subject to both coherent (one-photon) and parametric (two-photon) driving. We determine the structure and properties of the complete phase diagram by defining proper dimensionless parameters. Rich phases and phase transitions exist in the phase diagram, including phases with two or four critical energies above the ground-state level and first-order quantum phase transitions. We characterize the ESQPTs by singularities in the density of states, which show universal logarithmic divergence or discontinuity at the critical energies. Furthermore, we find that observables such as the photon number also exhibit singular behaviors at the critical energies and obey scaling laws. These results may extend our understanding of the driven Kerr oscillator and ESQPTs in finite-component systems.
A quantum phase transition (QPT) is a point of singularity in the evolution of the ground state properties as a function of pairing strength. However, a QPT may also be reflected in the properties of excited states and the nature of the low-lying excitation spectrum. This contribution examines excited-state scaling properties in the vicinity of the QPT for bosonic and fermionic two-level pairing models. It is found that the numerically-determined power laws for the excited-state scaling properties can be understood in terms of a simple semiclassical description.
Background: Composed systems have become of great interest in the framework of ground-state quantum phase transitions (QPTs) and many of their properties have been studied in detail. However, in these systems, the study of the so-called excited-state quantum phase transitions (ESQPTs) has not received so much attention.Purpose: A quantum analysis of the ESQPTs in the two-fluid Lipkin model is presented in this work. The study is performed through the Hamiltonian diagonalization for selected values of the control parameters in order to cover the most interesting regions of the system phase diagram.Method: A Hamiltonian that resembles the consistent-Q Hamiltonian of the interacting boson model (IBM) is diagonalized for selected values of the parameters. Properties such as the density of states, the Peres lattices, the nearest-neighbor spacing distribution, and the participation ratio are analyzed.Results: An overview of the spectrum of the two-fluid Lipkin model for selected positions in the phase diagram has been obtained. The location of the excited-state quantum phase transition can be easily singled out with the Peres lattice, with the nearest-neighbor spacing distribution, with Poincar\'e sections, or with the participation ratio.Conclusions: This study completes the analysis of QPTs for the two-fluid Lipkin model, extending the previous study to excited states. The ESQPT signatures in composed systems behave in the same way as in single ones, although the evidences of their presence can be sometimes blurred. The Peres lattice turns out to be a convenient tool to look into the position of the ESQPT and to define the concept of phase in the excited states realm.
The interplay between quantum criticality and ergodicity breaking constitutes a central challenge in complex quantum many-body systems. Here, we investigate ground-state quantum phase transitions (QPTs) and excited-state quantum phase transitions (ESQPTs), as well as ergodic-nonergodic transition in the cavity Heisenberg spin-chain (CHS) model. By combining quantum information measures with semiclassical fixed-point analysis, we identify a deformed phase that interpolates between the normal and superradiant phases, featuring a logarithmic nonanalyticity in the density of states (DoS) and two additional jump discontinuities. We further elucidate the spectral-structure mechanism underlying the connection between ESQPTs and the ergodic--nonergodic transition (ENET) via level statistics, participation ratios, and multifractal analysis. Our results provide a general framework for characterizing phase structures and spectral features in light--matter quantum many-body systems.
The Rabi model, a two-level atom coupled to a harmonic oscillator, can undergo a second-order quantum phase transition (QPT) [M.-J. Hwang et al., Phys. Rev. Lett. 115, 180404 (2015)]. Here we show that the Rabi QPT accompanies critical behavior in the higher-energy excited states, i.e., the excited-state QPT (ESQPT). We derive analytic expressions for the semiclassical density of states, which show a logarithmic divergence at a critical energy eigenvalue in the broken symmetry (superradiant) phase. Moreover, we find that the logarithmic singularities in the density of states lead to singularities in the relevant observables in the system such as photon number and atomic polarization. We corroborate our analytical semiclassical prediction of the ESQPT in the Rabi model with its numerically exact quantum mechanical solution.