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AIP conference proceedings

Quantum phase transitions and excited-state scaling in bosonic and fermionic pairing models

M. A. Caprio

Published Jan 1, 2019• 1 citation
Publisher

Abstract

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.

DOI
10.1063/1.5124588

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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.
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.
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.
We study the nonintegrable Dicke model and its integrable approximation, the Tavis-Cummings model, as functions of both the coupling constant and the excitation energy. Excited-state quantum phase transitions (ESQPT) are found analyzing the density of states in the semiclassical limit and comparing it with numerical results for the quantum case in large Hilbert spaces, taking advantage of efficient methods recently developed. Two different ESQPTs are identified in both models, which are signaled as singularities in the semiclassical density of states; one static ESQPT occurs for any coupling, whereas a dynamic ESQPT is observed only in the superradiant phase. The role of the unstable fixed points of the Hamiltonian semiclassical flux in the occurrence of the ESQPTs is discussed and determined. Numerical evidence is provided that shows that the semiclassical results describe very well the tendency of the quantum energy spectrum for any coupling in both models. Therefore, the semiclassical density of states can be used to study the statistical properties of the fluctuation in the spectra, a study that is presented in a companion paper.
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...