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arXiv:2412.04240 · Journal of Physics A Mathematical and Theoretical

Excited-state quantum phase transitions in constrained systems

Jakub Novotný, Pavel Stránský, Pavel Cejnar

Published Feb 11, 2025• 2 citations
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Abstract

Abstract We extend the standard semiclassical theory of excited-state quantum phase transitions (ESQPTs), based on a classification of stationary points in the classical Hamiltonian, to constrained systems. We adopt the method of Lagrange multipliers to find all stationary points and their properties directly from the Hamiltonian constrained by an arbitrary number of integrals of motion, and demonstrate the procedure on an algebraic u ( 3 ) boson model with two independent constraints. We also elaborate the Holstein–Primakoff (HP) mapping, used to eliminate one degree of freedom in bosonic systems constrained by a conserved number of excitations, and address the fact that this mapping leads, in the classical limit, to a compact phase space with singular behaviour that conceals some stationary points at the phase space boundary. It is shown that the HP method reveals all ESQPTs only after constructing a complete atlas of different HP mappings.

DOI
10.1088/1751-8121/adb4b2

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•Sep 18, 2018•0 citations
Abstract We extend the standard semiclassical theory of excited-state quantum phase transitions (ESQPTs), based on a classification of stationary points in the classical Hamiltonian, to constrained systems. We adopt the method of Lagrange multipliers to find all stationary points and their properties directly from the Hamiltonian constrained by an arbitrary number of integrals of motion, and demonstrate the procedure on an algebraic u ( 3 ) boson model with two independent constraints. We also elaborate the Holstein–Primakoff (HP) mapping, used to eliminate one degree of freedom in bosonic systems constrained by a conserved number of excitations, and address the fact that this mapping leads, in the classical limit, to a compact phase space with singular behaviour that conceals some stationary points at the phase space boundary. It is shown that the HP method reveals all ESQPTs only after constructing a complete atlas of different HP mappings.
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.
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.
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.
In the thesis we investigate and classify critical phenomena in the extended Dicke model (EDM) which describes the interaction between two-level atoms and a single-mode bosonic field (schematic model for cavity quantum electrodynamics). The model belongs to the class of so-called finite models, which keep the number of degrees of freedom f constant independently on the size of the system N . The important property of these systems is that the thermodynamic limit N → ∞ coincides with the classical limit ħ → 0. This allows us to study various quantum critical phenomena, in particular the ground-state quantum phase transitions (QPTs) and the excited-state quantum phase transitions (ESQPTs), by means of semiclassical methods. Using the semiclassical approach we identify and classify the QPTs and ESQPTs in various settings of the EDM and make a link to thermal phase transitions. We study the entanglement properties of both the ground state and the excited states as a function of the atom-field interaction strength. In the integrable version of the EDM we make a link between the ESQPT and monodromy, and discuss its effect on classical dynamics. The fate of monodromy under a non-integrable perturbation is observed. The dynamical consequences of the ESQPTs are examined using quantum quenches. The influence of the...