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Anharmonicity-induced excited-state quantum phase transition in the two-dimensional limit of the vibron model

Jamil Khalouf-Rivera, Francisco Pérez-Bernal, Miguel Carvajal

Published Jun 21, 2021• 0 citations
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Abstract

As already stated in Phys. Rev. A. 81 050101 (2010), the inclusion of anharmonicity into the 2D vibron model in the form of a quadratic bosonic number operator results in a second excited-state quantum phase transition. Such transition, in the classical limit of the model, can be associated with the lowering of the asymptotic energy of the energy functional. We delve into the nature of the new excited-state quantum phase transition and we characterize it in the previously overlooked symmetric phase of the model (also known as cylindrical oscillator or linear phase).

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As already stated in Phys. Rev. A. 81 050101 (2010), the inclusion of anharmonicity into the 2D vibron model in the form of a quadratic bosonic number operator results in a second excited-state quantum phase transition. Such transition, in the classical limit of the model, can be associated with the lowering of the asymptotic energy of the energy functional. We delve into the nature of the new excited-state quantum phase transition and we characterize it in the previously overlooked symmetric phase of the model (also known as cylindrical oscillator or linear phase).
In most cases, excited-state quantum phase transitions can be associated with the existence of critical points (local extrema or saddle points) in a system's classical limit energy functional. However, an excited-state quantum phase transition might also stem from the lowering of the asymptotic energy of the corresponding energy functional. One such example occurs in the two-dimensional (2D) limit of the vibron model, once an anharmonic term in the form of a quadratic bosonic number operator is added to the Hamiltonian. This case has been studied in the broken-symmetry phase [P\'erez-Bernal and \'Alvarez-Bajo, Phys. Rev. A 81, 050101 (2010)]. In the present work we delve further into the nature of this excited-state quantum phase transition and we characterize it in the symmetric phase of the model, making use of quantities such as the effective frequency, the expected value of the quantum number operator, the participation ratio, the density of states, and the quantum fidelity susceptibility. In addition to this, we extend the usage of the quasilinearity parameter, introduced in molecular physics, to characterize the phases in the spectrum of the anharmonic 2D limit of the vibron model and a practical analysis is included with the characterization of the critical energies for the linear isomers HCN and HNC.
We study the effect of the inclusion of the U(2) Lie algebra second-order Casimir operator in a model Hamiltonian of the vibron model two-dimensional limit. We extend previously published results, showing that the inclusion of the additional term does not alter the ground-state quantum phase transition between the symmetric and deformed limits of the model. Nevertheless, the inclusion of this operator implies deep changes on the excitation spectrum and on the ensuing excited-state quantum phase transitions.
The basic Lipkin-Meshkov-Glick model displays a second-order ground-state quantum phase transition and an excited-state quantum phase transition (ESQPT). The inclusion of an anharmonic term in the Hamiltonian implies a second ESQPT of a different nature. We characterize this ESQPT using the mean field limit of the model. The alternative ESQPT, associated with the changes in the boundary of the finite Hilbert space of the system, can be properly described using the order parameter of the ground-state quantum phase transition, the energy gap between adjacent states, the participation ratio, and the quantum fidelity susceptibility.
The standard Lipkin-Meshkov-Glick (LMG) model undergoes a second-order ground-state quantum phase transition (QPT) and an excited-state quantum phase transition (ESQPT). The inclusion of an anharmonic term in the LMG Hamiltonian gives rise to a second ESQPT that alters the static properties of the model [Gamito et al., Phys. Rev. E 106, 044125 (2022)2470-004510.1103/PhysRevE.106.044125]. In the present work, the dynamical implications associated to this new ESQPT are analyzed. For that purpose, a quantum quench protocol is defined on the system Hamiltonian that takes an initial state, usually the ground state, into a complex excited state that evolves on time. The impact of the new ESQPT on the time evolution of the survival probability and the local density of states after the quantum quench, as well as on the Loschmidt echoes and the microcanonical out-of-time-order correlator (OTOC) are discussed. The anharmonity-induced ESQPT, despite having a different physical origin, has dynamical consequences similar to those observed in the ESQPT already present in the standard LMG model.
We study in detail the relation between the two-dimensional Euclidean dynamical E(2) symmetry and the quantum phase transition in the two-dimensional limit of the vibron model, called the U(3) vibron model. Both geometric and algebraic descriptions of the U(3) vibron model show that structures of low-lying states at the critical point of the model with a quartic potential as its classical limit can be approximately described by the E(2) symmetry. We also fit the finite-size scaling exponent of the energy levels and E1 transition rates in the F(2) model, which is exactly the E(2) model but with truncation in its Hilbert subspace, as well as those at the critical point in the U(3) vibron model. The N-scaling power law around the critical point shows that the E(2) symmetry is well preserved even for cases with finite number of bosons. In addition, two kinds of experimentally accessible effective order parameters, such as the energy ratios E21/E11, E31/E 11 and E1 transition ratios B(E1;21→1 1)/B(E1;11→11), B(E1;02→ 11)/B(E1;11→01), are proposed to identify the second-order phase transition in such systems. Possible empirical examples exhibiting approximate E(2) symmetry are also presented. © 2010 IOP Publishing Ltd.
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.