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arXiv:2106.11044 · Physical Review A

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

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

Published Mar 30, 2022• 14 citations
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Abstract

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.

DOI
10.1103/physreva.105.032215

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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.
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).
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.
We characterize excited state quantum phase transitions in the two dimensional limit of the vibron model with the quantum fidelity susceptibility, comparing the obtained results with the information provided by the participation ratio. As an application, we locate the eigenstate closest to the barrier to linearity and determine the linear or bent character of the different overtones for particular bending modes of six molecular species. We perform a fit and use the optimized eigenvalues and eigenstates in three cases and make use of recently published results for the other three cases.
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.
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 quantum phase transition in the U(2) vibron model is investigated in large-N limit with the coherent state method and for finite N in a numerical scheme Through analyzing the potential structure and the corresponding low-lying spectrum, it is found that the dynamical characters at the critical point can be approximately described by the E(1) critical point symmetry, but such an approximation becomes poor with the increasing of the excitation energy. In addition, the scaling behavior of the spectrum at the critical point has been also investigated in a numerical way. The results indicate that the scaling law of the spectrum at the critical point of the 2nd order phase transition may be independent of the dimension of system.