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arXiv:2102.12335 · SciPost Physics

Quantum fidelity susceptibility in excited state quantum phase transitions: application to the bending spectra of nonrigid molecules

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

Published Jan 4, 2022• 13 citations
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Abstract

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.

DOI
10.21468/scipostphys.12.1.002

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arXiv:1709.07111•Physical Review A•Jan 26, 2018•35 citations
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.
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.
The fidelity approach to the Gaussian transitions in spin-one XXZ\mathit{XXZ}XXZ spin chains with three different values of Ising-type anisotropy $\ensuremath{\lambda}$ is analyzed by means of the density matrix renormalization group technique for systems of large sizes. We find that, despite the success in the cases of $\ensuremath{\lambda}=2.59$ and 1, the fidelity susceptibility fails to detect the Gaussian transition for $\ensuremath{\lambda}=0.5$. Thus our results demonstrate the limitation of the fidelity susceptibility in characterizing quantum phase transitions, which was proposed recently in general frameworks.
Quantum phase transitions that take place between two distinct topological phases remain an unexplored area for the applicability of the fidelity approach. Here, we apply this method to spin systems in two and three dimensions and show that the fidelity susceptibility can be used to determine the boundary between different topological phases particular to these models, while at the same time offer information about the critical exponent of the correlation length. The success of this approach relies on its independence on local-order parameters or breaking symmetry mechanisms, with which nontopological phases are usually characterized. We also consider a topological insulator-superconducting phase transition in three dimensions and point out the relevant features of fidelity susceptibility at the boundary between these phases.
This thesis, comprising three publications, explores the efficacy of novel generalization of the fidelity susceptibility and their numerical application to the study of frustrated quantum phase transitions in two and one dimensions. Specifically, they will be used in exact diagonalization studies of the various limiting cases of the anisotropic next-nearest neighbour triangular lattice Heisenberg model (ANNTLHM). These generalized susceptibilities are related to the order parameter susceptibilities and spin stiffness and are believed to exhibit similar behaviour although with greater sensitivity. This makes them ideal for numerical studies on small systems. Additionally, the utility of the excited-state fidelity and twist boundary conditions will be explored. All studies are done through numerical exact diagonalization. In the limit of interchain couplings going to zero the ANNTLHM reduces to the well studied J1−J2J_1-J_2J1​−J2​ chain with a known, difficult to identify, BKT-type transition. In the first publication of this work the generalized fidelity susceptibilities introduced therein are shown to be able to identify this transition as well as characterize the already understood phases it straddles. The second publication of this work then seeks to apply these generalized fidelity susceptibilities, as well as the excited-state fidelity, to the study of the general phase diagram of the ANNTLHM. It is shown that the regular and excited-state fidelities are useful quantities for the mapping of novel phase diagrams and that the generalized fidelity susceptibilities can provide valuable information as to the nature of the phases within the mapped phase regions. The final paper sees the application of twisted boundary conditions to the anisotropic triangular model (next-nearest neighbour interactions are zero). It is demonstrated that these boundary conditions greatly enhance the ability to numerically explore incommensurate physics in small systems.
The Husimi distribution is proposed for a phase-space analysis of quantum phase transitions in the two-dimensional U(3) vibron model for NNN-size molecules. We show that the inverse participation ratio and Wehrl's entropy of the Husimi distribution give sharp signatures of the quantum (shape) phase transition from linear to bent. Numerical results are complemented with a variational approach using parity-symmetry-adapted U(3) coherent states, which reach the minimum Wehrl entropy N(3+2N)(N+1)(N+2)\frac{N(3+2N)}{(N+1)(N+2)}(N+1)(N+2)N(3+2N)​, in the rigidly linear phase, according to a generalized Wehrl-Lieb conjecture. We also propose a characterization of the vibron-model quantum phase transition by means of the zeros of the Husimi distribution.
Quantum criticality usually occurs in many-body systems. Recently it was shown that the quantum Rabi model, which describes a two-level atom coupled to a single model cavity field, presents quantum phase transitions from a normal phase to a superradiate phase when the ratio between the frequency of the two-level atom and the frequency of the cavity field extends to infinity. In this work, we study quantum phase transitions in the quantum Rabi model from the fidelity susceptibility perspective. We found that the fidelity susceptibility and the generalized adiabatic susceptibility present universal finite-size scaling behaviors near the quantum critical point of the Rabi model if the ratio between frequency of the two-level atom and frequency of the cavity field is finite. From the finite-size scaling analysis of the fidelity susceptibility, we found that the adiabatic dimension of the fidelity susceptibility and the generalized adiabatic susceptibility of fourth order in the Rabi model are 4/34/34/3 and 2, respectively. Meanwhile, the correlation length critical exponent and the dynamical critical exponent in the quantum critical point of the Rabi model are found to be 3/23/23/2 and 1/31/31/3, respectively. Since the fidelity susceptibility and the generalized adiabatic susceptibility are the moments of the quantum noise spectrum which are directly measurable by experiments in linear response regime, the scaling behavior of the fidelity susceptibility in the Rabi model could be tested experimentally. The simple structure of the quantum Rabi model paves the way for experimentally observing the universal scaling behavior of the fidelity susceptibility at a quantum phase transition.