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Physical Review A

Extended projector method: Application to Jahn-Teller and pseudo-Jahn-Teller systems

Marc Vanhimbeeck, Hans A. De Raedt, D. Schoemaker

Published Nov 1, 1988• 0 citations
Publisher

Abstract

Within the Trotter-Suzuki approximations on ${e}^{\mathrm{\ensuremath{-}}\ensuremath{\beta}H}$, the extended projector method [Phys. Rev. A 36, 4612 (1987)] previously introduced by us is shown to be a valuable technique in investigating the general class of few-level systems coupled both linearly and quadratically to a selected number of harmonic oscillators. As an illustration we compare eigenvalues and Ham-reduction factors with the literature for the cubic linear Jahn-Teller systems T\ensuremath{\bigotimes}\ensuremath{\epsilon} and T\ensuremath{\bigotimes}${\ensuremath{\tau}}_{2}$ as well as for the system E\ensuremath{\bigotimes}\ensuremath{\epsilon} for which a quadratic warping interaction is included. For the latter system a symmetry projector is additionally developed, dealing with the transformation properties of nonlinear combinations of representational basis functions. As another illustration, the low-energy eigenstates of a two-level system quadratically coupled to a single harmonic oscillator are investigated. Within the ground state, the system becomes unstable at high coupling strengths and through the evaluation of time- and frequency-dependent correlation functions we found substantially different ground-state dynamic properties compared with those of a linearly coupled system.

DOI
10.1103/physreva.38.5021

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Within the Trotter-Suzuki approximations on e\ensuremath−\ensuremathβH{e}^{\mathrm{\ensuremath{-}}\ensuremath{\beta}H}e\ensuremath−\ensuremathβH, the extended projector method [Phys. Rev. A 36, 4612 (1987)] previously introduced by us is shown to be a valuable technique in investigating the general class of few-level systems coupled both linearly and quadratically to a selected number of harmonic oscillators. As an illustration we compare eigenvalues and Ham-reduction factors with the literature for the cubic linear Jahn-Teller systems T\ensuremath{\bigotimes}\ensuremath{\epsilon} and T\ensuremath{\bigotimes}\ensuremathτ2{\ensuremath{\tau}}_{2}\ensuremathτ2​ as well as for the system E\ensuremath{\bigotimes}\ensuremath{\epsilon} for which a quadratic warping interaction is included. For the latter system a symmetry projector is additionally developed, dealing with the transformation properties of nonlinear combinations of representational basis functions. As another illustration, the low-energy eigenstates of a two-level system quadratically coupled to a single harmonic oscillator are investigated. Within the ground state, the system becomes unstable at high coupling strengths and through the evaluation of time- and frequency-dependent correlation functions we found substantially different ground-state dynamic properties compared with those of a linearly coupled system.
Symmetrized Trotter approximations to ${e}^{\mathrm{\ensuremath{-}}\ensuremath{\beta}H}$ are employed to compute the low-energy eigenstates of a two-level system linearly coupled to a harmonic oscillator. At high coupling strengths the ground state is nearly doubly degenerate and the projectors become very inefficient. It is demonstrated that the projection technique in combination with the variational principle yields a new efficient method to calculate simultaneously all nearly degenerate eigenvalues and eigenstates. The dynamic properties of the system in the ground state are studied by evaluating the time- and frequency-dependent correlation functions. Depending on the ratio of the tunneling frequency relative to the oscillator frequency, the system behaves differently under increasing coupling strength.
A recursive method is developed to treat electrons coupled to phonons. It is applied to small systems with $E\ensuremath{\bigotimes}\ensuremath{\beta}$ Jahn-Teller coupling. Two cases are considered, a model with one electron and two orbitals on a single site (related to the Rabi Hamiltonian) and a model with two electrons on two sites. The corresponding Green's functions are represented by rational functions. It is found that the spectra change substantially when one phonon couples to the electron but are relatively robust under an increasing number of phonons.
We compare and contrast the entanglement in the ground state of two Jahn-Teller models. The $E\ensuremath{\bigotimes}\ensuremath{\beta}$ system models the coupling of a two-level electronic system, or qubit, to a single-oscillator mode, while the $E\ensuremath{\bigotimes}\ensuremath{\epsilon}$ models the qubit coupled to two independent, degenerate oscillator modes. In the absence of a transverse magnetic field applied to the qubit, both systems exhibit a degenerate ground state. Whereas there always exists a completely separable ground state in the $E\ensuremath{\bigotimes}\ensuremath{\beta}$ system, the ground states of the $E\ensuremath{\bigotimes}\ensuremath{\epsilon}$ model always exhibit entanglement. For the $E\ensuremath{\bigotimes}\ensuremath{\beta}$ case we aim to clarify results from previous work, alluding to a link between the ground-state entanglement characteristics and a bifurcation of a fixed point in the classical analog. In the $E\ensuremath{\bigotimes}\ensuremath{\epsilon}$ case we make use of an ansatz for the ground state. We compare this ansatz to exact numerical calculations and use it to investigate how the entanglement is shared between the three system degrees of freedom.
In this work we have applied the optimal coupled-cluster approximation to study the ground state of the E⊗(b1+b2) Jahn–Teller effect. The effectiveness of the optimal coupled-cluster approximation has been investigated for the whole range of the asymmetry parameter and various coupling strengths. It is shown that our results up to the third level of approximation are in very good agreement with the exact numerical diagonalization results and are better than those from earlier variational treatments. Furthermore, unlike previous variational treatments, the optimal coupled-cluster approximation has the advantage that the accuracy of both the ground state energy and wavefunction estimates is being taken care of and can be systematically improved. Since the mathematical treatment in this work is simple, the optimal coupled-cluster approximation could be easily extended to the studies of other fermion-boson interacting systems, e.g. the extended Jahn–Teller system.