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Physical Review B

Berry phase and entanglement in the icosahedral H⊗(g⊕2h) Jahn-Teller system with trigonal minima

Erwin Lijnen, Arnout Jozef Ceulemans

Published Jan 25, 2005• 15 citations
Publisher

Abstract

The $H\ensuremath{\bigotimes}(g\ensuremath{\bigoplus}2h)$ Jahn-Teller problem in the trigonal regime offers an example of a dynamic ground-state crossover from a degenerate to a singlet state. This crossover is connected to a change in tunneling paths which results in a loss of Berry phase. Previous studies tried to explain the entanglement which lies at the basis of this phenomenon using two-dimensional (2D) cuts through coordinate space but have reached opposite interpretations. In this paper we introduce a method which scans the topology of the full coordinate space using a net of triangular cross sections between the ten ${D}_{3d}$ minima. It is shown that the Berry phase of every loop on the potential energy hypersurface can be expressed in terms of these 2D triangular cross sections. Using this method we show that the absence of Berry phase for the cycles containing five trigonal wells originates from two seams of conical intersections going through these cycles. The connection between this analysis and previous treatments is established and the apparent dilemma of interpretation is resolved.

DOI
10.1103/physrevb.71.014305

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The H\ensuremath⨂(g\ensuremath⨁2h)H\ensuremath{\bigotimes}(g\ensuremath{\bigoplus}2h)H\ensuremath⨂(g\ensuremath⨁2h) Jahn-Teller problem in the trigonal regime offers an example of a dynamic ground-state crossover from a degenerate to a singlet state. This crossover is connected to a change in tunneling paths which results in a loss of Berry phase. Previous studies tried to explain the entanglement which lies at the basis of this phenomenon using two-dimensional (2D) cuts through coordinate space but have reached opposite interpretations. In this paper we introduce a method which scans the topology of the full coordinate space using a net of triangular cross sections between the ten D3d{D}_{3d}D3d​ minima. It is shown that the Berry phase of every loop on the potential energy hypersurface can be expressed in terms of these 2D triangular cross sections. Using this method we show that the absence of Berry phase for the cycles containing five trigonal wells originates from two seams of conical intersections going through these cycles. The connection between this analysis and previous treatments is established and the apparent dilemma of interpretation is resolved.
The topology of the electronic structure for an avoided crossing and a conical intersection (CI) is different and is characterized by the presence of the geometric phase in the electronic wavefunction in the latter. Using the linear Jahn-Teller model, we show that an avoided crossing can be created from the conical intersection while preserving the nontrivial topology of the latter by adding a Pauli σyσ_yσy​ term to the Hamiltonian. Analogously to solid state systems, we derive a half-integer topological invariant as the integral of the Berry curvature over the CI nuclear branching space. We investigate the influence of electronic topology on chemical dynamics by conducting fewest-switches surface hopping simulations and find distinct hopping rates on identical eigensurfaces but with different topologies. Our work extends the influence of topology on molecular excited state dynamics beyond the Berry phase that can be practically realized through electron-nuclear and spin-orbit coupling.
Jahn-Teller systems and the Jahn-Teller effect are discussed in terms of cavity QED models. By expressing the field modes in a quadrature representation, it is shown that certain setups of a two-level system interacting with a bimodal cavity are described by the Jahn-Teller $E\ifmmode\times\else\texttimes\fi{}\ensuremath{\epsilon}$ Hamiltonian. We identify the corresponding adiabatic potential surfaces and the conical intersection. The effects of a nonzero geometrical Berry phase, governed by encircling the conical intersection, are studied in detail both theoretically and numerically. The numerical analysis is carried out by applying a wave packet propagation method, more commonly used in molecular or chemical physics, and analytic expressions for the characteristic time scales are presented. It is found that the collapse-revival structure is greatly influenced by the geometrical phase and as a consequence, the field intensities contain direct information about this phase. We also mention the link between the Jahn-Teller effect and the Dicke phase transition in cavity QED.
A proof is given for the Jahn-Teller problem of an orbital doublet (E\ensuremath{\bigotimes}\ensuremath{\epsilon}) that the order of the lowest vibronic levels is fixed by the requirement that the vibrational part of the wave function change sign under 2\ensuremath{\pi} rotation in the vibrational coordinates. This sign change in turn is a consequence of the sign change in the electronic part of the wave function, a special case of Berry's geometrical phase. Experimental confirmation of this sign change and thus of Berry's phase is available from studies of Jahn-Teller defects in crystals that reveal the sequence of these lowest levels.
We demonstrate that Berry phases may greatly affect the dynamics of spin-orbit-coupled Bose-Einstein condensates. The effective model Hamiltonian under consideration is shown to be equivalent to the $E\ifmmode\times\else\texttimes\fi{}\ensuremath{\epsilon}$ Jahn-Teller model first introduced in molecular physics. The corresponding conical intersection is identified and the Berry phase acquired for a wave packet encircling the intersection is studied. It is found that this phase manifests itself in the density profile of the condensate, making it a directly measurable quantity via time-of-flight detection. Moreover, the non-Abelian gauge structure of the system is addressed and we verify how it affects the dynamics.