Impact of phonons on quantum phase transitions in nanorings of coupled quantum dots

CDW-to-CDW and CDW-to-SDW (CDW/SDW=charge-spin-density wave) quantum phase transitions have been recently reported for finite rings described by an extended Hubbard model [I. Bâldea, H. Köppel, and L. S. Cederbaum, Eur. Phys. J. B 20, 289 (2001)]. We present exact (Lanczos) diagonalization results which demonstrate that these transitions survive in the presence of a dynamic Su-Schrieffer-Heeger (SSH) electron-phonon coupling. The two transitions are affected in a different way discussed in detail. By treating the SSH electron-phonon coupling dynamically, two levels of different symmetries do cross, allowing one to define precisely a critical point. Because the ground-state symmetry changes at the critical points, we suggest to study the quantum phase transitions by optical methods. Molecular rings such as polyenes (annulenes) turn out to be too far from the critical points, where the most interesting phenomena occur. However, we present an analysis revealing that, if metallic quantum dots of the type already fabricated can be assembled in nanorings, model parameters can be tuned and quantum phase transitions can become observable.