Phonons

The CPT can also be applied to phonon-mediated interactions, again provided that the phonon interaction be local, which restricts us to the Holstein or the Holstein-Hubbard model. Zhao et al. [35] applies CPT to the one-dimensional Holstein model, which involves dispersionless phonons only:

H = -i E (4ci(T C H.c.) C !o - y !>i(bj C ¿/), (8.100)

i i i where b annihilates a phonon at site i, !0 is the phonon frequency and y the electron-phonon coupling. From a computational point of view, the difficulty with phonons is that they require an infinite-dimensional Hilbert space, and thus some truncation is needed. Zhao et al. [35] uses an optimized phonon approach [36-38] for that purpose. At fixed !0, this systems undergoes a transition from a normal state to a charge-density state wave (CDW) at a certain critical coupling gc. The CDW state has a gap clearly visible in the CPT spectrum [35]. This is a case of spontaneous breaking of a discrete symmetry that occurs at the same time in the phonon and electron system.

Adding a Hubbard interaction to (8.100) yields the Holstein-Hubbard model. As shown in [39], the electron-phonon interaction has a sizeable effect on the electron spectral function, eventually suppressing the holon peak visible in the absence of phonons, while keeping the spin peak relatively intact. The authors argue that the ARPES spectra of the quasi-one-dimensional organic conductor TTF-TCNQ reported in [40] cannot be correctly interpreted with the Hubbard model alone, but requires the introduction of phonons.

Acknowledgments The author would like to thank the following people for discussions which, over the years, have strengthened and widened his understanding of quantum cluster methods: M. Civelli, G. Kotliar, B. Kyung, M. Jarrell, Th. Maier, S. Okamoto, D. Plouffe, M. Potthoff, AM. Tremblay, and C. Weber. Computational resources for this review were provided by RQCHP and Compute Canada.

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