Phonon renormalization and Pomeranchuk instability in the Holstein model

Abstract

The Holstein model with dispersionless Einstein phonons is one of the simplest models describing electron-phonon interactions in condensed matter. A naive extrapolation of perturbation theory in powers of the relevant dimensionless electron-phonon coupling ${\lambda }_0$ suggests that at zero temperature the model exhibits a Pomeranchuk instability characterized by a divergent uniform compressibility at a critical value of ${\lambda }_0$ of order unity. In this work, we re-examine this problem using modern functional renormalization group (RG) methods. For dimensions $d>3$ we find that the RG flow of the Holstein model indeed exhibits a tricritical fixed point associated with a Pomeranchuk instability. This non-Gaussian fixed point is ultraviolet stable and is closely related to the well-known ultraviolet stable fixed point of ${\varphi }^3$-theory above six dimensions. To realize the Pomeranchuk critical point in the Holstein model at fixed density both the electron-phonon coupling ${\lambda }_0$ and the adiabatic ratio ${\omega }_0/{\epsilon }_F$ have to be fine-tuned to assume critical values of order unity, where ${\omega }_0$ is the phonon frequency and ${\epsilon }_F$ is the Fermi energy. On the other hand, for dimensions $d\le 3$ we find that the RG flow of the Holstein model does not have any critical fixed points. This rules out a quantum critical point associated with a Pomeranchuk instability in $d\le 3$.

17 pages, 7 figures

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