  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 $\phi^3$-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 $d \leq 3$ 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 $d \leq 3$.

Strongly Correlated Electrons