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Phonon renormalization and Pomeranchuk instability in the Holstein model

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 $λ_0$ suggests that at zero temperature the model exhibits a Pomeranchuk instability characterized by a divergent uniform compressibility at a critical value of $λ_0$ 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 $ϕ^3$-theory above six dimensions. To realize the Pomeranchuk critical point in the Holstein model at fixed density both the electron-phonon coupling $λ_0$ and the adiabatic ratio $ω_0 / ε_F$ have to be fine-tuned to assume critical values of order unity, where $ω_0$ is the phonon frequency and $ε_F$ 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 $d \leq 3$.