Phonon renormalization and Pomeranchuk instability in the Holstein model

TL;DR

This study uses functional renormalization group methods to analyze the Holstein model, revealing a Pomeranchuk instability in dimensions greater than three but ruling out quantum critical points in three or fewer dimensions.

Contribution

It demonstrates the existence of a tricritical fixed point related to Pomeranchuk instability in high dimensions using modern RG techniques.

Findings

Pomeranchuk instability exists in dimensions > 3.

No critical fixed points for d ≤ 3.

Ultraviolet stable fixed point related to φ^3-theory.

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