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 ${\lambda }_0$ 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. However, 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 $d\le 3$.

Figure 1

Perturbation series for the phonon self-energy $\mathrm{\Delta }\left(Q\right)$ in the Holstein model. Wavy lines represent the phonon propagator $D_0\left(Q\right)$, solid arrows represent the electron propagator $G_0\left(K\right)$, and black dots represent the bare electron-phonon vertex ${\gamma }_0\propto \sqrt{{\lambda }_0}$. Diagram (a) represents the leading-order contribution $-{\gamma }_0^2{\mathrm{\Pi }}_0\left(Q\right)$ to the phonon self-energy, diagrams (b) and (c) are generated by the Hartree corrections to the electronic self-energy, diagrams (d) and (e) are due to the exchange corrections to the electronic self-energy, and diagram (f) represents the leading vertex correction.

Figure 2

Graphical representation of exact FRG flow equations for the irreducible vertices of effective phonon action (2.11). (a) Flow equation (3.20) for the free energy; (b) flow equation (3.22) for the one-point vertex; (c) flow equation (3.24) for the two-point vertex; (d) flow equation (3.26) for the three-point vertex, where “$+$ 2 perm.” denotes two additional diagrams obtained by exchanging the labels $(Q_1\leftrightarrow Q_2)$ and $(Q_1\leftrightarrow Q_3)$ of the external legs attached to the three-point vertex. The thick wavy lines represent the regularized phonon propagator ${\tilde{D}}_{\mathrm{\Lambda }}\left(Q\right)$, while the slash represents the regulator insertion, $-{\partial }_{\mathrm{\Lambda }}{R}_{\mathrm{\Lambda }}\left(Q\right)$. A cross inside a loop corresponds to a sum where each propagator of the loop is once slashed according to the product rule (3.25). A colored circle labeled by the number $n$ represents an $n$-point vertex ${\tilde{\mathrm{\Gamma }}}_{\mathrm{\Lambda }}^{\left(n\right)}$. The dots above the vertices denote a scale derivative.

Figure 3

RG flow of the effective phonon action (2.11) in the ${\tilde{g}}_l\text{−}{\tilde{r}}_l$ plane in $d=1,2,3$ obtained from the numerical solution of the flow equations (3.40). The arrows indicate the direction of the RG flow toward the infrared. For $d\le 3$ the dimensionless rescaled flow equations have only a trivial, i.e., Gaussian, fixed point $G$, which is represented by a purple dot. A subset of initial values, which exhibit a runaway flow, tend to ${\tilde{r}}_{*}=-1$ which is marked as an orange dashed line.

Figure 4

RG flow of the effective phonon action (2.11) in the ${\tilde{g}}_l\text{−}{\tilde{r}}_l$ plane in $d=4$ obtained from the numerical solution of the flow equations (3.40). In addition to the Gaussian fixed point $G$ there are two nontrivial fixed points $P^{-}$ and $P^{+}$, which are denoted by red dots, corresponding to a Pomeranchuk instability. The two Pomeranchuk fixed points are tricritical, as there are three relevant couplings ${\tilde{h}}_l, {\tilde{r}}_l$, and ${\tilde{g}}_l$ whose initial values have to be fine-tuned to reach the fixed point. The eigenvectors of the linearized flow around the Pomeranchuk fixed point are represented by red arrows. Note only the eigenvectors which lie in the ${\tilde{g}}_l\text{−}{\tilde{r}}_l$ plane are shown. The thick blue line emanates from Pomeranchuck fixed point and flows into the Gaussian fixed point.

Figure 5

RG flow of the effective phonon action (2.11) in $d=4$ obtained from the numerical solution of the flow equations (3.40). Left: RG flow in the ${\tilde{g}}_l\text{−}{\tilde{r}}_l$ plane at $\tilde{r}={\tilde{r}}_{*}$. Middle: RG flow in the ${\tilde{r}}_l\text{−}{\tilde{h}}_l$ plane at ${\tilde{g}}_l={\tilde{g}}_{*}^{+}$. Right: RG flow in the ${\tilde{r}}_l\text{−}{\tilde{h}}_l$ plane at ${\tilde{g}}_l={\tilde{g}}_{*}^{-}$.

Figure 6

Schematic ground state phase diagram of the Holstein model in $d=3$ in the plane spanned by the electron-phonon coupling ${\lambda }_0$ and the adiabatic ratio ${\omega }_0/{\epsilon }_F$ for fixed but not too small densities. We have ignored a possible superfluid phase, assuming that the weak-coupling phase is a normal Fermi liquid (FL). In the adiabatic regime ${\omega }_0/{\epsilon }_F\ll 1$ we expect a first-order transition to a phase-separated (PS) inhomogeneous state when ${\lambda }_0$ exceeds a certain threshold of order unity. However, in the antiadiabatic regime ${\omega }_0/{\epsilon }_F\gg 1$ there is a first-order transition to a state with charge-density-wave (CDW) order. In $d\le 3$ the intersection of the three phase boundaries (red dot) is a noncritical triple point. For $d>3$ the triple point transforms into a critical point while the phase boundary between the FL and the PS phases remains first order.

Figure 7

Diagrammatic representation of Dyson-Schwinger equations for the Holstein model: (a) electronic self-energy, (b) phonon self-energy, and (c) electron-phonon vertex (represented by a green triangle). The solid arrows represent the exact electron propagator $G\left(K\right)$ and the thick wavy lines represent the exact phonon propagator $D\left(Q\right)$. The thin wavy line in the tadpole contribution to the electronic self-energy represents the bare static phonon propagator $D_0\left(0\right)=1/{\omega }_0^2$. The bare phonon vertex ${\gamma }_0$ is represented by a black dot and the irreducible fermionic two-body interaction ${\mathrm{\Gamma }}^{\overline{c}\overline{c}cc}(K+Q,K^{\prime },K^{\prime }+Q,K)$ is represented by a blue square.