Pomeranchuk and Density-Wave Instabilities

A Pomeranchuk instability is a zero-wavevector deformation of the Fermi surface in a forward-scattering eigenchannel. Charge- and spin-density waves instead break translations at finite wavevector. Enhanced susceptibility in either channel is a tendency; established order additionally requires thermodynamic scaling, symmetry breaking, and exclusion of competing explanations.

Required background. Use Landau theory and Fermi-liquid response. Helpful background. Channel-resolved fRG can compare the tendencies in a controlled weak-coupling truncation.

Zero-wavevector shape deformations

For an isotropic three-dimensional Fermi liquid, parameterize the surface displacement by $u(\hat k)=\sum_{\ell m}u_{\ell m}Y_{\ell m}(\hat k)$. In the chapter’s Landau convention,

$$\delta E=\frac{N(0)v_F^2}{2} \sum_{\ell m}\left(1+\frac{F_\ell^{s,a}}{2\ell+1}\right)|u_{\ell m}^{s,a}|^2$$

up to the chosen normalization of $u$. A coefficient crossing zero destabilizes the symmetric surface. The $\ell=0$ symmetric channel is phase separation or compressibility failure; $\ell=1$ is constrained by boosts in a Galilean fluid; an $\ell=2$ symmetric distortion is the continuum nematic example. On a crystal, angular momentum is replaced by irreducible representations of the point group.

Oganesyan, Kivelson, and Fradkin analyze the two-dimensional nematic transition and its overdamped collective modes in Oganesyan, Kivelson, and Fradkin 2001, §§ II–IV.

Finite-wavevector density waves

Define charge and spin susceptibilities $\chi_{c,s}(\mathbf q,\omega)$ from the corresponding densities. In a simple ladder approximation,

$$\chi_\alpha(\mathbf q,0) =\frac{\chi_0(\mathbf q,0)}{1-g_\alpha(\mathbf q)\chi_0(\mathbf q,0)},$$

with the sign absorbed into the declared channel coupling $g_\alpha$. A peak of $\chi_0$ at a nesting vector enhances a finite-$q$ instability. The denominator criterion is approximation-dependent; long-range Coulomb forces strongly penalize uniform charge separation and reshape the charge channel.

The distinction is operational: a nematic order parameter changes rotational or point-group symmetry without enlarging the translation cell, whereas a density wave produces Bragg structure at $\mathbf Q\ne0$ and folds the Brillouin zone. The forward and finite-wavevector channels have different low-energy kinematics Shankar 1994, §§ IV–VI, pp. 145–177. Composite or vestigial nematicity can arise from fluctuating density waves, so coincident onsets do not establish causality.

Evidence checks

State the angular or lattice basis, normalize susceptibilities consistently, follow size and temperature scaling, include domains and external strain, and compare charge, spin, pairing, and structural channels. A finite-size peak or softened mode is evidence for proximity, not proof of order.

Exercises

Classify a distortion proportional to $\cos k_x-\cos k_y$ at $q=0$ and a density modulation proportional to $\cos(Qx)$.

Solution

The first changes the relative $x$ and $y$ shape of the Fermi surface without breaking translations and is a lattice nematic/Pomeranchuk channel. The second carries finite wavevector $Q$ and is a density wave; it enlarges the unit cell when commensurate.

References

• Vadim Oganesyan, Steven A. Kivelson, and Eduardo Fradkin, “Quantum Theory of a Nematic Fermi Fluid,” Physical Review B 64 (2001) 195109, doi:10.1103/PhysRevB.64.195109, Open PDF.
• Ramamurti Shankar, “Renormalization-Group Approach to Interacting Fermions,” Reviews of Modern Physics 66 (1994) 129–192, doi:10.1103/RevModPhys.66.129.