Kohn Anomaly in Electron-Phonon Systems

• Kohn anomaly is a pronounced softening in phonon dispersion curves caused by a divergence in electronic susceptibility at critical wavevectors.
• It manifests as observable lattice instabilities and often signals the onset of charge density waves in materials like quasi-1D metals and TMDs.
• Recent studies incorporate detailed band structures and electron-phonon coupling metrics to refine models predicting phonon anomalies in complex systems.

A Kohn anomaly is a singularity or pronounced softening in the phonon dispersion curve at specific wavevectors, resulting from a non-analyticity in the electronic susceptibility. Predicted originally by Walter Kohn, this effect is fundamentally tied to the interplay between the electron-phonon interaction and the Fermi surface topology, leading to anomalous features in the vibrational spectrum—most notably near wavevectors that connect large, similarly oriented portions of the Fermi surface ("nesting vectors"). The Kohn anomaly is central to understanding the formation and stability of charge density waves (CDWs) and provides a microscopic basis for the lattice instabilities observed in metals and semimetals.

1. Physical Origin of the Kohn Anomaly

The Kohn anomaly arises from the dielectric response of itinerant electrons subjected to a periodic lattice distortion. Consider the bare electronic susceptibility (Lindhard function): $$\chi_0(q) = \sum_{k} \frac{f(\epsilon_k) - f(\epsilon_{k+q})}{\epsilon_{k+q} - \epsilon_k + i0^+}$$ In perfectly nested systems, such as a 1D chain at half filling, $\chi_0(q)$ exhibits a logarithmically divergent peak at %%%%1%%%%. This reflects strong electronic screening at that wavevector; the screening modifies the restoring force on phonons, leading to a softening (minimum) in the phonon dispersion $\omega(q)$ near $q=2k_F$. In higher dimensions, if substantial parallel portions of the Fermi surface can be mapped onto each other by a wavevector $q$, a pronounced anomaly remains but is typically less singular (Zhu et al., 2015).

2. Phonon Softening: Manifestation in Dispersion

The general form for the renormalized phonon frequency is: $$\omega^2(q) = \omega_0^2(q) - 2\omega_0(q)|g(q)|^2 \operatorname{Re}\chi_0(q)$$ where $\omega_0(q)$ is the bare (uncoupled) phonon frequency, $g(q)$ the electron-phonon coupling matrix element, and $\operatorname{Re}\chi_0(q)$ the electronic susceptibility. The Kohn anomaly is visible as a pronounced dip—sometimes a cusp or discontinuity in slope—in $\omega(q)$ at a $q$ where $\operatorname{Re}\chi_0(q)$ is maximized due to Fermi surface geometry. This effect can be directly tracked via inelastic neutron or x-ray scattering.

The anomaly need not lead to full phonon softening ($\omega(q)\rightarrow0$), but in some cases, typically with strong electron-phonon coupling, the softening drives an instability toward a modulated ground state—most notably, a CDW.

3. Kohn Anomaly and Charge Density Wave Instabilities

The connection between a Kohn anomaly and CDW formation is established within the mean-field Peierls scenario:

• In 1D, the divergence of $\chi_0(q)$ at $2k_F$ leads to a complete softening of the phonon, opening a gap at the Fermi level and dimerizing the lattice.
• In real materials, this is generalized: a sufficiently strong anomaly at a characteristic $q_{CDW}$ may, if $|g(q)|^2\operatorname{Re}\chi_0(q)$ is large enough, drive the modulus of $\omega(q_{CDW})$ to zero at a critical temperature, resulting in the onset of CDW order described by $\rho(r) = \rho_0 + \Delta\rho \cos(q_{CDW}\cdot r)$ (Zhu et al., 2015, Alidoosti et al., 2020, Long et al., 2016).

An explicit instability condition is given by: $1 - V\,\chi_0(q_{CDW}) = 0$ where $V$ is an effective electron-electron or electron-phonon interaction parameter (Long et al., 2016). If this is satisfied, the phonon softens completely, leading to a phase transition.

4. Case Studies and Experimental Evidence

The prototypical Kohn anomaly is observed in quasi-1D conductors and transition-metal dichalcogenides (TMDs):

2H-NbSe$_2$: No strong Fermi surface nesting is seen at the experimental $q_{CDW}$; instead, the momentum dependence of the electron-phonon matrix elements $|g(q)|^2$ dominates. Both phonon softening (Kohn-like anomaly) and increased linewidth are observed at $q_{CDW}$; extracted $|g(q)|^2$ peaks at the same $q$, confirming the anomaly's electron-phonon coupling origin (Zhu et al., 2015).

Graphene: At high electronic doping, the Fermi level approaches a van Hove singularity at the $M$ point. Here, $\chi_0(q)$ diverges logarithmically at the corresponding nesting vector. Phonon measurements show a splitting of the Raman G-peak below the CDW transition temperature, providing direct evidence for the phonon anomaly-driven lattice distortion and CDW formation (Long et al., 2016).

Monolayer InSe: Above a Lifshitz transition, Fermi surface topology enhances $\chi_0(q)$ at a symmetry-specific wavevector, resulting in a pronounced Kohn anomaly and a high-temperature CDW phase (Alidoosti et al., 2020).

5. Beyond the Simple Peierls/Kohn Paradigm

The classic view that Kohn anomalies and diverging $\chi_0(q)$ (i.e., Fermi surface nesting) are necessary and sufficient for CDW or related instabilities has been refined. Many systems, including TMDs and cuprates, show that the momentum-dependence of the electron-phonon coupling $g(k,q)$ and details of the Fermi surface beyond nesting—such as multiple bands or strong correlations—must be taken into account. Large $|g(q)|^2$ at a given $q$ can induce a strong Kohn anomaly even when $\chi_0(q)$ does not peak, and, conversely, nesting without sufficient EPC does not guarantee softening (Zhu et al., 2015, Yilmaz et al., 2022).

For example, in Bi$_2$Sr$_2$CaCu$_2$O$_8$, large EPC is measured but phonon softening is insufficient to drive a CDW: charge order in cuprates must then be ascribed to other mechanisms (magnetic, Coulombic), not a Kohn anomaly (Zhu et al., 2015, Caprara et al., 2016).

6. Kohn Anomalies in Modern Electronic Models and Materials

The interplay of Kohn anomalies with dimensionality, band structure, and external parameters:

• Extended $t$–$V$ models in 1D show a sequence of commensurate/incommensurate CDW phases traced to the underlying Fermi surface, with strong Kohn anomalies at corresponding wavevectors (Szyniszewski, 2015).
• Finite-temperature disorder and correlation effects can fill in the Kohn-anomaly-driven CDW gap (as in the Falicov-Kimball model), leading to a stable, gapless regime with pronounced subgap density of states (Žonda et al., 2019).
• Surface-sensitive ARPES in 1T-VSe$_2$ reveals that multi-band Fermiology may obscure or eliminate traditional Kohn anomaly signatures at the surface, demanding re-interpretation of earlier CDW evidence (Yilmaz et al., 2022).

Umklapp effects and commensuration further enrich CDW phenomenology. Landau expansions with umklapp/commensuration terms ($r^n \cos n\phi$) can yield first-order transitions, tricritical points, and additional anomalies in phonon spectra, often detected experimentally as hysteretic jumps in CDW amplitude or latent heat at the CDW transition (Rozhkov, 2024).

7. Summary Table: Diagnostic Features of a Kohn Anomaly

Manifestation	Physical Mechanism	Typical Materials/Context
Cusp/dip in $\omega(q)$	Divergent/peaked $\chi_0(q)$ (FSN)	Quasi-1D metals, TMDs
Full $\omega(q)=0$ softening	Critical $|g(q)|^2\operatorname{Re}\chi_0(q)$	Strongly coupled CDW systems
Splitting in Raman/IR modes	Lifting phonon degeneracy at $q_{CDW}$	Graphene, TMDs
Absence of Kohn anomaly	Flat $\chi_0(q)$ and/or small EPC	Many 2D/3D metals, cuprates
Hysteresis/latent heat	Umklapp-driven first-order transition	IrTe$_2$, Lu$_5$Ir$_4$Si$_{10}$, EuTe$_4$

Kohn anomalies are universal features of metallic systems, governing the electron-phonon coupling landscape and enabling (but not guaranteeing) the formation of CDW and related modulated states. Their experimental signatures, theoretical mechanisms, and relationship with electronic instabilities remain central to contemporary condensed matter physics (Zhu et al., 2015, Long et al., 2016, Alidoosti et al., 2020, Rozhkov, 2024, Yilmaz et al., 2022).