Van Hove Singularities in Crystals

• Van Hove singularities are critical points in a crystal's band structure where the density of states diverges due to saddle points and higher-order dispersions.
• They play a fundamental role in electronic instabilities, influencing superconductivity, charge density waves, and nematic order, as revealed by ARPES, STM, and transport experiments.
• Engineering and tuning of VHSs via twist angles, strain, or chemical doping enable control over electronic phases in materials like graphene, kagome lattices, and topological systems.

A Van Hove singularity (VHS) is a critical point in the electronic band structure of a crystalline material where the density of states (DOS) diverges. In two-dimensional (2D) systems, VHSs typically appear at saddle points of the band dispersion, leading to logarithmic or, in some cases, stronger algebraic divergences in the DOS. These singularities have profound consequences for electronic correlations, transport, and the emergence of ordered phases, including superconductivity, charge density wave (CDW), and nematic instabilities. The typology, physical mechanisms, and experimental signatures of VHSs have evolved significantly with recent advances in band-engineering, angle-resolved photoemission spectroscopy (ARPES), scanning tunneling microscopy/spectroscopy (STM/STS), and topological materials research.

1. Classification and Mathematical Structure of Van Hove Singularities

The canonical VHS in a 2D crystal emerges at saddle points of the dispersion $E(\vec{k})$ where %%%%1%%%% and the Hessian matrix has eigenvalues of opposite sign. The local expansion reads $$E(\vec{k}) \simeq E_{\text{VHS}} + \alpha k_x^2 - \beta k_y^2$$ ($\alpha, \beta > 0$). The DOS near this point diverges logarithmically:

$$\rho(E) = \frac{1}{2\pi^2 \sqrt{\alpha \beta}} \ln \left| \frac{\Lambda}{E - E_\text{VHS}} \right|$$

where $\Lambda$ is an ultraviolet cutoff. These saddle-point singularities represent "ordinary" or "first-order" VHSs (Piriou et al., 2011, Sanchez et al., 2021).

A higher-order VHS occurs when one or more quadratic curvatures vanish, so the expansion includes higher-degree terms, e.g. $E(q) = E_v + a\,q_x^p - b\,q_y^q$ ($p, q > 2$). The associated DOS displays power-law divergences:

$$\rho(E) \propto |E - E_v|^{-\mu}, \quad \text{with} \quad \mu = 1 - \left(\frac{1}{p} + \frac{1}{q}\right)$$

Such high-order VHSs can appear naturally in kagome lattices, twisted bilayers, or engineered by tuning parameters such as strain, chemical potential, or symmetry-breaking fields (Patra et al., 2024, Markiewicz et al., 2021, Chandrasekaran et al., 2023).

2. Physical Origin and Experimental Identification

Van Hove singularities are topological features of the band structure, manifesting wherever the gradient of energy vanishes and the curvature is indefinite (saddle point), extremal (minimum/maximum), or, in higher-order cases, degenerate up to quartic or higher terms (Sanchez et al., 2021, Chandrasekaran et al., 2023).

Experimental techniques for detecting VHSs include:

• ARPES: Directly images band dispersions and saddle points; resolves both ordinary and high-order VHSs near Fermi energy, distinguishing sublattice or orbital character in multi-orbital systems (Hu et al., 2021, Lan et al., 4 Jan 2026).
• STM/STS: Probes the local DOS, observing logarithmic peaks (VHS) and their evolution with doping, disorder, or external fields (Piriou et al., 2011, Li et al., 2017).
• Transport (Hall, thermoelectric): The divergence in DOS is reflected in carrier density, resistivity, Nernst effect, and susceptibility measurements (Kim et al., 2023, Chapai et al., 2024, Shibata et al., 2023).

Characterization of VHSs by symmetry and orbital content (e.g., sublattice-pure vs. sublattice-mixed) has been enabled by polarization-dependent ARPES, particularly in kagome metals and topological materials (Lan et al., 4 Jan 2026, Hu et al., 2021).

3. Tuning and Engineering of VHSs

The control and tuning of VHSs—either their energy position relative to the Fermi level, their order, or their orbital character—unlock rich correlated phenomena:

• Twisted Bilayers: In graphene, the twist angle $\theta$ determines the energy separation $$\Delta E_\text{VHS} \simeq \hbar v_F 2K \sin(\theta/2)$$ between VHSs, permitting alignment near the Fermi level and facilitating correlated states (Yan et al., 2012, Zhao et al., 2021).
• Electric Field: Alkali-metal deposition and ionic-liquid gating can shift VHSs through the Fermi level in ultrathin oxides, inducing Lifshitz transitions and modulating the divergent DOS (Kim et al., 2023).
• Chemical Substitution and Doping: In kagome and chiral metals, substitution (e.g., Ta for V in CsV$_3$Sb$_5$) precisely aligns VHSs with the Fermi energy, controlling superconducting $T_c$ and suppressing competing CDW instabilities (Luo et al., 2023).
• External Pressure and Strain: STM-induced mechanical pressure or uniaxial strain tunes interlayer coupling or symmetry-breaking anisotropies, engineering transitions from ordinary to higher-order VHSs (Zhao et al., 2021, Chandrasekaran et al., 2023).
• Topological and Floquet Systems: Non-Hermitian interface engineering (Floquet drives, gain/loss effects) can induce saddle-point merging and realize higher-order VHSs with distinct power-law DOS (Banerjee et al., 2023).

4. Impact on Electronic Instabilities

The enhanced DOS at a VHS amplifies particle-particle and particle-hole scattering, often driving correlated phases:

• Superconductivity: Alignment of the Fermi level with a VHS boosts $T_c$ via the increased DOS, facilitating both boson-mediated and purely electronic pairing mechanisms. High-order VHSs often correlate positively with higher $T_c$ across families of cuprates and kagome superconductors (Markiewicz et al., 2021, Patra et al., 2024, Luo et al., 2023).
• Charge and Spin Density Waves (CDW/SDW): VHS-enhanced susceptibility at specific nesting vectors can drive CDW or SDW order. Sublattice-pure VHSs often favor bond-density wave or nematic instabilities due to suppressed on-site interaction channels (Lan et al., 4 Jan 2026, Patra et al., 2024).
• Nematicity: High-order or anisotropic VHSs, as in CsTi$_3$Bi$_5$, suppress translational symmetry-breaking but promote electronic nematic distortions via Pomeranchuk instabilities; the algebraic DOS divergence magnifies nematic susceptibility (Patra et al., 2024).
• Metamagnetism and Lifshitz Transitions: Tuning of VHSs through the Fermi level can trigger sharp changes in Fermi surface topology, DOS jumps, and metamagnetic transitions, especially in ruthenates and similar oxide systems (Marques et al., 2023).

5. Theoretical Frameworks and Transfer Matrix Perspective

The connection between VHSs and exceptional points (EPs) of the transfer matrix has been established for lattice models: any $p^\mathrm{th}$-order VHS coincides with an EP of order $p$ of the non-Hermitian transfer matrix, with detailed recipe for engineering such points by constraining the derivatives of the band dispersion (Saha et al., 2024). This unifies the treatment of VHSs across quantum, photonic, and non-Hermitian platforms.

Tables are useful to organize the correspondence between VHS type, DOS divergence, material context, and key theoretical formulas:

VHS Type	DOS Divergence	Representative Systems
Saddle (ordinary)	$\ln|E-E_{VHS}|$	graphene, cuprates, SrRuO$_3$
High-order (quartic, etc.)	$|E-E_{VHS}|^{-\mu}$, $\mu=\frac{2}{n+m}-1$	kagome, engineered lattices
Non-Hermitian	$|E-E_{VHS}|^{-\mu}$, topology controlled	Floquet semimetals

6. Experimental Signatures and Applications

Direct observation of VHSs is manifest in ARPES (band flatness/saddle character), STM/STS (DOS peaks), transport (carrier density divergence or sign change), and thermoelectric (enhanced Nernst coefficients). The sensitivity of correlated states to VHS alignment enables functionality for:

• Tunable Superconductors and Correlated Insulators: via band-engineering in twisted bilayers and oxides (Yan et al., 2012, Kim et al., 2023).
• Topological Phases and Berry Physics: helicoid-arc VHSs in chiral conductors link VHSs to surface state topology and novel pairing channels (Sanchez et al., 2021).
• Photonic Metamaterials: “photonic VHSs” arise at slow-light points in hyperbolic metamaterials, enabling near-field LDOS enhancement for quantum emitters, nano-lasing, and biosensing (Cortes et al., 2013).

7. Outlook: Unified Perspective, Future Directions, and Unresolved Issues

Recent work demonstrates that the nature (order, sublattice, orbital content) and tunability of Van Hove singularities are central organizing principles for electronic instability in low-dimensional and topological systems. Open directions include:

• Engineering of arbitrary-order VHSs via symmetries, non-Hermitian effects, and Floquet drives (Chandrasekaran et al., 2023, Banerjee et al., 2023, Saha et al., 2024).
• Direct comparison of high-order VHS power-law exponents in different material classes and their link to superconducting or nematic phase boundaries (Patra et al., 2024, Markiewicz et al., 2021, Markiewicz et al., 2023).
• Quantitative mastery of DOS amplitude and coupling strength in optimal superconductors—balancing VHS strength and competing orders (Markiewicz et al., 2023).
• Extension of the VHS–EP correspondence to bosonic lattices, photonic crystals, and artificial quantum matter (Cortes et al., 2013, Saha et al., 2024).

A comprehensive understanding of VHSs—ordinary and high-order, in equilibrium and driven systems—affords predictive control of collective phenomena and materials design in quantum matter.