Van Hove singularity

A Van Hove singularity occurs where the gradient of a crystal band, E(k), vanishes in reciprocal space. These stationary points may be minima, maxima, or saddle points; they cause the density of states to vary nonanalytically with energy.¹ The effect is a property of the band structure rather than a literal divergence of the number of particles at one energy.

In a crystal, allowed electronic states are organized into bands over the Brillouin zone. The density of states counts states per energy interval, and its structure reflects how rapidly the bands disperse. Near a stationary point, many neighboring wave vectors have similar energies, concentrating states and producing a peak, cusp, or divergence. Van Hove's original work established the general connection between critical points of the dispersion relation and spectral singularities.¹

The dimensionality of the crystal determines how a Van Hove singularity appears. In one dimension, a simple band extremum produces an inverse-square-root divergence in the density of states. In two dimensions, an ordinary extremum gives a step-like edge, while a saddle point produces a logarithmic divergence. In three dimensions, the anomaly is generally weaker, often appearing as a finite cusp or a square-root variation.

The two-dimensional saddle point is especially prominent because it can lie close to a Fermi level and strongly amplify interaction effects. Real materials round the ideal mathematical behavior through temperature, disorder, finite sample size, electron interactions, and experimental energy resolution. Consequently, a measured peak is usually finite even when an idealized band model predicts a divergence. The classification is part of the broader theory of critical points in band dispersions.

A Van Hove singularity can magnify electronic instabilities because response functions receive unusually large contributions from states near the singular energy. When the Fermi level approaches such a feature, relatively weak interactions may favor ferromagnetism, charge-density waves, nematic order, or superconductivity; the outcome depends on band geometry, interaction type, and competing scattering channels.

The concept has been used extensively in discussions of high-temperature superconducting cuprates, where saddle-point features near the Fermi surface motivated the “Van Hove scenario” for enhanced transition temperatures. Related analyses of two-dimensional correlated-electron models show that the singularity can alter renormalization behavior and make several ordering tendencies compete rather than selecting one universal phase.² Spectroscopy, photoemission, tunneling, and quantum-oscillation measurements can reveal these features indirectly or directly.

Not every sharp density-of-states feature is a Van Hove singularity: the defining requirement is a critical point of the band energy, not merely a large peak caused by impurities or a narrow flat band. In multiband materials, several critical points may overlap, producing an apparently broad or asymmetric anomaly that is difficult to assign to one band.

Small changes in carrier concentration, pressure, strain, or chemical composition can move a Fermi level through a singularity and trigger a Lifshitz transition, in which the topology of the Fermi surface changes without necessarily breaking a conventional symmetry. The idea also applies beyond ordinary parabolic bands: saddle points in graphene-like lattices and engineered two-dimensional materials can occur at symmetry-related momenta and interact with sublattice, orbital, or spin structure.³ Thus the singularity is both a geometric feature of reciprocal space and a possible amplifier of many-body physics.

Density of states

The number of quantum states available per unit energy, often resolved for a material, band, momentum, or spin.

Brillouin zone

The primitive cell of a crystal's reciprocal lattice, containing the distinct crystal momenta used to describe band structure.

Saddle point

A stationary point whose curvature has opposite signs along different momentum-space directions.

Fermi level

The chemical potential commonly used to identify the energy separating mostly occupied and mostly unoccupied electronic states.

Lifshitz transition

A change in the topology of a Fermi surface caused by varying a control parameter such as carrier density or pressure.