High-Order van Hove Singularities

• High-order van Hove singularities are critical points where the Hessian vanishes, leading to dominant higher-order terms and power-law density-of-states divergences.
• They are characterized by the (n, m) classification that bridges conventional quadratic saddles with quasi-flat bands and supports phenomena like Lifshitz transitions and topological responses.
• The enhanced DOS at these singularities boosts electronic interaction susceptibilities, promoting ordered phases such as unconventional superconductivity, magnetism, and nematic states in engineered systems.

A high-order van Hove singularity is a momentum-space critical point of a band dispersion at which the usual quadratic saddle-point description fails because the Hessian becomes degenerate or vanishes, so that the leading nonzero terms in the local expansion occur at cubic, quartic, or still higher order. In two dimensions this promotes the familiar logarithmic van Hove divergence of the density of states to a power law, thereby strongly enhancing the phase space for interaction effects. Recent work places high-order van Hove singularities at the intersection of several themes: quasi-flat and exactly flat bands, Lifshitz transitions, moiré minibands, kagome and ruthenate physics, topological Chern bands, and engineered synthetic platforms (Classen et al., 2024).

1. Definition and local classification

For a conventional two-dimensional van Hove singularity, one expands the band energy around a critical momentum $k_0$ with $\nabla \varepsilon(k_0)=0$ and nonvanishing Hessian determinant. In appropriate local coordinates,

$$\varepsilon(k)-\varepsilon(k_0)\approx \alpha k_x^2-\beta k_y^2,$$

which yields the standard logarithmic DOS singularity. A high-order van Hove singularity instead occurs when the quadratic description collapses: in the formulation emphasized in the 2024 review, all second derivatives vanish at $k_0$, $H=0$, and the first nonzero terms in the Taylor expansion appear at higher orders $n,m>2$ (Classen et al., 2024).

A convenient classification labels the singularity by the pair $(n,m)$ in

$$\varepsilon(k)=\varepsilon_{\mathrm{VHS}}+\alpha_x k_x^n+\alpha_y k_y^m+\text{(higher-order cross terms)}.$$

The ordinary saddle is recovered as $(n,m)=(2,2)$, while larger $n$ and $\nabla \varepsilon(k_0)=0$0 encode progressively flatter local dispersion. This $\nabla \varepsilon(k_0)=0$1 language is especially useful for relating HOVHS to quasi-flat bands and for comparing different materials platforms (Classen et al., 2024).

A complementary classification distinguishes two mechanisms. In the type-I case, both quadratic coefficients vanish and the expansion begins at cubic order, so that three or more Fermi sheets meet at a multicritical saddle point. In the type-II case, exactly one Hessian eigenvalue vanishes while the other remains finite; then the low-energy expansion takes the form

$\nabla \varepsilon(k_0)=0$2

and the high-order point is reached at $\nabla \varepsilon(k_0)=0$3, where two Fermi sheets touch tangentially rather than crossing at a finite angle (Yuan et al., 2019). This distinction is operationally important because type-II HOVHS arise naturally by tuning a single control parameter in moiré graphene and related systems (Yuan et al., 2019).

2. Density of states and singular exponents

The DOS near a HOVHS follows from

$\nabla \varepsilon(k_0)=0$4

Using the local form $\nabla \varepsilon(k_0)=0$5, one obtains the scaling

$\nabla \varepsilon(k_0)=0$6

The conventional saddle $\nabla \varepsilon(k_0)=0$7 sits at the marginal case $\nabla \varepsilon(k_0)=0$8, corresponding to the logarithmic divergence, whereas $\nabla \varepsilon(k_0)=0$9 or $$\varepsilon(k)-\varepsilon(k_0)\approx \alpha k_x^2-\beta k_y^2,$$0 with $$\varepsilon(k)-\varepsilon(k_0)\approx \alpha k_x^2-\beta k_y^2,$$1 already produce algebraic singularities, and $$\varepsilon(k)-\varepsilon(k_0)\approx \alpha k_x^2-\beta k_y^2,$$2 as $$\varepsilon(k)-\varepsilon(k_0)\approx \alpha k_x^2-\beta k_y^2,$$3 (Classen et al., 2024).

Several canonical exponents recur across the literature. A type-II HOVHS with $$\varepsilon(k)-\varepsilon(k_0)\approx \alpha k_x^2-\beta k_y^2,$$4 in the expansion above yields

$$\varepsilon(k)-\varepsilon(k_0)\approx \alpha k_x^2-\beta k_y^2,$$5

with an intrinsically particle-hole-asymmetric peak in twisted bilayer graphene (Yuan et al., 2019). A cubic or “monkey-saddle” dispersion, such as $$\varepsilon(k)-\varepsilon(k_0)\approx \alpha k_x^2-\beta k_y^2,$$6, gives $$\varepsilon(k)-\varepsilon(k_0)\approx \alpha k_x^2-\beta k_y^2,$$7, which appears in multiple settings including honeycomb models, twisted trilayer graphene, Haldane-type models, and kagome-derived band structures (Lee et al., 2024). Quartic HOVHS with $$\varepsilon(k)-\varepsilon(k_0)\approx \alpha k_x^2-\beta k_y^2,$$8-symmetric dispersion can produce $$\varepsilon(k)-\varepsilon(k_0)\approx \alpha k_x^2-\beta k_y^2,$$9, while anisotropic $k_0$0 saddles give $k_0$1 (Zervou et al., 2022). In mirror-symmetric twisted trilayer graphene, an additional topological Lifshitz transition leads to an anomalous $k_0$2 singularity when the ratio $k_0$3 reaches unity (Guerci et al., 2021).

Local form	DOS behavior	Representative setting
$k_0$4 quadratic saddle	$k_0$5	ordinary VHS
$k_0$6 or $k_0$7 anisotropic HOVHS	$k_0$8	extended saddles, distorted kagome
cubic “monkey saddle”	$k_0$9	moiré graphene, Haldane, kagome
quartic isotropic HOVHS	$H=0$0	$H=0$1-symmetric quartic saddles
special Lifshitz critical point	$H=0$2	mirror-symmetric twisted trilayer graphene

This hierarchy of exponents formalizes the notion that HOVHS are not a single universality class but a family of critical points whose low-energy singularity is set by the first nonzero homogeneous term in the dispersion (Classen et al., 2024).

3. Relation to flat bands, Lifshitz structure, and topology

The 2024 review makes the flat-band connection explicit: an exactly flat band corresponds formally to the limit $H=0$3, where all derivatives vanish to infinite order and $H=0$4 in a neighborhood of $H=0$5. In practice, finite but large $H=0$6 and $H=0$7 produce a quasi-flat local pocket. Quantitatively, the local bandwidth shrinks as

$H=0$8

while the DOS peak height grows as

$H=0$9

This places HOVHS as an interpolation between ordinary saddles and true flat bands rather than as a disconnected phenomenon (Classen et al., 2024).

Because a van Hove singularity marks a Lifshitz transition, HOVHS typically encode the merging, annihilation, or higher-order tangency of Fermi-surface sheets. In twisted bilayer graphene, the type-II scenario describes the coalescence of ordinary VHS into tangential touching points as a single tuning parameter is varied (Yuan et al., 2019). In Bernal bilayer and rhombohedral trilayer graphene, a one-parameter interpolation from three separate ordinary VHS to a single cubic HOVHS captures the Lifshitz transition from three pockets to one larger pocket, with distinct parquet-RG fixed-point structures on the two sides of the crossover (Lee et al., 2024).

Topological band structures supply further variants. Mirror-symmetric twisted trilayer graphene hosts a zero-energy HOVHS with exponent $n,m>2$0, protected by $n,m>2$1 rotation and a combined mirror-particle-hole symmetry, and tunable by twist angle and perpendicular electric field (Guerci et al., 2021). Topological moiré surface states on three-dimensional topological insulators realize $n,m>2$2 Chern bands whose valley saddles merge into a cubic HOVHS, and the low-temperature intrinsic anomalous Hall response satisfies

$n,m>2$3

so the Hall anomaly inherits the HOVHS power law (Pullasseri et al., 2024). Kagome topological bands with complex next-nearest-neighbor hopping similarly exhibit HOVHS with exponents $n,m>2$4, alongside Chern numbers ranging from $n,m>2$5 to $n,m>2$6 (Wang et al., 2024). This suggests that HOVHS often mediate between band flattening and topological response rather than belonging exclusively to either category.

4. Interaction effects and ordered phases

The central many-body consequence of a HOVHS is the enhancement of any susceptibility that is weighted by the DOS. In the review formulation, even a simple mean-field criterion such as $n,m>2$7 becomes easier to satisfy when $n,m>2$8 diverges with $n,m>2$9. Candidate orders include ferro- and antiferromagnetism, charge- and spin-density waves, Pomeranchuk distortions, and unconventional superconductivity; for $(n,m)$0, the pairing susceptibility diverges as $(n,m)$1 in the normal state, which strongly boosts weak-coupling pairing tendencies (Classen et al., 2024).

More detailed treatments corroborate that general picture. In magic-angle twisted bilayer graphene, the intervalley density-wave susceptibility diverges faster than $(n,m)$2, and weak strain or intervalley interactions can split the VHS peak into two, consistent with scanning-tunneling observations of peak splitting under doping (Yuan et al., 2019). In a honeycomb-lattice extended Hubbard model, determinant and constrained-path quantum Monte Carlo locate the HOVHS at the critical hopping relation

$(n,m)$3

where the quadratic mass tensor collapses in one direction and the DOS becomes $(n,m)$4. Near the corresponding filling, $(n,m)$5 for $(n,m)$6, $(n,m)$7, $(n,m)$8, the system exhibits a crossover between ferromagnetic and antiferromagnetic fluctuations, and long-distance $(n,m)$9-wave correlations dominate the pairing sector; nearest-neighbor Coulomb interactions suppress that pairing approximately in proportion to $$\varepsilon(k)=\varepsilon_{\mathrm{VHS}}+\alpha_x k_x^n+\alpha_y k_y^m+\text{(higher-order cross terms)}.$$0 (Wei et al., 15 May 2025).

Renormalization-group studies reach similar conclusions from a different direction. For a model with nested Fermi-surface patches plus a quartic HOVHS patch, the additional power-law bubbles can raise the spin-density-wave critical temperature from $$\varepsilon(k)=\varepsilon_{\mathrm{VHS}}+\alpha_x k_x^n+\alpha_y k_y^m+\text{(higher-order cross terms)}.$$1 to $$\varepsilon(k)=\varepsilon_{\mathrm{VHS}}+\alpha_x k_x^n+\alpha_y k_y^m+\text{(higher-order cross terms)}.$$2 for the quoted bare couplings, i.e. by roughly two orders of magnitude, and can also favor charge-density-wave order for other couplings (Zervou et al., 2022). On the distorted kagome surface of Co$$\varepsilon(k)=\varepsilon_{\mathrm{VHS}}+\alpha_x k_x^n+\alpha_y k_y^m+\text{(higher-order cross terms)}.$$3Sn$$\varepsilon(k)=\varepsilon_{\mathrm{VHS}}+\alpha_x k_x^n+\alpha_y k_y^m+\text{(higher-order cross terms)}.$$4S$$\varepsilon(k)=\varepsilon_{\mathrm{VHS}}+\alpha_x k_x^n+\alpha_y k_y^m+\text{(higher-order cross terms)}.$$5, a quartic HOvHS pinned to the Fermi energy is reported to precipitate a Pomeranchuk instability and a cascade of nematic states over an energy shell of about $$\varepsilon(k)=\varepsilon_{\mathrm{VHS}}+\alpha_x k_x^n+\alpha_y k_y^m+\text{(higher-order cross terms)}.$$6 meV, without generating additional translational symmetry breaking (Nag et al., 2024). More specialized analyses further argue that an $$\varepsilon(k)=\varepsilon_{\mathrm{VHS}}+\alpha_x k_x^n+\alpha_y k_y^m+\text{(higher-order cross terms)}.$$7 quartic HOVHS with $$\varepsilon(k)=\varepsilon_{\mathrm{VHS}}+\alpha_x k_x^n+\alpha_y k_y^m+\text{(higher-order cross terms)}.$$8 can support triplet superconductivity with $$\varepsilon(k)=\varepsilon_{\mathrm{VHS}}+\alpha_x k_x^n+\alpha_y k_y^m+\text{(higher-order cross terms)}.$$9, with an upper estimate of about $(n,m)=(2,2)$0 mK for Sr$(n,m)=(2,2)$1Ru$(n,m)=(2,2)$2O$(n,m)=(2,2)$3 in the quoted parameter set (Sanjeevappa et al., 23 Mar 2026).

5. Materials platforms and experimental signatures

The materials survey in the 2024 review identifies twisted bilayer and multilayer graphene near magic angles, ABC-stacked graphene trilayers under displacement field, moiré transition-metal dichalcogenides, kagome metals, and photonic or cold-atom simulators as established or promising HOVHS platforms (Classen et al., 2024). In magic-angle twisted bilayer graphene, scanning tunneling spectroscopy observes an asymmetric DOS peak whose log-log plot shows two parallel lines of slope $(n,m)=(2,2)$4, with peak width of tens of meV; tuning by twist angle or hydrostatic pressure moves the system through the critical regime where the type-II HOVHS emerges (Yuan et al., 2019). The same work gives a critical coupling $(n,m)=(2,2)$5 for $(n,m)=(2,2)$6, corresponding to $(n,m)=(2,2)$7–$(n,m)=(2,2)$8 for the quoted continuum-model parameters (Yuan et al., 2019).

ABC-stacked trilayer graphene aligned with hBN provides a field-tuned type-I realization: at $(n,m)=(2,2)$9 eV the quadratic terms vanish and three Fermi pockets meet at $n$0 (Yuan et al., 2019). Mirror-symmetric twisted trilayer graphene hosts a zero-energy HOVHS tunable by twist angle and perpendicular field; for realistic corrugation $n$1 and $n$2, one finds $n$3 V/nm, and scanning tunneling spectroscopy is predicted to distinguish the $n$4 and $n$5 regimes (Guerci et al., 2021).

Kagome-derived systems show several distinct mechanisms. The review notes that purely two-dimensional kagome dispersion can support $n$6 or higher saddles at high-symmetry points (Classen et al., 2024). On the distorted Co$n$7Sn$n$8S$n$9 surface, scanning tunneling spectroscopy reveals a sharp zero-bias peak well fitted by $\nabla \varepsilon(k_0)=0$00 after subtracting a constant background, and quasiparticle interference at the Fermi energy retains only one $\nabla \varepsilon(k_0)=0$01–$\nabla \varepsilon(k_0)=0$02 scattering branch, consistent with a nematic reconstruction driven by the HOvHS (Nag et al., 2024). Bilayer kagome borophene has been proposed to host both a conventional and a monkey-saddle VHS in the same band: the conventional one lies $\nabla \varepsilon(k_0)=0$03 eV below the Fermi level, the high-order one $\nabla \varepsilon(k_0)=0$04 eV below, and the adjacent Dirac-like cone reaches a Fermi velocity of $\nabla \varepsilon(k_0)=0$05 m/s (Gao et al., 2023).

Synthetic settings broaden the scope beyond solid-state systems. Photonic and cold-atom lattices can realize tailored $\nabla \varepsilon(k_0)=0$06 dispersions with tunable exponents (Classen et al., 2024). In a checkerboard optical-lattice insulator designed for cavity polaritons, the interband gap edge realizes an effective HOVHS in the joint DOS, with the divergence engineered without sub-gap absorption by maintaining a full direct gap and using spin-polarized noninteracting fermions (Gianardi et al., 19 Sep 2025). Such examples indicate that the relevant singular object need not be only the electronic DOS; analogous constructions can target the joint DOS or cavity self-energy.

6. Detection, engineering, and open directions

A general method for diagnosing HOVHS in multiband Hamiltonians is provided by the extended Feynman-Hellmann framework. Rather than diagonalizing a model analytically near every candidate momentum, one computes derivatives of the band eigenvalue along a path $\nabla \varepsilon(k_0)=0$07 using recursive formulas for $\nabla \varepsilon(k_0)=0$08. This yields the Taylor expansion directly and supports a concrete workflow: identify critical points from $\nabla \varepsilon(k_0)=0$09, inspect the Hessian, test whether the third-order tensor has support on null directions, and continue to quartic or higher order when necessary (Chandrasekaran et al., 2022). The same formalism incorporates tuning parameters by promoting the path to $\nabla \varepsilon(k_0)=0$10, so that the coefficient whose zero defines the HOVHS can be solved as a function of strain, bias, staggered potential, or other control parameters (Chandrasekaran et al., 2022).

The engineering program already has concrete case studies. In the Haldane model, tuning the staggered potential to

$\nabla \varepsilon(k_0)=0$11

produces a monkey saddle at $\nabla \varepsilon(k_0)=0$12 with

$\nabla \varepsilon(k_0)=0$13

and small detuning restores quadratic curvature and splits the multicritical point into ordinary extrema and saddles (Chandrasekaran et al., 2022). In the surface layer of Sr$\nabla \varepsilon(k_0)=0$14RuO$\nabla \varepsilon(k_0)=0$15, tight-binding models constrained by ARPES and QPI show how octahedral rotation and weak nematicity move an ordinary VHS toward an $\nabla \varepsilon(k_0)=0$16-type HOVHS; the bare DFT model yields $\nabla \varepsilon(k_0)=0$17, while the experimentally refined surface lies close to the critical regime (Chandrasekaran et al., 2023). Non-Hermitian Floquet interfaces offer a different route: when an exceptional ring grazes the Fermi line, paired ordinary VHS can coalesce into a single higher-order DOS peak with power-law exponents $\nabla \varepsilon(k_0)=0$18 and $\nabla \varepsilon(k_0)=0$19, a mechanism explicitly distinguished from Hermitian saddle merging (Banerjee et al., 2023).

The present literature therefore frames HOVHS as a unifying concept rather than a niche singularity class. It links local band flattening to experimentally tunable Lifshitz transitions, provides a systematic route from ordinary saddles to quasi-flat bands, and organizes a broad range of interaction-driven phenomena across moiré graphene, kagome systems, ruthenates, Chern bands, and synthetic lattices (Classen et al., 2024). A plausible implication is that future progress will depend less on identifying isolated examples and more on building controlled classification, detection, and tuning schemes across multiband and topological settings.