Type-II van Hove Singularity

• Type-II van Hove singularity is defined by saddle points at non-TRIM locations, impacting symmetry constraints on electron pairing.
• It enhances the density of states via logarithmic divergences, which can trigger unusual ferromagnetic and triplet superconducting fluctuations.
• Realizations in square-lattice Hubbard models, BC₃, and Ni-based trichalcogenides reveal diverse lattice environments affecting low-energy electronic behavior.

Searching arXiv for papers on Type-II van Hove singularities and closely related higher-order/unconventional VHS. Type-II van Hove singularity denotes, in standard contemporary usage, a van Hove singularity whose saddle-point momentum is not a time-reversal-invariant momentum (TRIM). In the language of saddle-point momenta $\mathbf K$, type-I corresponds to $\mathbf K=-\mathbf K\;(\mathrm{mod}\ \mathbf G)$, while type-II corresponds to $$\mathbf K\neq -\mathbf K\;(\mathrm{mod}\ \mathbf G)$$, with $\mathbf G$ a reciprocal lattice vector (Yao et al., 2013). In surface-state settings the same distinction is phrased as a type-I singularity located at a surface high-symmetry/TRIM point and a type-II singularity located at an arbitrary momentum in the surface Brillouin zone (Sanchez et al., 2021). The distinction is consequential because it changes the symmetry constraints on pairing and on the low-energy states that inherit the van Hove-enhanced density of states (DOS) (Meng et al., 2014).

1. Definition and taxonomy

The standard type-I/type-II distinction is a momentum-space classification, not a DOS classification. Both types are saddle-point singularities, but the saddle-point location differs. For type-I, the saddle lies at a TRIM; for type-II, it lies away from TRIM, so the van Hove momentum is not mapped to itself by time reversal (Yao et al., 2013). In the surface formulation used for helicoid-arc states in topological chiral crystals, the same distinction becomes: type-I at a surface high-symmetry point, type-II at a generic momentum in the surface Brillouin zone (Sanchez et al., 2021).

This location-based taxonomy matters because odd-parity pairing obeys

$\Delta(-\mathbf k)=-\Delta(\mathbf k).$

At a type-I saddle point, where $\mathbf K=-\mathbf K$, this enforces $\Delta(\mathbf K)=0$, so the odd-parity order parameter is suppressed exactly where the DOS is largest. At a type-II saddle point, $\mathbf K$ and $-\mathbf K$ are distinct, so that specific kinematic suppression is removed (Yao et al., 2013). This is the core reason the modern literature repeatedly connects type-II van Hove singularities to spin-triplet odd-parity superconductivity (Meng et al., 2014).

The terminology is not universal across all unconventional van Hove literature. Several later works study higher-order, exceptional, Mexican-hat, or line-extended singularities that are clearly related to nonstandard DOS enhancement but are not type-II in this standard sense (Akashi, 2023, Yuan et al., 2019).

2. Saddle-point geometry and density of states

In two dimensions, a conventional van Hove singularity arises from a saddle point of the band dispersion, and the DOS diverges logarithmically. In the weak-coupling patch treatment of a 2D van Hove singularity, the DOS behaves as

$\rho(\omega)\sim \log\frac{E_0}{\omega},$

with $\mathbf K=-\mathbf K\;(\mathrm{mod}\ \mathbf G)$0 a bandwidth-scale cutoff (Yao et al., 2013). In the topological chiral-crystal setting, the implicit local structure is the usual saddle-point form

$\mathbf K=-\mathbf K\;(\mathrm{mod}\ \mathbf G)$1

which again yields a logarithmic DOS enhancement in two dimensions (Sanchez et al., 2021).

Type-II does not alter the basic saddle-point origin of the singularity; it relocates the saddle to generic $\mathbf K=-\mathbf K\;(\mathrm{mod}\ \mathbf G)$2. That relocation changes which symmetry operations constrain the low-energy states and therefore changes the symmetry analysis of interaction-driven instabilities. In practice, the distinction is especially important when the Fermi surface is not sufficiently nested and the dominant particle-hole fluctuations are ferromagnetic or ferromagnetic-like rather than antiferromagnetic (Meng et al., 2014).

The distinction can also be phrased through surface-state contour evolution. In topological chiral crystals, the constant-energy helicoid arcs can approach, touch, and reconnect, or a single arc can self-touch. At the touching energy $\mathbf K=-\mathbf K\;(\mathrm{mod}\ \mathbf G)$3, the surface dispersion becomes a saddle point, and the associated DOS enhancement is therefore a surface van Hove singularity (Sanchez et al., 2021).

3. Lattice and model realizations

A canonical microscopic realization is the square-lattice Hubbard model with hopping up to third neighbors,

$\mathbf K=-\mathbf K\;(\mathrm{mod}\ \mathbf G)$4

and single-particle dispersion

$\mathbf K=-\mathbf K\;(\mathrm{mod}\ \mathbf G)$5

For $\mathbf K=-\mathbf K\;(\mathrm{mod}\ \mathbf G)$6, $\mathbf K=-\mathbf K\;(\mathrm{mod}\ \mathbf G)$7, and $\mathbf K=-\mathbf K\;(\mathrm{mod}\ \mathbf G)$8, the four van Hove saddle points are

$\mathbf K=-\mathbf K\;(\mathrm{mod}\ \mathbf G)$9

and the corresponding filling is $$\mathbf K\neq -\mathbf K\;(\mathrm{mod}\ \mathbf G)$$0 (Meng et al., 2014). These momenta are not TRIM and therefore realize a type-II van Hove singularity in the precise sense of the taxonomy above.

Monolayer BC$$\mathbf K\neq -\mathbf K\;(\mathrm{mod}\ \mathbf G)$$1 provides a honeycomb-lattice realization in which the relevant conduction band is modeled with

$$\mathbf K\neq -\mathbf K\;(\mathrm{mod}\ \mathbf G)$$2

and the van Hove filling is

$$\mathbf K\neq -\mathbf K\;(\mathrm{mod}\ \mathbf G)$$3

The paper emphasizes that the corresponding van Hove points are not time-reversal invariant, and hence are of type-II (Wang et al., 2017).

Ni-based transition-metal trichalcogenides furnish a multi-orbital honeycomb realization. In that setting the authors identify six type-II van Hove points along $$\mathbf K\neq -\mathbf K\;(\mathrm{mod}\ \mathbf G)$$4-M and $$\mathbf K\neq -\mathbf K\;(\mathrm{mod}\ \mathbf G)$$5-K, and a DOS peak roughly $$\mathbf K\neq -\mathbf K\;(\mathrm{mod}\ \mathbf G)$$6 eV above the Fermi level in the undoped band structure. The relevant electron-doped regime is around $$\mathbf K\neq -\mathbf K\;(\mathrm{mod}\ \mathbf G)$$7 to $$\mathbf K\neq -\mathbf K\;(\mathrm{mod}\ \mathbf G)$$8 electrons per Ni (Li et al., 2021).

System	Saddle-point location	Representative parameter
Square-lattice Hubbard	$$\mathbf K\neq -\mathbf K\;(\mathrm{mod}\ \mathbf G)$$9, $\mathbf G$0	$\mathbf G$1 (Meng et al., 2014)
BC$\mathbf G$2	Non-time-reversal-invariant van Hove points	$\mathbf G$3 (Wang et al., 2017)
Ni-based trichalcogenides	Six points along $\mathbf G$4-M and $\mathbf G$5-K	$\mathbf G$6 eV above $\mathbf G$7 (Li et al., 2021)

These realizations already show that type-II van Hove singularity is not tied to a single lattice class. Square, honeycomb, and multi-orbital honeycomb settings all realize it, with the common ingredient being the displacement of the saddle points away from TRIM.

4. Topological and surface-state realizations

Type-II van Hove singularities also arise in topological surface bands. In topological chiral crystals RhSi and CoSi, the projected bulk chiral charges $\mathbf G$8 produce two helicoid Fermi arcs terminating at each projected pocket. Because the arc endpoints wind in opposite senses around $\mathbf G$9 and $\Delta(-\mathbf k)=-\Delta(\mathbf k).$0 as binding energy is varied, arc touching becomes unavoidable. In RhSi, the observed singularity is type-I and occurs through inter-helicoid-arc touching at the surface high-symmetry point $\Delta(-\mathbf k)=-\Delta(\mathbf k).$1, with

$\Delta(-\mathbf k)=-\Delta(\mathbf k).$2

relative to the Fermi level of Ni$\Delta(-\mathbf k)=-\Delta(\mathbf k).$3Rh$\Delta(-\mathbf k)=-\Delta(\mathbf k).$4Si. In CoSi, the more unusual type-II singularity is intra-helicoid-arc: a single arc contracts, touches at a generic momentum, and pulls away again at

$\Delta(-\mathbf k)=-\Delta(\mathbf k).$5

about $\Delta(-\mathbf k)=-\Delta(\mathbf k).$6 meV below the Fermi level (Sanchez et al., 2021).

Pt$\Delta(-\mathbf k)=-\Delta(\mathbf k).$7HgSe$\Delta(-\mathbf k)=-\Delta(\mathbf k).$8 is a different topological realization in which the naturally cleaved (001) surface hosts topological surface states with saddle-like dispersion and type-II van Hove singularities. With spin-orbit coupling, the bulk is characterized by

$\Delta(-\mathbf k)=-\Delta(\mathbf k).$9

and the relevant (001) surface little group at $\mathbf K=-\mathbf K$0 is $\mathbf K=-\mathbf K$1, so saddle-like dispersion is symmetry-allowed. The symmetry-based $\mathbf K=-\mathbf K$2 Hamiltonian

$\mathbf K=-\mathbf K$3

has saddle-like dispersion for $\mathbf K=-\mathbf K$4, and the mirror-protected crossings occur at

$\mathbf K=-\mathbf K$5

Because the saddle points are displaced away from the surface TRIM, the paper identifies them as type-II van Hove singularities (Ghosh et al., 2019).

These topological examples broaden the meaning of type-II beyond ordinary bulk-band saddle points. In both cases the singularity is still a saddle-point DOS enhancement at generic momentum, but the saddle arises from topological surface-state connectivity rather than from a conventional bulk band extremum.

5. Correlation physics and ordered phases

Weak-coupling renormalization-group work established the basic interaction-driven distinction between the two VHS types. For type-I van Hove singularities, weak repulsive interactions generically induce unconventional singlet pairing. For type-II van Hove singularities, weak repulsive interactions favor triplet pairing, such as $\mathbf K=-\mathbf K$6-wave, when the Fermi surface is not sufficiently nested. In tetragonal systems this naturally leads to either chiral $\mathbf K=-\mathbf K$7 pairing or time-reversal-invariant $\mathbf K=-\mathbf K$8 $\mathbf K=-\mathbf K$9 pairing (Yao et al., 2013).

The square-lattice Hubbard analysis was subsequently extended beyond weak-coupling patch RG. Using RPA, DMFT+Parquet, and DCA, a wide doping range centered around the type-II van Hove filling was found to support a two-fold degenerate, spin-triplet, odd-parity $\Delta(\mathbf K)=0$0-wave pairing state when the Fermi surface is not sufficiently nested. In RPA the leading pairing symmetry is the doubly degenerate $\Delta(\mathbf K)=0$1-wave channel for

$\Delta(\mathbf K)=0$2

around $\Delta(\mathbf K)=0$3, while exactly at the VHS the divergent DOS drives a ferromagnetic divergence that prevents a meaningful superconducting RPA analysis there (Meng et al., 2014).

BC$\Delta(\mathbf K)=0$4 exhibits a sharper competition between magnetism and superconductivity. Functional renormalization group calculations show that at the exact type-II VHS filling

$\Delta(\mathbf K)=0$5

the SDW channel dominates, with $\Delta(\mathbf K)=0$6 and critical scale

$\Delta(\mathbf K)=0$7

Slightly below or above the singularity, at $\Delta(\mathbf K)=0$8 and $\Delta(\mathbf K)=0$9, the leading instability becomes $\mathbf K$0-wave superconductivity. A small nearest-neighbor Coulomb repulsion,

$\mathbf K$1

enhances the superconducting scale (Wang et al., 2017).

In Ni-based transition-metal trichalcogenides, approaching the type-II vHs strengthens the $\mathbf K$2 spin susceptibility and therefore the ferromagnetic fluctuation. Within multi-orbital RPA this suppresses the even-parity $\mathbf K$3 $\mathbf K$4 state in favor of the odd-parity $\mathbf K$5 $\mathbf K$6 state, producing a singlet-to-triplet pairing transition near $\mathbf K$7–$\mathbf K$8 (Li et al., 2021).

A common implication across these studies is that type-II geometry reallocates the van Hove-enhanced phase space into channels compatible with triplet odd-parity pairing. The papers do not claim that ferromagnetism is absent; rather, the central issue is the competition between ferromagnetic or ferromagnetic-like fluctuations and the triplet superconductivity those same fluctuations can mediate (Wang et al., 2017, Meng et al., 2014).

6. Related generalizations and terminological boundaries

Type-II van Hove singularity is not a catch-all label for every nonstandard DOS singularity. Several closely related but distinct categories appear in the recent literature.

A three-dimensional nearly free-electron analysis of Bragg-plane intersections studies “high-order van Hove singularities,” “continuous saddle points,” “extended saddle points,” and “saddle loops,” but explicitly does not use the standard type-I/type-II taxonomy. Its central mechanism is the line intersection of three Bragg planes in 3D, which produces multi-branch hybridization, continuous saddle structures, and companion Dirac nodal lines. In H$\mathbf K$9S, the sharp DOS peak is attributed to a saddle loop associated with the Bragg intersection $-\mathbf K$0, not to a standard type-II van Hove singularity (Akashi, 2023).

The paper “Magic of high order van Hove singularity” introduces a different internal nomenclature in which “type-II high-order VHS” means a degenerate saddle with one vanishing Hessian eigenvalue. In that usage the local critical DOS behaves as

$-\mathbf K$1

and the term “type-II” refers to Hessian degeneracy structure rather than to the TRIM-versus-generic-momentum distinction of standard type-II van Hove singularity (Yuan et al., 2019).

Other nearby categories include the higher-order VHS in mirror-symmetric twisted trilayer graphene, where the singularity arises from the merging of ordinary VHS and Dirac cones at zero energy and has exponent $-\mathbf K$2, explicitly beyond recent single-band classifications (Guerci et al., 2021); the “exceptional” van Hove singularity in pseudogapped metals, where interaction-driven Fermi-surface splitting causes two reconstructed sheets to touch near a second-order VHS (Paul et al., 2022); and the Mexican-hat inverted-band VHS in Sn-doped Bi$-\mathbf K$3Sb$-\mathbf K$4Te$-\mathbf K$5S, where the singularity is tied to finite-$-\mathbf K$6 band-edge structure rather than to an identified saddle-point type-II VHS (Jiang et al., 2021).

This terminological boundary is important. In standard modern usage, type-II van Hove singularity means a saddle-point singularity at generic, non-TRIM momentum. Many unconventional singularities share the same phenomenology of enhanced DOS or Lifshitz reconstruction, but they should not be conflated with the standard type-II classification unless their defining momentum-space criterion is the same (Sanchez et al., 2021, Yao et al., 2013).