High-order Van Hove singularities and their connection to flat bands

TL;DR

High-order Van Hove singularities (HOVHS) generalize standard VHS by incorporating higher-order degeneracies of critical points, yielding power-law divergences in the DOS in 2D and strong interaction effects. The authors outline a catastrophe-theory–based classification (corank, codimension, determinacy, winding) and connect tuning strategies (strain, twist, fields) to the realization of HOVHS and their relation to flat bands. They analyze interaction effects via bare susceptibilities, ladder resummations, and RG, revealing potential non-Fermi-liquid fixed points (supermetal) and rich competition among orders. Experimental evidence from strontium ruthenates, kagomé metals, and graphene/moiré systems supports the relevance of HOVHS for correlated states, tunable band structure, and emergent topological phenomena, with implications for designing materials hosting novel quantum phases.

Abstract

The flattening of single-particle band structures plays an important role in the quest for novel quantum states of matter due to the crucial role of interactions. Recent advances in theory and experiment made it possible to construct and tune systems with nearly flat bands, ranging from graphene multilayers and moire' materials to kagome' metals and ruthenates. While theoretical models predict exactly flat bands under certain ideal conditions, evidence was provided that these systems host high-order Van Hove points, i.e., points of high local band flatness and power-law divergence in energy of the density of states. In this review, we examine recent developments in engineering and realising such weakly dispersive bands. We focus on high-order Van Hove singularities and explore their connection to exactly flat bands. We provide classification schemes and discuss interaction effects. We also review experimental evidence for high-order Van Hove singularities and point out future research directions.

Figures (4)

• Figure 1: Triangular-lattice energy dispersion at $(k_x,2\pi/\sqrt{3})/a$ going through Van Hove point $M_1$ at $k_x=0$ for three values of the next-nearest-neighbor hopping $t'/t$ and corresponding Fermi surface at Van Hove filling. A HOVHS occurs for $t'=t/9$ (red solid line). For small $t'\leq t/9$ (rose, dotted), there are three Van Hove points at the $M_i$ points of the Brillouin zone (black dots). Each splits into two (gray dots) located along $K'-M-K$ for $t'>t/9$ (dashed, orange). The density of states (DOS) shows a stronger power-law singularity in the case of a HOVHS.
• Figure 2: (a) The $n$=2 2D regular VHS in the form of a saddle point. The DOS $\nu$ diverges as $\ln\mu$. Red lines indicate Fermi surfaces above and below the singularity. The critical Fermi surface is shown by the dotted black line. (b) The $n$=3 singularity that can occur at 3-fold symmetric points. (c) The $n=4$ singularity at a 4-fold symmetric point, it occurs very close to the Fermi energy in the case of Sr$_3$Ru$_2$O$_7$. (d) The thermodynamic phase diagram of Sr$_3$Ru$_2$O$_7$ where the different regions (red, blue, mixed) correspond to different behavior of the specific heat C/T as a function of an applied magnetic field. The regions A and B correspond to two spin-density-wave phases. (e) Schematic of the quasi-2D FS of Sr$_3$Ru$_2$O$_7$ in the $k_z$=0 plane at the Fermi energy (left hand side) and at $\mu_\textrm{C}$ (right side). The crucial bands that are close to the $n=4$ multicritical point are highlighted in red. The central pocket is a small perturbation. In order to emphasise the characteristic clover leaf Fermi surface we show an extended $k$-space picture beyond the BZ boundaries. Figure adapted from Efremov_PRL_2019.
• Figure 3: (a) The lattice structure of kagome metals CsV$_3$Sb$_5$. (b) Real space structure of the kagome vanadium planes; red, blue, and green coloring indicate the three sublattices. (c) Two distinct types of sublattice contributions to van Hove points in CsV$_3$Sb$_5$: p-type (sublattice pure, left panel) and m-type (sublattice mixing, right panel). (d) Schematics of dispersion around ordinary and HOVHS. The gray curves indicate constant energy contours that show markedly flat features along the $k_y$ direction in the HOVHS case as highlighted by the black arrow. (e) Experimental band dispersion along $\Gamma-K-M-Gamma$ direction. Dashed lines indicate the energy position of the Fermi level, Dirac cone and VHS$_3$. (f) Calculated bands along $\Gamma-K-M-Gamma$ direction. The red arrows in (e, f) mark multiple VHS. (g) Fitting to the measured dispersion along $M-K$ by $E=M-b_2 k_y^4$ (solid black line). The red dots represent the experimental data shown in (e). Adapted from Ref. Hu2022.
• Figure 4: (a) Measured tunneling conductance $G$ (open circles) of twisted bilayer graphene with angle $1.10^\circ$ and theoretical fit (solid lines). Dashed lines indicate peak positions. (b) Conductance minus a background offset $G_c=57.6$ nS around the peak at $E_V=16.72$ meV on a logarithmic scale. Red dashed lines correspond to power-law scaling $|E-E_V|^{-1/4}$ on both sides of the peak with asymmetric ratio $\eta$. Figures 4(a) and 4(b) are taken from Ref. Yuan2019. (c) Fermi-surface topologies in biased bilayer graphene as function of displacement field $\delta$ and Fermi level $E_F$. They are separated by band-edge (dash-dotted) or ordinary Van Hove transitions (solid). There is a HOVH point at the crossing of these lines. In the gray area, the Fermi level lies in the band gap. Taken from Ref. Shtyk_2017.