Symmetry-Induced Lines of Critical Points

• Symmetry-induced lines of critical points are continuous families of singular points in electronic band structures, defined by crystal symmetries and scaling laws.
• They feature high-order, non-analytic dispersions with power-law divergent density of states and vanishing electron velocities.
• Tuning parameters like strain, twist angle, and gating allows precise control over these features, enabling the exploration of novel correlated quantum phases.

Symmetry-induced lines of critical points refer to the occurrence of continuous families—or higher multiplicity sets—of singular points in the energy band structure of crystalline materials, whose properties, stability, and physical consequences are globally dictated by symmetry principles, scaling relations, and topological considerations. Beyond the classic van Hove scenario of ordinary (quadratic) band criticalities, these symmetry-induced lines encompass a broad taxonomy of high-order critical points where the electron velocity vanishes and the density of states (DOS) acquires non-analytic or strongly divergent character. The structure, location, and multiplicity of these points are controlled by the underlying crystal symmetries and can be tuned by a limited set of external parameters, making them central to understanding singular phenomena in correlated electron systems.

1. Taxonomy of Critical Points and Symmetry Classes

Critical points of energy bands, defined as momenta in the Brillouin zone (BZ) at which the group velocity vanishes, are divided into:

• Ordinary (van Hove) Critical Points: Generic quadratic saddle points with nonzero Hessian eigenvalues, e.g.,

$E = \alpha p_x^2 + \beta p_y^2$

These appear generically (codimension zero) with no fine-tuning or demanding symmetries beyond those generic in the crystal structure.

• High-Order Critical Points: Arise when the dispersion deviates from quadratic at a particular symmetry-enforced point or under fine-tuning. At least one Hessian eigenvalue vanishes, and the local expansion involves higher degree monomials:

$E = \alpha p_x^2 + \gamma p_x p_y^2 + \kappa p_y^4$

Their realization typically requires tuning of external parameters or protection by crystal symmetries, allowing for more exotic local band topology and DOS singularity.

Symmetry dictates the types and multiplicity of these critical points:

• Generic points (no symmetry): Classes such as $A_n$, characterized by simple Hessian degeneracies.
• Mirror-invariant and rotation-invariant points: More restrictive, host classes such as $D_n$, $E_n$, and $C_n$, which are stabilized by symmetry constraints forbidding certain terms in the Taylor expansion.
• Lines or planes of critical points: Non-generic except in presence of nonsymmorphic or accidental symmetries, requiring infinite fine-tuning; physical realizations are rare.

2. Scaling, Canonical Forms, and Topological Classification

The classification of high-order critical points systematically leverages scaling and homogeneity:

• Scaling property near a critical point:

$$E(\lambda^a p_x, \lambda^b p_y) = \lambda E(p_x,p_y)$$

Compatible integer monomials $p_x^m p_y^n$ furnish canonical forms, leading to "principal classes" such as:

$$\{p_x^m,p_y^n\},\quad\{p_x^m p_y, p_y^n\},\quad\{p_x^m p_y, p_x p_y^n\},$$

with explicit relations between degree $(m,n)$ and scaling exponents $(a, b)$ (see Eqs. (34)-(36) in the original paper).

The symmetry classification translates into a set of allowed energy expansions at high-symmetry momenta:

• Mirror symmetry: Forces absence of certain odd-power terms.
• Rotational symmetry $C_n$: Restricts the form to homogenous polynomials symmetric under $n$-fold rotation (e.g., $(p_x + i p_y)^n$).
• Topological characteristics: The multiplicity $\mu$ (number of degenerate points that split under infinitesimal breaking of symmetry) and the global arrangement of singularities.

3. Symmetry’s Influence on Location and Multiplicity

Symmetry determines not only what critical points are possible, but also where in the BZ they are realized and how many parameters require tuning:

• High-symmetry points (e.g., $\Gamma$, $K$, $M$) often host high-order critical points with reduced tuning cost.
• Symmetry-forbidden terms: Increase the codimension (number of tuning parameters needed to reach the singularity), but can also enforce degeneracy or protected lines.
• Parameter tuning: Twist angle, strain, pressure, and external fields serve as practical means to bring high-order critical points to the Fermi level or to split lines into discrete points.

Explicit examples:

• Mirror points: Single-parameter tuning can access a $p_x^4 - h p_x^2 + p_y^2$ dispersion.
• $C_3$ symmetry: Enables cubic band crossings relevant to moiré lattice systems.
• $C_4$ symmetry: Allows dispersions such as $\mathrm{Re}[(p_x+i p_y)^4]$ at high-symmetry points.

4. Physical Consequences: DOS Power Laws and Particle-Hole Asymmetry

The nature of the critical points dictates singularities in the electronic DOS:

• Ordinary (van Hove) points: Logarithmic divergences in the DOS.
• High-order points: Power-law divergent DOS,

$\rho(\varepsilon) \propto |\varepsilon|^\nu$

with nontrivial exponents (e.g., for $E = p_x^2 + p_x p_y^2 + p_y^4$, $\nu = -1/4$).

• Particle-hole asymmetry is generic in high-order critical points, as symmetry breaking in the local expansion gives rise to

$$\eta = \frac{\rho(-|\varepsilon|)}{\rho(|\varepsilon|)} = \frac{\cos\theta}{\cos(\theta + \nu\pi)}$$

(Eq. (43)), with the asymmetry parameter $\theta$ set by details of the class and underlying mass or hybridization terms. For certain classes, $\eta$ can take irrational values (e.g., $\sqrt{2}$ or $\sqrt{3}$).

These singularities can stabilize or amplify interaction-driven phenomena, making high-order critical points fertile ground for unconventional superconductivity, magnetism, and strong-coupling instabilities.

5. External Parameter Tuning and Realization

Implementing symmetry-induced or high-order critical points in a material requires tuning a specific set of experimentally accessible parameters:

• Twist angle (magic-angle graphene and related material stacks): Moves critical points through the Fermi energy.
• Strain/pressure: Modulates bandwidths, potentially merges or splits degenerate points.
• Out-of-plane electric fields or gating: Tunes sublattice-polarization-induced gaps or shifts high-order points.
• Number of required parameters: Set by the multiplicity of the critical point—higher symmetry or protection reduces tuning demands.

6. Summary Table of Symmetry Classes and Physical Properties

Class	Canonical Dispersion	Multiplicity ($\mu$)	DOS Exponent ($\nu$)	Particle-Hole Asymmetry ($\eta$)	Symmetry Type
$A_n$	$p_x^{n+1} \pm p_y^2$	$n$	$1/(n+1)-1/2$	See Table S	Generic, Mirror
$C_n$	$\mathrm{Re}[(p_x+ip_y)^n] + r(p_x^2+p_y^2)^{n/2}$	$(n-1)^2$	$2/n-1$	See Fig. 8, Table S	$C_n$ Rotation

(Refer to Tables S and T for details on other classes and explicit formulas.)

7. Applications, Implications, and Control

Symmetry-induced high-order critical points:

• Enhance electronic correlations by producing stronger DOS singularities.
• Induce robust, tunable physical responses: When close to the Fermi energy, these points can be linked to the emergence of superconductivity and other correlated phases in moiré and topologically nontrivial materials.
• Manifest strong particle-hole asymmetry that fundamentally affects thermoelectric and transport phenomena.
• Are not generic as lines: Generic perturbation or symmetry breaking splits lines into isolated points—lines only persist as a result of unlifted symmetry protection or fine-tuning.

Conclusion

Symmetry, topology, and scaling together furnish a universal and predictive framework for classifying, realizing, and controlling high-order critical points in electronic band structures. These symmetry-induced lines or families of critical points stand at the center of modern studies of quantum materials, yielding both singular electronic properties and a roadmap for material design through symmetry engineering and parameter tuning. The presented classification not only extends van Hove’s theory but articulates a landscape of electronic singularities with broad impact on correlated matter, topology, and emergent physical phenomena.