Quadratic Band Touchings (QBTs)

• Quadratic band touchings (QBTs) are band-crossing points where electronic bands meet with a k² dispersion, fundamentally altering density of states and topological properties.
• Low-energy Hamiltonians constrained by crystalline symmetries yield a pseudospin winding number of 2, directly impacting Berry phase and Landau quantization features.
• Interaction-driven instabilities in QBT systems lead to emergent phases such as quantum anomalous Hall, quantum spin Hall, and nematic orders, detectable via STM and spectroscopic techniques.

A quadratic band touching (QBT) refers to a band-crossing point in the electronic structure of a crystal where two or more bands meet with quadratic, rather than linear, dispersion. In contrast to Dirac points (with linear $k$-dependence), QBTs exhibit a $k^2$ dispersion of the form $$E_{\pm}(\mathbf{k}) = E_0 \pm \alpha |\mathbf{k} - \mathbf{k}_0|^2$$ around a high-symmetry momentum $\mathbf{k}_0$. QBTs fundamentally alter low-energy density of states, instabilities, and topological response, and arise in a range of two- and three-dimensional systems including Bernal-stacked bilayer graphene, kagome and checkerboard lattices, pyrochlore iridates, and Luttinger semimetals.

1. Low-Energy Hamiltonians, Symmetry, and Topology

The minimal QBT Hamiltonian in two dimensions is a $2\times2$ model expanded to quadratic order: $$H(\mathbf{k}) = d_x(\mathbf{k})\,\sigma_x + d_y(\mathbf{k})\,\sigma_y + d_z(\mathbf{k})\,\sigma_z + d_0(\mathbf{k})\,I$$ with $d_x(\mathbf{k}) \propto k_x^2 - k_y^2$, $d_y(\mathbf{k}) \propto 2k_x k_y$, and $d_z$ representing possible mass terms or symmetry-breaking perturbations (Ghadimi et al., 6 Feb 2026, Wang et al., 25 Dec 2025, Jung et al., 2023).

Crystalline point group symmetry enforces the quadratic dispersion and determines the possible winding structure:

• For $C_{4}$ or $C_{6}$, only even powers in momentum appear, stabilizing QBTs.
• Time reversal, inversion, and mirror symmetries set constraints on the allowed couplings, e.g., the pseudospin winding number.

A central invariant is the topological winding number $w$, which counts the winding of the pseudospin $\mathbf{d}(\mathbf{k})$ as $\mathbf{k}$ traces a loop around the QBT. For canonical QBTs, $w=2$, resulting in a $2\pi$ Berry phase. In three-dimensional Luttinger semimetals ($j=3/2$), similar quadratic touchings occur at zone center with nontrivial spinor structure (Tchoumakov et al., 2019, Cheng et al., 2017). Extensions to triply-degenerate quadratic crossings (TQBCs) with additional flat bands are realized in specific lattice models and correspond to higher pseudospin representations (Liu et al., 2023).

2. Quantum Geometry, Wavefunction Texture, and Landau Quantization

The Bloch eigenfunctions of QBTs encode a nontrivial "elliptic" geometry on the Bloch sphere characterized by two diameters $d_1$, $d_2$ and orientation $\phi$:

• The product $d_1 d_2$ determines the Berry phase $\gamma \sim -\pi d_1 d_2$.
• The shape and anisotropy of the projection determine Landau level (LL) spectra via Onsager quantization; a constant shift in LL energies is set by $d_1 d_2$, while their ratio controls LL splitting (Jung et al., 2023).
• In the presence of QBTs, LL quantization exhibits nontrivial features: zero modes, LL splitting, and, for TQBCs, infinite ladders of near-zero LLs, yielding anomalies in quantum Hall response (Liu et al., 2023).

The pseudospin texture underlying these features is observable in quasiparticle interference (QPI) as wavefront dislocations (WDs) in STM, which directly count the pseudospin winding rather than the topological vorticity itself (Ghadimi et al., 6 Feb 2026).

3. Interaction-Driven Instabilities and Correlated Phases

QBTs in two dimensions are marginally unstable to interactions due to the finite density of states at the Fermi level. Weak repulsive interactions generically induce symmetry-breaking states, classified by order parameter structure:

• Spin nematic: breaks spin and/or lattice $C_n$ rotation; splits the QBT into Dirac point pairs.
• Quantum anomalous Hall (QAH): time-reversal breaking, opens a topological Chern gap, yields quantized Hall response.
• Quantum spin Hall (QSH): spin-dependent gap, results in helical edge modes and nontrivial $\mathbb{Z}_2$ index.

Functional and conventional renormalization group (FRG/RG) approaches, as well as unbiased quantum Monte Carlo, establish that for generic two-orbital models (Uebelacker et al., 2011, Shah et al., 2020, Liu et al., 21 Jul 2025):

• Repulsive interactions are marginally relevant: any small $U$ can drive instabilities.
• The phase diagram as a function of intra- and inter-orbital repulsion exhibits transitions from nematic to QAH to QSH states as $U'/U$ increases.
• No fine-tuned longer-range or spin-orbit interactions are necessary; the k-dependent "orbital makeup" of Bloch eigenstates is essential.
• In 3D Luttinger semimetals, marginal Coulomb repulsion modifies plasmon and mass renormalization but can be cut off by finite doping (Tchoumakov et al., 2019, Cheng et al., 2017).

At intermediate couplings and particularly in symmetry-protected settings (checkerboard, kagome lattices), nontrivial metallic phases such as bond-nematic Dirac semimetals—with split Dirac nodes and broken $C_4$ symmetry—can intervene between QAH and site-nematic insulating phases (Liu et al., 21 Jul 2025, Dóra et al., 2016). Out-of-equilibrium, time evolution following an interaction quench can show BCS-like dynamical phase transitions in gapped (QAH) channels, while nematic order can persist with low-energy power-law decay (Dóra et al., 2016).

Interaction Type	Leading Instability (2D)	Topological Character
Repulsive Short-Range	QAH, QSH, spin-nematic	Chern/$\mathbb{Z}_2$/Nematic
Long-Range (Coulomb, 3D)	Mass renormalization, screening	Suppressed instabilities
Infinite-Range	Coexistence QAH/nematic	Second/third order transitions

4. Floquet and External-Field Engineering of QBTs

Time-dependent driving by electromagnetic fields enables tunable control of QBTs:

• Circularly polarized light can open Floquet-induced Chern gaps at QBTs via high-order photon processes (gap $\sim A_0^4/\Omega$); in modeled oxide heterostructures and kagome lattices, this yields light-induced Chern insulators (Du et al., 2016, Du et al., 2016).
• Linearly polarized light splits a QBT into Dirac points, with splitting controlled by the field amplitude and orientation, but independent of drive frequency.
• Optical Hall conductivity and Kerr/Faraday response track the Floquet Chern number and reveal sideband resonance structure—enabling direct detection via pump-probe and ARPES experiments.

High-frequency drives can engineer effective Hamiltonian parameters (nearest, next-nearest neighbor hoppings in oxide bilayers or kagome networks), effecting band inversion, gap opening/splitting, and Berry curvature redistribution (Du et al., 2016, Du et al., 2016). Strain or other symmetry-lowering perturbations can continuously interpolate between QBT, Dirac, and flat-band regimes (Wang et al., 3 Oct 2025).

5. Realization, Detection, and Experimental Probes

QBTs are realized in Bernal-stacked bilayer graphene (AB), kagome and Lieb lattices, pyrochlore iridates, and artificial molecular graphene via non-Abelian gauge engineering (Juan, 2013). Characteristic signatures include:

• LDOS plateau at $E=0$ and sharp step/kink at the QBT energy cutoff, measurable by site-resolved STM/STS (Juan, 2013).
• Terahertz and optical spectroscopy: Large dielectric constant, anomalous Hall response, and plasmaron features, as demonstrated in Pr$_2$Ir$_2$O$_7$, HgTe, and related half-Heusler materials (Tchoumakov et al., 2019, Cheng et al., 2017).
• Bloch oscillations and tunneling: QBTs enhance Landau-Zener nonlinearities, give rise to multiple frequency components in oscillatory response, and modify tunneling transparency thresholds, Fano resonance shifts, and shot noise properties (Wang et al., 3 Oct 2025, Gregefalk et al., 2023).
• STM wavefront dislocations: Channel- and texture-resolved mapping of the underlying pseudospin winding, enabling the distinction between topological vorticity and wavefunction texture upon QBT annihilation or splitting (Ghadimi et al., 6 Feb 2026).

6. Generalized Symmetries, Topological Response, and Conformal Structure

QBTs provide a platform for nontrivial generalized symmetries and realize connections to exotic quantum field-theoretic structures:

• The conservation of a generalized total angular momentum $J_z = L_z + (w/2)\,\sigma_z$, where $w$ is the QBT winding, has been demonstrated in theory and photonic experiments (Wang et al., 25 Dec 2025). This mediates selection rules for pseudospin–orbital–angular momentum conversion.
• For multi-band touchings (e.g., pseudospin-1 or "triply-degenerate" points), the total winding is preserved under various perturbations, and results in infinite ladders of quantum Hall plateaus (Liu et al., 2023).
• The equal-time correlators of free-fermion QBT Hamiltonians correspond to those of $d$-dimensional symplectic fermion CFT with central charge $c=-2$ (for $d=2$), exhibiting logarithmic operator structure, Jordan blocks under $2\pi$ rotation, and topological ground-state degeneracy associated with anyonic defects—a realization of a "logarithmic conformal quantum critical point" (Masaoka, 20 Nov 2025).

7. Open Directions and Theoretical Challenges

Despite significant progress, several nontrivial questions remain open:

• The competing interplay between topology, interaction-driven instabilities, and quantum geometry in realistic material platforms.
• The persistence of topological and nematic phases in the presence of disorder, strain, and finite doping.
• The experimental resolution of quantum geometry and pseudospin winding via channel-resolved STM or photonic probes.
• The non-Fermi liquid regimes and emergent universalities beyond the Fermi-liquid stabilization at high carrier densities in 3D QBT systems.
• Extension to time-reversal symmetry-breaking or non-equilibrium engineered topological quantum criticality.

Quadratic band touchings thus constitute a central organizing concept at the interface of topological band theory, many-body physics, and quantum geometry, with wide relevance in both synthetic/emergent and naturally occurring correlated matter (Uebelacker et al., 2011, Jung et al., 2023, Ghadimi et al., 6 Feb 2026, Liu et al., 21 Jul 2025, Wang et al., 25 Dec 2025, Liu et al., 21 Jul 2025).