Pseudo-Landau Levels in Dirac Materials

• Pseudo-Landau Levels are quantum states induced by strain in graphene and related Dirac materials, mimicking Landau quantization without an external magnetic field.
• They arise from nonuniform strain that creates valley-dependent pseudo-gauge fields, preserving time-reversal symmetry and leading to a robust zeroth level.
• Experimental observations via STM, STS, and transport measurements highlight their potential for applications in valleytronics, artificial lattices, and correlated electron physics.

Searching arXiv for relevant papers on pseudo-Landau levels to ground the article. arXiv query: "pseudo-Landau levels graphene strain" Pseudo-Landau levels (PLLs) are Landau-quantized states generated without a real magnetic field, most prominently through strain-induced pseudo-gauge fields that act on low-energy Dirac quasiparticles in graphene and related systems. In graphene, nonuniform strain enters the continuum Hamiltonian as a valley-dependent vector potential, produces a pseudomagnetic field of opposite sign in the two valleys, and yields a Landau-like spectrum with a robust zeroth level while preserving global time-reversal symmetry (Liu et al., 2021, Li et al., 2018). More generally, PLL physics has been extended to spatially modulated Dirac systems, higher-dimensional nodal semimetals, nodal superconductors, Bernal bilayer graphene, and, in a distinct usage of the term, patterned-gate GaAs 2DEGs where Landau-level-like flat bands arise from superlattice confinement rather than a gauge field (Kariyado, 2017, Köhler et al., 2023, Massarelli et al., 2017, Pantaleon et al., 2024).

1. Gauge-field origin and Landau quantization

In monolayer graphene, low-energy electrons near a given valley are described by

$H_0 = v_F\,\boldsymbol{\sigma}\cdot \mathbf{p},$

while nonuniform strain modifies nearest-neighbor hoppings and is encoded in a pseudo-gauge field,

$$H = v_F\,\boldsymbol{\sigma}\cdot(\mathbf{p}+e\,\mathbf{A}_{\mathrm{ps}}),\qquad \mathbf{B}_{\mathrm{ps}}=\nabla\times \mathbf{A}_{\mathrm{ps}}.$$

For each valley separately, the resulting spectrum is Landau-like,

$$E_n=\mathrm{sgn}(n)\,v_F\sqrt{2e\hbar |n B_{\mathrm{ps}}|},$$

or equivalently $$E_n\propto \mathrm{sgn}(n)\sqrt{|n|B_{\mathrm{ps}}}$$ (Liu et al., 2021, Li et al., 2018). The decisive distinction from a real magnetic field is valley antisymmetry: $\mathbf{B}_{\mathrm{ps}}$ changes sign between $K$ and $K'$, so there is no real magnetic flux or Lorentz force and global time-reversal symmetry is preserved.

This gauge-field perspective is not restricted to graphene. In spatially modulated Dirac systems, a position-dependent shift of the Dirac node, $$\boldsymbol{k}\to \boldsymbol{k}\pm \boldsymbol{k}_0(\mathbf{r})$$, is mathematically equivalent to a vector potential $$\boldsymbol{A}^{(\pm)}=\mp (\hbar/e)\boldsymbol{k}_0$$, with an effective field $B\sim (\hbar/e)\Delta k/L$ for a node shift $$H = v_F\,\boldsymbol{\sigma}\cdot(\mathbf{p}+e\,\mathbf{A}_{\mathrm{ps}}),\qquad \mathbf{B}_{\mathrm{ps}}=\nabla\times \mathbf{A}_{\mathrm{ps}}.$$0 over a length $$H = v_F\,\boldsymbol{\sigma}\cdot(\mathbf{p}+e\,\mathbf{A}_{\mathrm{ps}}),\qquad \mathbf{B}_{\mathrm{ps}}=\nabla\times \mathbf{A}_{\mathrm{ps}}.$$1 (Kariyado, 2017). In that setting the maximum observable Landau index obeys the counting formula

$$H = v_F\,\boldsymbol{\sigma}\cdot(\mathbf{p}+e\,\mathbf{A}_{\mathrm{ps}}),\qquad \mathbf{B}_{\mathrm{ps}}=\nabla\times \mathbf{A}_{\mathrm{ps}}.$$2

and anisotropy enhances PLL formation through

$$H = v_F\,\boldsymbol{\sigma}\cdot(\mathbf{p}+e\,\mathbf{A}_{\mathrm{ps}}),\qquad \mathbf{B}_{\mathrm{ps}}=\nabla\times \mathbf{A}_{\mathrm{ps}}.$$3

(Kariyado, 2017). In higher-dimensional hyperdiamond nodal semimetals, suitably designed strain produces sharp relativistic PLLs through dimensional reduction: each point on a $$H = v_F\,\boldsymbol{\sigma}\cdot(\mathbf{p}+e\,\mathbf{A}_{\mathrm{ps}}),\qquad \mathbf{B}_{\mathrm{ps}}=\nabla\times \mathbf{A}_{\mathrm{ps}}.$$4-dimensional nodal manifold generates an effective two-dimensional Landau problem with

$$H = v_F\,\boldsymbol{\sigma}\cdot(\mathbf{p}+e\,\mathbf{A}_{\mathrm{ps}}),\qquad \mathbf{B}_{\mathrm{ps}}=\nabla\times \mathbf{A}_{\mathrm{ps}}.$$5

(Köhler et al., 2023).

2. Zeroth level, sublattice structure, and validity limits

The zeroth PLL is the most robust and experimentally accessible member of the ladder. In strained graphene it sits at the local Dirac point, shifted by doping, and is typically the most intense spectroscopic feature, whereas higher $$H = v_F\,\boldsymbol{\sigma}\cdot(\mathbf{p}+e\,\mathbf{A}_{\mathrm{ps}}),\qquad \mathbf{B}_{\mathrm{ps}}=\nabla\times \mathbf{A}_{\mathrm{ps}}.$$6 PLLs are weaker and less sharply defined (Liu et al., 2021). A central limitation is that the pseudo-magnetic-field description is exact only near charge neutrality; at higher energies the mapping deteriorates, which suppresses and broadens higher-index peaks in both theory and experiment (Liu et al., 2021, Li et al., 2018).

Theoretical treatments of graphene PLLs classically emphasize sublattice polarization of the $$H = v_F\,\boldsymbol{\sigma}\cdot(\mathbf{p}+e\,\mathbf{A}_{\mathrm{ps}}),\qquad \mathbf{B}_{\mathrm{ps}}=\nabla\times \mathbf{A}_{\mathrm{ps}}.$$7 level. In transport calculations for strained ribbons, the $$H = v_F\,\boldsymbol{\sigma}\cdot(\mathbf{p}+e\,\mathbf{A}_{\mathrm{ps}}),\qquad \mathbf{B}_{\mathrm{ps}}=\nabla\times \mathbf{A}_{\mathrm{ps}}.$$8 pseudo-Landau level is sublattice polarized and localized near the strain maxima at the ribbon ends, and this polarization serves as a fingerprint distinguishing it from Fabry–Pérot resonances or generic edge states (Gradinar et al., 2013). In disordered nanobubble superlattices, the zeroth PLL likewise exhibits a low-energy sublattice polarization that is unique to pseudomagnetic fields preserving time-reversal symmetry (Settnes et al., 2016). However, this property is not universal across all geometries: in suspended strained graphene monolayers with periodically rippled channels, localized zeroth-PLL states appear on both sublattices, which the authors identify as a loss of sublattice polarization caused by the specific strain geometry and the coupling of many ripples (Liu et al., 2021).

Low-index PLL structure also depends strongly on lattice and band topology. In the $$H = v_F\,\boldsymbol{\sigma}\cdot(\mathbf{p}+e\,\mathbf{A}_{\mathrm{ps}}),\qquad \mathbf{B}_{\mathrm{ps}}=\nabla\times \mathbf{A}_{\mathrm{ps}}.$$9-$$E_n=\mathrm{sgn}(n)\,v_F\sqrt{2e\hbar |n B_{\mathrm{ps}}|},$$0 lattice, the first pLL is sublattice polarized on only two of the three sublattices, and under nonuniform uniaxial strain it becomes dispersive because the Fermi velocity is renormalized (Sun et al., 2022). In Bernal-stacked bilayer graphene, the zeroth and first pseudo-Landau levels are dispersionless and sublattice-polarized, reflecting the massive rather than massless character of the low-energy carriers (Liu et al., 2024).

3. Experimental manifestations in graphene

Scanning tunneling microscopy and spectroscopy have provided the most direct access to PLLs in graphene. In periodically strained suspended monolayer channels fabricated by hydrogen and metallic nanoparticle etching, STM reveals quasi-periodic ripples, while STS records four pronounced peaks near the Dirac point that are fitted by Dirac Landau quantization; the robust zeroth PLL appears throughout the strained region with a full width of about $$E_n=\mathrm{sgn}(n)\,v_F\sqrt{2e\hbar |n B_{\mathrm{ps}}|},$$1 eV, and the extracted pseudo-magnetic field oscillates with the same period as the ripple superstructure (Liu et al., 2021). In strained graphene on Rh foils, STS at $$E_n=\mathrm{sgn}(n)\,v_F\sqrt{2e\hbar |n B_{\mathrm{ps}}|},$$2 likewise resolves PLLs, while finite real magnetic field produces valley-dependent effective fields $$E_n=\mathrm{sgn}(n)\,v_F\sqrt{2e\hbar |n B_{\mathrm{ps}}|},$$3, thereby splitting $$E_n=\mathrm{sgn}(n)\,v_F\sqrt{2e\hbar |n B_{\mathrm{ps}}|},$$4 Landau levels and enabling direct observation of valley polarization and valley inversion across a ripple where $$E_n=\mathrm{sgn}(n)\,v_F\sqrt{2e\hbar |n B_{\mathrm{ps}}|},$$5 changes sign (Li et al., 2018).

Transport reveals a complementary set of signatures. In suspended armchair nanoribbons clamped by metallic contacts, strong strain inhomogeneity near the contacts generates localized $$E_n=\mathrm{sgn}(n)\,v_F\sqrt{2e\hbar |n B_{\mathrm{ps}}|},$$6 PLL states that appear as an isolated quadruplet of low-energy conductance resonances around $$E_n=\mathrm{sgn}(n)\,v_F\sqrt{2e\hbar |n B_{\mathrm{ps}}|},$$7 meV and $$E_n=\mathrm{sgn}(n)\,v_F\sqrt{2e\hbar |n B_{\mathrm{ps}}|},$$8 meV; their spatial localization and sublattice polarization identify them as pseudo-magnetic zero modes rather than ordinary cavity resonances (Gradinar et al., 2013). In ordered networks of graphene nanobubbles with pseudomagnetic fields up to about $$E_n=\mathrm{sgn}(n)\,v_F\sqrt{2e\hbar |n B_{\mathrm{ps}}|},$$9 T, real-space Kubo calculations show sharp DOS peaks following the $$E_n\propto \mathrm{sgn}(n)\sqrt{|n|B_{\mathrm{ps}}}$$0 Dirac law, minima of the elastic mean free path at PLL energies, and an anomalous regime near the zeroth PLL in which the mean free path increases with disorder because disorder weakens the low-energy sublattice confinement created by the $$E_n\propto \mathrm{sgn}(n)\sqrt{|n|B_{\mathrm{ps}}}$$1 state (Settnes et al., 2016).

Strong bending provides a third route. For circularly bent zigzag nanoribbons, the combination of nonuniform Fermi velocity and nonuniform pseudomagnetic field produces analytically tractable but dispersive PLLs. The exact solution predicts ARPES-visible PLL bands, Shubnikov–de Haas-like oscillations in the complete absence of magnetic fields, and negative strain-resistivity associated with the valley anomaly (Liu et al., 2021).

4. Coupled PLLs as building blocks

A distinctive feature of pseudo-Landau levels, absent in ordinary extended Landau levels, is their nanoscale locality. This makes them effective orbitals that can be coupled into artificial lattices. In suspended graphene channels with periodic ripples, the localized zeroth PLLs on successive ripple heads hybridize into a one-dimensional metallic state that is well described by an SSH chain,

$$E_n\propto \mathrm{sgn}(n)\sqrt{|n|B_{\mathrm{ps}}}$$2

with

$$E_n\propto \mathrm{sgn}(n)\sqrt{|n|B_{\mathrm{ps}}}$$3

For the symmetric ripple structures realized experimentally, $$E_n\propto \mathrm{sgn}(n)\sqrt{|n|B_{\mathrm{ps}}}$$4 meV and the zeroth-PLL-derived band is metallic; weakening the strain near the ripple head broadens the band and increases the effective coupling to about $$E_n\propto \mathrm{sgn}(n)\sqrt{|n|B_{\mathrm{ps}}}$$5 meV while preserving the metallic regime (Liu et al., 2021). The corresponding LDOS forms a serpentine pattern that follows the tilted ripple crests.

This building-block viewpoint also underlies the transport quadruplet of end-localized $$E_n\propto \mathrm{sgn}(n)\sqrt{|n|B_{\mathrm{ps}}}$$6 PLLs in suspended ribbons, where left and right pseudo-magnetic puddles hybridize into nearly degenerate pairs (Gradinar et al., 2013). In Bernal bilayer graphene zigzag ribbons, the strain-induced guiding center coincides with the domain wall of a two-legged SSH model, and the lowest PLLs can be viewed as domain-wall states of a coupled-chain problem (Liu et al., 2024).

The same locality supports strongly correlated constructions. In graphene nanobubbles, the degenerate zeroth-PLL orbitals act as “Landau sites”; under Coulomb repulsion they form an effective $$E_n\propto \mathrm{sgn}(n)\sqrt{|n|B_{\mathrm{ps}}}$$7 pseudospin coupled to a bath via exchange interaction, generating an $$E_n\propto \mathrm{sgn}(n)\sqrt{|n|B_{\mathrm{ps}}}$$8 Kondo effect and, in two-bubble geometries, a flavor-frozen Kondo regime (Chen et al., 2024). This establishes a route from strain-defined PLL orbitals to programmable impurity physics.

5. Generalizations beyond monolayer graphene

PLLs extend beyond the canonical monolayer Dirac problem. In the $$E_n\propto \mathrm{sgn}(n)\sqrt{|n|B_{\mathrm{ps}}}$$9-$\mathbf{B}_{\mathrm{ps}}$0 lattice, nonuniform uniaxial and triaxial strain generate analytical pLL spectra that interpolate between graphene and dice-lattice limits through the parameter $\mathbf{B}_{\mathrm{ps}}$1. Triaxial strain yields oscillatory DOS peaks and flat-band-adjacent pLLs, whereas uniaxial strain produces dispersive pLLs and a valley-polarized current because the Fermi velocity becomes position dependent (Sun et al., 2022).

In Bernal-stacked bilayer graphene, strain-induced pseudo-Landau quantization survives even though the low-energy excitations are massive. Near the strain-induced guiding center, the lattice model reduces to an exactly solvable coupled Dirac problem whose zeroth and first PLLs are dispersionless and sublattice-polarized; interactions projected onto these two levels produce a global antiferromagnetic order (Liu et al., 2024). This extends PLL physics from massless to massive chiral quasiparticles.

Nodal superconductors supply a further generalization. For Bogoliubov quasiparticles in a $\mathbf{B}_{\mathrm{ps}}$2-wave superconductor, slow spatial variation of hopping amplitudes or of an extended $\mathbf{B}_{\mathrm{ps}}$3-wave pairing component enters the low-energy BdG Hamiltonian as a vector potential, producing pseudo-LLs with

$\mathbf{B}_{\mathrm{ps}}$4

at the nodes that feel the pseudo-field (Massarelli et al., 2017). A notable distinction from graphene is symmetry: in a time-reversal-invariant spin-singlet superconductor, scalar gauge potentials are forbidden, so the pseudo-LL structure is not destabilized by the deformation-potential mechanism that can collapse graphene PLLs (Massarelli et al., 2017).

In higher-dimensional hyperdiamond nodal semimetals, tailored $\mathbf{B}_{\mathrm{ps}}$5-axial strain yields sharp relativistic PLLs through dimensional reduction rather than a single global minimal-coupling field; motion near each point of the nodal manifold reduces to a two-dimensional Landau problem with $\mathbf{B}_{\mathrm{ps}}$6-independent level spacing (Köhler et al., 2023). By contrast, the patterned-gate GaAs 2DEG uses the term “pseudo-Landau levels” in a different sense: there the low-energy flat bands emerge from local harmonic confinement in a scalar superlattice potential, and the paper explicitly does not derive an emergent gauge field for that system (Pantaleon et al., 2024). This broader usage is conceptually useful but should not be conflated with valley-antisymmetric strain gauge fields in Dirac materials.

6. Misconceptions, limitations, and research directions

Several recurring misconceptions have been clarified by the literature. PLLs do not require real magnetic flux, and they do not imply a real Lorentz force; their origin is modified hopping amplitudes or, more generally, spatially modulated Dirac data (Liu et al., 2021, Kariyado, 2017). Nor is every strain-localized state a PLL: in the periodically rippled suspended channels, large-scale tight-binding calculations explicitly ruled out the possibility that the observed serpentine LDOS arises from simple ripple-localized states unconnected to pseudo-magnetic quantization (Liu et al., 2021). Finally, the zeroth PLL is often, but not always, sublattice polarized; coupled-ripple geometries can destroy that simple pattern (Liu et al., 2021).

The principal theoretical limitation is energetic. The pseudo-magnetic-field description is controlled near charge neutrality and becomes unreliable at higher energies, which is why higher-index PLLs are often weak, broadened, or absent in spectroscopy (Li et al., 2018, Liu et al., 2021). A separate instability is produced by the scalar deformation potential. For some strain configurations in graphene, the associated effective electric field satisfies the same relativistic condition as real crossed electric and magnetic fields, so that PLLs collapse when the effective field approaches $\mathbf{B}_{\mathrm{ps}}$7; this mechanism explains why similar nanobubbles can display markedly different STM signatures (Castro et al., 2016).

Open directions remain extensive. Experimentally, deterministic strain programming for valley devices, dynamic control of pseudo-fields, and systematic studies of disorder and intervalley scattering are unresolved (Li et al., 2018). Conceptually, coupling PLL orbitals into designed one- and two-dimensional artificial lattices, extending dimensional-reduction constructions to broader nodal semimetals, and exploiting interaction-dominated zeroth PLLs for Kondo physics, magnetism, and possibly more exotic correlated phases remain active themes (Liu et al., 2021, Köhler et al., 2023, Chen et al., 2024). A related but distinct nomenclature also persists in bilayer graphene under a real magnetic field, where the interaction-split $\mathbf{B}_{\mathrm{ps}}$8 manifold is termed “pseudo-zero-mode Landau levels” and acquires a Lamb-shift-like orbital splitting from Coulombic quantum fluctuations of the Dirac sea (Shizuya, 2012). That usage underscores a broader point: low-index Landau manifolds in multicomponent Dirac materials are unusually sensitive to vacuum structure, symmetry breaking, and interaction effects, and PLLs provide one of the most versatile routes for engineering such manifolds without external fields.