Valley-Asymmetric Landau Levels

• Valley-asymmetric Landau-level structure is defined by the splitting of degenerate Landau levels into valley-specific energies due to strain, fields, or spin–orbit coupling.
• Experimental techniques such as STM/STS and magnetotransport have quantified energy splits in graphene and TMDs, validating theoretical predictions with observed ΔE values up to 5.5 meV.
• Insights into valley-resolved LLs enable advancements in valleytronics, quantum Hall studies, and the design of devices that exploit valley-selective transport and optical phenomena.

Valley-asymmetric Landau-level structure refers to the lifting of degeneracy between Landau levels (LLs) associated with inequivalent valleys in multivalley quantum systems—most notably in 2D materials such as graphene, transition metal dichalcogenides (TMDs), and other Dirac/Weyl semimetals—due to symmetry-breaking effects such as strain, applied electric fields, pressure, or intrinsic spin-orbit coupling. This phenomenon produces valley-polarized LLs with distinct energies in each valley, which fundamentally alters quantum Hall physics, valley-selective transport, and the design of valleytronic devices.

1. Fundamental Theory of Valley-Asymmetric Landau Levels

At low energy, the electronic states in systems such as monolayer graphene and TMDs are described by multiple valleys—inequivalent minima in momentum space—each with Dirac- or Dirac-like dispersion. In the presence of a perpendicular magnetic field, quantization yields Landau levels. When valley degeneracy is preserved, LLs in different valleys are energetically degenerate. However, when additional symmetry breaking is present, these energies split in a valley-dependent fashion.

In strained graphene, for example, the effective Hamiltonian for valley $\tau=\pm1$ acquires a pseudo-gauge field:

$$H^{(\tau)} = v_F\,\sigma\cdot[p + e\,\tau\,A_\mathrm{ps}(\mathbf{r})],$$

where $A_\mathrm{ps}$ encodes the strain-induced pseudo-magnetic field $B_\mathrm{ps} = \nabla\times A_\mathrm{ps}$, which enters with opposite sign in the $K$ and $K'$ valleys. In the presence of a real magnetic field $B_\mathrm{ext}$ and a valley-odd pseudo-field $B_{\mathrm{ps}}$, the effective field seen by each valley is $B^{(\tau)} = B_\mathrm{ext} + \tau B_\mathrm{ps}$, directly lifting the valley LL degeneracy (Li et al., 2015).

A general formula for the valley-dependent relativistic Landau level spectrum (e.g., in graphene) is:

$$E_n^{(\tau)} = \operatorname{sgn}(n)\,v_F\,\sqrt{2e\hbar|n|\left|B_\mathrm{ext} + \tau B_\mathrm{ps}\right|},$$

where $n$ is the Landau index, and the valley asymmetry appears whenever $B_\mathrm{ps}\neq0$.

Similar valley asymmetry arises from other single-particle perturbations (e.g., interlayer bias, pressure, spin–orbit coupling, or strain), or through valley-specific many-body effects.

2. Experimental Manifestations and Measurement Protocols

Valley-asymmetric Landau-level structure has been observed across various platforms using techniques such as scanning tunneling microscopy/spectroscopy (STM/STS), magnetotransport, and valley-resolved optical spectroscopy:

• Strained Graphene: STM/STS on CVD graphene grown on Rh foil with periodic ripples directly resolves the splitting of LLs (primarily $n=-1$), with the valley splitting $\Delta E$ scaling as predicted by theory and directly extractable from $dI/dV$ spectra (Li et al., 2015).
• ZrSiS Surface States: High-field STM in strained ZrSiS visualizes LLs separated into valley-polarized sublevels; by intra-unit-cell analysis and Fourier mapping, valley-specific dispersions and Berry phases can be determined, providing model-independent access to valley-resolved LL structure (Butler et al., 2024).
• TMD Monolayers (e.g., MoS₂, WSe₂): Valley and spin polarization of LLs are inferred via optical selection rules and magneto-optical spectroscopy, directly observing valley- and spin-polarized LLs, and resolving their energies via dopant and field tuning (Szulakowska et al., 2018, Wang et al., 2017). Valley Zeeman splittings are quantified from inter-LL optical transitions.
• Bilayer and Trilayer Graphene: In Bernal-stacked bilayer graphene, application of a perpendicular electric field creates a valley-dependent gap in the $n=0$ LLs, observable as split delta-like DOS peaks in STM and as valley-polarized quantum Hall plateaus. In ABA-stacked trilayer graphene, Coulomb interactions and hopping parameters yield enhanced valley and electron–hole asymmetries observable in quantum Hall and cyclotron resonance experiments (Shizuya, 2012, Shizuya, 2014).

3. Model Systems: Symmetry Breaking Mechanisms

Valley asymmetry in LL spectra arises from multiple mechanisms, each producing characteristic splitting patterns and experimental signatures:

Mechanism	Model Hamiltonian Features	Valley Splitting Signature
Strain (Graphene, ZrSiS)	Pseudo-gauge field $A_\mathrm{ps}$; $B_\mathrm{ps}$ with $\tau$-odd coupling	LL energies depend on $B_\mathrm{ext}\pm B_\mathrm{ps}$, splitting all $|n|\geq1$ levels (Li et al., 2015, Butler et al., 2024)
Spin–orbit coupling (TMDs)	$H_{SO}=\lambda\sigma_z$ or more complex with $\tau$ and $s$	$n=0$ LLs at different band edges in K and K'; sizable valley Zeeman shifts (Szulakowska et al., 2018, Wang et al., 2017)
Interlayer bias (BLG)	Finite $\Delta$ term, $H^\text{eff}=\Delta\sigma_z$	$n=0$ LLs at $E=\pm\Delta$ in K/K', observable as split DOS peaks (Kawarabayashi et al., 2011, Shizuya, 2012)
Pressure (BLG), hopping t₃, t₄	Multi-cone splitting, valley-dependent band offsets	Four Dirac cones (one at K, three at T); LL fans with valley-sensitive offset; valley Hall conductivity with 3:1 step ratio (Munoz et al., 2016)
Mirror symmetry breaking (TLG)	Layer-symmetric and antisymmetric hybridization, $\gamma_3$ warping	LL anticrossings and valley splitting controlled by electric field and interlayer coupling (Shimazaki et al., 2016, Shizuya, 2014)
Many-body exchange/correlation	LL-dependent self-energy, vacuum polarization	Enhanced or filling-dependent valley splitting, orbital Lamb shift (Szulakowska et al., 2018, Shizuya, 2014, Shizuya, 2012)

4. Quantitative Outcomes and Comparison to Theory

Measurements of valley LL splitting quantitatively match theoretical predictions across a range of systems:

• In strained graphene on Rh, valley splitting of the $n=-1$ LL reaches $\Delta E\sim2.7$–$4.5$ meV, corresponding to $B_\mathrm{ps}\sim0.45$–$0.75$ T, with spatial uniformity on the ripple scale (Li et al., 2015).
• In ZrSiS, valley energy offsets $\Delta E$ as small as $<1$ meV are resolvable, corresponding to uniaxial strain $\epsilon<0.025\%$; splittings up to $5.5$ meV are observed with $0.1\%$ strain (Butler et al., 2024).
• For monolayer MoS₂, valley Zeeman splittings in the $n=0$ and $n=1$ LLs are a few meV at $B\lesssim30$ T, enhanced by exchange; experimental $g_v$-factors can greatly exceed DFT predictions, owing to strong correlation (Szulakowska et al., 2018, Pisoni et al., 2018).
• In biased BLG, the valley-split $n=0$ LLs shift by $\pm\Delta$ with remarkable robustness against smooth bond disorder; these levels are observed as delta-function-like peaks in local DOS (Kawarabayashi et al., 2011).
• In trilayer graphene, splittings between zero modes are $10$–$40$ meV due to the combined effect of nonleading hopping ($\gamma_2,\gamma_5,\Delta'$) and the level-dependent vacuum Fock self-energy ("orbital Lamb shift") (Shizuya, 2014).

5. Valleytronics, Quantum Oscillations, and Functional Consequences

The valley-asymmetric Landau-level structure enables manipulation and detection of valley degrees of freedom:

• Valley-polarized quantum Hall states: Selective population of LLs in a given valley by tuning strain, electric field, or chemical potential, realizing quantum Hall plateaus with valley symmetry breaking (Kawarabayashi et al., 2011, Shizuya, 2014, Munoz et al., 2016).
• Valley-selective transport and optics: Realization of valley filters, switches, or valves, where only LLs in a chosen valley participate in transport or optical transitions. Typical schemes exploit strain ($B_\mathrm{ps}$), bias ($\Delta$), or pressure (cone splitting).
• Quantum oscillations in valley-polarized spin currents: In TMDC/ferromagnet hybrid systems, quantum oscillations of valley-polarized spin current directly trace the underlying valley-resolved LL quantization, evident in spin Seebeck experiments (Hu et al., 6 Feb 2026).
• Fundamental insight into topological robustness: Valley splitting of LLs reveals the presence or absence of topological protection. For example, in massive Dirac systems, the $n=0$ LL is topologically protected against off-diagonal disorder, while three-band TMD models support only partial robustness due to the absence of true chiral symmetry (Ding et al., 24 Nov 2025).

6. Role of Many-Body Corrections and Disorder

The bare valley splitting can be substantially modified by interactions:

• Many-body (exchange/correlation) corrections generate enhanced valley splitting, new anticrossing gaps, and strong density- or filling-factor dependence of LL energies, as observed in both TMDs and graphene-based systems (Szulakowska et al., 2018, Pisoni et al., 2018, Shizuya, 2014).
• Disorder effects: For example, in graphene, the $n=0$ LL remains sharply peaked even under strong bond disorder, unless intervalley scattering dominates, producing a characteristic three-peak structure in the density of states (Malla et al., 2018). In TMDs, chiral protection is weaker or absent, but the three-band cyclic selection rules grant some resilience to specific off-diagonal perturbations (Ding et al., 24 Nov 2025).

7. Outlook and Prospects

The study and exploitation of valley-asymmetric Landau-level structures enable advanced valley-based device concepts and provide new platforms to study correlated quantum phenomena:

• Strain engineering: Precise control of local strain enables valley-selective LL engineering and sensitive strain detection down to $<0.025\%$ (Butler et al., 2024).
• Valley-polarized edge modes: Coexistence of valley splitting and quantum Hall phases yields robust chiral modes for valleytronics (Li et al., 2015).
• Probing band topology: The functional dependence of valley LL splitting on symmetry-breaking perturbations encodes information about underlying band topology, symmetry obstruction, and quantum geometry (Ding et al., 24 Nov 2025).
• Future materials: The outlined principles generalize to multi-valley, multi-orbital systems, including topological semimetals, artificial lattice systems, and engineered heterostructures.

The valley-asymmetric Landau-level structure is established as a universal and tunable feature of multivalley quantum materials, shaping both their fundamental quantum Hall response and their potential for valley-selective quantum technologies (Li et al., 2015, Butler et al., 2024, Szulakowska et al., 2018, Shizuya, 2014, Wang et al., 2017).