Fig. 3

Theoretical model of spin, valley, and orbital anisotropies in the ZLL. a Single-particle energy spectrum of the ZLL at B = 31 T derived from a four-band tight binding model²⁹ (see also Eq. (3)). The interlayer potential u couples to the layer polarization of each state as ξα _N u, differing in sign for the two valleys and magnitude for the two orbitals; the spin degeneracy is lifted by the Zeeman energy E _Z ≈ 3.6 meV; and the N = 0 and 1 orbitals are split by the band structure parameter Δ₁₀ ≈ 9.7 meV. b–d Level filling schematic for \(\frac{{{p_0}}}{c} = - 100\) mV. Within the Hartree-Fock approximation, we calculate the energy to add an additional electron to level σξN given the current filling {ν _σξN }, generating eight curves ε _σξN (ν) which change with the total filling ν. Colors indicate the ξN index of the level, while solid vs. dashed indicates the spin. The bold portion indicates the range of ν over which the level is coincident with the Fermi energy. As isospin σξ fills, both of its N = 0 and 1 orbitals decrease in energy due to favorable Coulomb correlations, while the components of the opposite valley (i.e., layer) decrease slightly in energy due to the capacitance of the bilayer. The relative magnitude of these effects, combined with the single-particle splittings, determines the filling order, shown here for three interactions strengths parameterized by the boron nitride dielectric constant, \(\epsilon _{{\rm{BN}}}^{||}\). Large \(\epsilon _{{\rm{BN}}}^{{\rm{||}}} = 18\) (e) corresponds to negligible Coulomb interactions, \(\epsilon _{{\rm{BN}}}^{{\rm{||}}} = 6.6\) (f) corresponds to intermediate Coulomb interactions, and \(\epsilon _{{\rm{BN}}}^{||} = 1\) (g) corresponds to maximally strong Coulomb interactoins. e–g Hartree-Fock phase diagram in the three interaction strength regimes. Colors blue, cyan, red, and orange indicate whether levels of type ξN = −0, −1, +0, +1 are filling, so that the result should mimic the observed C _A. The intermediate interaction regime shows good agreement with the experimental C _A data, while interactions which are too weak (e) or strong (g) do not reproduce the observed filling sequences. The black dashed line indicates a cut at p ₀/c = −100 meV corresponding to the particular filling sequence shown in b–d