Fig. 2: Modeling normal-state fermiology in double-layer two-valley electron liquids with 30^∘ twist.

a Real-space triangular lattices for two layers of spin-triplet valley-singlet (STVS) superconductors stacked with angular twist θ. Dots in red (blue) denote lattice sites R (\(\tilde{{{{{{{{\bf{R}}}}}}}}}\)) in layer 1 (2) with lattice constant a = 2.46 Å. Hopping parameter t = 1 eV in all figures. b Fermi surface (FS) contours around ± K points of two decoupled layers at θ = 30^∘ for chemical potential 2.4t < μ < 3t. G₂, G₃ denote reciprocal lattice vectors in layer 1. (c) Dual-momentum space lattice points k_m (red dots) and \({\tilde{{{{{{{{\bf{p}}}}}}}}}}_{n}\) (blue dots) for a fixed momentum k₀ in units of 2π/a (see subsection “Dual momentum-space tight-binding model" in Methods). Red (blue) hexagons denote Wigner-Seitz cells in reciprocal lattice of layer 1 (2). Note that the original location of each k_m in the first Brillouin zone is exactly the location measured from the center of the red hexagon that contains it. Bloch momenta near ± K and \(\pm \tilde{K}\) in (b) are covered by m, n = 1, 2, . . . , 6 (encircled dots) as k₀ varies in momentum-space. Inset: Fourier transform t_⊥(q) of inter-layer coupling, t_⊥(K) extrapolated at Ka/π = 4/3. (d) Band structure of \({{{{{{{{\mathcal{H}}}}}}}}}_{0,{{{{{{{\rm{eff}}}}}}}}}\) with topmost bands indexed by p = 1, 2, 3. Dashed lines indicate chemical potentials of FS contours shown in (e–h). Units of k_0,x, k_0,y axes are in Å⁻¹. Red (blue) ± symbols indicate signs of pairing in layer 1 (2).