Fig. 1: String-based chargon pairing in mixD bilayers.

We introduce an efficient pairing mechanism for distinguishable chargons with flavours μ = ± connected by a string Σ. a,b, The string pairing mechanism can be realized in a model of d-dimensional bilayers with spin-1/2 particles and strong repulsive interactions for d = 1 (a) and d = 2 (b). We consider mixD systems, where the hopping t_⊥ between the layers is suppressed by a gradient Δ, while the superexchange J_⊥ is kept intact and J_⊥ ≫ J_∣∣. The strong interlayer superexchange leads to the formation of rung singlets (dimers indicated by grey ellipses with red and blue spheres), depicted here for d = 1 (a) and d = 2 (b). The motion of doped holes (grey spheres) via t_∥ within different layers μ = ± tilts the singlets along their paths, which corresponds to the formation of Σ. The potential energy associated with the string increases with its length as more singlets are tilted. Grey (green) shading corresponds to vertical (tilted) singlets. c,d, E_B obtained by comparing the energies of spinon–chargon and chargon–chargon mesons in the GSP model, and the ground-state energy for a system with one versus two holes in DMRG as described in the text. We assumed that J_∥/J_⊥ = 0.01 and varies t_∥/J_⊥ in equation (4). In the DMRG, weak interlayer tunnelling of strength t_⊥/J_⊥ = 0.01 was added to ensure convergence. The shading indicates the tight binding and strong coupling regimes. c, E_B for a 40 × 2 ladder (d = 1). d, E_B for a bilayer system on a 12 × 4 cylinder.