Fig. 1. Schematic illustration of spinon hoppings up to second neighbors on the square lattice. (a) The zero-flux QSL with a uniform nearest-neighbor spinon hopping coefficient t_1,ij = t_1,ji = t₁ and next-nearest-neighbor spinon hopping coefficient t_2,ij = t_2,ji = t₂. (b) The π-flux QSL with a gauge fixing such that the red thick lines stand for negative spinon hopping coefficient t_1,ij = t_1,ji = –t₁, while the meaning of other lines remains unchanged.

Fig. 2. Calculated dynamical spin structure factor S(q,ω) along the high symmetry line Γ–M–Γ–X–M in the first Brillouin zone, (a) zero-flux spinon Fermi surafce QSL with V-shape character around the Γ and (b) π-flux Dirac QSL with clear low-energy cone features around the high symmetry points. Contour plot of the upper edge of S(q,ω) in the first Brillouin zone for (c) zero-flux spinon Fermi surafce QSL and (d) π-flux Dirac QSL. (e) Original Brillouin zone (outer black square) and the folded Brillouin zone (light gray square) of square lattice. The parameters adopted in the calculation are t₂/t₁ = 0.2 with zero temperature k_BT/t₁ = 0.

Fig. 3. (a) Dynamic spin structure factor for zero-flux QSL with t₂/t₁ = 0.2 and z-direction magnetic field B_z / t₁ = 4. (b) Schematic illustration of the particle–hole excitations with small momenta. Such excitations for each q are degenerate at zero field, while the two-fold degeneracy is lifted soon when the Zeeman field is turned on.

Fig. 4. (a) Schematic illustration of the spinon hopping matrix involving the complex second neighbor hopping coefficients, hopping along the arrows corresponds to ϕ, while hopping oppositely the arrows corresponds to –ϕ. Contour plot of Berry curvatures calculated when t₂/t₁ = 0.3 and ϕ = π/2 for (b) the lower two bands and (c) the upper two bands. (d) Representive spinon bands calculated when t₂/t₁ = 0.2, ϕ = π/3, and B_z/t₁ = 0.4, the corresponding Chern numbers from the lowest band to the highest one are –1, –1, +1, +1, respectively. (e) The evolution of thermal Hall conductivity with temperature for different phase ϕ, where the magnetic field is fixed at B_z/t₁ = 0.4, and the unit of κ_xy/T here is πkB2/6ℏ.