Fig. 6: The spectrum of the two-column transfer operator TT^t for the single layer spin model.

With the periodic boundary condition, we calculate the spectrum of the single layer spin model with the circumference L_y = 12, where λ_j is the (j + 1)th dominant eigenvalue and θ is the spin angle. a The red lines represent \(-\mathrm{log}\,(| {\lambda }_{j}/{\lambda }_{0}| )\), where each red line corresponds to an eigenvalue, and the spectrum becomes nearly continuous in the filled areas. The black dashed lines correspond to the phase transition points L and A in the phase diagram Fig. 4b. b The blue dots are the arguments of the spectrum \(\arg ({\lambda }_{j})\) that become nearly continuous in the filled areas. The vertical dashed lines indicate two exactly solvable points E and A in the phase diagram Fig. 4b.