For the Abelian phase ϵ$_{n}$ has a gap since λ$_{n}$ ( k$_{y}$ ) does not cross 1 / 2. If the gap is E$_{g}$ = min( | ϵ$_{n}$ ( k$_{y}$ ) | ), the asymptotic behavior of C$_{E}$ at t → 0 is C$_{E}$ ≃ e − E$_{g}$$^{/t}$/t $^{2}$. For the non-Abelian phase, as shown in Fig. 2(b) there is a gapless branch of λ$_{n}$ ( k$_{y}$ ) crossing 1 / 2, which corresponds to ϵ$_{n}$ ( k$_{y}$ ) crossing 0. Near k$_{y}$ = 0 the asymptotic behavior of | ϵ$_{n}$ ( k$_{y}$ ) | is | ϵ$_{n}$ ( k$_{y}$ ) | ≃ v | k$_{y}$ | , which gives C$_{E}$ ∝ t/v . In the same way as the heat capacity in a thermodynamic system, C$_{E}$/t is constant at t → 0 limit and is proportional to the density of state 1 /v at “maximally entangled limit” ϵ$_{n}$ → 0 or λ$_{n}$ → 1 / 2. 

