Novel Exciton Condensation and Lorenz Number in Graphene

We present a novel condensation of boson (exciton = electron + hole) in 2 dimension at the temperature Tc = (6/π² )^½ T_F (fermi temperature). At Tc, the condensed bosons are to become fermions, electrons to holes, and vice versa. We found the Lorenz number peak value (LNPV) in the case of the elastic scatterings, L(b)/L(0) - 1 = 2bg/3, L(0) = (9/4) ζ(3)/ln2 = 3.895 = 1.183 (π²/3) in the unit of (k_B/e)² , where b = boson/fermion = 2gp/n, g and n are the fermion degeneracy and density, respectively, p the optimum density for the condensation, with the temperature full width (the intrinsic nature) for the half LNPV, Tw = 2(1 - w) Tc, w = [(1 + b)/(2 + b)]^½ . Calculated LNPV and Tw for g = 4, account very well, without any parameter, for the data in graphene by Crossno et al, Science 351, 1058 (2016). We found the optimum 2bg/3 = 64/3, p = 4 x 10⁹ /cm² , v_F = 1.03 x 10⁸ cm/s , Tc = 45 K, and Tw = 4.5 K for graphene. We predict the Hall resistance (Rh) peak value and the plasmon frequency square (Ws) dip value,
Rh(b)/Rh(0) - 1 = b = Ws(0)/Ws(b) - 1 , respectively, with the same Tw as before.