4 0 50 100 150 200 250 Thermopower [ V / K ] 0 2 4 6 8 10 T /| t | 0 50 100 150 200 250 Thermopower [ V / K ] 0 2 4 6 8 10 T /| t | S * S$_{Kelvin}$ S$_{MH}$ (a)  x  = 0.67   t  > 0   J  = 0 (c)  x  = 0.75   t  > 0   J  = 0 (b)  x  = 0.67   t  < 0   J  = 0 (d)  x  = 0.75   t  < 0   J  = 0 t ? (e) FIG. 1: Thermopower vs. T for the t - J model (with J = 0) corresponding to NCO in the Curie-Weiss metallic phase near x ∼ 0 . 7. (a) and (c) correspond to x = 0 . 67 and x = 0 . 75 for the NCO system ( t > 0) while in (b) and (d) the sign of the hopping has been switched to investigate the enhance- ment expected for frustrated systems. The black  red (light gray)  and blue (dark gray) lines are S $^{∗}$  S Kelvin   and S MH . Finite size effects at low T are treated in the spirit described previously $^{2}$. At each x   for T below an appropriately cho- sen cutoff temperature T 0 = 0 . 5 | t |   the thermopower is fit to S ( T ) → aT + bT 2 where a and b are obtained from the com- puted S ( T 0 ) and S $^{′}$( T 0 ) providing a sensible extrapolation to low T and plotted as dashed lines. The inset figure (e) depicts the 12-site unit cell. which is off by an important factor of two from S$_{Kelvin}$ (Eq. (11)). The formulations (Mott-Heikes and Kelvin) would be identical if µ ∝ T   which occurs if the system possesses a ground state degeneracy  and in the classical regime. The high frequency approximation gives a better result than all these and  again in the low temperature limit  to O ( T $^{3}$)  Other than the neglect of the energy derivative of τ   this is the same as the exact result. Hence  ranking the ther- mopower approximations for non-interacting electrons we have  from worst to best  S$_{MH}$   S$_{Kelvin}$   and S ∗ with the exact result being S$_{Mott}$ . For intermediate coupling models  the relative rankings of the various approximations can be different. In partic- ular  S$_{Kelvin}$ can be superior to S $^{∗}$  since the effect of cor- relations is diluted in the latter by making the assump- tion of ω ≫ U   whereas S$_{Kelvin}$ retains ω ≪ U . The sign of the true (i.e. transport) thermopower and the trans- port Hall constant are expected to flip as we approach half filling in the Hubbard or t - J models due to the on- set of correlations (carriers become holes measured from half filling rather than from a completely filled band). In the case of the t - J model  the high frequency Hall con- stant R ∗ H and S ∗ do display this behavior $^{11}$. However  for the Hubbard model  R ∗ H and S ∗ do not display a sign change $^{16 17}$. S$_{Kelvin}$ on the other hand  does appear to show the expected change in sign $^{16 18}$. Further discussion concerning the relative merits of S$_{Kelvin}$ and S ∗ will be reported later $^{16}$. S ∗ adequately describes the physics of NCO for x > 0 . 5 and  in particular the so-called Curie-Weiss metallic phase 2 near x ∼ 0 . 7. The subject of this work  however  is S$_{Kelvin}$ . We see in Figs. 1a and c and 2a  similar to S$_{MH}$   S$_{Kelvin}$ does a good job capturing the physics with minimal computational effort. However  S$_{Kelvin}$ does seem to overestimate the thermopower for inter- mediate temperatures and high dopings as compared to S$_{MH}$ . Near x ≈ 0 . 7  S$_{Kelvin}$ and S$_{MH}$ are similar but as x is decreased the two formulae diverge and for low dop- ings  S$_{Kelvin}$ better captures the physics as it is closer to the more accurate high frequency limit S $^{∗}$. To show the usefulness of S$_{Kelvin}$   we apply it to NCO since (i) we have previously investigated 2 this system while benchmarking S $^{∗}$  (ii) the system is intrinsically interesting $^{1}$  and (iii) we can compare different ther- mopower formulations on equal footing. As discussed $^{2}$  the action in NCO takes place primarily in the cobalt ox- ide planes where d -shell spin-1/2 electrons live on a trian- gular lattice and these strongly interacting 2D electrons can be modeled with the t - J model. Hence  we exactly di- agonalize the t - J model on a GLYPH<suppress>L = 12 site two-dimensional triangular lattice with periodic boundary conditions (cf. Fig. 1e). Note that we only show results for the t - J model with zero super-exchange interaction ( J = 0)  as the re- sults only weakly depend on J . To map the t - J model to NCO we follow Refs. 2 and 4 and give results as a function of electron doping x = | 1 − n | away from half filling ( n is electron number density). An interesting property of the triangular lattice un- derlying the physics of NCO is its geometrical frustra- tion $^{3}$  cf. inset Fig. 1e. It was predicted 2 4 that if the sign of the hopping amplitude were flipped to t < 0 the thermopower would be enhanced at low to intermediate T . We have considered this situation in Figs. 1b and d and 2b. Since the thermopower enhancement for t < 0 compared to t > 0 is largely a consequence of electron- electron interaction it is important to determine whether this effect is captured by S$_{Kelvin}$ . We see this enhance- ment is captured to some extent by S$_{Kelvin}$ and S$_{Kelvin}$ is better than S$_{MH}$ in the large doping region where the enhancement is the greatest  but is missing some of the electron-electron physics at very low T that is captured by S ∗ (as is S$_{MH}$ ). C. NCO and the t - J model B. Hubbard model S ∗ = T π $^{2}$k2 B 3 q$_{e}$ d dµ ln [ ρ$_{0}$ ( µ ) 〈 ( v x $_{p}$) $^{2}$〉$_{µ}$ ] |$_{µ}$ → µ$_{0}$ . (13)