Spin-charge split pairing in underdoped cuprate superconductors: support from low-$T$ specific heat

We calculate the specific heat of a weakly interacting dilute system of bosons on a lattice and show that it is consistent with the measured electronic specific heat in the superconducting state of underdoped cuprates with boson concentration $\rho \sim x\mathrm{/}2$, where $x$ is the hole (dopant) concentration. As usual, the $T^3$ term is due to Goldstone phonons. The zero-point energy, through its dependence on the condensate density ${\rho }_0(T)$, accounts for the anomalous $T$-linear term. These results support the split-pairing mechanism, in which spinons (pure spin) are paired at $T^{\ast }$ and holons (pure charge) form real-space pairs at $T_p<T^{\ast }$, creating a gauge-coupled physical pair of charge $+2e$ and concentration $x\mathrm{/}2$ which Bose condenses below $T_c$, accounting for the observed phases.