Bethe ansatz and bosonization procedures are used to describe the thermodynamics of the strong-coupled Hubbard chain in the textit{spin-incoherent} Luttinger liquid (LL) regime: $J(equiv 4t^2/U)ll k_B Tll E_F$, where $t$ is the hopping amplitude, $U(gg t)$ is the repulsive on-site Coulomb interaction, and $k_B T (E_Fsim t)$ is the thermal (Fermi) energy. We introduce a fractional Landau LL approach, whose $U=infty$ fixed point is exactly mapped onto an ideal gas with two species obeying the Haldane-Wu textit{exclusion} fractional statistics. This phenomenological approach sheds light on the behavior of several thermodynamic properties in the spin-incoherent LL regime: specific heat, charge compressibility, magnetic susceptibility, and Drude weight. In fact, besides the hopping (mass) renormalization, the fractional Landau LL parameters, due to quasiparticle interaction, are determined and relationships with velocities of holons and spinons are unveiled. The specific heat thus obtained is in very good agreement with previous density matrix renormalization group (DMRG) simulations of the $t$-$J$ model in the spin-incoherent regime. A phase diagram is provided and two thermodynamic paths to access this regime clarifies both the numerical and analytical procedures. Further, we show that the high-$T$ limit of the fractional Landau LL entropy and chemical potential exhibit the expected results of the $t$-$J$ model, under the condition $Ugg k_B T$. Lastly, finite-temperature Lanczos simulations of the single-particle distribution function confirm the characteristics of the spin-incoherent regime and the high-$T$ limit observed in previous DMRG studies.
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