Fractional exclusion statistics and thermodynamics of the Hubbard chain in the spin-incoherent Luttinger liquid regime

TL;DR

This paper develops a fractional Landau Luttinger liquid approach to describe the thermodynamics of the strongly interacting Hubbard chain in the spin-incoherent regime, aligning well with numerical simulations and providing new insights into quasiparticle interactions.

Contribution

It introduces a fractional Landau LL framework mapped to an ideal gas with fractional statistics, elucidating thermodynamic properties in the spin-incoherent regime of the Hubbard chain.

Findings

Good agreement with DMRG simulations of the $t$-$J$ model

Derivation of fractional Landau parameters and velocities of holons and spinons

Confirmation of high-$T$ behavior through Lanczos simulations

Abstract

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: $J(\equiv 4t^2/U)\ll k_B T\ll E_F$, where $t$ is the hopping amplitude, $U(\gg t)$ is the repulsive on-site Coulomb interaction, and $k_B T (E_F\sim t)$ 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 $U\gg k_B T$. 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.