We calculate the Wilson ratio of the one-dimensional Fermi gas with spin imbalance. The Wilson ratio of attractively interacting fermions is solely determined by the density stiffness and sound velocity of pairs and of excess fermions for the two-com
ponent Tomonaga-Luttinger liquid (TLL) phase. The ratio exhibits anomalous enhancement at the two critical points due to the sudden change in the density of states. Despite a breakdown of the quasiparticle description in one dimension, two important features of the Fermi liquid are retained, namely the specific heat is linearly proportional to temperature whereas the susceptibility is independent of temperature. In contrast to the phenomenological TLL parameter, the Wilson ratio provides a powerful parameter for testing universal quantum liquids of interacting fermions in one, two and three dimensions.
We propose an exactly solvable model of one-dimensional anyons with competing $delta$-function and derivative $delta$-function interaction potentials. The Bethe ansatz equations are derived in terms of the $N$-particle sector for the quantum anyonic
field model of the generalized derivative nonlinear Schr{o}dinger equation. This more general anyon model exhibits richer physics than that of the recently studied one-dimensional model of $delta$-function interacting anyons. We show that the anyonic signature is inextricably related to the velocities of the colliding particles and the pairwise dynamical interaction between particles.
We investigate magnetic properties and statistical effects in 1D strongly repulsive two-component fermions and in a 1D mixture of strongly repulsive polarized fermions and bosons. Universality in the characteristics of phase transitions, magnetizatio
n and susceptibility in the presence of an external magnetic field $H$ are analyzed from the exact thermodynamic Bethe ansatz solution. We show explicitly that polarized fermions with a repulsive interaction have antiferromagnetic behavior at zero temperature. A universality class of linear field-dependent magnetization persists for weak and finite strong interaction. The system is fully polarized when the external field exceeds the critical value $H^F_capprox frac{8}{gamma}E_F$, where $E_F$ is the Fermi energy and $gamma$ is the dimensionless interaction strength. In contrast, the mixture of polarized fermions and bosons in an external field exhibits square-root field-dependent magnetization in the vicinities of H=0 and the critical value $H=H^M_capprox frac{16}{gamma}E_F$. We find that a pure boson phase occurs in the absence of the external field, fully-polarized fermions and bosons coexist for $0<H<H^M_c$, and a fully-polarized fermion phase occurs for $Hge H_c^M$. This phase diagram for the Bose-Fermi mixture is reminiscent of weakly attractive fermions with population imbalance, where the interacting fermions with opposite spins form singlet pairs.
We investigate the low temperature behaviour of the integrable 1D two-component spinor Bose gas using the thermodynamic Bethe ansatz. We find that for strong coupling the characteristics of the thermodynamics at low temperatures are quantitatively af
fected by the spin ferromagnetic states, which are described by an effective ferromagnetic Heisenberg chain. The free energy, specific heat, susceptibility and local pair correlation function are calculated for various physical regimes in terms of temperature and interaction strength. These thermodynamic properties reveal spin effects which are significantly different than those of the spinless Bose gas. The zero-field susceptibility for finite strong repulsion exceeds that of a free spin paramagnet. The critical exponents of the specific heat $c_v sim T^{1/2}$ and the susceptibility $chi sim T^{-2}$ are indicative of the ferromagnetic signature of the two-component spinor Bose gas. Our analytic results are consistent with general arguments by Eisenberg and Lieb for polarized spinor bosons.
