Dynamical Properties of the One-Dimensional Spin-$1/2$ Bose-Hubbard Model near a Mott-Insulator to Ferromagnetic-Liquid Transition

We investigate the dynamics of the one-dimensional strongly repulsive spin-$1/2$ Bose-Hubbard model for filling $\nu \le 1$. While at $\nu =1$ the system is a Hubbard-Mott insulator exhibiting dynamical properties of the Heisenberg ferromagnet, at $\nu <1$ it is a ferromagnetic liquid with complex spin dynamics. We find that close to the insulator-liquid transition the system admits for a complete separation of spin and density degrees of freedom valid at all energy and momentum scales within the $t\mathrm{\text{−}}J$ approximation. This allows us to derive the propagator of transverse spin waves and the shape of the magnon peak in the dynamic spin structure factor.

Figure 1

Shown are the parametric regimes (A), (B), and (C) corresponding to different relative values of the parameters $q_F=\pi \mu$, $J/t_h$, and $E_F/t_h$.

Figure 2

Shown is the spectral function $A(k,\omega )$ as a function of $\omega$ for several values of $k$: $3\pi /8$ (red), $5\pi /8$ (green), $3\pi /4$ (blue), and $\pi$ (pink). The curves are obtained from the numerical analysis of the Fredholm determinant in Eq. (16). The positions of the singularities following from the asymptotic expression (18) are indicated with vertical dotted lines. The parameters of the model used for getting the plot are $J{t}_h^{-1}=0.04$ and $q_F=0.1$, which implies $E_F/2J=0.125$.