Binding of Holons and Spinons in the One-Dimensional Anisotropic $t\mathrm{\text{−}}J$ Model

We study the binding of a holon and a spinon in the one-dimensional anisotropic $t\mathrm{\text{−}}J$ model using a Bethe-Salpeter equation approach, exact diagonalization, and density matrix renormalization group methods on chains of up to 128 sites. We find that holon-spinon binding changes dramatically as a function of anisotropy parameter $\alpha =J_{\perp }/J_z$: it evolves from an exactly deducible impuritylike result in the Ising limit to an exponentially shallow bound state near the isotropic case. A remarkable agreement between the theory and numerical results suggests that such a change is controlled by the corresponding evolution of the spinon energy spectrum.

Figure 1

A hole in the Ising AF (circle) moved by four sites from origin. The spinon is indicated by the dashed box.

Figure 2

Lowest energies $E_k-E_0$ vs $k$ in an $L=21$ chain with (a) zero and (b) one hole, $J_z=4.0$. Solid lines in (a) are the spinon dispersion from BA [9]; in (b) the lines are guides to the eye.

Figure 3

Theoretical (lines) and numerical results by method $A$ (circles) and method $B$ (diamonds) for $\Delta$ vs $\alpha$, $J_z=4.0$. Inset: $|\Delta |$ for $\alpha \ge 0.4$ on a semilog plot.

Figure 4

The 4-even (circles) and 4-odd (squares) branches for (a) holon and (b) pair energies vs $1/L$. $J_z=4.0$ and $\alpha =0.4$ ($\alpha =0.25$, insets). Dashed curves are the fits described in text.

Figure 5

${\Delta }_L$ data vs $1/L$ in method $B$, $J_z=4.0$ and $\alpha =0.4$. Branches are defined in Table . Dashed lines are guides to the eye, solid line is the 4th order polynomial fit of the $B2$ branch.