Bound States in the Continuum Realized in the One-Dimensional Two-Particle Hubbard Model with an Impurity

We report a bound state of the one-dimensional two-particle (bosonic or fermionic) Hubbard model with an impurity potential. This state has the Bethe-ansatz form, although the model is nonintegrable. Moreover, for a wide region in parameter space, its energy is located in the continuum band. A remarkable advantage of this state with respect to similar states in other systems is the simple analytical form of the wave function and eigenvalue. This state can be tuned in and out of the continuum continuously.

Figure 1

The three continuum bands revealed by using the two quantities of $D=|x_1|+|x_2|$ and $P_{\mathrm{out}}^R$. The three horizontal lines are clearly visible in each panel. The eigenstates and eigenvalues $E_n$ are solved by exact diagonalization on a $(2N+1)$-site ($N=20$) lattice and with open boundary condition. The values of the parameters are $(V,U)=(-1.5,6)$.

Figure 2

The $x_1$-$x_2$ lattice decomposed into six regions by the $x_1$ axis, $x_2$ axis, and $x_1=x_2$ line. Note that adjacent regions share the boundary between them. Especially, the origin belongs to all the six regions.

Figure 3

In the region labeled with (ii)-BIC, the state with energy $E_{b1}$ [see Eq. (6)] is a bound state embedded in the [$-4$, $+4$] continuum. In the region labeled with (ii)-BOC, this state is a bound state below the [$-4$, $+4$] continuum band and outside of all the three continuum bands. The boundary between the two regions is determined by the condition $E_{b1}=-4$. In the region labeled with (i), the odd-parity bound state with energy $E_{b2}$ exists, which is below all the continuum bands.

Figure 4

Images of the squared wave function of the bound state in the continuum, with (a) $(V,U)=(-2,-0.4)$, $E_{b1}=-0.2672$, (b) $(V,U)=(-2,-1.5)$, $E_{b1}=-0.7669$, and (c) $(V,U)=(-2,-1.8)$, $E_{b1}=-0.8185$. These states are obtained in a finite lattice by exact diagonalization with 100 sites on each side of the impurity and open boundary condition. They agree with the analytic expressions in Eqs. (3, 4).