Topological two-body bound states in the interacting Haldane model

We study the topological properties of the two-body bound states in an interacting Haldane model as a function of interparticle interactions. In particular, we identify topological phases where the two-body edge states have either the same or the opposite chirality as compared to single-particle edge states. We highlight that in the moderately interacting regime, which is relevant for the experimental realization with ultracold atoms, the topological transition is affected by the internal structure of the bound state, and the phase boundaries are consequently deformed.

Figure 1

(a) Graphic representation of the Haldane model in the honeycomb lattice. The black and grey dots represents the $A$ and $B$ sites forming the two different sublattices. The phase $\varphi$ of the next-nearest-neighbor hopping is taken to be positive along the directions set by the purple arrows. Examples are given of a next-nearest-neighbor hopping process for the doublon at (b) second and (c) third order in perturbation theory.

Figure 2

(a) Doublon energy spectrum as a function of the center-of-mass momentum $k_y$, with periodic boundary conditions along $y$ and open boundary conditions along $x$ (bearded termination on both ends). The dots are obtained from an exact diagonalization of the two-body Hamiltonian with the parameters $N_x=78$, $N_y=51$, $J^{\prime }/J=0.2$, $U/J=30$, $\varphi =\pi /5$, and $\mathrm{\Delta }=0$. The solid lines are obtained from the effective model in Eq. (3) with the same parameters. Particle densities per site $\rho$ of the states highlighted with a red circle in (a) are projected on the $x$ axes in (b) and (c). In (b) we show the topological states labeled with L and R, respectively, localized on the left (L) and on the right (R) edges. In (c) we show the Tamm-Shockley states labeled with TL and TR, respectively, localized on the left (TL) and on the right (TR) edges. Solid lines are obtained from the effective model whereas the open circles are the exact diagonalization results.

Figure 3

Gap between the two doublon bands as a function of the next-nearest-neighbor hopping phase $\varphi$ and the onsite $A-B$ imbalance $\mathrm{\Delta }$ in units of $|J_{\infty }^{\prime }|=2J^{\prime 2}/U$ for $J^{\prime }/J=0.2$, $U/J=500$ (a) and $U/J=30$ (b). The diagram is obtained from the exact diagonalization of a system with periodic boundary conditions ($N_x=60$ and $N_y=51$). The gap closing points track the topological phase transition. This transition is compared with the predictions of the phase boundaries at second-order (black dashed line) and third-order (white solid line) perturbation theory. (c) Critical phase at $\mathrm{\Delta }=0$ as a function of $U$. Open circles are the gap closing points estimated with exact diagonalization. The dashed and the solid line correspond to the prediction of the effective model at second and third order, respectively. The cross corresponds to the parameters shown in Fig. 2. (d) Comparison between the single-particle topological phase diagram (blue dashed line) and the doublon topological phase diagram (red solid line) in units of the single-particle next-nearest-neighbor hopping $J^{\prime }$, indicating the regions of nonvanishing Chern numbers.

Figure 4

(a)–(d) Energy spectra of the two-body model as a function of the center-of-mass momentum $k_y$ with different interactions, respectively, $U/J=5,3.5,2.7,2.5$. The spectra are obtained from exact diagonalization of the two-body Hamiltonian with periodic boundary conditions along $y$ and open boundary conditions along $x$, with a bearded termination on both ends. Parameters are $N_x=78$, $N_y=51$, $J^{\prime }/J=0.2$, $\varphi =\pi /3$, and $\mathrm{\Delta }/J=0.01$. Only the first $3N_x$ eigenvalues from the top are shown. The grey area indicates the region of energy of the scattering continuum.

Figure 5

A honeycomb lattice with bearded terminations on both ends. The on-site energies of the effective model for the doublon are specified for the left and the right edges, for both sublattices.

Figure 6

Energy spectrum of the doublons as a function of the center-of-mass momentum $k_y$ along the $y$ direction, with periodic boundary conditions along $y$ and open boundary conditions along $x$ with a bearded termination on both ends. The spectra are obtained from the effective model in Eq. (3) of the main text, for $N_x=100$, $J^{\prime }/J=0.2$, $U/J=100$, $\varphi =\pi /3$, and $\mathrm{\Delta }/J=0.001$ and with additional on-site energy shifts ${\epsilon }_{\mu }$ on the edge sites. The energy of the site $\mu$ on the $\nu$ edge is increased by (a) ${\epsilon }_{\mu ,\nu }=0.25\delta E_{\mu ,\nu }$, (b) ${\epsilon }_{\mu ,\nu }=0.5\delta E_{\mu ,\nu }$, (c) ${\epsilon }_{\mu ,\nu }=0.75\delta E_{\mu ,\nu }$, and (d) ${\epsilon }_{\mu ,\nu }=\delta E_{\mu ,\nu }$, see notation in the text. The last case corresponds to the uniform situation where there is no energy difference between the edge sites and the bulk sites.