Bosonic orbital Su-Schrieffer-Heeger model in a lattice of rings

We study the topological properties of interacting and noninteracting bosons loaded in the orbital angular momentum states $l=1$ in a lattice of rings with alternating distances. At the single-particle level, the two circulation states within each site lead to two decoupled Su-Schrieffer-Heeger lattices with correlated topological phases. We characterize the topological configuration of these lattices in terms of the alternating distances, as well as their single-particle spectrum and topologically protected edge states. Second, we add on-site interactions for the two-boson case, which lead to the appearance of multiple bound states and edge bound states. We investigate the doublon bands in terms of a strong-link model and we analyze the resulting subspaces using perturbation theory in the limit of strong interactions. All analytical results are benchmarked against exact diagonalization simulations.

Figure 1

Representation of the considered one-dimensional lattice of rings. Each unit cell consists of two sites, $A_m$ and $B_m$, formed by identical ring potentials where $({\rho }_{j_m},{\phi }_{j_m})$ are the local radial and azimuthal coordinates at each site. The distance between adjacent sites alternates between $d$ for consecutive sites within a unit cell and $d^{\prime }$ for the sites in adjacent unit cells.

Figure 2

Couplings $t_a^{(\prime )}$ and $t_s^{(\prime )}$ as a function of the separation distance $d^{(\prime )}$ between rings for ${\rho }_0=2.5\sigma$ and ${\rho }_0=5\sigma$, where $\sigma =\sqrt{\hslash /M\omega }$ is the harmonic oscillator length.

Figure 3

Phase diagram of the (a) symmetric, ${\widehat{\mathcal{H}}}_s$, and (b) antisymmetric, ${\widehat{\mathcal{H}}}_a$, chains as a function of the separation distances between rings, $d^{(\prime )}$, for ${\rho }_0=5\sigma$, where $\sigma =\sqrt{\hslash /M\omega }$ is the harmonic oscillator length. Color represents the ratios (a) $t_s/t_s^{\prime }$ and (b) $t_a/t_a^{\prime }$; the black line separates the trivial phases with ${\mathcal{Z}}_{1,2}=0$ (warm colors) and the topological phases with ${\mathcal{Z}}_{1,2}=\pi$ (cold colors). The dashed blue lines and dotted yellow lines bound the nearly dimerized regimes of the trivial and topological phases, respectively.

Figure 4

Single-particle energy spectrum for a chain of $N_c=20$ unit cells with ${\rho }_0=5\sigma$, distances $d^{\prime }=3.6\sigma , d=4\sigma$ (black dots) and $d=5\sigma$ (blue crosses). The inset shows the inner bands given by ${\widehat{\mathcal{H}}}_a$ and the zero-energy edge states.

Figure 5

Edge states of (a),(c) ${\widehat{\mathcal{H}}}_s$ and (b),(d) ${\widehat{\mathcal{H}}}_a$. The distance is $d=4\sigma$, (a) and (b), and $d=5\sigma$, (c) and (d), with $d^{\prime }=3.6\sigma , {\rho }_0=5\sigma$, and $N_c=20$ for all cases. (i) Amplitude of the left (filled black bars) and right (empty blue bars) edge states of the symmetric and antisymmetric states (13) at each site. (ii) Real space density, ${\left|\mathrm{\Psi }\right|}^2$ of the first 15 sites of the left edge state. The edge states arise as the symmetric and antisymmetric superpositions of their hybridized counterparts.

Figure 6

(a) Two-particle energy spectrum for a chain of $N_c=15$ unit cells with ${\rho }_0=5\sigma$ and distances $d^{\prime }=3.6\sigma , d=5\sigma$ obtained through exact diagonalization (gray lines) and eigenvalues of the strong-link Hamiltonian, Eq. (18), for $J_3^{\prime }=J$ in color. (b)–(e) Numerical results for various sections of the spectrum with the color indicating (b) the expectation value of the bound state population ${\mathcal{N}}_b$; (c) and (e), the expectation value of the average distance to the nearest edge ${\mathcal{N}}_e$, and (d) parity $\mathrm{\Pi }$. The circle indicates a crossing between different doublon bands and the rectangles indicate the sections of the spectrum depicted in (b)–(e).

Figure 7

Adjacency graph of the strong-link Hamiltonian ${\widehat{\mathcal{H}}}_{SL}$, Eq. (18). Each vertex represents one of the basis states in Eq. (17) and the edges represent the off-diagonal (solid blue lines) and diagonal (solid blue loops) matrix elements of ${\widehat{\mathcal{H}}}_{SL}$. The dashed lines indicate the reflection symmetries with respect to the $oxz$ (yellow), $oyz$ (pink), and $oxy$ planes, where the states $|B_m^{+}{B}_m^{-}\rangle$ and $|A_s^{+}{A}_s^{-}\rangle$ are located in the out-of-plane axis $z$ (gray).

Figure 8

Two-particle energy spectrum for $N_c=15$ unit cells with ${\rho }_0=5\sigma$, distances $d^{\prime }=3.6\sigma , d=4.5\sigma$, and $U/t_s^{\prime }=20$ for the (a) $\mathcal{B}$ and (b) $\mathcal{A}$ subspaces. (i) Energies as a function of the edge on-site potential correction $V$ in units of $V_{\mathcal{A}\left(\mathcal{B}\right)}$ and (ii) energies for the exactly compensated on-site potential mismatch, (aii) $V=V_{\mathcal{B}}$ and (bii) $V=V_{\mathcal{A}}$.