Topological Bogoliubov Quasiparticles from Bose-Einstein Condensate in a Flat Band System

For bosons with flat energy dispersion, condensation can occur in different symmetry sectors. Here, we consider bosons in a kagome lattice with $\pi$-flux hopping, which, in the presence of mean-field interactions, exhibit degenerate condensates in the $\mathrm{\Gamma }$ and the $K$ point. We analyze the excitation above both condensates and find strikingly different properties: For the $K$-point condensate, the Bogoliubov–de Gennes (BdG) Hamiltonian has broken particle-hole symmetry and exhibits a topologically trivial quasiparticle band structure. However, band flatness plays a key role in breaking the time-reversal symmetry of the BdG Hamiltonian for a $\mathrm{\Gamma }$-point condensate. Consequently, its quasiparticle band structure exhibits nontrivial topology, characterized by nonzero Chern numbers and by the presence of edge states. Although quantum fluctuations energetically favor the $K$-point condensate, the interesting properties of the $\mathrm{\Gamma }$-point condensate become relevant for anisotropic hopping. The topological properties of the $\mathrm{\Gamma }$-point condensate get even richer in the presence of extended Bose-Hubbard interactions. We find a topological phase transition into a topological condensate characterized by high Chern number and also comment on the realization and detection of such excitations.

Figure 1

Kagome lattice with three different sublattices in the left panel and single-particle dispersion relation of the tight-binding Hamiltonian in Eq. (1) for $t>0$ in the right panel. Here, ${\delta }_i$ represents the distance between two different nearest-neighbor sublattices, and $b_i$ stands for the reciprocal lattice vectors in momentum space.

Figure 2

Bogoliubov dispersions of quasiparticle and quasiholes, for different values of nearest-neighbor interaction $V$ and fixed on-site interaction $U/t=3$. The value of nearest-neighbor interaction is $V/t=0.5$ in panels (a)–(d) and $V/t=4$ in panel (b). In panels (a) and (b), condensation occurs at $k_{cp}=\mathrm{\Gamma }$. In this case, particle-hole symmetry leads to $\omega (k_{+})=\omega (k_{-})$. In panels (c) and (d), condensation occurs at $k_{cp}=K$. In this case, particle-hole symmetry is broken, and we depict the dispersion of quasiparticles and quasiholes separately. In all panels, we have chosen $t=\rho =1$.

Figure 3

The Chern number of Bogoliubov mode bands in addition to the energy gap between different bands for different values of the nearest-neighbor interaction $V$ is plotted. The energy gap between the lowest (highest) and middle energy band, i.e., ${\mathrm{\Delta }}_{12}=|{\omega }_1-{\omega }_2|$ (${\mathrm{\Delta }}_{23}=|{\omega }_2-{\omega }_3|$), is shown by a red circle (blue up triangle) line marker. The corresponding Chern numbers from lowest to highest energy band are specified as $C_1$, $C_2$, and $C_3$ with green diamond, pink square, and cyan down triangle line markers, respectively. Here, we supposed $U/t=3$ and $\rho =1$.

Figure 4

Bulk dispersion relation of Bogoliubov quasiparticles above the $\mathrm{\Gamma }$ condensate, in panels (a) and (b). Dispersion relation of Bogoliubov quasiparticles above the $\mathrm{\Gamma }$ condensate, obtained from the bulk Hamiltonian in panels (a) and (b) and for the slab Hamiltonian in panels (c) and (d). The panels on the left (a),(c) are for $V=0$ and on the right are for $V/t=0.6$, while in all panels $U/t=3$ is assumed. In the bulk Hamiltonian, all bands in different colors are energetically separated from each other, and Chern numbers are well defined (given in the boxes), although between the first and second bands in (b) the indirect band gap becomes zero. Noting the opposite group velocities of red and blue edge states, seen from the dispersion in (c) or (d), we interpret the pair as one chiral mode. Note that the chirality of the mode analyzed in the left panel is opposite to the chirality of the mode analyzed in the right panel. The chirality change is due to the topological transition at finite $V$, rendering the Chern number of the central band to 2 in the right panels.