Topological Bogoliubov excitations in inversion-symmetric systems of interacting bosons

On top of the mean-field analysis of a Bose-Einstein condensate, one typically applies the Bogoliubov theory to analyze quantum fluctuations of the excited modes. Therefore, one has to diagonalize the Bogoliubov Hamiltonian in a symplectic manner. In our article we investigate the topology of these Bogoliubov excitations in inversion-invariant systems of interacting bosons. We analyze how the condensate influences the topology of the Bogoliubov excitations. Analogously to the fermionic case, here we establish a symplectic extension of the polarization characterizing the topology of the Bogoliubov excitations and link it to the eigenvalues of the inversion operator at the inversion-invariant momenta. We also demonstrate an instructive but experimentally feasible example that this quantity is also related to edge states in the excitation spectrum.

Figure 1

(a) Sketch of the Hamiltonian (1). (b) Corresponding phase diagram distinguishing between different wave functions of the condensate for the parameters ${\nu }_{\mathrm{so}}/\omega =0.2$ and ${\chi }_{\sigma ,{\sigma }^{\prime }}={\delta }_{\sigma ,{\sigma }^{\prime }}\chi$. (c) Instances of the corresponding wave functions for ${\nu }_0/\omega =0.1$. The labels I, II, and III denote phases with a localized wave function, condensation at zero momentum, and condensation at finite momentum $k>0$, respectively. (d) For phase II, we depict some Bogoliubov dispersion relations. The corresponding parameters are marked with arrows in the phase diagram. As explained in the main text, one can distinguish the topology of the Bogoliubov excitations depending on the symplectic polarization $P_s$. For this reason, one can split phase II into a trivial phase IIa and a topological phase IIb. The phase boundary is strongly influenced by the condensate due to the interactions between the particles. In Sec. 4f we show that topologically protected edge states appear in the topological phase IIb.

Figure 2

(a) and (b) Spectra of the Bogoliubov excitations with fixed boundary conditions for $M=30$ sites in phases IIa and IIb, respectively. Before performing the Bogoliubov diagonalization, we first minimize the Gross-Pitaevskii functional. The resulting wave function of the condensates close to the boundaries is depicted in the insets. Due to the fixed boundary condition, the density of the condensate is lower at the boundaries. Using the condensate wave function, we then perform the Bogoliubov diagonalization. The excitation energies are all positive and real valued, as we have prepared the wave function in its ground state. One clearly identifies the midgap states. The wave function of one of these midgap states is depicted in (c).