Interacting quantum fluid in a polariton Chern insulator

We study Bogoliubov excitations of a spinor Bose-Einstein condensate in a honeycomb periodic potential, in the presence of a Zeeman field and of a spin-orbit coupling specific for photonic systems, which is due to the energy splitting between TE and TM polarized eigenstates. We also consider spin-anisotropic interactions typical for cavity polaritons. We show that the nontrivial topology of the single-particle case is also present for the interacting system. At low condensate density, the topology of the single-particle bands is transferred to the bogolon dispersion. At a critical value, the self-induced Zeeman field at the Dirac points of the dispersion becomes equal to the real Zeeman field and then exceeds it. The gap is thus closed and then reopened with inverted Chern numbers. This change of topology is accompanied by a change of the propagation directions of the one-way edge modes. This result demonstrates that the chirality of a topological insulator can be reversed by collective effects in a Bose-Einstein condensate.

Figure 1

(a) TE-TM field pattern in the reciprocal space. (b) TE-TM field for an elastic circle (fixed energy).

Figure 2

(a) Polaritonic micropillar scheme. (b) Polaritonic graphene scheme.

Figure 3

(a) Texture of the TE-TM effective field in reciprocal space (white arrows); the colors show the corresponding in-plane polarization domains. (b) TE-TM Dresselhaus-like field for an elastic circle at $K$ and $K^{\prime }$ points.

Figure 4

Berry curvatures around $K$. (a,b) Berry curvatures of the first and second “valence” branches, respectively, for $\delta J/J=0.2<0.5$. (c,d) Similar images for $\delta J/J=0.7>0.5$. Zeeman field is $\mathrm{\Delta }=0.1J$ in both cases.

Figure 5

Band structures and polarization textures of Bogoliubov excitations without and with applied magnetic field. (a,b) Energy dispersions without applied magnetic field: ${\alpha }_2=0,-0.2{\alpha }_1$, respectively. (c) Dispersions with applied magnetic field $B<B_C$. (d) Dispersions with applied magnetic field $B>B_C$. (e–h) Corresponding polarization and pseudospin textures. The colors represent the linear polarization degree while the white arrows show the corresponding pseudospin orientation of the lowest-energy state. (Set of parameters: $\delta J=0.2J, {\alpha }_{1}n=J$.)

Figure 6

Gap map and qualitative phase diagram. (a) Plot of the gap value under a constant magnetic field as a function of the density. The gap closes at the critical density $n=n_S$. (Parameters $\delta J=0.2J, \mathrm{\Delta }=0.6J$, black solid line is ${\alpha }_2=-0.2{\alpha }_1$, red dashed line is ${\alpha }_2=-0.5{\alpha }_1$.) (b) The Chern numbers of the conduction band with a constant applied magnetic field. The insets illustrate the corresponding edge states.