Exciton Condensation in Quantum Hall Bilayers at Total Filling $\nu_T=5$

We study the coupled quantum Hall bilayers each at half-filled first excited Landau levels with varying the layer distance. Based on numerical exact diagonalization on torus, we identify two distinct phases separated by a critical layer distance $d_c$. From $d_c$ to infinite layer distance, the topological phase is smoothly connected to a direct tensor product of two Moore-Read states, while the interlayer coherence emerges at $d<d_c$ characterized by the $xy$ easy-plane ferromagnetic energy spectra, gapless pseudospin excitations and the finite exciton superfluid stiffness, corresponding to the exciton superfluid state. More interestingly, the results of the ground state fidelity, the evolution of energy spectra, and the superfluid stiffness indicate a possible continuous transition. Theoretically it can be interpreted as a topological phase transition which simultaneously changes the topology of ground state and breaks symmetry, providing an interesting example of transitions beyond Landau paradigm.