The geometric impact of the quantum Hall interface on a cone

Recently, quantum Hall interface has become a popular subject of research; distinct from that of the quantum Hall edge, which is constrained by external background confinement, the interface has the freedom to move, likely towards a string-like state. In disk geometry, it was known that the interface energy has an extra correction due to its curvature which depends on the size of the disk. In this work, we analytically calculate the energy of the integer quantum Hall interface on a cone surface which has the advantage that its curvature is more easily adjustable. By tuning the length and curvature of the interface by the cone angle parameter $\beta$, we analyze the dependence of the quantum Hall interface energy on the curvature and verify this geometric correction. Moreover, we find that the tip of the cone geometry has an extra contribution to the energy that reflects on the $u_2,u_4$ term.