Probing topological superconductors with emergent gravity

Topological superconductors are characterized by topological invariants that describe the number and nature of their robust boundary modes. These invariants must also have observable consequences in the bulk of the system, akin to the quantized bulk Hall conductivity in the quantum Hall effect, but such consequences are made elusive by the spontaneous breaking of $U(1)$ symmetry in the superconductor. Here we focus on 2+1 dimensional spin-less $p$-wave superconductors and show that emergent gravity serves as a natural bulk probe for their topological invariant. This emergent gravity is due to the same attractive interaction between fermions that leads to superconductivity, and is therefore built into topological superconductors. The bulk response of a topological superconductor to the emergent gravitational field is encoded in a gravitational Chern-Simons term, and is related to the existence of robust boundary modes via energy-momentum conservation, or gravitational anomaly inflow. The gravitational Chern-Simons term implies a universal relation between variations in the superconducting order parameter and the energy-momentum currents and densities that they induce. The spontaneous breaking of $U(1)$ symmetry in the superconductor leads to additional bulk responses, encoded in a gravitational pseudo Chern-Simons term. Although not of topological nature, these carry surprising similarities to the topological responses of the gravitational Chern- Simons term. We show how these two types of responses can be disentangled.