Low-energy effective theory in the bulk for transport in a topological phase

We construct a low-energy effective action for a two-dimensional nonrelativistic topological (i.e., gapped) phase of matter in a continuum, which completely describes all of its bulk electrical, thermal, and stress-related properties in the limit of low frequencies, long distances, and zero temperature, without assuming either Lorentz or Galilean invariance. This is done by generalizing Luttinger's approach to thermoelectric phenomena, via the introduction of a background vielbein (i.e., gravitational) field and spin connection a la Cartan, in addition to the electromagnetic vector potential, in the action for the microscopic degrees of freedom (the matter fields). Crucially, the geometry of spacetime is allowed to have timelike and spacelike torsion. These background fields make all natural invariances-under U(1) gauge transformations, translations in both space and time, and spatial rotations-appear locally, and corresponding conserved currents and the stress tensor can be obtained, which obey natural continuity equations. On integrating out the matter fields, we derive the most general form of a local bulk induced action to first order in derivatives of the background fields, from which thermodynamic and transport properties can be obtained. We show that the gapped bulk cannot contribute to low-temperature thermoelectric transport other than the ordinary Hall conductivity; the other thermoelectric effects (if they occur) are thus purely edge effects. The coupling to reduced spacelike torsion is found to be absent in minimally coupled models, and, using a generalized Belinfante stress tensor, the stress response to time-dependent vielbeins (i.e., strains) is the Hall viscosity, which is robust against perturbations and related to the spin current, as in earlier work.