Topological Field Theory Far from Equilibrium

The observable properties of topological quantum matter are often described by topological field theories. Here, we demonstrate that this principle extends beyond thermal equilibrium. To this end, we construct a model of two-dimensional driven open dynamics with a Chern insulator steady state. Within a Keldysh field theory approach, we show that under mild assumptions—particle number conservation and purity of the stationary state—an abelian Chern-Simons theory describes its response to external perturbations. As a corollary, we predict chiral edge modes stabilized by a dissipative bulk.

Figure 1

Left: Visualizing the action of the jump operators $\widehat{L}$. The operators ${\widehat{L}}_{1,2}$ annihilate particles in the upper band to either recreate them in the lower band or redistribute them in the upper band. Similarly, ${\widehat{L}}_{3,4}$ create particles in the lower band by transfer from the upper band or redistribution from the lower band. The stationary state of this process is a fully occupied lower band. Right: A mean field decoupling $⟨{\psi }_{\alpha }{\psi }_{\alpha }^{†}⟩\to n$ reduces the quartic field polynomials $L^{†}L\to l^{†}l$ to quadratic ones. The presence of the wavy line indicates that the $L$ fields carry a nontrivial gauge representation, upgrading them to current operators in the presence of an external vector potential, cf. Eq. (7).

Figure 2

Structure of the diagrams appearing in the expansion in $A$. Both diagrams are individually ultraviolet divergent in momentum space. The divergences cancel and the remaining contribution is identical to the triangle structure on the right, where the empty dot represents the expansion of the propagators $G$ to first order in the external frequency, $\omega$. This gives a contribution $\sim \omega A_i{A}_j$ which becomes one part of the Chern-Simons action.