Chiral magnetic and vortical effects in higher dimensions at weak coupling

Chiral magnetic effect and chiral vortical effect are parity odd transport phenomena originating from chiral anomaly, and have generalizations to all even dimensional space-time higher than four dimensions. We attempt to compute the associated P-odd retarded response functions in the weak coupling limit of chiral fermion theory in all even dimensions, using the diagrammatic technique of real-time perturbation theory. We also clarify the necessary Kubo formula relating the computed P-odd retarded correlation functions and the associated anomalous transport coefficients. We speculate on the 8-fold classification of topological phases.

Figure 1

The Schwinger-Keldysh contour appropriate for computing real-time retarded response functions at finite temperature.

Figure 2

The diagrams responsible for the retarded response of the current $J^{\mu }$ to one external gauge potential.

Figure 3

One of the diagrams responsible for the retarded response of the current $J^{\mu }$ to $(m-1)$ number of external gauge potentials. There are $m$ number of such diagrams with different positions of $G_{rr}$ propagator.

Figure 4

Diagrams for $J^{\mu }$ with $n_1$ Type I, $n_2$ Type II, and $(n-1-s)$ current vertices. We have $n_1+2n_2=s$, and have to sum over all possible $n_2$ ranging from 0 to $[s/2]$.

Figure 5

Diagrams for $T_{II}^{0i}$ contribution to the total $T^{0i}$ response function. We have a constraint $n_1+2n_2=(s-1)$, and have to sum over all $n_2$ ranging from 0 to $[(s-1)/2]$.