Degeneracy-Projected Polarization Formulas for Hall-Type Conductivities

Kubo formulas for Hall, transverse thermoelectric, and thermal Hall conductivities are simplified into on-shell commutators of degeneracy projected polarizations. The new expressions are computationally economical, and apply to general Hamiltonians without a gap restriction. We show that Hall currents in open boundaries are carried by gapless chiral excitations. Extrapolation of finite lattice calculations to the dc-thermodynamic limit is demonstrated for a disordered metal.

Figure 1

Extrapolation of numerical Hall conductivities of the square lattice Hamiltonian, Eq. (18). Disorder averaged ${\sigma }_{xy}$ are plotted versus $ϵ$, for a sequence of linear dimensions $L_x$. Stars mark the values of $\overline{ϵ}$ as defined in Eq. (17). The disorder strength is fixed at $w=3$. The temperature, Fermi energy, and magnetic field are $T=0.3$, ${\epsilon }_F=-1$, and $B=0.025$, respectively. Inset: the dc limit ${\sigma }_{xy}^{\mathrm{dc}}$ (marked by black arrows) for three values of disorder strength.