A path integral derivation of the equations of anomalous Hall effect

Abstract

A path integral (Lagrangian formalism) is used to derive the effective equations of motion of the anomalous Hall effect with Berry's phase on the basis of the adiabatic condition $|E_{n\pm 1}-E_n|\gg 2\pi \hslash /T$, where $T$ is the typical time scale of the slower system and $E_n$ is the energy level of the fast system. In the conventional definition of the adiabatic condition with $T\to \mathrm{large}$ and fixed energy eigenvalues, no commutation relations are defined for slower variables by the Bjorken-Johnson-Low prescription except for the starting canonical commutators. On the other hand, in a singular limit $|E_{n\pm 1}-E_n|\to \infty$ with specific $E_n$ kept fixed for which any motions of the slower variables $X_k$ can be treated to be adiabatic, the non-canonical dynamical system with deformed commutators and the Nernst effect appear. In the Born-Oppenheimer approximation based on the canonical commutation relations, the equations of motion of the anomalous Hall effect is obtained if one uses an auxiliary variable $X_k^{(n)}=X_k+{\mathcal{A}}_k^{(n)}$ with Berry's connection ${\mathcal{A}}_k^{(n)}$ in the absence of the electromagnetic vector potential $e{A}_k(X)$ and thus without the Nernst effect. It is shown that the gauge symmetries associated with Berry's connection and the electromagnetic vector potential $e{A}_k(X)$ are incompatible in the canonical Hamiltonian formalism. The appearance of the non-canonical dynamical system with the Nernst effect is a consequence of the deformation of the quantum principle to incorporate the two incompatible gauge symmetries.

arXiv comment29 pages. The manuscript including Abstract has been substantially reorganized. This modified version is to appear in Phys. Rev. B

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Cited by in corpus (2)

• Berry's phase and chiral anomalies
• Geometric transport signatures of strained multi-Weyl semimetals