Path-integral derivation of the equations of the anomalous Hall effect

  title={Path-integral derivation of the equations of the anomalous Hall effect},
  author={Kazuo Fujikawa and Koichiro Umetsu},
  journal={Physical Review B},
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 ± 1 − E n | ≫ 2 π ~ /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 → large and fixed energy eigenvalues, no commutation relations are defined for slower variables by the Bjorken-Johnson-Low… 



No anomalous canonical commutators induced by Berry’s phase

Quantal phase factors accompanying adiabatic changes

  • M. Berry
  • Physics
    Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1984
A quantal system in an eigenstate, slowly transported round a circuit C by varying parameters R in its Hamiltonian Ĥ(R), will acquire a geometrical phase factor exp{iγ(C)} in addition to the familiar

Characteristics of Chiral Anomaly in View of Various Applications

In view of the recent applications of chiral anomaly to various fields beyond particle physics, we discuss some basic aspects of chiral anomaly which may help deepen our understanding of chiral


The conventional formulation of the nonadiabatic (Aharonov–Anandan) phase is based on the equivalence class {eiα(t)ψ(t,x)} which is not a symmetry of the Schrodinger equation. This equivalence class

Lensing of Dirac monopole in Berry’s phase

Berry’s phase, which is associated with the slow cyclic motion with a finite period, looks like a Dirac monopole when seen from far away but smoothly changes to a dipole near the level crossing point

Second quantized formulation of geometric phases

The level crossing problem and associated geometric terms are neatly formulated by the second-quantized formulation. This formulation exhibits a hidden local gauge symmetry related to the

Geometric phases and hidden local gauge symmetry

The analysis of geometric phases associated with level crossing is reduced to the familiar diagonalization of the Hamiltonian in the second quantized formulation. A hidden local gauge symmetry, which


By using a second quantized formulation of level crossing, which does not assume adiabatic approximation, a convenient formula for geometric terms including off-diagonal terms is derived. The

The Adiabatic Theorem of Quantum Mechanics

We prove the adiabatic theorem for quantum evolution without the traditional gap condition. We show that the theorem holds essentially in all cases where it can be formulated. In particular, our

Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase

It is shown that the "geometrical phase factor" recently found by Berry in his study of the quantum adiabatic theorem is precisely the holonomy in a Hermitian line bundle since the adiabatic theorem