Scalling theory of quantum ratchet in low temperature limit

Directionality of responses in noncentrosymmetric materials is a central issue in condensed matter physics. The quantum dynamics of particles in an asymmetric washboard potential with dissipation is a typical system.
Previous study on the 2nd order mobility have shown the rich physics in this system, the sign reversal of the rectified current and quantum to classical crossover, for instance. However, the relations with the quantum phase transition taking place at certain strength of the dissipation above which the system goes to a localized state, is still a nontrivial issue.

Here, we clarified the detailed low temperature behavior of the steady velocity and mobility of a particle under the ratchet potential both analytically and numerically. The exponent in the power law dependence for arbitrary order mobility is determined and summarized in simple scaling forms. The relations to the quantum phase transition is also discussed.