Current Operator and Particle Pumping

Part of the Lecture Notes in Physics book series (LNP, volume 919)


In the previous chapter, we described quantized adiabatic pumping of particles in a one-dimensional lattice in an intuitive and visual fashion, using the concepts of the control-freak pumping cycle and the time evolution of the Wannier centers. Here, we provide a more formal description of the same effect. Based on Ehrenfest’s theorem, we identify the current operator describing the flow of probability density through a cross section of the one-dimensional lattice, and find that the momentum- and time-resolved current in a given filled band of the lattice is proportional to the Berry curvature associated to that band. Naturally, this leads to the same conclusion as we have seen before: that the number of particles adiabatically pumped through a cross section of the crystal is given by the Chern number of the corresponding filled band, and therefore it is an integer.


Current Operator Berry Phase Adiabatic Limit Chern Number Arbitrary Cross Section 
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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Wigner Research Centre for PhysicsHungarian Academy of SciencesBudapestHungary
  2. 2.Department of Physics of Complex SystemsEötvös Loránd UniversityBudapestHungary
  3. 3.Department of Materials PhysicsEötvös Loránd UniversityBudapestHungary
  4. 4.Department of PhysicsBudapest University of Technology and EconomicsBudapestHungary

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