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Galilean invariant superfluid :

Galilean invariant superfluid :. Magnus force. Lorentz force. Helmholtz ’ theorem. Examples of superfluids with broken Galilean invariance:. Superfluid on lattices: Josephson junction array Bose – Hubbard model. Continuous approximation for a lattice superfluid. Lagrangian:.

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Galilean invariant superfluid :

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  1. Galilean invariant superfluid: Magnus force Lorentz force

  2. Helmholtz’ theorem

  3. Examples of superfluids with broken Galilean invariance: Superfluid on lattices: Josephson junction array Bose – Hubbard model

  4. Continuous approximation for a lattice superfluid Lagrangian: Canonical pair of conjugate variables: Broken Galilean invariance:

  5. Noether’s theorem Gauge invariance (particle number conservation) Translational invariance (?momentum? conservation) Without Galilean invariance it is not the momentum conservation law!

  6. True momentum True momentum conservation law follows (approximately) from the equations of motion:

  7. Bosons in a periodic potential true momentum quasimomentum

  8. Lorentz force is determined from the balance of the Noether momentum (quasimomentum):

  9. Magnus force is determined from the balance of the true momentum:

  10. Two limits: Particle – hole symmetry Ballistic motion of vortices Van der Zant et al., Europhys. Lett. (1992) Connection with the Hall effect:

  11. Bose – Hubbard model Mean-field approach: Parameters of the continuous model:

  12. Superfluid – Mott insulator transition Near a beak between the lobes N and N+1:

  13. A beak between the lobes N and N+1: The amplitude of the Magnus force (Hall conductivity); -1/2 1/2 Changes a sign at the line Lindner, Auerbach, and Arovas (Phys. Rev. B, 2010)

  14. Conclusions Calculation of the Magnus and the Lorentz forces on a vortex in lattice superfluids requires the analysis of two balances, for the true momentum of particles (Magnus force) and for the quasimomentum (Lorentz force) similar to that in the Bloch theory of particles in the periodic potential. This yields the expression for the Magnus force, which agrees with the exact results for the Josephson junction array and for the Galilean invariant liquid. The Magnus force (Hall conductivity) was found for the beaks (extreme areas of the superfluid phase between the Mott – insulator phases) on the phase diagram. There is area where the Magnus force has a sign opposite to that determined by the velocity circulation around the vortex.

  15. Thank you!

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