Measuring and characterizing the quantum metric tensor

1. Talk to me about: - Thermalization and dephasing in Kibble-Zurek - Real-time dynamics from non- equilibrium QMC Measuring and characterizing the quantummetric tensor Michael Kolodrubetz, Physics Department, Boston University Equilibration and Thermalization Conference, Stellenbosh, April 17 2013 In collaboration with:AnatoliPolkovnikov (BU) and Vladimir Gritsev (Fribourg)

2. Outline • Definition of the metric tensor • Measuring the metric tensor • Noise-noise correlations • Corrections to adiabaticity • Classification of quantum geometry • XY model in a transverse field • Geometric invariants • Euler integrals • Gaussian curvature • Classification of singularities • Conclusions

3. Fubini-study metric

4. Fubini-study metric Berry connection

5. Fubini-study metric Berry connection Metric tensor

6. Fubini-study metric Berry connection Metric tensor Berry curvature

7. Measuring the metric

8. Measuring the metric Generalized force

9. Measuring the metric Generalized force

10. Measuring the metric Generalized force

11. Measuring the metric Generalized force

12. Measuring the metric

13. Measuring the metric

14. Measuring the metric

15. Measuring the metric

16. Measuring the metric

17. Measuring the metric • For Bloch Hamiltonians, Neupert et al. pointed out relation to current-current noise correlations • [arXiv:1303.4643]

18. Measuring the metric • For Bloch Hamiltonians, Neupert et al. pointed out relation to current-current noise correlations • [arXiv:1303.4643] • Generalizable to other parameters/non-interacting systems

19. Measuring the metric • For Bloch Hamiltonians, Neupert et al. pointed out relation to current-current noise correlations • [arXiv:1303.4643] • Generalizable to other parameters/non-interacting systems

20. Measuring the metric

21. Measuring the metric REAL TIME

22. Measuring the metric REAL TIME IMAG. TIME

23. Measuring the metric REAL TIME IMAG. TIME

24. Measuring the metric REAL TIME IMAG. TIME

25. Measuring the metric • Real time extensions:

26. Measuring the metric • Real time extensions:

27. Measuring the metric • Real time extensions:

28. Measuring the metric • Real time extensions:

29. Measuring the metric • Real time extensions: (related the Loschmidt echo)

30. Visualizing the metric

31. Visualizing the metric Transverse field Anisotropy Global z-rotation

32. Visualizing the metric Transverse field Anisotropy Global z-rotation

33. Visualizing the metric

34. Visualizing the metric h- plane

35. Visualizing the metric h- plane

36. Visualizing the metric h- plane

37. Visualizing the metric - plane

38. Visualizing the metric - plane

39. Visualizing the metric No (simple) representative surface in the h- plane - plane

40. Geometric invariants • Geometric invariants do not change under reparameterization • Metric is not a geometric invariant • Shape/topology is a geometric invariant • Gaussian curvature K • Geodesic curvature kg http://cis.jhu.edu/education/introPatternTheory/additional/curvature/curvature19.html http://www.solitaryroad.com/c335.html

41. Geometric invariants Gauss-Bonnet theorem:

42. Geometric invariants Gauss-Bonnet theorem:

43. Geometric invariants Gauss-Bonnet theorem:

44. Geometric invariants Gauss-Bonnet theorem: 1 0 1

45. Geometric invariants - plane

46. Geometric invariants - plane

47. Geometric invariants Are these Eulerintegrals universal? YES!Protected by criticalscaling theory - plane

48. Geometric invariants Are these Eulerintegrals universal? YES!Protected by criticalscaling theory - plane

49. Singularities of curvature -h plane

50. Integrable singularities Kh Kh h h