1. MEASURING AND CHARACTERIZING THE QUANTUM METRIC TENSOR Michael Kolodrubetz, Physics Department, Boston University Equilibration and Thermalization Conference, Stellenbosh, April 17 2013 In collaboration with: Anatoli Polkovnikov (BU) and Vladimir Gritsev (Fribourg) Talk to me about: - Thermalization and dephasing in Kibble-Zurek - Real-time dynamics from non- equilibrium QMC

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. 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. Generalized force

9. MEASURING THE METRIC Generalized force

10. MEASURING THE METRIC Generalized force

11. MEASURING THE METRIC Generalized force

12. MEASURING THE METRIC

17. 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. 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. Transverse field Anisotropy Global z-rotation

32. VISUALIZING THE METRIC Transverse field Anisotropy Global z-rotation

33. VISUALIZING THE METRIC

34. 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 k g 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  -  plane Are these Euler integrals universal? YES! Protected by critical scaling theory

48. GEOMETRIC INVARIANTS  -  plane Are these Euler integrals universal? YES! Protected by critical scaling theory

49. SINGULARITIES OF CURVATURE  -h plane

50. INTEGRABLE SINGULARITIES KhKh h h KhKh

51. CONICAL SINGULARITIES

52. Same scaling dimesions (not multi-critical)

53. CONICAL SINGULARITIES Same scaling dimesions (not multi-critical)

54. CURVATURE SINGULARITIES

55. CONCLUSIONS Measuring the metric tensor  Proportional to integrated noise-noise correlations  Leading order non-adiabatic corrections to generalized force Classification of quantum geometry  Geometry is characterized by set of invariants  Gaussian curvature (K)  Geodesic curvature (k g )  Singularities of XY model are classified as  Integrable  Conical  Curvature  Singularities and integrals are protected