MEASURING AND CHARACTERIZING THE QUANTUM METRIC TENSOR Michael Kolodrubetz, Physics Department, Boston University Equilibration and Thermalization Conference,

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Presentation transcript:

MEASURING AND CHARACTERIZING THE QUANTUM METRIC TENSOR Michael Kolodrubetz, Physics Department, Boston University Equilibration and Thermalization Conference, Stellenbosh, April 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

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

FUBINI-STUDY METRIC

Berry connection

FUBINI-STUDY METRIC Berry connection Metric tensor

FUBINI-STUDY METRIC Berry connection Metric tensor Berry curvature

MEASURING THE METRIC

Generalized force

MEASURING THE METRIC Generalized force

MEASURING THE METRIC Generalized force

MEASURING THE METRIC Generalized force

MEASURING THE METRIC

For Bloch Hamiltonians, Neupert et al. pointed out relation to current-current noise correlations  [arXiv: ]

MEASURING THE METRIC For Bloch Hamiltonians, Neupert et al. pointed out relation to current-current noise correlations  [arXiv: ] Generalizable to other parameters/non-interacting systems 

MEASURING THE METRIC For Bloch Hamiltonians, Neupert et al. pointed out relation to current-current noise correlations  [arXiv: ] Generalizable to other parameters/non-interacting systems 

MEASURING THE METRIC

REAL TIME

MEASURING THE METRIC REAL TIME IMAG. TIME

MEASURING THE METRIC REAL TIME IMAG. TIME

MEASURING THE METRIC REAL TIME IMAG. TIME

MEASURING THE METRIC Real time extensions:

MEASURING THE METRIC Real time extensions:

MEASURING THE METRIC Real time extensions:

MEASURING THE METRIC Real time extensions:

MEASURING THE METRIC Real time extensions: (related the Loschmidt echo)

VISUALIZING THE METRIC

Transverse field Anisotropy Global z-rotation

VISUALIZING THE METRIC Transverse field Anisotropy Global z-rotation

VISUALIZING THE METRIC

h-  plane

VISUALIZING THE METRIC h-  plane

VISUALIZING THE METRIC h-  plane

VISUALIZING THE METRIC  -  plane

VISUALIZING THE METRIC  -  plane

VISUALIZING THE METRIC No (simple) representative surface in the h-  plane  -  plane

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 additional/curvature/curvature19.html

GEOMETRIC INVARIANTS Gauss-Bonnet theorem:

GEOMETRIC INVARIANTS Gauss-Bonnet theorem:

GEOMETRIC INVARIANTS Gauss-Bonnet theorem:

GEOMETRIC INVARIANTS Gauss-Bonnet theorem: 1 0 1

GEOMETRIC INVARIANTS  -  plane

GEOMETRIC INVARIANTS  -  plane

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

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

SINGULARITIES OF CURVATURE  -h plane

INTEGRABLE SINGULARITIES KhKh h h KhKh

CONICAL SINGULARITIES

Same scaling dimesions (not multi-critical)

CONICAL SINGULARITIES Same scaling dimesions (not multi-critical)

CURVATURE SINGULARITIES

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