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 nonequilibrium 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
FUBINI-STUDY METRIC
Berry connection
FUBINI-STUDY METRIC
Berry connection
Metric tensor
FUBINI-STUDY METRIC
Berry connection
Metric tensor
Berry curvature
MEASURING THE METRIC
MEASURING THE METRIC
Generalized force
MEASURING THE METRIC
Generalized force
MEASURING THE METRIC
Generalized force
MEASURING THE METRIC
Generalized force
MEASURING THE METRIC
MEASURING THE METRIC
MEASURING THE METRIC
MEASURING THE METRIC
MEASURING THE METRIC
MEASURING THE METRIC
For Bloch Hamiltonians, Neupert et al. pointed out relation to current-current noise correlations
 [arXiv:1303.4643]
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

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

MEASURING THE METRIC
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
VISUALIZING THE METRIC
Transverse field
Anisotropy
Global z-rotation
VISUALIZING THE METRIC
Transverse field
Anisotropy
Global z-rotation
VISUALIZING THE METRIC
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
http://cis.jhu.edu/education/introPatternTheory/
additional/curvature/curvature19.html
Geodesic curvature kg
http://www.solitaryroad.com/c335.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
Are these Euler
integrals universal?
YES!
Protected by critical
scaling theory
- plane
GEOMETRIC INVARIANTS
Are these Euler
integrals universal?
YES!
Protected by critical
scaling theory
- plane
SINGULARITIES OF CURVATURE
-h plane
INTEGRABLE SINGULARITIES
Kh
Kh
h
h
CONICAL SINGULARITIES
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 (kg)
 Singularities of XY model are classified as
 Integrable
 Conical
 Curvature
 Singularities and integrals are protected