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Geometric aspects
of closed and open quantum systems
Mikhail Pletyukhov
RWTH Aachen University
SPTCM School, 28-29.03.2019
Outline
Lecture 1: Magnetic monopole
– First touch on geometry and topology in field theory
– Nice illustration for cohomology and fiber bundle theories
– Topological charge quantization
Lecture 2: Berry phase
– Geometric description of adiabatic evolution
– Spin in magnetic field: Monopole in parameter space
– Berry-Zak phase in band structure of solids
Outline
Lecture 3: Berry connection and topological invariants
– From geometry to topology: (symmetry protected) topological phases of solids
– Topological invariants: winding number, Chern number
– Edge states and bulk-boundary correspondence
Lecture 4: Geometry of open quantum systems
– Lindblad master equation
– Sarandy-Lidar connection for non-Hermitian operators
– Geometric description of metastability and hysteresis
Outline
Lecture 5: Classical and quantum Fisher information
– Metric in spaces of probability distributions and density matrices
– Classical and quantum (Fisher-Rao) bounds in estimation theory
– Quantum metric approach to quantum phase transitions
if time permits...
REASON: No magnetic charges in Nature (at least none has been found)
Symmetry is lost, if we add electric charge and current distributions
Consequence: B-field is divergence-free, all B-field lines are closed
Contrast between E- and B-fields: electric charge is a source and a sink of
E-field lines
Poisson equation
Integral form (Gauss law)
the field
satisfies
b) Apply integral form to justify the occurrence of -function in rhs
Exercise 1: a) Verify that at
the field
satisfies
b) Apply integral form to justify the occurrence of -function in rhs
Solution: a)
Using leads to
Exercise 1: a) Verify that at
Exercise 1: a) Verify that at the field
satisfies
b) Apply integral form to justify the occurrence of -function in rhs
Solution: b) Integrate over imaginary surface of small radius
We obtain on both sides
Exercise 1: a) Verify that at the field
satisfies
b) Apply integral form to justify the occurrence of -function in rhs
Solution: b) Integrate over imaginary surface of small radius
We obtain on both sides
Point-like charge makes
impossible a contraction
of a sphere to a point
P. Dirac (1931): What if a magnetic charge existed?
Modify the Maxwell’s equation
By analogy with electric field
Magnetic flux through a closed surface enclosing the magnetic charge
Vector potential formulation: what changes?
In conventional electrodynamics, both fields
are expressed in terms of scalar and vector potentials globally
In presence of , a global representation
is no longer possible:
Dirac’s problem: find a local representation
Topological aspect: distinguishing between global and local equivalence
Short excursion into the cohomology theory (subject of differential topology)
– differentiable manifold
– tangent space is a linear space
– differential n-forms (rank-n antisymmetric tensors)
Examples: 0-form is scalar
1-form is analogue of a vector (covector)
2-form is antisymmetric tensor
Exterior product
Exterior derivative(generalization of )
Short excursion into the cohomology theory (subject of differential topology)
For each there is a (pseudo-)vector
such that
Then is an expression for
By construction analogue of
Short excursion into the cohomology theory (subject of differential topology)
Important objects: Exact and closed n-forms
Exact n-form :
Closed n-form :
(n-1)-form A, such that
it satisfies
Exact closed
Converse is NOT always true
Short excursion into the cohomology theory (subject of differential topology)
Important objects: Exact and closed n-forms
Exact n-form :
Closed n-form :
(n-1)-form A, such that
it satisfies
Exact closed
Converse is NOT always true
Cohomology theory:
classifies closed forms, which are not exact
Our attempt (following Dirac):
for a non-closed form
find a form A such that locally
(globally this is impossible)
Dirac proposed a solution:
Ill-defined along the ray (string) [in spherical coordinates : at South Pole]Ill-defined along the ray (string)
Well-defined at North Pole (explains superscript)
Written as 1-form
Dirac monopole
vector potential
it holdsExercise 2: a) Check that at
where
b) Find B-field inside the Dirac string
[it must be a singular contribution!]
Exercise 2: b) Find B-field inside the Dirac string
Solution: it must be a singular contribution
Total field
Stokes theorem:spherical cap
LHS:
RHS:
Out-flux of point-like monopole = in-flux through string
Is the solution unique? What is special in the ray ?
This solution is NOT unique.
Dirac string can be directed along ANY ray (point on a sphere).
Choose . New solution by inversion
Ill-defined along the ray (string)
[in spherical coordinates : at North Pole]
Ill-defined along the ray (string)
Well-defined at South Pole (explains superscript)
it holdsExercise 3: a) Check that at
where
b) Find including singular string contribution
Solution: a) It follows from
b)
and
Out-flux of point-like monopole = in-flux through string
Wu-Yang monopole
Wu, Yang (1975): description of Dirac monopole in terms of
fiber bundle theory (subject of differential topology)
– several coordinate charts (patches)
– locally defined connections (vector potentials)
– transition functions to relate different charts in overlap regions
Two charts: and
In overlap (e.g. on equator): 1) both and are well-defined;
2) generate the same
Hence, they must be related by a gauge (gradient) transformation!
Wu-Yang monopole
For
Wu-Yang description: potential creates field
Southernhemisphere
Northernhemisphere
Flux through whole sphere equals !
Charge quantization
Consider (quantum) electron in field of (classical) magnetic monopole
Perform gauge transformation :
On equator:single-valuedness
(QM postulate!)
henceConsequences:
– all electric charges are quantized
– g must be large (since e is small)