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...

Lecture 1: Magnetic monopole

Magnetic monopole

Classical electrodynamics: Maxwell’s equations in vacuum

Symmetric under change

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!]

it holdsExercise 2: a) Check that at

where

Solution: a) Represent

Calculate

it holdsExercise 2: a) Check that at

where

Solution: a) Represent

Calculate

it holdsExercise 2: a) Check that at

where

Solution: a) Represent

Calculate

Use

it holdsExercise 2: a) Check that at

where

Solution: a) Represent

Calculate

Use

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

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 !

Wu-Yang monopole

Check by Stokes theorem:

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)

Summary of this lecture

● Magnetic monopole: richer geometric and topological structure than in

conventional electrodynamics

● Cohomology theory: exact and closed forms

● Topological aspect: global vs local properties

● Wu-Yang monopole and fiber bundles; charge quantization