Zhi Han
Definition of topological space: 20th century
First problem: 17th century
Point set topology is a disease from which the human race will soon recover.
Henri Poincaré
Is topology fundamental? Absolutely.
Is topology useful?
"for theoretical discoveries of topological phase transitions and topological phases of matter"
Grid of electrons
Quantum Hall effects: Current on the boundary of quantum matter; No current on the inside
When the physics is different on the bulk vs. the edge.
(What in the world is topological quantum matter? - Fan Zhang)
(Griffths Introduction to QM)
\(\Phi\): Magnetic flux
The accumulated phase does not depend on the path taken. Geometric phase:
(Griffths Introduction to QM)
Why?
Winding numbers of a loop can be defined as the number of times the loop goes around a "hole" counterclockwise.
Integers: \(\mathbb{Z}\)
A curve has winding number zero if we can contract it to a point.
Fundamental group
Topology looks at invariant properties of objects under continuous transformations.
The converse is true: objects that share the same topological invariants are the same topologically.
(Wikipedia)
Outside solenoid:
Plot of \(\mathbf{A}\): magnetic vector potential in a solenoid
Topological reason: \(\mathbf{A}\) has a hole.
de Rham cohomology, which (roughly speaking) measures precisely the extent to which the fundamental theorem of calculus fails in higher dimensions and on general manifolds. (Terence Tao)
this failure is related to the possible existence of "holes" in the manifold. (Wikipedia)
Residue theorem.
"\(f\) is \(2\pi i \times \) winding number \(\times\) number of holes \(\times\) residue."
Rewrite as quadratic form
Vector of creation and annhilation operators
Bogoliubov-de Gennes
Hamiltonian. \(H_{BdG}\)
Topological properties?
Particle hole symmetry
The Kitaev chain is just \(n\) electrons in a row:
Let's saw an electron in half!
No unpaired Majoranas, and each Majorana pair has energy \(\mu/2\) (ansatz):
Or like this, each Majorana pair has an energy \(t\):
Rewrite as fermions:
Rewrite as fermions: Obtain Hamiltonian of a superconductor.
\(\mu\) = chemical potential
\(t\) = kinetic energy
\(\Delta\) = superconducting band gap/Cooper pairs
Topological
phase transition at \(\mu = 2t\)
Topological \( |\mu|<2t\)
Trivial \( |\mu|\geq2t\)
Topological
phase transition at \(\mu = 2t\)
Topological \( |\mu|<2t\)
Trivial \( |\mu|\geq2t\)
The Majoranas are a topological property of the system. If one electron falls off the chain, a new Majorana will appear.
Store quantum information in the Majorana?
Main idea behind topological quantum computing.
We can do quantum gates by braiding the Majoranas.
Great quantum memory!
You and your buddy were made in a pair,
Then wandered around, braiding here, braiding there.
You'll fuse back together when braiding is through
Well bid you adieu as you vanish from view.
Alexei exhibits a knack for persuading
That someday we'll crunch quantum data by braiding, With quantum states hidden where no one can see, Protected from damage through topology.
Anyon, anyon, where do you roam?
Braid for a while ... before you go home.
John Preskill
Classification of Topological Superconductors
Bulk-edge correspondence
Topological Phases
Holes
Topological quantum computation
[1]Topology in Condensed Matter Course (Main reference)
[2]M. Leijnse and K. Flensberg, Introduction to Topological Superconductivity and Majorana Fermions, Semicond. Sci. Technol. 27, 124003 (2012).
[3]C. W. J. Beenakker, Search for Majorana Fermions in Superconductors, Annu. Rev. Condens. Matter Phys. 4, 113 (2013).
[4]B. Field and T. Simula, Introduction to Topological Quantum Computation with Non-Abelian Anyons, Quantum Sci. Technol. 3, 045004 (2018).