Interacting
topological phases of matter
Morten Ib Munk-Nielsen
Advisor: Karsten Flensberg
Overview
A brief introduction to the ideas of topological phases.
An even briefer intro to Majoranas
What I am trying to do.
Condensed matter
“More is different”
Philip Anderson
Emergent phenomena:
Properties that cannot be
explained from properties of
individual constituents.
Phase transitions
Thermodynamic process. Example: Magnet.
Symmetry broken state
Symmetric state
Turns out not all phases may be described by symmetry breaking.
“Topological phases”
Topology
http://rioranchomathcamp.com/topology.asp
Topology
Euler characteristic:
“Topological invariant”.
Manifolds without boundary:
Topology
Torus:
Topology in quantum mechanics
Periodic potential (lattice): k-space
Eigenstates:
Brillouin zone (BZ): Set of physically distinct
crystal momenta k.
Chern number:
Berry curvature:
is an integer!
Physics?
Quantum Hall effect: Hall conductivity proportional to chern number.
Topology?
The Chern number is a topological invariant: Hamiltonian may be smoothly
deformed in any way as long as gap is not closed.
Very brief intro to Majorana bound states
Zero-energy states found at interfaces between different topological regions.
They are their own antiparticle:
Cannot make sense to occupancy of a single Majorana, but of two Majoranas
constitute a fermion.
Topologically non-trivial,
superconductor,
Topologically trivial
Experimental progress
Fusion rules
Outcome depends on state of the combined Majoranas.
Aasen et al: Milestones toward Majorana-based quantum computing, April 2016.
Fusion rules
Aasen et al: Milestones toward Majorana-based quantum computing, April 2016.
Interactions?
Existence of Majoranas topologically protected, but the states of the Majoranas
are not.
Electrostatic charging => photon-electron interaction.
The general question I’m trying to answer
What happens when interactions are added to topological phases.
In particular, what happens to the Majoranas under electron-boson interaction.
Concretely
Two Majoranas constitute a fermionic state. This defines a two-level system.
Can define Pauli operators on the system, for instance
Thank you for the attention!