Anyons and Topological Order

This module is devoted to the theory of topological order and anyons. These topics are introduced both at a mathematical level, by explaining homotopy groups, braids, fusion and braiding rules, and at a physical level, by explaining seminal models and physical systems wheretopological order and anyons arise. Important parts of this module are:

• Quantum braiding statistics: beyond bosons and fermions. Path integral definition of quantum    statistics. The configuration space of identical particles. The homotopy group. Classification of closed paths in different three-dimensional and two-dimensional manifolds. The braid group. Properties of braids. Braids and knots. Abelian and non-Abelian anyons. Physical representation of an Abelian anyon as a fractional charge attached to a flux. Aharonov-Bohm effect.

• Topological order vs symmetry breaking order. Landau theory. Symmetry breaking order and

local order parameter. Global order vs local order. Manifestations of topological order:

topological degeneracy, topological entanglement entropy, exotic braiding statistics of

elementary excitations.

• The toric code model as a seminal model for topological order and anyons. The string

language. The ground state as a quantum loop condensate. Topological entanglement entropy of ground state. Topological degeneracy. Wilson loops and string order. Anyonic character of the excitations.

• Fractional quantum Hall systems. Landau levels. Projection onto the lowest Landau level. Hilbert space of polynomials. Many-particle interacting Hamiltonian. Laughlin liquids. Properties of Laughlin liquids. Plasma analogy. Topological order in Laughin liquids. Charge fractionalization. Fractional statistics of quasiparticles. Chern-Simons theory. Composite fermions. Flux-charge composites. 

• Topological insulators and topological superconductors. The SSH (Su-Schrieffer-Heeger) model: a seminal model for a topological insulator. Topological invariants. Topological edge modes. Particle fractionalization in one-dimension. The Kitaev chain: a seminal model for a topological superconductor. Majorana fermions. Non-Abelian braiding statistics of Majorana fermions.

• Topological qubits and topological protection. The toric code model as a topological quantum code. The topological qubit. Topological protection in the ground state subspace.  • The mathematics of anyon models. Anyon models as tensor categories. Fusion rules. Braiding rules. Pentagon and Hexagon equations. Modular matrices. Seminal examples of anyon models: Z_n, SU(2)_k, Ising, Fibonacci, Boson-Lattice construction of anyon models.

• Topological quantum computation

After participation in the Module the student is able to:

1. Understand and apply the concept of quantum brading statistics, configuration space, and homotopy group. Classify closed paths is a given manyfold. Decide whether anyons (beyond bosons and fermions) can exist or not in a given manyfold. 

2. Understand braids and knots. Decompose braids in elementary braids. Decide whether two braids are or not equivalent.

3. Understand the concept of charge-flux composite. Obtain the fractional statistics of a given charge-flux composite.

4. Understand and characterize the concept ot topological order. Understand the difference between symmetry breaking order and topological order. 

5. Solve the toric code model, obtaining the ground states and excitations. Apply the string language to characterize the eigenstates of the model. Understand the concept of topological degeneracy and topological entanglement entropy. Calculate and characterize the ground state subspace on a torus. Obtain the braiding and fusion properties of the anyonic excitations.

6. Solve the problem of a particle in a magnetic field without specifing a gauge. Characterize many-body wave functions in the lowest Landau level. Project the Coulomb interaction onto the Lowest Landau Level. Obtain the Laughlin liquids as exact eigenstates of a truncated Coulomb interaction. Calculate the density of the Laughlin liquid by using the plasma analogy. Obtain the wave functions of an elementary excitation. Prove charge fractionalization and anyonic character of the excitations. 

7. Solve the SSH model. Understand and apply the concept of Zak phase as a topological invariant. Obtain the topological protected edge modes. Solve the Kitaev chain. Obtain the Majorana modes. Calculate their non-Abelian braiding statistics.

8. Understand the concept of topological qubit and topological protection. Use the toric code model as a topological quantum memory.

9. Understand the concepts of fusion rules, braiding rules, pentagon and hexagon equations. Characterize seminal anyon models, such as Z_n, Ising, Fibonacci, SU(2)_k.

10. Understand the concept of topological quantum computation. Generate simple quantum gates with braids.

No prerequisites beyond the requirements for the Master’s program in Quantum Science and Technology.

The module consists of a lecture series (4 SWS) and exercise classes (2 SWS), comprising two lecture sessions and one exercise session per week.

Lectures: My teaching methods are the result of years of exploring educational means and interacting with students. Important points are:

Transmit my own thrill and motivation, Connect with students background and interests, Give the general picture of the subject, Set specific goals for the understanding of theoretical concepts, Explain important concepts in a clear and simple way using my drawings, Make sure concepts are understood.

Create an interactive environment: I succeed in creating a dynamical and participative atmosphere during the lectures by combining my explanations with questions that help students reflect on what they are learning. After a few lectures, students actively participate, posing new questions that influence the following lectures, thus becoming active part of the module.

I present the material on the blackboard, combining it with keynote presentations in which I show the students drawings and animations to highlight important concepts.

Exercises: During the exercise classes I assign exercises that make students progress towards the modules goals, giving them feedback about their advances by, for instance, letting them explain their results on the blackboard and discussing the solutions with them.

Presentations: In addition, during the exercise classes a round of presentations is made in which each student presents a result from a more advanced theme related to the module.

Participation in the exercise classes is strongly recommended, since the exercises are aids for acquiring a deeper understanding of the core concepts of the course and for practicing to solve typical exam problems.

Power point and One Note presentation.

Standard textbooks, for example:

• Quantum field theory of many-body systems: from the origin of sound to an origin of light, Xiao-Gang Wen, Oxford University Press, 2004.

• Fractional statistics and quantum theory, Avinash Kahre, World Scientific, 2005

• Topological quantum computation, Lecture notes by John Preskill, 2004

• Anyons in an exactly solvable model and beyond, Alexei Kitaev, Annals of Physics 321 (2006)

• Non-Abelian anyons and topological quantum computation, Chetan Nayak et al. Rev. Mod. Phys. 80, 1083 (2008)

• The quantum Hall effects: fractional and integral, T. Chakrabortym P. Pietiläinen, Springer series in Solid State Sciences (1995)

• The quantum Hall effect: novel excitations and broken symmetries, S. M. Girvin, arXiv:cond-mat/9907002

There will be a written exam of 180 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using conceptual questions and computational tasks.

For example an assignment in the exam might be:

• Anyons and braiding statistics. Discuss and prove whether anyons can or not exist in different non-trivial manyfolds (the surfase of a cylinder, the surface of sphere, the interiour of a torus)
• Toric code model. Calculate properties of ground states and excitations. Obtain certain unitary gates in the topological protected ground state subspace.
• Fracional quantum Hall systems. Obtain bosonic Laughlin liquids as exact eigenstates of a contact interaction projected to the Lowest Landau level. Obtain Landau levels for a neutral particle in a harmonic rotating trap.
• Topological insulators and superconductors. Obtain properties of edge states in the SSH model. Obtain braiding properties of Majorana fermions in the Kitaev chain.
• Charge-flux composites. Derive the fractional statistics of a given charge-flux composite.