Topological Condensend Matter Physics

Learning goals:
- We know the basic phenomenology of the quantum Hall effect (QHE)
- We know the structure of the lowest Landau level (LLL)
- We understand the role of disorder for the QHE.

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Learning goals:
- We know the pumping argument of Laughlin and the concept of spectral flow.
- We know that there is always a delocalized state in each LL.
- We know that σxy is given by the Chern number.
- We understand why the Chern number is an integer.

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Learning goals:
- We know the Su-Schrieffer-Herger model.
- We understand its bulk-boundary correspondence.
- We can characterize its topology through winding number invariants and Wilson loops.
- We understand the connection between Wilson loops, polarization, and the position operator.

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Learning goals:
- We know Dirac fermions.
- We know what a Chern insulator is.
- We know the BHZ model.
- We can explain the idea of “pair-switching”.

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Learning goals:
- We understand the Bogoliubov-de-Gennes representation of a mean-field superconducting Hamiltonian and its relation to a Majorana fermion representation.
- We know one-dimensional topological superconductors, their topological invariant, boundary modes and topological classification.
- We understand how interactions reduce the topological classification from Z to Z 8 in one-dimensional topological superconductors.

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Learning goals:
- We know the chiral p-wave superconductor in two dimensions and can argue why it has bound states in vortices.
- We understand the non-Abelian nature of vortex bound states.
- We can motivate Kitaev’s 16-fold way classification for 2D superconductors.

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Learning goals:
- We know the three symmetries on which the table of topological insulators is based.
- We know how, in principle, one can build the table.
- We know how to derive the indices for each symmetry group.
- We know how to make use of the table in real life.

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Learning goals:
- We understand how the topological classification of insulators and the bulk-boundary correspondence is enhanced by including crystalline symmetries.
- We know how topological invariants such as mirror-graded winding numbers and the mirror Chern number are defined.
- We have an understanding of higher-order topological insulators.

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Learning goals:
- We understand what we mean by Wannierizablity.?
- We know how to think of a band as a representation of the space group.
- We know how to construct elementary band representations.
- We can use the Bilbao server to analyze bands according to their symmetry properties.

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Learning goals:
- We know Weyl semimetals, their Fermi arc surface states and chiral anomaly.
- We have an overview of other types of symmetry-enforced degeneracies in band structures, including point-like degeneracies of several bands and nodal lines.

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Learning goals:
- We know the physical motivation, Hamiltonian, and phase diagram of Kitaev’s honeycomb model.
- We understand how some phases reduce to the toric code Hamiltonian.
- We know how to rewrite the ground state and low-lying excitations in terms of Majorana degrees of freedom.
- We know the toric code model Hamiltonian and understand its ground state manifold.
- We know the emergent excitations above the ground states, and how to derive their statistics.

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Learning goals:
- We are acquainted with the basic phenomenology of the fractional quantum Hall effect.
- We know the Laughlin wave function.
- We can explain the mutual statistic of Laughlin quasi-particles

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Learning goals:
- We know what a coherent state path integral is.
- We know the concept of a composite fermion.
- We know how to get from composite fermions to a Chern-Simons theory.

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