Topological Condensend Matter Physics

Notes of a lecture series held by Titus Neupert and Sebastian Huber in spring 2021 at UZH and ETHZ.

The fascinating world of topological aspects of condesned matter systems is exposed in a 13 weeks lecture series. The course starts with the introduction of the most celebrated topological phase: the quantum Hall effect discovered in 1980. In the following chapters we develop the theoretical concepts that underpin the field of topological condensed matter physics. All topics are explained with the combination of abstract concepts and tangible illustatrations using the standard toy models. We finish the course by using our acquired knowledge to approach the magic of the fractional quantum Hall phases.

Below you find the chapter titles together with the respective learning goals and the pdf of the lecture notes for each week.

The entire Downloadlecture notes as a single file (PDF, 7.8 MB).

Accompaning exerices are available upon request.

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.

DownloadNotes (PDF, 1.4 MB)

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.

DownloadNotes (PDF, 528 KB)

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”.

DownloadNotes (PDF, 1.3 MB)

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.

DownloadNotes (PDF, 414 KB)

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.

DownloadNotes (PDF, 893 KB)

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.

DownloadNotes (PDF, 885 KB)

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.

DownloadNotes (PDF, 1.4 MB)

The lecture notes of an older version of the course by Sebastian Huber can be found here.
Further lectures notes on the topic by Titus Neupert can be obtained external pagehere.