Talks

I will give the basics on property(T), and explain the characterization in terms of actions on median spaces. I will discuss CAT(0) cubical complexes as examples of median spaces, and discuss groups acting on those objects as having a strong negation of property(T).
Possible problems for 2nd week:

Read ‘Spectral interpretations of property(T)’ by Yann Ollivier, with a generalization in mind. http://www.yann-ollivier.org/rech/publs/aut_spec_T.pdf
Study the orbits of the action of discrete cocompact subgroup P in SL^(2,R) on a median space (viewing P as a subgroup of the mapping class group of a surface prevents a proper action on a CAT(0) space). This will involve reading ‘Geometries of 3-manifolds’ by Peter Scott https://deepblue.lib.umich.edu/bitstream/handle/2027.42/135276/blms0401....

Pre-requisites: Bridson-Haefliger Part I, Part II Sections 1, 2 (ideally also 6,8,10), Part III H

The planar Hall effect in 3D systems is an effective probe for their Berry curvature, topology, and electronic properties. However, the Berry curvature-induced conventional planar Hall effect is forbidden in 2D systems as the out-of-plane Berry curvature cannot couple to the band velocity of the electrons moving in the 2D plane. Here, we demonstrate a unique 2D planar Hall effect (2DPHE) originating from the hidden planar components of the Berry curvature and orbital magnetic moment in quasi-2D materials. We identify all planar band geometric contributions to 2DPHE and classify their crystalline symmetry restrictions. Using gated bilayer graphene as an example, we show that in addition to capturing the hidden band geometric effects, 2DPHE is also sensitive to the Lifshitz transitions. 

Reference: arXiv 2405.00379 (2024)

In this talk, I will present a detailed experimental study of the particle- hole (PH) symmetry of the the abelian and a putative non-abelian Fractional Quantum Hall (FQH) state about half filling in a multiband system. Specifically, we focus on the lowest Landau level of the monolayer-like band of Bernal stacked trilayer graphene (TLG). In pristine TLG, the excitation energy gaps, Lande g factor, effective mass, and disorder broadening of the FQH states is the same as their hole-
conjugate counterpart. This precise PH symmetry stems from the lattice mirror symmetry that precludes Landau-level mixing. Introducing a non- zero displacement field D disrupts this mirror symmetry, facilitating the interaction and hybridization between the NM = 0 of monolayer-like and NB = 2 bilayer-like Landau levels. This band hybridization eventually leads to a violation of the particle-hole symmetry of the FQH states. We find the one-third and two-fifth FQH states to be more robust against Landau level m...

Transition metal oxides exhibit a rich variety of collective phenomena already in the bulk due to a strong interplay of lattice, charge, spin and orbital degrees of freedom. Oxide interfaces open new possibilities for applications in electronics devices or for energy conversion due to the emergence of novel electronic phases that are not available the bulk constituents. In this talk I will review the insight obtained from density functional theory (DFT) calculations with an on-site Coulomb term how this novel functionality can be steered by a set of control parameters. Several examples will be discussed: (i) the prediction of topological Chern insulating phases and competition with Mott insulating states in oxide superlattices with a honeycomb [1] and dice pattern [2]; the emergence of a spin-polarized two-dimensional electron gas at the EuTiO 3 (001) surface and in LaAlO 3 /EuTiO 3 /SrTiO 3 (001) [3] (ii) the role of the film geometry [4] and the interface structure [5] in infinite-la...