Experimental demonstrations of tunable correlation effects in magic-angle twisted bilayer graphene have put two-dimensional moiré quantum materials at the forefront of condensed-matter research. Bilayers of transition metal dichalcogenides (TMDs) have further enriched the opportunities for analysis and utilization of correlations in such systems. Recently, within the latter material class, the relevance of many-body interactions with an extended range has been demonstrated. Interestingly, the interaction, its range, and the filling can be tuned experimentally by twist angle, substrate engineering and gating. Moiré hetero-bilayer TMDs can be accurately modelled by an effective extended Hubbard model on the triangular superlattice, which defines a starting point for quantum many-body approaches. In my tutorial session, I will review how to accurately model hetero-bilayer TMDs with an extended Hubbard model on the triangular lattice and display some recent experimental results. Then, I will introduce the functional renormalization group for correlated fermion systems as a suitable quantum many-body method and discuss the Fermi surface instabilities and resulting correlated phases of hetero-bilayer TMDs. The results from this approach suggest that hetero-bilayer TMDs are unique platforms to realize topological superconductivity with high winding number which reflects in pronounced experimental signatures such as enhanced quantum Hall features. I will also show how the functional renormalization group can be further developed to be suitable many-body method for the exploration of a wide class of correlated moiré materials.
In the 1st lecture, I will focus on one spatial dimension (1d), to introduce a few physical arguments for Lieb-Schultz-Mattis (LSM) theorem and its generalizations, where certain systems forbid a gapped symmetric ground state. I will then introduce matrix product states (MPS) and the group cohomology classification of one-dimensional symmetry protected topological (SPT) phases, which naturally leads to a proof of the LSM theorems in 1d.
In the 2nd lecture, I will introduce SPT phases in 2d and their classification. After presenting a few examples of the LSM theorems for SPT phases in 1d and 2d, I will discuss the general construction of these theorems.
In the 3rd lecture, I will go beyond SPT phases to discuss symmetry enriched topological (SET) phases in 2d and their classifications. Using toric code as an example, I will discuss possible experimental signatures that can sharply distinguish SET phases.