Quantized Hall Response in the Twisted Graphene Systems with and without Magnetic Fields

Abstract

Quantized Hall response is one of the most well-known manifestations of topological orders in quantum systems. An amenable platform to study the quantized Hall response is the twisted graphene system because of its tunability. The first part of the thesis focuses on the integer Hall response in graphene systems. A nonvanishing Hall response requires a broken time-reversal symmetry (TRS). Here, we focus on two ways of breaking TRS. The first way is by external magnetic fields. For untwisted graphene systems, the Landau level is sufficient to capture the physics, while the situation is much more complicated in the twisted systems because of the interplay between magnetic fields and moiré potential, which causes the Hofstadter butterfly. The common approach is to construct a complete basis using the wave function of the Landau level. However, we construct the basis merely from the commutation relation between the canonical momentum operators and position operators, which relieves us from the cumbersome calculations of the explicit expression of the wave function. Moreover, we also generalize the Streda formula to the gapless case and theoretically calculate 𝜕𝑛/𝜕𝐵 in the noninteracting twisted graphene systems. This quantity is directly related to the experimental measurements of magnetization because 𝜕𝑛/𝜕𝐵 = 𝜕𝑀/𝜕𝜇. The second way is by spontaneous symmetry breaking, and we particularly study the integer quantum anomalous Hall (IQAH) effect recently observed in rhombohedral pentalayer graphene (RPG). In most twisted graphene systems, the IQAH effect is realized by the exchange energy, which lifts the spin-valley degeneracy. However, the Stoner-like mechanism cannot explain the IQAH effect in RPG because the single-particle band structure does not possess an isolated Chern band. We theoretically discuss and analyze the convergence of the Hartree-Fock (HF) calculation with various reference fields. Utilizing the proper reference field, we find that the interaction opens up a correlating gap and leads to an anomalous Hall crystal at filling one.

The second part of the thesis focuses on the fractional Hall response, and we primarily try to theoretically explain the two puzzles of the experimentally observed fractional quantum anomalous Hall (FQAH) effect in RPG. The first puzzle is how the FQAH state forms in RPG. The puzzle is mainly caused by the numerical accessibility of exact diagonalization (ED) because ED can only calculate systems with one or two bands. Even the density matrix renormalization group calculation is limited by the number of bands in the system. In the conventional fractional Chern insulator (FCI) theory, the system is projected to the flat band because the theory assumes a very large single-particle gap, which is absent in the RPG. We resolve these problems by introducing a variational ansatz that regards the quasiparticle bands as the variables, which coherently combines the HF and ED calculations. Utilizing this ansatz, we find an FQAH state with energy lower than the other solutions. The second puzzle is that the FQAH effect at some fractional fillings only exists at finite temperatures and crosses over to the IQAH effect at lower temperatures. The phenomenon is seemingly counterintuitive because the FQAH effect is expected to be stable at low temperatures but unstable against thermal activations. We believe that the crossover may arise from the competition between the energy penalty for thermal excitations and the increase in entropy. We consider a toy model with attractive impurities, and thus, the particles will be trapped by the impurities at low temperatures. If there are too many particles being trapped, there are not enough itinerant particles to form an FCI state. However, thermal activation may allow the particles to escape from the impurities to the extent of forming an FCI state. This mechanism is numerically verified in a toy model that consists of a flat Chern band and impurities.

Date of Award	4 Sept 2025
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Xiao LI (Supervisor)