Numerical studies of (fractional) quantum anomalous Hall systems on lattice models

Abstract

The quantum Hall effect is one of the most celebrated phenomena in condensed matter physics and has evolved into a large field with continuously devoted theoretic and exper- imental efforts. Despite its fast development at the material level, there still exist many unresolved problems even at the microscopic model level, that keep attracting broad interest. To solve and understand these problems, the numerical quantum many-body simulations stand out among the most powerful tools in the model studies of especially fractional quantum Hall effect when many-body interactions play important roles. In this thesis, I will present some numerical works related to the quantum Hall physics on lattice models, starting with some general introductions in Chapter 1 and then moving on to the results of the integer cases. One of the earliest and most famous interaction-driven quantum anomalous Hall (QAH) state is from the quadratic-band- touching (QBT) semimetal on checkerboard lattice, but the competition between QAH states and other symmetry-breaking states remains unclear for long until our clarifica- tion. In Chapter 2, I will show the updated quantum phase diagram for this QBT model, and it will be shown that the QAH state can also be driven by interactions from Dirac semimetals. Then I will move to the fractional case. In Chapter 3, I will first introduce the thermodynamic study of the symmetric fractional QAH (FQAH) states on topolog- ical flat-band models, where we have revealed the low-energy collective excitations of such states and found that it is the neutral gap instead of the charge gap that determines the onset temperature of incompressibility in such states. Further, I will introduce our discovery of a new and generic type of FQAH state that coexists with a smectic charge order with spontaneously translation symmetry breaking, of which both ground-state and thermodynamic properties are studied. In Chapter 4, I will introduce some inter- esting quantum phase transitions out of FQAH states. The first one is a numerical demonstration of a long-sought continuous topological transition from a FQAH state to a superfluid. The rest of the transitions in this chapter are related to the softening of collective modes in FQAH states, which could lead to various exotic cases apart from the trivial charge density waves, including the aforementioned FQAH+smectic state, and a gapless as well as charge-ordered but non-superfluid bosonic state, as an intermediate phase between an FQAH state and a supersolid.