• topology
• topological marker
• Chern insulator
• current dressing
• topological insulators
• Chern number
• quantum mechanics
• quantum electrodynamics
• Haldane model
• Haldane
• topological quantum matter

Topology plays a pivotal role in condensed matter physics and provides an essential framework for understanding the properties of quantum matter.
For example, the topological description of electrons in crystals has led to the explanation of fundamental physical phenomena such as the quantum Hall effect.
In recent years, there has been an intensive research activity focused on the search for novel topological phases of matter.
Besides their fundamental relevance, topological states of matter promise application in a broad range of fields ranging from novel electronic devices to quantum computing.
In this thesis, we focused on the quantum electrodynamics of the Haldane model, a prototypical theoretical model of a topological quantum system.
The Haldane model provides the simplest description of a Chern insulator, i.e. a topological insulator (TI) characterized by the Chern number as a topological marker.
In particular, we investigated finite-size corrections to the quantized Hall conductance.
Moreover, we provided a description of the spatial distribution of the ground state's persistent currents in relation to the topological phase diagram of the model.
This work was originally motivated by the possibility of externally controlling topological properties.
So far, topological effects have mostly been studied in relation to electronic properties.
However, one can think that these properties are in fact entangled to other degrees of freedom, not necessarily of electronic origin.
For example, Chern insulators break time-reversal symmetry (TRS), which means the ground state generally exhibits persistent currents. These currents act as sources of magnetic flux
which in turn can have feedback on the ground state currents.
As a result, one must think of the actual ground state as an entangled ground state which entails the mutual coupling between the ground state currents and the self-generated magnetic flux, which could potentially modify the topological properties, for example by changing the phase diagram.
In this respect, to gain insight into the possible modifications of topological properties, one should analyze the ground state of the coupled system composed of both the electrons and the electromagnetic field.
To achieve this goal, we devised a methodical approach to determine the anomalous Hall state by direct evaluation of the transverse conductance in finite-size systems.
Furthermore, we developed a systematic understanding of the spatial distribution of the ground state currents, both for the infinite periodic case and for a cylindrical configuration with finite width.
Both research directions turned out to be fascinating and rich subjects on their own, becoming the primary focus of the thesis.
This research led to original results related to
(i) finite-size effects on the quantum anomalous Hall effect, and
(ii) the connection between the topological phase diagram and the spatial distribution of ground state currents.
Specifically, starting from the topological phase of the model, we described a phase transition between a zero and a finite Hall conductance for a critical value of the transverse size of the system.
As the size of the system approaches infinity, the Hall conductance tends towards its quantized value following a power regression.
We related this phenomenon to the competition between two characteristic length
scales, namely the localization length of the edge states and the size of the system.
Eventually, we described a nontrivial behavior of the ground state's persistent current distribution as a function of the position in the topological phase diagram. In particular, as expected from broken TRS, the ground state's persistent currents can be non-zero even outside the topological region of the phase diagram.
The study of the net current transfer revealed the regions of the parameters space in which one may expect significant modifications to the topological properties due to the aforementioned
dressing of the ground state by self-generated flux.
These regions lie close to the boundaries of the topological phase, thus suggesting the possibility of observing anomalous Hall effect outside the boundaries of the purely electronic topological nontrivial region.