Topological phases are phases of matter that are characterized by discrete quantities known as topological invariants. This thesis explores two such phases, the static Chern insulator phase (symmetry class A) in two dimensions and the dynamical Floquet chiral phase (symmetry class AIII) in one dimension.

A Chern insulator is a gapped single particle system on a lattice with a non-zero first Chern number for some of its energy-momentum bands. We consider subjecting a Chern insulator to a non-uniform external electric field. The response is band geometric which means it is robust to deformations of the energy bands which do not cause band touchings. We find a connection between this response and previous work on band geometric quantities.

A Floquet insulator is a unitary time evolved system defined by a time periodic Hamiltonian. We first describe an existing model of a 1D chain with 2D onsite Hilbert space and chiral symmetry. Then we introduce a disordered model of the system and look at how its topological properties are robust to the disorder. We find a power law scaling of $\nu = 2$ for the localization-delocalization transition of the eigenstates of the unitary operator as the drive evolves towards the midway point of its evolution.

Integer and fractional quantum Hall phases are prototypical examples of topologically ordered phases of matter. A standard setting for studying these phases is in continuum two-dimensional electron fluids with broken time-reversal symmetry, where the single particle bands are familiar Landau levels. In this thesis, we explore aspects of quantum Hall fluids in the presence of periodic lattice potential.

We focus mostly on the perturbative, Harper-Hofstadter regime of low magnetic flux density, where the single-particle bands are effectively mixtures of Landau levels. We study the stability of strongly-correlated fractional quantum Hall fluids in relation to the geometry of single-particle Chern bands in tight-binding models. We first treat the case of the Hofstadter model and show the convergence of the geometry of Hofstadter bands to that of Landau levels in the limit of small flux density. We then introduce and study a new series of tight-binding models which do not necessarily converge to Landau levels but still host fractional phases. In each case, we observe correlations between band geometry and many-body gaps consistent with the “geometric stability hypothesis.”

Finally, we give a perturbative calculation of the current response of Chern bands to inhomogeneous electric fields. In continuum quantum Hall fluids with galilean symmetry, the O(q2) contribution to this response is related to the Hall viscosity, which measures the linear response to variations in spatial geometry in time-reversal odd two-dimensional fluids.

This dissertation focus on the classification of topological matters in static and periodically driven systems.

First, we build a complete topological classification of local unitary operators in free fermionic systems. This result can be encoded in a periodic table with a period of eight, similar to the periodic table of topological insulators. In our classification, we define two local unitaries in a certain symmetry class to be topologically equivalent if they can be connected via finite time evolutions (locally generated unitaries) without breaking the symmetries of the symmetry class while maintaining locality. In this way, the classification we derive will allow us to distinguish locally generated unitaries from those that cannot be locally generated.Besides, we also how to find possible generating Hamiltonians for locally generated unitaries. These results can be used to study the edge behaviors of Floquet systems.

Second, we propose bulk invariants and edge invariants that are locally computable and improve existing topological invariants by being applicable to systems with the disorder. Afterward, we set up a rigorous connection between bulk and edge invariants for Floquet systems belonging to Class AIII and class AII of the Altland-Zirnbauer symmetry classification.

Last, after treating a static system as a Floquet system generated by a constant Hamiltonian, we employ the tools we developed in Floquet systems to classify the edge behaviors in static systems.

Grid-connected battery energy storage systems (BESS) are a promising technology to decrease peak load. In this paper, a linear programming (LP) optimization model of a BESS that minimizes demand charge in real-time is developed. The model is employed to study the effect of two temporal resolutions of load profiles, 15~min and 1~hour, on peak load shaving and optimal sizing of BESS. In contrast to previous studies with perfect forecast, realistic forecast errors are prescribed; a 24~hour persistence forecasts and a recurrent neural network (RNN) model are used for short-term load forecasts. The LP routine was coupled with model predictive control (MPC) to obtain optimal BESS dispatch schedules and to reduce the deviations due to load forecast errors in real time. The performance of the proposed algorithm is simulated for one month on a grid-connected building. Compared to perfect load forecasts, the RNN forecast error causes a monthly average of 1.3~kW higher optimized net load, and corresponding \$27 higher optimal demand charge (ODC). Correcting persistence forecast with more accurate RNN forecast yield better peak shaving performance with a monthly average of 28.1% reduction in optimized net load. The temporal load resolution has a significant influence on the ODC. The difference of ODC for two temporal resolutions depend strongly on the load profile characteristics. When the battery capacity is above a certain power and energy threshold limit, the 1~hour load profile is found to be sufficient for determining ODC. However, when the power and/or energy is constrained, the difference in the ODC between 15 min and 1 hour resolution ranges from $14.7 to $123.1 for 28 days.

The topological properties of electronic band structures are closely related to the degree of localization possible for the associated wavefunctions. In this dissertation, we investigate certain aspects of this interplay between topology and localization in the context of static as well as driven (Floquet) topological insulators.

The first part of this dissertation is motivated by Landau levels, the energy levels of electrons in a two dimensional plane that are subject to a perpendicular magnetic field. Landau levels form a key element of theoretical models of the quantum Hall effect, which inspired the study of topological insulators. Each Landau level is highly degenerate or flat, and is topologically non-trivial. Motivated by Landau levels, we study the topological properties of tight-binding Hamiltonians which only have flat energy levels. We find that the spectral projectors of such Hamiltonians are strictly local. In chapters 2 and 3, we show that in one dimension, compact Wannier functions (and their analogs in the absence of lattice translational invariance) can be constructed if and only if the subspace they span is described by a strictly local projector. Using this insight, in Chapter 4, we present and prove a no-go theorem which says that if a strictly local tight-binding Hamiltonian in two dimensions only has flat bands, then each of the bands must have a Chern number of zero. All results are proven without the requirement of lattice translational invariance. The role of an inequality relating the number of energies of the Hamiltonian and the system size is also clarified.

In the second part of this dissertation (Chapter 5), we present some results concerning a delocalization transition that arises in a certain class of Floquet topological insulators.Specifically, we study chiral Floquet topological insulators in one dimension, and show that the localization lengths of eigenstates of the time evolution operator diverge with a universal exponent of two as the time approachs a special point in drive.

This dissertation comprises two parts, each of which details an investigation within topological matter.

In the first part, we investigate the real space localization properties of static topological insulators in two dimensions. We develop three numerical procedures each of which removes an extensive fraction of the local degrees of freedom. We probe the real space properties of the residual projection operator, showing that the maximally localized length scale ξ supported in the residual Hilbert space diverges as the fraction of states remaining approaches 0. The power-law exponent ν = 1/2 characterizing the singular behavior of ξ appears universal across class A insulators with non-zero Chern number.

In the second part, we explore a class of Floquet topological phases in three dimensions. These phases have translational invariance but no other symmetry, and exhibit anomalous transport at a boundary surface. We show that the boundary behavior of such phases falls into equivalence classes up to local 2D unitary evolution. This provides a classification of the 3D bulk, which we conjecture to be complete. We demonstrate that such phases may be generated by exactly solvable “exchange drives” in the bulk. The edge behavior of a general exchange drive in two or three dimensions is shown to derive from the geometric properties of its action in the bulk, a form of bulk-boundary correspondence.