Blagoje  Oblak

We consider a time-dependent small quantum system weakly coupled to an environment, whose effective dynamics we address by means of a Lindblad equation. We assume the Hamiltonian part of the Lindbladian is slowly varying in time and the dissipator part has small amplitude. With an intent to assess the validity of adiabatic quantum control in presence of dissipation, we study the properties of the evolved state of the small system as the adiabatic parameter and coupling constant both go to zero, in various asymptotic regimes.

This talk aims to provide an introduction to the notion of Generalized Landau Levels (GLLs) from a geometric point of view. GLLs are Bloch bands that generalize the standard notion of Landau levels of a charged particle in a uniform magnetic field. Beginning with a foundational introduction to quantum geometry—the differential geometry of families of quantum states—we delve into the specific case of Bloch bands, unraveling the inequalities that emerge relating the quantum metric and the Berry curvature, the saturation of which implies holomorphicity and gives rise to the concept of Kähler band. A Kähler band can then be understood as a regular holomorphic curve in complex projective space. The geometry of holomorphic curves shares many properties with that of real curves in Euclidean space. In particular, there is a distinguished moving frame along the curve, the Frenet-Serret frame (unique up to a global phase), whose elements are the GLLs. The frame satisfies the so-called Frenet-Serret equations which, together with the Maurer-Cartan structure equation, allow us not only to derive the quantum geometry of each GLL but also to establish geometric recursion relations among them. The content of these recursion relations is a manifestation of Calabi’s rigidity theorem for Kähler immersions into projective space that, in this language, not only establishes the uniqueness, up to a momentum-independent unitary transformation, of a Kähler band with a given Berry curvature profile, but also completely determines the quantum geometry of the GLLs. As a natural consequence, the quantum volume of the quantum metric of the nth GLL is exactly quantized to 2n+1. The discussion finds direct applications to moiré materials, where the 0th GLL, the Kähler band, and the 1st GLL are bands which can stabilize fractional Abelian and non-Abelian, respectively, fractional Chern insulating phases.

The behavior of electrons in a metal presents a wide variety of emergent behavior including a number of phase transitions. The mean-field scaling limit acts as a simplified model capturing part of this complexity. In this limit, results going beyond the precision of Hartree-Fock theory have recently been obtained by approximate bosonization methods as a mathematically rigorous version of the random phase approximation. I will review the expansion of the ground state energy and present results extending to the dynamics and momentum distribution of excitations.

The bulk-edge correspondence is a cornerstone theorem in condensed matter physics, linking a topological index of an infinitely extended system (the bulk) to the number of chiral edge modes at the boundary of a half-infinite sample (the edge). Meanwhile, the massive Dirac Hamiltonian serves as a quintessential model for topological insulators, widely employed as a low-energy effective theory. Surprisingly, this correspondence breaks down for the Dirac Hamiltonian: even with a properly regularized bulk, the number of edge modes depends on the choice of boundary conditions. This talk aims to uncover the origins of this anomaly, which is tied to the presence of a "ghost" topological charge in the asymptotic part of the spectrum. We will present a generalized bulk-edge correspondence theorem for unbounded operators, employing scattering theory and a relative form of Levinson's theorem. Additionally, we will provide a complete classification of anomalous boundary conditions, revealing that such anomalies are not rare but ubiquitous and typical.