This dissertation studies interacting electron models for quantum matter from two perspectives with different approaches respectively. The first part of this text focuses on rigorously proving Lieb-Robinson bounds for continuous space integer quantum Hall systems perturbed by short range two-body interactions. A Lieb-Robinson bound is an operator norm inequality that implies a finite rate of information propagation in a quantum system, a low energy analogue to the relativistic limit given by the speed of light. Many results of this nature have been obtained for lattice systems, but over the last several years, progress has been made in identifying classes of continuous space models that admit propagation estimates. In this thesis, we prove Lieb-Robinson bounds for two-dimensional continuum fermions in a constant, perpendicular magnetic field using the strategy of Gebert et al. \cite{gebert:2020}, and use this result to prove strong continuity of the Heisenberg dynamics in the limit where inter-particle interactions occur in infinite volume. We then shift to a different framework, namely that of lattice-localized frames, to represent a class of interacting quantum Hall systems in continuous space, as introduced by Bachmann and De Nittis, \cite{bachmann2025lieb}. With this framework, we define a family of parametrized Hamiltonians and construct a continuum analogue of the Hastings-Wen quasi-adiabatic evolution, i.e. dynamics in parameter space that satisfy Lieb-Robinson bounds themselves. Moreover, we then directly apply this to prove quantization of Hall conductance for these models. The new results of these sections are based on works in preparation, \cite{hingorani2026lieb,hingorani2026quantization}. The second portion of this dissertation focuses on two specific materials and phases: magic angle twisted bilayer graphene (MATBG) and superconducting lanthanum nickel gallium-2 (LaNiGa
2). MATBG has garnered a tremendous amount of attention since works such as \cite{MB} and \cite{tarnopolsky2019origin}, which first suggested the existence of flat bands in the electronic band structure of twisted bilayer graphene: bilayer graphene with a relative twist between the layers. Astonishingly, this phenomenon occurs for specific twist angles dubbed magic angles. This quenching of kinetic energy sets the stage for electron-electron interactions to dominate the behavior of the material. The onset of Mott insulating states at several carrier densities as seen through peaks in inverse compressibility \cite{expts-b} drew our interest. In collaboration with Oitmaa and Singh, \cite{hingorani2022onset}, we study the SU(4) Fermi-Hubbard model as a candidate effective model for describing the low energy behavior of MATBG, where we uncover the onset of charge incompressibility and Mott gaps, indicating the existence of the insulating phase.Additionally, we investigate the superconducting pairing mechanism of LaNiGa
2, which has been highlighted as a potential time-reversal symmetry breaking (TRSB) material below its critical temperature due to experimental evidence for internal magnetization, \cite{Ghosh2020}, \cite{Hillier2012}, \cite{Sundar2024}. In our recent work, \cite{tsnb-1mxc}, we report the existence of a Hebel-Slichter (HS) peak in the nuclear-spin relaxation rate, an effect attributed to the divergence in the electronic density of states as the superconducting gap opens. Furthermore, we study models with triplet and singlet spin pairing respectively to ascertain a mechanism that reproduces a HS peak. In doing so, we find that two distinct superconducting gaps and TRSB may not coexist, which raises questions about whether LaNiGa
2 really does break time reversal symmetry or whether a different model is required to fully characterize this material in its superconducting phase.
Lieb-Robinson bounds demonstrate the emergence of locality in many-body quantum systems. Intuitively, Lieb-Robinson bounds state that, with local or exponentially decaying interactions, the correlation that can be built up between two sites separated by distance r after a time t decays as exp ( v t − r ) , where v is the emergent Lieb-Robinson velocity. In many problems, it is important to also capture how much of an operator grows to act on r d sites in d spatial dimensions. Perturbation theory and cluster expansion methods suggest that, at short times, these volume-filling operators are suppressed as exp ( − r d ) . We confirm this intuition, showing that, for r > v t , the volume-filling operator is suppressed by exp ( − ( r − v t ) d / ( v t ) d − 1 ) . This closes a conceptual and practical gap between the cluster expansion and the Lieb-Robinson bound. We then present two very different applications of this new bound. Firstly, we obtain improved bounds on the classical computational resources necessary to simulate many-body dynamics with error tolerance ε for any finite time t : as ε becomes sufficiently small, only ε − O ( t d − 1 ) resources are needed. A protocol that likely saturates this bound is given. Secondly, we prove that disorder operators have volume-law suppression near the “solvable (Ising) point” in quantum phases with spontaneous symmetry breaking, which implies a new diagnostic for distinguishing many-body phases of quantum matter.
We rigorously show that a local spin system giving rise to a slow Hamiltonian dynamics is stable against generic, even time-dependent, local perturbations. The sum of these perturbations can cover a significant amount of the system's size. The stability of the slow dynamics follows from proving that the Lieb-Robinson bound for the dynamics of the total Hamiltonian is the sum of two contributions: the Lieb-Robinson bound of the unperturbed dynamics and an additional term coming from the Lieb-Robinson bound of the perturbations with respect to the unperturbed Hamiltonian. Our results are particularly relevant in the context of the study of the stability of Many-Body-Localized systems, implying that if a so called ergodic region is present in the system, to spread across a certain distance it takes a time proportional to the exponential of such distance. The non-perturbative nature of our result allows us to develop a dual description of the dynamics of a system. As a consequence we are able to prove that the presence of a region of disorder in a ergodic system implies the slowing down of the dynamics in the vicinity of that region.
Study how quantum information propagates through spacetime manifold provides a means of identifying, distinguishing, and classifying novel phases of matter fertilized by many-body effects in strongly interacting systems in and out of equilibrium. Via a fuller characterization of key aspects regarding dynamic behaviors of information, we perform such an analysis on constrained many-body localization -- a newly proposed fully localized state under infinite-interaction limit -- in quasirandom Rydberg blockade spin chain models using thermal out-of-time-order commutators (OTOCs). The OTOC light cones predict a hitherto unknown Lieb-Robinson bound for constrained many-body localization, which is qualitatively different from that of unconstrained many-body Anderson insulators stabilized at weak-interaction limit. Our corroborated numeric and analytic study suggests that constrained many-body localization is a distinct dynamical eigenstate phase whose nonergodicity is beyond local-integral-of-motion phenomenology. Together, these findings consolidate the hierarchy of unconventional quantum dynamics encompassing constrained, unconstrained, and diagonal many-body-localized regimes.
This thesis analyses several locality properties of extended quantum lattice systems, considering both spins and fermions. We begin by assuming locality of the Hamiltonian and then discuss how this translates into locality of the time evolution and the thermal states. Starting from a local Hamiltonian, it is well known that the Heisenberg time evolution satisfies Lieb-Robinson bounds, which quantify its locality. We strengthen these bounds for long-range interacting fermions and for commuting interactions. Furthermore, we establish the first Lieb-Robinson bounds for Hamiltonians whose local terms can grow linearly away from the origin. We also prove stability of Lieb-Robinson bounds with respect to local perturbations in the Hamiltonian. For the thermal states we focus on three types of locality known as decay of correlations, the local perturbations perturb locally (LPPL) principle and local indistinguishability. As a key tool we use spectral filters, which were introduced by Hastings in the 2000s for various applications. It enable us to use Lieb-Robinson bounds to obtain information about locality properties of states. In particular, we use quantum belief propagation to prove an equivalence of decay of correlations, LPPL and local indistinguishability for Gibbs states at any positive temperature. Using the improved Lieb-Robinson bounds mentioned before, we obtain improvements for decay of correlations in the ground state, the locality of the spectral flow and LPPL for long-range interacting systems and commuting Hamiltonians. Furthermore, we adjust the filter functions to achieve exponential decay for certain applications that were previously limited to stretched exponential decay. Spectral filters are also used to rigorously justify response theory, which usually requires a gapped system to begin with. To relax this assumption, we introduce various notions of local gaps, discuss their relations and use one to justify response theory and Kubo’s formula in the absence of a global gap. In another result we use local indistinguishability as an input to prove bulk-edge correspondence at finite temperature, in the sense that the bulk magnetisation is equal to the edge current. This result has mostly been studied for non-interacting models or at zero temperature before. Finally, we present a class of weakly interacting spin systems with an on-site gap and prove that the ground state satisfies a strong form of LPPL. Moreover, we show that these systems satisfy decay of correlations and local indistinguishability uniformly in temperature.
We review several aspects of Many‐Body Localization‐like properties exhibited by the disordered XY chains: localization properties of the energy eigenstates and thermal states, propagation bounds of Lieb‐Robinson type, decay of correlation functions, absence of particle transport, bounds on the bipartite entanglement, and bounded entanglement growth under the dynamics. We also prove new results on the absence of energy transport and Fock space localization. All these properties are made accessible to mathematical analysis due to the exact mapping of the XY chain to a system of quasi‐free fermions given by the Jordan‐Wigner transformation. Motivated by these results we discuss conjectured properties of more general disordered quantum spin and other systems as possible directions for future mathematical research.