This dissertation presents new results on two problems concerning the dynamics of certainclasses of interacting many-body systems in quantum mechanics. Chapter 1 is an introduction,\nwhich includes mathematical preliminaries and a summary of the results.\nIn Chapter 2 we introduce the notion of transmission time to study the dynamics of disordered\nquantum spin chains and prove results relating its behavior to many-body localization properties.\nWe also study two versions of the so-called Local Integrals of Motion (LIOM) representation of\nspin chain Hamiltonians and their relation to dynamical many-body localization. We prove that\nuniform-in-time dynamical localization expressed by a zero-velocity Lieb-Robinson bound implies\nthe existence of a LIOM representation of the dynamics as well as a weak converse of this statement.\nWe also prove that for a class of spin chains satisfying a form of exponential dynamical localization,\nsparse perturbations result in a dynamics in which transmission times diverge at least as a power\nlaw of distance, with a power for which we provide lower bound that diverges with increasing\nsparseness of the perturbation.\nIn Chapter 3 we introduce a class of UV-regularized two-body interactions for fermions in Rd\nand prove a Lieb-Robinson estimate for the dynamics of this class of many-body systems. As a\nstep toward this result, we also prove a propagation bound of Lieb-Robinson type for Schrodinger\noperators. We apply the propagation bound to prove the existence of infnite-volume dynamics as\na strongly continuous group of automorphisms on the CAR algebra.
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.
The Lieb-Robinson (LR) bound rigorously shows that in quantum systems with short-range interactions, the maximum amount of information that travels beyond an effective "light cone" decays exponentially with distance from the light-cone front, which expands at finite velocity.Despite being a fundamental result, existing bounds are often extremely loose, limiting their applications.We introduce a method that dramatically and qualitatively improves LR bounds in models with finite-range interactions.Most prominently, in systems with a large local Hilbert space dimension D, our method gives a LR velocity that grows much slower than previous bounds with D as D → ∞.For example, in the Heisenberg model with spin S, we find v ≤ const.compared to the previous v ∝ S, which diverges at large S, and in multiorbital Hubbard models with N orbitals, we find v ∝ √ N instead of previous v ∝ N , and similarly in the Nstate truncated Bose-Hubbard model and Wen's quantum rotor model.Our bounds also scale qualitatively better in some systems when the spatial dimension or certain model parameters become large, for example in the d-dimensional quantum Ising model and perturbed toric code models.Even in spin-1/2 Ising and Fermi-Hubbard models, our method improves the LR velocity by an order of magnitude with typical model parameters, and significantly improves the LR bound at large distance and early time.
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.
In this work, we investigate how quickly local perturbations propagate in interacting boson systems with Bose-Hubbard-type Hamiltonians. In general, these systems have unbounded local energies, and arbitrarily fast information propagation may occur. We focus on a specific but experimentally natural situation in which the number of bosons at any one site in the unperturbed initial state is approximately limited. We rigorously prove the existence of an almost-linear information-propagation light cone, thus establishing a Lieb-Robinson bound: the wave front grows at most as t log^{2}(t). We prove the clustering theorem for gapped ground states and study the time complexity of classically simulating one-dimensional quench dynamics, a topic of great practical interest.
In this paper, we prove a family of Lieb-Robinson bounds for discrete spin systems with long-range interactions. Our results apply for arbitrary
k-body interactions, so long as they decay with a power law greater than
kd, where
d is the dimension of the system. More precisely, we require that the sum of the norm of terms with diameter greater than or equal to
R, acting on any one site, decays as a power law $1/{R}^{\ensuremath{\alpha}}$, with $\ensuremath{\alpha}>d$. These bounds allow us to prove that, at any fixed time, the spatial decay of a time evolved operator follows arbitrarily closely to $1/{r}^{\ensuremath{\alpha}}$. Moreover, we introduce an alternative light cone definition for power-law interacting quantum systems which captures the region of the system where changing the Hamiltonian can affect the evolution of a local operator. In short-range interacting systems, this light cone agrees with the conventional definition. However, in long-range interacting systems, our definition yields a stricter light cone, which is more relevant in most physical contexts.