Noncommutative Riemannian geometry is an area that has seen intense activity over the past 25 years. Despite this, noncommutative complex geometry is only now beginning to receive serious attention. The theory of quantum groups provides a large family of very interesting potential examples, namely the quantum flag manifolds. Thus far, only the irreducible quantum flag manifolds have been investigated as noncommutative complex spaces. In a series of papers, Heckenberger and Kolb showed that for each of these spaces, there exists a q-deformed Dolbeault double complex. In this thesis a comprehensive framework for noncommutative complex geometry on quantum homogeneous spaces is introduced. The main ingredients used are covariant differential calculi and Takeuchi's categorical equivalence for faithfully at quantum homogeneous spaces. A number of basic results are established, producing a simple set of necessary and sufficient conditions for noncommutative complex structures to exist. It is shown that when applied to the quantum projective spaces, this theory reproduces the q-Dolbeault double complexes of Heckenberger and Kolb. Furthermore, the framework is used to q-deform results from Borel{Bott{ Weil theory, and to produce the beginnings of a theory of noncommutative Kahler geometry.

This thesis studies matter emergent as topological excitations of quantum geometry in quantum gravity models. In these models, states are framed four-valent spin networks embedded in a topological three manifold, and the local evolution moves are dual Pachner moves. We first formulate our theory of embedded framed four-valent spin networks by proposing a new graphic calculus of these networks. With this graphic calculus, we study the equivalence classes and the evolution of these networks, and find what we call 3-strand braids, as topological excitations of embedded four-valent spin networks. Each 3-strand braid consists of two nodes that share three edges that may or may not be braided and twisted. The twists happen to be in units of 1/3. Under certain stability condition, some 3-strand braids are stable. Stable braids have rich dynamics encoded in our theory by dual Pachner moves. Firstly, all stable braids can propagate as induced by the expansion and contraction of other regions of their host spin network under evolution. Some braids can also propagate actively, in the sense that they can exchange places with substructures adjacent to them in the graph under the local evolution moves. Secondly, two adjacent braids may have a direct interaction: they merge under the evolution moves to form a new braid if one of them falls into a class called actively interacting braids. The reverse of a direct interaction may happen too, through which a braid decays to another braid by emitting an actively interacting braid. Thirdly, two neighboring braids may exchange a virtual actively interacting braid and become two different braids, in what is called an exchange interaction. Braid dynamics implies an analogue between actively interacting braids and bosons. We also invent a novel algebraic formalism for stable braids. With this new tool, we derive conservation laws from interactions of the braid excitations of spin networks. We show that actively interacting braids form a noncommutative algebra under direction interaction. Each actively interacting braid also behaves like a morphism on non-actively interacting braids. These findings reinforce the analogue between actively interacting braids and bosons. Another important discovery is that stable braids admit seven, and only seven, discrete transformations that uniquely correspond to analogues of C, P, T, and their products. Along with this finding, a braid's electric charge appears to be a function of a conserved quantity, effective twist, of the braids, and thus is quantized in units of 1/3. In addition, each $CPT$-multiplet of actively interacting braids has a unique, characteristic non-negative integer. Braid interactions turn out to be invariant under C, P, and T. Finally, we present an effective description, based on Feynman diagrams, of braid dynamics. This language manifests the analogue between actively interacting braids and bosons, as the topological conservation laws permit them to be singly created and destroyed and as exchanges of these excitations give rise to interactions between braids that are charged under the topological conservation rules. Additionally, we find a constraint on probability amplitudes of braid interactions. We discuss some subtleties, open issues, future directions, and work in progress at the end.

This thesis studies matter emergent as topological excitations of quantum geometry in quantum gravity models. In these models, states are framed four-valent spin networks embedded in a topological three manifold, and the local evolution moves are dual Pachner moves. We first formulate our theory of embedded framed four-valent spin networks by proposing a new graphic calculus of these networks. With this graphic calculus, we study the equivalence classes and the evolution of these networks, and find what we call 3-strand braids, as topological excitations of embedded four-valent spin networks. Each 3-strand braid consists of two nodes that share three edges that may or may not be braided and twisted. The twists happen to be in units of 1/3. Under certain stability condition, some 3-strand braids are stable. Stable braids have rich dynamics encoded in our theory by dual Pachner moves. Firstly, all stable braids can propagate as induced by the expansion and contraction of other regions of their host spin network under evolution. Some braids can also propagate actively, in the sense that they can exchange places with substructures adjacent to them in the graph under the local evolution moves. Secondly, two adjacent braids may have a direct interaction: they merge under the evolution moves to form a new braid if one of them falls into a class called actively interacting braids. The reverse of a direct interaction may happen too, through which a braid decays to another braid by emitting an actively interacting braid. Thirdly, two neighboring braids may exchange a virtual actively interacting braid and become two different braids, in what is called an exchange interaction. Braid dynamics implies an analogue between actively interacting braids and bosons. We also invent a novel algebraic formalism for stable braids. With this new tool, we derive conservation laws from interactions of the braid excitations of spin networks. We show that actively interacting braids form a noncommutative algebra under direction interaction. Each actively interacting braid also behaves like a morphism on non-actively interacting braids. These findings reinforce the analogue between actively interacting braids and bosons. Another important discovery is that stable braids admit seven, and only seven, discrete transformations that uniquely correspond to analogues of C, P, T, and their products. Along with this finding, a braid's electric charge appears to be a function of a conserved quantity, effective twist, of the braids, and thus is quantized in units of 1/3. In addition, each $CPT$-multiplet of actively interacting braids has a unique, characteristic non-negative integer. Braid interactions turn out to be invariant under C, P, and T. Finally, we present an effective description, based on Feynman diagrams, of braid dynamics. This language manifests the analogue between actively interacting braids and bosons, as the topological conservation laws permit them to be singly created and destroyed and as exchanges of these excitations give rise to interactions between braids that are charged under the topological conservation rules. Additionally, we find a constraint on probability amplitudes of braid interactions. We discuss some subtleties, open issues, future directions, and work in progress at the end.

This dissertation focuses on one question: how should one drive an experimentally prepared state of a generic quantum system into a different target-state, simultaneously minimizing energy dissipation and maximizing the fidelity between the target and evolved-states? We develop optimal adiabatic driving protocols for general quantum systems, and show that these are geodesic paths. Geometric ideas have always played a fundamental role in the understanding and unification of physical phenomena, and the recent discovery of topological insulators has drawn great interest to topology from the field of condensed matter physics. Here, we discuss the quantum geometric tensor, a mathematical object that encodes geometrical and topological properties of a quantum system. It is related to the fidelity susceptibility (an important quantity regarding quantum phase transitions) and to the Berry curvature, which enables topological characterization through Berry phases. A refined understanding of the interplay between geometry and topology in quantum mechanics is of direct relevance to several emergent technologies, such as quantum computers, quantum cryptography, and quantum sensors. As a demonstration of how powerful geometric and topological ideas can become when combined, we present the results of an experiment that we recently proposed. This experimental work was done at the Google Quantum Lab, where researchers were able to visualize the topological nature of a two-qubit system in sharp detail, a startling contrast with earlier methods. To achieve this feat, the optimal protocols described in this dissertation were used, allowing for a great improvement on the experimental apparatus, without the need for technical engineering advances. Expanding the existing literature on the quantum geometric tensor using notions from differential geometry and topology, we build on the subject nowadays known as quantum geometry. We discuss how slowly changing a parameter of a quantum system produces a measurable output of its response, merely due to its geometric nature. Next, we topologically characterize different classes of Hamiltonians using the Berry monopole charges, and establish their topological protection. Finally, we explore how such knowledge allows one to access topologically forbidden regions by adiabatically breaking and reestablishing symmetries.

This thesis is based on two papers by the author and consists of two parts. We review the recent developments in the theory of multiple M2-branes and 3-algebras leading to multiple D2-brane theories. The inclusion of flux terms for the supersymmetric BLG and ABJM theories of closed M2-branes is discussed and then generalised to open M2-branes. Here the boundary condition is derived and different BPS configurations are examined where we find a mass deformed Basu-Harvey equation for the M2-M5 system. The Lorentzian 3-algebra is then employed for obtaining a theory of D2-branes in a flux background, we then obtain the new fuzzy funnel solution of the system of D2-D4 branes in a flux. Matrix theories and their compactifications as well as noncommutative geometry and noncommutative gauge theories are reviewed with a discussion of their generalisations to three dimensions to be used to describe the M-theory three form potential. A new feature of string theory is then obtained called the quantum Nambu geometry (QNG). It is found by considering the action for D1-strings in a RR flux background and we demonstrate that there is a large flux double scaling limit where the action is dominated by a Chern-Simons-Myers coupling term. A classical solution to this is the quantised spacetime known as the quantum Nambu geometry. Various matrix models are obtained from this action, these are the large flux dominated terms of the full actions for the corresponding matrix models. The QNG gives rise to an expansion of D1-strings to D4-branes in the IIA theory, so we obtain an action for the large flux terms for this action which is verified by a dimensional reduction of the PST action describing M5-branes. We make a generalisation of the D4-brane action to describe M5-branes using a duality. We are describing the 3-form self-dual field strength of a non-abelian generalisation of the PST action.

In recent decades enormous progress has been made in studying the geometrical structure of the quantum state space. Far from an abstraction, this geometric struc- ture is defined operationally in terms of the distinguishability of states connected by parameterizations that can be controlled in a laboratory. This geometry is manifest in the kinds of response functions that are measured by well established experimen- tal techniques, such as inelastic neutron scattering. In this thesis we explore the properties of the state space geometry in the vicinity of the ground state of two paradigmatic models of low dimensional magnetism. The first model is the spin-1 anti-ferromagnetic Heisenberg chain, which is a central example of symmetry pro- tected topological physics in one dimension, exhibiting a non-local string order, and symmetry protected short range entanglement. The second is the Kitaev honeycomb model, a rare example of an analytically solvable quantum spin liquid, characterized by long range topological order. In Chapter 2 we employ the single mode approximation to estimate the genuine multipartite entanglement in the spin-1 chain as a function of the unaxial anisotropy up to finite temperature. We find that the genuine multipartite entanglement ex- hibits a finite temperature plateau, and recove the universality class of the phase transition induced by negative anisotropy be examining the finite size scaling of the quantum Fisher information. In Chapter 4 we map out the zero temperature phase diagram in terms of the QFI for a patch of the phase space parameterized by the anisotropy and applied magnetic field, establishing that any non-zero anisotropy en- hances that entanglement of the SPT phase, and the robustness of the phase to finite temperatures. We also establish a connection between genuine multipartite entanglement and state space curvature. In Chapter 3 we turn to the Kitaev honeycomb model and demonstrate that, while the QFI associated to local operators remains trivial, the second derivative of such quantities with respect to the driving parameter exhibit divergences. We characterize the critical exponents associated with these divergences. / Thesis / Doctor of Philosophy (PhD) / Systems composed of many bodies tend to order as their energy is reduced. Steam, a state characterized by the complete disorder of the constituent water molecules, condenses to liquid water as the temperature (energy) decreases, wherein the water molecules are organized enough for insects to walk atop them. Water freezes to ice, which is so ordered that it can hold sleds and skaters. Quantum mechanics allows for patterns of organization that go beyond the solid-liquid-gas states. These patterns are manifest in the smallest degrees of freedom in a solid, the electrons, and are responsible for fridge magnets and transistors. While quantum systems still tend to order at lower energies, they are characterized by omni-present fluctuations that can conceal hidden forms of organization. One can imagine that the states of matter live in a vast space, where each point represents a different pattern. In this thesis we show that by probing the geometry of this space, we can detect hidden kinds of order that would be otherwise invisible to us.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1999. / Includes bibliographical references (p. 121-123).

In this thesis, research starts by questioning Berry curvature dipole's role in electronic properties in solids. Strongly inspired by the recent studies, we discover a more profound interpretation of the Berry curvature dipole. It is demonstrated that the anomalous correction to the electron acceleration is proportional to the Berry curvature dipole and is responsible for the Non-linear Hall effect recently discovered in materials with broken inversion symmetry. This allows uncovering a deeper meaning of the Berry curvature dipole as a non-linear version of the Drude weight that serves as a measurable order parameter for broken inversion symmetry in metals. Later, we introduce the Quantum Rectification Sum Rule in time-reversal invariant materials is derived by showing that the integral over frequency of the rectification conductivity depends solely on the Berry connection and not on the band energies or relaxation rates. In the final part of the thesis, we use the Keldysch-Floquet formalism to obtain non-perturbative predictions of the optical responses in solids, mainly focusing on the clean limit response of systems with broken time-reversal symmetry.

[pt] A investigação da geometria quântica em semicondutores e isoladores tornou-se significativa devido às suas implicações nas características dos materiais. A noção de geometria quântica surge considerando a métrica quântica do estado de Bloch da banda de valência, que é definido a partir da sobreposição dos estados de Bloch em momentos ligeiramente diferentes. Ao integrar a métrica quântica em toda a zona de Brillouin, introduzimos uma quantidade que chamamos de número de fidelidade, que significa a distância média entre estados de Bloch adjacentes. Além disso, apresentamos um formalismo para expressar o número de fidelidade como um marcador de fidelidade local no espaço real que pode ser definido em qualquer sítio da rede. O marcador pode ser calculado diretamente diagonalizando o hamiltoniano da rede que descreve o comportamento das partículas na rede. Posteriormente, o conceito de número e marcador de fidelidade é estendido para temperatura finita utilizando a teoria de resposta linear, conectando-os a medições experimentais que envolvem analisar o poder de absorção óptica global e local quando o material é exposto à luz linearmente polarizada. Particularmente para materiais bidimensionais, a opacidade do material permite a determinação direta do número de fidelidade espectral, permitindo a detecção experimental do número de fidelidade. Finalmente, um marcador de fidelidade não local é introduzido considerando a divergência da métrica quântica. Este marcador é postulado como um indicador universal de transições de fase quântica, assumindo que o momento cristalino permanece um número quântico válido. Este marcador não local pode ser interpretado como uma função de correlação dos estados de Wannier, que são funções de onda localizadas que descrevem estados eletrônicos em um cristal. A generalidade e aplicabilidade destes conceitos são demonstradas através da investigação de vários isoladores topológicos e transições de fase topológicas em diferentes dimensões. Essas descobertas elaboram o significado dessas quantidades e sua conexão com vários fenômenos fundamentais na física da matéria condensada. / [en] The investigation of quantum geometry in semiconductors and insulators has become significant due to its implications for material characteristics. The notion of quantum geometry arises by considering the quantum metric of the valence-band Bloch state, which is defined from the overlap of the Bloch states at slightly different momenta. By integrating the quantum metric through-out the Brillouin zone, we introduce a quantity that we call fidelity number, which signifies the average distance between adjacent Bloch states. Furthermore, we present a formalism to express the fidelity number as a local fidelity marker in real space that can be defined on every lattice site. The marker can be calculated directly by diagonalizing the lattice Hamiltonian that describes particle behavior on the lattice. Subsequently, the concept of the fidelity number and marker is extended to finite temperature using linear-response theory, connecting them to experimental measurements which involves analyze the global and local optical absorption power when the material is exposed to linearly polarized light. Particularly for two-dimensional materials, the material s opacity enables straightforward determination of the fidelity number spectral, allowing for experimental detection of the fidelity number. Finally, a nonlocal fidelity marker is introduced by considering the divergence of the quantum metric. This marker is postulated as a universal indicator of quantum phase transitions, assuming the crystalline momentum remains a valid quantum number. This nonlocal marker can be interpreted as a correlation function of Wannier states, which are localized wave functions describing electronic states in a crystal. The generality and applicability of these concepts are demonstrated through the investigation of various topological insulators and topological phase transitions across different dimensions. These findings elaborate the significance of these quantities and their connection to various fundamental phenomena in condensed matter physics.