We discuss the appearance of quantum orders in the Hubbard model for interacting electrons, at half-filling. Such phases do not have local order parameters and need to be characterized by the quantum mechanical properties of their ground state. On one hand, we study the Mott transition from a metal to a spin liquid insulator in two dimensions, of potential relevance to some layered organic compounds. The correlation-driven transition occurs at fixed filling and involves fractionalization of the electron: upon entering the insulator, a Fermi surface of neutral spinons coupled to an internal gauge field emerges. We focus on the transport properties near the quantum critical point and find that the emergent gauge fluctuations play a key role in determining the universal scaling. Second, motivated by a class of three-dimensional transition metal oxides, the pyrochlore iridates, we study the interplay of non-trivial band topology and correlations. Building on the strong spin orbit coupling in these compounds, we construct a general microscopic Hubbard model and determine its mean-field phase diagram, which contains topological insulators, Weyl semimetals, axion insulators and various antiferromagnets. We also discuss the effects many-body correlations on theses phases. We close by examining a fractionalized topological insulator that combines the two main themes of the thesis: fractionalization and non-trivial band topology. Specifically, we study how the two-dimensional protected surface states of a topological Mott insulator interact with a three-dimensional emergent gauge field. Various correlation effects on observables are identified.

We discuss the appearance of quantum orders in the Hubbard model for interacting electrons, at half-filling. Such phases do not have local order parameters and need to be characterized by the quantum mechanical properties of their ground state. On one hand, we study the Mott transition from a metal to a spin liquid insulator in two dimensions, of potential relevance to some layered organic compounds. The correlation-driven transition occurs at fixed filling and involves fractionalization of the electron: upon entering the insulator, a Fermi surface of neutral spinons coupled to an internal gauge field emerges. We focus on the transport properties near the quantum critical point and find that the emergent gauge fluctuations play a key role in determining the universal scaling. Second, motivated by a class of three-dimensional transition metal oxides, the pyrochlore iridates, we study the interplay of non-trivial band topology and correlations. Building on the strong spin orbit coupling in these compounds, we construct a general microscopic Hubbard model and determine its mean-field phase diagram, which contains topological insulators, Weyl semimetals, axion insulators and various antiferromagnets. We also discuss the effects many-body correlations on theses phases. We close by examining a fractionalized topological insulator that combines the two main themes of the thesis: fractionalization and non-trivial band topology. Specifically, we study how the two-dimensional protected surface states of a topological Mott insulator interact with a three-dimensional emergent gauge field. Various correlation effects on observables are identified.

Interactions between the microscopic constituents of a solid---a many-body system--- can lead to novel phases and exotic physical phenomena like fractionalization, topological order, quantum spin liquids, emergent gauge field, etc.. The concept of frustration provides a ground for such exotic phenomena. Frustration can prevent a many-body system from establishing long-range order down to the lowest temperatures due to competing interactions. Instead, competing interactions may result in disordered and liquid-like phases of matter that provide the vacuum for fractional excitations. The absence of any order parameter in strongly frustrated systems---due to not breaking any symmetry spontaneously--- immediately raises the question about possible experimental probes of spin-liquids and their fractional excitations. Dynamic probes, like inelastic neutron scattering or Raman scattering, provide an experimental method to detect signatures of fractionalised quasiparticles. The energy and momentum transferred in a scattering event is split between the fractional quasiparticles. On the theory side, computing such dynamical signatures beyond one spatial dimension is generally a difficult task. In this thesis, numerical methods like density matrix renormalisation group and matrix product states are used to study strongly frustrated magnets and their dynamics in a non-perturbative way. This thesis covers two physical models in the context of frustration and emergent gauge fields. Firstly, the Kitaev model of spin-1/2 degrees of freedom subject to strongly anisotropic spin exchange. The Kitaev model features quantum spin liquid ground states with fractionalization of spins into Majorana fermions and Z_2-fluxes---the visons of an emergent Z_2 gauge theory. The main questions addressed here concern the stability of the quantum spin liquid phase upon adding perturbations relevant in magnetic compounds such as Heisenberg or the symmetric-offdiagonal Gamma exchange. Applying a magnetic field drives the Kitaev model into a topologically ordered phase. The excitations and dynamical signatures within the spin liquid, the topologically ordered phase, and within ordered phases are studied. Secondly, a classical minimal model of the proton configuration in water ice is studied. The ice rules, a local constraint describing the low energy manifold, result in emergent Maxwell's equation. Upon applying an external electric field along certain axis, a polarization plateau occurs in which the remaining degrees of freedom can be described by dimers on two-dimensional lattices.