| We study the role of lattice frustration, competing interactions and quantum fluctuations in stabilizing non-trivial states of matter in strongly correlated systems. Our analysis focuses on three types of physical phenomena: magnetism in Mott insulators, superconductivity in repulsive fermion systems and multiferroicity in complex oxides.;In the context of frustrated magnets, we propose a real-space mean-field framework, which combines exact diagonalization in finite clusters and variational calculation of the state of an infinite system, thus capturing local correlations and providing a controlled and unbiased approximation scheme. This method is applied to several models of quantum magnetism, such as the square-lattice Heisenberg antiferromagnet with competing first and second neighbor exchange interactions. Using a single variational ansatz for the ground state, we compute the zero-temperature phase diagram of this model, which includes a quantum paramagnetic state. We show that this state has a correlated plaquette nature and breaks translational invariance, but preserves lattice point-group symmetries. Next, we study the phenomenon of magnetization plateaux in the orthogonal dimer compound SrCu2(BO3)2, described by the Shastry-Sutherland model. We demonstrate that plateaux are stabilized in certain spin patterns, satisfying local commensurability conditions, which we also derive.;Lattice frustration usually hinders the existence of a long-range order. However, in some cases frustration can be beneficial for stabilizing an ordered state, even in a strongly interacting system. We illustrate this mechanism, by considering the Hubbard model with modulated electron hoppings. Within a controlled approximation, we demonstrate how magnetic fluctuations lead to a d-wave superconducting state for arbitrarily strong fermion repulsion. We also discuss the possibility to observe this phenomenon in cold atom experiments.;Another class of systems, where frustration and quantum fluctuations serve as prerequisites for a complex ordered state, are multiferroics with ferroelectricity due to charge ordering. Using the rare-earth oxide LuFe 2O4 as an example, we present a theory of multiferroic behavior, caused by the lattice frustration and order-from-disorder physics. Using this theory we explicitly demonstrate that the double exchange mechanism leads to a significant coupling between electric and magnetic orders. |
