Emergent quantum and classical phases from competing interactions

In this thesis we explore two distinct settings where the interplay of interactions leads to an emergence of new phases of matter. In the first part, we focus on the competition between the Kosterlitz-Thouless scenario of vortex unbinding with the onset of discrete relative order. Many two-dimensional physical systems ranging from atomic-molecular condensates to low dimensional superconductors and liquid-crystal films are described by coupled XY models. Such coupled U(1) systems further introduce rich physics, bringing topology into contact with fractionalization and deconfinement. Motivated by a hidden-order phase transition in isotropic liquid-crystal 54COOBC films, we study the finite-temperature phase diagram of a minimalist hexatic-nematic XY model. We identify a small region of composite Potts order above the vortex-binding transition; this phase is characterized by relative hexatic-nematic ordering though both variables are disordered. We propose that the Potts order results from a confinement of fractional vortices into extended nematic defects and discuss the broader implications of fractional vortices and composite ordering in the wider class of coupled XY condensates. In the second part, we are motivated by the relevance of Hund’s coupling in the context of multiorbital superconductors, and revisit the problem of a multiorbital Kondo and Anderson impurity with Hund’s interaction. Using dynamical large-N techniques, we are able to follow the ground state, dynamic, and thermodynamic properties of these systems over many decades of temperature. In the Hund-Kondo setting, we capture the emergence of large moments and the resulting exponentially suppressed Kondo temperature. We focus our analysis on the intermediate finite temperature phase which presents an alternate paramagnetic state due to the emergent large moment. Delving further into this unconventional paramagnetic regime, we study the Hund-Anderson model, where the impurity's electronic occupancy can be explicitly tuned. We show that valence fluctuations drastically impede the development of a large fluctuating moment over a wide range of temperatures and energy, characterized by quenched orbital degrees of freedom and a singular logarithmic behavior of the spin susceptibility, closely resembling power-law scaling. Such singular spin fluctuations in this unconventional phase are suspected to play an important role in future models of Hund's driven Cooper pairing. We then expand the use of the large-N Schwinger boson approach from the setting of Kondo and Anderson impurity models to the single band Hubbard model. After fractionalizing the local degrees of freedom in terms of holons and Schwinger bosons, we obtain a set of self-energy equations for the infinite-U Hubbard model and the t-J model. A plethora of dynamical and thermodynamical quantities can be extracted, and we present preliminary results for the phase diagram of the infinite-U Hubbard model upon hole doping. Finally, we close by briefly attacking the problem of superconductivity in metals with strong Hund's coupling. Taking advantage of the pre-entangled states formed through the joint action of spin-orbit and Hund's coupling within the multiorbital local setting, we propose an extension of the large-N framework to include pairing correlations.Ph.D.Includes bibliographical reference