Multicomponent superconductors described by several complex matter fields have properties radically different from those of their single-component counterparts. Examples include partially ordered phases and spontaneous breaking of time-reversal symmetry due to frustration between Josephson-coupled components. Recent experimental results make such symmetry breaking a topic of central interest in superconductivity. Multicomponent gauge theories appear as effective theories e.g. for quantum antiferromagnets, and are thus of interest well beyond superconductivity. The nature of the phase transitions in these models is of great importance in modern physics, and yet remains poorly understood. These models and phenomena are studied theoretically in this thesis, mainly using large-scale Monte Carlo simulations.Superconducting s+is states have recently been described for superconductors with N = 3 components. The novelty of these states is that they break time-reversal symmetry due to frustration of interband couplings. In the first paper, we consider whether there can be new states in N-component Ginzburg-Landau models with bilinear Josephson couplings when N >= 4. We find that these models have new states associated with accidental continuous ground-state degeneracies. Also, we show that the possible combinations of signs of the couplings can for any N be divided into equivalence classes in a way that is related to the graph-theoretical concept of Seidel switching.In the second paper, we consider fluctuation effects in models of SU(N) symmetric superconductors. We demonstrate that there is a novel type of paired phase that is given by proliferation of non-topological vortices for N = 3 and 4, and that despite the absence of topologically stable vortices these systems form vortex lattices in external magnetic field; these lattices involve structures that are not simply hexagonal and differ between components.In the third paper, we consider fluctuation effects in London models of U(1)^N symmetric superconductors. These models are of central interest due to the theory of deconfined quantum criticality, according to which such gauge theories may describe phase transitions beyond the Ginzburg-Landau-Wilson paradigm. The direct transitions from fully ordered to fully disordered phases have been reported to be continuous for N = 1 and N = 183, and discontinuous for N = 2. The nature of the phase transitions for small N is an outstanding open question. We demonstrate that the degree of discontinuity of the direct transitions increases with N, at least for small N, and that the transitions from paired phases to fully disordered phases can be discontinuous. Both these results are in contrast to previous expectations.In the fourth and final paper, we report the first experimental observation of a state of matter that has an order parameter that is fourth order in fermionic fields: a bosonic Z_2 metal, in which time-reversal symmetry is broken due to partial ordering of Cooper pairs despite superconducting order being absent. By considering fluctuation effects in phase-frustrated three-component Ginzburg-Landau models, we place constraints on the models used to describe the material in question. Also, we give an example of this anomalous state occurring in a type-2 Ginzburg-Landau model in external magnetic field, despite it not occurring in this model in the absence of external field.

We classify ground states and normal modes for n-component superconductors with frustrated intercomponent Josephson couplings, focusing on n=4. The results should be relevant not only to multiband superconductors, but also to Josephson-coupled multilayers and Josephson-junction arrays. It was recently discussed that three-component superconductors can break time-reversal symmetry as a consequence of phase frustration. We discuss how to classify frustrated superconductors with an arbitrary number of components. Although already for the four-component case there are a large number of different combinations of phase-locking and phase-antilocking Josephson couplings, we establish that there are a much smaller number of equivalence classes where properties of frustrated multicomponent superconductors can be mapped to each other. This classification is related to the graph-theoretical concept of Seidel switching. Numerically, we calculate ground states, normal modes, and characteristic length scales for the four-component case. We report conditions of appearance of new accidental continuous ground-state degeneracies.