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Abstract

In this thesis we examine universal scaling properties of strongly-correlated systems near and far from equilibrium. We discuss quantum phase transitions at vanishing temperature, multicritical and dynamic critical behavior near thermal equilibrium, and scaling properties of nonequilibrium steady states. We employ nonperturbative methods including the functional renormalization group as well as Monte Carlo simulations. A general outline of the functional renormalization group is given in the introductory chapters.

In the first part of this thesis, we investigate spinless fermions on the honeycomb lattice interacting via short-range repulsive interactions. Such a system can be seen as a simple model for suspended graphene. The short-range interactions control the ground state properties of the system that may lead to a chiral phase transition from the semimetal to the charge density wave (CDW)/Kekulé ordered state. We determine the universal scaling properties at the chiral transition, and establish the presence of large anomalous dimensions indicating the importance of strong fluctuations.

The competition of two nonvanishing order parameters and their corresponding multicritical behavior are investigated in the subsequent chapter. We characterize the bicritical and tetracritical behavior in the purely bosonic O(N_1) + O(N_2) symmetric model and comment on possible applications to condensed-matter and high-energy physics.

In the following chapter we discuss the long-time relaxational behavior at criticality of an order parameter with O(N) symmetry coupled to an additional conserved density. We find an anomalous diffusion phase with new dynamic scaling properties. Using the functional renormalization group we determine the complete dynamic critical behavior of the model in 2 < d < 4 dimensions and compare our results to experiments.

Finally, we investigate the scaling properties of stationary states far from equilibrium. At the example of the one-dimensional Burgers’ equation we develop a novel approach to hydrodynamic turbulence using lattice Monte Carlo methods. We apply these techniques to determine the statistical properties of small-scale fluctuations in this model and identify the anomalous scaling behavior.