Abstract

The Landau-Ginzburg-Wilson (LGW) paradigm is the backbone of the modern understanding of critical phenoma. It rests on the assumption that a continuous phase transition can be described solely in terms of a fluctuating order parameter. In this sandwich thesis, we analyze several quantum phase transitions in two-dimensional systems where this basic assumption is violated. Our analysis is structured in three parts: In the first part, two ordering transitions in metals are studied, where the presence of gapless fermionic modes invalidates the pure order parameter description. Instead, we apply a renormalization group approach which retains both fermionic and bosonic (order paramater) degrees of freedom, using the deviation from an upper critical dimension of d = 5/2 as a control parameter. We first apply this technique to describe the quantum phase transition between a normal metal and an inhomogeneous (FFLO) superconductor. Our analysis rigorously confirms the mean field expectation that this transition is continuous, and shows that interesting non-Fermi liquid physics can arise at the critical point, manifesting itself in unusual scaling of various observables. The second case study is the onset of incommensurate 2kF charge density wave order. While non-Fermi liquid features are less pronounced at this transition, we find a strong dynamical nesting of the Fermi surface, which stabilizes the density wave formation and results in a continuous transition, opposed to early theoretical claims. In the second part, we present a novel case study of “deconfined criticality” for a quantum magnet: Here, symmetries are broken on both sides of the phase transition, which would be first order within a conventional two-order-parameter LGW description. By contrast, the scenario of deconfined criticality predicts a continuous transition, driven by condensation of topological defects which carry quantum numbers of the opposite phase. Its paradigmatic application is the transition from a SU (2)-Neél state to a valence-bond solid (a singlet covering configuration that breaks spatial symmetries) on the square lattice. Here, we propose a novel extension and study a SU (3) antiferromagnet on a triangular lattice, which supports a transition between a magnetic, three-sublattice color-ordered phase and a trimerized SU (3) singlet phase. We provide a critical theory in terms of fractional bosonic degrees of freedom (and with a topological defect interpretation), and study its fixed point properties with functional renormalization group. This yields a critical fixed point in a suitable large-N -limit, implying a continuous transition in accordance with the deconfined criticality framework. In the third and final part, we study the “polaron”-problem of a single impurity coupled to a majority Fermi sea, which is particularly interesting when formation of a bound state (“molecule”) between impurity and majority is allowed for. Starting from the exactly solvable limit of infinite impurity mass, we present a controlled computation of impurity spectra (both single- and two-particle) for heavy impurities based on Feynman diagram techniques, with an eye for experiments on doped semiconductors and ultracold gases. Furthermore, we discuss various aspects of the “molecule-to-polaron” transition that occurs in these systems, which also defies a simple LGW description in the single-impurity limit.