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In this thesis, we study the low-energy effective theory for the antiferromagnetic quantum critical metal in two dimensions. The theory has been the subject of intense study for more than twenty years, due to the novel physics of non-Fermi liquid metals and its potential relevance to high-temperature superconductors and heavy-fermion compounds. In the first part of the thesis, we present the perturbative study of the theory in 3 minus epsilon space dimensions by extending the earlier one-loop analysis to higher-loop orders. We show that the expansion is not organized by the standard loop expansion, and a two-loop graph becomes as important as one-loop graphs even in the small epsilon limit due to an infrared singularity caused by an emergent quasilocality. This qualitatively changes the nature of the infrared fixed point, and the epsilon expansion is controlled only after the two-loop effect is taken into account. Furthermore, we show that a ratio between velocities emerges as a small parameter, which suppresses a large class of diagrams. We show that the critical exponents do not receive quantum corrections beyond the linear order in epsilon in the limit that the ratio of velocities vanishes. In the second part of the thesis, we present a nonperturbative solution to the theory in two dimensions based on an ansatz that is inspired by the perturbative analysis. Being a strongly coupled theory, it can still be solved reliably in the low-energy limit as quantum fluctuations are organized by the ratio of velocities that dynamically flows to zero in the low-energy limit. We predict the exact critical exponents that govern the universal scaling of physical observables at low temperatures. / Thesis / Doctor of Philosophy (PhD)

Thesis (doctoral)--Universiteit van Amsterdam, 1989. / Summary also in Dutch. Includes bibliographical references.

Thesis (M.F.A.)--Rochester Institute of Technology, 2007. / Typescript. Includes bibliographical references (leaves 64-65).

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Polymer electrolytes are indispensable in numerous electrochemical systems. Existing polymer electrolytes rarely meet all technical demands by their applications (e.g., high ionic conductivity and good mechanical strength), and new types of polymer electrolytes continue to be developed. In this dissertation, the molecular structure and dynamics of three emerging types of polymer electrolytes featuring Coulombic liquids, i.e., polymerized ionic liquids (polyILs), nanoscale ionic materials (NIMs), and polymeric ion gels, were investigated using molecular dynamics (MD) simulations to help guide their rational design. First, the molecular structure and dynamics of a prototypical polyILs, i.e., poly(1-butyl-3-vinylimidazolium hexafluorophosphate), supported on neutral and charged quartz substrates were investigated. It was found that the structure of the interfacial polyILs is affected by the surface charge on the substrate and deviates greatly from that in bulk. The mobile anions at the polyIL-substrate interfaces diffuse mainly by intra-chain hopping, similar to that in bulk polyILs. However, the diffusion rate of the interfacial mobile anions is much slower than that in bulk due to the slower decay of their association with neighboring polymerized cations. Second, the structure and dynamics of polymeric canopies in the modeling NIMs where the canopy thickness is much smaller than their host nanoparticle were studied. Without added electrolyte ions, the polymeric canopies are strongly adsorbed on the solid substrate but maintain modest in-plane mobility. When electrolyte ion pairs are added, the added counter-ions exchange with the polymeric canopies adsorbed on the charged substrate. However, the number of the adsorbed electrolyte counter-ions exceeds the number of desorbed polymeric canopies, which leads to an overscreening of the substrate's charge. The desorbed polymers can rapidly exchange with the polymers grafted electrostatically on the substrate. Finally, the molecular structure and dynamics of an ion gel consisting of PBDT polyanions and room-temperature ionic liquids (RTIL) were studied. First, a semi-coarse-grained model was developed to investigate the packing and dynamics of the ions in this ion gel. Ions in the interstitial space between polyanions exhibit distinct ordering, which suggests the formation of a long-range electrostatic network in the ion gel. The dynamics of ions slow down compared to that in bulk due to the association of the counter-ions with the polyanions' sulfonate groups. Next, the RTIL-mediated interactions between charged nanorods were studied. It was discovered that effective rod-rod interaction energy oscillates with rod-rod spacing due to the interference between the space charge near each rod as the two rods approach each other. To separate two rods initially positioned at the principal free energy minimum, a significant energy barrier (~several kBT per nanometer of the nanorod) must be overcome, which helps explain the large mechanical modulus of the PBDT ion gel reported experimentally. / Ph. D. / Polymer electrolytes are an indispensable component in numerous electrochemical devices. However, despite decades of research and development, few existing polymer electrolytes can offer the electrochemical, transport, mechanical, and thermal properties demanded by practical devices and new polymer electrolytes are continuously being developed to address this issue. In this dissertation, the molecular structure and dynamics of three emerging novel polymer electrolytes, i.e., polymerized ionic liquids (polyILs), nanoscale ionic materials (NIMs), and polymeric ion gels, are investigated to understand how their transport and mechanical properties are affected by their molecular design. The study of polyILs focused on the interfacial behavior of a prototypical polyILs supported on neutral and charged quartz substrates. It was shown that the structure and diffusion mechanism of the interfacial polyILs are sensitive to the surface charges of the substrate and can deviate strongly from that in bulk polyILs. The study of NIMs focused on how the transport properties of the dynamically grafted polymers are affected by electrolyte ion pairs. It was discovered that the contaminated ions can affect the conformation the polymeric canopies and the exchange between the “free” and “grafted” polymers. The study of polymeric ion gels focused on the molecular and mesoscopic structure of the ionic liquids in the gel and the mechanisms of ion transport in these gels. It was discovered that the ions exhibit distinct structure at the intermolecular and the interrod scales, suggesting the formation of extensive electrostatic networks in the gel. The dynamics of ions captured in simulations is qualitatively consistent with experimental observations.