Universal Non-Debye Scaling in the Density of States of Amorphous Solids.

At the jamming transition, amorphous packings are known to display anomalous vibrational modes with a density of states (DOS) that remains constant at low frequency. The scaling of the DOS at higher packing fractions remains, however, unclear. One might expect to find a simple Debye scaling, but recent results from effective medium theory and the exact solution of mean-field models both predict an anomalous, non-Debye scaling. Being mean-field in nature, however, these solutions are only strictly valid in the limit of infinite spatial dimension, and it is unclear what value they have for finite-dimensional systems. Here, we study packings of soft spheres in dimensions 3 through 7 and find, away from jamming, a universal non-Debye scaling of the DOS that is consistent with the mean-field predictions. We also consider how the soft mode participation ratio evolves as dimension increases.

Moduli Space of (0,2) Conformal Field Theories

In this thesis we study aspects of (0,2) superconformal field theories (SCFTs), which are suitable for compactification of the heterotic string. In the first part, we study a class of (2,2) SCFTs obtained by fibering a Landau-Ginzburg (LG) orbifold CFT over a compact K\"ahler base manifold. While such models are naturally obtained as phases in a gauged linear sigma model (GLSM), our construction is independent of such an embedding. We discuss the general properties of such theories and present a technique to study the massless spectrum of the associated heterotic compactification. We test the validity of our method by applying it to hybrid phases of GLSMs and comparing spectra among the phases. In the second part, we turn to the study of the role of accidental symmetries in two-dimensional (0,2) SCFTs obtained by RG flow from (0,2) LG theories. These accidental symmetries are ubiquitous, and, unlike in the case of (2,2) theories, their identification is key to correctly identifying the IR fixed point and its properties. We develop a number of tools that help to identify such accidental symmetries in the context of (0,2) LG models and provide a conjecture for a toric structure of the SCFT moduli space in a large class of models. In the final part, we study the stability of heterotic compactifications described by (0,2) GLSMs with respect to worldsheet instanton corrections to the space-time superpotential following the work of Beasley and Witten. We show that generic models elude the vanishing theorem proved there, and may not determine supersymmetric heterotic vacua. We then construct a subclass of GLSMs for which a vanishing theorem holds.

Nonlinear parameter retrieval from three- and four-wave mixing in metamaterials

We present a generalized nonlinear susceptibility retrieval method for metamaterials based on transfer matrices and valid in the nondepleted pump approximation. We construct a general formalism to describe the transfer matrix method for nonlinear media and apply it to the processes of three- and four-wave mixing. The accuracy of this approach is verified via finite element simulations. The method is then reversed to give a set of equations for retrieving the nonlinear susceptibility. Finally, we apply the proposed retrieval operation to a three-wave mixing transmission experiment performed on a varactor loaded split ring resonator metamaterial sample and find quantitative agreement with an analytical effective medium theory model. © 2010 The American Physical Society.

Fermion Mass Generation without Spontaneous Symmetry Breaking

The conventional mechanism of fermion mass generation in the Standard Model involves Spontaneous Symmetry Breaking (SSB). In this thesis, we study an alternate mechanism for the generation of fermion masses that does not require SSB, in the context of lattice field theories. Being inherently strongly coupled, this mechanism requires a non-perturbative approach like the lattice approach.

In order to explore this mechanism, we study a simple lattice model with a four-fermion interaction that has massless fermions at weak couplings and massive fermions at strong couplings, but without any spontaneous symmetry breaking. Prior work on this type of mass generation mechanism in 4D, was done long ago using either mean-field theory or Monte-Carlo calculations on small lattices. In this thesis, we have developed a new computational approach that enables us to perform large scale quantum Monte-Carlo calculations to study the phase structure of this theory. In 4D, our results confirm prior results, but differ in some quantitative details of the phase diagram. In contrast, in 3D, we discover a new second order critical point using calculations on lattices up to size $ 60^3$. Such large scale calculations are unprecedented. The presence of the critical point implies the existence of an alternate mechanism of fermion mass generation without any SSB, that could be of interest in continuum quantum field theory.