My doctoral work focused on theories of strongly-correlated electron systems in which the interactions between electrons lead to exotic (i.e. not electron-like) low-energy excitations. In particular, I have explored ideas of ``spin-charge separation'' in the cuprate materials which display high-temperature superconductivity and abnormal non-superconducting phases. My research has emphasized both experimental consequences of these ideas and the physical consequences of powerful theoretical tools such as duality. I intend to further develop these lines of inquiry in new directions and to begin explorations of other interesting condensed matter phenomena. Not only do these lines of research promise exciting developments, but they are particularly adaptive to undergraduate involvement.

In one-dimensional systems, the existence of exotic, spin-charge separated excitations has both a firm theoretical grounding and experimental confirmations. The ubiquity of this striking physics stems from special properties of electrons confined to move in one spatial dimension: phase space restrictions make interactions between electrons of primary importance. In contrast, the stunning success of Fermi Liquid theory in describing three-dimensional electron systems tells us that once these kinetic restrictions are removed, electron interactions become mostly irrelevant. The middle ground posed by electrons confined to move in two dimensions has been gaining attention in recent years, perhaps due to the excitement generated by the discovery of the quasi-two-dimensional high-temperature superconductors. The guiding notion of my doctoral research is that the elementary low-energy excitations of some strongly-correlated 2d electron systems could carry fractions of the quantum numbers of the electron (e.g. an excitation with charge $e$ and a separate excitation with spin $1/2$). Finding a model (and thence to a set of physical circumstances) which actually displays this property has proven difficult. Recent work of my doctoral advisor, Matthew Fisher, and his collaborator, Todadri Senthil, has laid out a model which meets this goal. Standing on the firm footing of a well-defined model which exhibits these exotic properties has allowed us to explore the
physical consequences of these exciting ideas within a well-defined framework.

Two of my recent papers worked from a phenomenological model for these exotic excitations and derived experimental consequences. In both papers I found robust features which are not dependent on the details of the model used but rather bring into focus the qualitative and physical distinctions between this ``exotic'' phase and more conventional phases. For instance, the spectral function, which is the primary object measured in angle-resolved photoemission experiments, displays properties which are markedly different from those expected from any Fermi Liquid. This experiment allows one to access the spectrum of electron excitations in the system. In a Fermi Liquid, the elementary excitations of the system are electrons at low energies, hence there is some window of energy and momentum where the electron is ``long-lived.'' If instead the system's low-lying excitations are separate spin- and charge-carrying particles, then an electron
injected into the system ``decays'' into these constituents. The corresponding spectral functions for these two scenarios illustrates physically an essential distinction between these two possible phases. In the second paper, I consider the ramifications of spin-charge separation for a system with magnetic order. The low-lying excitations of any simple anti-ferromagnet are the spin waves dictated by the broken symmetry of the ground state (just as a solid has its phonons). However, probing to higher energies one may ask what other spin-carrying excitations emerge. In a system where there exists a well-defined excitation with zero charge and spin $1/2$, probing the spin physics will eventually reveal the existence of these strange particles, usually called spinons in the literature. We have derived a form for their response to neutron scattering as well as laying out a theory which allows one to systematically address the question of how these spin-arrying excitations interact with the spin waves. In the short term, my plans for continuing research in this area include understanding the consequences of spin-charge separation for quantities like the electrical and thermal conductivity. The Wiedemann-Franz law gives us a relationship between these two quantities which would seem to be violated if the system had excitations which carried heat but no charge (such as the spinons described above).

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In both of these papers, I have taken an exciting theoretical idea and worked out robust and measurable consequences. I believe that this is an extremely important aspect of exploring any particular
model, because the physics is contained in the physical response of the system to external probes. In the future, I would like to continue making contact between exciting theoretical developments and experimental consequences. One aspect of this is dialog with experimental groups. While working on these papers, I was afforded the opportunity to talk at length with experimenters doing angle resolved photoemission spectroscopy (ARPES) and neutron scattering on these and other interesting systems at various places around the country. This collaboration was extremely helpful to me in understanding the details of the experiments and in most cases led me to explore aspects of the theory which I would have otherwise overlooked. For the most part, this dialog was conducted over email and phone conversations and I plan to continue conversations with experimental groups. I feel that particularly for beginning scientists, a firm grounding in the real world is an essential research tool. By exploring the physical consequences of models,
undergraduate researchers gain first-hand working knowledge of the scientific method.

Above one-dimension, one might imagine that for a system with exotic excitations there exists some range of parameters where the electron regains its status as the fundamental excitation of the system. This has lead me and my collaborators to consider an extremely interesting sort of phase transition where these spin-charge separated excitations are ``glued back together'' to form electrons. Here, having a model which
describes both types of phases is key. Numerical work on the model of Senthil and Fisher indicates that there is such a transition, but thorough characterization of this transition will be difficult. In earlier work, I have attempted to access such a transition in a simplified model. Using field theory techniques, I derived the
two-loop renormalization group equations for the transition in this new universality class. Additionally, using powerful duality techniques it was possible to characterize the different transitions from superconductor to
spin-charge confined insulator at the mean field level. This work taught me the power of particle-vortex duality in two dimensions. The transition of bosons from their disordered phase to their superfluid phase can be accessed via the usual routes. The concept of duality allows one to characterize the nominally disordered phase as the ordered phase of a new particle--the vortex, thereby making the transition from the superfluid to the insulator an {ordering}transition for the vortices. Duality thereby provides a framework for understanding this phase transition from both sides. In general, theories of defects in ordered phases (for instance, some liquid crystals phases) are extremely powerful. This work can also be applied to models of
interacting bosons in two dimensions. Recent numerical work reveals many phases of these models with confusing and poorly-characterized transitions between them. In future work, I plan to more fully understand the correspondence between physical quantities expressed in terms of the original bosons and in terms of
their duals, the vortices. For instance, if the bosons are charged, can one derive a formula for the electrical conductivity in terms of the vortices? Once these relationships are mapped out and understood, one is able
to apply all the usual calculational techniques in otherwise-unexplored phases.

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The recent focus of my work has been many-body wavefunctions of strongly-correlated electron systems. Condensed matter has two wildly successful examples of many-body wavefunctions that capture the essential physics of the system under study: the BCS wavefunction of superconductivity and Laughlin's
wavefunction for the quantum hall effect. In both cases these wavefunctions provide abundant physical insight into these amazing cooperative phenomena. They can also be used as jumping-off points for
physical situations where the same sorts of cooperation might occur. Two concrete examples are high-temperature superconductors and two-dimensional electron gases at low density. In the $T>T_c$ phase of underdoped high-temperature superconductors, there is abundant evidence that the system is not behaving as a Fermi Liquid. The question is whether some of the key elements of the BCS state are surviving to high temperatures despite the loss of superconductivity. Physically, this is very similar to the spin-charge separation detailed above. The physical picture of electrons which are in some sense ``already paired'' but lack phase coherence and hence superconductivity is an extremely powerful one. Understanding this phenomenon from a many-body wavefunction point of view would allow one to calculate many physical quantities and would give enormous physical insight. In recent experiments on two-dimensional electron gases at low density, a puzzling phase emerges as electron interactions (and possibly disorder) destroy the metallic state. Does an analog of the Laughlin wavefunction, where the electrons avoid each other at a cost in kinetic energy, become favorable in some regime? In both of these systems, variational computations on trial wavefunctions have a chance to illuminate some of the strange physics occurring.

Wavefunction analysis involves only a rudimentary knowledge of quantum mechanics and the variational minimization can be accomplished with minimal computer time and sophistication. In some cases, an
analytic understanding of the trade-offs in energy can be obtained with minimal ``fancy math''. I believe this line of inquiry is ideal for undergraduate involvement since the maximal physical insight is given from the
minimal mathematical sophistication.

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In the longer view, in five years I plan to have branched out into the realm of soft condensed matter, doing modeling of systems such as glasses. The low-temperature properties of these systems remain largely a mystery, due in part to the fact that they ``fall out of equilibrium'' and do not conform to many of the oldest and most worked-out paradigms of condensed matter. The ``transition'' from a liquid to a glass represents a new frontier for those interested in phase transitions and statistical mechanics. This proposed line of research, which involves numerical modeling and involves no quantum mechanics, is particularly accessible to undergraduates who hope to do exciting and relevant physics research.

Condensed matter offers undergraduates in particular a rich playfield from which to explore physics. Doing theoretical research gives them the opportunity to use their course work to gain insight into actual physical systems, thereby practicing the scientific method. In many cases, some of the most exciting new physical developments and systems (such as strongly correlated electrons) can be tackled using rather elementary methods (such as wavefunction analysis). In this way, the student can make a concrete contribution to a current field of research. My highest research priority is creating an accessible starting point for undergraduates to do science in this exciting field. I believe my interests and experience place me in a unique position to mentor their explorations.

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