### Research

I am primarily a quantum condensed matter physicist with a focus on
"strongly correlated" systems, although inevitably I also work on
problems in classical statistical mechanics as do all members of my
tribe. The term strongly correlated refers to materials in which the
interactions among the electrons are important, even for a qualitative
understanding of their behavior. This is in contrast to the textbook
examples of metals, semiconductors and insulators, whose behavior is
largely explicable in terms of independent electrons.

Currently, I am interested in the following problems:

Topological Phases: Phases of matter are
defined by their boundaries which are loci of non-analyticity of the
thermodynamic functions. The classical explanation of why there is more
than one phase of a system with a given microscopics, due to Landau, is
that the thermodynamic state of the system can break one or more
symmetries. Via Landau's notion of an order parameter this leads to
non-analyticities in the thermodynamics at a symmetry-breaking phase
transition. A while back Wegner constructed the first spin models with
phase transitions but no breaking of symmetries. His models were,
in subsequent parlance, lattice gauge theories with local symmetries.
Starting with work on the fractional quantum Hall effect and on the
cuprate superconductors there has been a growing recognition that
condensed matter systems can give rise to an "emergent" gauge structure
at low energies, i.e. unlike in Wegner's models it does not have to be
built in at the ground level. With this insight it is clear that
condensed matter systems can generally exhibit phases and phase
transitions whose raison d'etre is not symmetry breaking. While the list
of such "topological phases" in experimental systems is currently
small (fractional quantum Hall states, protons in ice obeying the
Bernal-Fowler rules and superconductivity with electromagnetic
interactions are examples) the list of models with such phases is
growing rapidly. My own interest has centered around the "resonating
valence bond" (RVB) state proposed by Anderson at about the same time as
Wegner's work which turns out to be the simplest of the topological
phases with an emergent Ising gauge field. In a series of papers my
collaborators and I have demonstrated its existence in quantum dimer
models and spin models and elucidated its description by a topological
gauge theory.

High temperature Superconductivity:
There is still no consensus on the microscopic origin of high
temperature superconductivity in the cuprates and it remains an area of
intense and active research. These are perhaps the most comprehensively
studied materials in history and at issue is giving a coherent
explanation of their entire phase diagram inclusive of
antiferromagnetism, superconductivity, the "pseudogap" region in
between and a host of interesting temperature dependences. While my work
on RVB theory bears on general theoretical questions in this area, my
direct involvement has grown out of experiments carried out by my
colleague Phuan Ong's group and is focused on understanding the effect
of superconducting fluctuations on the physics of the cuprates outside
the superconducting region. Specifically, my collaborators and I have
provided a theoretical account of such fluctuations in the Nernst effect
and in magnetization measurements.

Frustrated Magnetism: Classical
frustrated magnets are primarily identified by large ground state
degeneracies, the Ising antiferromagnet on a triangular lattice being
the canonical example of this phenomenon and of the subclass of *
geometrically* frustrated magnets. There is an increasing list of
laboratory realizations of such systems, especially of the geometrically
frustrated variety on lattices such as the kagome and pyrochlore. The
challenge for theorists is to work out the statistical mechanics which
is sensitive to the precise lattices and interactions at issue and can
exhibit the phenomenon of "order by disorder" in which a non-trivial
ordering is selected for entropic reasons. Further, interesting new
physics can arise when quantum dynamics is introduced into these large
ground state manifolds on account of the singular nature of the
perturbation. Understanding quantum effects is important for laboratory
realizations with small spins. My recent work in this area has covered
the coupling between lattice and magnetic degrees of freedom in the
pyrochlore system, the low temperature correlations of the same, the
statistical mechanics of spin ice and the study of quantum Ising models
on various frustrated lattices.

Quantum Phase Transitions: These are
continuous phase transitions that take place at absolute zero, i.e., in
the ground state of the system, when some parameter other than the
temperature is varied. They are very interesting, for quantum effects
are intrinsically important to them, most importantly in scrambling the
dynamics with the thermodynamics. This leads to a host of new features,
relative to their classical counterparts, which are an area of active
research. I have worked on such transitions between quantum Hall states
and most recently on the theory of non-linear transport in their
proximity which ends up involving the Schwinger-Zener mechanism of
vacuum breakdown .

Previously I have worked a fair amount on the fascinating physics of
the quantum Hall effect and I retain an interest in that problem.

Quantum Hall Effect: Two-dimensional
electron gases placed in high magnetic fields exhibit the quantum Hall
Effect, which reflects an underlying intricate set of novel phases. The
excitations in these phases have been of great interest for they are
believed to carry fractional quantum numbers, i.e., charge and
statistics. I've been interested in various aspects of these
excitations--whether they carry a third fractional quantum number (an
intrinsic spin), under what circumstances the various quantum numbers
can be measured in the laboratory, and their internal structure in
various limits.

For example, interesting variants of these excitations arise when the
spin of the electrons can fluctuate. In some cases the excitations
develop topologically nontrivial spin order ("skyrmions") as a
consequence of geometric, or Berry, phases in the system. This led to
questions about the role of these geometric phases near the edges of
quantum Hall systems where yet another class of fascinating excitations
lives ("edge states"), and other electronic systems where local spin
order and conduction coexist.

As this page will not always be completely up-to-date, the best way to
get a sense of my current research interests is to look up the recent
ones among my papers posted on the
cond-mat archive.

Some selected papers are listed here:

Skyrmions and the crossover from the integer to fractional quantum Hall effect
at small Zeeman energies (1993)

Off-Diagonal Long Range Order and Scaling in a Disordered Quantum Hall System
(1994)

Evidence for Charge-Flux Duality near the Quantum Hall Liquid-to-Insulator Transition
(1996)

Continuous quantum phase transitions (tutorial, 1997)

Resonating Valence Bond Phase in the Triangular Lattice Quantum Dimer Model
(2001)

Phase structure of non-commutative scalar field theories (2001)

Gaussian Superconducting Fluctuations, Thermal Transport, and the Nernst Effect (2002)

Dipolar Spin Correlations in Classical Pyrochlore Magnets (2004)

SU (2)-invariant spin-1/2 Hamiltonians with RVB and other valence bond phases
(2005)