This award supports theoretical research and education in condensed matter physics. The two central research topics are (i) strongly fluctuating quantum magnetism, typically in low dimensions, and (ii) transport and interfacial physics in graphene and graphite. The major objective in the magnetism work is to characterize the physics of quantum-disordered phases using several approaches. One such approach is through the use of quantum entanglement spectra, which have been shown to yield detailed information regarding the edge excitations of fractional quantum Hall effect states. This line of analysis exploits some deep connections between the fractional quantum Hall effect and spin chains. Partitioning the fractional quantum Hall effect wavefunctions in angular momentum space is equivalent to a reciprocal momentum space partitioning of the corresponding spin chain wavefunctions, and reveals information about the bulk excitation spectra in certain gapless systems. Another direction of this research is in the generalization of valence bond solid states and an investigation of their properties, such as the "hidden" string order in the S=1 Affleck, Kennedy, Lieb, and Tasaki chain, which is emblematic of the Haldane phase. Generalizations to SU(N) spins and to singlets extended over N-site simplices will be investigated, along with corresponding higher-dimensional fractional quantum Hall effect wavefunctions.
The work on graphene will focus on junctions and interfaces in graphene and graphite, including junctions between monolayer and bilayer graphene, and also with graphane. The effects of periodic potentials will be studied, both with and without external fields. In graphite, the c-axis transport of the turbostratic material will be modeled, along with the electronic structure of crystalline dislocations.
The educational elements of this work include the training of a graduate student and the development of detailed, book-quality, lecture notes for graduate students and advanced undergraduates.
NONTECHNICAL SUMMARY
This award supports theoretical research and education in condensed matter physics. The research effort will focus on two main topics: (i) magnetism at the microscopic level, and (ii) graphene, which is a one-atom-thick planar sheet of carbon atoms. A common thread is that many of the systems studied are intrinsically one-dimensional or two-dimensional.
Low-dimensional quantum magnetism has provided the condensed matter community with some remarkable and compelling paradigms of "quantum-disordered" phases - states of matter where quantum mechanical fluctuations, even at the lowest possible temperatures, lead to a "melting" of classical order. At issue is how to characterize these disordered phases. A recent approach uses entanglement, an intrinsically quantum-mechanical concept, to characterize such states. Deep connections between one-dimensional quantum magnets and two-dimensional electron gases in a strong magnetic field will also be exploited and investigated.
Graphene has an unusual electronic structure, in which charge carrying excitations behave as if they are massless, reminiscent of photons, the quanta of light. The research effort here will focus on large-scale inhomogeneities in graphene, including interfaces between single and multilayer graphene, and disruptions in the stacking pattern of graphite. In both cases, the consequences for electronic conduction will be investigated.
This is fundamental research that contributes to the intellectual foundations of future device technologies. Graphene, in particular, has unique properties that make it a promising material for various applications including future electronic devices.
The educational aspects of this proposal include the training of a PhD student in physics, and the further development of an extensive collection of detailed, book-quality lecture notes for advanced undergraduates and physics graduate students.
This project had three main outcomes over the report submission period. The first set of results concerned what is known as the Higgs mode in condensed matter systems. The "Higgs boson" has been in the public eye since its discovery at CERN in 2012. As has been widely reported, the Higgs boson is of fundamental significance to our entire universe, as it gives rise to the masses of elementary particles. Mathematically, the Higgs boson is realized in systems with a "spontaneously broken continuous symmetry". This very fundamental and important concept can be apprehended by considering a highly simplified model of a little ball sliding around inside the "Mexican hat" shape depicted in Figure 1. The lowest energy state for the ball is anywhere along the bottom trough. When the ball picks a particular spot along the trough, it spontaneously breaks the continuous rotational symmetry of the Mexican hat. Now there are two types of motions the ball can perform. First, it could move along the trough in either direction (the green arrows in the figure). Since all points along the trough are equivalent, such motion involves only kinetic energy, which can be arbitrarily small. This excitations corresponds to a "Goldstone boson". Radial oscillations, however, change the ball's potential energy, resulting in a finite frequency oscillation known as the Higgs. It is important to emphasize that the condensed matter Higgs mode is not an elementary particle like that discovered at CERN, but rather is an elementary *excitation* of a system, such as a magnet or a superconductor, which can be observed in a laboratory. One very important aspect is the dimension of space in which our system lives. For example, surfaces or interfaces are two-dimensional, in contrast with bulk three-dimensional materials. One important aspect to our work was to understand how the condensed matter Higgs mode could be observed in systems in two space dimensions. Previously it had been thought that the Higgs was "overdamped" and impossible to observe, but our work showed that if one does the proper type of experiment, the Higgs mode can be observed, and indeed it was recently found in beautiful experiments on trapped cold atomic gases. Our most recent work on the Higgs focused on the electrical conductivity of charged systems exhibiting broken continuous symmetry, such as superconductors. We studied the special properties of such systems just as they are on the cusp of transitioning between superconducting and insulating states - a very delicate state of affairs known as a quantum phase transition. Another significant outcome from the review period was work on topological characterizations of thermal and dissipative systems. Over the past 25 years, the role of topology - the mathematical study of properties unchanged by smooth deformations - has risen to the fore in condensed matter physics. Discrete "topological quantum numbers" are used to describe distinct phases of matter, and transitions between different topological phases in matter are described by discrete jumps (such as from 1 to 2). Typically such classifications are only possible at zero temperature. With my graduate student, I showed how such a characterization could be defined mathematically even at finite temperature. Turning now to summarize the remainder of the entire award, much of the work was done with my graduate students Z. Huang and Y. Kiselev. Their training constituted one of the main broader impacts of my work. Huang's early research was on the quantum entanglement properties of topological phases. One common feature of topological phases is the existence of edge or surface states. These states have the potential of being quite useful - they may carry dissipationless electrical current. for example. When one investigates how these states are entangled at the quantum level, one finds that "internal" versions of these edge states play an important role. Huang and I also investigated, in collaboration with A. Balatsky, the physics of so-called "Weyl semimetals", which are states of matter interpolating between topological and nontopological phases. In particular, we studied the effects of impurities on the WS phase. With Kiselev, we examined some exotic magnetic models known as "simplex solids", which I had first described in 2008. We showed, using a variety of techniques, how these systems could either exhibit somewhat conventional classical order, or could instead be in a "quantum disordered" phase, where the classical order is melted by quantum fluctuations. I also worked, with collaborators, on graphene, which is an atomically thin two-dimensional honeycomb lattice of carbon atoms. Two projects here were to study (i) the effects of irradiating graphene nanoribbons, and (ii) to analyze experiments on trilayers of graphene sheets exposed to a large magnetic field. For example, we showed how we could theoretically infer the stacking order of the trilayer from its electrical conductivity. All our major results have been communicated in peer-reviewed journals.
