Honeycomb Lattice Potentials and Dirac Points

We prove that the two-dimensional Schroedinger operator with a potential having the symmetry of a honeycomb structure has dispersion surfaces with conical singularities (Dirac points) at the vertices of its Brillouin zone. No assumptions are made on the size of the potential. We then prove the robustness of such conical singularities to a restrictive class of perturbations, which break the honeycomb lattice symmetry. General small perturbations of potentials with Dirac points do not have Dirac points; their dispersion surfaces are smooth. The presence of Dirac points in honeycomb structures is associated with many novel electronic and optical properties of materials such as graphene.Comment: To appear in Journal of the American Mathematical Society; 54 pages, 2 figures [note: earlier replacement was original version

Wave packets in Honeycomb Structures and Two-Dimensional Dirac Equations

In a recent article [10], the authors proved that the non-relativistic Schr\"odinger operator with a generic honeycomb lattice potential has conical (Dirac) points in its dispersion surfaces. These conical points occur for quasi-momenta, which are located at the vertices of the Brillouin zone, a regular hexagon. In this paper, we study the time-evolution of wave-packets, which are spectrally concentrated near such conical points. We prove that the large, but finite, time dynamics is governed by the two-dimensional Dirac equations.Comment: 34 pages, 2 figure

Waves in Honeycomb Structures

We review recent work of the authors on the non-relativistic Schr\"odinger equation with a honeycomb lattice potential, $V$. In particular, we summarize results on (i) the existence of Dirac points, conical singularities in dispersion surfaces of $H_V=-\Delta+V$ and (ii) the two-dimensional Dirac equations, as a large, but finite time, effective description of $e^{-iH_Vt}\psi_0$, for data $\psi_0$, which is spectrally localized at a Dirac point. We conclude with a formal derivation and discussion of the effective large time evolution for the nonlinear Schr\"odinger - Gross Pitaevskii equation for small amplitude initial conditions, $\psi_0$. The effective dynamics are governed by a nonlinear Dirac system.Comment: 11 pages, 2 figures, 39 \`emes Journ\'ees EDP - Biarretz. arXiv admin note: text overlap with arXiv:1212.607

\u27Friend to the Martyr, a Friend to the Woman of Shame\u27: Thinking About the Law, Shame and Humiliation

This paper considers the intersection between law, humiliation and shame, and how the law has the capacity to allow for, to encourage, or (in some cases) to remediate humiliation, or humiliating or shaming behavior. The need for new attention to be paid to this question has increased exponentially as we begin to also take more seriously international human rights mandates, especially – although certainly not exclusively – in the context of the recently-ratified United Nations Convention on the Rights of Persons with Disabilities, a Convention that calls for “respect for inherent dignity,” and characterizes discrimination against any person on the basis of disability [as] a violation of the inherent dignity and worth of the human person... Humiliation and shaming, we believe, contravene basic fundamental human rights and raise important constitutional questions implicating the due process and equal protection clauses. Humiliation and shaming practices include “scarlet letter”-like criminal sanctions, police stop-and-frisk practices, the treatment of persons with mental disabilities in the justice system, and the use of sex offender registries. Humiliation and shame are detrimental in the ways that lead to recidivism, inhibit rehabilitation, discourage treatment, and injure victims. They also directly contravene the guiding principles of therapeutic jurisprudence, especially in the context of its relationship to the importance of dignity in the law, and potentially violate international human rights law principles as well.In this paper, we will explore how humiliation and shaming are bad for all participants in the legal system, and bad for the law itself. We will urge that humiliating and shaming techniques be banned, and that, this ban will enhance dignity for the entire legal system and society as a whole. First, we consider the meaning of shame and humiliation. Then, we briefly discuss principles of therapeutic jurisprudence (TJ) and its relationship to the significance of dignity, and then consider recent developments in international human rights law, both of which are valuable interpretive tools in this conversation. Next, we consider how the United States Supreme Court has considered these concepts in recent cases. Following this, we consider several relevant areas of law and policy from theperspective of how overt shaming is employed: scarlet letter punishments, use of the police power, treatment of institutionalized persons with mental disabilities and elders, and sex offender registry law. We then, using a TJ filter and drawing on international human rights law principles, examine why these shaming tactics are contrary to bedrock principles of the legal system: the mandates to honor dignity, to minimize recidivism, and to enhance rehabilitation