Multiparticle equations for interacting Dirac fermions in magnetically confined graphene quantum dots

We study the energy of quasi-particles in graphene within the Hartree-Fock approximation. The quasi-particles are confined via an inhomogeneous magnetic field and interact via the Coulomb potential. We show that the associated functional has a minimizer and determines the stability conditions for the N-particle problem in such a graphene quantum dot

On the Maximal Excess Charge of the Chandrasekhar-Coulomb Hamiltonian in Two Dimensions

We show that for the straightforward quantized relativistic Coulomb Hamiltonian of a two-dimensional atom -- or the corresponding magnetic quantum dot -- the maximal number of electrons does not exceed twice the nuclear charge. It result is then generalized to the presence of external magnetic fields and atomic Hamiltonians. This is based on the positivity of |\bx| T(\bp) + T(\bp) |\bx| which -- in two dimensions -- is false for the non-relativistic case T(\bp) = \bp^2, but is proven in this paper for T(\bp) = |\bp|, i.e., the ultra-relativistic kinetic energy

Localization of two-dimensional massless Dirac fermions in a magnetic quantum dot

We consider a two-dimensional massless Dirac operator $H$ in the presence of a perturbed homogeneous magnetic field $B=B_0+b$ and a scalar electric potential $V$. For $V\in L_{\rm loc}^p(\R^2)$, $p\in(2,\infty]$, and $b\in L_{\rm loc}^q(\R^2)$, $q\in(1,\infty]$, both decaying at infinity, we show that states in the discrete spectrum of $H$ are superexponentially localized. We establish the existence of such states between the zeroth and the first Landau level assuming that V=0. In addition, under the condition that $b$ is rotationally symmetric and that $V$ satisfies certain analyticity condition on the angular variable, we show that states belonging to the discrete spectrum of $H$ are Gaussian-like localized

Ground state properties of graphene in Hartree-Fock theory

We study the Hartree-Fock approximation of graphene in infinite volume, with instantaneous Coulomb interactions. First we construct its translation-invariant ground state and we recover the well-known fact that, due to the exchange term, the effective Fermi velocity is logarithmically divergent at zero momentum. In a second step we prove the existence of a ground state in the presence of local defects and we discuss some properties of the linear response to an external electric field. All our results are non perturbative.Comment: 27 page

Orbital ferromagnetism in interacting few-electron dots with strong spin-orbit coupling

We study the ground state of $N$ weakly interacting electrons (with $N\le 10$) in a two-dimensional parabolic quantum dot with strong Rashba spin-orbit coupling. Using dimensionless parameters for the Coulomb interaction, $\lambda\lesssim 1$, and the Rashba coupling, $\alpha\gg 1$, the low-energy physics is characterized by an almost flat single-particle dispersion. From an analytical approach for $\alpha\to \infty$ and $N=2$, and from numerical exact diagonalization and Hartree-Fock calculations, we find a transition from a conventional unmagnetized ground state (for $\lambda<\lambda_c$) to an orbital ferromagnet (for $\lambda>\lambda_c$), with a large magnetization and a circulating charge current. We show that the critical interaction strength, $\lambda_c=\lambda_c(\alpha,N)$, vanishes in the limit $\alpha\to \infty$.Comment: 15 pages, 9 figures; (v2) more discussion added, fig.8 correcte

Electron states in the field of charged impurities in two-dimensional Dirac systems

We review the theoretical and experimental results connected with the electron states in two-dimensional Dirac systems paying a special attention to the atomic collapse in graphene. Two-electron bound states of a Coulomb impurity are considered too. A rather subtle role of a magnetic field in the supercritical charge problem in graphene is discussed. The electron states in the field of two equally charged impurities are studied and the conditions for supercritical instability to occur are determined. It is shown that the supercriticality of novel type is realized in gapped graphene with two unlikely charged impurities. For sufficiently large charges of impurities, it is found that the wave function of the occupied electron bound state of the highest energy changes its localization from the negatively charged impurity to the positively charged one as the distance between the impurities increases. The specifics of the atomic collapse in bilayer graphene is considered and it is shown that the atomic collapse in this material is not related to the phenomenon of the fall-to-center.Comment: Review Article, 39 pages, 14 figures. arXiv admin note: text overlap with arXiv:1311.0064, arXiv:1401.5992, arXiv:1611.05221, arXiv:1510.02890, arXiv:1406.5770 by other author

Lieb–Thirring and Cwickel–Lieb–Rozenblum inequalities for perturbed graphene with a Coulomb impurity

We study the two dimensional massless Coulomb–Dirac operator restricted to its positive spectral subspace and prove estimates on the negative eigenvalues created by electromagnetic perturbations

Nanoscopic systems driven by magnetic and electric AC fields

Tesis doctoral inédita, leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Física de la Materia Condensada. Fecha de lectura: 15-06-201

Ground state properties of graphene in Hartree-Fock theory

Abstract. We study the Hartree-Fock approximation of graphene in infinite volume, with instantaneous Coulomb interactions. First we construct its translation-invariant ground state and we recover the well-known fact that, due to the exchange term, the effective Fermi velocity is logarithmically divergent at zero momentum. In a second step we prove the existence of a ground state in the presence of local defects and we discuss some properties of the linear response to an external electric field. All our results are non perturbative. c 2012, by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes