Graphene's honeycomb lattice structure is quite remarkable in the sense that it leads, in the long wavelength limit, to a massless Dirac equation description of nonrelativistic quasiparticles associated with electrons and holes present in the two dimensional crystallite. In the case of cold bosonic atoms trapped in a honeycomb optical lattice, Haddad and Carr (2009) have recently shown, by taking into account binary contact interactions, that the dynamics of these Bose-Einstein condensates is governed by a nonlinear Dirac equation (NLDE). In this paper, we study exact stationary solutions of such a NLDE. After proving that the energy eigenvalues are real, we show that the sum of orbital angular momentum and pseudospin angular momentum normal to the crystal commutes with the nonlinear Hamiltonian whenever magnitudes of the pseudospin components do not depend on the polar angle $ϕ$ . We obtain some exact stationary and localized solutions of the NLDE.Graphene's honeycomb lattice structure is quite remarkable in the sense that it leads, in the long wavelength limit, to a massless Dirac equation description of nonrelativistic quasiparticles associated with electrons and holes present in the two dimensional crystallite. In the case of cold bosonic atoms trapped in a honeycomb optical lattice, Haddad and Carr (2009) have recently shown, by taking into account binary contact interactions, that the dynamics of these Bose-Einstein condensates is governed by a nonlinear Dirac equation (NLDE). In this paper, we study exact stationary solutions of such a NLDE. After proving that the energy eigenvalues are real, we show that the sum of orbital angular momentum and pseudospin angular momentum normal to the crystal commutes with the nonlinear Hamiltonian whenever magnitudes of the pseudospin components do not depend on the polar angle $ϕ$. We obtain some exact stationary and localized solutions of the NLDE.

arXiv:1012.0976Dec 5, 20101 citation

The Pauli Equation and the Dirac Operator

Peter Woit

Springer eBooksJan 1, 20171 citation

Quantum states in rotating electromagnetic fields

Boris V. Gisin

We describe a new class of exact square integrable solutions of the Pauli and Dirac equation in rotating electromagnetic fields. Solutions obtained by putting equations in the stationary form with help of a coordinate transformation corresponding to the transition into a rotating frame. The transformation is assumed to be Galilean one however a non-Galilean transformation is of particular interest for such solutions. Obtained solutions, especially of Dirac's equation, are valid for arbitrary values of parameters and may be tested experimentally.We describe a new class of exact square integrable solutions of the Pauli and Dirac equation in rotating electromagnetic fields. Solutions obtained by putting equations in the stationary form with help of a coordinate transformation corresponding to the transition into a rotating frame. The transformation is assumed to be Galilean one however a non-Galilean transformation is of particular interest for such solutions. Obtained solutions, especially of Dirac's equation, are valid for arbitrary values of parameters and may be tested experimentally.

arXiv:1011.2622Nov 11, 20101 citation

Model Wavefunctions For Non-Abelian Quasiparticles

Bogdan Andrei Bernevig, F. D. M. Haldane

Bulletin of the American Physical SocietyMar 10, 20080 citations

On the physical applications of hyper-Hamiltonian dynamics

Giuseppe Gaeta, Miguel A. Rodrı́guez

An extension of Hamiltonian dynamics, defined on hyper-Kahler manifolds ('hyper-Hamiltonian dynamics') and sharing many of the attractive features of standard Hamiltonian dynamics, was introduced in previous work. In this paper, we discuss applications of the theory to physically interesting cases, dealing with the dynamics of particles with spin 1/2 in a magnetic field, i.e. the Pauli and the Dirac equations. While the free Pauli equation corresponds to a hyper-Hamiltonian flow, it turns out that the hyper-Hamiltonian description of the Dirac equation, and of the full Pauli one, is in terms of two commuting hyper-Hamiltonian flows. In this framework one can use a factorization principle discussed here (which is a special case of a general phenomenon studied by Walcher) and provide an explicit description of the resulting flow. On the other hand, by applying the familiar Foldy?Wouthuysen and Cini?Tousheck transformations (and the one recently introduced by Mulligan) which separate?in suitable limits?the Dirac equation into two equations, each of these turn out to be described by a single hyper-Hamiltonian flow. Thus the hyper-Hamiltonian construction is able to describe the fundamental dynamics for particles with spin.An extension of Hamiltonian dynamics, defined on hyper-Kahler manifolds ('hyper-Hamiltonian dynamics') and sharing many of the attractive features of standard Hamiltonian dynamics, was introduced in previous work. In this paper, we discuss applications of the theory to physically interesting cases, dealing with the dynamics of particles with spin 1/2 in a magnetic field, i.e. the Pauli and the Dirac equations. While the free Pauli equation corresponds to a hyper-Hamiltonian flow, it turns out that the hyper-Hamiltonian description of the Dirac equation, and of the full Pauli one, is in terms of two commuting hyper-Hamiltonian flows. In this framework one can use a factorization principle discussed here (which is a special case of a general phenomenon studied by Walcher) and provide an explicit description of the resulting flow. On the other hand, by applying the familiar Foldy?Wouthuysen and Cini?Tousheck transformations (and the one recently introduced by Mulligan) which separate?in suitable limits?the Dirac equation into two equations, each of these turn out to be described by a single hyper-Hamiltonian flow. Thus the hyper-Hamiltonian construction is able to describe the fundamental dynamics for particles with spin.

Journal of Physics A Mathematical and TheoreticalApr 15, 20085 citations

ϕ

. We obtain some exact stationary and localized solutions of the NLDE.