Abstract
In the present paper we introduce a modified Dirac operator and solve the associated Dirichlet boundary value problem in \({\mathbb{R}^3}\) for functions which are q–monogenic when the Clifford algebra depends on parameters. By determining the compatibility conditions we establish theorems of existence and uniqueness of solutions to the modified Dirac equation, and the associated Laplace equation, such that they correspond to a scalar operator. We also discuss the problem using fixedpoint methods and analyze a physical realization were the solutions to the Laplace equation can be interpreted as a set of electric potentials coupled to each other. Our solutions could be relevant in the analysis of new exotic phases of nature, such as topological insulators were the Dirac nature of the charge carriers implies new physical properties which go beyond the standard description of conventional charge carriers in electronic systems by means of the Schrödinger equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. Arnold, Ordinary Differential Equations. Springer–Verlag, New York– Berlin, (2006).
Balderrama C., Di Teodoro A., Infante A.: Some integral representation for meta–monogenic functions in Clifford algebras depending on parameters. Adv. Appl. Clifford Algebras 23, 793–813 (2013)
A. Di Teodoro and C. Vanegas, A Dirichlet Problem for the First Order Inhomogeneous Meta–Monogenic Equation in Parameter Depending Clifford Algebras. Complex Analysis and Operator Theory, June 2014, (Online First).
F.Brackx, R. Delanghe and F. Sommen, Clifford Analysis. Pitman Research Notes in Mathematics Vol.76, London, (1982).
H. Begehr, Integral representations in complex, hypercomplex and Clifford analysis. Integral transform and Special Functions 13 (2002), 223241.
Castro Neto A.H. et al.: Electronic properties of graphene. Rev. Mod. Phys. 81, 109 (2009)
Chen L. et al.: Evidence for Dirac Fermions in a Honeycomb Lattice Based on Silicon. Phys. Rev. Lett. 109, 056804 (2012)
R. Feynman, R. Leighton and M. Sands, The Feynman Lectures on Physics: Definitive and Extended Edition. Addison Wesley, (2005).
Fleurence A. et al.: Experimental Evidence for Epitaxial Silicene on Diboride Thin Films. Phys. Rev. Lett. 108, 245501 (2012)
Geim A.K., Novoselov K.S.: The rise of graphene. Nature Mater. 6, 183 (2007)
R.P. Gilbert and J. Buchanan, First Order Elliptic Systems. A Function Theoretic Approach. Mathematics in Science and Engineering, vol. 163. Academic Press, Massachusetts (1983)
Guzmán–Verri G.G., LewYan Voon L.C.: Electronic structure of siliconbased nanostructures. Phys. Rev. B 76, 075131 (2007)
J. Jackson, Classical Electrodynamics. Wiley, (1998).
Kane C.L., Mele E.J.: Quantum Spin Hall Effect in Graphene. Phys. Rev. Lett. 95, 226801 (2005)
C. Miranda, Partial differential equations of elliptic type. Second revised edition. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 2. Springer–Verlag, New York–Berlin, (1970).
Morimoto T., Furusaki A.: Topological classification with additional symmetries from Clifford algebras. Phys. Rev. B 88, 125129 (2013)
A.S.A. Mshimba and W. Tutschke (eds.), Functional Analytic Methods in Complex Analysis and Applications to Partial Differential Equations. Proceedings of the Second Workshop held at the ICTP in Trieste, January 25–29, 1993. World Scientific (1995).
Novoselov K.S. et al.: Electric Field Effect in Atomically Thin Carbon Films. Science 306, 666 (2004)
Peres N.M.R.: Scattering in one–dimensional heterostructures described by the Dirac equation J. Phys.: Condens. Matter 21, 095501 (2009)
E. Obolashvili, Partial Differential Equations in Clifford Analysis 1 ed., Chapman and Hall/CRC, (1999).
Takeda K., Shiraishi K.: Theoretical possibility of stage corrugation in Si and Ge analogs of graphite. Phys. Rev. B 50, 14916 (1994)
W. Tutschke, An elementary approach to Clifford analysis. Contained in the Proceedings [17], 402–408.
Tutschke W.: Generalized analytic functions in higher dimensions. Georgian Mathematical Journal vol. 14(no. 3), 581–595 (2007)
W. Tutschke and C.J. Vanegas, Clifford algebras depending on parameters and their applications to partial differential equations. Contained in Some topics on value distribution and differentiability in complex and p–adic analysis. Science Press Beijing (2008), 430–449.
W. Tutschke and C.J. Vanegas, A boundary value problem for monogenic functions in parameter–depending Clifford algebras. Complex Variables and Elliptic Equations vol. 56, issue 1–4 (2011), 113–118.
W. Tutschke and C.J. Vanegas, General algebraic structures of Clifford type and Cauchy–Pompeiu formulae for some piecewise constant structure relations. Adv. Appl. Clifford Algebras 21 4 (2011), 829–838.
Vogt P. et al.: Silicene: Compelling Experimental Evidence for Graphenelike Two–Dimensional Silicon. Phys. Rev. Lett. 108, 155501 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to professor Tutschke on the occasion of his 80th birthday
Rights and permissions
About this article
Cite this article
Teodoro, A.D., Franquiz, R. & López, A. A Modified Dirac Operator in Parameter–Dependent Clifford Algebra: A Physical Realization. Adv. Appl. Clifford Algebras 25, 303–320 (2015). https://doi.org/10.1007/s00006-014-0510-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00006-014-0510-0