Brazilian Journal of Physics

, Volume 47, Issue 2, pp 215–230 | Cite as

Introduction to Topological Phases and Electronic Interactions in (2+1) Dimensions

  • Leandro O. Nascimento
Condensed Matter


A brief introduction to topological phases is provided, considering several two-band Hamiltonians in one and two dimensions. Relevant concepts of the topological insulator theory, such as: Berry phase, Chern number, and the quantum adiabatic theorem, are reviewed in a basic framework, which is meant to be accessible to non-specialists. We discuss the Kitaev chain, SSH, and BHZ models. The role of the electromagnetic interaction in the topological insulator theory is addressed in the light of the pseudo-quantum electrodynamics (PQED). The well-known parity anomaly for massless Dirac particle is reviewed in terms of the Chern number. Within the continuum limit, a half-quantized Hall conductivity is obtained. Thereafter, by considering the lattice regularization of the Dirac theory, we show how one may obtain the well-known quantum Hall conductivity for a single Dirac cone. The renormalization of the electron energy spectrum, for both small and large coupling regime, is derived. In particular, it is shown that massless Dirac particles may, only in the strong correlated limit, break either chiral or parity symmetries. For graphene, this implies the generation of Landau-like energy levels and the quantum valley Hall effect.


Condensed matter physics Graphene Field theory methods 



This work was partly supported by: the Ministry of Science, Technology and Innovation of Brazil (MCTI-Brazil); the Ministry of Education and Culture of Brazil (MEC-Brazil); and the program “Science without Borders” of National Council for Scientific and Technological Development (CNPQ-Brazil). I am grateful to E. C. Marino, V. S. Alves, C. Morais Smith, T. Macri, R. G. Pereira, and L. Fritz for very interesting and stimulating discussions.


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Copyright information

© Sociedade Brasileira de Física 2017

Authors and Affiliations

  1. 1.International Institute of PhysicsCampus Universitário-Lagoa-NovaNatalBrazil
  2. 2.Universidade Federal do Pará, Faculdade de Ciências NaturaisBrevesBrazil

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