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

## Abstract

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.

## Keywords

Condensed matter physics Graphene Field theory methods## Notes

### Acknowledgments

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.

## References

- 1.K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubong, I.V. Grigorieva, A. A. Firsov, Electric field effect in atomically thin carbon films. Science.
**306**, 666 (2004)ADSCrossRefGoogle Scholar - 2.B. Lalmi, H. Oughaddou, H. Enriquez, A. Kara, S. Vizzini, B. Ealet, B. Aufray, Epitaxial growth of a silicene sheet. Appl. Phys. Lett.
**97**, 223109 (2010)ADSCrossRefGoogle Scholar - 3.L. Li, Y. Yu, G.J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X.H. Chen, Y. Zhang, Black phosphorus field-effect transistors. Nat. Nanotech.
**9**, 372 (2014)ADSCrossRefGoogle Scholar - 4.X.-S. Ye, Z.-G. Shao, H. Zhao, L. Yang, C.-L. Wang, Intrinsic carrier mobility of germanene is larger than graphene’s: first-principle calculations. RSC Adv.
**4**, 21216–21220 (2014)CrossRefGoogle Scholar - 5.Q.H. Wang, K.K. Zadeh, A. Kis, J.N. Coleman, M.S. Stran, Electronics and optoelectronics of two-dimensional transition metal dichalcogenides. Nat. Nanotech.
**7**, 699 (2012)ADSCrossRefGoogle Scholar - 6.B.A. Bernevig, T. Hughes,
*Topological insulators and topological superconductors*. Princeton University Press (2013)Google Scholar - 7.S.-Q. Shen,
*Topological insulators Dirac equation in condensed matter*. Spring Series in Solid-State Sciences (2012)Google Scholar - 8.M.Z. Hasan, C.L. Kane, Colloquium: topological insulators. Rev. Mod. Phys.
**82**, 3045 (2010)ADSCrossRefGoogle Scholar - 9.A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, The electronic properties of graphene. Rev. Mod. Phys.
**81**, 109 (2009)ADSCrossRefGoogle Scholar - 10.A.Y. Kitaev, Unpaired Majorana fermions in quantum wires. Phys.-Usp.
**44**, 131 (2001)ADSCrossRefGoogle Scholar - 11.W.P. Su, J.R. Schrieffer, A.J. Heeger, Solitons in polyacetylene. Phys. Rev. Lett.
**42**, 1698 (1979)ADSCrossRefGoogle Scholar - 12.C.L. Kane, E.J. Mele, Quantum spin Hall effect in graphene. Phys. Rev. Lett.
**95**, 226801 (2005)ADSCrossRefGoogle Scholar - 13.E.C. Marino, L.O. Nascimento, V.S. Alves, C. Morais Smith, Interaction induced quantum valley Hall effect in graphene. Phys. Rev. X.
**5**, 011040 (2015)Google Scholar - 14.F.D.M. Haldane, Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett.
**61**, 2015 (1988)ADSCrossRefGoogle Scholar - 15.T. Appelquist, M.J. Bowick, E. Cohler, L.C.R. Wijewardhana, Chiral-symmetry breaking in 2+1 dimensions. Phys. Rev. L.tt.
**55**, 1715 (1985)ADSCrossRefGoogle Scholar - 16.C.D. Roberts, A.G. Williams, Dyson-Schwinger equations and their application to hadronic physics. Prog. Part. Nucl. Phys.
**33**, 477 (1994)ADSCrossRefGoogle Scholar - 17.P. Maris, Influence of the full vertex and vacuum polarization on the fermion propagator in (2+1)-dimensional QED. Phys. Rev. D.
**54**, 4049 (1996)ADSCrossRefGoogle Scholar - 18.E.C. Marino, Quantum electrodynamics of particles on a plane and the Chern-Simons theory. Nucl. Phys. B.
**408**, 551 (1993)ADSMathSciNetCrossRefMATHGoogle Scholar - 19.A. Kovner, B. Rosenstein, Kosterlitz-Thouless mechanism of two-dimensional superconductivity. Phys. Rev. B.
**42**, 4748 (1990)ADSCrossRefGoogle Scholar - 20.M. Atala, M. Aidelsburger, J.T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, I. Bloch, Direct measurement of the Zak phase in topological Bloch bands. Nat. Phys.
**9**, 795 (2013)CrossRefGoogle Scholar - 21.D. Sticlet, F. Piéchon, J.-N. Fuchs, P. Kalugin, P. Simon, Geometrical engineering of a two-band Chern insulator in two dimensions with arbitrary topological index. Phys. Rev. B.
**85**, 165456 (2012)ADSCrossRefGoogle Scholar - 22.X.-L. Qi, Y.-S. Wu, S.-C. Zhang, Topological quantization of the spin hall effect in two-dimensional paramagnetic semiconductors. Phys. Rev. B.
**74**, 085308 (2006)ADSCrossRefGoogle Scholar - 23.R. Jackiw, Fractional charge from topology in polyacetylene and graphene. AIP Conf. Proc.
**939**, 341 (2007)ADSCrossRefGoogle Scholar - 24.A. Coste, M. Luscher, Parity anomaly and fermion- boson transmutation in 3-dimensional lattice QED. Nucl. Phys. B.
**323**, 631 (1989)ADSMathSciNetCrossRefGoogle Scholar - 25.B.A. Bernevig, T.L. Hughes, S.-C. Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science.
**15**, 314 (2006)Google Scholar - 26.M. König, S. Wiedmann, C. Brune, A. Roth, H. Buhmann, L.W. Molenkamp, X.-L. Qi, S.-C Zhang, Quantum spin Hall insulator state in HgTe quantum wells. Science.
**2**, 318 (2007)Google Scholar - 27.K.S. Novoselov, Z. Jiang, Y. Zhang, S.V. Morozov1, H.L. Stormer, U. Zeitler, J.C. Maan, G.S. Boebinger, P. Kim, A.K. Geim1, Room-temperature quantum Hall effect in graphene. Science.
**9**, 315 (2007)Google Scholar - 28.A.W.W. Ludwig, M.P.A. Fisher, R. Shankar, G. Grinstein, Integer quantum Hall transition: an alternative approach and exact results. Phys. Rev. B.
**50**, 7526 (1994)ADSCrossRefGoogle Scholar - 29.K. Ziegler, On the minimal conductivity in graphene. Phys. Rev. B.
**75**, 233407 (2007)ADSCrossRefGoogle Scholar - 30.X. Du, I. Skachko, A. Barker, E.Y. Andrei, Approaching ballistic transport in suspended graphene. Nat. Nanotech.
**3**, 491 (2008)ADSCrossRefGoogle Scholar - 31.D.C. Elias, R.V. Gorbachev, A.S. Mayorov, S.V. Morozov, A.A. Zhukov, P. Blake, L.A. Ponomarenko, I.V. Grigorieva, K.S. Novoselov, F. Guinea, A.K. Geim, Dirac cones reshaped by interaction effects in suspended graphene. Nat. Phys.
**8**, 172 (2011)CrossRefGoogle Scholar - 32.E. Zohar, J.I. Cirac, B. Reznik, Quantum simulation of gauge theories with ultracold atoms: local gauge invariance from angular-momentum conservation. Phys. Rev. A.
**88**, 023617 (2013)ADSCrossRefGoogle Scholar