**Antonio Sergio Teixeira Pires, one of JPCM’s esteemed Board Members and Professor of Physics at Universidade Federal de Minas Gerais, shares with us his comments on his research highlight from the journal last year. Published in April 2014, the paper by Y-X Zhu et al 2014 J. Phys.: Condens. Matter 26 175601 studies two-dimensional magnetic topological insulators.**

A topological insulator (TI) is a material that behaves as an insulator in the bulk, but has conducting states on its edge or surface. These surface states are symmetry protected by particle number conservation and time reversal symmetry. TI’s could potentially be used in computers and other electronic devices.

A two dimensional band structure can be described as a mapping from the crystal momentum, defined on a torus, to the Hamiltonian written in the form of a Bloch Hamiltonian H(k). We can classify topologically the gapped band structures by considering the equivalence classes of H(k) that can be continuously deformed into one another without closing the energy gap. These classes are distinguished by an integer

Cthat is topologically invariant (this means that it does not change when the Hamiltonian varies smoothly) called the Chern invariant. (Crepresents the number of gapless edge modes).Stimulated by the interest in TI, researchers started studying the properties of correlated electron systems TI’s. There are two typical correlated TI’s: The so called interacting spinful Haldane model and the interacting Kane-Mele model (KM). The first model breaks time-reversal symmetry while it keeps spin rotation symmetry. It can be described by the Chern number

C, and is therefore called a Chern insulator; the second model breaks spin-rotation symmetry while it keeps time-reversal symmetry and to label this class of TI Kane and Merle proposed aZ_{2}topological invariant. The full properties of both models are still under investigation.The authors study both types of two dimensional magnetic topological insulators in a honeycomb lattice at a finite temperature. For the Chern insulator, the ground state can be understood as a new type of topological quantum state — the topological spin density-wave state (TSDW). They found that the thermal-fluctuation induced magnetic topological insulator (MTI) appears in the intermediate interaction region. By contrast, for the correlated Z

_{2}topological insulator, the TSDW and the MTI do not exist.For free electrons, in the first model, there are two phases: the Chern insulator and the normal band insulator state. As the interaction increases, the correlated Chern insulator becomes unstable against an antiferromagnetic spin-density wave. Using the mean-field approach, they obtain the self consistent equations for the staggered magnetization (order parameter) by minimizing the free energy. They found that four phases exist at zero temperature. Using the Kubo formalism they have shown that there is no true ‘topological’ phase transition at finite temperature.

For the KM model, only one phase transition at zero temperature exists: the magnetic phase transition between a magnetic ordered state and a non-magnetic state. In the non-magnetic state, the ground state is the Z

_{2}topological insulator with quantum spin Hall effect. In the magnetic state the quantum spin Hall effect disappears. At finite temperature, the phase transition turns into a cross-over, and there is no true long-range magnetic order.To treat topological orders, the authors introduce the concepts of ‘mass-gap-competition’ and ‘mass-gap-coexistence’ to characterize the difference between the two types of TI.

You can read the original article by Y-X Zhu *et al* 2014 *J. Phys.: Condens. Matter* **26** 175601 here. For more papers, reviews and fast track communications on topological insulators, check out JPCM’s subject collection.

This work is licensed under a Creative Commons Attribution 3.0 Unported License.

Image taken and adapted from Y-X Zhu *et al* 2014 *J. Phys.: Condens. Matter* **26** 175601.

Categories: Journal of Physics: Condensed Matter