Kramers theorem tells us that for a system with an Hamiltonian that obeys a time reversal symmetry [H, T] = 0, where T is the time-reversal operator, all its eigenstates are at least twofold degenerate. The crossing point (Dirac point (k=0)) is located at a time reversal-invariant point. Then gap opening  violates the Kramers theorem, since the state at k = 0 becomes non-degenerate.

Suppose you apply magnetic field to break the TRS, you usually observe that the Dirac point e.g. at the Gamma point opens a gap, i.e. leading to an insulating surface.

Dear Gupta,

We usually claim that the topological insulator could be characterized by time reversal (TR) symmetry since the topological states of topological insulators are protected by TR symmetry. In details, these topological states come out with Kramers doublet (Actually it’s a consequence of TR invariance). Generally, the Kramers doublet is located at different momentum points like k and –k, however, for some special points like Dirac points, -k is k itself, hence the energy levels in these points are double degenerate. This degeneracy, is the well-known Kramers degeneracy, which combines with R symmetry ensures the crossing of the energy levels at special points in the Brillouin zone (BZ). That’s why we can see well-defined Dirac points/cones at the Fermi level and gapless surface/edge states.

If we apply magnetic field to the system, the Kramers degeneracy will be removed and TR is broken, resulting in the vanishing of gapless surface/edge states and an open gap as said by Prof. Binghai Yan. Those are the effects brought by magnetic field.

You should know that magnetic field usually breaks the TM symmetry as the emergent ferromagnetism are always closely related to the breaking of TR symmetry. For example, 2D quantum hall state belongs to topological class, but it is not a topological state because it explicitly breaks time-reversal (TR) symmetry. In general, the realization of 2D quantum hall state which firstly discovered by Klitzing usually needs to be applied a quite high external magnetic field.

However, spin orbital coupling (SOC) and crystal field will not break the TR invariance, that’s why people started to look for a physical realization of a new topological class with an intrinsic spin Hall effect since the year of 2003 (Murakami et al., 2003; 2004; Sinova et al., 2004; Kane and Mele (2005a); Bernevig, Hughes, and Zhang (2006)).

Dear All,

Thank you very much for your response.

Dear Fang,

Thank you very much for your nice explanation.

I am slightly confused by below statement.

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The simplest topologically ordered state, the quantum Hall state, occurs when a strong magnetic field is applied to a 2D electron gas between two semiconductors [8, 9]. Due to the resulting Lorentz force the electrons will enter quantized orbits within the material. However, at the edges, these orbits will bounce and lead to an electronic state that conducts in one direction [10].

Is that magnetic field or spin-orbit coupling required for topological insulator?

Hi, Gupta,

Quantum Hall (QH) effect is a quantum version of general spin Hall effect realized in 2D systems.  Assuming a 2D semiconductor junction are placed in a large magnetic field, the longitude conductance will be zero while the quantum plateau at νe2/h of the Hall conductance will come out. Here, ν is filling factor. That’s just integer quantum hall effect. Nowadays, we usually explain this phenomenon using two physical pictures. One is derived from quantum mechanics explanation, for an electron in a uniform magnetic field, the motion could be treated as a harmonic oscillator, their energy levels are quantized. Generally, we call the energy levels as “Landau levels”. The other one is a semi-classical picture, for an electron in a uniform magnetic field, the particle cycles around the magnetic flux fast due to the Lorentz forces. And the radius of the cycle is quantized caused by the magnetic field. If a particle travels in proximity to the boundary, the particle will bounce back to the rigid boundary and skip along the boundary forward. As a result, conduction channels along the boundary are formed (These states are insensitive to the geometry of the structure and other impurities). While, the velocity of the particles in the bulk is much slower than that of electrons near the boundary, and they appear to be very localized or pinned. Now, you can see that the so called edge states in 2D systems are formed. Thus, these 2D systems featuring QH edge states belong to topological phase (but not topological insulators). One thing you’d better remember is that these QH states are dissipationless states, their potential applications are huge, which I think motivates many scientists, including Prof. Shou-Cheng Zhang of  Stanford Univ. and Qi-Kun Xue of Tsinghua Univ, to explore and study the quantum anomalous Hall effect.

As mentioned in my former reply, the topological edge/surface states of general topological insulators preserve TR symmetry, hence they don't need to apply a magnetic field, and otherwise the edge/surface states will vanish. However, if you want to observe quantum anomalous Hall effect in some topological insulators, you have to drive the magnetism to couple the topological states and finally break the TR symmetry. That’s what Prof. Qi-Kun Xue’s team did in this prominent work (DOI: 10.1126/science.1234414).

The nontrivial characteristics of topological insulators results from spin orbital couplings (SOC), so one can see that SOC is strongly needed in a topological insulator. For example, if we suppose that there exists strong SOC in graphene, then a gap will appear at Dirac point. But it still preserve TR invariance because the boundaries induce edge states with opposite spin and opposite directions of electron motion (a kind of quantum spin Hall effect). Hence, we can safely say that if the SOC in graphene is large enough, it may become TI, but actually the SOC of graphene in reality is pretty small, hence it’s not thought to be a kind of general topological insulator. In fact, the work concerning the topological states of graphene has been started since 2005 by Prof. Kane.

Thanks Dear Fang

Electrical insulators as a new phase state of quantum matter with some bulk gap and some odd number of Dirac formations have recently been entangled with a two-dimensional model by us through an external magnetic field. The magnetic field used has its own impurities so much so that the eccentricity resulting from the turning point of magnetic critical insulation leads to aspects of compilation that have never been revealed by the “approximate approximation calculus” mostly used in a run under such circumstances.

Crystal and electronic structures have given us the topology that have made it likely for us to depend more on over-stuck service rather than on projections that go into manifolds and submanifolds of various dimensions. In additions, fulton heaters in this specific setting of complex systems where technique does not exist for efficient approximation, has led us to find even lower than submanifolds: something like up-submanifolds and down-submanifolds whereby the cloesed set of combining topological insulators itself has become the target in insulating the whole bulk without any regard for the gapless and/or edge services states. This means that the topological insulators that we are working on have been both gapless at surface and without such states so that they might be able to carry a pure spin current. However, this pure spin current is the point where we have made our own initiative provided through the closure of those areas where points cannot satisfy every path from one origin-set of the insulator to another point of it simulacrum. Therefore, we are actually depending on, so to speak, closure models for creating metainsulators, giving yet more models of simulations of magnetic insulators.

The surface of state of spin resolved ARPES works so that the focus of one direction is in Riemannian contrast to the zones that consequently appear throughout the folding holes as borderlines between submanifolds. For this, without any doubt, means that the dimensions of submanifolds where we are working on our different froms have been entangled with ergodicity as the totality of anisotropy that rises from insulation through coupling strength. The simplest form is, of course, happening within a two-dimensional partially anisotropic plane. In the most complex forms, they can even go into the quantum mechanic levels such as N-Hilbert dimension.

Roya Attarzadeh Don Senior Prof. Reza Sanaye

As we know that external electric field on topological insulator breaks the inversion symmetry. but i want to know back end phenomenon?