Towards, and In The Metallic Regime

If we drive our system into the metallic regime by shifting the Fermi level into the bands, we will get additional non-topological contributions which can be calculated straightforwardly using the formalism described in section 3.1.5. In a quasi-stationary situation, the density matrix is essentially the equilibrium one,

¯

ρ_eq =fD(E−E_F) δ(E−(σzζ+ ¯HM)), (3.214) and we can integrate over the gauge invariant energy, since it plays no role in the present investigation (we essentially evaluate all transport quantities at the Fermi level E_F, like the relaxation times), so that

¯

ρ_eq(t,R,P) =fD(σzζ(t,R,P) + ¯H^M−E_F)

= ¯ρ₀+ ¯ρ₁=fD(σzζ−E_F) +σzH¯M∂ζρ¯₀ , (3.215)

which in the zero temperature approximation simply becomes a delta function,f_D⁰(ζσz−E_F) =

−δ(ζσ_z −E_F), and which naturally is zero when |E_F| ≤ |M_z|, i.e. when the Fermi level is inside the gap.

The presence of impurities leads to momentum relaxation which in the simplest model can be incorporated phenomenologically by introducing the relaxation time τ, or as in section 2.2.1, by calculating the self-energy for the impurity potentials which leads to the appearance of the collision integral in the transport equation. Then, assuming low-energy transport and isotropic scattering, one can reduce the effect of the collision integral to a single parameter, the scattering time evaluated at the Fermi-level τ =τ(E_F). Either way, and neglecting gradient corrections to the collision integral, one arrives at the Boltzmann equation in relaxation time approximation which in the rotated frame takes the form (see eqn. (3.125))

With the help of expressions (3.103) and (3.104), one can solve this equation for the non-equilibrium part of the density matrix, ¯ρ= ¯ρ_neq+ ¯ρ_eq which in linear response reads

¯

We should emphasize that the persistent current contributions in (3.119) are crucial for the correct density, since they cancel terms that would be divergent upon momentum integration.

In particular, when the Fermi level is inside the gap, these dipolar terms give finite contribu-tions which exactly compensate the terms due to Θ^rp and B^(p) in equation (3.119) together with (3.103). This is also the reason why we needed to consider only the effective electric field E^(p) for the topological contribution of the current (3.207). Let us therefore bring the expression for the current density into a more convenient form.

According to result (3.99), we have the simple relation Ω =σzζΘ in the case of a two band model, and splitting off the magnetic dipole energy from the density matrix ¯ρ_eq = ¯ρ₀+ ¯ρ₁=

¯

ρ0+σzH¯M∂ζρ¯0, we find that many terms cancel in the expression for the equilibrium current density (3.119), so that using identity (3.116), and after some algebra, one eventually arrives at In this expression, we can simply substitute the primed quantities from (3.160)-(3.164) in order to obtain the influence of an external magnetic field. The second integral in this result now gives only contributions that are on-shell, due to the appearance of derivatives of the density matrix, which become sharply peaked around the Fermi level E_F for sufficiently low temperatures.

Let us first consider the equilibrium current, i.e. in absence of time-dependence and the electric field, then ¯ρ0 = fD(σzζ −EF) and due to ^∂_∂R^ρ¯0 = σzf_D⁰(σzζ−EF)_∂R^∂ζ, contributions

from the two bands have opposite signs. Since ¯ρ₀ is isotropic in momentum space, only the last integral in (3.218) contributes to the equilibrium current, and we obtain

j_eq(R) =X

σz

~v²

2 σ_z(ˆe_z×∇)M_z

Z d²P (2π~)²

1

ζ f_D⁰(σ_zζ−E_F)

=− 1

2hΘ(|EF| − |Mz|) sgn(EF) (ˆez×∇)Mz , (3.219) where we used the zero-temperature approximation f_D⁰() = −δ() and the sum over σ_z is equal to taking the trace over the bands. Since ∇·(ˆez×∇) = 0, this equilibrium current (3.219) is divergence-free and appears at gradients of the mass termMz, and in particular, we find the current flow along domain boundaries that lead to circular currents. This equilibrium current is independent ofE_F, however, of opposite sign for the two bands and vanishes when E_F is inside the gap. We will consistently recover this behavior later in section 4.5, when studying domain walls on strong topological insulators, or in section 4.3, where the zero energy bound states at a domain-wall are found to be chiral. Furthermore, we will find later that this current is responsible for an anisotropic exchange interaction contribution for the magnetizationM.

We can now also study what happens to the topological contributions to the densities (3.207) and (3.208), when the Fermi level moves into the band,

n(R) =−X

σz

Z d²P

(2π~)² fD(σzζ−E_F)TrΘ^rp= Mz

2hv|E_F|(∇·M+vqB(R)ˆez) , (3.220)

j(R) =−X

σz

Z d²P

(2π~)² fD(σzζ−E_F) E^(p)+qE×B^(p)

=− Mz

2hv|E_F|(∂tM⊥+vq eˆz×E) . (3.221) If the Fermi-level is inside the gap, we need to substitute |E_F| = |Mz| and recover results (3.207) and (3.208) and when it moves into either of the two bands, it monotonously decreases as the Fermi level moves further away from the gap.

We have seen that the response due to the external electromagnetic field and the ferromagnetic exchange field M always appeared in the combined form B^eff, so one might conclude that both can be simply mapped onto each other, and it is sufficient to consider only either of them, however, this conclusion is too premature. Of course, such a mapping would only make sense for theM_x and M_y components since there is no momentum inz-direction which could absorb Mz, so we forget about the Mz-component for the moment. At best, it would be possible to treat this question within the full bulk description of the STI mentioned earlier.

Indeed, on the one side, the topologically induced charge and currents due to a magnetization structure appears on equal footing as the external electromagnetic field and likewise for the Hamiltonian (see result (3.203)), so the energy spectrum can be simply mapped onto each other by virtue of (3.213). Yet, on the other side, the kinetic variables differ, which results in different equations of motion that cannot be mapped onto each other easily, and which is best seen at equation of motion for the kinetic momentum (3.167), where the external magnetic field enters as expected, whereas the action ofMdoesnotenter as∇·M, asB^effwould suggest.

Instead, it enters asB^(r) according to result (3.198), which is rather different. Furthermore, we will find an explicit dependency on the combination ∇×M later in result (3.233). In conclusion, it means that despite the identical band structure, we have a different motion of the electrons. The situation of spin-momentum locking due to the Rashba spin-orbit term in (3.188) is actually quite different from the study of ferromagnetic conductors with momentum isotropic dispersion∝p²1, where the diagonalization transformation depends only on position [73], so that the momentum operator turns into its gauge invariant counterpart, analogously to an external magnetic field. Thus, both magnetic and exchange fields can be mapped onto each other, at least as long as transitions between spin-up and -down bands can be neglected.

This system describing a topological state of matter has the property that zero bound states may emerge under certain conditions, which will be investigated in section (4.3). However, since these states lie within the gap, they are not straightforwardly accessible within this formulation, as they involve coherent excitations shared between two bands.