F-F Interface on Top of a Topological Insulator

Without loss of generality, we assume that the interface lies along theeˆydirection so that we can expect a bound state in theeˆ_x direction, while the transverse wave-vectorqalong theeˆ_y direction is a good quantum number. For this geometry, we have k=−i∂_xeˆ_x+qeˆ_y, so that the Hamiltonian has the form (here, we set~= 1)

H=ivσ_y∂_x+qvσ_x−MLσ−∆Mθ(x)σ−µL−∆µ θ(x) , (4.37) where ∆M = MR−ML is the difference between the magnetization on the left and right side, and likewise for the chemical potential ∆µ = µR−µL. The situation is illustrated in Figure 4.3.

On either side of the interface, we anticipate a solution of the form

|ψ_Li=e^iqy+λLx|χi , |ψ_Ri=e^iqy−λRx|χi (4.38) with the two inverse localization lengths λL and λR, while the spinor |χi has to be equal on both sides due to the boundary condition for the wave function at the interfacex= 0, which requires the wave function to be continuous. The two solutions obey the eigenvalue equation forx <0 andx >0, respectively

(ivλLσy+qvσx−MLσ−µLσ₀)|χi=E|χi , (4.39) (−ivλRσy+qvσx−MRσ−µRσ₀)|χi=E|χi , (4.40)

with energy eigenvalue E. When multiplied by ∓iσy from the left, these equations yield by virtue of (a·σ)(b·σ) = (a·b)σ₀+iσ·(a×b) the following expression

(vλL/R±i(eˆy·ML/R))|χi= (±vqσz±σ·(eˆy×ML/R)∓i(µL/R−E)σy)|χi . (4.41) Adding both yields the simple eigenvalue equation

(v(λL+λR)−i(ˆey·∆M))|χi=−((ˆey×∆M)·σ−i∆µσy)|χi , (4.42) from which we immediately identify the bound state to be an eigenstate of the 2×2 matrix on the right-hand side. For ∆µ6= 0, these two states are not orthogonal since this operator is not Hermitian in that case, but we will see that only one of those two states is a bound one.

At any rate, the normalized stateshχ±|χ±i= 1 are characterized by the following expectation value:

hχ±|σ|χ±i= sin(ϕ/2)∆M_⊥

|∆M_⊥|±cos(ϕ/2)eˆy×∆M_⊥

|∆M_⊥| . (4.43)

Here, we introduced the angle sin(ϕ/2)≡ −_|∆M^∆µ_⊥| 6= 0 that parametrizes the potential barrier

∆µ, and ∆M_⊥ is simply component of ∆M within thex-z plane.

The inverse localization lengths of the bound state on either side of the contact are explicitly given by

vλL=−iˆe_y·ML+qv(sin(ϕ/2)ˆe_z∓cos(ϕ/2)ˆe_x)· ∆M_⊥

|∆M_⊥|+ 1

|∆M_⊥|(±cos(ϕ/2) ∆M_⊥·ML+ sin(ϕ/2)ˆe_y·(ML×MR)) (4.44)

λR=λ^∗_L|M_L↔M_R , (4.45)

where the upper/lower sign corresponds to |χ±i, and which has to be chosen such that ReλL,ReλR >0. Comparing the terms in the above expressions, we find almost all terms to be of opposite sign inλL versusλR, which implies that the increase of these terms makes the bonding weaker on one side and stronger on the other. The only term that contributes to a binding on both sides is cos(ϕ/2) ∆M_⊥·ML.

For the moment, we neglect the influence of finite q with the additional assumption that ϕ= 0. Then this gives rise to the constraint sgn(M_L²−ML·MR) = sgn(M_R²−ML·MR), or

|ML|

|MR|−cosθ |MR|

|ML|−cosθ

>0 , (4.46)

where θis the angle between ML and MR. Thus, the existence of a bound state depends on the relative angle θ and the relative magnitude of the magnetization vectors, ^|M_|M^R|

L|. A phase diagram for the existence of a bound states is shown in figure 4.4(a).

In the end, we find that the bound state is always given by|χi ≡ |χ−i. This does not change, even whenϕ6= 0 orq6= 0 due to the above observation that all other relevant terms inλLand λR are equal, but of opposite sign. Also, note that theM_y-component plays no major role for the bound state other than giving rise to an oscillatory modulation of the bound state wave function, but it does not alter energies or any other expectation values of physical relevance.

ML

MR

ˆ ex

ˆ ez

θ

(a) ∆µ= 0

ML

MR

ˆ ex

ˆ ez

θ

(b) sin(ϕ/2)≡ −|∆M^∆µ_⊥| = 1

Figure 4.4.: Phase diagrams showing the existence of a bound state (indicated by the shading) at the interface between two ferromagnets with magnetizations MR and ML. Only the relative orientation of MR and ML is relevant for the existence of a bound state, andnotits absolute orientation in thex-zplane. Therefore, without loss of generality we can align MR along the x-direction. The boundaries are defined by a circle with center MR/2 and radius MR/2 and the vertical line tangent to the circle atMR. Plot (b) depicts the case of non-vanishing chemical potential difference ∆µ, parametrized by the angleϕwhich here is chosen to be ϕ=π/2. Note that changing the sign of ϕ simply yields a diagram that is the mirror image with respect to the directionMR.

This is not surprising, since according to section 4.2.2, this component can be eliminated by virtue of a gauge transformation.

In the case of a finite potential difference ∆µ = −|∆M_⊥|sin(ϕ/2), we first note that for

|∆µ|>|∆M_⊥|, the important contribution toλ becomes imaginary and thus, a bound state is not possible. For|ϕ| ≤π, we have the following constraints for the exsistance of a bound state,

cos(ϕ/2)

|ML|

|MR|−cosθ

−sin(ϕ/2) sinθ >0 and

cos(ϕ/2)

|MR|

|ML| −cosθ

+ sin(ϕ/2) sinθ >0.

In Figure 4.4(b), we show how finite ϕmodifies the bound state phase diagram.

The energy of the bound states is then readily obtained by taking the expectation value of

|ψi with respect to the Hamiltonian,

E(q) =hψ|H|ψi=E₀+vDWq (4.47) where the constant part of the energy reads

E₀ =−sin(ϕ/2)(MR)_⊥² −(ML)_⊥²

2|∆M_⊥| −cos(ϕ/2) eˆ_y·(ML×MR)

|∆M_⊥| , (4.48)

(4.49)

x z ML

M_R α+^θ₂ α−^θ₂

Figure 4.5.: Definitions of the anglesθ and α in thex-z plane. The interface defined by the domain wall is perpendicular to this plane, i.e. along they direction.

and we assume without loss of generality thatµL=−µR. The group velocity for propagation along the domain wall is

vDW =vhψ|σˆex|ψi=v[sin(ϕ/2)ˆex−cos(ϕ/2)ˆez]· ∆M_⊥

|∆M_⊥| . (4.50) Note that the energy shift E₀ of the bound state changes sign when we exchange the two magnetizationsMR ↔ML, and it is zero only whenMLandMRare anti-parallel. Essentially, the dispersion with the transverse momentumqis linear, so that the group velocity is constant with its sign determining the chirality of the bound state. Of course, what one sees here is the chirality of the underlying metallic states on the surface of the topological insulator, which manifests itself in the chirality of the bound states at a domain wall.

Finally, in the case|ML|=|MR|=M, we give a simple expression for the energy dispersion of the bound state

E(q) =−Mcos(ϕ/2) cos(θ/2) sgn(θ)−vqcos(ϕ/2 +α) , (4.51) where θ is the angle between ML and MR in the x-z plane.² α describes the absolute orientation of ∆M_⊥ in thex-z plane (see Fig. 4.5). We can see that the group velocity does not depend on the relative angleθ, only on the sumϕ/2 +α which combines the strength of the potential barrierϕ and the absolute position α of the two vectors in thex-z plane. On the other side, the energy shiftE0 is essentially controlled by the relative angleθ.