A Rashba Green’s function in position-space

# Universality of low-energy Rashba scattering

## Abstract

We investigate the scattering of a quantum particle with a two-dimensional (2D) Rashba spin-orbit coupled dispersion off of circularly symmetric potentials. As the energy of the particle approaches the bottom of the lowest spin-split band, i.e., the van Hove singularity, earlier work has shown that scattering off of an infinite circular barrier exhibits a number of features unusual from the point of view of conventional 2D scattering theory: the low-energy -matrix is independent of the range of the potential, all partial waves contribute equally, the differential cross section becomes increasingly anisotropic and 1D-like, and the total cross section exhibits quantized plateaus. Via a nonperturbative determination of the -matrix and an optical theorem which we prove here, we show that this behavior is universal for Rashba scattering off of any circularly symmetric, spin independent, finite-range potential. This is relevant both for impurity scattering in the noninteracting limit as well as for short-range two-particle scattering in the interacting problem.

###### pacs:
71.10.Ca, 71.70.Ej, 72.10.-d

## I Introduction

Long considered a small relativistic correction of little qualitative importance to condensed matter physics, spin-orbit coupling has come to the fore of this field in the past ten years or so owing to the discovery of a rich phenomenology associated with it, including the spin Hall Sinova et al. (2015) and quantum spin Hall Maciejko et al. (2011) effects, three-dimensional (3D) topological insulators Hasan and Kane (2010); Qi and Zhang (2011), Weyl semimetals Armitage et al. (2017), and spin-orbit coupled Mott insulators Witczak-Krempa et al. (2014), to name a few. In 2D crystals with broken inversion symmetry, the spin degeneracy of the electronic band structure may be lifted by Rashba spin-orbit coupling Rashba (1960); Yu. A. Bychkov and Rashba (1984). A similar type of spin-orbit coupling can also be engineering synthetically via laser-atom interactions, as recently demonstrated in an ultracold gas of K fermionic atoms Huang et al. (2016).

The spin-split dispersion in a 2D Rashba system is described in terms of two distinct helicity bands, but below a threshold energy (Dirac point), particles are confined to one of these. At the bottom of this lower band, the density of states is enhanced to form a van Hove singularity. In particular, this is the relevant regime for a dilute spin-orbit coupled 2D electron gas, which has been shown to host a variety of exotic phases in the presence of electron-electron interactions Berg et al. (2012); Silvestrov and Entin-Wohlman (2014); Ruhman and Berg (2014); Bahri and Potter (2015). We showed in earlier work Hutchinson and Maciejko (2016) that in this limit, single-particle scattering from a hard disk potential (i.e., an infinite circular barrier) exhibits a variety of unusual behaviors. The -matrix, which is a matrix as there are two same-helicity degenerate scattering channels below the Dirac point, was found in the low-energy limit to be purely off-diagonal with off-diagonal elements equal to  Hutchinson and Maciejko (2017) for every partial wave As a result, the low-energy differential scattering cross section is extremely anisotropic, with scattering at all angles highly suppressed except forward scattering () and backscattering (), which have the same amplitude. This stands in stark contrast with the usual dominance of the isotropic -wave scattering channel at low energies in conventional systems in both 2D and 3D. Finally, instead of the usual smooth divergence (moderated by a logarithm) of the total cross section as the energy in conventional 2D systems with a parabolic dispersion Friedrich (2013), in the Rashba hard-disk problem the total cross section (which in 2D has units of length) was found to increase in quantized steps of magnitude where is the wavenumber of the degenerate Rashba ground state manifold. Surprisingly, all these features are independent of the radius of the barrier, and were found to hold also for a delta-shell potential of arbitrary radius. This led us to conjecture in Ref. Hutchinson and Maciejko (2016) that these peculiar features are a universal property of low-energy scattering in the Rashba system and should hold for arbitrary spin-independent, circularly symmetric, finite-range potentials.

In the present work we establish that this conjecture is true via a nonperturbative solution of the Lippmann-Schwinger equation for an arbitrary potential satisfying the requirements listed above. In Sec. II we briefly review the basics of Rashba spin-orbit coupling and establish our notation. In Sec. III we formulate the Lippmann-Schwinger equation for our problem, introduce the -matrix and establish its relation to the -matrix, then relate the and matrices to the differential and total scattering cross sections, deriving a new optical theorem for low-energy Rashba scattering in the process. In Sec. IV we present a nonperturbative solution of the Lippmann-Schwinger equation, obtaining the -matrix in the low-energy limit. Our solution relies on the application of a momentum cutoff around the degenerate low-energy Rashba ring of states. As expected, the low-energy -matrix is universal and exhibits a distinct 1D character. Using the relation between the and matrices derived earlier, we obtain the universal off-diagonal -matrix of Ref. Hutchinson and Maciejko (2016). Our optical theorem allows us to show that the quantized plateaus seen in our previous work for the hard disk potential are indeed a generic feature of the low-energy total cross section, independent of the details of the potential. Finally, in Sec. V we illustrate these results with a number of example potentials. We conclude in Sec. VI, and derive a number of technical results in Appendices A-C.

## Ii Rashba spin-orbit coupling

We begin with the unperturbed Rashba Hamiltonian in two dimensions Yu. A. Bychkov and Rashba (1984),

 H(k)=k22m+λ^z⋅(σ×k), (1)

where is the electron wave vector confined to the - plane (we work in units of ), is the vector of Pauli matrices, and is the Rashba coupling. This Hamiltonian is readily diagonalized to give the spin-split spectrum

 E±(k)=k22m±λ|k|, (2)

and eigenspinors

 η±(θk)=1√2(1∓ieiθk). (3)

There is a degenerate ring of states for each wave vector of magnitude . Since the spin expectation value in the corresponding eigenstates is locked orthogonally to the wave vector, this spectrum consists of two bands of opposite helicity, designated by the subscripts. We are exclusively interested in the lower of these two bands, and so it is useful to write all quantities in terms of the ground state energy , and the ground state wave vector magnitude . These are the only quantities that are controlled by the Rashba coupling in our problem. Along this vein, we parameterize the electron scattering energy by the dimensionless quantity .

For any given energy and wave vector angle , there exist two degenerate negative-helicity states of different wave vector magnitude. One has a wave vector whose magnitude is greater than , while the other is less than . We denote these magnitudes by

 k≷=k0(1±δ). (4)

## Iii Scattering Quantities

Roughly speaking, the -matrix is the portion of the -matrix in which some scattering occurs. Since Rashba scattering involves some subtleties, it is worth deriving the exact relation between these objects in the negative energy regime, elucidating various scattering quantities along the way. The natural starting point is the Lippmann-Schwinger equation,

 ψkσ(r;E) = ψinkσ(r;E) +∑σ′∫d2r′G+σσ′(r,r′;E)V(r′)ψkσ′(r′),

where is the retarded position-space Green’s function of the unperturbed Hamiltonian, is the scattering potential, and is a spin index. The incident wavefunction is chosen to be a negative helicity plane wave with wavevector oriented at an angle with respect to the -axis,

 ψink(r;E)=eik⋅rη−(θk). (6)

We can relate this to the -matrix through the defining relation

 T|i⟩=V|ψ⟩, (7)

where is the initial state, and is the scattering state. In terms of wavefunctions, we write

 V|ψ⟩=∑σ′∫d2r′T|r′σ′⟩ψinkσ′(r′;E), (8)

or equivalently,

 V(r)ψkσ(r;E)=∑σ′∫d2r′Trr′σσ′eik⋅r′η−σ′(θk). (9)

We will need to Fourier transform the -matrix to momentum-space,

 Trr′σσ′=∫d2k′(2π)2∫d2~k(2π)2Tk′~kσσ′eik′⋅re−i~k⋅r′. (10)

Substituting (10) into (9) and (9) into (III), we obtain a modified Lippman-Schwinger equation,

 ψkσ(r;E) = ψinkσ(r;E) +∑σ′σ′′∫d2r′∫d2k′(2π)2G+σσ′(r,r′;E) ×Tk′kσ′σ′′eik′⋅r′η−σ′′(θk).

To proceed any further requires knowing the position-space Green’s function. This is derived in Appendix A: see Eq. (81) for and Eq. (86) for . To match with the -matrix, we must consider the asymptotic wavefunction, which for a finite range potential, amounts to imposing , and in the Green’s function, where denotes a unit vector in the direction of . Using the asymptotic form of the Hankel function for large argument,

 H±l(x)≈√2πxe±i(x−lπ/2−π/4), (12)

we obtain the asymptotic Green’s function

 G+σσ′(r,r′;E)≈−mk++k−√i2πr∑j=+,−gjσσ′(r)e−ikj⋅r′, (13)

where and is given in Eq. (77) and (78). We have also defined the matrix

 gj(r) ≡ √kjeikjr(1ie−iθrj−ieiθrj1) (14) = 2√kjeikjrηj(θr)ηj(θr)†.

Since the dependence of the Green’s function has been isolated, we can now evaluate the integrals in (LABEL:eq:Lipp2) to get the asymptotic wavefunction,

 ψk(r;E) ≈ ψink(r;E)−m(k++k−)√2iπr ×∑j=+,−√kjeikjrηj(θr)ηj(θr)†Tkjkη−(θk).

### iii.1 Relation between T and S matrices

At this point we orient the -axis along the incident wave direction () and recognize that for any negative energy, the magnitude of the corresponding wavevector is either or using the notation in (4). We write this as , where .

To connect the -matrix to the -matrix (or equivalently the scattering amplitude), we use the definition of the -matrix as the unitary transformation from asymptotic ingoing to asymptotic outgoing states. Schematically,

 ψ>(r;E) ∼ ψin>+S>>ϕout>+S<>ϕout<, (16) ψ<(r;E) ∼ ψin<+S><ϕout>+S<<ϕout<. (17)

In Ref. Hutchinson and Maciejko (2016), the form of the -matrix for lower-helicity scattering off of a finite range, circularly symmetric potential was obtained. Using a slightly modified notation, we summarize these results by writing the asymptotic wavefunction outside such a potential as

 ψμ(r;E) ≈ ψinμ(r;E) +2m√ikμ∑ν=>,

Here, the indices indicate the magnitude of the wavevector as discussed above, , and . The common spinor factor is formally equivalent to the definition (3) due to the fact that the group velocity is oppositely directed for the state (see Ref. Hutchinson and Maciejko (2016) for details). The factor of in front of the sum is chosen to make consistent with the conventional scattering amplitude in two dimensions Adhikari (1986). With these conventions, the scattering amplitude has the following relation to the -matrix expanded in partial waves,

 fμν(θr)=e−iπ4(1+sν)4m√2π∞∑l=−∞eil(θr+π2(1−sν))(Slμν−Iμν). (19)

The strategy now is to simply equate (III) and (III.1). To do this, we need a sum over wave vector magnitudes in (III) rather than helicity index . This is accomplished by noting from (4) and (78), the mathematical relation

 k±=∓k≶, (20)

valid for any negative energy. Eq. (III) then reads

 ψμ(r;E) ≈ ψinμ(r;E)−mk>−k<√2iπr ×(√k>eik>rη−(θr)η−(θr)†Tk>kη−(0) +i√k

where . For the term, we simply note that since and ,

 η−(θr)†Tk>kη−(0)=Tk>k−−, (22)

which is the component of the helicity transform of involving only transitions within the negative helicity state. For the term, we use the fact that to write the eigenspinors as

 η+(θr)=1√2(1−ieiθr)=1√2(1iei(θr+π))=η−(θ−k<), (23)

which makes it clear that

 η+(θr)†T−k

The Lippman-Schwinger equation finally reads

 ψμ(r;E) ≈ ψinμ(r;E)+me−iπ4k<−k>√2iπr ×∑ν√kνeisν(kνr+1)Tsνkνkμ−−ηsν(θr)e−iπ4sν.

Comparing (LABEL:eq:Lipp3) to (III.1), we may simply read off the relation between the -matrix and scattering amplitude:

 Tsνkνkμ−−=√2π(k<−k>)√kμkνe−iπ4(1+sν)fμν(θr), (26)

or, in terms of the -matrix written in (19),

 Tkνkμ−−=imk0δ√kμkν∞∑l=−∞eilθ(Slμν−Iμν), (27)

using , and letting . Rotational symmetry of the Hamiltonian allows us to expand the -matrix in partial wave components as well, so that we may invert (27) to get

 Slμν=Iμν−imk0δ√kμkνTl(kν,kμ), (28)

where . The above result can be shown to be equivalent to the usual definition of the -matrix (see, e.g., Ref. Sakurai (1994)),

 Sfi=δfi−2πiδ(Ef−Ei)Tfi, (29)

with the appropriate change of basis (see Appendix B for details).

### iii.2 Cross section and optical theorem

To complete our scattering formalism we determine the differential cross section. Beginning with Fermi’s golden rule, the transition rate is connected to the -matrix via

 wμ→νdθ=2π|Tkμkν−−|2ρ(Eν)dθ, (30)

where is the density of final states in the channel within an angle of :

 ρ(Eν)=∫∞0dkk(2π)2δ(Eν−E(k))=m(2π)2kνk0δ. (31)

Furthermore, the differential cross section in this channel is simply the transition rate divided by the incident flux,

 dσdθ∣∣∣μν = wμ→ν|jμ| (32) = m22πkνk20δ2|Tkμkν−−|2 = 12πkμ∣∣ ∣∣∞∑l=−∞eilθ(Slμν−Iμν)∣∣ ∣∣2.

This last expression was denoted in Ref. Hutchinson and Maciejko (2016). Integrating over angles and summing over scattering channels gives the total cross section for an incident wave,

 σμ = ∫2π0∑ν12πkμ∣∣ ∣∣∞∑l=−∞eilθ(Slμν−δμν)∣∣ ∣∣2 (33) = 1kμ∞∑l=−∞(∣∣Slμμ−1∣∣2+∣∣Slμ,−μ∣∣2) (34) = 1kμ∞∑l=−∞(2−(Slμμ+Sl∗μμ)) (35) = 2kμ∞∑l=−∞(1−Re(Slμμ)), (36)

where denotes the off-diagonal component with first index , and we used the unitarity condition of the -matrix () in line (35). The final form of this cross section makes it clear that the diagonal part of the -matrix in (27) obeys an optical theorem, since

 ImTkμkμ−−(θ=0) = −k0δmkμ∞∑l=−∞(1−Re(Slμμ)) (37) = −k0δ2mσμ.

## Iv Rashba T-matrix

With this scattering formalism at hand, we may compute any scattering observable in a Rashba system with , provided we know the -matrix . In a conventional 2D system without spin-orbit coupling, the -matrix takes on a form at low energies that is dominated by the s-wave term,

 Tkk′≈T0(E)∼1/mi−1πln(E/Ea), (38)

where is a parameter that encodes the potential , and is related to the scattering length (see, e.g., Ref. Friedrich (2013); Randeria et al. (1990)). Before doing any calculation, we can already see that the Rashba -matrix must have a different energy dependence than (38), simply by looking at the Lippmann-Schwinger equation (LABEL:eq:Lipp3). Since the coefficient of the scattered wavefunction goes as for low energies, the -matrix must at least be linear in in order to keep the probability density finite. We now make this explicit by deriving the low-energy Rashba -matrix for any circularly symmetric, spin-independent potential of finite range.

First, we impose a momentum cutoff

 k0−~Λ

to avoid ultraviolet divergences. This amounts to keeping only the low-energy modes in our model, similar to the momentum shell renormalization group approach in the many-body problem Shankar (1994); Yang and Sachdev (2006). The appropriate dimensionless quantity corresponding to this cutoff is , so that we will always enforce the following hierarchy of scales:

 δ≪Λ≪1. (40)

In the helicity basis denoted by , any central spin-independent potential may be written as

 Vij(k,k′) = ∫d2rei(k−k′)⋅rV(r)ηi(θk′)†ηj(θk) = 12∞∑l=−∞Vl(k,k′)eil(θk′−θk)(1+ijei(θk−θk′)) = 12∞∑l=−∞(Vl(k,k′)+ijVl+1(k,k′))eilθk′−k,

where , and in the second line, we introduced the partial wave component

 Vl(k,k′)=∫2π0dθk′−k2π∫∞0drrV(r)J0(|k−k′|r)eilθk′−k, (42)

where is the zeroth order Bessel function of the first kind.

Now the -matrix is defined by the Born series

 T=V+VG+T. (43)

We write this in the momentum-helicity basis in which the Green’s function is diagonal,

 Tkνkμji = Vji(kν,kμ) (44) +∑n=+,−∫d2q(2π)2Vjn(kν,q)G+nn(q)Tqkμni.

We want to expand the potential about the ground state wavevector . More precisely, let us examine the components given by (42). For the on-shell terms in (44), the argument of this Bessel function is

 |kμ−kν|r = r√k2μ+k2ν−2kμkνcosθk′−k (45) = √2k0r√1−cosθk′−k+O(δ).

The off-shell components in the integral of (44) may also be expanded about . The argument of the Bessel function becomes

 |kν−q|r=√2k0r√(1+ϵ)(1−cosθk′−k)+O(δ), (46)

where we have changed the integration variable using . Thus, to order in the potential, we can approximate the on-shell terms as , and the off-shell terms as . This is a crucial approximation. Since now the right hand side of (44) is independent of the magnitude , the -matrix is independent of this magnitude as well

 Tkνkμij≈Tij(^kν,kμ). (47)

We will argue in Appendix C that the error in this approximation is . Writing the -matrix in partial wave components just as we did with the potential, the Born series simplifies to

 ∞∑l=−∞Tlji(kμ)eilθ = ∞∑l=−∞12[Vl(k0,k0)+ijVl+1(k0,k0)]eilθ +∑n=+,−∞∑l=−∞∫∞0dqq4π(Vl(k0,q) +jnVl+1(k0,q))G+nn(q)Tlni(kμ)eilθ.

Equation (IV) may be solved algebraically for each partial wave component. Since the diagonal parts of the potential are equal, this equation decouples into two pairs of coupled equations. For the lower helicity band, the relevant pair is

 Tl−−(kμ) ≈ 12[Vl(k0,k0)+Vl+1(k0,k0)] (49) +Il−Tl−−(kμ)+Jl−Tl+−(kμ), Tl+−(kμ) ≈ 12[Vl(k0,k0)−Vl+1(k0,k0)] (50) +Il+Tl−−(kμ)+Jl+Tl+−(kμ),

where we have defined the integrals

 Il± = ∫∞0dqq4π[Vl(k0,q)∓Vl+1(k0,q)]G+−−(q), (51) Jl± = ∫∞0dqq4π[Vl(k0,q)±Vl+1(k0,q)]G+++(q). (52)

Using the fact that , we may solve for to get

 Tl−− ≈ 12(1−Il−(1−2Jl+)−Jl+) (53) ×[Vl(k0,k0)(1−Jl++Jl−) +Vl+1(k0,k0)(1−Jl+−Jl−)].

The integrals correspond to transitions between different helicity bands, and these are expected to have a negligible contribution to the low energy scattering. Indeed, one can show that and so

 Tl−−≈12[Vl(k0,k0)+Vl+1(k0,k0)]1−Il−. (54)

The energy dependence of the -matrix is entirely determined by the integral of Eq. (51). We claim that to leading order in , this integral is approximated by

 Il− = −m2(iδ+2πΛ)[Vl(k0,k0)+Vl+1(k0,k0)] (55) +O(δ)+O(Λ),

so that the -matrix is

 Tl−− = 12[Vl(k0,k0)+Vl+1(k0,k0)]1+m2(iδ+2πΛ)[Vl(k0,k0)+Vl+1(k0,k0)]+O(δ2).

The detailed derivation of this result is left for Appendix C. It is convenient to define a new dimensionless parameter

 δ∗l≡m2(Vl(k0,k0)+Vl+1(k0,k0)), (57)

such that to leading order in , we can write the -matrix as

 Tl−− ≈ 1mδ∗l1+iδ∗l/δ (58) = −iδm+O(δ2). (59)

The low-energy limit of the -matrix (28) is thus simply

 Sl=(0−1−10), (60)

as was found in Ref. Hutchinson and Maciejko (2016) for an infinite circular barrier. Equation (60) is the main result of this work. It establishes that the low-energy -matrix for circularly symmetric potentials in Rashba systems is completely universal, as conjectured in our earlier work: it is independent of any details of the potential, provided the latter has finite range. Thus all the conclusions drawn in Ref. Hutchinson and Maciejko (2016) from the particular form (60) of the -matrix, such as the extreme anisotropy of the differential cross section, are equally universal.

We now draw our attention to the peculiar energy dependence of the -matrix in Eq. (59). Firstly, the -matrix scales as the square root of the difference between the scattering energy and the ground state energy, in contrast with the inverse logarithm dependence found in conventional 2D systems (38). Furthermore, it does not depend on the details of the potential (its range or strength), as already mentioned. Lastly, the partial wave components of the low-energy -matrix are independent of partial wave number . The usual intuition of low-energy physics being dominated by -wave scattering does not apply to the Rashba system.

The energy dependence in (59) is very telling. Suppose we were to look for the universal form of a low-energy -matrix in a 1D scattering problem with a conventional quadratic dispersion. We could follow the same reasoning used above. A finite-range on-shell potential in momentum space can be approximated by a constant at low energy,

 V(k,k′)≈limk,k′→0V(k,k′)=∫∞−∞dxV(x)≡V. (61)

The momentum-space -matrix must again be independent of in this approximation, so that

 Tk′k ≈ T(k) = V+⎛⎜⎝∫∞−∞dx∫∞−∞dq2πeiqxV(x)E−q22m+iη⎞⎟⎠T(k) = V+(mi√2mE∫∞−∞dxV(x)ei√2mEx)T(k).

If we only consider the lowest order terms in , and make use of the fact that the potential is short-ranged, we get the following approximate solution for ,

 T ≈ V1−im√2mE∫∞−∞dxV(x)ei√2mEx (63) = im√2mE+O(E).

Thus, provided we identify the 1D parameter with , we get the same -matrix as in the low-energy 2D Rashba case1. This connection suggests once again that low-energy Rashba scattering has a fundamental 1D character, independent of the details of the potential. Indeed, it was shown in Ref. Hutchinson and Maciejko (2016) that Eq. (60) implies only forward and backward scattering are allowed at very low energies. In other words, the wavefunction behaves like that of a particle scattering in a 1D system.

One might notice that (63) and (59) differ by a minus sign. For the Rashba case, this sign ensures that the scattering cross section is positive in the optical theorem (37). More importantly, it has interesting implications for the -matrix. Looking at (28) we see that this sign guarantees that the diagonal part of the -matrix vanishes as approaches zero.

The form of the low-energy -matrix has interesting consequences for the cross section. First note that the total cross section becomes infinite at the threshold energy . This result is typical of 2D scattering, though the reasons for it are not. Using the optical theorem (37), our -matrix approximation gives a low-energy cross section of

 σ≈2k0∞∑l=−∞δ∗2l/δ21+δ∗2l/δ2. (64)

Qualitatively speaking, there is a threshold parameter for each partial wave . As we lower the energy, and thereby , we pass through these points one by one. Each time the condition is satisfied an additional two partial waves (one for and one for ) contribute to the scattering, and the cross section increases by , tending to infinity stepwise as . This is unlike the conventional 2D case in which the prefactor blows up while the partial wave sum remains finite. Thus there generically is a series of jumps and plateaus in the cross section as a function of (see, e.g., Fig. 6). However, because decays as increases, these plateaus become smaller and smaller as we approach the ground state energy. The precise location of the jumps depends on the details of the potential via Eq. (57), but the magnitude , of the plateaus in the cross section is universal.

## V Example Potentials

### v.1 Delta function potential

The simplest finite-range potential we can consider is the delta function

 V(r)=V0rδ(r)δ(θ), (65)

which has partial wave components , from (42). Since this is independent of the momenta and , the T matrix is as well, and there is no need for an approximation at this level. Instead, the -matrix exactly satisfies the equations

 T0−−=V0/21−(I0+J0)=T+−, (66)

where we have made use of the fact that , and for the delta potential. The integral may be ignored since (using )

 J0=2mV0∫Λ−Λdϵ4π1+ϵδ2−4(ϵ+1)−ϵ2∼O(Λ). (67)

The other integral evaluates to

 I0 = 2mV0∫Λ−Λdϵ4π1+ϵδ2−ϵ2+iη (68) ≈ mV02π(−iπδ+2Λ),

so that

 T0−−=V0/21+m2(iδ+2πΛ)V0≈1/miδ+2πΛ, (69)

in agreement with (IV). We emphasize that the lowest order contributions in are independent of the cutoff scale. This is in stark contrast with the conventional 2D case where the contact -matrix satisfies

 T0 = V0