# Ferromagnetic order of nuclear spins coupled to conduction electrons: A combined effect of electron-electron and spin-orbit interactions

Phys. Rev. B 85, 115424
##### I. INTRODUCTION

Spontaneous nuclear spin polarization in semiconductor heterostructures at finite but low temperatures has recently attracted a considerable attention both on the theoretical and experimental 5 sides. Apart from a fundamental interest in the new type of a ferromagnetic phase transition, the interest is also motivated by an expectation that spontaneous polarization of nuclear spins should suppress decoherence in single-electron spin qubits caused by the hyperfine interaction with the surrounding nuclear spins,1,2 and ultimately facilitate quantum computing with single-electron spins.6,7

Improvements in experimental techniques have lead to extending the longitudinal spin relaxation times in semiconductor quantum dots (QDs) to as long as 1 s. The decoherence time in single electron GaAs QDs has been reported to exceed $1μ$s in experiments using spin-echo techniques at magnetic fields below 100 mT,11,12 whereas a dephasing time of GaAs electron-spin qubits coupled to a nuclear bath has lately been measured to be above $200μ$s. 13 Still, even state-of-the-art dynamical nuclear polarization methods allow for merely up to $60%$ polarization of nuclear spins, 18 whereas polarization of above $99%$ is required in order to extend the electron spin decay time only by one order of magnitude. 17 Full magnetization of nuclear spins by virtue of a ferromagnetic nuclear spin phase transition (FNSPT), if achieved in practice, promises a drastic improvement over other decoherence reduction techniques.

The main mechanism of the interaction between nuclear spins in the presence of conduction electrons is the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. 19 The effective Hamiltonian of the RKKY interaction between on-site nuclear spins of magnitude $I$,

$HRKKY=−12∑r,r′Jij(r,r′)Ii(r)Ij(r′),$
###### (1.1)

is parameterized by an effective exchange coupling

$Jij(r,r′)=A24ns2χij(r,r′),$
###### (1.2)

where $A$ is the hyperfine coupling constant, $ns$ is the number density of nuclear spins, and

$χij(r,r′)=−∫01/Tdτ〈TτSi(r,τ)Sj(r′,0)〉$
###### (1.3)

is the (static) correlation function of electron spins. [Hereafter, we will refer to $χij(r,r′)$—and to its momentum-space Fourier transform—as to “spin susceptibility,” although it is to be understood that this quantity differs from the thermodynamic susceptibility, defined as a correlation function of electron magnetization, by a factor of $μB2,$ where $μB$ is the Bohr magneton.] It is worth emphasizing that $χij(r,r′)$ contains all the effects of the electron-electron interaction;1,2 this circumstance has two important consequences for the RKKY coupling. First, the electron-electron interaction increases the uniform spin susceptibility, which should lead to an enhancement of the critical temperature of the FNSPT, at least at the mean-field level. Second, stability of the nuclear-spin ferromagnetic order is controlled by the long-wavelength behavior of the magnon dispersion $ω(q̃)$ which, in its turn, is determined by $χij(q̃)$ at $q̃→0$. In a spin-isotropic and translationally invariant system,

$ω(q̃)=A24nsI[χ(0)−χ(q̃)],$
###### (1.4)

with $χij=δijχ$, while the magnetization is given by

$M(T)=μNIns−∫q̃∈BZdDq̃(2π)D1eω(q̃)/T−1,$
###### (1.5)

where $μN$ is the nuclear-spin magneton (we set $kB=ℏ=1$ throughout the paper). The second term in Eq. (1.5) describes a reduction in the magnetization due to thermally excited magnons. In a free two-dimensional electron gas (2DEG), $χ(q̃)$ is constant for $q̃≤2kF$, and thus the magnon contribution to $M(T)$ diverges in the $q̃→0$ limit, which means that long-range order (LRO) is unstable. However, residual interactions among the Fermi-liquid quasiparticles lead to a nonanalytic behavior of the spin susceptibility; for $q̃≪kF$, $χ(q̃)=χ(0)+Cq̃$, where both the magnitude and the sign of $C$ depend on the strength of the electron-electron interaction.20,21 In two opposite limits, i.e., at weak coupling and near the Stoner instability, 22 the prefactor $C$ is positive which, according to Eqs. (1.4) and (1.5), means that LRO is unstable. However, $C$ is negative [and thus the integral in Eq. (1.5) is convergent] near a Kohn-Luttinger superconducting instability;3,23 also, in a generic Fermi liquid with neither strong nor weak interactions, $C$ is likely to be negative due to higher-order scattering processes in the particle-hole channel.

The spin-wave-theory argument presented above is supported by the analysis of the RKKY kernel in real space. A linear-in- $q̃$ term in $χ(q̃)$ corresponds to a dipole-dipole like, $1/r3$ term in $χ(r)$ (see Sec. III). If $C>0$, the dipole-dipole interaction is repulsive, and the ferromagnetic ground state is unstable and vice versa, if $C<0$, the dipole-dipole attraction stabilizes the ferromagnetic state.

It is worth noting here that even finiteness of the magnon contribution to the magnetization does not guarantee the existence of LRO. Although the Mermin-Wagner theorem 27 in its original formulation is valid only for sufficiently short-range forces and thus not applicable to the RKKY interaction, it has recently been proven 28 that magnetic LRO is impossible even for the RKKY interaction in $D≤2$. From the practical point of view, however, the absence of LRO in 2D is not really detrimental for suppression of nuclear-spin induced decoherence. Indeed, nuclear spins need to be ordered within the size of the electron qubit (a double QD system formed by gating a 2DEG) as well as its immediate surrounding such that there is no flow of magnetization. Since fluctuations grow only as a logarithm of the system size in 2D, it is always possible to achieve a quasi-LRO at low enough temperatures and on a scale smaller that the thermal correlation length. In addition, spin-orbit interaction (SOI)—which is the main subject of this paper, see below—makes a long-range order possible even in 2D. 28

The electron spin susceptibility in Eq. (1.4) was assumed to be at zero temperature. First, since the nuclear spin temperature is finite, the system as a whole is not in equilibrium. However, a time scale associated with “equilibration” is sufficiently long to assume that there is no energy transfer from the nuclear- to electron-spin system. Second, if the electron temperature is finite, the linear $q̃$ scaling of $χ(q̃)$ is cut off at the momentum of order $T/vF≡1/LT$. For $q̃≪1/LT$, $χ(T,q̃)∝T+O(vF2q̃2/T)$ such that $ω(q̃)∝q̃2$ and, according to Eq. (1.5), spin waves would destroy LRO. However, at low enough temperatures, the thermal length $LT$ is much larger than a typical size of the electron qubit $LQ$. (For example, $LT∼1$ mm at $T∼1$ mK.) Therefore $q̃≳1/LQ≫1/LT=T/vF$ and, indeed, the electron temperature can be assumed to be zero.

In practically all nuclear-spin systems of current interest, such as GaAs or carbon-13 nanotubes, spin-orbit interaction (SOI) plays a vital role. The main focus of the paper is the combined effect of the electron-electron and spin-orbit (SO) interactions on the spin susceptibility of 2DEG and, in particular, on its $q̃$ dependence, and thus on the existence/stability of the nuclear-spin ferromagnetic order.

The interplay between the electron-electron and SOIs is of crucial importance here. Although the SOI breaks spin-rotational invariance and thus may be expected to result in an anisotropic spin response, this does not happen for the Rashba and Dresselhaus SOIs alone: the spin susceptibility of free electrons is isotropic [up to $exp(−EF/T)$ terms] as long as both spin-orbit-split subbands remain occupied. 4 The electron-electron interaction breaks isotropy, which can be proven within a Fermi-liquid formalism generalized for systems with SOI. 29 Specific models adhere to this general statement. In particular, $χzz>χxx=χyy$ for a dense electron gas with the Coulomb interaction. 30

In this paper, we analyze the $q̃$ dependence of the spin susceptibility in the presence of the SOI. The natural momentum-space scale introduced by a (weak) Rashba SOI with coupling constant $α$ ( $|α|≪vF$) is the difference of the Fermi momenta in two Rashba subbands:

$qα≡2m*|α|,$
###### (1.6)

where $m*$ is the band mass of 2DEG. Accordingly, the dependence of $χij$ on $q̃$ is different for $q̃$ above and below $qα$; in the latter case, it is also different for the out-of-plane and in-plane components. To second order in electron-electron interaction with potential $U(q)$, the out-of-plane component is independent of $q̃$ for $q̃≤qα$:

$δχzz(q̃,α)=2χ0u2kF2|α|kF3EF.$
###### (1.7a)

On the other hand, the in-plane component scales linearly with $q̃$ even for $q̃≤qα$:

$δχxx(q̃,α)=δχyy(q̃,α)=χ0u2kF2|α|kF3EF+49πvFq̃EF,$
###### (1.7b)

In Eqs. (1.7a) and (1.7b), $uq≡m*U(q)/4π$, $kF$ is the Fermi momentum, $EF=kF2/2m*$ is the Fermi energy, $χ0=m*/π$ is the spin susceptibility of a free 2DEG, and $δχij$ denotes a nonanalytic part of $χij$. For $qα≪q̃≪kF$, the spin susceptibility goes back to the result of Ref. 20 valid in the absence of the SOI:

$δχij(q̃,α=0)=δij23πχ0u2kF2vFq̃EF.$
###### (1.8)

Note that the subleading term in $q̃$ in Eq. (1.7b) differs by a factor of $2/3$ from the leading term in $q̃$ in Eq. (1.8). There is no contradiction, however, because Eqs. (1.8) and (1.7b) correspond to the regions of $q̃≤qα$ and $q̃≫qα$, respectively.

Equations (1.7a) and (1.7b) show that the uniform spin susceptibility is anisotropic: $δχzz(0,α)=2δχxx(0,α)$. This implies that the RKKY coupling is stronger if nuclear spins are aligned along the normal to the 2DEG plane, and thus the nuclear-spin order is of the Ising type. In general, a 2D Heisenberg system with anisotropic exchange interaction is expected to have a finite-temperature phase transition. 31 In an anisotropic case, the dispersion of the out-of-plane spin-wave mode is given by2,32

$ω(q̃)=A24nsI[χzz(0)−χxx(q̃)],$
###### (1.9)

with $q̃⊥ẑ$. Ising-like anisotropy implies a finite gap in the magnon spectrum. In our case, however, the situation is complicated by the positive slope of the linear $q̃$ dependence of the second-order result for $χxx(q)$, which according to Eq. (1.9) translates into $ω(q̃)$ decreasing with $q̃$. Combining the asymptotic forms of $χij$ from Eqs. (1.7a), (1.7b), and (1.8) together, as shown in Fig. 1, we see that $ω(q̃)$ is necessarily negative in the interval $qα≪q̃≪kF$, and thus LRO is unstable. Therefore anisotropy alone is not sufficient to ensure the stability of LRO: in order to reverse the sign of the $q̃$ dependence, one also needs to invoke other mechanisms, arising from higher orders in the electron-electron interaction. We show that at least one of these mechanisms—renormalization in the Cooper channel—is still operational even for $q̃≪qα$ and capable of reversing the sign of the $q̃$ dependence is the system is close to (but not necessarily in the immediate vicinity of) the Kohn-Luttinger instability.

We note that the dependences of $δχij$ on $q̃$ in the presence of the SOI is similar to the dependences on the temperature and magnetic field, 4 presented below for completeness:

$δχzz(T,α)=2χ0u2kF2|α|kF3EF+O(T3),δχzz(Bz,α)=2χ0u2kF2|α|kF3EF+OΔz2,δχxx(T,α)=χ0u2kF2|α|kF3EF+TEF+O(T3),δχxx(Bx,α)=χ0u2kF2|α|kF3EF+163π|Δx|EF.$
###### (1.10a)
Here, $Δi=gμBBi/2$ and $T,Δi≪|α|kF$. As Eqs. (1.7a), (1.7b), and (1.10a) demonstrate, while nonanalytic scaling of $δχzz$ with all three variables ( $q̃$, $T$, $B$) is cut off by the scale introduced by SOI, scaling of $δχxx$ continues below the SOI scale. This difference was shown in Ref. 4 to arise from the differences in the dependence of the energies of particle-hole pairs with zero total momentum on the magnetic field: while the energy of such a pair depends on the SO energy for $B||ẑ$, this energy drops out for $B⊥ẑ$.

In addition to modifying the behavior of $χij$ for $q̃≤qα$, SOI leads to a new type of the Kohn anomaly arising due to interband transitions: a nonanalyticity of $χij(q̃,α)$ at $q̃=qα$. The nonanalyticity is stronger in $χzz$ than in $χxx$: $δχzz(q̃≈qα)∝(q̃−qα)3/2Θ(q̃−qα)$, while $δχxx(q̃≈qα)∝(q̃−qα)5/2Θ(q̃−qα)$, where $Θ(x)$ is the step function. Consequently, the real-space RKKY interaction exhibits long-wavelength oscillations $χzz(r)∝cos(qαr)/r3$ and $χxx(r)∝sin(qαr)/r4$ in addition to conventional Friedel oscillations behaving as $sin(2kFr)/r2$. It is worth noting that the long-wavelength Friedel oscillations occur only in the presence of both electron-electron and SO interactions.

This paper is organized as follows. In Sec. II, we derive perturbatively the electron spin susceptibility of interacting 2DEG with the SOI as a function of momentum; in particular, Secs. II A,II B,II C,II D outline the derivation of all relevant second-order diagrams, Sec. II E is devoted to Cooper renormalization of the second order result, and in Sec. II F, we show that, in contrast to the spin susceptibility, the charge susceptibility is analytic at small $q̃$ (as it is also the case in the absence of SOI). In Sec. III, we derive the real-space form of the RKKY interaction and show that it exhibits long-wavelength oscillations with a period given by the SO length $2π/qα$. Details of the calculations are delegated to Appendices AD. In particular, the free energy in the presence of the SOI is derived beyond the random phase approximation in Appendix D. The summary and discussion of the main results are provided in Sec. IV.

##### II. SPIN SUSCEPTIBILITY OF INTERACTING ELECTRON GAS

Dynamics of a free electron in a two-dimensional electron gas (2DEG) in the presence of the Rashba spin-orbit interaction with a coupling strength $α$ is described by the following Hamiltonian:

$H=p22m*+α(pxσy−pyσx),$
###### (2.1)

where $p=(px,py)$ is the electron momentum of an electron and $σ$ is a vector of Pauli matrices. The interaction between electrons will be treated perturbatively. For this purpose, we introduce a Green's function:

$G(P)=1iωp−H−EF=∑sΩs(p)gs(P)$
###### (2.2)

with

$Ωs(p)=121+sp(pyσx−pxσy)$
###### (2.3)

and

$gs(P)=1iωp−εp−sαp,$
###### (2.4)

where $P≡(ωp,p)$ with $ωp$ being a fermionic Matsubara frequency, $εp=p2/2m*−EF$, and $s=±1$ is a Rashba index.

The nonanalytic part of a spin susceptibility tensor to second order in electron-electron interaction is given by seven linear response diagrams depicted in Figs. 2–7. Due to symmetry of the Rashba SOI, $χij(q̃)=χii(q̃)δij$ and $χxx=χyy≠χzz$.

In the following sections, we calculate all diagrams that contribute to nonanalytic behavior of the out-of-plane, $χzz$, and in-plane, $χxx=χyy$, components of the spin susceptibility tensor for small external momenta ( $q̃≪kF$) and at $T=0.$ In the absence of SOI, the nonanalytic contributions to the spin susceptibility from individual diagrams are determined by “backscattering” or “Cooper-channel” processes,20,24 in which two fermions with initial momenta $k$ and $p$ move in almost opposite directions, such that $k≈−p$. Backscattering processes are further subdivided into those with small momentum transfer, such that $(k,−k)→(k,−k)$, and those with momentum transfers near $2kF$, such that $(k,−k)→(−k,k)$. In the net result, all $q=0$ contributions cancel out and only $2kF$ contributions survive. We will show that this is also the case in the presence of the SOI. In what follows, all “ $q=0$” diagrams are to be understood as the $q=0$ channel of the backscattering process.

###### 1. General formulation

The first diagram is a self-energy insertion into the free-electron spin susceptibility, see Fig. 2. There are two contributions to the nonanalytic behavior: (i) from the region of small momentum transfers, i.e., $q≪kF$,

$χ1,q=0ij(q̃)=2U2(0)∫Q∫K∫PTr[G(P)G(P+Q)]×Tr[G(K+Q̃)σiG(K)G(K+Q)G(K)σj]$
###### (2.5a)

and (ii) from the region of momentum transfers close to $2kF$, i.e., $|k−p|≈2kF$ and $q≪kF$,

$χ1,q=2kFij(q̃)=2U2(2kF)∫Q∫K∫PTr[G(K+Q)G(P+Q)]×Tr[G(K+Q̃)σiG(K)G(P)G(K)σj].$
###### (2.5b)

Here, $K≡(ωk,k)$ and $∫K≡(2π)−3∫dωkd2k$ (and the same for other momenta). The time component of $Q̃=(Ω̃,q̃)$ is equal to zero throughout the paper. Since the calculation is performed at $T=0,$ there is no difference between the fermionic and bosonic Matsubara frequencies. A factor of 2 appears because the self-energy can be inserted either into the upper or the lower arm of the free-electron susceptibility. As subsequent analysis will show, a typical value of the momentum transfer $q$ is on the order of either the external momentum $q̃$ or the “Rashba momentum” $qα$ [cf. Eq. (1.6)], whichever is larger. In both cases, $q≪kF$ while the momenta of both fermions are near $kF$, thus we neglect $q$ in the angular dependences of the Rashba vertices: $Ωs(k+q)≈Ωs(k+q̃)≈Ωs(k)=[1+s(sinθkq̃σx−cosθkq̃σy)]/2$ with $θab≡∠(a,b)$. [The origin of the $x̂$ axis is arbitrary and can be chosen along $q̃.$] Also, we impose the backscattering correlation between the fermionic momenta: $k=−p$ in the $2kF$ part of the diagram. With these simplifications, we obtain

$χ1,q=0ij(q̃)=2U2(0)∫dΩ2π∫dθkq̃2π∫qdq2πalmnrijbst×Ilmnr(Ω,θkq̃,q,q̃)Πst(Ω,q),$
###### (2.6a)

$χ1,q=2kFij(q̃)=2U2(2kF)∫dΩ2π∫dθkq̃2π∫qdq2πãlmsrijb̃nt×Ilmnr(Ω,θkq̃,q,q̃)Πst(Ω,q),$
###### (2.6b)

where summation over the Rashba indices is implied,

$almnrij≡Tr[Ωl(k)σiΩm(k)Ωn(k)Ωr(k)σj],$
###### (2.7a)

$bst≡Tr[Ωs(p)Ωt(p)]=(1+st)/2,$
###### (2.7b)

$ãlmsrij≡Tr[Ωl(k)σiΩm(k)Ωs(−k)Ωr(k)σj],$
###### (2.7c)

$b̃nt≡Tr[Ωn(−p)Ωt(p)]=(1−nt)/2,$
###### (2.7d)

$Ilmnr(Ω,θkq̃,q,q̃)≡∫dθkq2π∫dωk2π∫dεk2π×gl(ωk,k+q̃)gm(ωk,k)gn(ωk+Ω,k+q)gr(ωk,k),$
###### (2.7e)

and, finally, the partial components of the dynamic particle-hole bubble are given by

$Πst(Ω,q)≡∫dθpq2π∫dωp2π∫dεp2π×gs(ωp,p)gt(ωp+Ω,p+q)=m2π|Ω|vF2q2+[Ω+i(t−s)αkF]2.$
###### (2.7f)

For the derivation of the particle-hole bubble, see, e.g. Ref. 4. Calculation of other common integrals is presented in Appendix A.

The main difference between the out-of-plane and in-plane components is in the structure of the “quaternion,” defined by Eq. (2.7e) and calculated explicitly in Appendix A [cf. Eq. (A3)]. The dependence of $Ilmnr$ on the external momentum $q̃$ enters only in a combination with the SOI coupling as $vFq̃cosθkq̃+(s−s′)αkF$, where $s,s′∈{l,m,n,r}$. Combinations of indices $l,m,n,r$ are determined by the spin vertices $σi,j$ and are, therefore, different for the out-of-plane and in-plane components. The out-of-plane component contains only such combinations ${l,m,n,r}$ for which the coefficient $s−s′$ is finite. Therefore the SOI energy scale is always present and, for $q̃≪qα$, one can expand in $q̃/qα$. The leading term in this expansion is proportional to $|α|$ but any finite-order correction in $q̃/qα$ vanishes. In fact, one can calculate the entire dependence of $χ1zz$ on $q̃$ (which is done in Appendix B) and show that $χ1zz$ is indeed independent of $q̃$ for $q̃≤qα$ (and similar for the remaining diagrams). On the other hand, some of the quaternions that enter the in-plane component have $s=s′$ and thus do not contain the SOI, which means that one cannot expand in $q̃/qα$ anymore. These quaternions provide linear-in- $q̃$ dependence of $χ1xx$ even for $q̃≤qα$, where the slope of this dependence is $2/3$ of that in the absence of the SOI. This is the origin of the difference in the $q̃$ dependences of $χzz$ and $χxx$, as presented by Eqs. (1.7a) and (1.7b).

The evaluation of the out-of-plane and in-plane part of diagram 1 is a subject of the next two sections.

###### 2. Diagram 1: out-of-plane component

We begin with the out-of-plane component of the spin susceptibility, in which case $almnrzz=[1+mr+n(m+r)−l(m+n+r+mnr)]/8$ and $ãlmsrzz=[1+mr−s(m+r)+l(s−m−r+mrs)]/8$. Summation over the Rashba indices yields

$χ1,q=0zz=4U2(0)∫dΩ2π∫dθkq̃2π∫qdq2π(I+−−−+I−+++)Π0$
###### (2.8a)

and

$χ1,q=2kFzz=2U2(2kF)∫dΩ2π∫dθkq̃2π∫qdq2π×[(I+−−−+I−+++)Π0+I+−+−Π+−+I−+−+Π−+],$
###### (2.8b)

where $Π0=Π++=Π−−$.

As we explained in Sec. II A 1, the quaternions in Eqs. (2.8a) and (2.8b) contain $q̃$ only in combination with $qα$. Therefore, for $q≪qα$, the leading term is obtained by simply setting $q̃=0$, upon which the remaining integrals can be readily calculated. The results are given by Eqs. (A7) and (A8), so that

$χ1,q=0zz=u02χ0|α|kF3EF$
###### (2.9a)

and

$χ1,q=2kFzz=u2kF2χ0|α|kF3EF.$
###### (2.9b)

In fact, it is shown in Appendix B that Eqs. (2.9a) and (2.9b) hold for any $q≤qα$ rather than only for $q̃=0$.

###### 3. Diagram 1: in-plane component

The in-plane component of the spin susceptibility differs substantially from its out-of-plane counterpart due the angular dependence of the traces $almnrij$ and $ãlmsrij$ which, for the in-plane case, read

$almnrxx=18[1+mr+n(m+r)−l(m+n+r+mnr)cos2θk],ãlmsrxx=18[1+mr−s(m+r)+l(s−m−r+mrs)cos2θk].$
###### (2.10)

(For the sake of convenience, we choose the $x$ axis to be perpendicular to $q̃$ when calculating all diagrams for $χxx$.) Summing over the Rashba indices, one arrives at

$χ1,q=0xx=4U2(0)∫dΩ2π∫dθkq̃2π∫qdq2π×[sin2θkq̃(I+−−−+I−+++)Π0+cos2θkq̃(I+++++I−−−−)Π0]$
###### (2.11a)

and

$χ1,q=2kFxx=2U2(2kF)∫dΩ2π∫dθkq̃2π∫qdq2π×[sin2θkq̃(I+−−−+I−+++)Π0+cos2θkq̃(I+++++I−−−−)Π0+sin2θkq̃(I+−+−Π+−+I−+−+Π−+)+cos2θkq̃(I++−+Π−++I−−+−Π+−)].$
###### (2.11b)

Details of the calculation are given in Appendix. A2b; here, we present only the results for $q̃≤qα$:

$χ1,q=0xx=12χ1,q=0zz+u02χ029πvFq̃EF=u02χ0|α|kF6EF+29πvFq̃EF,$
###### (2.12a)

$χ1,q=2kFxx=12χ1,q=2kFzz+u2kF2χ029πvFq̃EF=u2kF2χ0|α|kF6EF+29πvFq̃EF.$
###### (2.12b)

Notice that the linear-in- $q̃$ dependence survives in the in-plane component of the spin susceptibility even for $q̃≤qα$. Similar behavior was found in Ref. 4 for the temperature dependence of the uniform spin susceptibility in the presence of the SOI.

###### B. Diagram 2

Diagram 2, shown in Fig. 3, is a vertex correction to the spin susceptibility. As in the previous case, there are two regions of momentum transfers relevant for the nonanalytic behavior of the spin susceptibility: the $q=0$ region, where

$χ2,q=0ij=U2(0)∫Q∫K∫PTr[G(P)G(P+Q)]×Tr[G(K+Q̃)G(K+Q+Q̃)σiG(K+Q)G(K)σj],$
###### (2.13a)

and the $2kF$-region, where

$χ2,q=2kFij=U2(2kF)∫Q∫K∫PTr[G(K+Q)G(P+Q)]×Tr[G(K+Q̃)G(P+Q̃)σiG(P)G(K)σj].$
###### (2.13b)

Explicitly,

$χ2,q=0ij=U2(0)∫dΩ2π∫dθkq̃2π∫qdq2πclmnrijbst×Jlmnr(Ω,θkq̃,q,q̃)Πst(Ω,q),$
###### (2.14a)

$χ2,q=2kFij=U2(2kF)∫dΩ2π∫dθkq̃2π∫qdq2πc̃lrsmijb̃nt×Ilmn(Ω,θkq̃,q,q̃)Irst(Ω,θkq̃,q,−q̃),$
###### (2.14b)

where

$clmnrij≡Tr[Ωl(k)Ωm(k)σiΩn(k)Ωr(k)σj],$
###### (2.15a)
$c̃lrsmij≡Tr[Ωl(k)Ωr(−k)σiΩs(−k)Ωm(k)σj],$
###### (2.15b)
$Jlmnr(Ω,θkq̃,q,q̃)≡∫dθkq2π∫dωp2π∫dεk2πgl(ωk+Ω,k+q)×gm(ωk+Ω,k+q+q̃)gn(ωk+Ω,k+q)gr(ωk,k),$
###### (2.15c)
$Ilmn(Ω,θkq̃,q,q̃)≡∫dθkq2π∫dωp2π∫dεk2π×gl(ωk,k+q̃)gm(ωk,k)gn(ωk+Ω,k+q).$
###### (2.15d)
As before, summation over the Rashba is implied. Integrals (2.15c) and (2.15d) are derived in Appendix A.

Traces entering the $q=0$ part of the out-of-plane and in-plane components are evaluated as

$clmnrzz=1+nr−m(n+r)+l(m−n−r+mnr)8,clmnrxx=(1+lm)(1+nr)+(l+m)(n+r)cos2θkq̃8.$
###### (2.16)

Summing over the Rashba indices and using the symmetry properties of $Ilmnr$ and $Jlmnr$, it can be shown that the $q=0$ parts of diagrams 1 and 2 cancel each other

$χ2,q=0ij=−χ1,q=0ij,$
###### (2.17)

which is also the case in the absence of the SOI. 20 Therefore we only need to calculate the $2kF$ part of diagram 2.

###### 1. Diagram 2: out-of-plane component

Summation over the Rashba indices with the coefficient $c̃lrsmzz=[1+mr−s(m+r)+l(s−r−m+mrs)]/8$ for the out-of-plane part gives

$χ2,q=2kFzz=U2(2kF)∫dΩ2π∫dθkq̃2π∫qdq2π×[I+−+(Ω,θkq̃,q,q̃)I−+−(Ω,θkq̃,q,−q̃)+I+−−(Ω,θkq̃,q,q̃)I−++(Ω,θkq̃,q,−q̃)+(q̃→−q̃)],$
###### (2.18)

where $(q̃→−q̃)$ stands for the preceding terms with an opposite sign of momentum. Integrating over $q$ and $Ω$ at $q̃=0$, yields [cf. Eq. (A9)]

$χ2,q=2kFzz=u2kF2χ0|α|kF3EF.$
###### (2.19)

Again, an exact calculation at finite $q̃$ proves that this results holds for any $q̃≤qα$.

###### 2. Diagram 2: in-plane component

The in-plane component comes with a Rashba coefficient $c̃lmsrzz=[(1−lr)(1−ms)(l−r)(m−s)cos2θkq̃]/8$, such that

$χ2,q=2kFxx=U2(2kF)∫dΩ2π∫dθkq̃2π∫qdq2π×{sin2θkq̃[I+−+(Ω,θkq̃,q,q̃)I−+−(Ω,θkq̃,q,−q̃)+I+−−(Ω,θkq̃,q,q̃)I−++(Ω,θkq̃,q,−q̃)]+cos2θkq̃[I+++(Ω,θkq̃,q,q̃)I−−−(Ω,θkq̃,q,−q̃)+I++−(Ω,θkq̃,q,q̃)I−−+(Ω,θkq̃,q,−q̃)]+(q̃→−q̃)}.$
###### (2.20)

The first part, proportional to $sin2θkq̃$, contains the SOI coupling $α$. In this part, $q̃$ can be set to zero, and the resulting linear-in- $|α|$ part equals half of that for the out-of-plane component due to the integral over $sin2θkq̃$. On the other hand, in the term proportional to $cos2θkq̃$, the dependence on $|α|$ drops out upon integration over $q$, and the final result for $q̃≤qα$ reads as [cf. see Eq. (A12)]

$χ2,q=2kFxx=12χ2,q=2kFzz+u2kF2χ029πvFq̃EF=u2kF2χ0|α|kF6EF+29πvFq̃EF.$
###### C. Diagrams 3 and 4

We now turn to “Aslamazov-Larkin” diagrams, Fig. 4, which represent interaction via fluctuational particle-hole pairs. Without SOI, these diagrams are identically equal to zero because the spin vertices are averaged independently and thus vanish. With SOI, this argument does not hold because the Green's functions now also contain Pauli matrices and, in general, diagrams 3 and 4 do not vanish identically. Nevertheless, we show here that the nonanalytic parts of diagrams 2 and 3 are still equal to zero.

Diagrams 3 and 4 correspond to the following analytical expressions:

$χ3ij=∫Q∫K∫PU2(|q|)Tr[G(P−Q̃)G(P−Q)G(P)σi]×Tr[G(K+Q̃)G(K+Q)G(K)σj],$
###### (2.22a)

$χ3ij=∫Q∫K∫PU2(|q|)Tr[G(P)G(P+Q)G(P+Q̃)σi]×Tr[G(K+Q̃)G(K+Q)G(K)σj].$
###### (2.22b)

Note that the second trace is the same in both diagrams. In what follows, we prove that

$χ3ij=χ4ij=0$
###### (2.23)

for both small and large momentum transfers $q$.

###### 1. Diagrams 3 and 4: out-of-plane components

The out-of-plane case is straightforward. Evaluating the second traces in Eqs. (2.22a) and (2.22b), one finds that they vanish:

$dlnmz≡Tr[Ωl(k)Ωn(k)Ωm(k)σz]=0,$
###### (2.24)

for the $q=0$ case, and

$d̃lnmz≡Tr[Ωl(k)Ωn(−k)Ωm(k)σz]=0,$
###### (2.25)

for the $q=2kF$ case. Therefore $χ3zz=χ4zz=0$.

###### 2. Diagrams 3 and 4: in-plane components

For the in-plane part of the spin susceptibility, the proof is more complicated as the traces do not vanish on their own. To calculate the $q=0$ part, we need the following two objects:

$dlnmx≡Tr[Ωl(k)Ωn(k)Ωm(k)σx]=cosθkq̃(l+m+n+lmn)/4$
###### (2.26)

and

$Ilmn′(Ω,θkq̃,q,q̃)≡m*2π∫dωk∫dεkgl(ωk,k+q̃)×gm(ωk,k)gn(ωk+Ω,k+q)=im*ΩiΩ−vFqcosθkq+vFq̃cosθkq̃+(l−n)αkF×1iΩ−vFqcosθkq+(m−n)αkF.$
###### (2.27)

The prime over $I$ denotes that integration over the angle $θkq$ is not yet performed as compared to $Ilmn(Ω,θkq̃,q,q̃)$ defined by Eq. (2.15d).

Summing over the Rashba indices, one finds

$∑lmndlnmxIlmn′(Ω,θkq̃,q,q̃)=0$
###### (2.28)

and, therefore, the in-plane component at small momentum transfers vanishes.

The trace for the $q=2kF$ case turns out to be the same as for the $q=0$ one because

$d̃lnmx≡Tr[Ωl(k)Ωn(−k)Ωm(k)σx]=dlnmx.$
###### (2.29)

However, in order to see the vanishing of the $2kF$ part, the integral over $εk$ has to be evaluated explicitly with $q=2kF$, i.e.,

$Ilmn′′(Ω,θkq̃,q=2kF,q̃)=m*2π∫dεkgl(ωk,k+q̃)gm(ωk,k)gn(ωk+Ω,k+q)=im*[1−Θ(ωk)−Θ(ωk+Ω)][i(2ωk+Ω)−vFq̃cosθkq̃−vFq−vFkFφ2−(m+n)αkF][i(2ωk+Ω)−vFq−vFkFφ2−(l+n)αkF],$
###### (2.30)

where we used an expansion of $εk+q$ around $q=2kF$: $εk+q≈−εk+vF(q−2kF)+vFkFφ2,whereφ≡π−θkq$. Summing over the Rashba indices, we obtain

$∑lmnd̃lnmxIlmn′′(q≈2kF,q̃)=0$
###### (2.31)

and, therefore, the $2kF$ part of the in-plane components of diagrams 3 and 4 is also equal to zero.

###### D. Remaining diagrams and the final result for the spin susceptibility

The remaining diagrams can be expressed in terms of the those we have already calculated.

Diagram 5 in Fig. 5 reads

$χ5ij=−4U(0)U(2kF)∫Q∫K∫PTr[G(K+Q̃)σiG(K)×G(K+Q)G(P+Q)G(P)G(K)σj]=−4U(0)U(2kF)∫dΩ2π∫dθkq̃2π∫qdq2πflmntsrijIlmnrΠst$
###### (2.32)

with

$flmntsrij≡Tr[Ωl(k)σiΩm(k)Ωn(k)×Ωt(−k)Ωs(−k)Ωr(k)σj]$
###### (2.33)

and $q≪|k−p|=2kF$. A factor of 4 appears because the “sunrise” self-energy can be inserted into either the lower or the upper arm of the bubble while each of the interaction lines can carry momentum of either $q=0$ or $q=2kF$. A minus sign is due to an odd number of fermionic loops. Upon summation over the Rashba indices, we obtain

$χ5ijU(0)U(2kF)=−χ1,q=0ijU2(0).$
###### (2.34)

Diagrams 6 and $7b$ in Figs. 6 and 7, correspondingly, are related as well. Explicitly, diagram 6 reads

$χ6ij=−2U(0)U(2kF)∫Q∫K∫PTr[G(K+Q̃)σiG(K)×G(K+Q)G(P−Q̃)σjG(P)G(P−Q)]=−2U(0)U(2kF)∫dΩ2π∫dθkq̃2π∫qdq2πglmnrstij×Ilmn(Ω,θkq̃,q,q̃)Irst(−Ω,θkq̃,−q,q̃)$
###### (2.35)

with

$glmntsrij≡Tr[Ωl(k)σiΩm(k)Ωn(k)Ωr(−k)σjΩs(−k)Ωt(−k)].$
###### (2.36)

For diagram $7b$, we obtain

$χ7bij=−2U(0)U(2kF)∫Q∫K∫PTr[G(K+Q̃)σiG(K)×G(K+Q)G(P+Q)G(P)σjG(P+Q̃)]=−2U(0)U(2kF)∫dΩ2π∫dθkq̃2π∫qdq2πh̃lmntsrij×Ilmn(Ω,θkq̃,q,q̃)Irst(Ω,θkq̃,q,−q̃)$
###### (2.37)

with

$h̃lmntsrij≡Tr[Ωl(k)σiΩm(k)Ωn(k)Ωt(−k)Ωs(−k)σjΩr(−k)].$
###### (2.38)

In both cases, $q≪|k−p|=2kF$. Using the symmetry property $Irst(−Ω,θkq̃,−q,−q̃)=−I−r−s−t(Ω,θkq̃,q,q̃)$ in $χ4ij$, summing over the Rashba indices, and noticing that $I+++(Ω,θkq̃,q,q̃)=I−−−(Ω,θkq̃,q,q̃)$, we arrive at

$χ6ij=χ7bij.$
###### (2.39)

Finally, diagram $7a$ shown in Fig. 7 is related to diagram 2 at small momentum transfers. Indeed,

$χ7aij=−2U(0)U(2kF)∫Q∫K∫PTr[G(K+Q+Q̃)σi×G(K+Q)G(P)G(P+Q)G(K)σjG(K+Q̃)]=−2U(0)U(2kF)∫dΩ2π∫dθkq̃2π∫qdq2πhlmnstrijJlmnrΠst$
###### (2.40)

with

$hlmnstrij≡Tr[Ωl(k)Ωm(k)σiΩn(k)Ωs(−k)Ωt(−k)Ωr(k)σj],$
###### (2.41)

where again $q≪|k−p|=2kF$. After summation over the Rashba indices, this diagram proves related to the small-momentum part of diagram 2 as

$χ5aijU(0)U(2kF)=−χ2,q=0ijU2(0).$
###### (2.42)

The results of this section along with Eq. (2.17) show that the sum of all diagrams proportional to $U(0)U(2kF)$ cancel each other:

$χ5ij+χ6ij+χ7aij+χ7bij=0.$
###### (2.43)

Therefore, as in the absence of SOI, the nonanalytic part of the spin susceptibility is determined only by the Kohn anomaly at $q=2kF$.

Summing up the contributions from diagrams $1--3$, we obtain the results presented in Eqs. (1.7a) and (1.7b).

###### E. Cooper-channel renormalization to higher orders in the electron-electron interaction

An important question is how the second-order results, obtained earlier in this section, are modified by higher-order effects. In the absence of SOI, the most important effect—at least within the weak-coupling approach—is logarithmic renormalization of the second-order result by the interaction in the Cooper channel. As it was shown in Ref. 3, this effect reverses the sign of the $q̃$ dependence due to proximity to the Kohn-Luttinger superconducting instability; the sign reversal occurs at $q̃=e2TKL/vF≈7.4TKL/vF$, where $TKL$ is the Kohn-Luttinger critical temperature. For momenta below the SO scale ( $qα$), $χzz$ ceases to depend on $q̃$ but $χxx$ still scales linearly with $q̃$. What is necessary to understand now is whether the linear-in- $q̃$ term in $χxx$ is renormalized in the Cooper channel. The answer to this question is quite natural. The $|α|$ and $q̃$ terms in the second-order result for $χxx$ [see Eq. (1.7b)] come from different parts of diagram: the $|α|$ term comes from $q̃$ independent part and vice versa. Starting from the third order and beyond, these two terms acquire logarithmic renormalizations but the main logarithm of these renormalizations contains only one energy scale. In other words, the $|α|$ term is renormalized via $ln|α|$ while the $q̃$ is renormalized via $lnq̃$. For example, the third-order result for the $2kF$ part of diagram 1 (see Fig. 2) reads [for simplicity, we assume here a contact interaction with $U(q)=const$]

$χ1,q=2kFxx=−u32χ03|α|kFEFlnΛ|α|kF+23πvFq̃EFlnΛvFq̃,$
###### (2.44)

where $u=m*U/4π$ and $Λ$ is the ultraviolet cutoff. Details of this calculation are given in Appendix C. It is clear already from this result that the logarithmic renormalization of the $q̃$ term in $χxx$ remains operational even for $q̃≪qα$, with consequences similar to those in Ref. 3.

###### F. Charge susceptibility

In the absence of SOI, nonanalytic behavior as a function of external parameters $q̃$, $T$, $H$ is present only in the spin but not charge susceptibility.20,33,34 An interesting question is whether the charge susceptibility also becomes nonanalytic in the presence of SOI. We answer this question in the negative: the charge susceptibility remains analytic. To show this, we consider all seven diagrams replacing both spin vertices by unities. The calculation goes along the same lines as before, thereby we only list the results for specific diagrams; for $q̃≪qα$,

$δχ1c=−δχ4c=χ03πu02+u2kF2vFq̃EF,δχ2c=−δχ3c=χ03πu2kF2−u02vFq̃EF,δχ5c=−δχ6c=−χ03πu0u2kFvFq̃EF,$
###### (2.45)

whereas $χ7c=0$ on its own ( $χ7ac=−χ7bc$). First, we immediately notice that SOI drops out from every diagram even in the limit $q̃≪qα$. Second, the sum of the nonanalytic parts of all the charge susceptibility diagrams is zero, $δχc=0$, as in the case of no SOI.

##### III. RKKY INTERACTION IN REAL SPACE

A nonanalytic behavior of the spin susceptibility in the momentum space leads to a power-law decrease of the RKKY interaction with distance. In this section, we discuss the relation between various nonanalyticities in $χij(q)$ and the real-space behavior of the RKKY interaction. We show that, in addition to conventional $2kF$ Friedel oscillations, a combination of the electron-electron and SO interactions leads to a new effect: long-range Friedel-like oscillations with the period given by the SO length.

###### A. No spin-orbit interaction

First, we discuss the case of no SOI, when the spin susceptibility is isotropic: $χij(q̃)=δijχ(q̃)$. For free electrons, the only nonanalyticity in $χ0(q̃)$ is the Kohn anomaly at $q̃=2kF,$ which translates into Friedel oscillations of the RKKY kernel; in 2D and for $kFr≫1$, 35

$χ0(r)=χ02πsin2kFrr2.$
###### (3.1)

One effect of the electron-electron interaction is a logarithmic amplification of the Kohn anomaly (which also becomes symmetric about the $q̃=2kF$ point): $χ(q̃≈2kF)∝|q̃−2kF|ln|q̃−2kF|$. 36 Consequently, $χ(r)$ is also enhanced by a logarithmic factor compared to the free-electron case: $χ(r)∝sin(2kFr)ln(kFr)/r2$.

Another effect is related to the nonanalyticity at small $q̃$: $χ(q̃)=χ0+Cq̃$. 20 To second order in the electron-electron interaction [cf. Eq. (1.8)],

$C=C2≡4χ03πkFu2kF2;$
###### (3.2)

however, as we explained in Sec. I, both the magnitude and sign of $C$ can be changed due to higher-order effects. (Cooper channel renormalization leads also to multiplicative $lnq̃$ corrections to the linear-in- $q̃$ term; those correspond to multiplicative $lnr$ renormalization of the real-space result and are ignored here.)

In 2D, $χ(r)$ is related to $χ(q̃)$ via

$χ(r)=12π∫0∞dq̃q̃χ(q̃)J0(q̃r).$
###### (3.3)

Power-counting suggests that the $q̃$ term in $χ(q̃)$ translates into a dipole-dipole-like $1/r3$ term in $χ(r)$. To see if this is indeed the case, we calculate the integral

$A=∫0Λdq̃q̃2J0(q̃r)$
###### (3.4)

with an arbitrary cutoff $Λ$, and search for a universal, $Λ$-independent term in the result. If such a term exists, it corresponds to a long-range component of the RKKY interaction. Using an identity $xJ0(x)=ddx[xJ1(x)]$ and integrating by parts, we obtain

$A=1r3(Λr)2J1(Λr)−∫0ΛrdxxJ1(x)=1r3{(Λr)2J1(Λr)−πΛr2[J1(Λr)H0(Λr)−J0(Λr)H1(Λr)]},$
###### (3.5)

where $Hν(x)$ is the Struve function. The asymptotic expansion of the last term in the preceding equation indeed contains a universal term

$πΛr2J1(Λr)H0(Λr)−J0(Λr)H1(Λr)|Λr→∞=1+⋯,$
###### (3.6)

where $⋯$ stands for nonuniversal terms. A corresponding term in $χ(r)$ reads

$χ(r)=−C2πr3.$
###### (3.7)

As a check, we also calculate the Fourier transform of the $q̃$-independent term in $χij$. The corresponding integral

$Ã=∫0Λdq̃q̃J0(q̃r)=ΛrJ1(Λr)$
###### (3.8)

does not contain a $Λ$-independent term and, therefore, a constant term in $χ(q̃)$ does not produce a long-range component of the RKKY interaction, which is indeed the case for free electrons.

Equation (3.7) describes a dipole-dipole-like part of the RKKY interaction that falls off faster than Friedel oscillations but is not oscillatory. (Incidentally, it is the same behavior as that of a screened Coulomb potential in 2D, which also has a $q̃$ nonanalyticity at small $q̃$. 37)

In a translationally invariant system, $HRKKY=−A28ns2∑r,r′χ(|r−r′|)IriIr′j$. Therefore, if $C>0$, i.e., $χ(q̃)$ increases with $q̃$, the dipole-dipole interaction is repulsive for parallel nuclear spins and attractive for antiparallel ones. Since the $1/r3$ behavior sets in only at large distances, the resulting phase is a helimagnet rather than an antiferromagnet. Vice versa, if $C<0,$ the dipole-dipole interaction is attractive for parallel spins. This corresponds precisely to the conclusions drawn from the spin-wave theory: a stable FM phase requires that $ω(q̃)>0$, which is the case if $C<0$.

###### C. Free electrons

In a free-electron system, the SOI splits the Fermi surface into two surfaces corresponding to two branches of the Rashba spectrum with opposite helicities. Consequently, both components of the spin susceptibility in the momentum space have two Kohn anomalies located at the momenta $2kF±=2kF∓qα$ with $qα=2m*α.$ To see this explicitly, we evaluate the diagonal components of $χij(q̃)$ for $q̃≈2kF$,

$χ0ii(q̃)=−∑s,t∫K|〈k,s|σi|k+q̃,t〉|2gt(ω,k+q̃)gs(ω,k).$
###### (3.9)

For $q̃≈2kF$, the matrix elements of the spin operators in the helical basis reduce to

$|〈k+q̃,t|σx|k,s〉|2=|〈k+q̃,t|σz|k,s〉|2=12(1+st).$
###### (3.10)

Therefore $χii(q̃)$ contains only contributions from intraband transitions,

$χ0xx(q̃)=χzz(q̃)=−∫Kg+(ω,k+q̃)g+(ω,k)−∫Kg−(ω,k+q̃)g−(ω,k).$
###### (3.11)

Each of the two terms in Eq. (3.11) has its own Kohn anomaly at $q̃=2kFs$, $s=±$. In real space, this corresponds to beating of Friedel oscillations with a period $2π/qα.$

This behavior needs to be contrasted with that of Friedel oscillations in the charge susceptibility, where, to leading order in $α$, the Kohn anomaly is present only at twice the Fermi momenta in the absence of SOI. 38 Consequently, the period of Friedel oscillations is the same as in the absence of SOI. (Beating occurs in the presence of both Rashba and Dresselhaus interactions. 39) This is so because, for $q̃$ near $2kF,$ the matrix element entering $χc(q̃)$ reduces to

$〈k+q̃,t|k,s〉2=12(1−st),$

which implies that $χc$ contains only contributions from interband transitions:

$χ0c(q̃)=−2∫Kg+ω,k+q̃g−ω,k.$
###### (3.12)

The Kohn anomaly in $χ0c$ corresponds to the nesting condition $εk+q̃+=−εk−$, which is satisfied only for $q̃=2kF.$

###### 1. Interacting electrons

The electron-electron interaction is expected to affect the $2kF$-Kohn anomalies in $χxx$ and $χzz$ in a way similar to that in the absence of SOI. However, a combination of the electron-electron and SO interaction leads to a new effect: a Kohn anomaly at the momentum $qα≪2kF$. Consequently, the RKKY interaction contains a component that oscillates with a long period given by the SO length $λSO=2π/qα$ rather than the half of the Fermi wavelength.

To second order in the electron-electron interaction, the full dependence of the electron spin susceptibility on the momentum is shown in Appendix B to be given by

$δχxx(q̃)=2C2q̃3+C2q̃2Re{131−qαq̃22+qαq̃2+qαq̃arcsinqαq̃},$
###### (3.13a)

$δχzz(q̃)=C2q̃Re1−qαq̃2+qαq̃arcsinqαq̃.$
###### (3.13b)

Equations (3.13a) and (3.13b) are valid for an arbitrary value of the ratio $q̃/qα$ (but for $q̃≪kF$). For $q̃≫qα$, both $δχxx$ and $δχzz$ scale as $q̃$. For $q̃≪qα$, $δχxx$ continues to scale as $q̃$ (but with a smaller slope compared to the opposite case), while $δχzz$ is $q̃$ independent. The crossover between the two regimes is not continuous, however, because certain derivatives of both $δχxx$ and $δχzz$ diverge at $q̃=qα$. Expanding around the singularity at $q̃=qα$, one finds

$δχxx=2C2q̃3+C2̃2{Θ(qα−q̃)+Θ(q̃−qα)1+2b5q̃qα−15/2},$
###### (3.14a)

$δχzz=C2̃Θ(qα−q̃)+Θ(q̃−qα)1+bq̃qα−13/2,$
###### (3.14b)

where $Θ(x)$ is the step function, $C2̃=πC2qα/2$ and $b=42/3π$. The $q̃$ dependences of $δχxx$ and $δχzz$ are shown in Fig. 8.

The singularity is stronger in $δχzz∝(q̃−qα)3/2$ whose second derivative diverges at $q̃=qα$, whereas it is only third derivative of $δχxx∝(q̃−qα)5/2$ that diverges at this point. Both divergences are weaker than the free-electron Kohn anomaly $χ∝(q̃−2kF)1/2.$

We now derive the real-space form of the RRKY interaction, starting from $χzz(r)$. Substituting Eq. (3.14b) into Eq. (3.3) and noting that only the part proportional to $(q̃/qα−1)3/2$ contributes, we arrive at the following integral:

$χzz(r)=C2̃b2π∫qαΛdq̃q̃J0(q̃r)q̃qα−13/2,$
###### (3.15)

where $Λ$ is an arbitrarily chosen cutoff that does not affect the long-range behavior of $χzz(r)$. Replacing $J0(x)$ by its large- $x$ asymptotic form and $q̃$ by $qα$ in all nonsingular and nonoscillatory parts of the integrand, we simplify the previous expression to

$χzz(r)=C2̃b2π2qαπr∫0Λdq̃q̃qα3/2cos(q̃+qα)r−π4.$
###### (3.16)

Integrating by parts twice and dropping the high-energy contribution, we arrive at an integral that converges at the upper limit. The final result reads

$χzz(r)=−χ023π2u2kF2kFcosqαrr3.$
###### (3.17)

Equation (3.17) describes long-wavelength Friedel-like oscillations that fall off with $r$ faster than the usual $2kF$ oscillations. Notice that Eq. (3.17), while valid formally only for $qαr≫1$, reproduces correctly the dipole-dipole term [see Eq. (3.7) with $C=C2$] in the opposite limit of $qαr≪1$. Therefore Eq. (3.17) can be used as an extrapolation formula applicable for any value of $qαr$.

In addition to the Kohn anomaly at $q̃=qα$, the in-plane component also contains a nonoscillatory but nonanalytic term proportional to $q̃$. As it was also the case in the absence of SOI, this term translates into a dipole-dipole part of the RKKY interaction. Analysis of Sec. III fully applies here; we just need to replace the prefactor $C$ in Eq. (3.7) by $2C2/3$, where $C2$ is defined by Eq. (3.2). The role of the cutoff $Λ$ in Eq. (3.4) is now being played by $qα$, therefore, $C→2C2/3$ for $r≫qα−1$. For $r≪qα−1$, the prefactor is the same as in the absence of SOI. Summarizing, the dipole-dipole part of the in-plane RKKY interaction is

$χd−dxx(r)=−23π2u2kF2χ01/r3,forqαr≪1,2/3r3,forqαr≫1.$
###### (3.18)

The oscillatory part of $χxx(r)$ is obtained by the same method as for $χzz(r)$ with the only difference that one needs to integrate by parts three times in order to obtain a convergent integral. Consequently, $χxx(r)$ falls off with $r$ as $1/r4$. The $r$ dependence of $χxx(r)$, resulting from the SOI, is given by a sum of the nonoscillatory and oscillatory parts:

$χxx(r)=χd-dxx(r)+χ013π2u2kF2qαkFsin(qαr)r4.$
###### (3.19)

Finally, the conventional, $2kF$ Friedel oscillations should be added to Eqs. (3.17) and (3.19) to get a complete $r$ dependence. The dipole-dipole part and long-wavelength Friedel oscillations fall off faster then conventional Friedel oscillations. In order to extract the long-wavelength part from the data, one needs to average the measured $χij(r)$ over many Fermi wavelengths. Recently, $2kF$ oscillations in the RKKY interaction between magnetic adatoms on metallic surfaces have been observed directly via scanning tunneling microscopy. 40 Hopefully, improvements in spatial resolution would allow for an experimental verification of our prediction for the long-wavelength component of the RKKY interaction.

As a final remark, we showed in Sec. II F that the charge susceptibility does not exhibit small- $q$ nonanalyticities. This result also implies that the long-wavelength oscillations are absent in the charge susceptibility; therefore Friedel oscillations produced by nonmagnetic impurities contain only a conventional, $2kF$ component.

##### IV. SUMMARY AND DISCUSSION

We have studied the nonanalytic behavior of the electron spin susceptibility of a two-dimensional electron gas (2DEG) with SOI as a function of momentum $q̃=|q̃|$ in the context of a ferromagnetic nuclear-spin phase transition (FNSPT). Similarly to the dependence on temperature and magnetic field, 4 the combined effect of the electro-electron and spin-orbit interactions affects two distinct components of the spin susceptibility tensor differently. For $q̃≤2m*|α|$, where $m*$ is the effective electron mass and $α$ is the spin-orbit coupling, the out-of-plane component of the spin susceptibility $χzz(q̃,α)$ does not depend on momentum (in other words, the momentum-dependence is cut off by the SOI), [cf. Eq. (1.7a)], whereas its in-plane counterparts, $χxx(q̃,α)=χyy(q̃,α)$, scale linearly with $q̃$ even below the energy scale given by the SOI [cf. Eq. (1.7b)].

Beyond second order in electron-electron interaction, renormalization effects in the Cooper channel, being the most relevant channel in the weak-coupling regime, start to play a dominant role. As we have shown in Sec. II E, the leading linear-in- $|α|$ term becomes renormalized by $ln|α|$, while the subleading linear-in- $q̃$ term acquires additional $lnq̃$ dependence. This behavior is a natural consequence of the separation of energy scales in each of the diagrams and suggests that, in general, $χ(n)({Ei})∝Un∑iEilnn−2Ei$, where $Ei$ stands for a generic energy scale (in our case $Ei={|α|kF,vFq̃}$ but temperature or the magnetic field could be included as well).

Our analysis of the spin susceptibility gives important insights into the nature of a FNSPT. First, the SOI-induced anisotropy of the spin susceptibility implies that the ordered phase is of an Ising type with nuclear spins aligned along the $z$ axis since $χzz>χxx$. Second, the ferromagnetic phase cannot be stable as long as the higher-order effects of the electron-electron interaction are not taken into account. In this paper, we focused only on one type of those effects, i.e., renormalization in the Cooper channel. Without Cooper renormalization, the slope of the magnon dispersion is negative, even though the magnon spectrum is gapped at zero momentum, cf. Fig. 1. This implies that spin-wave excitations destroy the ferromagnetic order. Only inclusion of higher-order processes in the Cooper channel, similarly to the mechanism proposed in Ref. 3, leads to the reversal of the slope of the spin susceptibility in the (not necessarily immediate) vicinity of the Kohn-Luttinger instability, and allows for the spin-wave dispersion to become positive at all values of the momentum. This ensures stability of the ordered phase at sufficiently low temperatures.1,2

We have also shown that a combination of the electron-electron and SO interactions leads to a new effect: a Kohn anomaly at the momentum splitting of the two Rashba subbands. Consequently, the real-space RKKY interaction has a long-wavelength component with a period determined by the SO rather than the Fermi wavelength.

Another issue is whether the SOI modifies the behavior of the charge susceptibility which is known to be analytic in the absence of the SOI.20,33,34 As our calculation shows, the answer to this question is negative.

One more comment on the spin and charge susceptibilities is in order: despite the fact that we considered only the Rashba SOI, all our results are applicable to systems where the Dresselhaus SOI with coupling strength $β$ takes place of Rashba SOI, i.e., $β≠0$, $α=0$; in this case, the Rashba SOI should be simply replaced by the Dresselhaus SOI ( $α→β$).

Finally, we analyzed the nonanalytic dependence of the free energy, $F$, in the presence of the SOI and at zero temperature beyond the RPA. This analysis is important in the context of interacting helical Fermi liquids that have recently attracted considerable attention. In contrast to the RPA result, 42 which predicts that the free energy scales with $α$ as $α4ln|α|$, our result shows that the renormalization is stronger, namely, $F∝U2|α|3C(Uln|α|)$, where $C(x→1)∼x2$ and $C(x→∞)∼1/x2$.

## References

1. P. Simon and D. Loss, Phys. Rev. Lett. 98, 156401 (2007).
2. P. Simon, B. Braunecker, and D. Loss, Phys. Rev. B 77, 045108 (2008).
3. S. Chesi, R. A. Żak, P. Simon, and D. Loss, Phys. Rev. B 79, 115445 (2009).
4. R. A. Żak, D. L. Maslov, and D. Loss, Phys. Rev. B 82, 115415 (2010).
5. A. C. Clark, K. K. Schwarzwälder, T. Bandi, D. Maradan, and D. M. Zumbühl, Rev. Sci. Instr. 81, 103904 (2010).
6. D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998).
7. R. A. Żak, B. Röthlisberger, S. Chesi, and D. Loss, Riv. Nuovo Cimento 33, 345 (2010).
8. M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, Nature (London) 432, 81 (2004).
9. J. M. Elzerman, R. Hanson, L. H. W. van Beveren, B. Witkamp, L. M. Vandersypen, and L. P. Kouwenhoven, Nature (London) 430, 431 (2004).
10. S. Amasha, K. MacLean, I. P. Radu, D. M. Zumbühl, M. A. Kastner, M. P. Hanson, and A. C. Gossard, Phys. Rev. Lett. 100, 046803 (2008).
11. J. R. Petta, A. C. Johnson, J. M. Taylor, E. Laird, A. Yacoby, M. D. Lukin, and C. M. Marcus, Science 309, 2180 (2005).
12. F. H. L. Koppens, K. C. Nowack, and L. M. K. Vandersypen, Phys. Rev. Lett. 100, 236802 (2008).
13. H. Bluhm, S. Foletti, I. Neder, M. Rudner, D. Mahalu, V. Umansky, and A. Yacoby, Nat. Phys. 7, 109 (2010).
14. G. Burkard, D. Loss, and D. P. DiVincenzo, Phys. Rev. B 59, 2070 (1999).
15. A. V. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. Lett. 88, 186802 (2002).
16. A. V. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. B 67, 195329 (2003).
17. W. A. Coish and D. Loss, Phys. Rev. B 70, 195340 (2004).
18. A. S. Brackner, A. Stinaff, D. Gammon, E. Ware, G. Tischler, A. Shabaev, L. Efros, D. Park, D. Gershoni, L. Korenev, and A. Merkulov, Phys. Rev. Lett. 94, 047402 (2005).
19. C. Kittel, Quantum Theory of Solids (Wiley, New York, 987).
20. A. V. Chubukov and D. L. Maslov, Phys. Rev. B 68, 155113 (2003).
21. A. V. Chubukov, C. Pépin, and J. Rech, Phys. Rev. Lett. 92, 147003 (2004); J. Rech, C. Pépin, and A. V. Chubukov, Phys. Rev. B 74, 195126 (2006).
22. In a quantum-critical region near the Stoner instability, the $\stackrel{̃}{q}$ term in the spin susceptibility transforms into a ${\stackrel{̃}{q}}^{3/2}$ one, cf. Ref. 21.
23. A. Shekhter and A. M. Finkel'stein, Phys. Rev. B 74, 205122 (2006).
24. D. L. Maslov, A. V. Chubukov, and R. Saha, Phys. Rev. B 74, 220402(R) (2006); D. L. Maslov and A. V. Chubukov, ibid. 79, 075112 (2009).
25. Strictly speaking, the nonanalytic behavior of $\chi$ in the generic FL regime was analyzed as a function of the temperature and of the magnetic field rather than as a function of $\stackrel{̃}{q}$. However, in all cases studied so far, the nonanalytic dependence of $\chi \left(\stackrel{̃}{q},H,T\right)$ has always been found to be symmetric in all variables, i.e., $\chi \left(\stackrel{̃}{q},H,T\right)=\chi \left(0,0,0\right)+\mathrm{max}\left\{{C}_{\stackrel{̃}{q}}\stackrel{̃}{q},{C}_{H}H,{C}_{T}T\right\}$, with ${C}_{\stackrel{̃}{q},H,T}$ being of the same sign. It is likely that the same also holds true in the generic FL regime.
26. A. Shekhter and A. M. Finkel'stein, Proc. Natl. Acad. Sci. USA 103, 15765 (2006).
27. N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).
28. D. Loss, F. L. Pedrocchi, and A. J. Leggett, Phys. Rev. Lett. 107, 107201 (2011).
29. A. Ashrafi and D. L. Maslov (unpublished).
30. S. Chesi, Ph.D. thesis, Purdue University, 2007.
31. A. O. Caride, C. Tsallis, and S. I. Zanette, Phys. Rev. Lett. 51, 145 (1983); 51, 616 (1983); M. Kaufman and M. Kardar, ibid. 52, 483 (1984).
32. N. W. Ashcroft and N. D. Mernin, Solid State Physics (Saunders College, Philadelphia, 1976).
33. D. Belitz, T. R. Kirkpatrick, and T. Vojta, Phys. Rev. B 55, 9452 (1997).
34. G. Y. Chitov and A. J. Millis, Phys. Rev. Lett. 86, 5337 (2001).
35. G. F. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University Press, Cambridge, 2005).
36. I. G. Khalil, M. Teter, and N. W. Ashcroft, Phys. Rev. B 65, 195309 (2002).
37. T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982).
38. M. Pletyukhov and V. Gritsev, Phys. Rev. B 74, 045307 (2006).
39. S. M. Badalyan, A. Matos-Abiague, G. Vignale, and J. Fabian, Phys. Rev. B 81, 205314 (2010).
40. L. Zhou et al., Nat. Phys. 6, 187 (2010).
41. A. Agarwal, S. Chesi, T. Jungwirth, J. Sinova, G. Vignale, and M. Polini, Phys. Rev. B 83, 115135 (2011).
42. S. Chesi and G. F. Giuliani, Phys. Rev. B 83, 235308 (2011).
43. S. Chesi and G. F. Giuliani, Phys. Rev. B 83, 235309 (2011).

Daniel Loss received his Diploma (1983) and Ph.D. (1985) in theoretical physics at the University of Zürich. From 1989 to 1991 he worked as a postdoc at the University of Illinois at Urbana-Champaign with A. J. Leggett, and from 1991 to 1993 at the IBM T.J. Watson Research Center, NY. In 1993 he joined the faculty of Simon Fraser University in Vancouver, and then returned to Switzerland in 1996 to become Professor of Physics at the University of Basel. His research interests include quantum coherence and spin physics in semiconducting and magnetic nanostructures, and quantum computing. In 2000 he became an APS Fellow and in 2005 he received the Humboldt Research Prize.

## Related Articles

Biological Physics

### Synopsis: Proteins as Shock Absorbers

Proteins in nerve cells function like shock absorbers that protect the cells from mechanical stress. Read More »

Cosmology

### Viewpoint: Cosmic Clues from Mini Clumps of Dark Matter

Searches for ultracompact clumps of cold dark matter have come up empty, but these nondetections place new limits on the early expansion history of the Universe. Read More »

Superfluidity

### Synopsis: Doubling Up with Superfluids

Researchers mixed two superfluids of different atoms together and observed that vortices in one affected those in the other—evidence of mutual interaction between the two species. Read More »