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

  • Robert Andrzej Żak, Dmitrii L. Maslov, and Daniel Loss, Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland and Department of Physics, University of Florida, P.O. Box 118440, Gainesville, Florida 32611-8440, USA
Phys. Rev. B 85, 115424
(Left) A skeleton diagram for the free energy in the presence of the Cooper renormalization; Γ̃ is a renormalized Cooper vertex. (Middle) The effective scattering amplitude Γs1s2;s3s4(1)(k,k;p,p) in the chiral basis. (Right) A generic nth order ladder diagram in the Cooper channel, Γs1s2;s3s4(n)(k,k;p,p).(Left) A skeleton diagram for the free energy in the presence of the Cooper renormalization; Γ̃ is a renormalized Cooper vertex. (Middle) The effective scattering amplitude Γs1s2;s3s4(1)(k,k;p,p) in the chiral basis. (Right) A generic nth order lad... Show more
Diagram 1 to third order in electron-electron interaction at large momentum transfer; here, the in-plane component is shown.
The nonanalytic part of the electron spin susceptibility in units of (2/3π)u2kF2(|α|/vF)χ0 as a function of the momentum in units of qα=2m*|α|. i=x,z. Solid lines are the exact results (3.13a) and (3.13b). Dashed lines are the approximate results (3.14a) and (3.14b) valid near the singularity at q=qα.The nonanalytic part of the electron spin susceptibility in units of (2/3π)u2kF2(|α|/vF)χ0 as a function of the momentum in units of qα=2m*|α|. i=x,z. Solid lines are the exact results (3.13a) and (3.13b). Dashed lines are the approximate results (3.... Show more
Diagram 7a (upper figure) and diagram 7b (lower figure). The transferred momenta are q= and |kp|=2kF.Diagram 7a (upper figure) and diagram 7b (lower figure). The transferred momenta are q= and |kp|=2kF. Show more
Diagram 6. The momentum transfer q is close to zero and |kp|=2kF.Diagram 6. The momentum transfer q is close to zero and |kp|=2kF. Show more
Diagram 5. The momentum transfer q is close to zero and |kp|=2kF.Diagram 5. The momentum transfer q is close to zero and |kp|=2kF. Show more
Top: diagram 3. Bottom: diagram 4. The momentum transfer q in both diagrams can be either small or close to 2kF.Top: diagram 3. Bottom: diagram 4. The momentum transfer q in both diagrams can be either small or close to 2kF. Show more
Diagram 2. Top: small-momentum transfer part. Bottom: 2kF-momentum transfer part.Diagram 2. Top: small-momentum transfer part. Bottom: 2kF-momentum transfer part. Show more
Diagram 1. Top: small-momentum transfer part. Bottom: 2kF-momentum transfer part. Kands denotes a fermion from Rashba subband s=±1 with “four-momentum” K=(ωk,k).Diagram 1. Top: small-momentum transfer part. Bottom: 2kF-momentum transfer part. Kands denotes a fermion from Rashba subband s=±1 with “four-momentum” K=(ωk,k). Show more
A normalized dispersion of the out-of-plane spin-wave mode ω̃(q̃)=ω(q̃)/(A2Iχ0/4ns) as a function of the momentum. To second order in interaction (lower curve) ω(q̃) is necessarily negative for m|α|q̃kF, and thus LRO is unstable. Solid parts of the curves corresponds to actual calculations; dashed parts are interpolations between various asymptotic regimes. Renormalization effects in the Cooper channel reverse the slope of ω(q̃) (upper curve) and stabilize LRO.A normalized dispersion of the out-of-plane spin-wave mode ω̃(q̃)=ω(q̃)/(A2Iχ0/4ns) as a function of the momentum. To second order in interaction (lower curve) ω(q̃) is necessarily negative for m|α|q̃kF, and thus LRO is unstable. Solid parts of the... Show more

Spontaneous nuclear spin polarization in semiconductor heterostructures at finite but low temperatures has recently attracted a considerable attention both on the theoretical 1–4 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. 8–10 The decoherence time in single electron GaAs QDs has been reported to exceed 1μs 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 200μs. 13 Still, even state-of-the-art dynamical nuclear polarization methods 14–18 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,


is parameterized by an effective exchange coupling


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


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 q̃0. In a spin-isotropic and translationally invariant system,


with χij=δijχ, while the magnetization is given by


where μN is the nuclear-spin magneton (we set kB==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 q̃2kF, and thus the magnon contribution to M(T) diverges in the q̃0 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 q̃kF, 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. 24–26

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 D2. 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/vF1/LT. For q̃1/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, LT1 mm at T1 mK.) Therefore q̃1/LQ1/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:


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 q̃qα:


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


In Eqs. (1.7a) and (1.7b), uqm*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 qαq̃kF, the spin susceptibility goes back to the result of Ref. 20 valid in the absence of the SOI:


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 q̃qα and q̃qα, 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


with q̃ẑ. 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 qαq̃kF, 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 q̃qα 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:

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 Bẑ.

In addition to modifying the behavior of χij for q̃qα, 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(q̃qα)(q̃qα)3/2Θ(q̃qα), while δχxx(q̃qα)(q̃qα)5/2Θ(q̃qα), 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.


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:


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:






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=χyyχzz.

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 ( q̃kF) 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 kp. 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.

A. Diagram 1
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., qkF,


and (ii) from the region of momentum transfers close to 2kF, i.e., |kp|2kF and qkF,


Here, K(ωk,k) and K(2π)3dωkd2k (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, qkF 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(sinθkq̃σxcosθkq̃σy)]/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



where summation over the Rashba indices is implied,






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


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+(ss)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 ss is finite. Therefore the SOI energy scale is always present and, for q̃qα, 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 q̃qα (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 q̃qα, 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+mrs(m+r)+l(smr+mrs)]/8. Summation over the Rashba indices yields




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 qqα, 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




In fact, it is shown in Appendix B that Eqs. (2.9a) and (2.9b) hold for any qqα 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


(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




Details of the calculation are given in Appendix. A2b; here, we present only the results for q̃qα:



Notice that the linear-in- q̃ dependence survives in the in-plane component of the spin susceptibility even for q̃qα. 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


and the 2kF-region, where






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


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


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+mrs(m+r)+l(srm+mrs)]/8 for the out-of-plane part gives


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


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

2. Diagram 2: in-plane component

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


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 q̃qα reads as [cf. see Eq. (A12)]

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:



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


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:


for the q=0 case, and


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:




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


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


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.,


where we used an expansion of εk+q around q=2kF: εk+qεk+vF(q2kF)+vFkFφ2,whereφπθkq. Summing over the Rashba indices, we obtain


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




and q|kp|=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


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




For diagram 7b, we obtain




In both cases, q|kp|=2kF. Using the symmetry property Irst(Ω,θkq̃,q,q̃)=Irst(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


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




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


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:


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/vF7.4TKL/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]


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 q̃qα, 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 q̃qα,


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


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 kFr1, 35


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(q̃2kF)|q̃2kF|ln|q̃2kF|. 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)],


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


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


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


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


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


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


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=A28ns2r,rχ(|rr|)IriIrj. 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.

B. With spin-orbit interaction
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±=2kFqα with qα=2m*α. To see this explicitly, we evaluate the diagonal components of χij(q̃) for q̃2kF,


For q̃2kF, the matrix elements of the spin operators in the helical basis reduce to


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


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


which implies that χc contains only contributions from interband transitions:


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 qα2kF. 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



Equations (3.13a) and (3.13b) are valid for an arbitrary value of the ratio q̃/qα (but for q̃kF). For q̃qα, both δχxx and δχzz scale as q̃. For q̃qα, 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



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(q̃qα)3/2 whose second derivative diverges at q̃=qα, whereas it is only third derivative of δχxx(q̃qα)5/2 that diverges at this point. Both divergences are weaker than the free-electron Kohn anomaly χ(q̃2kF)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/qα1)3/2 contributes, we arrive at the following integral:


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


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


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 qαr1, reproduces correctly the dipole-dipole term [see Eq. (3.7) with C=C2] in the opposite limit of qαr1. 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, C2C2/3 for rqα1. For rqα1, the prefactor is the same as in the absence of SOI. Summarizing, the dipole-dipole part of the in-plane RKKY interaction is


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:


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.


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 q̃2m*|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})UniEilnn2Ei, 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., β0, α=0; in this case, the Rashba SOI should be simply replaced by the Dresselhaus SOI ( αβ).

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. 41–43 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, FU2|α|3C(Uln|α|), where C(x1)x2 and C(x)1/x2.


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  25. Strictly speaking, the nonanalytic behavior of χ in the generic FL regime was analyzed as a function of the temperature and of the magnetic field rather than as a function of q̃. However, in all cases studied so far, the nonanalytic dependence of χ(q̃,H,T) has always been found to be symmetric in all variables, i.e., χ(q̃,H,T)=χ(0,0,0)+max{Cq̃q̃,CHH,CTT}, with Cq̃,H,T being of the same sign. It is likely that the same also holds true in the generic FL regime.
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About the Authors

Image of Robert Andrzej Żak
Image of Dmitrii L. Maslov
Image of Daniel Loss

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.

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