Raman Scattering by a Two-Dimensional Fermi Liquid with Spin-Orbit Coupling

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by  Saurabh Maiti

Raman Scattering by a Two-Dimensional Fermi Liquid with Spin-Orbit Coupling
Saurabh Maiti and Dmitrii L. Maslov
Department of Physics, University of Florida, Gainesville, FL 32611
(Dated: January 5, 2017)
WepresentamicroscopictheoryofRamanscatteringbyatwo-dimensionalFermiliquid(FL)with
RashbaandDresselhaustypesofspin-orbitcoupling,andsubjecttoanin-planemagneticfield(B~).
In the long-wavelength limit, the Raman spectrum probes the collective modes of such a FL: the
chiralspinwaves. Thecharacteristicfeaturesofthesemodesarealinear-in-qterminthedispersion
7
and thedependenceof themodefrequency on thedirections of both ~q and B~. Allof thesefeatures
1
0 havebeen observed in recent Raman experiments on CdTe quantumwells.
2

n Ramanscatteringoflightisauniquetoolthatallowsto modes are linearly polarized with M~ being in the 2DEG
a
study dynamics of elementary excitations in solids both plane for two of the modes and along the normal for the
J
in space and time and to probe both single-particle and third one. If the in-plane magnetic field (B~) is applied,
4
collective properties of electron systems. It helps to un- themodewithM~||B~ remainslinearlypolarizedwhilethe

] derstand a variety of phenomena: from pairing mech- other two modes with M~ ⊥ B~ become elliptically polar-
l
l anisms in high-temperature superconductors[1] to spin ized[14]. Figure1adepictstheevolutionoftheexcitation
a
h waves for in spintronic devices[2]. The latter requires spectrum with B at q = 0. As the field increases, two
- probingdynamics of electronspins,which canbe done if outofthethreemodesrunintothecontinuumofspin-flip
s
e the incident and scattered light are polarized perpendic- excitations (SFE), while the third one merges with the
m ular to each other (cross-polarizedgeometry)[3]. continuumatB =B ,whenthespin-splitFermisurfaces
c
. Non-trivial spin dynamics is encountered in systems (FSs) become degenerate, and re-emergesto the right of
t
a withspontaneousmagneticorder,orinanexternalmag- this point. As the field is increased further, this mode
m netic field, or else in the presence of spin-orbit coupling transformsgraduallyintothe Silin-Leggett(SL) modeof
- (SOC).InthisLetter,wedevelopamicroscopictheoryof a partially polarized FL [15–17]. In the experiment, the
d
Raman scattering by a two-dimensional (2D) Fermi liq- effective Zeeman energy is larger than both Rashba and
n
o uid(FL)inthepresenceofanin-planemagneticfieldand Dresselhaussplittingswhich,accordingtoFig.1a,allows
c bothRashba[4]andDresselhaus[5]typesofSOC.Break- us to focus on the case of a single CSW adiabatically
[ ing spin conservation leads to a substantial modification connected to the SL mode. We show that, at small q,
1 of the Raman spectrum already at the non-interacting the dispersion of this mode can be written as
v level [6]. Unlike in the SU(2)-invariant case, the cross-
1 section of Raman scattering cannot be expressed only
04 in terms of charge and spin susceptibilities. We show, Ω(~q,B~)=Ω(~0,θB~)+w(θq~,θB~)q+Aq2, (1)
however, that the scattering cross-section for the cross-
1
0 polarized geometry in the long-wavelength limit is pa-
. rameterizedbycomponentsofthespinsusceptibilityten- whereθ~x is the azimuthalangleof~x. Symmetry dictates
1
0 sor. In a FL, these components containpoles that corre- that Rashba and Dresselhaus SOC contribute sin(θq~ −
7 spond to collective modes arising from an interplay be- θB~)andcos(θq~+θB~)termstow(θq~,θB~),correspondingly,
1 tweenelectron-electroninteraction(eei)anddynamicsof whilethe massterm,Ω(~0,θB~),andtheboundariesofthe
v: spins. SFE continuum are π-periodic function of θB~.
i
X Recently, a dispersing peak was observed in resonant In general, Raman scattering probes both charge and
r RamanscatteringonmagneticallydopedCdTequantum spin excitations. However, if the polarizations of the in-
a wells in the presence of an in-plane magnetic field [7– cident (~e ) and scattered (~e ) light are perpendicular to
I S
9]. The telling signs of the interplay between eei and eachother,theRamanvertexcouplesdirectlytotheelec-
SOC are the linear-in-q term in the dispersion and the tron spin [3, 6]. The Feynman diagrams for the differen-
π-periodic modulation of the spectrum as the magnetic tialscatteringcross-sectionareshowninFig.2. Thefirst
field is rotated in the plane of 2D electron gas (2DEG). diagramontheleft-handsideisthepurespinpart,while
Weidentifytheobservedpeakwithoneofthelongsought the spin-charge part arising due to SOC is represented
after chiralspin waves(CSWs) [10–13]. These wavesare by a sum of bubbles connected by the Coulomb interac-
collectiveoscillationsofthemagnetization(M~)thatexist tion (dashed line). In the long wavelengthlimit (~q →0),
even in the absence of the magnetic field. In zero field, the spin-charge part vanishes by charge conservation; at
therearethreesuchmodes[11–13]whicharemassive,i.e., small but finite ~q, the spin part is still larger than the
their frequencies are finite at q →0, and disperse with q spin-charge one. For non-interacting electrons the con-
on a characteristic scale set by spin-orbit splitting. The tributionoftheformertothescatteringcross-sectioncan
2
12
(b) 1.2 (c)
10 η=10
1
8
∆D 6 eV)0.8
Ω/ 4 η=5 Ω (m0.6
2 η=0.6 0.4
η=0 0.2
0
0 0.2 0.4 0.6 0.8 1 0
θB/2π 0 0.2 0.4φ/2π0.6 0.8 1
FIG. 1: Color online: a) Schematically: excitation spectrum (at q = 0) of a 2D Fermi liquid with spin-orbit coupling and
subject toin-planemagneticfieldB~. Shadedregion: continuumofsingle-particle spin-flipexcitations; lines: chiralspinwaves.
b) Continuum as a function of the angle θ between B~ and the (100) direction. ∆ /∆ =0.5, η =∆ /∆ , where ∆
B~ R D Z D D/R/Z
is Dresselhaus/Rashba/Zeeman splitting. c) Continuum (shaded) and collective-mode frequency at q =0 (line) as a function
of φ=θB~ −π/2 for Cd1−xMnxTe quantumwell.
be written as [6] modes. Ingeneral,equationsofmotionforthethreecom-
ponents of magnetization are coupled to each other, and
ddΩ2dAO ∝ |γµµ′|2Im[L0µµ′]; γµµ′ =h~n·~σˆeiq~·~riµµ′, (2) hweitnhcediaffllertehnetcroemsidpuoense.nts of χˆ have the same poles but
Xµµ′
To proceedfurther, we assume that the magnitudes of
where dO is the solid-angleelement,~σˆ are the Paulima- the Rashba (∆R), Dresselhaus (∆D), and Zeeman (∆Z)
trices, L0µµ′ ≡ [f(εµ)−f(εµ′)]/(Ω+εµ−εµ′ +i0+), εµ energy splittings are much smaller than the Fermi en-
ergy [22]. The non-interacting Hamiltonian for a (001)
is the energy of state µ, f(ε) is the Fermi function, and
quantum well can then be written as
~n =~e ×~e [18]. At small ~q, the off-diagonal (spin-flip)
I S
processesdominatethecross-section. Inthemostgeneral
case, when both time-reversal and inversion symmetries H~k =vF(k−kF)σˆ0+ siσˆi, (4)
are broken, the cross-sectionis given by [19] iX=1,2
d2A where σˆ0 is the identity matrix, the x1 axis is chosen
dΩdO ∝Im ninjχ0ij(~q,Ω)+n23χ033(~q,Ω), (3) alongthe(100)direction,vF andkF aretheFermiveloc-
i,j∈X{1,2} ity and momentum in the absence of SOC and magnetic
field, correspondingly,and
where χ0 (~q,Ω) is the spin-spin correlation function of
lm
non-interacting electrons (the x and x axes are in the 1
1 2 s = ∆ sinθ +∆ cosθ +∆ cosθ ,
2DEG plane and the x3 axis is along the normal). In 1 2 R ~k D ~k Z B~
(cid:0) (cid:1)
the absence of SOC, the cross-section contains only the 1
s = − ∆ cosθ +∆ sinθ −∆ sinθ . (5)
transverse (to the direction of B~) spin-spin correlation 2 2 R ~k D ~k Z B~
(cid:0) (cid:1)
function, χ0 = χ0 +χ0 [20, 21]. In the presence of
SOC,howev⊥er,the+r−esult−d+oesnotreducetoχ0. Instead, The corresponding Green’s function is given by
⊥
the partial components of χ0 contribute to the cross-
section with weights determilnmed by the direction of ~n. Gˆ0K = g++2 g−σˆ0+ g+2−|s|g− siσˆi, (6a)
iX=1,2
1
g = , ε =v (k−k )±|s|, (6b)
± iω −ε ~k,± F F
n ~k,±
where |s|= s2+s2 and K ≡(~k,iω ).
FIG. 2: Diagrams for the differential cross-section of Ra- 1 2 n
man scattering in cross-polarized geometry. Shaded bubbles As SOC pand magnetic field are weak, the interac-
denote vertex corrections due to eei. V(q) is the coulomb tion of two quasiparticles with momenta ~k and ~k′ can
interaction. be described by an SU(2)-invariant Landau function
Fˆ(ϑ) = σˆ σˆ′Fs(ϑ)+~σˆ ·~σˆ′Fa(ϑ), where ϑ = θ −θ .
0 0 ~k ~k′
In the presence of eei, the free-electron correlation Withinthisapproximation,thechargeandspincollective
function in Eq. (3) is replaced by the renormalized one: modes are decoupled, and we focus on the interaction in
χ0 → χ . The poles of χ correspond to collective the spin channel, parameterized by Fa(ϑ).
lm lm lm
3
To set the stage, we discuss the SL mode in the ab- coefficient of the sin2θ term must be on the order of
B~
sence of SOC. Kohn’s theorem protects the q = 0 term ∆ ∆ /∆ . Inadditiontotheanisotropicpart,thereare
R D Z
in the dispersion from renormalization by eei [23, 24]; also isotropic corrections of order ∆2/∆ and ∆2 /∆ .
R Z D Z
therefore, Ω(~0,B~) = ∆Z. The absence of the linear-in-q To lowest order in SOC, the form of the linear term
term is guaranteed by symmetry: the only rotationally- is determined by the symmetries of the Rashba (group
invariant combination of ~q and B~ is ~q · B~, which is a C )andDresselhaus(groupD )typesofSOC.Inboth
∞v 2d
pseudoscalar and therefore not allowed since Ω must be cases, we need to form a scalar (Ω) out of a polar vector
a scalar. For the quadratic term, although (~q · B~)2 is (~q) and a pseudovector (B~). In the C group, this is
∞v
allowed by symmetry, such a term is absent because ~q only possible by forming the Rashba invariant B q −
1 2
is the orbital momentum whereas B acts only on elec- B q ∝ sin(θ −θ ), which is the same term as in the
2 1 q~ B~
tron spins. The energy of a particle-hole pair formed by original Rashba Hamiltonian with ~σˆ →B~. Likewise, the
fermions with momenta ~k and ~k+~q and with opposite
only possible scalar in the D group is the Dresselhaus
spinsisε −ε =v qcos(θ −θ )+q2/2m±∆ . In- 2d
tegration~k+ovq~e,±rθ ~ke,l∓iminaFtesthe d~kepeq~ndence onθ ,Zhence invariant B1q1 −B2q2 ∝ cos(θq~ + θB~). The quadratic
~k q~ term in the high-field limit canbe takenthe same as the
only a q2 term is possible. Combining the symmetry ar-
quadratic term in the SL mode, Eq. (7) [26].
guments with dimensional analysis, we find
Combiningtogetherallthe argumentsgivenabove,we
Ω(~q,B~)=∆ +a ({Fa})(vFq)2, (7) arrive at the following form of the coefficients in Eq. (1)
Z 2
∆
Z
where a2 is a dimensionless function which depends Ω(~0,B~) = ∆ +a ({Fa})∆2R+∆2Z
on the angular harmonics of Fa(ϑ): a ({Fa}) = Z 0 ∆
2 Z
a (Fa,Fa,Fa...). The precise form of a is determined ∆ ∆
2 0 1 2 2 +a˜ ({Fa}) R D sin2θ ,
by a microscopic theory [15, 25]. 0 ∆ B~
Z
We now apply the same reasoning to CSWs. If
∆
only Rashba SOC is present, there is no preferred in- w(θq~,θB~) = vFa1({Fa})(cid:20)∆R sin(θq~−θB~)
Z
plane direction, hence a linear-in-q term is absent while
∆
the quadratic term is isotropic. Therefore, the disper- + D cos(θ +θ ) ,
sions of the three CSWs can be written as Ω (~q,~0) = ∆Z q~ B~ (cid:21)
α
a˜α({Fa})∆ +a˜α({Fa})(v q)2/∆ ,wherea˜α aresome v2
0 R 2 F R 0,2 A = a ({Fa}) F , (8)
other dimensionless functions of the FL parameters and 2 ∆
Z
α= 1...3. Since Kohn’s theorem does not apply in the
presence of SOC systems, a˜α 6= 1. Explicit forms of the where a , a˜ , a and a are dimensionless functions.
0 0 0 1 2
functions a˜α were found in Refs. [11–13]. If only Dres-
0,2
selhaus SOC is present, the Hamiltonian can be trans-
fTtohhremerHeeadfomtroeil,ttothhneeiaRCnaSssWhhabsvaaerfdoeritffmheerbesynatmreseypimlnacmbinoegtthriθec~ksa.→sesπa/lt2h−ouθg~kh. µm) 100(a) Ω(q=0) (meV)0000....23450 0 . 5 φ/2π 1 eV)0.004..545 B=2T (b)
preIfsebnto,ththReaqsh=ba0atnedrmDirnestshelehaduisspetyrspioesn oisf oSbOvCiouasrlye µw (eV−10 Ω (m0.35 B=−2T
isotropic,while a linear-in-q term is not allowedby sym- −20 0.3 φ=π/4
mq,etthrye.liInnedaeretde,rmsinccoeutlhdeodnilsypebresioofntmheusfotrbmeca1nqa1ly+ticc2qin2 −300 0.2 0.4φ/2π0.6 0.8 1 0.250 0.2 vFq 0(m.4eV) 0.6 0.8
with c = const, but such a form does not obey the
1,2
symmetries of the D group (rotation by π and reflec- FIG.3: a)VariationoftheCSWvelocityw[Eq.(1)]withthe
2d
tion about the diagonal plane). The quadratic term can angleφ=θB~−π/2,asdefinedinRefs.[7–9]. Inset: variation
of thefrequency at q=0 with φ. b) Dispersion of a CSW in
have an anisotropic part of the form q q ∝ sin2θ , but
1 2 q~ theCdMnTequantumwellina2T field(B~ ⊥~q)forφ=π/4.
the prefactor of such a combination should be propor-
The negative sign of the field implies flippingits direction.
tional to the product ∆ ∆ , and thus the anisotropic
R D
part is small compared to the isotropic, q2 part.
Finally, let both types of SOC and the magnetic field We confirm Eq. (8) by an explicit calculation within
be presentbut, in agreementwith the experimentalcon- thes-waveapproximation,inwhichthe Landaufunction
ditions of Refs. [7–9], ∆Z ≫ ∆R,∆D. The q = 0 term contains only the zeroth harmonic: Fa(θ) =F0a. In this
inthedispersionthendependsontheorientationofB~ in case, the FL theory is equivalent to the Random Phase
the plane; the D -symmetry forces this dependence to Approximation (RPA) in the spin channel [13, 14], and
2d
beofsin2θ form. SincetheanisotropicpartofΩ(~0,θ ) weadopttheRPAmethodforconvenience. Summingup
B~ B~
is non-zero only if both types of SOC are present, the the series ofladder diagrams,we obtain for the tensor of
4
the spin susceptibility the dispersion in Eqs. (1) and (8) is simplified to
χˆ(Q)= (gµB)2χˆ1(Q) Iˆ+ F0aχˆ1(Q) −1, (9) Ω(~q,B~) = ∆Z 1+a0(r2+d2)+a˜0rdsin2θB~
4 (cid:20) 2 (cid:21) (cid:2) v q (v q)2
F F
±a r−dsin2θ +a .
where Q = (~q,iΩn), g is the effective Land´e factor, µB 1(cid:0) B~(cid:1) ∆Z 2 ∆Z (cid:21)
is the Bohr magneton, Iˆ is the 3 × 3 identity matrix, (12)
χ1(Q) = − Tr σˆ Gˆ1 σˆ Gˆ1 , and G1 differs from
ij K i K j K+Q K where r = ∆ /∆ and d = ∆ /∆ . Up to the an-
G0K in Eq. (R6a) onhly in that the ibare Zeeman energy is gular dependeRnce oZf the mode mDass, tZhis form was con-
replaced by the renormalized one, ∆∗Z = ∆Z/(1+F0a), jectured in Refs. [8, 9] on phenomenological grounds.
while ∆R and ∆D are not renormalized in the s-wave The measured frequency of the mode at q = 0 gives
approximation[10, 27, 28]. The collective modes are the ∆ = 0.4meV at B = 2T. For the number density of
Z
roots of the equation n = 2.7×1011cm−2 and effective mass of m∗ = 0.1m
e
(m is the bare electron mass), k =1.3×10−2˚A−1 and
Fa e F
Det Iˆ+ 0 χˆ1(~q,iΩ →Ω+i0+) =0. (10) v = 1.0 eV·˚A. The range of v q is 0−0.6 meV. Using
n F F
(cid:20) 2 (cid:21)
theseparameters,weachievethebestagreementwiththe
ThedetailsonsolvingthisequationaregivenintheSup- dataforΩ(~0,B~)andw(±π/2+θB~,θB~)(Fig.3a)bychoos-
plementary Material (SM). Here, we only mention that ing F0a = −0.41, ∆R ≈ 0.05 meV, and ∆D ≈ 0.1 meV
the results coincide with those in Eq. (8), and explicit which corresponds to the Rashba and Dresselhaus cou-
expressionsforthe dimensionlessfunctions ofthe FL pa- pling constants of α = 1.9 meV·˚A and β = 3.8 meV·˚A,
rameter Fa read in agreement with the values found in Ref. [9]. The
0
screened Coulomb interaction with a dielectric constant
a˜0(F0a) = −(1+F02a)F(02a+F0a), oq-fd1e0p.0enfdoernaceCdoTfethqeuamnotudme (wFeigll.y3ibel)dsreFp0rao≈du−ce0s.3t5h.eTehxe-
(1+Fa)(2+3Fa) perimentallyobservedoneverywell. Basedonthisagree-
a (Fa) = 0 0 ,
0 0 4Fa ment between the theory and experiment, we argue that
0
thecollectivemodeobservedinRefs. [7–9]is,infact,one
(1+Fa)2 (4+Fa)(1+Fa)+(Fa)2
a (Fa) = − 0 0 0 0 , of the sought-after CSWs [10–13], probed in the regime
1 0 (cid:2) Fa(2+Fa)2 (cid:3)
0 0 ofastrongmagneticfield. Thesametypeofexperiments
a (Fa) = (1+F0a)2. (11) performed at lower fields on systems with stronger SOC
2 0 2Fa and/orhighermobilities,e.g.,onInGaAs/InAlAs,should
0
reveal the whole spectrum shown in Fig. 1a.
[The form of a and a coincides with the previous re-
2 0 The phenomenological model of Refs. [8, 9] describes
sults [14, 25] in the s-wave approximation.] For repul-
the data assuming very strong (up to a factor of 6.5)
sive eei, Fa < 0 and the quadratic term in the disper-
0 renormalization of SOC by many-body effects, which is
sionis alwaysnegative, while the signof the interaction-
not consistentwith the moderate (<2) values of param-
dependent prefactor of the linear term [a (F )] is posi-
1 0 eter r in Cd Mn Te. Our theory explains the data
s 1−x x
tive. However,theoverallsignofthelineartermdepends
with no such assumption. References [8] and [9] argued
on the signs of the Rashba and Dresselhaus couplings.
that the dispersion can be obtained by a linear shift of
The continuumof SFE correspondsto interbandtran-
the SL dispersion: Aq2 →A|~q+~q |2. As is evident from
0
sitions and is given by a set of Ω that satisfy Ω =
Eq. (11),however,thisformholdsonlyatweakcoupling
|ε −ε | for all states on the FS. Because ε vary
~k,+ ~k,− ± (|Fa|≪1).
0
around the FS, Ω varies between the minimum (Ω )
min In conclusion, we developed a microscopic theory of
and maximum (Ω ) values, which determine a finite
max Raman scattering by a 2D FL in the presence of both
width of the continuum even at q = 0 (see Fig. 1a). In
Rashba and Dresselhaus SOC, and subject to an in-
the presence of the B~, both Ω1 and Ω2 depend on θB~. plane magnetic field. An interplay between eei and
The anisotropic part of ε~k,± is given by ∆R∆Dsin2θ~k SOC leads to resonance peaks at frequencies of CSWs.
+∆R∆∗Zsin(θ~k −θB~)+∆D∆∗Zcos(θ~k +θB~). As one can All the features of one of those peaks observed recently
see, changing θB~ → θB~ + π can be compensated by on a CdTe quantum well are successfully accounted for
θ~k → θ~k + π, and thus Ωmin,max(θB~) have a period of by this theory. The formalism developed here can be
π. Figure1bshowsΩmin,max(θB~)forarangeofthemag- readily extended to other 2D systems with broken in-
netic field. versionsymmetry,suchas grapheneontransition-metal-
We are now in a position to apply our results to re- dichalcogenide substrates and surface states of topologi-
cent Raman data on Cd1−xMnxTe quantum well [7–9]. cal insulators/superconductors.
In these experiments, ~q and B~ are chosen to be perpen- The authors are grateful to Y. Gallais, A. Kumar, I.
dicular to eachother, i.e., θ −θ =±π/2. Accordingly, Paul, F. Perez, and C. A. Ullrich for stimulating discus-
B~ q~
5
sions and to M. Imran for his help at the initial stage of 3, 448 (1970).
this work. Note: After this manuscript was almost com- [17] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics
pleted, we learned of a recent preprint, Ref. [29], which Part II (Pergamon Press, New York,1980).
[18] Although in reality the experiment is often performed
also identifies a 2θ modulation of the mode frequency
B~ underresonanceconditions,whenthefrequencyoflightis
at q =0 as a second-order effect in SOC.
tunedtoaparticularinter-bandtransition,theresonance
affects mostly the overall magnitude of the cross-section
but not its dependenceson Ω and ~q [6]. In this case, the
sum over µ and µ′ in Eq. (2) includes only the states
within either theconduction or valence band.
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Rev.Lett. 117, 137204, (2016). czewski, T. Wojtowicz, “Spin precession and spin waves
[10] A.Shekhter,M.Khodas, andA.M.Finkelstein, “Chiral in a chiral electron gas: beyond Larmor’s theorem”,
spinresonanceandspin-Hallconductivityinthepresence arXiv:1612.04314 (2016).
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Fermi Liquids with Spin-Orbit Coupling”, Phys. Rev.
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6
SUPPLEMENTARY MATERIAL
RAMAN RESPONSE
The Raman scattering cross-section for a non-interacting two-dimensional electron gas (2DEG) in cross-polarized
geometry is given by [6]
d2A 2πe2 ZZ¯
dΩdO ∝ |γµµ′|2ImL0µµ′ − q Imǫ(q,Ω), (13a)
Xµµ′
γµµ′ = ih~n·~σˆeiq~·~riµµ′, (13b)
L0µµ′ = Ωf+(εεµµ)−−εfµ(′ε+µ′)iδ, (13c)
Z = γµµ′he−iq~·~riµµ′L0µµ′; Z¯ = γµ∗µ′heiq~·~riµµ′L0µµ′ (13d)
Xµµ′ Xµµ′
2πe2
ǫ(q,Ω) = ǫ+ q |heiq~.~riµµ′|2L0µµ′. (13e)
Xµµ′
Here, ǫ is the background dielectric constant, µ and µ′ refer to the quantum numbers of electrons which include the
momentum and spin/chirality: µ = {~k,ν}, µ′ = {~k′,ν′} with ν,ν′ = ±1, ε is the energy of a state with quantum
µ
numberµ,f(ε)istheFermifunction,andhXiµµ′ ≡ d~rψµ†(~r)X(~r)ψµ′(~r)isthematrixelementofX(~r)betweenstates
µ and µ′. Furthermore,~n=~eI×~eS, where~eI,S are Rthe polarizations of incident and scattered light, correspondingly.
The two terms in Eq. (13a) refer to two contributions in Fig. 2 of the Main Text (MT): the first term is the spin
part (given by the first diagram) while the second term is the mixed spin-charge part (given by the sum of diagrams
starting from the second one). The matrix elements are computed with respect to the eigenvectorsof Hamiltonian in
Eq. (4) of the MT
ψ (~r)=ei~k·~r iνe−iφ~k , (14)
µ (cid:18) 1 (cid:19)
where tanφ =−s /s and s are defined by Eq. (5) of the MT. Explicitly, we obtain:
k 1 2 1,2
γµµ′ = iδ~k−~k′−q~ n1 iν′e−iφ~k+q~−iνeiφ~k +n2 −ν′e−iφ~k+q~−νeiφ~k +n3 νν′e−i(φ~k+q~−φ~k)−1 ,
h n o n o n oi
heiq~.~riµµ′ = δ~k−~k′−q~ νν′e−i(φ~k+q~−φ~k)+1 . (15)
h i
We areinterestedinthe small-q limit, whenthe q-dependenttermsinthe dispersionsofchiralspinwaves(CSWs)are
corrections to the frequencies of these waves at q =0, which implies that v q ≪Ω with v being the Fermi velocity
F F
in absence of both the magnetic field and spin-orbit coupling (SOC). At the same time, both the magnetic field and
SOC are assumed to be weak: Ω≪v k . Combining the two inequalities above, we obtain
F F
q ≪Ω/v ≪k . (16)
F F
The firstpartof this inequality ensures thatthe diagonalpartsof L0 inEq.(13a) aresmallby chargeconservation:
µµ′
L0 ∝q. Therefore,the maincontributiontothe cross-sectioninthislimitcomesfromoff-diagonaltermswith
~k,±;~k+q~,±
νν′ = −1, which correspond to processes that flip spin/chirality. Furthermore, Eq. (16) also implies that q ≪ k ;
F
therefore, φ can be approximated by φ , upon which the off-diagonal matrix elements are simplified to
~k+q~ k
γµµ′ ≈ −2iδ~k−~k′−q~ in1νcosφ~k++in2νsinφ~k+n3 , (17a)
|γµµ′|2 ≈ 4δ~k−~k′−q~ (n(cid:2)1cosφ~k+n2sinφ~k)2+n23 , (cid:3) (17b)
heiq~·~riµµ′ ≈ νν′+1=(cid:2)0. (cid:3) (17c)
Bythesameargument,thediagonaltermsinZ andZ¯ inEq.(13a)aresmallasq,whiletheoff-diagonalcomponents
of heiq~·~riµµ′ are small by Eq. (17c). Therefore, the second term in Eq. (13a) can be neglected compared to the first
one, and the cross-sectionis reduced to
d2A d2k
∝ n2cos2φ +n2sin2φ +n n sin2φ +n2 (g g˜ +g g˜ ), (18)
dΩdO Z (2π)2 1 ~k 2 ~k 1 2 ~k 3 Z + − + −
(cid:2) (cid:3) ω
7
where g =1/(iω−ε ) and g˜ =1/(i{ω+Ω}−ε ). The various terms in the equation above can be expressed
ν ~k,ν ν ~k+q~,ν
via the components of the spin-spin correlation function in the chiral basis:
χ0 (Q) = − (g g˜ +g g˜ )cos2φ ,
11 Z~kZω + − + − ~k
χ0 (Q) = − (g g˜ +g g˜ )sin2φ ,
22 Z~kZω + − + − ~k
χ0 (Q) = − (g g˜ +g g˜ )sinφ cosφ ,
12 Z~kZω + − + − ~k ~k
χ0 (Q) = χ0 (Q),
21 12
χ0 (Q) = − (g g˜ +g g˜ ), (19)
33 Z~kZω + − + −
where ≡ d2k/(2π)2 and ≡ dω/2π. Making use of the definitions, we re-write Eq. (18) as
~k ω
R R R R
d2A
∝n2χ0 +n2χ0 +n n (χ0 +χ0 )+n2χ0 , (20)
dΩdO 1 11 2 22 1 2 12 21 3 33
which is the result given in Eq. (3) of the MT.
GeneralizationofEq.(20)forthecaseofinteractingelectronsamountssimplytoreplacingthefree-electronspin-spin
correlation function by the interacting one.
DISPERSION OF SPIN CHIRAL WAVES
The general form of the dispersion is established in Eq. 8 of the MT using symmetry and dimensional analysis.
Here, we confirm this form by an explicit calculation which yields the forms of functions a , a˜ , a , and a . In
0 0 1 2
the Random Phase Approximation (RPA) or, equivalently, in the s-wave approximation for the Landau interaction
function, the eigenmode equation is given by Eq. (10) the MT. Therefore, the problem reduces to finding expansions
ofthespin-spincorrelationfunctions, χ1 (Q),toorderq2 [asinthe maintext, Q≡(~q,iΩ )]. Unlessstatedotherwise,
ab n
all frequencies in the Supplementary Materialare the Matsubaraones. Whenever it does not lead to a confusion, the
Matsubara index n will be suppressed and the frequencies will be denoted simply as iΩ. Analytic continuation to
real frequencies will be done at the final step. We follow the general scheme developed in Refs. [13, 14] to find the
dispersions of the collective modes. For brevity, we will switch to notations where Fa =−u and χ1 =−Π .
0 ab ab
General properties of the polarization bubble
The polarization bubble is defined as
Π (Q)=− Tr σˆ Gˆ1 σˆ Gˆ1 , (21)
ab Z~kZω h a K+Q b Ki
where a,b ∈ {0...3} and the Green’s function G1 is obtained from G0 in Eq. (6a) of the MT by replacing ∆ →
P P Z
∆∗ =∆ /(1−u). This yields
Z Z
g g˜ +g g˜ s s˜ g g˜ +g g˜ s s˜
+ + − − i j + − − + i j
Π (Q) = 2δ +Tr[σˆ σˆ σˆ σˆ ] + 2δ −Tr[σˆ σˆ σˆ σˆ ]
ab ab a i b j ab a i b j
Z~kZω(cid:20) 4 (cid:26) |s||s˜|(cid:27) 4 (cid:26) |s||s˜|(cid:27)
g g˜ −g g˜ s s˜ g g˜ −g g˜ s s˜
+ + − − j j + − − + j j
Tr[σˆ σˆ σˆ ] +Tr[σˆ σˆ σˆ ] + Tr[σˆ σˆ σˆ ] −Tr[σˆ σˆ σˆ ] ,
b a j a b j b a j a b j
4 (cid:26) |s| |s˜|(cid:27) 4 (cid:26) |s| |s˜|(cid:27)(cid:21)
(22)
wherei,j ∈{1,2}andatildeaboveanyquantitymeansthatits(2+1)momentumisshiftedbyQwithrespecttothe
momentum of the correspondingquantity without a tilde. Because the magnetic field andSOC are weak, integration
over the actual (anisotropic) Fermi surface can be replaced by that over the circular Fermi surface of radius k .
F
Accordingly,
dθ
···=N dξ ···≡N ... (23)
Z~k F Z 2π Z ~k F ZθZξ~k
8
where N =k /2πv is the 2D density of states and ξ =v (k−k ). In the same approximation, ε =ξ +ν|s|,
F F F ~k F F ~k,ν ~k
where |s| = s2+s2 is evaluated at ξ = 0 but does depend on the azimuthal angle θ of ~k. Furthermore, we will
1 2
be neglectingpthe difference between |s| at ~k +~q and |s| at ~k: keeping such terms would amount to higher order
corrections. Accordingly, we approximateε =ξ +ν|s˜|≈ξ +~v ·~q+ν|s|, where we also neglectedthe O(q2)
~k+q~,ν ~k+q~ ~k F
term as being higher order in q/k .
F
The chiral Green’s functions at momentum ~k +~q need to be expanded to second order in ~q. Within the same
approximations as specified above,
1
g˜ν ≡gν(~k+~q,iω+iΩ)=g¯ν +∂jg¯νqj + 2∂j∂j′g¯νqjqj′ ≈g¯ν +g¯ν2~vF ·~q+g¯ν3(~vF ·~q)2, (24)
where g¯ ≡g (~k,iω+iΩ) and ∂ ≡∂/∂k
ν ν l l
We will need the following integrals
(ν−ν′)|s|
Z Z gνg¯ν′ = −iΩ+(ν−ν′)|s|,
ξ~k ω
iΩ
Z Z gνg¯ν2′ = (iΩ+(ν−ν′)|s|)2,
ξ~k ω
iΩ
Z Z gνg¯ν3′ = (iΩ+(ν−ν′)|s|)3, (25)
ξ~k ω
where ≡ dξ and ≡ dω/(2π). Using these integrals, we find
dξ~k ~k ω
R R R R
~v ·~q (~v ·~q)2
F F
g g˜ = + (26a)
Z Z + + iΩ (iΩ)2
ξ~k ω
g g˜ = g g˜ , (26b)
− − + +
Z Z Z Z
ξ~k ω ξ~k ω
2|s| iΩ iΩ
g g˜ = ∓ + ~v ·~q+ (~v ·~q)2. (26c)
Z Z ± ∓ iΩ±2|s| (iΩ±2|s|)2 F (iΩ±2|s|)3 F
ξ~k ω
When s˜ ≈ s in Eq. 22, the spin-charge components of the polarization operator, Π (Q) with a 6= 0, reduce to a
a a a0
combinationΠ ∝ (g g˜ −g g˜ )s which is equalto zero by Eq. 26b. Thus the spin sector is decoupled form
a0 ξ~k ω + + − − a
the charge sector inRtheRlimit of q/k →0. Restricting a,b∈{1,2,3},we obtain
F
Tr[σˆ σˆ σˆ ]s = iλ s
a b j j abj j
Tr[σˆ σˆ σˆ σˆ ]s s = −2δ |s|2+4s s , (27)
a i b j i j ab a b
where λ is te Levi-Civita tensor and it is understood that s =0. We thus obtain a compact form of the spin-spin
abc 3
correlation function:
k s s s s s
F a b a b c
Π (Q)= (g g˜ +g g˜ ) +(g g˜ +g g˜ ) δ − +i(g g˜ −g g˜ )λ , (28)
ab 2πvF ZθZξ~kZω(cid:20) + + − − |s|2 + − − + (cid:18) ab |s|2 (cid:19) + − − + bac|s|(cid:21)
where the integrals over ω and ξ are to be substituted from Eqs. (26a-26b). For further convenience, we also list
~k
explicit formulas for s and related quantities:
a
1
s = ∆ sinθ+∆ cosθ−∆∗ cosθ ,
1 2 R D Z B~
(cid:0) (cid:1)
1
s = −∆ cosθ−∆ sinθ−∆∗ sinθ ,
2 2 R D Z B~
(cid:0) (cid:1)
4s2 = ∆2 sin2θ+∆2 cos2θ+(∆∗)2cos2θ +∆ ∆ sin2θ−2∆ ∆∗ sinθcosθ −2∆ ∆∗ cosθcosθ ,
1 R D Z B~ R D R Z B~ D Z B~
4s2 = ∆2 cos2θ+∆2 sin2θ+(∆∗)2sin2θ +∆ ∆ sin2θ+2∆ ∆∗ cosθsinθ +2∆ ∆∗ sinθsinθ ,
2 R D Z B~ R D R Z B~ D Z B~
4|s|2 = ∆2 +∆2 +(∆∗)2+2∆ ∆ sin2θ−2∆ ∆∗ sin(θ−θ )−2∆ ∆∗ cos(θ+θ ),
R D Z R D R Z B~ D Z B~
4s s = −(∆2 +∆2 )sinθcosθ+(∆∗)2sinθ cosθ −∆ ∆ +∆ ∆∗ cos(θ+θ )+∆ ∆∗ sin(θ−θ ), (29)
1 2 R D Z B~ B~ R D R Z B~ D Z B~
9
where θ is the angle between B~ and the x axis.
B~ 1
We can now apply the general result, Eq. (28) to particular situations. In what follows, we will make v q and Ω
F
dimensionlessbyrescalingto ∆∗ =∆ /(1−u),anddefiner ≡∆ /∆∗ andd≡∆ /∆∗. Note thatthese definitions
Z Z R Z D Z
differ from those in the MT, where ∆ and ∆ are rescaled to ∆ .
R D Z
Silin-Leggett mode
To test our general formula, we apply it first to a simple case of the Silin-Leggett mode, the dispersion of which is
known [25]. The Silin-Leggett mode is the collective mode in the absence of SOC (r =d=0) and in the presence of
B~,whichweassumetobe alongthe x axis. Inthis case,|s|=∆∗ isindependentofangleθ. UsingEq.(28), wefind:
1 Z
Π (Q) (v q)2 sin2θ 6Ω4+3Ω2+1(v q)2
11 = F + B~ 1− F ,
−2N 2Ω2 Ω2+1 (cid:18) Ω2(Ω2+1)2 2 (cid:19)
F
Π (Q) (v q)2 cos2θ 6Ω4+3Ω2+1(v q)2
22 = F + B~ 1− F ,
−2N 2Ω2 Ω2+1 (cid:18) Ω2(Ω2+1)2 2 (cid:19)
F
Π (Q) 1 Ω6−3Ω4 (v q)2
33 F
= 1+ ,
−2N Ω2+1(cid:18) Ω2(Ω2+1)2 2 (cid:19)
F
Π (Q) sinθ cosθ 6Ω4+3Ω2+1(v q)2
12 = − B~ B~ 1− F ,
−2N Ω2+1 (cid:18) Ω2(Ω2+1)2 2 (cid:19)
F
Π (Q) Ωsinθ 3Ω2−1 (v q)2
13 = − B~ 1− F ,
−2N Ω2+1 (cid:18) (Ω2+1)2 2 (cid:19)
F
Π (Q) Ωcosθ 3Ω2+1 (v q)2
23 = B~ 1− F . (30)
−2N Ω2+1 (cid:18) (Ω2+1)2 2 (cid:19)
F
Π can be written more compactly in a matrix form as
ab
Πˆ(Q) κa+(B+κb)sin2θB~ −(B+κb)sinθB~ cosθB~ (C+κc)sinθB~
−2N = −(B+κb)sinθB~ cosθB~ κa+(B+κb)cos2θB~ −(C+κc)cosθB~ , (31)
F −(C+κ )sinθ (C+κ )cosθ (B+κ )
 c B~ c B~ g 
where
1
B =
Ω2+1
Ω
C = − . (32)
Ω2+1
The other definitions are apparent by a term-by-term comparison of expressions in Eq. 30 and the corresponding
entries in Eq. (31). All the z’s are the corrections that are small in v q. At v q = 0, the eigenmode equation
F F
Det(1+uΠˆ/2N ) = 0 has a solution Ω2+1 = s ≡ 2u−u2 such that Ω2 = −(1−u)2. Restoring the dimensional
F 0
frequency and continuing analytically to real frequencies, we find that the frequency at the q =0 is simply given by
Ω =∆ , in agreement with Kohn’s theorem. To find corrections to this result at small but finite v q, we look for a
Z F
solution of the form Ω2+1=s +κ . Expanding the eigenmode equation to linear order in all the κ’s, we obtain
0 0
1−2Bu+(B2+C2)u2 =u[(1−Bu)(κ +κ +κ )−2uCκ ]. (33)
a b g c
Solving for κ , we obtain
0
2(1−u)2(v q)2
F
κ = . (34)
0
u 2
Restoring the units and continuing analytically, we obtain the frequency of the mode at finite q as
(1−u)2(v q)2
F
Ω=∆ − . (35)
Z
2u ∆
Z
Relabeling u→−Fa, we obtain the coefficient a , as given by Eq. (11) of the MT.
0 2
10
Chiral spin mode in the high-field limit
The frequency at q=0
Inthis section,wederivethe frequencyofthe chiralspinwaveatq =0andinthehigh-fieldlimit, when∆ ,∆ ≪
R D
∆ or r,d ≪ 1 for dimensionless quantities. [Unless u is very close 1, i.e., the system is close to a ferromagnetic
Z
transition, there is no real difference between conditions ∆ ,∆ ≪ ∆ and ∆ ,∆ ≪ ∆∗.] With Π (iΩ) ≡
R D Z R D Z ab
Π (iΩ,~q=0), we obtain from Eq. (28)
ab
Π (iΩ) sin2θ r2+d2 r2+d2−2rdsin2θ 1r2+d2 2rdsin2θ
11 = B~ 1−3 +2 B~ + + B~, (36)
−2N Ω2+1 (cid:20) Ω2+1 (Ω2+1)2 (cid:21) 2 Ω2+1 (Ω2+1)2
F
Π (iΩ) cos2θ r2+d2 r2+d2−2rdsin2θ 1r2+d2 2rdsin2θ
22 = B~ 1−3 +2 B~ + + B~,
−2N Ω2+1 (cid:20) Ω2+1 (Ω2+1)2 (cid:21) 2 Ω2+1 (Ω2+1)2
F
Π (iΩ) 1 r2+d2 r2+d2−2rdsin2θ r2+d2 4rdsin2θ
33 = 1−3 +2 B~ + + B~,
−2N Ω2+1(cid:20) Ω2+1 (Ω2+1)2 (cid:21) Ω2+1 (Ω2+1)2
F
Π (iΩ) sinθ cosθ r2+d2 r2+d2−2rdsin2θ rd 2rd
12 = − B~ B~ 1−3 +2 B~ + − ,
−2N Ω2+1 (cid:20) Ω2+1 (Ω2+1)2 (cid:21) Ω2+1 (Ω2+1)2
F
Π (iΩ) Ωsinθ r2+d2 r2+d2−2rdsin2θ 2rdΩcosθ
13 = − B~ 1−2 +2 B~ − B~,
−2N Ω2+1 (cid:20) Ω2+1 (Ω2+1)2 (cid:21) (Ω2+1)2
F
Π (iΩ) Ωcosθ r2+d2 r2+d2−2rdsin2θ 2rdΩsinθ
23 = B~ 1−2 +2 B~ + B~. (37)
−2N Ω2+1 (cid:20) Ω2+1 (Ω2+1)2 (cid:21) (Ω2+1)2
F
In a matrix form,
Πˆ(iΩ) κa+(B+κb)sin2θB~ κf −(B+κb)sinθB~ cosθB~ (C+κc)sinθB~ +κdcosθB~
−2N = κf −(B+κb)sinθB~ cosθB~ κa+(B+κb)cos2θB~ −(C+κc)cosθB~ −κdsinθB~ , (38)
F −(C+κ )sinθ −κ cosθ (C+κ )cosθ +κ sinθ 2κ +(B+κ )
 c B~ d B~ c B~ d B~ a b 
whereB andC arethesameasinEq.(32)andtheκ’s,whicharesmallinrandd,areagaindefinedbyaterm-by-term
comparison of Eq. 37 and the entries in Eq. (38). We seek a solution of the form Ω2+1 = s +κ , where z is also
0 0
small in r and d. To linear order in the z’s, The eignemode equation reads
1−2Bu+(B2+C2)u2 =u (1−Bu)(2κ +3κ −κ sin2θ )−2uC(κ +κ sin2θ ) , (39)
b a f B~ c d B~
(cid:2) (cid:3)
Solving for κ we get:
0
(1−u)(2−3u) (1−u)(2−u)
κ =(r2+d2) −rdsin2θ . (40)
0 (cid:26) 2u (cid:27) B~(cid:26) u (cid:27)
Restoring the units and continuing analyticallyto realfrequencies, we obtain the coefficients a and a˜ in Eq. (11)of
0 0
the MT.
A linear-in-q term in the dispersion
Now, we are interested in all terms to linear order in r, d and v q. Accordingly, we need to express to expand the
F
quantities in Eq. (29) to linear order in these variables:
2s = ∆∗ −cosθ +rsinθ+dcosθ ,
1 Z B~
2s = ∆∗ (cid:0)−sinθ −rcosθ−dsinθ(cid:1),
2 Z B~
4s2 = (∆∗(cid:0))2 cos2θ −2rsinθcosθ (cid:1)−2dcosθcosθ ,
1 Z B~ B~ B~
4s2 = (∆∗)2(cid:0)sin2θ +2rcosθsinθ +2dsinθsinθ (cid:1),
2 Z B~ B~ B~
4|s|2 = (∆∗)2(cid:0)1−2rsin(θ−θ )−2dcos(θ+θ ) , (cid:1)
Z B~ B~
4s s = (∆∗)2(cid:0)sinθ cosθ +rcos(θ+θ )+dsin(cid:1)(θ−θ ) , (41)
1 2 Z B~ B~ B~ B~
(cid:0) (cid:1)