Intrinsic Damping of Collective Spin Modes in a Two-Dimensional Fermi Liquid with Spin-Orbit Coupling

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by  Saurabh Maiti

Intrinsic Damping of Collective Spin Modes
in a Two-Dimensional Fermi Liquid with Spin-Orbit Coupling
Saurabh Maiti1,2 and Dmitrii L. Maslov1
1Department of Physics, University of Florida, Gainesville, FL 32611 and
2National High Magnetic Field Laboratory, Tallahassee, FL 32310
(Dated: February 3, 2015)
A Fermi liquid with spin-orbit coupling (SOC) is expected to support a new kind of collective
modes: oscillationsofmagnetizationintheabsenceofthemagneticfield. Weshowthatthesemodes
are damped by the electron-electron interaction even in the limit of an infinitely long wavelength
5 (q=0). Thelinewidthofthecollective modeison theorderof∆¯2/E ,where∆¯ isacharacteristic
1 F
spin-orbit energy splitting and E is the Fermi energy. Such damping is in a stark contrast to
0 F
knowndampingmechanismsofbothchargeandspin collectivemodesin theabsenceofSOC,all of
2
which disappear at q = 0, and arises because none of the components of total spin is conserved in

n thepresence of SOC.
a
J

0 Electron systems with spin-orbit coupling (SOC) ex- electron systems with SOC.
3 hibit rich physics, some of which may have technological
applications;1,2 equally rich is the physics of cold-atom
]
l systems with synthetic SOC.3,4 Combining many-body
e
interactions with broken SU(2) symmetry, one obtains
-
r aspecial–“chiral”–kindofFermiliquid(FL)5–7 thatsup-
t
s portsanewtypeofcollectivemodes,“chiral-spinwaves”–
t. oscillationsofthe spindensityinzeromagnetic field.8–11
a
m In the absence of SOC, electron-electron interaction
- (eei)12 does not affect certain properties of an electron
d system given that some symmetries are preserved. For
n FIG.1: Left: SchematicsoftheSilin-Leggettmodeinapar-
example, the conductivity and cyclotron-resonance fre-
o tially spin-polarized FL. Right: The chiral-spin modes in a
quency of a Galilean-invariant system are not affected
c FL with Rashba spin-orbit coupling. The shaded regions de-
[ by eei; same is true for the de Haas-van Alphen (dHvA) note the particle-hole continua, ∆ is the Larmor frequency,
1 frequency in an isotropic system13 and for the Larmor ∆˜B is the quasiparticle Zeeman enBergy, and ∆min/max is the
frequency in the presence of an SU(2)-symmetric in- lower/upper boundary of the continuumat q=0.
v
teraction. Being a relativistic effect, SOC breaks both
7
2 Galilean (but not necessarily rotational) invariance and
0 SU(2)symmetryandthuslifts theprotectionensuredby
0 these symmetries. As a result, several physical quanti-
0 tiesbecomedependentoneei. Thelistofsuchquantities
2. includestheopticalconductivity,16 Drudeweight,17,and KK KK KK KK KK++PP
KK++PP
0 frequencies of collective spin modes, which play the role
5 of Larmor frequencies in zero magnetic field.8–11 KK++QQ KK++QQ KK++QQ++PP KK++QQ KK++QQ KK++QQ++PP
1 ((aa)) ((bb)) ((cc))
: In this Letter, we discuss another fundamentally new
v
effectinducedsolelybySOC:intrinsicdampingofcollec-
i
X tivespinmodesintheuniform(q =0)limit. Interaction-
(((ddd))) ((ee))
r induced damping of collective modes is not, by itself,
a a new effect. For example, plasmons in 3D,20 2D,21
FIG.2: Top: The ladder(RPA)series for thespin suscepti-
and 1D (Ref. 22) electron systems, the Silin-Leggett bility. The boxed wavy line is the static effective interaction
(SL) mode18,19 in a partially spin-polarizedFL,23,25 and Ux. Bottom: Diagramscontributingtodampingofthecollec-
magnons in a ferromagnetic FL24,25 are all damped by tivemodes. Thewavylineindiagramsa-edenotesadynamic
interaction processes involving excitations of multiple interaction, Veff(P).
particle-hole pairs. (This mechanism is different from
Landau damping which involves only a single particle- Two-dimensional (2D) electron systems with
holepair: dampingbymultiple pairsoccursevenoutside momentum-dependent SOC, e.g., of Rashba or Dres-
thesingle-particlecontinuum.) However,Galileaninvari- selhaus types, bear certain similarity to a partially
ance, in the case of charge modes,26 and conservation of spin-polarizedFermi gas. The latter has a transverseSL
thetotalspincomponentalongthe field(S ), inthecase mode in the spin sector (see Fig. 1, left),18,19 while the
3
of spin modes,27 ensure that this kind of damping van- former has three (two transverse and one longitudinal)
ishes atq =0. We showherethat this is notthe casefor chiral-spin modes (Ω ...Ω in Fig. 1, right),8–11 which
1 3
2
correspond to oscillations of the three components of spin-charge polarization matrix Π0 are given by
magnetization. Indices 1 3 label the Cartesian system
−
with the 3 axis along the normal to the plane of a 2D Π0(Q) = Tr[σ G(K)σ G(K+Q)], (3)
ij i j
electron gas. (Although SL-mode has been studied ZK
previously in 3D, the same mode should occur in 2D
as well.) Conservation of S3 ensures that the frequency where K ≡ T k0 (2dπ2k)2, i,j ∈ 0,1,2,3, and 0 corre-
of the SL mode at q = 0 coincides with the Larmor sponds to the charge component. At q = 0, all mixed
R P R
frequency in the absence of eei. In the presence of SOC, spin-charge susceptibilities, Π0 with j = 0, vanish by
0j 6
none of the three spin components is conserved. As a charge conservation, while the matrix of spin suscepti-
result, one obtains three distinct modes with frequencies bilities can always be transformed to a diagonal form.
renormalized by eei. In addition, as we show here, In general, there are three spin modes, whose frequen-
these modes have finite linewidth which, in order of cies are found from the equations 1+δ U Π0 = 0, with
j x ii
magnitude, is given by the inverse transport lifetime δ = δ = 1 and δ = 1/2. The Green’s functions in
1 2 3
of a quasiparticle with energy equal to the spin-orbit Π0 contain the self-energy parts. However, for the spe-
ij
splitting. That the SL mode at q = 0 is not affected by cialcaseofU =constant,theydropout(seebelow)and,
x
eei follows already from the exact equations of motion after analytic continuation (iq Ω+iδ), one obtains
0
→
for magnetization.23 Diagrammatically, this occurs due
tgorapahscafnorcetlhlaetisopninbseutswceeepntibtihliety.s2e3lfS-euncehrgaycaanncdellvaetritoenx, Π033(Ω)=2ν (Ω+i∆δ)~2k2F ∆2 , (4)
however, does not occur in the presence of SOC. * − ~kF +FS
The single-particle Hamiltonian of a 2D system with where we have alreadyassumedthat SOC is weak in the
SOC can be written as (we set ~=1) sense specified above, ν is the density of states per spin
projection, ... denotesaveragingovertheFermisurface
k = k2 µ σ0+λ~σ f~(~k), (1) (FS), ~kF =hkF~ik/k, and kF is the Fermi momentum in
H 2m − · the absence of SOC. In general, ∆ varies from ∆
(cid:18) (cid:19) ~k min
to ∆ along the FS. The continuum of inter-subband
max
whereµisthechemicalpotential,σ0 isthe2 2unitma- particle-hole excitations, where ImΠ0 = 0, is confined
trix, ~σ is the three-dimensional vector of Pau×li matrices, 33 6
to the interval ∆ Ω ∆ . At the boundaries
λistheSOCconstant,andf~(~k)= f~( ~k)isa2Dvector of the continuum,mRineΠ≤0 ha≤s squmaarxe root singularities,28
− − 33
thatdependsonthedetailsofSOC;e.g.,f~=(k2, k1,0) which guarantee a solution of the eigenmode equation
−
for linearRashbaSOC.The single-particleGreen’s func- 1+U Π0 /2 = 0 for Ω < ∆ even at weak coupling.
x 33 min
tion is given by If SOC is isotropic, ∆ = ∆ ∆, the two square-
min max
≡
root singularities merge into a single pole at Ω=∆, the
1
G(K)= Ω (~k)g (K), Ω (~k)= σ +sηˆ , (2) inter-subband continuum shrinks to a single point, and
s s s 2 0 ~k the mode frequency is given by Ω = ∆√1 u, where
Xs (cid:2) (cid:3) u U ν.11 3 −
x
≡
where g (K)=(ik k2 s∆ /2+µ)−1, K (ik ,~k), Renormalization of the mode frequency by eei in the
s 0− 2m − ~k ≡ 0 SOCcaseandthelackthereofintheSLcaseisanimpor-
s = 1 labels either spin projection or chirality, ηˆ
± ~k ≡ tant difference, which we discuss now as it will help us
~σ f~/f~,and∆ =2λf~ isthespin-orbitsplittingwhich,
· | | ~k | | to understand the differences in damping later on. This
in general, depends not only on the magnitude but also
difference occurs because the Greens’ functions in the
on the direction of~k. We will be primarily interested in
RPA series in Fig. 1 include chirality- or spin-dependent
the case of weak SOC, when ∆~k ≪ EF for any ~k. In shifts in the chemical potential, which are given by the
what follows, we will be comparing the Ω3 chiral-spin momentum-andfrequency-independentpartsoftheself-
mode to the (2D) SL mode, as both modes are trans- energy. Each rung of the ladder diagram (Π0 ) contains
ij
verse to the SOC-induced/Zeeman magnetic field. The a difference of the self-energies
latter can be described by the same Hamiltonian with
f~= (0,0,∆ /2λ), where ∆ gµ B, g is the effective
B B ≡ B δΣs =Σs Σ−s = Ux (Bs,s′ B−s,s′)gs′(P),
g−factor, µB is the Bohr magneton, and B~ is the mag- − − Xs′ ZP −
neticfieldchosentobealongthe3-direction. Theorbital (5)
effect of the field is not considered here. where Bs,s′ is the matrix element for the transition
Withinthe RandomPhaseApproximation(RPA),the s s′. Within a given rung, δΣ renormalizes the spin-
s
→
spin susceptibility tensor is given by the ladder series in splitting perturbation, be it SOC or the magnetic field.
Fig. 2, where the boxed wavy line is a short-range inter- Since s=s′ inthe SL case,the self-energyofanelectron
action, U , whichmimics the exchangeinteractioninthe with given spin is proportionalto the number density of
x
spinchannel. As showninRef.11, the frequenciesofthe electronswiththe same spin. Hence, δΣ is proportional
s
collective modes correspond to the roots of the equation tomagnetization[δΣ =su∆ /(1 u)],andeachrungof
s B
Det σ σ + UxΠ0 , where the elements of the 4 4 thediagramcontainstherenormal−izedZeemanenergyof
0⊗ 0 2 ×
(cid:0) (cid:1)
3
a quasiparticle, ∆˜B =∆B/(1 u). The boundary of the Π = g2(K+Q)g (K)Σ (K+Q)
continuumatq =0isshiftedf−rom∆B to∆˜B (cf. Fig.1, b ZK − + −
left) as can be seen, e.g., from the 11 component of the (cid:2)
+ ∆ ∆ ,
rung: Π0 (Ω) = ν 2∆˜2B . The Larmor theorem ~kF →− ~kF
11 − (Ω+iδ)2−∆˜2B (cid:16) g−(K+Q)g+(cid:17)i(K)
is effected via a cancellationbetween the self-energy and Πc = [Σ+(K) Σ−(K+Q)]
iq +∆ −
vertexcontributions:23 whenthe rungissubstituted into ZK(cid:18) 0 B
the eigenmode equation, the factor of 1 u cancels out + ∆ ∆ , (6)
− ~kF →− ~kF
and the frequency of the mode coincides with bare ∆ .
B h i(cid:17)
The SOC case is different in that chirality, in contrast
to spin, is not conserved by eei, and the sum over s′ in
Eq. (5) contains both the s = s′ and s′ = s terms. ± denote up/down spins, Q = (iq0,0), Σ±(K) =
− g (K+P)V (P), and V (P) is some dynamic in-
For Ux = const, this implies that the self-energy of an − P ± eff eff
teraction. Inthelastline ofEq.(6),weusedtheidentity
electron with given chirality is proportional to the total
R
number density, and thus ∆Σ = 0.6 The vertex part
s
is, however,non-zero. Therefore,there is nocancellation
betweentheself-energyandvertexcontributions,andthe g±(K +P) g∓(K+P +Q)
g (K+P+Q)g (K+P)= −
frequenciesofthemodesarerenormalizedbyeei. IfU = ± ∓ iq ∆
x 0 B
6 ±
const, one can show29,30 that δΣ does not contain the (7)
s
zerothangularharmonicoftheinteraction(whichiswhy and integrated over P. Because the denominator in
δΣ =0forU =const),whereasthevertexcontribution Eq.(7)doesnotdependonP,thislaststepproducedthe
s x
does. Thus, there is no cancellation between the two same self-energies, Σ±, as in diagrams a and b. Adding
contributions in the general case as well. up the three lines of Eq. (6), we arrive at
Damping of collective modes in the region of fre-
quenciesandmomentaoutsidethesingle-particlecontin-
uumoccursviagenerationofmultipleparticle-holepairs, 2∆
B
Π +Π +Π = (A A ), (8)
whichrequiresadynamicinteraction,e.g.,adynamically a b c q2+∆2 +− −
0 B
screened Coulomb potential. The self-energy of collec-
tive modes is depicted diagrammatically in Fig. 2 a-e.
The same set of diagrams has been encountered in the
where A g2(K)Σ (K). Recalling that V (P) is
analysis of various two-particle correlation functions in ± ≡ K ± ± eff
realontheMatsubaraaxisandchangingthevariablesas
the case of an RPA-type interaction.31–34 Although the k k anRd p p , we find that A = A∗. Thus
Aslamazov-Larkin (AL) diagrams d and e contain two 0 → − 0 0 → − 0 ± ±
thefrequency-independentprefactorinEq.(8),A A ,
wavy lines, they are of the same order in the bare cou- +− −
is real. Continuing iq to the real axis, we see that the
pling constant of the theory (the electron charge in our 0
imaginarypartofδχ comesonlyfromaresonanceatthe
case), as diagrams a-c. However, the contribution of the ⊥
bare Larmor frequency, Ω = ∆ , which coincides with
ALdiagramstodampingvanisheswithintheapproxima- B
thepoleofχ intheRPAapproximation. Theonlyeffect
tions made in this work.36,37 ⊥
of the interaction processes represented by diagrams a-
In the case of the SL mode, renormalization of the
c is thus to renormalize the amplitude of the SL mode
transverse spin susceptibility (χ χ +χ ) by dia-
grams a-c is given by δχ (Q) =⊥ ∼µ211(Π +22Π +Π ), without either shifting its frequency or smearing it.
⊥ − B a b c
where For the case of SOC, it is convenient to consider
renormalization of the out-of-plane spin susceptibility,
Π = g2(K)g (K+Q)Σ (K)+ ∆ ∆ , δχ (Q)= µ2 (Π +Π +Π ), where now
a − + − ~kF →− ~kF 33 − B a b c
ZKh (cid:16) (cid:17)i
1
Π = g2(K)g (K+Q) Σ (K)+Σ (K) +g2(K)g (K+Q) Σ (K)+Σ (K) ,
a 2 − + { +− −+ } + − { ++ −− }
ZK
1(cid:2) (cid:3)
Π = g2(K+Q)g (K) Σ (K+Q)+Σ (K +Q) +g2(K+Q)g (K) Σ (K+Q)+Σ (K+Q) ,
b 2 − + { +− −+ } + − { ++ −− }
ZK
(cid:2) (cid:3)
1
−+ +−− +− +−+ −+ −++ +− −+−
Π = V (P) N M + N M + N M + N M ,
c eff
−ZKZP 2 iq0+∆~k+p~ iq0+∆~k+p~ iq0−∆~k+p~ iq0−∆~k+p~ !
= [g (K+P) g (K+P +Q)] 1+scos φ φ , =g (K)g (K+Q). (9)
Mrts r − t ~k− ~k+p~ Nrt r t
h (cid:16) (cid:17)i
4
Here, r,t,s = label the spin-split bands, φ de- cancel out between the self-energy and vertex diagrams.
± ~k
pends on the azimuthal angle θ of ~k (and is equal
~k
to θ for linear Rashba SOC),38 and the partial
~k However, there are two realistic approximations,
self-energies are defined as Σ (K) = g (K +
rt − P r namely,ofalong-rangeinteraction(p kF)andofweak
≪
P)Veff(P) 1+tcos(φ~k−φ~k+p~) ,suchthattheRtotalself- SOC (∆~k ≪ EF), within which the momentum depen-
dence of ∆ can be neglected. Assuming that these
energies ofhthe Rashba subbandis are Σ+ = Σ+++Σ−− ~k+p~
and Σ = Σ +Σ . In the last line of Eq. (9), we twoconditionsaresatisfied,the vertexpartcanagainbe
− +− −+
only used the identity (7). In contrast to the SL case, rewritten in terms of the partial self-energies. Even in
however,the spin-orbit splittings in the denominators of this limit, however,there is no complete cancellationbe-
Π dependon~k+p~,andthusintegrationoverP doesnot, tween the self-energy and vertex diagrams. Namely, we
c
findthatΠ +Π +Π canberewrittenasΠ +Π ,where
in general, produce the self-energies. This already tells a b c R D
us that, in general, the imaginary part of δχ cannot Π = ∆~kF Σ (K)g2(K) Σ (K)g2(K) and
33 R K q02+∆~2kF + + − − −
R (cid:8) (cid:9)
∆
Π = ~kF [Σ (K+Q) Σ (K)]g (K)g (K+Q) ∆ ∆ . (10)
D − q2+∆2 −− − +− − + − ~kF →− ~kF
ZK 0 ~kF n (cid:16) (cid:17)o
The firstterm, Π , has the same structure as in Eq.(8); in ω2 which, for the case of a contact interaction with
R C
using the same arguments as before, we conclude that coupling U, should be replaced by (UνE )2.
F
∼
Π does not contribute to damping. In contrast, the Since the frequencies of chiral-spin modes are propor-
R
second term, Π , does have, in general, an imaginary tionaltothespin-orbitsplitting,onemightbetemptedto
D
part at all frequencies, which means damping. To cal- concludethatitisbettertolookforthesemodesinmate-
culate Π explicitly, one needs to specify the interac- rialswith strongSOC. Our resultin Eq.(11) shows that
D
tion, which we choose to be in the form of a dynami- the advantage of strong SOC has its limits. Indeed, the
cally screened Coulomb potential. Deferring the compu- ratio of the linewidth to the mode frequency, γ Γ¯/∆¯,
tationaldetailstoSec.IVoftheSupplementaryMaterial, scales as C∆¯/E , where Γ¯ and ∆¯ are the appr≡opriate
F
we quote here only the final result for χ ; near the res- angular averages of Γ and ∆ , correspondingly, and C
33 ~kF
onance at Ω=Ω , a dimensionless prefactor. In a materialwith sufficiently
3
strong eei, one should expect that C 1. If, in addi-
χ−331(Ω)=(2νµ2B)−1A Ω23−(Ω+iΓ/2)2 tion, SOC is also strong (∆¯ EF), the∼n γ 1 and the
∼ ∼
A=*(cid:18)∆∆~2k~kFF−ξ~kΩF23(cid:19)2+(cid:2)FS(cid:28)∆∆~2k~2kFF−ξ~k2ΩF23(cid:29)−FS2;(cid:3) muunnoiddqauemeispfeeoadvteu(rardteaqomf=pSeO0d)C.e;WvienenceiofmntpthrheaasZstei,zetemhteahnSaLtentmehroigsdyeeffbreeeccmtoamisineass
Γ= ωC2 ∆~kF 2, (11) comparable to the Fermi energy.
2EF EF In a particular model of the screened Coulomb poten-
(cid:16) (cid:17) tial, the effect of damping appears to be rather weak.
where ω2 = r2E2 lnr−1/12π, r = √2e2/v is the cou-
C s F s s F ForisotropicSOC,Eq.(11)yieldsC =r2lnr−1/12π. Us-
pling constant of the Coulomb interaction, and ξ2 is a s s
~kF ing parameters for an InGaAs/InAlAs quantum well, we
dimensionless form-factor which depends on the details thenfindγ 2 10−3∆¯/E foranelectronnumberden-
F
of SOC; for isotropic SOC, ξ~k2F =1. sity of 1.6×≈10×12 cm−2. On the other hand, chiral-spin
The damping rate Γ has an expected FL form. Notice waves are also damped by disorder via the Dyakonov-
though that the quasiparticle damping rate in 2D scales Perel’ mechanism.8 For a mobility of 2 105 cm2/V s,
× ·
asΓ Ω2lnΩ,asopposedtojust Ω2, witha prefactor damping due to disorder is stronger than that by eei by
qp
which d∝oes not depend on r .35 Being a gauge-invariant a factor of ten. One should not forget, however, that
s
quantity, Γ contains the differences of the single-particle Eq. (11) is valid only for r 1 and the actual numbers
s
≪
self-energies [see Eq. (26)], while Γ is related to the may differ from quoted above as r increases. Although
qp s
self-energy itself. The infrared singularity in the self- dampingfromdisorderappearstobethedominanteffect
energy,which gives rise to the lnΩ factor in Γ , cancels insolid-statesystems,dampingduetointeractionshould
qp
out in Γ. As a result, Γ is on the order of the transport be dominant in (fermionic) cold-atom systems with syn-
decay rate, which is much smaller than Γ . The FL thetic SOC,3,4 which have virtually no disorder. In this
qp
nature of the result for Γ indicates that it would not case, the interaction is short-ranged but, as we have al-
change substantially if, instead of a FL with Coulomb ready mentioned, this should only affect the prefactor in
interaction, we would consider a FL of neutral particles Eq. (11).
with short-range interaction. The only change would be In conclusion, we showed that eei in the presence of
5
SOCnotonlygivesrisetoanewtypeofcollectivemodes the National High Magnetic Field Laboratory, which is
but also leads to their damping. This damping occurs supported by the National Science Foundation via Co-
evenatq =0anditsratescalesasthesquareofthespin- operative agreement No. DMR-1157490, the State of
orbit splitting. This effect occurs because neither of the Florida, and the U.S. Department of Energy. DLM ac-
three components of magnetization is a good quantum knowledges support from the National Science Founda-
number in the presence of SOC. tion via grant NSF DMR-1308972.
We would like to thank M. Imran and V. Zyuzin for
usefuldiscussions. SMisaDiracPost-DoctoralFellowat
Supplementary Material
I. THRESHOLD SINGULARITIES IN EQUATION (4) OF THE MAIN TEXT (MT) FOR GENERIC
SPIN-ORBIT COUPLING (SOC)
Forageneric(butstillweakcomparedtotheFermienergySOC)thespin-orbitbandsplitting,∆ ,isanisotropic.
~kkF
Suppose that ∆ varies from ∆ to ∆ along the Fermi surface (FS). Quite generally,the angular dependence
~kkF min max
of ∆ near the extremalpoints is ∆ ∆ +β2θ2 and ∆ ∆ β2θ2. Near each of these extremalpoints,
~kkF min ≈ − + max ≈ +− −
the diverging part of the angular integral in Eq. (4) of MT can be written as
∆2 1 dθ ∆+ 1 , near maximum,
~kF 2 2π (Ω−∆++β+2θ2) ∝ √Ω−∆+
*Ω2−∆~2kF +FS ≈  12 R2dπθ (Ω−∆∆−−+β−2θ2) ∝−√∆1−−Ω, near minimum.
R
Notethattheangularaverageisimaginarywhen∆− <Ω<∆+,whichcorrespondtothecontinuumofinter-subband
particle-holeexcitations. Thesingularityintherealpartofthesusceptibilityatthelowerendofthecontinuumensures
that the collective modes exist evenfor infinitesimally weak electron-electroninteraction(eei). For an isotropicSOC,
e.g., either for Rashba or Dresselhaus SOC, ∆ =∆ ∆ and the two square-rootbranch cuts merge into a simple
+ −
≡
pole 1/(Ω ∆).
−
II. STATIC PART OF THE SELF-ENERGY FOR MOMENTUM-DEPENDENT INTERACTION
Inthe MT,wearguedthatthe difference ofthe self-energiesofchiralsubbandsdoes notcontainthe zerothangular
harmonic of the interaction potential. In this section, we prove this statement. The self-energy of a subband with
chirality s= 1 due to eei via a static but momentum-dependent potential, U , is equal to
|q~|
±
1
Σ (K)= g (K′)+g (K′)+s[g (K′) g (K′)]cosφ U . (12)
s −2ZK′ + − + − − ~k~k′ |~k−~k′|
(cid:8) (cid:9)
For the case of linear Rashba SOC, φ is the angle between ~k and k~′ ( θ ); for a general case, φ is some
~k~k′ ≡ ~k~k′ ~k~k′
function of θ such that cosφ dθ = 0. We first consider the case of Rashba SOC in some detail and then
~k~k′ ~k~k′ ~k~k′
point out the qualitative features that remain the same for arbitrary SOC. In addition, we also assume that Rashba
R
SOC is weak (λ v ), in which case the spin-orbit splitting can be approximated by its value projected on the FS
F
≪
in the absence of SOC: ∆ 2λk . The difference of the self-energies of the chiral subbands, δΣ Σ Σ , is now
F s s −s
≈ ≡ −
given by
∆ ∆
δΣ (K) = s n ε + n ε cosθ U , (13)
s − Z~k′(cid:20) F (cid:18) ~k′ 2(cid:19)− F (cid:18) ~k′ − 2(cid:19)(cid:21) ~k~k′ |~k−~k′|
where n (ǫ) is the Fermi function. Setting T =0 and expanding the Fermi functions in ∆, we find
F
dθ
δΣ (K)=sν∆ ~k~k′ cosθ U , (14)
s 2π ~k~k′ |~k−~kF′ |
Z
where~k′ =k ~k′/~k. To find the frequency of the collective mode, one needs to project~k on the FS upon which δΣ
F F s
becomes proportional to the first harmonic of the interaction potential.
A similar argumentcan be made for an arbitrary(but still weak)SOC. The Green’s function in Eq. (2) of MT can
be separated into symmetric (S) and asymmetric (A) parts: G = GS+GA. Any crystalline plane has at least a C
2
6
symmetry upon which ~k ~k. This guarantees that ∆ = ∆ and thus the subband Green’s functions, g , are
→ − −~k ~k ±
even on ~k ~k. Then GS is even while GA is odd on this operation. Consider the self-energy in the spin basis (a
→ −
2 2 matrix)
×
Σ(K)= U G(K′), (15)
−ZK′ |~k−~k′|
which can be also decomposed into symmetric and asymmetric parts: Σ=ΣS+ΣA. The zeroth angularharmonic of
the interaction potential, U{0}, survives only in the part of the integral associated with GS and enters the diagonal
elements of Σ as U{0}n, where n is the totalnumber density. When Σ is transformedto the (diagonal)chiralbasis,
the U{0}n term−s remain on the diagonal and cancel out in the difference Σ Σ .
+ −
− −
III. ASLAMAZOV-LARKIN (AL) DIAGRAMS
In the presence of a dynamic and long range interaction, the AL diagrams are of the same order as the self energy
and vertex diagrams (See Refs. 29-32 of MT). In the absence of SOC and for a spin-invariant interaction, however,
AL diagrams for the spin susceptibility vanish for a trivial reason: Tr[σ GGG]= 0, (i = 1,2,3) because G σ .
i 0
∝
In the presence of the magnetic field (along the 3-axis), G = aσ +bσ . The relevant quantity in this case in the
0 3
transverse spin susceptibility, which contains the trace Tr[σ GGG] with i = 1,2. Again, this trace is equal to zero.
i
The vanishing of the AL diagrams in both these cases is a consequence of conservation of either total spin (in the
former case) or its component along the magnetic field (in the latter case). In the presence of SOC, this reasoning
does not apply because G now has off-diagonal elements. However, some general statements about the AL diagrams
can still be made for q =0 case.
KK PP KK11 KK PP--QQ KK11
ii KK++QQ KK++PP PP--QQ KK11++PP KK11++QQ ii ii KK++QQ KK++QQ--PP PP KK11++PP KK11++QQ ii
((dd)) ((ee))
FIG. 3: Detailed AL diagrams from Fig. 2 of MT
For convenience,we presentthe AL diagramsfromFig. 2, dande ofMT herein Fig.3. Itsuffices to consider only
the diagonal components of the spin susceptibility, χ , with i=1...3, in which case the diagrams read
ii
Πd = Tr[σ G(K)G(K +P)G(K +Q)]V V Tr[σ G(K +Q)G(K +P)G(K )],
ii i P P−Q i 1 1 1
ZP ZK ZK1
Πe = Tr[σ G(K +Q)G(K+Q P)G(K)]V V Tr[σ G(K +Q)G(K +P)G(K )].
ii i − P P−Q i 1 1 1
ZP ZK ZK1
(16)
Here, Q has only the temporal component: Q=(iq ,~0). We can define
0
h (P,Q) Tr[σ G G G ],
i i K+Q K+P K
≡
ZK
h˜ (P,Q) Tr[σ G G G ],
i i K K+P K+Q
≡
ZK
f (P,Q) Tr[σ G G G ]. (17)
i i K+Q K+Q−P K
≡
ZK
Using the matrix structure of G(K), we observe that
h (P,Q)=+h˜ (P,Q); h (P,Q)= h˜ (P,Q)and f (P,Q)= h˜∗(P,Q) for i=1,2,3. (18)
1,2 1,2 3 − 3 i − i
7
With the help of these properties, the AL diagrams can now be written as
Πd = h (P,Q)h˜ (P,Q)V V = h2(P,Q)V V for i=1,2 and
ii i i P P−Q i P P−Q
ZP ZP
= h2(P,Q)V V for i=3, (19)
− i P P−Q
ZP
Πe = f(P,Q)h˜ (P,Q)V V = h (P,Q)2V V , (20)
ii i P P−Q − | i | P Q−P
ZP ZP
and thus their sum is reduced to
Πd +Πe = h2(P,Q) h (P,Q)2 V V , for i=1,2 and
ii ii i −| i | P P−Q
ZP
(cid:0) (cid:1)
= h2(P,Q)+ h (P,Q)2 V V , for i=3. (21)
− i | i | P P−Q
ZP
(cid:0) (cid:1)
Wenowstudytwoparticularcases–thatofRashbaSOCandofcombinedRashba-DresselhausSOC–andshowthat
the AL diagrams do not contribute to damping in either of these two cases.
Rashba SOC: For i=3,
i
h (P,Q)= b(a c) sin(θ θ )g (K+Q)g (K+P)g (K). (22)
3 −4 − ~k− ~k+p~ a b c
a,b,c=± ZK
X
Since Q has no spatial component, we can choose the direction of p~ as a reference. On reflecting vector~k about this
direction, both θ and θ change signs and so does the factor of sin(...). On the other hand, since the energy
~k ~k+p~
spectrumis isotropic,eachofthe three subbandGreen’s functionin the formulaaboveis a function ofthe magnitude
of the corresponding momentum. It is then easy to see that all the three Green’s function are even on reflection
k k , k k . Therefore, the integral over θ vanishes, and the AL diagrams give no contribution to χ .
1 → 1 2 →− 2 ~k 33
The same reasoning applies for i=1. In this case,
1
h (P,Q)= (a+c)sinθ +abcsin(2θ +θ )+bsinθ g (K+Q)g (K+P)g (K) (23)
1 4 ~k ~k ~k+p~ ~k+p~ a b c
a,bX,c=±ZKh i
contains again an angular factor, which is odd on reflection, and a combination of the Green’s functions, which is
even on reflection. Therefore, h (P,Q)=0 and there is no contribution to χ from the AL diagrams either.
1 11
The same argument cannot be used for i=2, because the angular factor in
1
h (P,Q)= (a+c)cosθ +abccos 2θ +θ +bcosθ ) g (K+Q)g (K+P)g (K). (24)
2 −4 ~k ~k ~k+p~ ~k+p~ a b c
a,bX,c=±ZKh (cid:16) (cid:17) i
is even on reflection. Nevertheless, in-plane rotational symmetry of the Rashba Hamiltonian ensures that χ =χ .
11 22
Therefore,one does not need to consider χ separately: if there is no damping of resonance associatedwith χ , the
22 11
same is true for χ .
22
Combined Rashba and Dresselhaus SOC: Inthe presence ofboth Rashba and DresselhausSOC, the only change in
Eqs. ((22-24)) is that θ needs to replaced by φ , as defined after Eq. (12). It is convenient to choose the Cartesian
~k ~k
system rotated by π/4 compared to the conventional one. In such a system, the combined Rashba plus Dresselhaus
Hamiltonian is described by in Eq. (1) of MT with λf~(k)=([β+α]k ,[β α]k ,0) and
2 1
−
(α β)cosθ
cosφ = − ~k,
~k − Λ
~k
(β+α)sinθ
sinφ = ~k, (25)
~k − Λ
~k
where Λ = α2+β2 2αβcos2θ . Note that φ = tan−1 α+β tanθ . The spin-orbit splitting ∆ = 2k Λ
~k − ~k ~k α−β ~k ~kF F ~k
is even on repflection about the k1-axis. As discussed in MT(cid:16)one can ap(cid:17)proximate ∆~k+p~ by ∆~kF for a long-range
interaction; therefore, the products of the Green’s functions are still even on reflection. Because of the relation
between φ and θ , the same symmetries that led to the vanishing of h and h in the case with only Rashba SOC
~k ~k 1 3
ensure that h and h vanish is the case when both Rashba and Dresselhaus SOC are present as well.
1 3
8
IV. DERIVATION OF EQUATION (11) OF THE MAIN TEXT
Inthissection,wepresentthe evaluationofΠ fromEq.(10)ofthe MT.Forconvenience,wepresentthis equation
D
below in an expanded form:
∆
Π = ~kF [Σ (K +Q) Σ (K)]g (K)g (K+Q) [Σ (K+Q) Σ (K)]g (K)g (K+Q) ,
D − q2+∆2 { −− − +− − + − +− − −− + − }
ZK 0 ~kF
(26)
where Σ (K) = g (K +P)V (P) 1+bcos(φ φ ) and the angle φ is defined after Eq. (12). Since Π
ab − P a eff ~k− ~k+p~ ~k D
is already proportioRnal to ∆~kF, we canhneglect the effect ofiSOC on the interaction potential and take it to be a
dynamically screened Coulomb potential in 2D:
2πe2
V (P)= , (27)
eff
p+κ 1 |p0|
(cid:18) − √p20+vF2p2(cid:19)
where κ = √2r k is the inverse Thomas-Fermi screening length and r = √2e2/v . The momentum transfer p~
s F s F
is decomposed into two components: along the normal to the FS at point ~k (p ) and along the tangent (p ). We
|| ⊥
assume and then verify that the quasiparticle interaction is determined by processes with typical p on the order
||
of p /v , whereas typical p are on the order of κ. In turn, typical p are on the order of the external frequency
0 F ⊥ 0
which, in our case, is fixed by spin-orbit splitting, ∆¯, averaged over the FS. We choose to work in a realistic regime
of ∆¯ v κ E . [Although the form of the potential in Eq. (27) is, strictly speaking, valid for r 1 which
F F s
≪ ≪ ≪
implies that κ k , these strong inequalities are never satisfied in real materials. Therefore, the interval of energies
F
in between v κ≪and E is never wide enough to consider a possibility of ∆¯ being within this interval.] Since
F F
p p /v p (28)
|| 0 F ⊥
| |∼| | ≪| |
inthisregime,wecansimplifytheLandau-dampingterminEq.(27)to p /v p andalsoexpandthedenominatorof
0 F ⊥
| | | |
this equationinthis parameter. Subtracting offthe static screenedpotentialwhichgivesno contributiontodamping,
we obtain for the dynamical (p -dependent) part of the interaction
0
2πe2 2 ν p
Udyn(P)= | 0|. (29)
eff p +κ v p
(cid:18)| ⊥| (cid:19) F | ⊥|
Thanks to the inequality (28), the integrals over the fermionic (K) and bosonic (P) momenta become separable. Let
us pick one of the terms in Eq. (26):
Σ (K +Q)g (K)g (K+Q)= Udyng (K+Q+P)g (K)g (K+Q) 1 cos φ φ . (30)
−− − + − eff − − + − ~k− ~k+p~
ZK ZKZP h (cid:16) (cid:17)i
Since it is expected that typical p κ k while ~k is expected to be near the FS, the angular factor can be
⊥ F
expanded as 1 cos(φ φ ) | ξ2|p∼2⊥ , w≪here ξ is a form-factor that depends on the details of the SOC.
− ~k− ~k+p~ ≈ ~k2kF2 ~k
For example, in the case of linear Rashba SOC, φ coincides with the angle θ that~k makes with, e.g., the 1-axis.
~k ~k
Also, due to rotationalsymmetry,the result of the integraloverP does not depend on the directionof~k and thus we
can choose θ =0 in 1 cos(θ θ ). An expansionof 1 cos(θ ) then yields p2⊥ , which means that ξ =1 for
~k − ~k− ~k+p~ − ~k+p~ 2kF2 ~k
a system with linear Rashba SOC. Notice that the condition p p implies that p~ is almost perpendicular to~k,
|| ⊥
| |≪| |
i.e., that θ θ π/2.
| p~− ~k|≈
In the presence of both Rashba and Dresselhaus SOC, φ is given by Eq. (25). Expanding in p and using that
~k
θ θ π/2, we get
| p~− ~k|≈
p2 γ 2 2αβ
1 cos(φ φ ) ⊥ 1+ , where γ = <1 (31)
− ~k− ~k+p~ ≈ 2kF2 " (cid:18)1−γcos2θ~k(cid:19) # α2+β2
2
Thus ξ2 =1+ γ for this system. We avoidthe special case of α = β , when γ =1 and ξ has a pole: it
~k 1−γcos2θ~k | | | | ~k
?
can be shown t(cid:16)hat collecti(cid:17)ve modes are completely covered by the continuum in this case.
9
Next, we write g (K)g (K +Q) as [g (K) g (K +Q)]/(iq ∆ ) and integrate the products of two Green’s
− + − − + 0− ~k
functions over ε , k and p ; for one of the products, we obtain
~k 0 ||
dp dk dp ip p
k 0 k 0 0
dε g (K+P)g (K)= = | |. (32)
Z 2π Z 2π Z ~k a b Z 2π ip0−vFpk− a−2b∆~k 2vF
The rest of the integrals are evaluated in the similar manner. Collecting all terms, we arrive at
2
Π = ∆~kFξ~kF 2ν kF dp⊥p2 ∞ dp0Udyn(P) q +p + q p 2p . (33)
D −* q02+∆~2kF ! +~k vFkF2 Z0 2π ⊥Z0 2π eff {| 0 0| | 0− 0|− | 0|}
The factor in ... constraints the range of integration over p to (0,q ). This leads to
0 0
{ }
2
Π = ν ∆~kξ~kF rs2 q0dp (q p )p kF dp p⊥ (34)
D − * q02+∆~2k! +~k 2πEF Z0 0 0− 0 0Z0 ⊥(p⊥+κ)2
To logarithmic accuracy, the integral over p gives lnk /κ lnr while the integral over p gives q3/6. Note that
⊥ F ≈− s 0 0
typical p q while typical p κ (in the logarithmic sense), and thus our initial assumption is verified.
0 0 ⊥
∼ ∼
After analytic continuation, we get
2
Π = iν ∆~kFξkF ω2 Ω 3, (35)
D − * Ω2−∆~2kF ! +~kF C(cid:18)EF(cid:19)
where ω2 = r2E2 lnr−1/12π and r = √2e2/v . This is the correction to Π0 that is responsible for damping.
C s F s s F 33
Recalling that the physical susceptibility is given by χ = µ2Π /(1+ UxΠ ) Π differs from Π0 due to the
33 − B 33 2 33 33 33
damping correction. To account for this correction, we may write
∆2 ξ2
Π =2ν ~kF ~kF , (36)
33 *∆2 (Ω+iΓ/2)2+
~kF − FS
where Γ= ωC2 ∆~kF 2. Using the fact that Ω is the pole in χ when Γ=0, we can eliminate the U dependence.
2EF EF 3 33 x
Writing ∆2 ((cid:16)Ω+i(cid:17)Γ/2)2 as ∆2 Ω2+Ω2 (Ω+iΓ/2)2 and expanding in Ω2 (Ω+iΓ/2)2 near the resonance,
~kF − ~kF − 3 3− 3−
we obtain Eq. (11) of MT.
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