Weak and strong coupling limits of the two-dimensional Fröhlich polaron with spin-orbit Rashba interaction

0.44 MB·English

by  C. Grimaldi

Weak and strong coupling limits of the two-dimensional Fr¨ohlich polaron with
spin-orbit Rashba interaction
C. Grimaldi
Max-Planck-Institut fu¨r Physik komplexer Systeme, N¨othnitzer Srt.38, D-01187 Dresden Germany
LPM, Ecole Polytechnique F´ed´erale de Lausanne, Station 17, CH-1015 Lausanne, Switzerland
The continuous progress in fabricating low-dimensional systems with large spin-orbit couplings
has reached a point in which nowadays materials may display spin-orbit splitting energies ranging
from a few to hundreds of meV. This situation calls for a better understanding of the interplay
between the spin-orbit coupling and other interactions ubiquitously present in solids, in particular
8 when the spin-orbit splitting is comparable in magnitude with characteristic energy scales such as
0
theFermi energy and thephonon frequency.
0
Inthisarticle,thetwo-dimensionalFr¨ohlichelectron-phononproblemisreformulatedbyintroduc-
2
ingthecouplingtoaspin-orbitRashbapotential,allowing foradescriptionofthespin-orbiteffects

n on the electron-phonon interaction. The ground state of the resulting Fr¨ohlich-Rashba polaron is
a studied in theweak and strong coupling limits of theelectron-phonon interaction for arbitrary val-
J ues of the spin-orbit splitting. The weak coupling case is studied within the Rayleigh-Schr¨odinger

5 perturbation theory, while the strong-coupling electron-phonon regime is investigated by means of
2 variationalpolaronwavefunctionsintheadiabaticlimit. Itisfoundthat,forbothweakandstrong
coupling polarons, the ground state energy is systematically lowered by the spin-orbit interaction,

] indicating that the polaronic character is strengthened by the Rashba coupling. It is also shown
l
e that, consistently with the lowering of the ground state, the polaron effective mass is enhanced
- compared to the zero spin-orbit limit. Finally, it is argued that the crossover between weakly and
r
t strongly coupled polarons can be shifted bythe spin-orbit interaction.
s
.
t PACSnumbers: 71.38.-k,71.38.Fp,71.70.Ej
a
m
- I. INTRODUCTION Concerning the el-ph problem in low dimensions,
d
recent progresses in developing high-quality low-
n
dimensionalsystems andinmaterialengineeringprovide
o The Fr¨ohlich Hamiltonian describing a single electron
c hints that, for a vast class of low-dimensional materi-
coupled to longitudinal optical phonons is a paradig-
[ als, the usual 2D Fr¨ohlich model, as considered in lit-
matic model of the electron-phonon (el-ph) interaction,1
erature, may be incomplete. This concern comes about
2 and has represented in the past, in addition to its in-
by considering 2D systems exhibiting strong spin-orbit
v terest for the solid-state physics, an ideal problem for
9 (SO) splitting of the electronic states due to the inver-
testing many mathematical methods in quantum field
2 sion asymmetry in the direction orthogonal to the con-
theory.2 Because of the coupling with the phonon field,
4 ductingplane(RashbaSOmechanism). Thissituationis
the resulting quasi-particle, the polaron, has an effec-
3 encounteredinsemiconductorquantumwellswithasym-
9. tive mass larger, and a ground state energy lower, than metricconfiningpotentials,18inthesurfacestatesofmet-
the free electron. These quantities have been investi-
0 als and semi-metals,19,20,21 and in surface alloys such as
gated for the three-dimensional (3D) case by means of
7 Li/W(110),22 Pb/Ag(111),23,24 and Bi/Ag(111),25 with
0 perturbation theory for the weak-coupling limit,3 and of SO splitting energies ranging from a few meV in GaAs
v: variationaltreatmentsfor the intermediate,4 andstrong- quantumwellstoabout0.2eVinBi/Ag(111).25 Inthese
coupling cases.5,6 The path-integral variational calcula-
i systems, therefore, the SO energy may be of the same
X tions of Feynman,7 and subsequent refinements of this
order or even much larger than the typical phonon fre-
r method,8 have provided a solid description for all val-
quency, rising the question of how such state of affair
a ues ofthe coupling,verifiedalso by improvedvariational
affects the el-ph interaction,in general,and the Fr¨ohlich
methods,9 and by quantum Monte-Carlo studies.10,11
coupling, in particular.
The interest aroused some time ago on semiconduc-
tor heterojunctions, or other low-dimensional systems, As pointed out in several works,26,27,28,29,30,31 the
promptedtomodifytheFr¨ohlichmodeltoaccountingfor Rashba interaction describing the SO coupling can have
two-dimensional(2D)andquasi-2Dsystems.12Byapply- profoundeffects onthe low energyproperties of the itin-
ingthesamemethodsderivedforthe3Dcase,theground erant electrons. Namely, in the low density regime, the
statepropertiesforthestrictly2Dcasewereevaluatedfor Rashba SO coupling induces a topological change of the
weak, strong and intermediate couplings,13,14,15,16 and Fermisurfaceofthefreeelectrons,leadingtoaneffective
the obtained systematic loweringof the groundstate en- reduction of the dimensionality in the electronic density
ergy and the enhancing of the effective mass compared of states (DOS). In this situation, a 2D low density elec-
tothe 3Dcasehaspointedoutthe roleofdimensionality tron gas would develop, in the presence of SO Rashba
in enhancing the polaronic character.12,17 coupling, a phenomenology similar to one-dimensional
2
(1D) systems, triggeredby the square-rootdivergence of with
the (effectively 1D) DOS at low energies.
Some interesting consequences of this scenario on
2π2α2 1/4
the el-ph problem have already been discussed in M =iω , (5)
0 0
Ref.[30], concerning the superconducting transition, and (cid:18) mω0 (cid:19)
inRef.[31]fortheeffectivemassandthespectralproper-
ties. The picture arising from these works, although be-
whereα=e2(ǫ−1 ǫ−1) m/2ω isthedimensionlessel-
inglimitedtothemomentum-independentHolsteinel-ph ∞ − 0 0
ph coupling constant, with e being the electron charge,
interactionand to weak-to-moderatecouplings,confirms
p
and ǫ and ǫ the high frequency and static dielectric
that, for sufficiently low electron densities, the coupling ∞ 0
constants, respectively.
tothephononsisamplifiedbytheSOinteractionthrough
the 1D-like divergence of the DOS. It is worth clarifying here the significance of the 2D
Notwithstanding the relevance of these results for the Fr¨ohlich interaction of Eq.(4) with respect to the char-
Holstein model, the use of a local el-ph interaction may acteristics of specific materials. For quantum wells and
however result inadequate in the extremely low elec- 2D heterostructures,where the electron wavefunction is
tron density regime, where the SO effects are more assumed here to be confined in a sheet of zero thickness,
evident.30,31 Indeed,thelackofeffectivescreeninginthis Eq.(4) describes the coupling of the electron to bulk LO
casewouldrathersuggestalong-rangeinteractionasbe- phonons,whilethecouplingtointerfacephononmodesis
ing a moreappropriatedescriptionofthe el-phcoupling. neglected. Theinclusionofsuchinterfacephononcontri-
It is therefore natural to consider the 2D Fr¨ohlich po- butions may be important in describing specific materi-
laron,and its coupling to the SO interaction, as a model als,butitisunnecessaryforthepresentstudy,wherethe
better describing the unscreened el-ph interaction in 2D focusis ontheSO effectsonthe unscreened(long-range)
Rashba systems in the low density limit. el-ph interaction, for which Eq.(4) is a paradigm for the
Inthisarticle,asingleelectronmovingwithaparabolic 2Dcase. Concerningthe el-phcouplingofelectronicsur-
dispersion in the two-dimensional x-y plane is coupled face states, Eq.(4) coincides (apart from a redefinition
simultaneously to the Rashba SO potential and to the of M0) with the coupling to 2D surface phonons when
phonon degrees of freedom through a Fr¨ohlich interac- the coupling to bulk phonons extending below the sur-
tion term. The total system is then described by the 2D face is negligible.32 Such approximationis coherent with
Fr¨ohlich-Rashba Hamiltonian H = H +H +H , the ideal 2D assumption for the electron wave function,
el ph el−ph
where (~=1) which is physically realized when the electronic surface
stateshavenegligiblecouplingtothebulk. Afurthermo-
p2 tivationofusingthe2DFr¨ohlichmodel(4)isthat,inthe
H = +Ω(p) σ (1)
el absence of SO interaction, the ground state polaron en-
2m ·
ergyE andeffectivemassm∗ havealreadybeenstudied
P
is the Hamiltonian for an electronwith mass m and mo- by several authors,12,13,14,15,16 and the exact results ob-
mentum operator p= i∇ with components (p ,p ,0), tainedfortheweak(α 1)andstrong(α 1)coupling
x y
σ is the spin-vector op−erator with components given by limits provide a useful≪reference for the effe≫ct of nonzero
thePaulimatrices,andΩ(p)istheSOvectorfieldwhich SO coupling.
in the case of Rashba coupling reduces to:
In the present work, the 2D Fr¨ohlich-Rashba Hamil-
tonian is studied by considering the weak and strong
p
− y coupling limits of the el-ph interaction, with arbitrary
Ω(p)=γ p , (2)
 x  strength of the SO coupling γ. For α 1 the polaron
0 energy E and the effective mass m∗ a≪re obtained from
P
  second order perturbation theory in Sec.II, where nu-
where γ is the SO coupling parameter. The phononpart
merical and exact analytical results are presented. It is
of the Hamiltonian is given by
shown that the effect of γ = 0 is qualitatively similar to
thatobservedintheHolste6inmodel,30,31 namely,theSO
Hph =ω0 a†qaq, (3) coupling enhances the effective coupling to the phonons.
q Inparticular,E isloweredbyγand,simultaneously,the
X P
effective mass m∗ is enhanced. InSec.III the strongcou-
where a†q (aq) is the creation (annihilation) operator for pling limit α 1 is treated by the variational method,
a phonon with momentum q = (qx,qy) and optical fre- providing a ri≫gorous upper bound of the ground state
quency ω0. The el-ph interaction Hamiltonian for the energy for arbitrary values of the SO interaction. As for
2Delectroncoupledtolongitudinaloptical(LO)phonons the weakel-phcouplingcase,itis foundthatE (m∗)is
P
is:12,14 lowered(enhanced)bythe SO interaction,implying that
theRashbacouplingalwaysamplifiesthepolaronicchar-
1 1
H = (M eiq·ra +M∗e−iq·ra†) (4) acter,regardlessofwhether the el-ph interactionis weak
el−ph √A √q 0 q 0 q or strong.
q
X
3
II. WEAK COUPLING because the second term of Eq.(11) vanishes after the
integrationoverk′. In this way,the resulting self-energy
In the presence of SO interaction, the electron wave reduces to:
function is a spinor and its Green’s function is conve-
niently represented by a 2 2 matrix in the spin sub- Σ(k,ω)=Σ (k,ω)1+Σ (k,ω)Ωˆ(k) σ, (12)
d o
× ·
space. For α = 0 the free electron Green’s function G
0
is readily obtained from Hel: where 1 is the unit matrix and Σd and Σo are, respec-
tively, the diagonaland off-diagonalcontributions to the
G (k,ω) = ω k2 Ω(k) σ −1 self-energy,bothdependingsolelyonthemodulusofk.33
0
− 2m − · Their explicit expressions are:
(cid:18) (cid:19)
1
= [1+sΩˆ(k) σ]Gs(k,ω), (6)
2 · 0 M0 2 dk′ 1 1
s=± Σ (k,ω)=| | ,
X d 2 (2π)2 k k′ ω ω E (k′)
Z s | − | − 0− s
where k is a 2D electron wavenumber, Ωˆ(k) = X (13)
Ω(k)/Ω(k) and
| | M0 2 dk′ 1 sk k′
Σ (k,ω)=| | · .
1 o 2 (2π)2 k k′ ω ω E (k′)
Gs(k,ω)= (7) Z s | − | − 0− s
0 ω k2/2m sγk X (14)
− −
isthefreeelectronpropagatorforthetwo(s= 1)chiral
± In the limit of zero SO coupling, since E (k) k2/2m,
states characterized by two distinct bands with shifted s
→
parabolic dispersions k2/2m γk. The lowest band has Σo(k,ω) vanishes because of the summation over s =
± 1 in Eq.(14). Notice also that, independently of γ,
its minimum value E at k =k , where k and E are
− 0 0 0 0 ±Σ (k,ω)=0 when the factor 1/k k′ in Eq.(14) is re-
the Rashba momentum and energy defined respectively o
| − |
placed by a constant, as in the momentum-independent
by:
Holstein el-ph coupling model.
m
k =mγ, E = γ2. (8) By using Eq.(12) the Dyson equation for the interact-
0 0
2 ing propagator G reduces to
For later convenience, it is useful to express the electron
energy relative to E0, so that the poles of Eq.(7) appear G−1(k,ω) = G−01(k,ω)−Σ(k,ω)
at energies: k2
= ω Σ (k,ω) E
d 0
1 − 2m − −
E±(k)= 2m(k±k0)2. (9) [γk+Σo(k,ω)]Ωˆ(k) σ, (15)
− ·
The free electron ground state is then given by the elec-
and the poles ω of G are then given by:
tron occupying the lower band at wavenumber k = k ±
0
with energy ω =0.
In the weak el-ph coupling limit (α 1) the ground ω± =E±(k)+Σd(k,ω±) Σo(k,ω±). (16)
≪ ±
state properties are obtained by the electron self-energy
evaluated in the second order perturbation theory. At Now, the Rayleigh-Schr¨odingerperturbation theory per-
zerotemperature,theresultingsingleelectronself-energy mits to evaluate the lower energy pole ω at the lowest
−
is therefore: order in the el-ph coupling α. This is accomplished by
replacing ω by the unperturbed energy E (k) in the
dk′ 1 − −
Σ(k,ω)= M 2 G (k′,ω ω ). (10) energy variables of Σ and Σ . In this way, the lower
| 0| Z (2π)2|k−k′| 0 − 0 pole reduces to ω− =dE−(k)+oΣ−(k)+ (α2), where
O
Because of the momentum dependence of the Fr¨ohlich
interaction, and contrary to the Holstein el-ph case con- Σ−(k)=Σd(k,E−(k)) Σo(k,E−(k)). (17)
−
sidered in Ref.[31], the self-energy is not diagonal in the
spin subspace. However, since the momentum depen- Finally, by expanding Σ (k) up to the second order in
−
dence enter only throughthe modulus ofthe momentum k k ,thepolarondispersioninthevicinityofk canbe
0 0
transfer,equation(10)canberewritteninaquitesimple w−ritten as:
form. By using (Ωˆ(k) σ)2 =1 and
·
1
(Ωˆ(k) σ)(Ωˆ(k′) σ)=kˆ kˆ′+(kˆ kˆ′)σxσy, (11) ω− =EP + 2m∗(k−k0∗)2, (18)
· · · ×
thenthequantityΩˆ(k′) σ appearinginEq.(10)through where the polaron ground-state energy E , the effective
· P
G (k′,ω ω )canbe replacedsimply by(Ωˆ(k) σ)kˆ kˆ′ mass m∗, and the effective Rashba momentum k∗ are
0 − 0 · · 0
4
given respectively by: the effective mass can be expressed as
π
E = αω f (ε ), (22)
m∗ P −2 0 EP 0
E =Σ (k ) Σ′ (k )2 m∗ π
P − 0 − 2 − 0 =1+ αfm∗(ε0), (23)
m 8
=Σ (k )+ (α2), (19)
− 0
O
m∗ where the factors fEP(ε0) and fm∗(ε0) contain all the
m =[1+mΣ′−′(k0)]−1 effects of the SO interaction and depend solely on the
dimensionless SO parameter
=1 mΣ′′(k )+ (α2), (20)
− − 0 O
k∗ m∗ E0 mγ2
k00 =1− k0 Σ′−(k0) ε0 ≡ ω0 = 2ω0 . (24)
m
=1 Σ′ (k )+ (α2). (21) IntheweakSOlimit,theself-energyterms(13)and(14)
− k − 0 O
0 canbeexpandedinpowersoftheSOinteraction,allowing
for an analytical evaluation of the integrals. In this way,
up to the linear order in ε , f (ε ) and f (ε ) are
0 EP 0 m∗ 0
Let us firstconsider E andm∗. Inthe zeroSO limit, found to be:
P
πEαqsω.0/(129a)nadndm(∗2/0m) a=t k01=+0παle/a8d, rwehspicehcticvoerlryestpooEndP t=o fEP(ε0)=1+ ε40 +O(ε20), (25)
the results already reported in Refs.[13,14,15]. For finite 9
f (ε )=1+ ε + (ε2), (26)
values of the SO coupling the ground state energy and m∗ 0 8 0 O 0
indicating that the polaronic character is strengthened
by the SO interactionsince, throughEqs. (22) and(23),
the polaron energy E is lowered and, simultaneously,
P
1.15 theeffectivemassm∗ isenhancedwhenε0 >0. Thisfea-
ture is not limited to the small ε limit, but holds true
(a) 0
forarbitrarystrengthsoftheSOcoupling. Thisisshown
in Fig. 1 where f (ε ) and f (ε ), obtained from a
EP 0 m∗ 0
1.10 numerical integration of Eqs.(13) and (14), are plotted
)0 1.6 asafunctionofε0 bysolidlinesandcomparedwithEqs.
(EP (25) and (26) (dashed lines). The same quantities calcu-
f
1.05 )f(0EP11..24 Flaitge.d1 faonrdacwonidfierrmrathnagtetohfeε0graoruenpdlostttaetde einnetrhgeyiEnsPetasnodf
the effective mass m∗ are continuous functions of ε and
0
are, respectively, further lowered and enhanced by the
1.0
0 2 4 6 8 10 SO coupling. In the strong SO limit ε 1, it is found
0 0 ≫
1.00 thatfEP(ε0)growsasln(ε0)whilefm∗(ε0)growslinearly.
200 It is interesting to note that the Holstein-Rashba model
5 (b)
studied in Ref.[31] predicts results qualitatively similar
)0150
(m* totheFr¨ohlichmodel,indicatingthattheSOinteraction
4 f100 strengthen the polaronic character independently of the
specific form of the el-ph interaction.34
f()m*0 3 50 el-IpnhacdoduiptiloinngtaonEdPthaenSdOmi∗n,tethraecitniotnerpmlaoydibfieetswaelesno tthhee
0 2 4 6 8 10
0 Rashba momentum k0 through Eq.(21). In the weak SO
limit, the effective quantity k∗ is found to be
2 0
k∗ π
0 1 αε , (27)
0
1 k0 ≃ − 32
0.0 0.2 0.4 0.6 0.8 1.0
0 indicatingareductionofthebareRashbamomentumk ,
0
confirmed also by the numerical calculation of Eq.(21)
FIG. 1: (a): ground state energy factor fEP(ε) as a function
of the SO parameter ε0 = E0/ω0. The solid line is the nu- reported in Fig. 2 by the solid line. As shown in the
mericalcalculation,whilethedashedlineistheweakSOlimit inset, for fixed el-ph coupling α, k∗ however does not
0
Eq.(25). Inset: fEP(ε) plotted for a wider range of ε0. (b): deviatemuchfromitsbarelimitk0,evenforlargevalues
the effective mass factor fm∗(ε0) from numerical integration of the SO parameter ε0.
(solid line) and from Eq.(26) (dashed line). Inset: fm∗(ε0) Let us compare now the present results with those
plotted for a wider range of ε0. appeared recently in literature. In Ref.[35] the ground
5
laronic character survives also for large α values, or it is

0.00 instead limited to the weak coupling limit. In the next
0.00
section, this problem is studied for the limiting case of
strong el-ph interaction α 1, providing therefore, to-
1)/ -0.01 gether with the weak coup≫ling results, a global under-
1)/-0.01 k*/k-00-0.02 standing of the SO effects on the Fr¨ohlich polaron.
*/k-0 (
(k0 -0.030 2 4 6 8 10 III. STRONG COUPLING
-0.02 0
It is well known that a perturbative scheme such that
employed in the previous section fails to describe the
-0.03 Fr¨ohlich polaron ground state when the el-ph coupling
0.0 0.2 0.4 0.6 0.8 1.0
is very large. This is due to the fact that for α 1
0
≫
the lattice polarization, and resulting “self-trapping” ef-
FIG. 2: Effective Rashba momentum k0∗ as a function of the fect experienced by the the electron,37 renders the plane
SO parameter ε0 = E0/ω0. The numerical integration of
waverepresentationoftheunperturbedelectroninappro-
Eq.(21)(solidline)iscomparedwiththeweakSOresult(27)
priate for obtaining the polaron ground state. Instead,
(dashed line). Inset: the same quantity plotted for a wider
range of ε0. asoriginallyproposedinRef.[5]andrigorouslyprovedin
Refs.[38,39], the asymptotic description of the polaron
wave function in the strong coupling limit α 1 is
≫
that of a product between purely electronic, ψ(r), and
state energyofa polaronneara polar-polarsemiconduc-
purely phononic, ξ , wave functions. Within such adi-
tor interface with Rashba SO coupling has been evalu- | i
abatic limit, the ground state energy and the effective
ated with the Lee-Low-Pines method.4 As a function of
mass of a 2D Fr¨ohlich polaron have been calculated in
the SO splitting, the polaron ground state is found to
Refs.[14,15] by using the variational method with differ-
be lowered, in qualitative agreement therefore with the
ent ansatz wave functions. From Ref.[14], one realizes
present results. A more quantitative comparisonis how-
thatexponential,gaussianandPekar-typewavefunctions
ever precluded by the different model of Ref.[35], where
provide increasinglybetter estimates of E with accura-
P
contributions from interface phonon modes and confin-
ciesrespectivelyof14%,0.3%,and0.03%withrespectto
ing potentials are considered as well. In another work,36 theexactgroundstateenergyE /ω = 0.40474α2,ob-
P 0
the Rayleigh-Schr¨odinger perturbation theory has been −
tained by a numerical solution of the integro-differential
applied to the polaron ground state of the 2D Fr¨ohlich- equation for the electron wave function.16 In the follow-
Rashba model, permitting therefore a direct comparison
ing, the variational method is used to evaluate the SO
with the analysis presented here. Despite that the au-
effects on the polaron ground state.
thors of Ref.[36] find that the polaron ground state is
lowered by ε , their values of E differ from those plot-
0 P
ted in Fig. 1(a). In Ref.[36], in fact, the ground state
A. trial wave functions
energy factor f is found to be f (ε ) = 1/√1 ε ,
EP EP 0 − 0
whichimpliesasmallε expansiondifferentfromEq.(25)
0
and, more importantly, a divergence of E at ε =1. In For the nonzero SO case, due to the presence of the
P 0
Fig. 1(a), instead, nothing of special happens at ε =1. Pauli matrices in Eq.(1), suitable ansatz wave functions
0
This discrepancy is easily traced back in the fact that in must take into account the electron spin degrees of free-
Ref.[36]theexpansionofΣ (k),Eq.(17),ismadearound dom. Hence, in full generality, the strong-coupling po-
−
k =0,insteadofk =k asdonehere,whichdoesnotcor- laron wave function may be represented as: Ψ,ξ =
0 | i
respondtoaperturbativecalculationoftheground-state Ψ(r)ξ , where Ψ(r) is a two-components spinor for the
| i
energy. electron. The correspondingexpectation value of the to-
tal Hamiltonian H is:
Theresultspresentedinthissectionhavebeenderived
by assuming a weak coupling to the phonons. However,
Ψ,ξ H Ψ,ξ = ΨH Ψ + ξ H ξ
asitisclearfromtheplotsinFig. 1,theenhancementof h | | i h | el| i h | ph| i
the polaroniccharacterdrivenbyε for fixed α unavoid- 1 1
0 + (M ρ(q) ξ a ξ +h.c.),
0 q
ably renders the perturbative approach invalid for suffi- √A √q h | | i
q
ciently large ε values. For example, from Eq.(23), the X
0 (28)
validity of the weak coupling results for m∗/m are sub-
jected to the condition αfm∗(ε0) 1, otherwise higher where
≪
orderel-phcontributionsshouldbe consideredfora con-
sistent description of the SO effects. The question re-
ρ(q)= Ψeiq·r Ψ = dreiq·r Ψ(r)2. (29)
mains therefore whether the SO enhancement of the po- h | | i | |
Z
6
TheformofEq.(28)permits tointegrateoutthe phonon From the above expression of V(r), the functional (30)
wave function in the usual way. Hence, by introducing can be rewritten as E[Ψ] = ΨH Ψ + V¯/2, where
el
the phonon coherent state ξ = ePqξqa†q 0 , where V¯ = ΨV(r)Ψ . Now, if ΨGhis t|he e|xaict ground state
is a normalization factor a|nid ξqNa variatio|nial paramNe- wavehfun|ction|, wiith ground state energy EG = E[ΨG],
ftuern,cmtioinniamlization of (28) with respect to ξq leads to the tthhaent,ǫf=romEG(3+2)V¯a/n2d, sEoGth=athΨEq|H.(e3l2|Ψ) rie+duV¯ce/s2,toit: is found
M 2 dq 1
E[Ψ]= ΨH Ψ | 0| ρ(q)2, (30)
h | el| i− ω (2π)2q| |
0
Z
where the continuum limit A−1 dq/(2π)2 has
q → H Ψ (r)+[V(r) V¯/2]Ψ (r)=E Ψ (r). (34)
been performed. By choosing an appropriate functional el G − G G G
form for Ψ(r), and by minimiziPng E[ΨR] with respect
to the variational parameters defining Ψ(r), an upper
bound for the ground state energy is then E[Ψ ], where
0
Ψ (r) is such that E[Ψ ] = min(E[Ψ]). As done in
0 0
theprevioussection,thepolaronenergyisthenobtained As noted in Ref.[29] (see also Refs.[40,41]), the ground-
from state wave function of a 2D electron subjected to a SO
Rashba interaction and to a 2D central potential (i.e. a
EP =E[Ψ0]+E0, (31) potential depending only upon r = r) is of the form
| |
whereE isthefree-electronSOenergydefinedinEq.(8).
0
Of course, the functional form of Ψ(r) is decisive for
obtaining accurate estimates of the ground state energy,
and a suitable choice must be guided by looking at the
properties of the true ground state spinor ΨG(r). These Ψ (r)= ψ1(r) , (35)
can be deduced by a formal minimization of the func- G ψ2(r)eiϕ
(cid:18) (cid:19)
tional E[Ψ] with respect to Ψ. By introducing the La-
grange multiplier ǫ to ensure that the wave function is
normalized to unity, minimization of (30) leads to:
H Ψ(r)+V(r)Ψ(r)=ǫΨ(r), (32)
el where ϕ is the azimuthal angle of r. Now, if Eq.(35)
where,byusingthedefinitionofρ(q)givenininEq.(29): is used in Eq.(33), the resulting self-consistent potential
depends only upon r, V(r) V(r), so that Eq.(35) is
→
2M 2 dq ρ(q)∗ consistentlyalsothecorrectformforthepolaronground-
V(r)=− |ω00| (2π)2 q eiq·r state wave function. Hence, passing to polar coordi-
Z nates, Eq.(34) can be rewritten as a system of integro-
M 2 Ψ(r′)2
= | 0| dr′| | . (33) differential equations for the spinor components ψ1 and
− πω0 Z |r−r′| ψ2:
1 d2 1 d d 1
+ +U(r) ψ (r) γ + ψ (r)=E ψ (r), (36)
−2m dr2 rdr 1 − dr r 2 G 1
(cid:20) (cid:18) (cid:19) (cid:21) (cid:18) (cid:19)
1 d2 1 d 1 d
+ +U(r) ψ (r)+γ ψ (r)=E ψ (r), (37)
−2m dr2 rdr − r2 2 dr 1 G 2
(cid:20) (cid:18) (cid:19) (cid:21)
where U(r)=V(r) V¯/2 and the polaron energy is ob- and (37) that the polaron ground state energy scales as
−
tained from E = E +E . By introducing the dimen-
P G 0 ε
sionless variable ρ=r/ℓP, where ℓP =1/α(mω0)1/2 is a EP =F α02 α2ω0, (38)
measure of the polaron spatial extension in the zero SO
(cid:16) (cid:17)
limit, and by noticing that EG does not depend on the where ε0 = E0/ω0 is the dimensionless SO energy intro-
sign of γ, it is straightforward to realize from Eqs.(36) duced in Eq.(24) and is a generic function. It is found
F
thereforefromEq.(38)thatthedependenceofE onthe
P
SO interaction is through the effective parameter ε /α2,
0
7
which is treated in the following as an independent vari- regime,for which U(r) can be treatedas a perturbation.
able. Although ε /α2 is then formally allowed to vary To remedy to this deficiency, the following alternative
0
from0 to ,it isneverthelessimportanttoestimate the form of the polaron ansatz is proposed:
range over∞which ε /α2 is expected to vary for reason-
0
able valuesofthe microscopicparametersE0,ω0, andα. Ψ(r)=f(r) J0(br) , (43)
To this end, it must be reminded that the strong cou- J1(br)eiϕ
(cid:18) (cid:19)
pling limit of a 2D Fr¨ohlich polaron (in the absence of
where, as before, b is a variational SO parameter. As it
SOinteraction)isappropriateonlyforα'5,12 andthat
willbeshownbelow,thelowestvalueofE isgiveneither
the typical phonon energy scale is of the order of few to P
byEq.(42)orbyEq.(43), depending onthe specific form
tens meV, sayω 5 10meV. The largestvalue ofthe
0
Rashba energy E≈rep−orted so far is of about 0.2 eV,25 consideredfor f(r) and on the value of the SO coupling.
0
so that ε0/α2 . 1 2 is a rather conservative estimate
−
compatiblewithmaterialparametersandwiththestrong
B. ground state energy
coupling polaron hypothesis.
Letusnowevaluatethebehaviorofψ (r)andψ (r)for
1 2
Toevaluatethepolarongroundstateenergy,threedif-
r ℓ and r ℓ . By requiring a regular solution at
P P
≪ ≫ ferent trial wave functions for f(r) are considered: ex-
theorigin,itturnsoutbyinspectionofEqs.(36)and(37)
ponential, Gaussian and Pekar-type. As shown below,
that the spinor components of (35) behave as ψ (r) =
1
the Gaussian ansatz will provide results comparable to
const. and ψ (r) r as r 0, while the behavior for
1
∝ → those coming from the exponential and Pekar functions,
r ℓ isobtainedfromthelarger limitofEqs.(36)and
P
≫ despite its faster decay for r compared to Eq.(41).
(37):
→∞
These three trial wave functions will be used in combi-
1 d2ψ (r) dψ (r) nation with the sinuisodal and the Bessel-type spinors
1 2
γ =Wψ (r), (39)
− 2m dr2 − dr 1 of Eqs.(42) and (43), respectively, giving a total of six
1 d2ψ (r) dψ (r) different ansatzes for the Fr¨ohlich-Rashba polaron wave
2 1
− 2m dr2 +γ dr =Wψ2(r), (40) function.
Exponential ansatz. Let us start by evaluating the
wherethe quantityW =E +V¯/2isnegativeforbound functionalE[Ψ],Eq.(30),byusingtheexponentialansatz
G
states. Solutions of Eqs.(39) and (40) which are finite f(r) = exp( ar), where a is a variational parameter
A −
for r are linear combination of exp( λ r) and and is anormalizationfactor,incombinationwiththe
exp( λ→r∞) with − + sinuiAsodal trial wave function (42). By introducing the
−
− dimensionlessquantitiesa˜=aℓ ,˜b=bℓ ,andγ˜ =k ℓ ,
P P 0 P
λ = 2m(E +V¯/2) ik , (41) for nonzero SO interaction the functional (30) evaluated
± P 0
− ± with the exponential-sinuisodal ansatz reduces to
q
implying an exponentialdecayofthe polaronwavefunc-
E[Ψ] 1 ˜b2
tion, accompanied by periodic oscillations of wavelength = a˜2+˜b2+a˜2ln 1+
2π/k0. α2ω0 2" a˜2!#
The informations gathered on the limiting behaviors a˜2 3πa˜
ofthegroundstatewavefunctionaresufficientforguess- γ˜˜b 1+ . (44)
− a˜2+˜b2 − 8√2
ing some appropriate trial wave functions to be used in (cid:18) (cid:19)
Eq.(30). ByassumingthatforzeroSOcouplingtheelec-
For weak SO couplings, Eq.(44) has its minimum at˜b=
tronisinaspin-upstate,thenasimpleansatzcompatible
γ˜ = √2ε /α and a˜ = 3√2π/16, so that the resulting
with the limits discussed above is 0
polaron energy E =E[Ψ ]+E becomes
P 0 0
cos(br)
Ψ(r)=f(r)(cid:18) sin(br)eiϕ (cid:19), (42) EP = 3π 2 ε0 + ε20 . (45)
α2ω − 16 − α2 O α4
0 (cid:18) (cid:19) (cid:18) (cid:19)
wherebisavariationalSOparametervanishingforγ =0
andf(r)isanexponentiallydecayingfunctionforr In the ε = 0 limit, Eq.(45) reduces to E /α2ω =
0 P 0
andsuchthatf(0)=0. TheadvantageofEq.(42)is→th∞at (3π/16)2 0.3469, recovering therefore the result of
6 − ≃ −
onecanuseexponentialorPekar-typefunctionsforf(r), Ref.[14], while for ε > 0 the polaron energy is lowered
0
automatically recovering therefore the known results for by the SO interaction, in qualitative analogy with the
the zero SO case.14 It should be noted, however, that weak electron-phonon behavior discussed in Sec.II. The
in the U(r) 0 limit Eq.(42) does not reproduce cor- loweringofE isconfirmedbyanumericalminimization
P
→
rectly the behavior of the exact ground state wave func- ofEq.(44)whoseresultsareplottedinFig.3(a)(opencir-
tion, which is instead given by Eq.(35) with ψ (r) and cles). For ε /α2 =1, the polaron energy has dropped to
1 0
ψ (r) proportional to the Bessel functions J (k r) and E /α2ω 0.65, that is about two times lower than
2 0 0 P 0
J (k r), respectively.40,41 Hence,Eq.(42)isnotexpected the zero S≃O −case. However, upon increasing ε /α2, E
1 0 0 P
toprovideareliablegroundstateenergyinthestrongSO displays a minimum at ε /α2 3.98 [inset of Fig.3(a)]
0
≃
8

-0.3 -0.3 -0.3
0.0 0.0 0.0
-0.5 -0.5 -0.5
-0.4 2 ) -0.4 2 ) -0.4 2 )
0 0 0
E/(P-1.0 E/(P-1.0 E/(P-1.0
-0.5 -1.5 -0.5 -1.5 -0.5 -1.5
-2.0 -2.0 -2.0
2 )-00.6 0.1 1 0/ 210 100 -0.6 0.1 1 0/ 210 100 -0.6 0.1 1 0/ 210 100
/(P
E
-0.7 -0.7 -0.7
-0.8 -0.8 -0.8
Exponential ansatz Gaussian ansatz Pekar-type ansatz
-0.9 -0.9 -0.9
sinuisodal sinuisodal sinuisodal
(a) Bessel (b) Bessel (c) Bessel
-1.0 -1.0 -1.0
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
2 2 2
0/ 0/ 0/
FIG. 3: Polaron ground state energy as a function of ε0/α2 for different trial wave functions for f(r). (a): exponential; (b):
Gaussian; (c): Pekar. Thesinuisodal and theBessel typeof anstazes aregiven respectively byEq.(42)and Eq.(43). Inset: the
polaron energy for a wider range of ε0/α2 values.
and for larger values of the SO interaction the polaron wave function, the Gaussian ansatz gives an overalllow-
energy increases. Eventually, for ε0/α2 & 14 the calcu- eringofthepolaronenergyforbothsinuisodalandBessel
lated ground state energy becomes larger than the zero forms of the spinors. In the ε /α2 1 limit, and inde-
0
SO value E /α2ω = (3π/16)2. Such upturn of E pendently of which particular spino≪r is used, the ground
P 0 P
−
for large ε stems from the inadequacy of the sinuiso- state polaron energy is found to be:
0
dal components of (42) in treating the oscillatory be-
havior in the strong SO regime which, as pointed out EP = π ε0 + ε20 , (48)
above, should instead be given by Bessel-type functions. α2ω −8 − α2 O α4
0 (cid:18) (cid:19)
Indeed when the exponential ansatz for f(r) is used in
Eq.(43), rather than in Eq.(42), not only the resulting confirminginthisregimethelineardependenceontheSO
E is lower than the previous case, but also the upturn couplingofEq.(45). ForlargervaluesoftheSOcoupling,
P
of E disappears, leading to a monotonous lowering of and contraryto the case shownin Fig. 3(a), the sinuiso-
P
the polaron energy as ε /α2 increases [filled circles in dalandBessel-typespinorsgivebasicallythesamevalues
0
Fdoige.s3(nao)t].dAecsreεa0s/eα2in→defi∞ni,tehloywbevuetr,ratthheerpoalpaprornoaecnheersgya voaflEueP,afosrfoarlltShOeccaosuepwliintghstuhpeetxopεo0n/eαn2ti≃al1w.aBveeyfuonndcttiohnis,
limiting value. Although an accurate numerical evalua- the polaron energy obtained from the sinuisodal ansatz
tionofE forε /α2 >100hasturnedouttobedifficult, becomeslargerthanthatobtainedfromtheBesselspinor
P 0
the asymptotic valueofE canneverthelessbe obtained and, as shown in the inset of Fig. 3(b), rapidly increases
P
analyticallyfromthe strongSO limit of the exponential- while the Gaussian-Bessel anstaz gives a monotonous
Bessel expression for E[Ψ]: lowering of EP. For ε0/α2 1, the Gaussian-Bessel
≫
energy functional has the same form of Eq.(46) with the
E[Ψ] a˜2+˜b2 π latter term substituted by 2.279a˜, which implies
= ˜bγ˜ a˜, (46) −
α2ω0 2 − − √2 EP
lim 2.579. (49)
whose minimum is at˜b=γ˜ and a˜=π/√2, leading to ε0/α2→∞α2ω0 ≃−
Pekar-type ansatz. Let us now evaluate E by using
P
EP π2 inEqs.(42)and(43)the Pekar-typeansatzf(r)= (1+
ε0/lαi2m→∞α2ω0 =− 4 ≃−2.467. (47) agi1vre+s Ea2r/2α)e2xωp(−ar).0.F4o0r46z,e1r4owSOhicchouisplianglo,wtheirseaAnnesragtyz
P 0
≃ −
Gaussian ansatz. The results obtained by using a than those obtained from the exponential and Gaussian
Gaussianwavefunctionoftheformf(r)= exp( a2r2) trialwavefunctionsandonly0.03%higherthantheexact
A −
are plotted in Fig. 3(b). Compared to the exponential result 0.40474 of Ref.[16]. As shown in Fig. 3(c), the
−
9
Pekar-type ansatz gives slightly better estimates of E is the electron wave packet with p being a variational
P 0
also for nonzero SO couplings, with an overall behavior momentum, Ψ(r) is the ansatz localized wave function,
similar to the previous cases. Namely, in the weak SO and ξ′ = ePqξq′a†q 0 . Minimization of (52) with re-
regime one finds spect| toi ξ′ Ngives now|thie functional
q
E ε ε2
P = 0.4046 0 + 0 , (50) Jυ[Ψ′]= Ψ′ Hel υ pΨ′
α2ω − − α2 O α4 h | − · | i
0 (cid:18) (cid:19) dq ρ(q)′ 2 1
and, as before, for stronger SO couplings the energy ob- −|M0|2 (2π)2| q | ω q Ψ′ υ Ψ′ ,
0
tained from the sinuisodal spinor increases indefinitely Z − ·h | | i
(54)
with ε /α2. However, contrary to the exponential and
0
Gaussian ansatzes, the Pekar-type wave function may wherep is the electronmomentumoperatorandρ(q)′ =
give a lower polaron energy when used in combination Ψ′ eiq·r Ψ′ . By using Eq.(53), it is easily shown that
with the sinuisodal spinor. This holds true as long as hJυ[Ψ| ′] re|ducies to
ε0/α2 . 2.72, while for stronger SO couplings it is the
Bessel-typespinorwhichgivesthelowerEP [insetofFig. Jυ[Ψ′]= ΨHel Ψ + p20 p0 υ
3(c)]. A numerical minimization of the asymptotic limit h | | i 2m − ·
of the Pekar-BesselfunctioEnal for ε0/α2 ≫1 gives −|M0|2 (2dπq)2|ρ(qq)|2ω0 1q υ, (55)
lim P 2.91, (51) Z − ·
ε0/α2→∞α2ω0 ≃− where ρ(q) = Ψeiq·r Ψ . Equation (55) is minimized
withrespecttohp |byse|ttiingp =mυand,byexpanding
whichislowerthantheasymptoticvaluesofEqs.(47)and 0 0
the last term of Eq.(55) up to the second orderin υ, the
(49).
corresponding minimum J [Ψ] becomes:42
The resultsplottedinFig. 3clearlydemonstratethat, υ
sincethevariationalmethodprovidesanupperboundfor
m 2M 2 dq (q uˆ)2
true groundstate polaronenergy, the lowering of E in- J [Ψ]=E[Ψ] υ2 1+ | 0| · ρ(q)2 ,
P υ − 2 mω3 (2π)2 q | |
ducedbytheSOcouplingisarobustfeatureofthestrong (cid:20) 0 Z (cid:21)
(56)
coupling Fr¨ohlich-Rashba polaron. Among the different
where E[Ψ] is given in Eq.(30). From the above expres-
ansatzesstudied,thelowerpolaronenergyisobtainedby
sion, it is clear that J [Ψ] differs from J [Ψ] at least to
usingaPekar-typewavefunctionforf(r)incombination υ 0
order υ2. Hence, if Ψ and Ψ are the wave functions
withthesinuisodalspinorforweaktomoderatevaluesof υ 0
which minimize J [Ψ] and J [Ψ], respectively, then the
ε /α2 orwiththeBessel-typespinorforstrongerSOcou- υ 0
0 differenceΨ Ψ isalsooforderυ2. Asaconsequence,
plings. Giventhat,asdiscussedabove,reasonablevalues υ− 0
theminimumof (56),J [Ψ ],differsfromJ [Ψ ]onlyto
of ε /α2 for strongly-coupled polarons fall in the range υ υ υ 0
0 order(Ψ Ψ )2 = (υ4)sothat,byneglectingtermsof
0 ≤ ε0/α2 . 1−2, the Pekar-sinuisodal wave function higherorυd−erth0anυ2O,minimizationof (56)isachievedby
provides therefore the best description of the Fr¨ohlich-
thebestwavefunctionwhichminimizesE[Ψ]. Therefore,
Rashba polaron in this regime.
by using E[Ψ ] = E E and evaluating Ψ PΨ ,
0 P 0 0 0
− h | | i
from Eqs.(52) and (56) it turns out that
C. effective mass m 2M 2 dq (q uˆ)2
EP(υ)=EP+ 2 υ2 1+ m| ω03| (2π)2 ·q |ρ0(q)|2 ,
As demonstrated in Sec.II, the effective mass m∗ of (cid:20) 0 Z (cid:21)
(57)
a weakly-coupled polaron is enhanced by the SO in-
permitting us to identify the quantity within square
teraction and, given the results above, the same phe- brackets as the mass enhancement factor m∗/m. By in-
nomenon is reasonably expected to occur also for the tegratingoverthe directionofqandbyusing (5), m∗/m
strong-coupling case. To quantify the polaron mass en-
becomes in the strong-coupling limit
hancement within the localized wave function formal-
ism, it is usefulto follow the approachof Refs.[42,43,44], m∗ √2πα ∞dq
= q2 Ψ eiq·r Ψ 2, (58)
briefly described below, where a moving wave packet is m (mω )3/2 2π |h 0| | 0i|
constructedfromthe localizedwavefunction. Thequan- 0 Z0
tity to minimize is which, by replacing the momentum variable by the di-
mensionlessquantityq˜=qℓ ,givesamassenhancement
Jυ[Ψ′,ξ′]= Ψ′,ξ′ H υ PΨ′,ξ′ , (52) proportionalto α4 in the zePro SO case. By using the ex-
h | − · | i
where υ is a Lagrange multiplier, which will turn out to ponential,Gaussian,andPekar-typeansatzesinEq.(58),
be the mean polaronvelocity,and P=p+ qa†a is theresultingmassenhancementfactorbecomesm∗/m=
tishegitvoetnalbmyotmheenptruomduocpteΨra′t(orr).ξT′hwehwearveefunPctiqon|Ψq′,qξ′i (a3n/d16m)3∗π/m4α4≃≃0.703.6α442,1rαe4sp,emct∗i/vmely=.45(π/4)2α4 ≃ 0.617α4,
| i The results for nonzero SO coupling are plotted in
Ψ′(r)=eip0·rΨ(r) (53) Fig.4 for the sinuisodal (open circles) and Bessel (filled
10

3.5 3.5 3.5
3.0 Exponential ansatz 3.0 Gaussian ansatz 3.0 Pekar-type ansatz
sinuisodal sinuisodal sinuisodal
2.5 Bessel 2.5 Bessel 2.5 Bessel
4 )2.0 2.0 2.0
m
*/(
m 60 60 60
1.5 1.5 1.5
50 50 50
4 )40 4 )40 4 )40
m m m
1.0 m*/(30 1.0 m*/(30 1.0 m*/(30
20 20 20
10 10 10
0.5 0.5 0.5
0 0 0
0.01 0.1 1 2 10 100 0.01 0.1 1 2 10 100 0.01 0.1 1 2 10 100
(a) 0/ (b) 0/ (c) 0/
0.0 0.0 0.0
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
2 2 2
0/ 0/ 0/
FIG. 4: Polaron mass enhancement m∗/m in units of α4 as a function of ε0/α2 for different ansatz wave functions. (a):
exponential; (b): Gaussian; (c): Pekar. Inset: m∗/mα4 is plotted for a wider range of SO values.
circles) spinors evaluated with exponential (a), Gaus- the el-ph coupling α for different ε values. For ε = 0,
0 0
sian (b), and Pekar-type (c) wave functions. For all the polaron energy follows E /ω πα/2 for small α
P 0
cases, m∗/m increases with ε /α2 without much quan- andE /ω 0.4046α2 for largeα≃. −These two limiting
0 P 0
≃−
titative differences between the various ansatzes as long behaviors are plotted in Fig. 5 by the uppermost curves
as ε0/α2 . 2. As shown in the insets of Fig. 4, for and compared with a numerical solutions of the Feyn-
largervaluesofthe SO couplingthe useofthe sinuisodal man variational path integral for the 2D polaron (filled
spinor largely overestimates the increase of the effective circles). The largest deviation of the path integral solu-
mass compared to the Bessel-type spinor results. How- tions fromthe weak andstrongcoupling approximations
ever, despite of the weaker enhancement of m /m, the falls in the range of intermediate values of α and signals
∗
Bessel-type spinors give nevertheless an infinite effective a region of crossover between the weakly and strongly
mass at ε /α2 = . Indeed, independently of the par- coupled polaron. A rough estimate of the crossover po-
0
ticular form of f(∞r), for ε /α2 the expectation sition is given by a “critical” coupling, say α∗, obtained
0
value Ψ eiq·r Ψ appearing in→Eq∞.(58) goes like a/q by equating the weak and strong coupling results. For
0 0
for q h |, ren|deriing the integral over q of Eq.(58) di- ε = 0 therefore one has πα/2 = 0.4046α2, which gives
0
vergen→t. ∞ α∗ 3.9. Now,asshowninFig. 5forε =5andε =20,
0 0
≃
theincreaseoftheSOinteractionsystematicallyreduces,
for fixed α, the polaron ground state energy and, at the
IV. DISCUSSION AND CONCLUSIONS sametime,shiftstheintersectionpointbetweentheweak
and strongcoupling curvestowardssmaller values of the
el-phinteraction. The“critical”valueα∗ ofthecrossover
The results presented in the previous sections consis-
is therefore reduced by the SO interaction. For ε = 5
tently show that, for both the weak and strong coupling 0
and ε = 20 it is found that α∗ 3.6 and α∗ 2.7,
limits of the el-ph interaction, the ground state energy 0 ≃ ≃
respectively. The systematic reduction of the crossover
E of the Fr¨ohlich-Rashba polaron is lowered by the
P
coupling by the SO interaction is made evident in the
SO interaction and the mass is enhanced, leading to
inset of Fig. 5, where α∗ is plotted as a function of ε .
the conclusion that the Rashba coupling amplifies the 0
From Fig. 5 it is also expected that, beside the reduc-
polaronic character. This scenario suggests also that a
tionofα∗,thecrossoverregionislikelytobenarrowedby
weak-coupling polaron at ε = 0 may be turned into a
0
ε . Indeed,theintersectionbetweentheweakandstrong
strong-coupling one for ε > 0 or, more generally, that 0
0
coupling solutions for ε = 20 is apparently smoother
the crossover between weakly and strongly coupled po- 0
thanthe caseforε =0,suggestingthatthetrueground
larons may be shifted by the SO interaction. This pos- 0
stateenergywoulddeviateless,andinanarrowerregion
sibility can be tested by looking at the curves plotted in
aroundα∗,fromthe weakandstrongcoupling solutions.
the main panel of Fig. 5, where the weak and strong
coupling results for E /ω are reported as a function of The scenario illustrated above, and in particular the
P 0