Reinterpretation of Scanning Tunneling Microscopy on an Adsorbed Magnetic Atom

0.27 MB·English

by  Jongbae Hong

Reinterpretation of Scanning Tunneling Microscopy on an Adsorbed Magnetic Atom
Jongbae Hong∗
Max-Planck-Institut fu¨r Physik Komplexer Systeme, D-01187 Dresden, Germany
Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
(Dated: November25, 2009)
9 The observation of the Kondo effect in meso- tunneling is another type of coherent transport channel
0 scopic systems under bias1,2 has opened a new in addition to the Kondo channel. Interference between
0
chapter inthe physics ofthe Kondo phenomenon. these two coherent scattering channels could yield Fano
2
VarioustypesofdI/dV, whereI andV denotecur- resonances18 at the positions of the RTL because the

v rent and source-drain(s-d) bias, respectively, line RTLs are discrete states created within the continuum
o shapes have been measured by scanning tunnel- spectrum of the Kondo peak. However, only dispersed
N ing microscopy (STM)1,3−11. However, explana- RTL peaks are observed in most STM line shapes in-
tionbysingleFanolineshape1,12−16 isnotrelevant stead of the sharp Fano interference pattern because no
5
2 and even misleading. Here, we provide consistent Kondo peak occurs except case (iii) mentioned above.
explanationsforvariousasymmetricandsymmet- We employ Tersoff-Hamann formula19 that gives the

] ric line shapes in terms of a microscopic theory differential conductance of an STM system as
l
l that shows the creation of two resonant tunneling
ha levels (RTLs) when bias is applied17. One side dI Γt ρa(V),
- Kondo coupling between adatom and substrate dV ∝ σ
σ
s does not create Kondo peak that appears only X
e
m when the system has an overall Kondo coupling where Γt denotes the featureless coupling function be-
including both substrate and tip. The structure tween adatom and STM tip, and ρa(V) is the local den-
. σ
t of an asymmetric line shape is mostly governed sity of states (LDOS) of the adatom at a steady state
a
m by the RTL peaks. Therefore, Kondo effect is of bias V. The spectral function of an up-spin electron
- negligible in most asymmetric line shapes. at the adatom is given by ρa↑(ω) = (1/π)Re[(M−r1)33],
where
d ThedI/dV lineshapesmeasuredbySTM1,3−12 canbe
n
classified into four different groups. Typical examples of iω˜ γ Us γ γ
o such groups are given as follows: (i) a Co adatom on a − − ss J− st J−
c
2 [ bnAiiuafi(sc1;a1(ni1ti))ly1aoaCrsyoCmaudm(a1et1to1rmi)c4−olin7neasusCrhfuaa(pc1ee070s,h8);osw(uiriiinf)agcaeaCsdhoiopawdnianetagoramzseiorgno- M = γUsss∗ −iUω˜s∗ UJisω+˜ γUJ+t∗ γUstt∗ ,
v aninsulatingCu NfilmcoveringaCu(100)surfaceshow- r  − J− − J+ − − J+ − J− 
2 2  
18 ianngda(isvh)arapsKysotnemdospimeaiklarwittoh(siyiim) mweitthricanshoFueldaedras9t,o1m0;  −γst −γJ+ UJt+ −iω˜ γtt 
 
2 showing a peculiar wedge and symmetric step-wise line  γ γ Ut γ iω˜ 
0. shape11.  − J− − st J− − tt − 
1 SincetheexistenceoftwoRTLsisthenewdiscoveryof where ω′ ω ǫa U na↓ 18. All the matrix ele-
≡ − − h i
9 ments, except the 8 U-elements, have additional self-
this study and plays a crucial role in explaining the line
0 energy terms, β [iΣs(ω)+iΣt(ω)]=2β ∆, for a flat
shape, we discuss it first. The occurrenceoftwo RTLs is mn 0 0 mn
: wideband,where∆ (Γs+Γt)/4. Weuse∆toindicate
v a unique characteristicof nonequilibrium steady state of
≡
i a mesoscopic system with two reservoirs. Because of the the unit of energy.
X
Wecalculatethe matrixelementsofM forthesingle-
backandforthtransitsbetweenthe Kondoimpurity and r
r impurity Anderson model with two metallic reservoirs
a the left and right reservoirs, two more basis vectors are
whose Hamiltonian is given by
neededtodescribethedynamicsofthesystemunderbias
comparedwith the system of single reservoir. These two
= ǫ cα†cα + ǫ c† c +Un n
additional basis vectors yield two RTLs. In equilibrium, H k kσ kσ a aσ aσ a↑ a↓
however, these additional basis vectors are not needed k,σX,α∈s,t Xσ
because the probabilities of back and forth transits are + (Vαcα†cα +Vα∗cα†cα ),
ka aσ kσ ka kσ aσ
thesame. Onlyrelevantnonequilibriumtheoriesthatcan
k,σ,α∈s,t
X
properly handle the Kondo phenomena in a steady state
can verify the existence of the two RTLs. This resonant where s, t, ǫa, and U denote the substrate, tip, energy
level of the adatom, and Coulomb interaction, respec-
tively. The detailed procedure to reach the matrix M
r
and its elements is given in Supplementary information
∗Electronicaddress: [email protected];@pks.mpg.de of this paper.
2
The matrix M contains all the information on the
r
transportpropertiesoftheSTM-Kondosystems. Wedis- V
cuss the roles of γ first and those of UJs,±t and the coeffi- dI/d a b c
cientsβ later. Theupper-left3 3blockdescribesthe 0.5
mn
× 0.4
single-reservoir Anderson model between the substrate 0.4
and the magnetic adatom, while the lower-right 3 3
×
blockrepresentsanothersingle-reservoirAndersonmodel 0.4
betweenthe tipandthe adatom. The 2 2blocksatthe 0.2 0.2
×
upper-right and lower-left corners couple the substrate
andthe tipthroughthe adatom. Therefore,γ (γ )rep-
ss tt 0.3
resents the strength of the one-side Kondo coupling be-
tweenthe substrate(tip) andthe adatom,andγ repre- -1 0 1 0.0 0.4 -1 0 1
st
sentsthecouplingmechanismbetweenthetworeservoirs. FIG.2: ChangeindI/dV lineshape. Theunitoftheordinate
is (4e2/h)Γt. We set U =8. The unspecified parameters are
On the other hand, the antidiagonal element γ , where
γ =γ γ , gives the effect of steady stateJ. thesameasthoseused inthepreviouspanel. a,β11 =0.252,
J− J+ ≡ J β12 = 0.246, β22 = 0.256, and other βmn = 0.25. γss =
The operators comprising γ, which are given in Sup- 0.12, γtt = 0.1, γst = 0.65, γJ = 0.35, Re[UJs+] = 2.4, other
ppdlreeesmctareitpniottinaorsnysfoionrffotγhrmienarFtoiilogen.o1of.feFtahicgihsurpγea.1paeWrs,hegopiwvresestcehlneeatorpneriec-itsnoitdreieard-l RβIm2e2[[UU=JJs+]0]=.=272,0,.γa3tn.tdD=Iams0h[,eUdγJss,∓lttin]=e=: γγ0Js.sb==,00β..151c,,=Rγes0t[U.2=Js5,∓5γt,]Jβ==1200.=.355,0aa.2nn4dd,
Kondo coupling of γss(tt); Fig. 1b shows the difference Im[UJs+]=0.8.
between the leftward and rightward motions of a down-
spinelectroninγ ;Fig. 1cshowstheoverallKondocou-
J
pling of γ , i.e., an up-spin electron in both reservoirs peaks of the spectral function at the adatom. The five
st
has Kondo coupling with a down-spin electron passing peaksarethe Kondopeakatω′ =0,twoHubbardpeaks
throughtheadatomviaexchangingorhoppingtogether. at ω′ U/2, and two RTL peaks at ω′ = [(γ2 +
≈ ± ± ss
Figures 1b and 1c also show the resonant tunneling of a γt2t)/2+(γst −γJ)2 +O(U−2)]1/2. The spectral weight
down-spinelectronif it does not makea Kondo coupling of the Kondo peak can also be obtained from Mr as
with an up-spin electron.
U2 γ2 +γ2 +2(γ γ )2 −1
In most of STM systems, no Kondo coupling exists Z = 1+ { ss tt st− J } , (1)
2R 8(γ γ +γ2 γ2)2
between the adatom and tip, i.e., γtt = 0. In this case, (cid:20) ss tt st− J (cid:21)
the system does not have anoverallKondocoupling and where the subscript2R indicates a two-reservoirsystem.
γst = γJ. This implies that only resonant tunneling is It is interesting to note that when γst = γJ and γtt =
allowedin Figs 1b andc. However,when the system has 0, which is the case of most STM-Kondo systems, Z
2R
the overall Kondo coupling, γst > γJ and both γss and vanishes and the gap between the two RTLs is √2γss.
γtt are nonvanishing. In fact, the values of γst and γJ at SincenoKondopeakexistsinthiscase,theKondoeffect
a steady state cannot be determined by present theory. only appears as a separation of RTL peaks. One can
However,wefindthatchoosingappropriateconstantsfor see this case in the experiments on Co atom adsorbed
γst and γJ gives rise to the dI/dV line shapes that are on Au(111)1 or Cu(111)4−7 surface. On the other hand,
well-agreedwith experiments. whenγ >γ ,i.e., γ =0,Z does notvanishandthe
st J tt 2R
6
Another important information is obtained from the gapbetweentwoRTLsbecomeswider. Thisimpliesthat
matrix M . The zeros of the determinant of M in the thetwoRTLsrecedebylevelrepulsionduetotheKondo
r r
atomic limit, i.e., ∆ = 0, gives us the positions of five peak at the Fermi level. One can see this case in the
experiments on Co atom adsorbed on Cu(100) surface
atom atom covering with Cu2N9−10.
(cid:3994)
We now show our theoretical dI/dV line shapes ex-
a s (cid:3994) (cid:827)(cid:3994) (cid:827) t
plicitly. Figure 2a, which is obtained by choosing γ >
(cid:3993) ss
γ > 0, γ > γ , and Im[Us ] = 0, clearly shows both
tt st J J+
b ss (cid:827) t (cid:84) s (cid:827) t the Kondo and Fano peaks. However, the Kondo and
Fano peaks disappear and only the RTL peaks remains
(cid:3993) (cid:3994) (cid:3994) (cid:3993) as shown in Fig. 2b if we choose γ = 0, γ = γ , and
tt st J
c s (cid:827) t (cid:82) s (cid:827) t Im[Us ] > 0, which are the usual case of STM-Kondo
J+
(cid:3994) (cid:3994) (cid:3994) (cid:3994) system. Figure 2c is obtained by using a larger asym-
FIG.1: Motionsofadown-spinelectronthatmakesaKondo metry parameter Im[UJs+]. The red lines in Figs. 2b and
couplingwithanup-spinelectron. Theprocessesindicatedby 2c correspond to the dI/dV line shapes observed in a
number two are those of exchanging and hopping together. Co adatom on a Cu(111)4 and a Au(111)1 substrate, re-
a, γss and γtt, b, γJ, and c, γst. We do not show up-spin spectively. A small shift in the dip from zero bias stems
movementsinb. bandcalsoshowtheresonanttunnelingof from the nonvanishing asymmetry parameter. The con-
a down-spin when Kondocoupling is not constructed. ditions γ = γ and γ = 0 eliminate the Kondo peak
st J tt
3
neededtoobtainFig.3c,whichisthelineshapereported
a b c
0 V.4 0.4 by Choi, et al.9 for the same system. In conclusion, no
d 0.4
dI/ substantial Kondo coherence exists in the system of a
0.2 Co/Cu(100) 0.2 Co/Cu(100) 0.3 Co/Cu(100) Co adatom on a Cu(100) surface7−9 that shows highly
-2 -1 0 1V/ -1 0 1 -0.2 0.0 0.2
0.1 asymmetric dI/dV line shapes.
d e f
A remarkable change in the line shape occurs when
0.2 0.2
the Cu(100) substrate is covered by an insulating cop-
0.0 Co/CuN/Cu 0.0 Co/CuN/Cu 0.0 pernitride film. The line shapebecomessymmetric,and
-1 0 1 -1 0 1 -1 0 1 a Kondo peak and side peaks9 or steps10 appear for a
0.2 g
0.2 h Co adatom. However, a peculiar wedge at zero bias and
0.1 multiple steps appear for an Fe adatom11. Covering the
0.0 0.0 Fe/CuN/Cu Cu(100) surface with an insulating copper nitride film
-1 0 1 -1 0 1
causes at least two effects. One is to strengthen the co-
FIG.3: VariousdI/dV lineshapesforU =8. Theunitofthe herenceinaxialdirectionconnectingCuinsubstrate,Cu
ordinateand thepresentation schemeare thesame as Fig. 2. in film, and Co adatom. The other is to suppress both
a, β11 = 0.265,β12 = 0.23,β22 = 0.28, and other βmn = fluctuation (δj+s)2 1/2 and activity j+s on the sub-
1/4. γss = γtt = 0, and γst = γJ = 0.25. Re[UJs,∓t] = 0.5, strate side.hThae↓forimer results in thhe aK↓oindo coupling
Im[Us ] = 0.4, and other Im[U ] = 0. b, Im[Us ] = 0.2.
c, γsJs+= 0.07 and Im[UJs+] = 0J.∓3. d, β11 = 0.2J5+1, β12 = abnetdwReeen[Utshe] aindcarteoamse.anTdhetilpatatenrdatlshoeclaauttseers amadkecerseaγssse
0γ.J24=9 a0n.4d. βR22e[=UJs0+.2]5=2.2γ.4s,s R=e[0U.3Jt,+γ]tt==0.08.,1,anγdstI=m[0U.5Js,,∓ta]n=d in h(δja+↓t)J2+i1/2. As a result, all the imaginary parts of
0. e, Re[UJs+] = 4. f, γtt = 0, γst = 0.4. g, Re[UJs,−t] = UJs,∓t vanish, and the line shape becomes symmetric.
0.8, Re[UJs+] = 2.66, and Re[UJt+] = 1.33. h, γss = 0.5, On the basis of this analysis, we obtain a symmetric
γst =γJ =0.7, Re[UJs,−t]=0.4, Re[UJs+]=2, and Re[UJt+]= line shape with the Kondo peak and two RTL peaks as
1. Fermi level is shifted 0.1 to the right only in c. In h, shown in Fig. 3d. This line shape corresponds to the
two additional parameters have been used to take the spin one reported by Choi et al.9. Selecting a relatively large
excitation into account. Re[Us ] changes the RTL peaks of Fig. 3d to shoulders,
J+
as shown in Fig. 3e, which has been reported by Otte
et al.10. An interesting change occurs when we simply
in Fig. 2a according to equation (1) and also changes choose γ = γ and γ = 0 in Fig. 3e to remove the
st J tt
the Fano resonance into a dispersed RTL peak because Kondo peak. A sharp wedge appears at zero bias as
the Kondospectrum, whichis necessaryforconstructing shown in Fig. 3f. The wedge reflects the trace of a miss-
Fano interference, has been removed. ing Kondo peak. Such a wedge has been observed by
The blue lines in Figs. 2b and c indicate the line Hirjibehedin et al.11 by adopting Fe adatom instead of
shapes without the Kondo coupling in both sides, i.e., Co. We show in Fig. 3g that a step-like feature can be
γ = γ = 0. The similarity between the red and blue obtained by changing Re[Us,t]. An additional step ap-
ss tt J±
linesindicatesthatthestructureofthedI/dV lineshape pears in the line shape of ref. 11. We understand that it
is mostly governed by the RTL spectrum. The Kondo stemsfromthespin-spininteractionatthepositionofthe
effect between a Co adatom and a substrate appears as adatom. If we add the spin interaction at the adatom,
a slight widening of the dip structure and a shift in the the matrix M becomes a 7 7 matrix and additional
r
×
minimum. The amount of widening may be considered steps appear as shown in Fig. 3h.
as an energy scale of the STM-Kondo system with dip In conclusion, we have shown that two RTLs can ex-
structure because no Kondo peak appears in this case. plain most of the experiments on STM-Kondo systems
We measure the increase δD in Fig. 2b, i.e., the differ- consistently. If the Kondo coherence exists only in be-
ence between the gaps in red and blue, as δD = 0.1D, tween a magnetic adatom and a substrate, the Kondo
whereD isthefulldistanceofthe gapinred. SinceD in peak is not observed. The energy scale of the Kondo
ref. 4isapproximately20meV,weobtainδD 2.0meV coherence in this case may be defined by the amount
for the system used by Manoharan et al.4. T≈his energy of gap elongation of the dip. Interestingly, the highly
corresponds to a temperature of 23 K that is consider- asymmetricline shapes aresimply the RTL spectra with
ably higher than the temperature at which usual STM asymmetry. We showed that the Kondo peak and steps
measurementsareperformed. Inordermeasuretheelon- appearing in ref. 10 are attributed to the overall Kondo
gatedgap,onemusthaveaschemetoremovetheKondo coupling and the suppression of the fluctuation on the
coupling between the adatom and substrate. substrate side. Finally, we understand that the sharp
Highlyasymmetriclineshapeshavebeenobservedina wedge for anFe adatomreflects the trace ofdisappeared
Co adatom on a Cu(100) surface7−9. We obtain similar Kondo peak and the additionalstep stems from the spin
line shapes observed by N´eel et al.8 in Fig. 3a and b interaction at the adsorbed atom.
by adopting γ = γ = 0, which means that no Kondo
ss tt
effectisinvolvedinobtainingtheselineshapes. However, 1. Madhavan,V., Chen, W., Jamneala,T., Crommie,
aweakKondocouplingbetweenadatomandsubstrateis M. F. & Wingreen, N. S. Science 280, 567 (1998).
4
2. Cronenwett, S. M., Oosterkamp, T. H. & Kouwen-
hoven, L. P. Science 281, 540 (1998).
3. Heinrich,A. J., Gupta, J.A., Lutz, C.P. & Eigler,
D. M. Science 306, 466 (2004).
4. Manoharan, H. C., Lutz, C. P. & Eigler, D. M.
Nature 403, 512 (2000).
5. Wahl, P. et al., Phys. Rev. Lett. 93, 176603
(2004).
6. Vitali, L. et al., Phys. Rev. Lett. 101, 216802
(2008).
7. Knorr, N., Schneider, M. A., Diekh¨oner, L., Wahl,
P.&Kern,K.Phys. Rev. Lett. 88,096804(2002).
8. N´eel,N.etal.,Phys. Rev. Lett. 98,016801(2007).
9. Choi, T., Ruggiero, C. D. & Gupta, J. A. J. Vac.
Sci. Technol. B 27, 887 (2009).
10. Otte, A. F. et al., Nat. Phys. 4, 847 (2008).
11. Hirjibehedin,C.F.etal.,Science317,1199(2007).
12. Schiller,A.&Hershfield,S.Phys. Rev. B61,9036
(2000).
13. U´js´aghy, O., Kroha, J., Szunyogh, L. & Zawad-
owski, A. Phys. Rev. Lett. 85, 2557 (2000).
14. Plihal, M. & Gadzuk, J. W. Phys. Rev. B 63,
085404(2001).
15. Madhavan, V., Chen, W., Jamneala, T., Cromie,
M.F. & Wingreen,N. S. Phys. Rev. B64,165412
(2001).
16. Luo, H. G., Xiang, T., Wang, X. Q., Su, Z. B. &
Yu, L. Phys. Rev. Lett. 92, 256602(2004).
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(1985).
Acknowledgements This work was supported by the
Korea Research Foundation, Grant No. KRF-2008-314-
C00140.
5
Supplementary Information:
The Procedure Reaching Matrix M and the Expressions of β
r mn
The spectral function of a magnetic atom adsorbed on a substrate is given by ρ (ω) = (1/π)ImG+ (ω). We
a↑ − aa↑
showbelowthatthespectralfunctionρ (ω)ortheretardedGreen’sfunctionG+ (ω)canbeexpressedbythematrix
a↑ aa↑
M−1 given in the text of the main article. Since the retarded Green’s function expressed in a resolvent form in the
r
Heisenberg picture is written as1
iG+ (z)= c (zI+iL)−1 c =(M−1) ,
aa↑ h a↑| | a↑i aa
where L is the Liouville operator defined by Lˆ = [ , ˆ] for the Hamiltonian , z = iω + 0+, c indicates
a↑
O H O H −
the annihilation of an up-spin electron at the adsorbed atom, and the matrix M is constructed by the elements
M =zδ + e iLe , where the vectors e and e are the elements of a complete set of orthonormal basis vectors
kℓ kℓ ℓ k ℓ k
h | i
e ℓ=1, , spanning the Liouville space.The operator c is one of the members. The inner product is defined
ℓ a↑
{ | ··· ∞}
by e e e ,e† , where the angular and curly brackets denote the expectation value and anticommutator,
h k| ℓi ≡ h{ k ℓ}i
respectively.
Therefore, the first step to obtain the retarded Green’s function G+ (ω) is to find a complete set of orthonormal
aa↑
basisvectorsofthe Liouvillespace. Inref. 2givenbelow,wehaveshownabasisspanningthereducedLiouvillespace
that is effective for describing the dynamics of the system in the Kondo regime as
(cs , δn cs , δj+sc , δj−sc , c , δj−tc , δj+tc , δn ct , ct )
k↑ a↓ k↑ a↓ a↑ a↓ a↑ a↑ a↓ a↑ a↓ a↑ a↓ k↑ k↑
withnormalizationfactors (δn )2 −1/2 and (δj∓s,t)2 −1/2 forthecorrespondingbasisvectors,whereδ = ,
h a↓ i h a↓ i O O−hOi
j∓s,t = (i) (Vs,tc† cs,t V∗s,tc†s,tc ), where (i) is deleted for j+s,t, and k = 1,2, , indicates the quantum
a↓ k ka a↓ k↓ ∓ ka k↓ a↓ a↓ ··· ∞
states of the substrate or the tip.
P
WeconstructthefullmatrixMusingthesebasisvectorsinanordergivenabove. Then,Misgivenbythefollowing
nine-block matrix,
M M 0
ss as
M= M M M , (S1)
sa a ta
 
0 M M
at tt
 
where the blocks M , M and M , and M and M are 5 5, 5 , and 5 matrices, respectively. Since
a as at sa ta
× ×∞ ∞×
no direct coupling exists between the substrate and tip, null blocks occur at the two corners. The block M is
ss(tt)
constructed by the basis vectors cs(t) and δn cs(t). It is written as
k↑ a↓ k↑
M 0
M = 11 ,
ss(tt) 0 M
11
(cid:20) (cid:21)
where
z+iǫ 0 0
1
···
M11 = 0 ... 0 .
 
 
 0 0 z+iǫ 
 ··· ∞ 
 
On the other hand, M = M† and the 5 block M has the form
sa − as ×∞ as
0 0 C 0 0
M = ka ,
as Cs Cs 0 Ct Ct
(cid:20) nj− nj+ nj+ nj− (cid:21)
where the columns C , Cs,t , and Cs,t have elements iV∗, (ξ−s,tV )∗, and (ξ+s,tV )∗, respectively, where
ka nj− nj+ ka − a ka − a ka
ξ∓s,t =(1/2)[ i[n ,j∓s,t](1 2n ) +i(1 2 n ) j∓s,t ][ (δj∓s,t)2 (δn )2 ]−1/2.
a h a↓ a↓ − a↑ i − h a↓i h a↓ i h a↓ ih a↓ i
6
Thefirsttermispurelyrealandthesecondtermispurelyimaginary. Thedetailedprocessofobtainingξ∓s,tV from
a ka
iLc δn c δj∓ , which is the element of M , is given in Appendix A below.
−h k↑ a↓| a↑ a↓i sa
The 5 5 block M at center is given by
a
×
iω˜ γ Us γ γ
− − ss J− st J−
γ iω˜ Us γ γ
M = Usss∗ −Us∗ Jiω+˜ UJ+t∗ Ustt∗ ,
a − γJ− −γJ+ −Ut − iJω˜+ −γJ−
 −γst −γJ+ UJt+ −γ ittω˜ 
 − J− − st J− − tt − 
 
where ω˜ ω ǫ U n . γ and Us,t are given by
≡ − a− h a↓i J∓
γ = i(V∗cs +V∗ct )c† [j−s(t),j+s(t)] [ (δj−s(t))2 (δj+s(t))2 ]−1/2,
ss(tt) h ka k↑ ka k↑ a↑ a↓ a↓ ih a↓ ih a↓ i
k
X
γ = i(V∗ cs +V∗ct )c† [j−s,j+t] [ (δj−s)2 (δj+t)2 ]−1/2,
st h ka k↑ ka k↑ a↑ a↓ a↓ ih a↓ ih a↓ i
k
X
γ = i(V∗cs +V∗ct )c† [j∓s,j∓t] [ (δj∓s)2 (δj∓t)2 ]−1/2,
J∓ h ka k↑ ka k↑ a↑ a↓ a↓ ih a↓ ih a↓ i
k
X
and
Us,t = iU h[na↓,ja∓↓s,t](1−2na↑)i+hja∓↓s,ti(1−2hna↓i) .
J∓ 2 " (δj∓s,t)2 1/2 #
h a↓ i
The detailed process of calculation of the matrix elements is given in Appendix A in which the Hamiltonian for the
single reservoir Anderson model is used.
Theinfinite-dimensionalmatrixMofequation(S1)canbereducedtoanequivalent5 5-dimensionalmatrixusing
×
Lo¨wdin’s partitioning technique3,4. For this purpose, we consider an eigenvalue equation MC = 0, where C and 0
are column vectors. We partition the column vector C into three parts, i.e., C = (C C C )T, where T means the
s a t
transpose and C , C , and C correspond to rows (cs , δn cs ), (δj+sc , δj−sc , c , δj−tc , δj+tc ),
s a t k↑ a↓ k↑ a↓ a↑ a↓ a↑ a↑ a↓ a↑ a↓ a↑
and (δn ct , ct ), respectively. After eliminating C and C , we reach an equation (M M M−1M
a↓ k↑ k↑ s t a − sa ss as −
M M−1M )C M C =0. Then, we have the expression of the reduced matrix M given in the text, i.e.,
ta tt at a ≡ r a r
iω˜+iΣ (ω) γ +iΣ (ω) Us γ +iΣ (ω) γ +iΣ (ω)
−γ +iΣ11(ω) −iωs˜s+iΣ12(ω) UJs− γst +iΣ14(ω) γJ−+iΣ15(ω)
M = ss Us∗21 − Us∗22 iω˜+JiΣ+ (ω) J+ Ut∗24 st Ut∗25 ,
r γ −+iJΣ− (ω) γ −+iJΣ+ (ω) − Ut 33 iω˜−+iJΣ− (ω) γ +−iΣJ+(ω)
 − st 41 − J+ 42 J+ − 44 tt 45 
 γ +iΣ (ω) γ +iΣ (ω) Ut γ +iΣ (ω) iω˜+iΣ (ω)
− J− 51 − st 52 J− − tt 54 − 55 
 
where iΣ (ω) = β (iΣs(ω)+iΣt(ω)) = 2β ∆ for a flat wide-band, where ∆ (∆s +∆t)/2 = (Γs +Γt)/4.
∆ is usedmans a unit mofnene0rgy. The0coefficientsmβn are symmetric. They are given≡by β = ξ−s 2, β = ξ+s 2,
β =1, β = ξ+t 2, β = ξ−t 2, β =ξ+s∗ξ+t,mβn =ξ−s∗ξ+s, β =ξ−s∗ξ+t, β =ξ−1s1∗ξ−t|, βa |=ξ2+2s∗ξ−|t,aan|d
β33 =ξ+t∗4ξ4−t.|Tahe| coe5ffi5cien|tas β| 2a4re coampleax wh1e2n ma=n,awhil1e4β aareareal.15The daetaialed e2x5pressaionsaof β
45 a a mn 6 mm mn
are given in Appendix B.
Appendix A
CalculationofthematrixelementsofMforthe single-impurityAndersonmodelwithonereservoirisshownbelow.
Since calculation for the elements of M is very simple, we skip it.
ss(tt)
(1) Matrix elements of the block M :
sa
Nontrivial elements of M are iL(c δn ),δj∓c† , i.e.,
sa −h{ k↑ a↓ a↓ a↑}i
i[H,c δn ],j∓c† = i[H,c ]δn +c i[H,n ],δj∓c†
h{ k↑ a↓ a↓ a↑}i h{ k↑ a↓ k↑ a↓ a↓ a↑}i
= iǫ c δn ,δj∓c† iV c δn ,δj∓c† + c j−,δj∓c† . The first term
− kh{ k↑ a↓ a↓ a↑}i − kah{ a↑ a↓ a↓ a↑}i h{ k↑ a↓ a↓ a↑}i
7
( iǫ ) c δn ,δj∓c† = ( iǫ ) [n ,j∓]c c† should vanish because the up-spin dynamics of this form
− k h{ k↑ a↓ a↓ a↑}i − k h a↓ a↓ k↑ a↑i
is not allowed. The third term also vanishes because c j−,δj∓c† = j−δj∓ c ,c† =0. The second term,
h{ k↑ a↓ a↓ a↑}i h a↓ a↓{ k↑ a↑}i
however, must be calculated rigorously, since it is a hybridization term that is related to the Kondo process. The
operator c δn ,δj∓c† is expanded as c [δn ,δj∓c† ]+[δj∓c† ,c ]δn +2c δj∓c† δn . First two terms
{ a↑ a↓ a↓ a↑} a↑ a↓ a↓ a↑ a↓ a↑ a↑ a↓ a↑ a↓ a↑ a↓
are rewritten as c [δn ,δj∓c† ]=c [n ,j∓]c† =c c† [n ,j∓]=(1 n )[n ,j∓] and
a↑ a↓ a↓ a↑ a↑ a↓ a↓ a↑ a↑ a↑ a↓ a↓ − a↑ a↓ a↓
[δj∓c† ,c ]δn = δj∓[c† ,c ]δn = δj∓(1 2c c† )δn , while the third term is cancelled by the last
a↓ a↑ a↑ a↓ a↓ a↑ a↑ a↓ a↓ − a↑ a↑ a↓
expression 2c δj∓c† δn . Hence, iL(c δn ),j∓c† = iV (1 n )[n ,δj∓] + δj∓δn =
− a↑ a↓ a↑ a↓ −h{ k↑ a↓ a↓ a↑}i kah − a↑ a↓ a↓ a↓ a↓i
iV 1 n [n ,j∓] + 1[n ,j∓] + δj∓δn = iVka (1 2n )[n ,j∓] + δn ,δj∓ = iVka (1
kah 2 − a↑ a↓ a↓ 2 a↓ a↓ a↓ a↓i 2 h − a↑ a↓ a↓ { a↓ a↓}i 2 {h −
2n )[n ,j∓] +(1 2 n ) j∓ .
a↑(cid:0) a↓ a↓(cid:1)i − h a↓i h a↓i}
Therefore, the matrix elements h{iL(ck↑δna↓),δja∓↓c†a↑}i =V ξ∓, where
− ||ck↑δna↓||×||ca↑δja∓↓|| ka a
ξ∓ = 1 i(1 2n )[n ,j∓] +i(1 2 n ) j∓ (δn )2 −1/2 (δj∓)2 −1/2.
a 2{h − a↑ a↓ a↓ i − h a↓i h a↓i}h a↓ i h a↓ i
(2) Matrix elements of the block M :
a
(2-1) Diagonal elements:
By using the commutator expression i[H,ca↑] = −i kVk∗ack↑ −iǫaca↑ −iUca↑na↓, the diagonal elements of the
block M are given by
a
(A) −h{i[H,ca↑],c†a↑}i=i kVk∗ah{ck↑,c†a↑}i+iǫah{ca↑P,c†a↑}i+iUh{ca↑na↓,c†a↑}i
=iǫ +iU c n ,c† =iǫ +iU c ,c† n =iǫ +iU n , and
a h{ a↑ a↓ a↑}i P a h{ a↑ a↑} a↓i a h a↓i
(B) i[H,c δj∓],(c δj∓)† = i[H,c ]δj∓,δj∓c† c i[H,δj∓],δj∓c† . The first term of (B) is
−h{ a↑ a↓ a↑ a↓ }i −h{ a↑ a↓ a↓ a↑}i−h{ a↑ a↓ a↓ a↑}i
rewritten as
h{i[H,ca↑]δja∓↓,δja∓↓c†a↑}i=h{(−i kVk∗ack↑−iǫaca↑−iUca↑na↓)δja∓↓,(ca↑δja∓↓)†}i
= iǫ c δj∓,δj∓c† iU c n δj∓,δj∓c† . Applyingthedecouplingapproximation,n δj∓ = n δj∓,
− ah{ a↑ a↓ a↓ a↑}i− h{Pa↑ a↓ a↓ a↓ a↑}i a↓ a↓ h a↓i a↓
to the U-term above gives rise to a form of squared norm for the first term of (B), i.e., i[H,c ]δj∓,δj∓c† =
h{ a↑ a↓ a↓ a↑}i
[ iǫ iU n ] c δj− 2. The second term of (B), however, cannot be written in the form of a squared norm,
− a− h a↓i ×|| a↑ a↓||
because [H,j∓] j±. Therefore, we neglect it.
a↓ ∝ a↓
(2-2) Matrix elements U :
J∓
These are given by the inner products iLc ,δj∓c† , i.e., i[H,c ],δj∓c†
−h{ a↑ a↓ a↑}i −h{ a↑ a↓ a↑}i
=iǫahδja∓↓ih{ca↑,c†a↑}i+iUh{ca↑na↓,δja∓↓c†a↑}i+i kVk∗ahδja∓↓ih{ck↑,c†a↑}i
=iU c n ,δj∓c† . However, c δn ,δj∓c† = c [δn ,δj∓c† ]
h{ a↑ a↓ a↓ a↑}i h{ a↑ a↓ a↓Pa↑}i h a↑ a↓ a↓ a↑ i
+ [δj∓c† ,c ]δn + 2c δj∓c† δn , where the operator in the first term is rewritten as
h a↓ a↑ a↑ a↓i h a↑ a↓ a↑ a↓i
c [δn ,δj∓c† ] = c [n ,j∓]c† = c c† [n ,j∓] = (1 n )[n ,j∓], while the second term is given by
a↑ a↓ a↓ a↑ a↑ a↓ a↓ a↑ a↑ a↑ a↓ a↓ − a↑ a↓ a↓
[δj∓c† ,c ]δn = δj∓[c† ,c ]δn = δj∓δn 2δj∓c c† δn . The last one cancels the third term
h a↓ a↑ a↑ a↓i h a↓ a↑ a↑ a↓i h a↓ a↓i − h a↓ a↑ a↑ a↓i
of c δn ,δj∓c† above. Hence, c δn ,δj∓c†
h{ a↑ a↓ a↓ a↑}i h{ a↑ a↓ a↓ a↑}i
= (1 n )[n ,j∓]+δj∓δn = 1 n [n ,j∓]+ 1[δn ,δj∓]+δj∓δn
h − a↑ a↓ a↓ a↓ a↓i h 2 − a↑ a↓ a↓ 2 a↓ a↓ a↓ a↓i
= 1 (1 2n )[n ,j∓]+ δn ,δj∓ = 1 (1 2n )[n ,j∓] +(1 2 n ) j∓ .
2h − a↑ a↓ a↓ { a↓ a↓}(cid:0)i 2{h(cid:1) − a↑ a↓ a↓ i − h a↓i h a↓i}
Therefore, the matrix elements U that are defined by h{iLca↑,δja∓↓c†a↑}i are given by
J∓ − h(δj∓)2i1/2
a↓
U
U = [i (1 2n )[n ,j∓] +i(1 2 n ) j∓ ] (δj∓)2 −1/2
J∓ 2 h − a↑ a↓ a↓ i − h a↓i h a↓ih a↓ i
Ifwedefine Re[U ] U/2τ,wegetτ =2by theatomiclimit analysisforthetwo-reservoirAndersonmodel5. Then,
we have a relatioJn∓, 1/≡τ =i (1 2n )[n ,j∓] (δj∓)2 −1/2, which will be used in Appendix B.
h − a↑ a↓ a↓ ih a↓ i
(2-3) Matrix elements γ :
ss(tt)
The inner products iL(c δj∓),δj±c† gives rise to the expressions for γ . Since
−h{ a↑ a↓ a↓ a↑}i ss(tt)
iL(c δj∓),δj±c† = i[H,c ]δj∓,δj±c† + c i[H,j∓],δj±c† , the first term can be expanded
h{ a↑ a↓ a↓ a↑}i h{ a↑ a↓ a↓ a↑}i h{ a↑ a↓ a↓ a↑}i
as i[H,c ]δj∓,δj±c† = iǫ c δj∓,δj±c† iU c n δj∓,δj±c† i V∗ c δj∓,δj±c† .
h{ a↑ a↓ a↓ a↑}i − ah{ a↑ a↓ a↓ a↑}i − h{ a↑ a↓ a↓ a↓ a↑}i − k kah{ k↑ a↓ a↓ a↑}i
P
8
The first and second terms of this expression should vanish because they are not written in the form of a squared
norm. Since the third term describes the Kondo process, we treat it rigorously and have the following expression:
i V∗ c δj∓,δj±c† = i V∗ δj∓δj±c c† δj±δj∓c c†
− k kah{ k↑ a↓ a↓ a↑}i − k kah{ a↓ a↓ k↑ a↑− a↓ a↓ k↑ a↑}i
= i V∗ [δj∓,δj±]c c† = i V∗ [j∓,j±]c c† .
−P k kah a↓ a↓ k↑ a↑i −P k kah a↓ a↓ k↑ a↑i
The second term of iL(c δj∓),δj±c† , i.e., c i[H,j∓],δj±c† , is rather complicate because it contains
P h{ a↑ a↓ Pa↓ a↑}i h{ a↑ a↓ a↓ a↑}i
the commutator [H,j∓]. This commutator is expanded as i[H,j∓] = ǫ j± (i) ǫ (V c† c V∗c† c )
a↓ a↓ ± a a↓ ∓ k k ka k↓ a↓ ± ka a↓ k↓ ±
Un j± (i) ′ (V V c† c V∗V∗c† c ), where (i) is applied only to the lower signs and the prime in sum
a↑ a↓ ± k,ℓ ka ℓa k↓ ℓ↓± ka ℓa ℓ↓ k↓ P
denotesk =ℓ. Thefirsttwoarecancelledifweassumeǫ ǫ ,andtheU-termcanbeneglectedintheKondoregime.
a k
6 P ≈
The last term describes the round trip of a down-spin electron between the adatom and the substrate (tip), which
is equivalent to (j±)2. Since we consider only single trip of a down-spin electron in this work, we neglect the term
a↓
involving [H,j∓]. Thus, the matrix element M21/12 is given by −h{iL(ca↑δja∓↓),δja±↓c†a↑}i = i kVk∗ahck↑c†a↑[ja∓↓,ja±↓]i γ
a↓ a h(δj∓)2i h(δj±)2i h(δj∓)2i h(δj±)2i ≡±
a↓ a↓ P a↓ a↓
for the single-reservoirAnderson model. For the two-reservoirAnderson model, similar calculation gives rise to
p p p p
i V∗ cs c† +ct c† )[j−s(t),j+s(t)]
γ = k kah k↑ a↑ k↑ a↑ a↓ a↓ i.
ss(tt) P (δj−s(t))2 1/2 (δj+s(t))2 1/2
h a↓ i h a↓ i
Appendix B: β
mn
We obtain the expressions of β in terms of the definitions and the expressions of ξ∓ given in Appendix A.
mn a
The coefficients β in front of the self-energy function, i.e., iΣ (ω) = β (iΣs(ω)+iΣt(ω)), are symmetric in
mn mn mn 0 0
exchanging their indices. We have obtained an expression 1/τ = i (1 2n )[n ,j∓] (δj∓)2 −1/2 in Appendix A.
h − a↑ a↓ a↓ ih a↓ i
By using the operator identity [n ,j∓]= ij±, one can obtain a different expression for τ, i.e.,
a↓ a↓ ∓ a↓
1/τ =[ j±s,t 2 j±s,tn ]/ (δj∓s,t)2 ,
±h a↓ i∓ h a↓ a↑i h a↓ i
q
which will be used in the expressions of Re[β ]. The final forms of Re[β ] are given by using (δn )2 1/4 and
mn mn a↓
h i≈
+s(t) −s(t)
τ = 2. Since j < 0 and j > 0, an inequality, Re[β ] < Re[β ] < Re[β ], is applied to the substrate
h a↓ i h a↓ i 12 11 22
side in which Kondo coupling exists. As for Im[β ], we neglect all of them because Im[β ] contain the difference
mn mn
of relative fluctuations, which we assume they are the same.
We obtain the expressions of β in terms of ξ∓ given in Appendix A as follows:
mn i
Real parts of β :
mn
Re[β ]=Re[ξ−∗ξ−]= ( j+s 2 n j+s )2+(1 2 n )2 j−s 2 [4h(δna↓)2i]−1
11 s s h a↓i− h a↑ a↓i − h a↓i h a↓i h(δj−s)2i
a↓
= 1 + (1−2hna↓i)2hja−↓sih2 1 1 + (1−2hna↓i)2hja−↓si2 . i
(cid:20)τ2 h(δja−↓s)2i (cid:21)4h(δna↓)2i ≈(cid:20)4 h(δja−↓s)2i (cid:21)
Only the index s changes to t for Re[β ].
55
Re[β ]=Re[ξ+∗ξ+]= ( j−s 2 n j−s )2+(1 2 n )2 j+s 2 [4h(δna↓)2i]−1
22 s s h a↓i− h a↑ a↓i − h a↓i h a↓i h(δj+s)2i
a↓
= 1 + (1−2hna↓i)2hja+↓sih2 1 1 + (1−2hna↓i)2hja+↓si2 . i
(cid:20)τ2 h(δja+↓s)2i (cid:21)4h(δna↓)2i ≈(cid:20)4 h(δja+↓s)2i (cid:21)
Here, again, the index s changes to t for Re[β ].
44
Re[β ]=Re[ξ−∗ξ+]=Re[β ]
12 s s 21
= ( j+s 2 n j+s )( j−s +2 n j−s ) +(1 2 n )2 j−s j+s [4h(δna↓)2i]−1
h a↓i− h a↑ a↓i −h a↓i h a↑ a↓i − h a↓i h a↓ih a↓i h(δj−s)2i h(δj+s)2i
a↓ a↓
h i
= 1 + (1−2hna↓i)2hja−↓sihja+↓si 1 1 + (1−2hna↓i)2hja−↓sihja+↓si , p p
τ2 h(δj−s)2i h(δj+s)2i 4h(δna↓)2i ≈ 4 h(δj−s)2i h(δj+s)2i
(cid:20) a↓ a↓ (cid:21) (cid:20) a↓ a↓ (cid:21)
Re[β ]=Re[ξ−∗ξ+]=Re[β ]
14 p s tp 41 p p
= ( j+s 2 n j+s )( j−t +2 n j−t ) +(1 2 n )2 j−s j+t [4h(δna↓)2i]−1
h a↓i− h a↑ a↓i −h a↓i h a↑ a↓i − h a↓i h a↓ih a↓i h(δj−s)2i h(δj+t)2i
h i a↓ a↓
= 1 + (1−2hna↓i)2hja−↓sihja+↓ti 1 1 + (1−2hna↓i)2hja−↓sihja+↓ti , p p
τ2 h(δj−s)2i h(δj+t)2i 4h(δna↓)2i ≈ 4 h(δj−s)2i h(δj+t)2i
(cid:20) a↓ a↓ (cid:21) (cid:20) a↓ a↓ (cid:21)
Re[β15]=pRe[ξs−∗ξt−p]=Re[β51] p p
= ( j+s 2 n j+s )( j+t 2 n j+t ) +(1 2 n )2 j−s j−t [4h(δna↓)2i]−1
h a↓i− h a↑ a↓i h a↓i− h a↑ a↓i − h a↓i h a↓ih a↓i h(δj−s)2i h(δj−t)2i
a↓ a↓
h i
p p
9
= 1 + (1−2hna↓i)2hja−↓sihja−↓ti 1 1 + (1−2hna↓i)2hja−↓sihja−↓ti ,
τ2 h(δj−s)2i h(δj−t)2i 4h(δna↓)2i ≈ 4 h(δj−s)2i h(δj−t)2i
(cid:20) a↓ a↓ (cid:21) (cid:20) a↓ a↓ (cid:21)
Re[β24]=pRe[ξs+∗ξt+p]=Re[β42] p p
= ( j−s +2 n j−s )( j−t +2 n j−t ) +(1 2 n )2 j+s j+t [4h(δna↓)2i]−1
−h a↓i h a↑ a↓i −h a↓i h a↑ a↓i − h a↓i h a↓ih a↓i h(δj+s)2i h(δj+t)2i
a↓ a↓
h i
= 1 + (1−2hna↓i)2hja+↓sihja+↓ti 1 1 + (1−2hna↓i)2hja+↓sihja+↓ti , p p
τ2 h(δj+s)2i h(δj+t)2i 4h(δna↓)2i ≈ 4 h(δj+s)2i h(δj+t)2i
(cid:20) a↓ a↓ (cid:21) (cid:20) a↓ a↓ (cid:21)
Re[β ]=pRe[ξ+∗ξ−p]=Re[β ] p p
25 s t 52
= ( j−s +2 n j−s )( j+t 2 n j+T ) +(1 2 n )2 j+s j−t [4h(δna↓)2i]−1
−h a↓i h a↑ a↓i h a↓i− h a↑ a↓ i − h a↓i h a↓ih a↓i h(δj+s)2i h(δj−t)2i
a↓ a↓
h i
= 1 + (1−2hna↓i)2hja+↓sihja−↓ti 1 1 + (1−2hna↓i)2hja+↓sihja−↓ti , p p
τ2 h(δj+s)2i h(δj−t)2i 4h(δna↓)2i ≈ 4 h(δj+s)2i h(δj−t)2i
(cid:20) a↓ a↓ (cid:21) (cid:20) a↓ a↓ (cid:21)
Re[β ]=pRe[ξ+∗ξ−p]=Re[β ] p p
45 t t 54
= ( j−t +2 n j−t )( j+t 2 n j+t ) +(1 2 n )2 j+t j−t [4h(δna↓)2i]−1
−h a↓i h a↑ a↓i h a↓i− h a↑ a↓i − h a↓i h a↓ih a↓i h(δj+t)2i h(δj−t)2i
h i a↓ a↓
= 1 + (1−2hna↓i)2hja+↓tihja−↓ti 1 1 + (1−2hna↓i)2hja+↓tihja−↓ti . p p
τ2 h(δj+t)2i h(δj−t)2i 4h(δna↓)2i ≈ 4 h(δj+t)2i h(δj−t)2i
(cid:20) a↓ a↓ (cid:21) (cid:20) a↓ a↓ (cid:21)
Imaginapry partspof β : p p
mn
Im[β ]=Im[ξ−∗ξ+]
12 s s
= (1−2hna↓i) hja+↓si−2hna↑ja+↓si hja+↓si −hja−↓si+2hna↑ja−↓si hja−↓si
4h(δna↓)2i h(δj−s)2i h(δj+s)2i − h(δj+s)2i h(δj−s)2i
(cid:20) a↓ a↓ a↓ a↓ (cid:21)
= (1−2hna↓i)1 phja+↓si phja−↓si p p
4h(δna↓)2i τ h(δj+s)2i − h(δj−s)2i
(cid:20) a↓ a↓ (cid:21)
Im[β ]=Im[ξ−∗ξ+]
14 spt p
= (1−2hna↓i) hja+↓si−2hna↑ja+↓si hja+↓ti −hja−↓ti+2hna↑ja−↓ti hja−↓si
4h(δna↓)2i h(δj−s)2i h(δj+t)2i − h(δj+t)2i h(δj−s)2i
(cid:20) a↓ a↓ a↓ a↓ (cid:21)
= (1−2hna↓i)1 phja+↓ti phja−↓si p p
4h(δna↓)2i τ h(δj+t)2i − h(δj−s)2i
(cid:20) a↓ a↓ (cid:21)
Im[β ]=Im[ξ−∗ξ−]
15 spt p
= (1−2hna↓i) hja+↓si−2hna↑ja+↓si hja−↓ti hja+↓ti−2hna↑ja+↓ti hja−↓si
4h(δna↓)2i h(δj−s)2i h(δj−t)2i − h(δj−t)2i h(δj−s)2i
(cid:20) a↓ a↓ a↓ a↓ (cid:21)
= (1−2hna↓i)1 phja−↓ti phja−↓si , p p
4h(δna↓)2i τ h(δj−t)2i − h(δj−s)2i
(cid:20) a↓ a↓ (cid:21)
Im[β ]= (1−2hnpa↓i) −hja−↓si+p2hna↑ja−↓si hja−↓ti hja+↓ti−2hna↑ja+↓ti hja+↓si
25 4h(δna↓)2i h(δj+s)2i h(δj−t)2i − h(δj−t)2i h(δj+s)2i
(cid:20) a↓ a↓ a↓ a↓ (cid:21)
= (1−2hna↓i)1 hja−↓ti p hja+↓si p, p p
4h(δna↓)2i τ h(δj−t)2i − h(δj+s)2i
(cid:20) a↓ a↓ (cid:21)
Im[β ]= (1−2hnpa↓i) −hja−↓ti+p2hna↑ja−↓ti hja−↓ti hja+↓ti−2hna↑ja+↓ti hja+↓ti
45 4h(δna↓)2i h(δj+t)2i h(δj−t)2i − h(δj−t)2i h(δj+t)2i
(cid:20) a↓ a↓ a↓ a↓ (cid:21)
= (1−2hna↓i)1 hja−↓ti p hja+↓ti p, and p p
4h(δna↓)2i τ h(δj−t)2i − h(δj+t)2i
(cid:20) a↓ a↓ (cid:21)
Im[β ]= (1−2hnpa↓i) −hja−↓si+p2hna↑ja−↓si hja+↓ti −hja−↓ti+2hna↑ja−↓ti hja+↓si
24 4h(δna↓)2i h(δj+s)2i h(δj+t)2i − h(δj+t)2i h(δj+s)2i
(cid:20) a↓ a↓ a↓ a↓ (cid:21)
= (1−2hna↓i)1 hja+↓ti p hja+↓si p. p p
4h(δna↓)2i τ h(δj+t)2i − h(δj+s)2i
(cid:20) a↓ a↓ (cid:21)
References
1. Fulde, P. Elecptronic Correplations in Molecules and Solids (Springer-Verlag,Berlin, 1993).
2. Hong, J. cond-mat/0908.1345.
3. Lo¨wdin, P. O. J. Math. Phys. 3, 969 (1962).
4. Mujica, V., Kemp, M. & Ratner, M. A. J. Chem. Phys. 101, 6849 (1994).
5. Hong, J. & Woo, W. Phys. Rev. Lett. 99, 196801 (2007).