Linear and nonlinear Stark effect in triangular molecule

1.3 MB·English

by  Bogdan R. Bulka

Linear and nonlinear Stark effect in triangular molecule
Bogdan R. Bul ka1, Tomasz Kostyrko2 and Jakub L uczak1
1Institute of Molecular Physics, Polish Academy of Science,
ul. M. Smoluchowskiego 17, 60-179 Poznan´, Poland and
2Faculty of Physics, A. Mickiewicz University, ul. Umultowska 85,61-614 Poznan´, Poland
(Dated: January 18, 2011)
Weanalyze changes of theelectronic structureof atriangular molecule undertheinfluenceof an
electric field (i.e., theStark effect). The effectsof thefield are shown to beanisotropic and include
1
both a linear and a nonlinear part. For strong electron correlations, we explicitly derive exchange
1
couplings in an effective spin Hamiltonian. For some conditions one can find a dark spin state,
0
for which one of the spins is decoupled from the others. The model is also applied for studying
2
electronic transport through a system of three coherently coupled quantum dots. Since electron

n transfer rates are anisotropic, the current characteristics are anisotropic as well, differing for small
a and large electric field.
J

7 PACSnumbers: 73.23.-b,71.10.-w,73.63.Kv,75.50.Xx,33.57.+c
1

] I. INTRODUCTION leads to splitting of energy levels as well as to breaking
l
l of the symmetry of the system and changes the symme-
a
try of wave functions. We demonstrate that the electric
h In this paper we investigate electronic properties of
- a model of a triangular molecule in a presence of an field can induce significant changes in a spin arrange-
s ment. It also results in changes of coupling between
e electric field. Recently, similar models were considered
m for systems of three coherently coupled quantum dots spins and different characteristics of spin-spin correla-
tion functions with respect to an angle the electric field
. (QDs),1–8, for magnetic interactions in molecules,9–12 as
t forms with the median of the triangle. In particular we
a well as for complex phase orderings in strongly corre-
will show conditions for the appearance of the dark spin
m lated electronic materials with a triangular lattice (e.g.,
state. SecondwewillshowhowtheStarkeffectmanifests
- multiferroics13, cobaltates14, or organic compounds15).
d Thesesimplemodelsexhibitplentyofinterestingphysics. itselfinelectronictransport. We confine ourselvesto the
n caseinwhichatequilibriumthegroundstateissingletor
For example, in the system of quantum dots, one finds
o tripletandforanappliedbiasvoltageexcitedstateswith
complex charge and spin arrangements1, which can be
c threeelectrons(doublets andquadruplets)participatein
[ classified according a set of topological Hunds rules.
For some interference conditions a so-called dark state transport.
2
can occur, for which one of the QDs is decoupled from
v
the reservoirs and an electron can be trapped.4 Con-
4
2 sequently, electronic transport is changed; one can ob-
9 servearectificationeffect,negativedifferentialresistance, II. INFLUENCE OF ELECTRIC FIELD ON
5 and an enhancement of the shot noise.5 Moreover, one SPIN STATES
. can expect the Aharonov-Bohm effect, when a magnetic
2
1 flux penetrates the triangle. Moreover, one can expect
0 theAharonov-Bohm effect, when a magnetic flux pen-
1 etrates the triangle. The effect leads to a crossover
: from the singlet to the triplet ground state, which man-
v
i ifests itself in spin blockade in transport2 and interest-
X ing spin dynamics under two-electron-spin-resonance3.
r For the regime of coherent transport one can expect the
a
very rich phase diagram with many types of the Kondo
resonances8.
Our studies concern the Stark effect in the system
of strongly correlated electrons and they are addressed
mainly to coherently coupled quantum dots. For a small
number of electrons (n = 1 and n = 2 in the triangle)
the electric field induces a large polarization, and many FIG. 1: (Color online) The considered model of a triangular
aspects were already considered1–8. In this paper we fo- molecule placed in an electric field E.
cus on the situation with n = 3 electrons, for which the
induced polarization is minor, because the Coulomb in-
teractionsdominateandhinderapossibleshiftofanelec- A model of a triangular molecule in the presence of
tronic charge. First we will show that the electric field an electric field E is shown in Fig.1. The corresponding
2
Hamiltonian can be expressed as
H =t (c† c +h.c.)+U n n
M iσ jσ 0 i↑ i↓
iX<j,σ Xi
2π
+U1 niσnjσ′ +Eer cos[(θ+(i 1) ]niσ,(1)
− 3
i<Xj,σ,σ′ Xi,σ
where the first term describes electron hopping between
nearestneighborsites. Forthesakeofgeneralitywecon-
sidert<0aswellast>0,forwhichthemodeldescribes
the system of QDs with electrons or holes, respectively.
The secondandthe third terms correspondto insite and
intersite Coulomb interactions. The last term takes into
account the influence of the electric field E on the elec-
tronic polarization Pˆ = e r n , where r denotes
i,σ i iσ i
the vector pointing to the sPite i, e - a charge of an elec-
tron, θ - the angle between r and E, d - the length
1
of the arm of the triangle and r = d/√3. One can see FIG. 2: (Color online) Expectation values of the spin corre-
that the electric field modulates the local site energies: lators S1·S2,S2·S3 and S3·S1 (black,redand bluecurves,
ǫ =g cos[θ+(i 1)2π/3]. In further considerationswe respectively) calculated for the ground state as a function of
i E
−
take g =Eer as a coupling parameter. the angle θ of the electric field with respect to the triangle.
E
For n = 3 electrons in the system the wave functions TheplotsareobtainedfortheHubbardmodel(1)withalarge
are constructed from the singlet and the triplet states U0 = 30, U1 = 2, t = 1, gE = 0.3 (a), gE = 2.5 (b), gE = 5
by adding an electron (see [16]). There are quadruplet (c) and gE =10 (d). For comparison the dashed curves rep-
resent the spin correlators for the Heisenberg model (4) with
states(3/2Q )withS =3/2,S = 3/2, 1/2,andthe
Sz z ± ± the modulated exchange couplings Jij given by Eq.(5) – [for
corresponding wave functions are constructed from the
smaller gE ≤5 the dashed curves cover thesolid ones within
triplet states by adding an electron. The ground state
theplot resolution].
is, however, the doublet state (1/2D ) with S = 1/2,
Sz
S = 1/2, which can be formed from the states
z
±
QD2(3)andthesingletstateisatthe12(31)bond(the
1
D = c† (c† c† c† c† ) vac ,(2) electric field is then perpendicular to the bond). The
| Szi1 √2 1σ 2σ 3σ− 2σ 3σ | i
situationismorecomplexforalarge g >4t,whenthe
E
1 | | | |
D = [2c† c† c† c† (c† c† +c† c† )] vac (3) quadratic Stark effect begins to dominate and gradually
| Szi2 √6 1σ 2σ 3σ− 1σ 2σ 3σ 2σ 3σ | i changestheperiodofoscillationof S S aswellasthe
i j
h · i
configuration of the dark spin states.
andthestateswithdoublesiteoccupancyc†1σc†1σc†2σ|vaci, In order to understand the crossover from small to
c† c† c† vac , etc. The function D and D are largefields,weperformtheperturbativecanonicaltrans-
2σ 2σ 3σ| i | Szi1 | Szi2
constructed respectively from the singlet and the triplet formation of the Hubbard Hamiltonian (1) to an effec-
state at the 23–bondby adding anelectronto the QD 1. tiveHeisenbergHamiltonian,treatingtheintersiteterms
Fig.2presentsanevolutionofthe spin-spincorrelation (both the hopping andthe intersite Coulombinteraction
functions when the electric field increases, it manifests a terms) as small ones.17 To take into account both the
transition from the linear to the quadratic Stark effect. linear and non-linear Stark effect the perturbation ex-
For small fields the spin-correlation functions oscillate pansionshould be carriedout to the third order (see the
with the period 2π. At an intermediate value g =4t Appendix for details). For n = 3 electrons the effective
E
acrossoveroccurs,andforalargergthefunctio|ns |S S| | Hamiltonian reads
i j
h · i
show new components due to the quadratic Stark effect.
1
One can see in Fig.2a and 2b that at θ = 0 (where the H˜ = 3U + J S~ S~ , (4)
1 ij i j
electric field is perpendicular to the 23-bond and points (cid:18) · − 4(cid:19)
Xi<j
tothe1-stsite)thespincorrelators S S = S S =
1 2 3 1
0 and S2 S3 0.75. This mehans·thait thhe s·piniat with the exchange coupling
h · i ≈ −
QD1 is uncoupled from the spins forming the singlet at
the 23-bond, and the corresponding state is D . We 4t2 4t2(ǫ ǫ )2 8t3(2ǫ ǫ ǫ )
| Szi1 J = + j − i + m− i− j , (5)
call such a configuration the dark spin state, in contrast ij U U3 U3
0 0 0
to the dark states for n = 1 and n = 2 electrons in the
triangle molecule, when their properties are connected wheretheindicesi,j,mdenotethreedifferentsites. Here,
to a specific charge distribution.4,5 With rotation of the we also assumed that the on-site Coulomb interactions
electric field, the dark spin state occurs at θ = 2π/3 are stronger than the electric field, i.e. U ǫ . It is
0 i
≫
(θ = 4π/3), when the uncoupled spin is located at the seen that for the weak field the third order term [the
3
third term in Eq.(5)] depends linearly on the electric for which the corresponding wavefunctions are: D
| Szi1
field E, whereas the second order term behaves like a and D [given by Eq.(2) and (3)], respectively. Fig.3
| Szi2
second power of E and its period of oscillations is twice presents the plot of these eigenenergies as a function of
as large as the linear term. This result presents one of the electric field. In this case D is the dark spin
| Szi1
the main differences between the Stark effect in the sys- state, but its energy E < E for g <4t only. Fig.3
D1 D2 E
tem of strongly correlated electrons and that in atomic also shows the eigenergies ED1′ and ED2′ for the case
physics21. θ = π/3 (when the electric field is perpendicular to the
For a very large g , when the quadratic term domi- 13 bond and its direction is opposite to the 2-nd site).
E
nates, the dark spin state can occur for E parallel to a Now the corresponding eigenstates are
bond ofthe triangle. Thenthe singlet state is formed on
1
tInhisthbiosnldimaintdththeereuniscoaupdlierdecstpeinxcihsaantgteheproopcpeosss,itewQhiDch. |DSzi1′ = √2 c†2σ(c†3σc†1σ−c†3σc†1σ)|vaci(,10)
is symmetric with respect to exchange of spins between 1
QDs, and it does not depend on the orientation of the |DSzi2′ = √6 [2c†2σc†3σc†1σ−c†2σ(c†3σc†1σ+c†3σc†1σ)]|vaci(.11)
dipole (it is a quadratic dependence on E). Since the
exchange coupling Jij depends on the difference of site However in this case the dark spin state |DSzi1′ is the
energies [see Eq.(5)], its maximal value is at the elec- excitedstate(ED1′ >ED2′). Inthe bothpresentedcases
tric field parallel to the bond. For this case two other thestates|DSzi1,|DSzi1′ areconstructedfromthesinglet
exchange couplings are equal, and the dark spin state states, whereas |DSzi2, |DSzi2′ are constructed from the
appears. triplet states.
III. TRANSPORT IN SEQUENTIAL
TUNNELING REGIME
Letusnowanalyzeelectronictransportinasequential
tunneling regime through our system with the left and
the rightelectrode connected to the QD 1 and2, respec-
tively. In our calculations we need transfer rates from
the L (R) electrode to the molecule18
ΓL(R)+ =γ ν c† ν 2f(∆E µ ),
ν2→ν3 L(R) |h 3| 1(2)σ| 2i| ν2ν3 − L(R)
Xσ
(12)
where ν denotes the initial state for n electrons,
n
| i
∆E = E E is the corresponding energy dif-
FIG.3: (Coloronline)Splittingofstatesbytheelectricfieldin ν2ν3 ν3 − ν2
ference,γ isanettransferratethroughthepotential
theeffectiveHeisenbergmodel(4). Thesolidcurvesrepresent L(R)
barierbetweentheelectrodeandthemolecule,f denotes
thestatesED1 andED2 forθ=0,whereasthedashedcurves
represent ED1′, ED2′ for θ=π/3 (t=1, U0 =30, U1 =2). the Fermi distribution function, the chemical potentials
are taken as µ = E , µ = E +eV, E is the Fermi
L F R F F
energy and V is a bias voltage. Similarly, one can write
The eigenvalues of the Hamiltonian (4) can be readily
obtained as ΓνL3(→R)ν−2 for the reverse tunneling process when the elec-
tron leaves the molecule.
EQ =3U1, (6) Since the quadruplet functions |QSzi are constructed
ED1,2 =3U1−(6J ±2∆)/4 (7) mfroamtrixtheeletmriepnletstsar|TeSoznil,y:therefore, the nonzero transfer
for quadruplet and doublet, respectively.
Here, J = (J + J + J )/3 and ∆ = Q c† T =√2 Q c† T
12 23 31 |h ±3/2| 1(2)σ| ±1i| |h ±1/2| 1(2)σ| 0i|
J2 +J2 +J2 J J J J J J . If the
pelec1t2ric fie23ld is3p1e−rpe1n2di2c3ul−ar 1t2o 3t1he−ij2-3bo3n1d, ǫi = ǫj, =√3|hQ±1/2|c†1(2)σ|T±1i|=|xT2(1)3| (13)
the exchange couplings J =J and we have the dark
im jm for σ = 1/2. Here, we use T (the singlet solution
spin state. In particular for θ = 0, J12 = J31 and the S )asa±linearcombinationof|triSpzliets(singlets)localized
eigenvalues are |onithe ij bonds, and xT (xS) denote the corresponding
ij ij
3J coefficients. The doublet wavefunction is a linear com-
ED1 =3U1− 34J−J23, (8) banindatxiD2on]aosfw|DelSlzais1thaendsta|DteSszwi2it[hwditohubtlheesictoeeofficcciuepnatnxcD1y.
ED2 =3U1− 4 −J12, (9) We have checked that for a large U0 the transfer matrix
4
elements for the states with double occupied sites play theelectricfieldparameterg canbecontrolledindepen-
E
a minor role in electronic transport, and thus they are dently of the bias voltage, e.g. by means of application
ignored. The corresponding elements are: of additional lateral or back gate electrodes. For small
fields (g = 0.1) one can see negative differential resis-
E
|hD±1/2|c†1σ|T±1i|=√2|hD±1/2|c†1σ|T0i| tance(NDR)(anincreaseandnextadropofI withV)at
θ 0 and θ π. The NDR effect is due to charge accu-
≈ 2/3|xD2 xT23|, (14) m≈ulationon≈thedarkstateandtheinterchannelCoulomb
|hD±1/2|c†2σ|T±1i|=√2|hpD±1/2|c†2σ|T0i| blockade5. We have also analyzed all current contribu-
≈|(xD1 /√2+xD2 /√6)xT13|, (15) ttihoenscuIrνr2e,νn3ttflhorwousgvhiavatrhieoussinegnleertgaynldevtehlse. bAotshexdpoeucbtleedt
|hD±1/2|c†1σ|Si|≈|xD1 xS23|, (16) states for t = −1 (top plots in Fig.4). Although the
D c† S 1/2(xD √3xD)xS . (17) otherstatesareinthevoltagewindow,theyplayaminor
|h ±1/2| 2σ| i|≈ | 1 − 2 13| roleandthe correspondingI areexponentially small
ν2,ν3
Next, we solve the corresponding master equation18 (these processesarethermally activated only). The situ-
ationforg =0.8(Fig.4b)isdifferent. Oneseesthatthe
E
dPνn = Γℓ+ P + Γℓ− P values of I are higher and the θ dependence is different.
dt νn−1→νn νn−1 νn+1→νn νn+1 At θ π/3 a new minimum appears. It is an evidence
ℓ,Xνn−1 ℓ,Xνn+1 ofthe≈Starkeffect, whichis alsomanifestedin activation
P ( Γℓ− + Γℓ+ ) (18) of the triplet and the quadruplet states. The transfer
− νn νn→νn−1 νn→νn+1
ℓ,Xνn−1 ℓ,Xνn+1 of electrons via the quadruplet state is now substantial
and its contribution I is larger than those from the
T,Q
to find the occupation probability Pνn of the eigenstates other states. This situation can be explained by ana-
|νniinthesteadylimit,i.e. fordPνn/dt=0. Thecurrent lyzing the activation energies ∆Eν2,ν3 in Fig.5a and the
in this limit reads corresponding transfer rates ΓL+ . Since the singlet is
ν2→ν3
the nondegenerate ground state, it shows the quadratic
I =e ΓL+ P ΓL− P . (19)
ν2→ν3 ν2 − ν3→ν2 ν3 Stark effect and∆ESD is a parabola. In contrastto that
νX2,ν3(cid:0) (cid:1) the activation energies from the triplet state ∆E and
TD
∆E show, for θ = 5π/3, a linear and a parabolic de-
TQ
pendence. For this case the triplet levels can be derived
explicitly as:
g
E
E =U +t ,
T 1
− 2
1
E =U + (g 2t 9g2 +12g t+36t2). (20)
T 1 4 E − ±q E E
We took the Fermi energy as E =5.1, thus, for a small
F
g , ∆E <∆E <E and the corresponding trans-
E TD TQ F
ferratesΓL+ andΓL+ [seeEq.(12)]areexponentially
T→D T→Q
small. For larger fields (g > 0.2) these energies are
E
above E and the transfer rates ΓL+ and ΓL+ are
F T→D T→Q
activated together with ΓL+ (the currentstarts to flow
S→D
through all these levels at a threshold voltage).
ThebottomrowofFig.4presentsthecurrentmapsfor
the case t = 1, for which the triplet is the ground state
at equilibrium. For a bias voltage larger than a thresh-
old one (for eV > 0.7), electrons are transferred via the
triplet and the d∼oublet states as well as the quadruplet
state. We can clearly see (at eV 0.9) the second step
FIG.4: (Coloronline)Mapsofthecurrentasafunctionofthe ≈
inthecurrent,whenthequadrupletstateisactivated. In
biasvoltageV andtheangleθoftheelectricfieldwithrespect
this case, the current maps are also different for a small
tothetriangleforgE =0.1(left column)andgE =0.8(right
and large g . This results from the dependence of the
column). TheFermilevelistakenEF =5.1,soatequilibrium E
activation energies ∆E (and the transfer rates) on
thesystemcontainstwoelectronsinthesinglet(triplet)state ν2,ν3
fort=−1(t=1)-top(bottom)plots, respectively. Wealso the electric field. Fig.5b shows that ∆ESD (almost) lin-
assumed that U0 = 30, U1 = 2, the temperature T = 0.05, early increases with gE and for large gE ∆ESD > EF.
γL=γR =γ0, and thecurrent is in units of eγ0. Therefore, the transfer rate ΓLS→+D is activated.
It is worth noticing that the degenerate states (the
The results of numerical calculations are presented in triplet E for t = 1 as well as the singlet E for t =
T S
−
Fig.4 as a map in the V-θ space. Here we assumed that 1) have different dependences on the electric field [see
5
doublet states: D and D . Moreover, we pre-
| Szi1 | Szi2
dict that the anisotropic Stark effect should be seen in
electronic transport.
The studied model is general and can be applied to
real molecules with the triangular symmetry, to study
their magnetic and optical features of interestfor molec-
ular spintronics. For example, we predict that spacial
anisotropy induced by the electric field in the effective
Heisenberg model will be manifested in molecular mag-
netism (e.g. in magnetization, magnetic susceptibility,
or ESR spectra)9,10,12. Moreover, the model can be the
paradigmformaterialswithstronglycorrelatedelectrons
on triangular lattices13–15. In our opinion multiferroics
FIG.5: (Coloronline)Electricfielddependenceoftheenergy
are the best candidates to observe the Stark effect, be-
difference ∆Eν2ν3 for the most relevant states participating cause in such materials local ferroelectric orderings can
in electronic transport. The plots a) and b) are for the case
t = −1 and t = 1, which correspond to the current maps in modifyexchangecouplingsbetweenspinsaswellasmag-
Fig.4 for the top and the bottom panels at θ = 5π/3 and netic orderings.
EF =5.1,whenatequilibriumthegroundstateisthesinglet
and thetriplet, respectively. Here, weomit theindices1 and
2for thedoubletstates, becausetheiractivation energies are Acknowledgments
very close toeach other.
We would like to thank Arturo Tagliacozzo for stimu-
lating discussions. This workwassupportedby Ministry
Eq.(20), the plots for ∆TD and ∆TQ in Fig.5a as well as of Science and Higher Education (Poland) from sources
for ∆SD in Fig.5b]. One of them is linear vs gE, whereas for science in years 2009-2012 and by the EU project
the second one is nonlinear. This is in contrast to the Marie Curie ITN NanoCTM.
Stark effect inatomic physics21, forwhich alldegenerate
states show a linear field dependence in a wide range.
Here we analyzed the electronic transport, in which Appendix: Canonical transformation of the Hubbard
two- and three-electron states (with transitions ν model with modulation of site energy
2
| i ↔
ν ) participated. By using the electron-hole symmetry
3
| i
of the model (1) one obtains the same results for tran- Here we apply a canonicalperturbation theory for the
sitions |ν3i ↔ |ν4i (between the states with three and Hamiltonian (1) for U1 =0, which we rewrite here as:
four electrons) - provided that one changes the sign of
the hopping t. H =W +T +T +T (A.1)
0 −1 +1
where W represent the sum of all the single site terms,
andT isacontributiontothehoppingpartoftheHamil-
IV. CONCLUSIONS n
tonian which increases by n the number of the double
occupiedsitesinthesystem. UsingHubbardoperators22
Summarizing, we considered the influence of the elec- Xαα′ α α′ (definedinterms ofthe exacteigenstates
tricfieldonstronglycorrelatedelectronsinthetriangular αi for≡a|niishola|ted site i) the contributions to the model
molecule(thelinearandthequadraticStarkeffect). The |(Ai.1) can be represented by:
orientation θ of E with respect to the molecule is im-
portant, because the electric field breaks the symmetry W = E Xαα
iα i
of the system and changes the symmetry of wave func- Xi,α
tions. For some θ one finds the dark states, responsible
T = t (Xσ0X0σ+X2σXσ2),
for negative differential resistance. For n = 3 electrons 0 ij i j i j
wederivedthekineticexchangecouplingJ betweenthe iX<j,σ
ij
spins, which showedquadratic and linear dependence on T = σt X2,−σX0σ, T =T† (A.2)
E. The spin-spin correlation functions exhibit different +1 iX<j,σ ij i j −1 +1
angleθ characteristicsforasmallandlargeE. Inpartic-
ular, we studied the dark spin states and their evolution The site energies: E 0,ǫ ,2ǫ +U correspond to
iα i i 0
withE. Themodelcanbeappliedtostudiesofentangle- the eigenstates α 0,∈, {,2 . The follo}wing analysis is
ment of three spin qubits19 with the electric field in its validfora limitof∈th{e s↑ma↓ll T} ,whenthe differencesbe-
n
specificrole. Inparticular,itcanbeappliedtoadescrip- tween energies E for single and double occupied sites
iα
tionofanexperimentjustrecentlyperformedinasystem are much larger than the hopping parameters. In the
of three QDs,20 which presented coherent spin manipu- derivationofthe effective Hamiltonianwe applythe per-
lation in a qubit with the logical basis formed from the turbationtheorywithrespecttothehoppingpart,which
6
can be formulated in terms of the recursive canonical where
transformation.17 With the use of the Hubbard opera-
torsthemethodcanbeeasilygeneralizedtoanarbitrary
J(2) =2t2(∆−1+∆−1). (A.7)
formoftheon–sitezero–orderHamiltonian,includingthe ij ij ij ji
site–dependent terms.23
The effective Hamiltonian can be rewritten in a more
familiar form with a help of the spin operators
1. The effective second order spin Hamiltonian
1
Up to the second order with respect to the hopping H˜ = W + J(2) S~ S~ . (A.8)
|C00 ij (cid:18) i· j − 4(cid:19)
part the transformed Hamiltonian reads Xi<j
1
H˜ =e−iSHeiS H + [iS,H ]. (A.3)
0 1
≈ 2
where H = W +T and H = T +T . Here, it is
0 0 1 +1 −1 2. The effective third order spin Hamiltonian
assumed that in the expansion the linear term with re-
specttotheoff–diagonalpartofhoppingvanishes,which
is guaranteed by a condition: Aderivationofthehigherordertermsinageneralcase
isbasedontherecursiveprocedure,17howeveritisrather
[iS,H ]= H . (A.4) involved and will be discussed in a separate paper. Here
0 1
−
weonlypresenttheextra3rdordertermprojectedtothe
From this condition we can derive in the explicit form C subspace
00
the transformation matrix
iS = iSij, H˜(3) = J(3) S~ S~ 1 . (A.9)
Xi<j (cid:12)(cid:12)C00 Xi<j ij (cid:18) i· j − 4(cid:19)
σt σt (cid:12)
iS = ijX−σ2Xσ0 ijXσ0X−σ2
ij (cid:18)∆ i j − ∆ i j
Xσ ij ji The exchange parameter reads:
σt σt
ijX0σX2−σ+ ijX2−σX0σ . (A.5)
−∆ji i j ∆ij i j (cid:19) J(3) =2t t t ∆−1∆−1+∆−1∆−1
ij ji im mj ij im ji jm
where ∆ij = ǫi +U0 −ǫj. After inserting the operator +∆−im1∆−jm1 −∆−ji1∆−m1i −∆(cid:0)−ij1∆−m1j −∆−m1i∆−m1j .(A.10)
S (A.5) into (A.3) we obtain the effective Hamiltonian (cid:1)
validuptosecondorderperturbationwithrespecttothe
Here,theindicesi,j,mdenotethreedifferentsites. Note,
hopping part. In a form projected to the subspace C ,
00
that the extra term vanishes for the uniform case ǫ =
definedassubspaceofmany-electronstateswithallsites 1
ǫ =ǫ .
singly occupied, i.e. for n = 3 electrons in the triangle, 2 3
the Hamiltonian reads: By expanding J(2) and J(3) [Eqs.(A.7) and (A.10)] in
ij ij
a series with respect to ǫ U we ontain the exchange
H˜ = W + 1 σαJ(2)X−σαXσ,−α (A.6) coupling Eq.(5). i ≪ 0
|C00 2 ij i j
i<Xj,σ,α
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