Table Of ContentCoexistence of Pairing Tendencies and Ferromagnetism
in a Doped Two-Orbital Hubbard Model on Two-Leg Ladders
J. C. Xavier,1 G. Alvarez,2 A. Moreo,3 and E. Dagotto3
1Instituto de F´ısica, Universidade Federal de Uberlaˆndia,
Caixa Postal 593, 38400-902 Uberlaˆndia, MG, Brazil
2Computer Science & Mathematics Division and Center for Nanophase Materials Sciences,
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
3Department of Physics, University of Tennessee, Knoxville,
TN 37996 and Materials Science and Technology Division,
Oak Ridge National Laboratory, Oak Ridge,TN 37831
0
1 (Dated: January 18, 2010)
0
UsingtheDensityMatrixRenormalizationGroupandtwo-legladders,weinvestigateanelectronic
2
two-orbitalHubbardmodelincludingplaquettediagonalhoppingamplitudes. Ourgoal istosearch
n for regimes where charges added to the undoped state form pairs, presumably a precursor of a
a superconducting state. For the electronic density ρ = 2, i.e. the undoped limit, our investigations
J show a robust (π,0) antiferromagnetic ground state, as in previous investigations. Doping away
8 fromρ=2andforlargevaluesoftheHundcouplingJ,aferromagneticregionisfoundtobestable.
1 Moreover, when the interorbital on-site Hubbard repulsion is smaller than the Hund coupling, i.e.
for U′ < J in the standard notation of multiorbital Hubbard models, our results indicate the
] coexistence of pairing tendencies and ferromagnetism close to ρ=2. These results are compatible
l
e withpreviousinvestigations usingonedimensionalsystems. Althoughfurtherresearch isneededto
- clarify iftherangeofcouplingsusedhereisofrelevanceforrealmaterials, suchassuperconducting
r
t heavy fermions or pnictides, our theoretical results address a possible mechanism for pairing that
s may beactive in thepresence of short-range ferromagnetic fluctuations.
.
t
a
PACSnumbers: 71.10.Fd,71.27.+a,74.20.-z
m
-
d
I. INTRODUCTION possibleregionfortherobustcoexistenceof(spintriplet)
n
pairing together with magnetic order is where ferromag-
o
netism develops. This is in qualitative agreement with
c
It is widely believed that magnetism is a fundamen-
[ previousinvestigationscarriedoutusingone-dimensional
tal ingredient to explain the origin of high-temperature systems.11Oureffortshouldbeconsideredsimplyaspro-
1 superconductivity in several materials. In fact, there
viding the first steps in relating pairing and magnetism
v is experimental evidence that the superconductivity in
in a complex two-orbital model via computational tech-
7
many heavy fermion (HF) compounds is mediated by
9 niquesonladdergeometries. Antiferromagneticorder,as
spin fluctuations.1–4 Mechanisms for superconductivity
0 found in the pnictides, could also be favorable for pair-
based on antiferromagnetism have been extensively dis-
3 ing tendencies in the spin singlet channel, as discussed
1. ctourssseadsfwoerllt.h5eRCecue-nbtalsye,dcohnisgihd-etreambpleeeraxtcuitreemseunptehrcaosnbdeuecn- recently.12,13
0
generatedbythediscoveryofhigh-temperaturesupercon-
0 Inprinciple,atheoreticalinvestigationbasedonmodel
1 ductivity in the ironpnictides.6 Except for the cuprates, Hamiltonians for strongly correlated materials starts
: the iron-basedsuperconductorsnowhavethe highestsu- with an effective tight-binding model, containing the
v
perconducting (SC) critical temperature T of any ma-
i c minimumingredientstodescribethephysicsofthemate-
X terial (see for example Ref. 7). As in HF systems and
rialsunder investigation. However,evenif a well-defined
r cuprate superconductors, in the pnictides there is also reasonable model is used, it is still highly non-trivial to
a evidence that the superconductivity is not mediated by
extractthegroundstatepropertiesoftheseeffectivemod-
the electron-phononinteraction.8,9
els in two or three dimensions using unbiased numerical
Magnetism and superconductivity can appear in dif- methods. In fact, at present there are no accurate tech-
ferent ways. In some HF compounds, superconductiv- niques to study Hubbard-like models in dimensions two
ity and antiferromagnetic (AFM) order co-exist,1 while and three. Thus, in order to get at least some insight
for the cuprates the superconductivity emerges after the on the ground states properties of theses models, it is
long-range AFM order is destroyed by doping.5 In some common practice to study the model Hamiltonians in
HF systems, it is the superconductivity and ferromag- quasi-one-dimensional geometries. In particular, a very
netism(FM) (asopposedto AFM order)that co-exist.10 popular route that has been used for several theoretical
Inthiswork,wewillbeinterestedindetectingaclearev- investigations is to study strongly correlated electronic
idence of pairing of extra charges that are added to the systems using“ladder”geometries.14 The N-leg ladders
undopedlimitwheremagneticorderexists. Usingatwo- consist of N chains of length L coupled by some param-
orbitalmodelandtwo-legladders,itwillbeshownthata eter (as, for example, fermionic hopping terms). The
2
two-dimensional system can in principle be obtained by II. MODEL
considering the limits of both N and L sent to infinity,
although in practice this is difficult to do. This ladder-
In these studies, we have considered the following
based procedure has been used to investigate models for
Hamiltonian defined on a two-leg ladder geometry
the high temperature superconductors15 and for the HF
systems.4,16 Some important results were obtained with H = tλ,λ′ d† d +H.c.
thismethod. Forexample,researchbasedonmicroscopic − X γ,γ′(cid:16) j,γσ,λ j+1,γ′σ,λ′ (cid:17)
models for the high T superconductors,5 as well as re- j,σ,γ,γ′,λ,λ′
c
mseeadrciahteodnbHyFanmtiofedrerlosm,4aignndeitciactfleutchtautatsiuonpserccaonndbuecsttivabitiy- − X t˜γ,γ′(cid:16)d†j,γσ,1dj,γ′σ,2+H.c.(cid:17)
j,σ,γ,γ′
lized, in agreement with several experiments. Thus, the
+ U ρ ρ +(U′ J/2) ρ ρ
use of ladders appears to be an important ingredient to X j,γ↑,λ j,γ↓,λ − X j,x,λ j,y,λ
unveil dominant ground state tendencies. Moreover, the j,γ,λ j,λ
hopping amplitudes that will be used in our investiga- 2J S S
tions below include next-nearest-neighbor diagonal hop- − X j,x,λ· j,y,λ
j,λ
pings that are only possible when plaquettes exist in the
+ J (d† d† d d +H.c), (1)
lattice under consideration. X j,x↑,λ j,x↓,λ j,y↓,λ j,y↑,λ
j,λ
Note that microscopic models that may present su-
perconductivity induced by antiferromagnetism, such as where d†j,γσ,λ creates an electron with spin projection σ
the one-orbital Hubbard model and the Kondo Lattice in the orbital γ = x,y (dxz and dyz, respectively) at
model, have been extensively studied by several au- the rung j and leg λ = 1,2, Sj,γ,λ is the electron spin
thors. However, microscopic models for superconductiv- density operator, ρ = d† d , and ρ =
j,γσ,λ j,γσ,λ j,γσ,λ j,γ,λ
ity in a ferromagnetic spin background have been much d† d .
less explored, with the exception of studies using one- Pσ j,γσ,λ j,γσ,λ
dimensionalchains.11 This may be causedin partby the The hopping amplitudes are: tλx,,xλ =t˜y,y =−t1, tλy,,yλ =
perception that superconductivity and ferromagnetism, t˜x,x = −t2, tγ1,,2γ = tγ2,,1γ = −t3, tγ1,,2γ′ = −tγ2,,1γ′ = −t4, if
as opposed to antiferromagnetism, can not coexist.17 γ =γ′,andzerootherwise. Toavoidaproliferationofex-
6
However,this perceptionhas beenchallengedby the dis- traparametersinouranalysis,wehavedecidedtofixthe
covery of superconductivity and FM in the HF com- values of these hoppings from considerations previously
pounds UGe 10 and URhGe.18 Moreover, SC and FM usedinthepnictidecontext. Ouruseofmodelsoriginally
2
were also observed19 in the d-band metal ZrZn . devised for pnictides is simply based on the pragmatic
2
observation that some pairing tendencies and ferromag-
Motivatedbythediscoveryofsuperconductivityinthe netic regions at large J were already observed in those
HFcompoundUGe ,afewyearsagoKarchevet al. pro- models.12,13 It should be clear though, that our research
2
posed a one-band model to study the coexistence of su- is mainly motivated by heavy fermion phenomenology.
perconductivity and ferromagnetism.20 However, other Following this strategy, then the hopping amplitudes
researchershavearguedthatthetreatmentusedtoinves- t1, t2, t3, and t4 are obtained using the Slater-Koster
tigatethatmodel,aswellasthemodelitself,werenotap- tight-binding scheme, and they are given by12,13,25
propriatetodescribethecoexistenceofthesephases.21–23
t = 2 b2 a2+g2 /∆
1 pd
− (cid:0) − (cid:1)
Duetothelackofstudiesofmicroscopicmodelsforsu- t = 2 b2 a2 g2 /∆
2 pd
perconductivity mediated by ferromagnetic fluctuations − (cid:0) − − (cid:1)
t = b2+a2 g2 /∆
beyond one-dimensional chains,11 in this work we have 3 −(cid:0) − (cid:1) pd
decided to investigate a microscopic ladder model where t4 = ab g2 /∆pd, (2)
−(cid:0) − (cid:1)
thecoexistenceofsuperconductivityandferromagnetism
appears possible. In fact, it will be shown below that where the Fe-As hopping amplitudes are a =
the two-orbital model that has been originally proposed 0.324(pdσ) 0.374(pdπ), b = 0.324(pdσ)+0.123(pdπ),
−
to describe the low-energy physics of the iron-based and g = 0.263(pdσ) + 0.31(pdπ). ∆ is the energy
pd
superconductors12,13,24 actually leads to the coexistence difference between the p and d levels. We have set
of pairing and spin ferromagnetic tendencies. Qualita- (pdσ)2/∆ = 1 to fix the energy scale, and we use
pd
tiveourresultsarecompatiblewiththosereportedusing pdπ/pdσ = 0.2, as previously discussed.12,13 Regard-
chains.11Althoughthemodelconsideredheremaynotbe ing the coup−lings U, J, and J′, note that they are not
a proper effective model for superconducting ferromag- independent, but they are assumed to satisfy the rela-
nets such as UGe , we believe that the mechanism that tion U = U′+2J, which is strictly valid within a cubic
2
bounds together the charges carriers (see section III.B) environment for the full t sector.26,27 For an explicit
2g
is so simple and generic that our calculations may also derivation of this relation in the case of manganites see
apply in a variety of other models as well. Ref. 28.
3
We have investigatedthe model defined above using a 10 (a)
two-legladder of size 2 L,by means ofthe Density Ma- 2x12 q =0
trix Renormalization G×roup (DMRG) technique,29 un- 8 2x12 qy=π
der open boundary conditions (OBC), and keeping up y
U =-0.5 2x16 q =0
to m=1400 states per block in the final DMRG sweep. ) 6 eff y
We have carried out 6 13 sweeps, and the discarded qx 2x16 q =π
owuerigDhtMwRaGs tpyproiccaeldlyur∼1e0,−t6h−e−c1e0n−te9rabtlothckesfianrael scwomeeppo.seInd S( 4 J=1.5 2x24 qyy=0
2x24 q =π
of 16 states. We have used a FORTRAN DMRG code y
to calculate most results, and a C++ DMRG code for 2
additional validation.30 We have also confirmed some of
our results against Lanczos Exact Diagonalization tech- 0
1 1.5 2
π
niques, when possible. q /
x
In this work, we will focus on the region of Hubbard
5
andHundparameterswhereUeff U 3J =U′ J <0, (b)
≡ − −
although a few results for positive values of U will
eff
be presented as well. As shown below, in the region 5
where Ueff < 0 there is a robust evidence of binding of ) 4
holes/electrons close to density ρ = 2. In this regime 0 4.96
,
U′ < J. This inequality may not seem realistic at first π
( U =-0.5
sight, since the on-site spin triplet formation favored by S eff 4.92 J=1.5
J (Hund’s rules) has its origin in the alleviation of the 3
2x8
Coulombic energy penalization caused by U′. However,
4.88
-2 -1 0 1 2
while in the full five-orbital model for Fe-based com-
U
poundsU′ <J isunphysicalforthereasonstatedabove, eff
2
the two-orbital model is an effective model and it is un- 0.5 1 1.5 2
J
clear how the main parameters are affected by the pro-
jection from five to two orbitals. Thus, the regime U′ 0.4
(c)
comparable to J may not be unrealistic. Clearly addi-
2x24
tional investigations are needed to analyze if this regime
0.2
of couplings is of relevance for real materials, such as
)
heavyfermions. Ab-initiocalculationsareneededforthis j
(
purpose (beyond the scope of the present analysis). pin 0
s
C
III. RESULTS -0.2
DMRG
J=1.5 U =-0.5
A. Magnetic Properties at ρ=2 eff fit
-0.4
0 5 10 15
j
Letusstartouranalysisbyfocusingonthedensityρ=
2. To investigate the magnetic order at this density, the
Fourier transform of the real-space spin-spin correlation Figure 1: (Color online). (a) Spin-structure factor S(qx) vs.
function was measured: qx for the two-leg ladder system with sizes L = 12,16, and
24, and for density ρ = 2, J = 1.5, and Ueff = −0.5. (b)
S(q)= 61L X eiqx(j−j′)eiqy(λ−λ′)hSj,λ·Sj′,λ′i, San(πd,U0e)ffa=s a−f0u.n5c.tTiohneoifnsJetfosrholawdsdtehrsewmiatghnliitnuedaerosfiztehsisLp=eak8
j,j′,λ,λ′ as a function of Ueff, for the coupling J =1.5. (c) Spin-spin
(3)
correlation Cspin(j) vs. j along the long ladder direction, for
where Sj,λ =Sj,x,λ+Sj,y,λ. In Fig. 1(a), the spin struc- a system with size L=24, and for thecouplings Ueff =−0.5
ture factor S(q) of the two-leg model is presented for and J =1.5. The dashed curveis a fit given by Eq.(5).
several system sizes. As can be observed in this figure,
there is a robust peak at wavevector q=(π,0), showing
thetendencytowardastripe-likeAFMorder. Thisisthe
analog of the magnetic order found in pnictides but us-
ing a two-leg ladder geometry. Similar results, obtained The results for the spin structure factor show that
with Exact Diagonalization on small clusters, were re- along the y(x)-axes the spins are aligned following a
portedbefore.12,13 Note thatthe peakincreaseswiththe ferromagnetic (AFM) order, at least at short distances.
system size, suggesting the development of a true long- Thisstripe-likeAFMstructureispresentinawiderange
rangemagneticorderinsystemswithahigherdimension. of parameters, as shown in Fig. 1(b) (inset), including
4
U > 0. Neutron scattering measurements for pnic- afinitecluster,andthiswouldbeindicativethateffective
eff
tides also show a similar spin order.31 We have observed attractive forces are present in the system.
that the stripe-like AFM structure appears only when Beforepresentingournumericalresultsforthebinding
the plaquette-diagonal hopping amplitudes (t and t ) energies,letusfirstconsiderthefollowingstrongcoupling
3 4
areofvalue similarasthose ofthe nearest-neighborhop- regime defined by U = 3J U = J U′ 1. As
eff
− − − ≫
ping amplitudes. If we force t = t =0 then the peak it will be argued below, and as it was discussed in pre-
3 4
in the spin structure factor S(q) appears at wavevector vious investigations for one-dimensionalchains,11 in this
q=(π,π). Note also that for the two-leg geometry, the regime the binding of hole/electrons is clearly present.
(π,0) AFM state is not, naturally, degenerate with the For completeness, let us address this limit in detail,
(0,π) AFM state. Due to this fact, a study in a ladder although it is clear that large J compared with U′ leads
geometry could make a better connection with the two- to aneffective attractiveinteraction.11 Letus denote the
dimensionalresultsofthepnictidematerialswhere(π,0) statesoftheone-siteproblemviathesymbol sy ,where
is favored over (0,π) by a lattice distortion. (cid:16)sx(cid:17)
the “arrows” s (s ) represent the electrons (with their
x y
For quasi-one-dimensional systems, a true long-range
spins projections) at the orbital x(y). For this one-site
magnetic order is replaced by a power-law decay of the
problemwithtwoelectronsandthelimitconsideredhere
spin-spin correlations. Thus, in order to analyze the
where J is large, the ground state energy is degener-
range of the magnetic order, we have also investigated
ate. Its value is e = U , and the three corresponding
2 eff
the spin-spin correlation function along one of the legs
eigenstates are ↓ , ↑ , and [ ↑ + ↓ ]/√2. Since
(say, leg 1) defined as (cid:16)↓(cid:17) (cid:16)↑(cid:17) (cid:16)↓(cid:17) (cid:16)↑(cid:17)
U 1, we can approximate the Hamiltonian as
eff
Cspin(l)= M1 X (cid:10)Siz,1Sjz,1(cid:11), (4) Ha−te)=grHo≫kuinnedticst+ateHiennter∼gyHoifntt,haentdwoth-leeg(hmigohdleyl dbeegcoenmeers-
|i−j|=l
E(0)=2L e =2L U .
2 eff
× ×
where M is the number of site pairs (i,j) satisfying l = Below, the dominant spin arrangement in the ground
state at density ρ=2 is shown to guide the discussion
i j . Inpractice,wehaveaveragedoverallpairsofsites
| − |
separated by distance l, in order to minimize boundary
↓ ↑ ↓ ↑ ↓
effects (afew sites atthe edgeswerealsodiscardedwhile (cid:16)↓(cid:17) (cid:16)↑(cid:17) (cid:16)↓(cid:17) (cid:16)↑(cid:17) (cid:16)↓(cid:17)
,
implementing this averagingprocedure). ↓ ↑ ↓ ↑ ↑
In Fig. 1(c), the spin correlation function C (j) is (cid:16)↓(cid:17) (cid:16)↑(cid:17) (cid:16)↓(cid:17) (cid:16)↑(cid:17) (cid:16)↑(cid:17)
spin
shown for the 2 24 cluster and using U = 0.5. The
× eff − using only the up and down projections of the spin one
dashedlineisafitofthenumericaldatawiththefunction
states at every site for simplicity. Such a state is to be
cos(πx) expected since for J 1 the alignment of the two spins
C˜spin =a x1/3 . (5) at the two orbitals i≫n the same site will occur, and as
U 2J 1 then having two electrons in the same or-
Similar results were found for the 2 16 cluster. The ob- bit∼al is n≫ot allowed. Of course, any other configuration
×
served power-law decay suggests that a two-dimensional (suchasa fully ferromagneticstate)is alsoequallylikely
system with the same model and parameters would de- as the one shown in the figure for U 1. However,
eff
− ≫
velop long-rangemagnetic order at zero temperature. the hopping terms will lift the large degeneracy,and our
numericalresultsfor the spin-spincorrelationspresented
before indicate that hopping terms favorthe (π,0) AFM
B. Doping with two holes or electrons configuration, for a wide range of couplings in the un-
doped limit. For this reason, here we have chosen to
Let us now consider the effect of doping with charges present the dominant spin arrangement of the ground
this ladder system. If tendencies toward the pairing of state as a stripe-like AFM state, but it could be fully
theextrachargesareunveiled,theywouldbeanindicator FM as well. Regardless of this detail, the arguments we
that this model could become superconducting in a two will use below to obtain the energies E(n) will hold true
dimensionalgeometry,for the couplings here considered. both for a stripe-like AFM state, as well as for a FM
Letus startwith the calculationofthe binding energy state.
of two doped holes/electrons. This binding energy is de- Let us now add two extra electrons to the undoped
fined as ∆ = E(2)+E(0) 2E(1), where E(n) is the system. In the limit being considered here, where the
b
−
groundstate energywith (4L+n) holes/electrons(4L is hopping amplitudes are negligible, the best way for the
the number of electrons corresponding to the“undoped” systemtominimizeitsenergyistohavethetwoelectrons
limit where there is anelectronper orbitalandper site). located on the same site, since in this way less on-site
On a finite system the binding energy ∆ > 0 is posi- ferromagnetic links are broken. The on-site Hubbard U
b
tive if the electrons/holes do not form a bound state,5 energy penalization is the same whether the doubly oc-
while in the thermodynamic limit ∆ should vanish in cupiedorbitalsareatthesamesiteornot,andsinceU′is
b
the absence of pairing. On the other hand, if the extra negligible with respect to J in the limit considered,then
holes/electronsformaboundstate,then∆ <0evenon the(effectivelyattractive)Hundcouplingdeterminesthe
b
5
location of the extra charge, leading to the double occu- 10
(a)
pation of both orbitals at the same site.11 In this case,
2x3 two doped electrons
the dominant spin arrangementof the ground state is 8
FM
6
↓ ↑ ↓↑ ↑ ↓
(cid:16)↓(cid:17) (cid:16)↑(cid:17) (cid:16)↓↑(cid:17) (cid:16)↑(cid:17) (cid:16)↓(cid:17). J
↓ ↑ ↓ ↑ ↑ 4
(cid:16)↓(cid:17) (cid:16)↑(cid:17) (cid:16)↓(cid:17) (cid:16)↑(cid:17) (cid:16)↑(cid:17)
Once again, a stripe-like AMF backgroundis used, since 2 ∆b<-0.1
for J 1 and close to the density ρ = 2 our numerical total spin changes continuously
∼
data show (see Fig. 4(a)) that the (π,0) AFM order is
0
the dominant one, but it could have been FM as well. 0 0.2 0.4 0.6
3J-U
In the limit of couplings considered here, the ground
state energy for the doped two-electronsystem is
(b)
E(2)=(2L+5)Ueff +8J. (6) 8 2x3 two doped holes
Note that if the two extra electrons that are in the state
6
↓↑ were to move in opposite directions (after consid- FM
(cid:16)↓↑(cid:17)
ering the presence of small but nonzero hopping terms) J4
theywouldbreaktwoferromagneticon-sitelinks. Sucha
∆ <-0.1
statewouldhavealargeenergyanditistherefore“forbid- b
2
den”for U 1. However, if the two extra electrons
eff
move“tog−ether≫”(i.e. ↓) in the same direction,forming a
↓ total spin changes continuously
spin triplet, then no other on-site FM links are broken. 0
Thus,to minimize the energythe two-electronsaddedto 0 0.2 0.4
3J-U
the systemmustformaboundstate,atleastinthe limit
where the hoppings amplitudes are very small compared
with the Hubbard and Hund couplings (the same spin (c)
0.4 binding of electrons U =-1.0 J =1.5
dominantpictureworkswheninsteadofaddingelectrons eff
binding of electrons U =-1.0 J=3.0
we remove two electrons, i.e. for the two-hole problem). eff
0.2 binding of holes U =0.0 J=5.0
Thisargument,knownfrompreviousinvestigationsusing eff
chains,11 explains the binding of electrons in the limit binding of holes Ueff=-0.5 J=3
0
Ueff 1. More explicitly, the binding energy of the b
− ≫ ∆
dopedtwo-electrons/holessysteminthelimitconsidered
here is given by -0.2
∆ = U . (7)
b −| eff| -0.4
Based on the discussion above we conclude that for 0 0.1 0.2 0.3
U 1,there isanindicationofpairing,andperhaps 1/L
eff
− ≫
superconductivity, in the two-orbitalmodel. In fact, the
on-site interorbital pairing state found here in this ex- Figure 2: (Color online). (a)-(b) Phase diagram (J vs.
treme regime results to be a spin triplet and transforms −Ueff =3J−U)forthe2×3clustershowingtheregionwhere
the binding energies of electrons (a) and holes (b) is smaller
according to the irreducible representation A of the
group D .32 Interestingly, Exact Diagonalizati2ogncalcu- than ∆b < −0.1, indicative of pairing. Above the red line,
4h the total spin saturates to its maximum value. Below this
lations in two-dimensional clusters, still in the FM state
line, the total spin changes continuously from the maximum
of the model but U′ > J, found indications of a pairing
value to zero at J=0. The blue lines show the region of J
state with the same characteristics and symmetry but where ∆b < −0.1 for Ueff = −0.5. These lines are the same
withelectronsatdistanceofonelatticespacingfromeach aspresentedinFig.3. (c)∆b vs. 1/Lforsomecouplings(see
other.12,13 The argument presented above for this pair- legend).
ing is actually valid in any dimension, and also it works
for the two doped holes case, as already mentioned. To
confirmtheseargumentations,∆ wascalculatednumer- relevance for real materials. The answer to this ques-
b
ically for large values of U . An excellent agreement tion appears to be positive. In fact, the binding of elec-
eff
−
between the numerical data and the analytic expression trons/holes has been observed numerically for U . 0
eff
(Eq. (7)) was found. andseveralvaluesofthecouplingJ,asshowninFig. 2.33
Nowthe crucialquestioniswhetherthereisbindingof More specifically, Figs. 2(a)-(b) show the region in the
holes/electrons for values of U that may be of more J (3J U)planewherethebindingenergiesofelectrons
eff
− − −
6
15 C. Phase Diagram
U =-0.5 2x2
eff 2x3 InFig.3thephasediagram(J vs. density)ofthetwo-
2x4
leg ladder for U = 0.5 and several system sizes (see
10 2x6 eff −
binding 2x8 legend) is presented. For ρ=2, it was observedthat the
FM 2x12 totalspinis zerofor L even,and thatit canbe 0,1, or 2
J
for L odd, depending of the values of J. Note that for L
FM odd and ρ = 2, the (π,0) AFM configuration (+-+-+-)
5
does not have the same number of + and - spins.
For 2 < ρ < 2.5 (1 < ρ < 2), a ferromagnetic phase
wasfound (the regionabovethe symbols)with magnetic
binding
moment per site given by m = 2 ρ/2 (m = ρ/2). The
01 1.5 ρ2 2.5 3 symbolsindicatethevalueofJc w−herethetotalspinsat-
urates. The criticalvalue J wasdeterminedby the level
c
crossingoftheenergiesinthesectorwithSz =Smax and
Figure3: (Coloronline). (a)Phasediagramofthetwo-leglad- total
Sz = Smax 1. Using this procedure, we were able to
dermodelHamiltoniandefinedinEq.1. Theregionabovethe total −
symbols is a fully-saturated ferromagnetic phase (see text). obtain Jc for large systems. For densities in the ranges
Thebluelinescorrespondtoregionswhere∆b <−0.1forthe ρ . 1.25 and ρ & 2.5, we have not found any trace of
cluster 2×3 with two doped electrons/holes, as presented in ferromagnetism. Below the FM region, it is very hard
Figs. 2(a)-(b). numerically to determine the total spin with good accu-
racy for large systems. However, our results for the 2 2
×
and2 3clusterswithtwodopedelectrons/holessuggest
×
thatthe totalspinchangescontinuously,frommaximum
value at J to zero at J=0. The total spin can be ex-
(Fig.2(a))andholes(Fig.2(b))arelessthan-0.1,forthe c
tractedfromthespinstructurefactoratq=(0,0). Fora
case of a 2 3 cluster. The region where ∆ < 0.1 was
× b − few setsofcouplingswe alsoobserved,throughthevalue
chosentoberepresented,asopposedto∆ =0,sincepre-
b of S(q=(0,0)), that in fact the total spinof the ground
vious experiencein the contextofthe cuprates5 suggests
state changes continuously for the 2 8 cluster as well.
that this procedure effectively takes into account size ef- ×
These results suggest that the total spin varies continu-
fects better. In practice, other values for this“cutoff”do
ouslybelowthe FMregionpresentinFig.3 foranyclus-
notalterourqualitativeconclusions. Similarresultswere
ter sizes. Overall, our results are qualitatively compati-
found also for the 2 2 cluster.
× ble with those found in one-dimensionalsystems.11 Note
also that tendencies to FM states at robust J were also
Thus far, only small clusters have been considered be-
reported via Exact Diagonalization methods on small
cause the numerical analysis of large clusters would be
clusters.12,13
too time-consuming, particularly with regards to calcu-
We believethe FMphaseis stabilizedby amechanism
latingthehundredsofpointsthatarerequiredtoextract
thathasthesamecharacteristicsastheDoubleExchange
comprehensive phase diagrams. However, larger system
(DE) mechanism.34 In the original DE scenario, there
sizes were consideredfor a few selected sets of couplings,
are mobile and localized degrees of freedom. In the DE
as shown in Fig. 2(c). In this figure, ∆ vs. 1/L for
b
mechanism these degrees of freedom are separated and
some couplings is presented. Here, it is clearly observed
well defined. Although we do not have localized degrees
that in the bulk limit the binding energiesof addedelec-
of freedom in our model, from the perspective of one
trons/holesconvergetononzerovaluesforsomecoupling
electronatagivenorbitalanelectronatthesamesitebut
sets. Close to ρ =2, these results strongly indicate that
the other orbital behaves in some respects as a localized
there are pairing tendencies for U . 0 . Thus, to the
eff
spin. For doped systems, when an electron moves from
extendthatfutureinvestigationsshowthatJ comparable
to U′ is a realistic regime for effective two-orbital mod- one site to the other, in order to minimize the kinetic
energy and the energy related with the Hund coupling,
els,thisprovidesapossiblemechanismforpairinginreal
all spins have to be aligned.
materials.
The blue lines in Fig. 3 are the same that were pre-
In Figs. 2(a)-(b), the magnetic phase diagram for the sented in Figs. 2(a)-(b). We expect that the region of
case of two doped electrons/holes is also presented. The binding extends beyond these lines up to the density
regionabove the red (bold) line is a ferromagneticphase ρ = 2, forming regions in parameter space where super-
withthemaximumtotalspinS =2L 1. Belowthis conductivity exists inside the phase diagram.
total
−
line, we have observed that the total spin changes con- We have also measured the spin structure factor S(q)
tinuously from zero, at J = 0, up to its maximum value away from the undoped density ρ = 2. In Fig. 4, S(q)
at J . As observedin these figures and for a largeregion is presented for some particular densities for the two-leg
c
ofcouplings,pairing(andpresumablysuperconductivity) laddermodelwithsizeL=16,J =1.5,andU = 0.5.
eff
−
co-exists with ferromagnetic tendencies. As can be observed in Fig. 4(a), there is still a peak at
7
7
(a) tendencies. Note that for the electron doped case, there
q =0 ρ=15/8 is a small peak at q=(0,π) for densities ρ&2.2.
6 y
q = π ρ=15/8
5 y
U =-0.5 q =0 ρ=33/16
)x4 eff qy= π ρ=33/16 IV. CONCLUSION
q
y
S( 3 J=1.5 q =0 ρ=17/8
y Using ladders, we have studied analytically and nu-
2 qy= π ρ=17/8 merically a two-orbital Hubbard model. Via the DMRG
technique we were able to investigate the model defined
1
onatwo-legladdergeometryforsystemswithlinearsizes
0 up to L = 24. Our spin structure factor data show that
1 1.5 2
π
q / forthe“undoped”densityρ=2,astripe-likeAFMorder
x
ispresent,asobservedinpreviousExactDiagonalization
3
(b) studies.12,13 We have also presented evidence for triplet
q =0 ρ=7/4
pairing tendencies of added electrons/holes close to the
y
Ueff=-0.5 q = π ρ=7/4 density ρ = 2, in some range of couplings, in qualitative
2 qy=0 ρ=9/4 agreement with previous investigations using chains,11
) J=1.5 y andwithExactDiagonalizationcalculationsina less ex-
qx qy= π ρ=9/4 treme FM regime of models for pnictides.12,13 More pre-
(
S cisely, we have found that pairing (and presumably su-
1 perconductivity)and ferromagnetismco-existfor a large
region of parameters in the regime U′ < J. Even for U′
comparable to J our results still indicate a (mild) ten-
dency to pairing. Whether this rangeofcouplings for U′
0 1 1.5 2 andJ isrealizedinrealmaterials,suchasheavyfermions
π
q /
or pnictides, is a matter to be decided via experiments,
x
or with the help of ab-initio computer simulations.
Figure 4: (Color online). Spin-structure factor S(qx) vs. qx
for the two-leg ladder system with size L=16, J =1.5, and
Ueff =−0.5. (a) S(qx) for the densities ρ=15/8, ρ=33/16, Acknowledgments
andρ=17/8(seelegend). (b)S(qx)forthedensitiesρ=7/4
and ρ=9/4.
This research was supported by the Brazilian agen-
cies FAPEMIG and CNPq, the National Science Foun-
q = (π,0) for densities close to ρ = 2. Note that these dation grant DMR-0706020, the Division of Materials
peaks have smaller intensity than those found for ρ = 2 Science and Engineering of the U.S. Department of En-
in Fig. 4(a), for the system with size L = 16. We have ergy, and the Center for Nanophase Materials Sciences,
also observed that the height of the peak at q = (π,0) sponsored by the Scientific User Facilities Division, Ba-
increases with the system sizes for the densities close to sic Energy Sciences, U.S. Department of Energy, under
ρ = 2. These results indicate that a stripe-like AFM contract with UT-Battelle. The authors are grateful to
magnetic order also exists for densities close to ρ = 2. Maria Daghofer for providing us with Exact Diagonal-
As shownin Fig. 4(b), this orderdoes notexistanymore ization data to compare our DMRG results against and
for ρ & 2.2 and ρ . 1.7, at least within the precision to Fernando Reboredo and Satoshi Okamoto for useful
of our calculations, and it is replaced by ferromagnetic comments.
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