Table Of ContentAbsence of a true long-range orbital order in a two-leg Kondo ladder
J. C. Xavier1 and A. L. Malvezzi2
1Instituto de F´ısica, Universidade Federal de Uberlˆandia,
Caixa Postal 593, 38400-902 Uberlˆandia, MG, Brazil
2Departamento de F´ısica, Faculdade de Ciˆencias,
Universidade Estadual Paulista, Caixa Postal 473, 17015-970 Bauru, SP, Brazil
(Dated: February6, 2008)
7
0
Weinvestigate,throughthedensity-matrixrenormalizationgroupandtheLanczostechnique,the
0
possibilityofatwo-legKondoladderpresentanincommensurateorbitalorder. Ourresultsindicatea
2
staggeredshort-rangeorbitalorderathalf-filling. Awayfromhalf-fillingourdataareconsistentwith
n an incommensurate quasi-long-range orbital order. We also observed that an interaction between
a thelocalized spinsenhances therung-rungcurrent correlations.
J
9 PACSnumbers: 71.10.Pm,75.10.-b,75.30.Mb
1
] I. INTRODUCTION t
l
e
-
r In 1985, it was observed that the heavy fermion su-
t J
s perconductor URu2Si2 presents a second order phase
. transition at 17.5K.1 This phase transition is character-
t
a ized by sharp features in the specific heat1 and several
m
others thermodynamic properties (see, e.g., Ref. 2 and
Localized Conduction
- References therein). The large entropy loss associated
d
in this phase transition is equivalent to an ordered mo-
n
o mentofabout0.5µB.However,the sizeofstaggeredmo- Figure 1: (Color online) A schematic representation of the
c ment measured by neutrons scattering measurements is two-leg Kondo ladder. It also shows the circulating currents
[ m ∼ 0.03µ .1 The order parameter associated with this around theplaquettes(in this examplea staggered one).
B
phase transition is, at the present moment, not estab-
2
v lished and it is challenging to discover the nature of the
0 hidden order behind the transition.
0 Manytheoreticalgroupshaveproposedseveralkindsof model (KLM). This model is the simplest one believed
1 hidden order.2,3,4,5,6,7 But, until now, experiments were to presentthe physics of heavy fermions materials10 (see
8
0 not able to establish which is the correctone. Certainly, next section). Our approach will be numerical, through
6 also from the theoretical point of view, more studies are thedensity-matrixrenormalizationgroup(DMRG)11and
0 needed to clarify the correct order associated with this the Lanczos technique.12 These techniques are non-
/ mysterious phase transition. In this front, we present perturbative, however limited by the system size. For
t
a here a numerical study of a microscopic model for the this reason, we consider the two-leg Kondo ladder (2-
m
heavy fermion systems. LKL), which is the simplest geometry able to present an
- Inthisworkwefocusontheorderparameterproposed orbital order.
d
afewyearsagobyChandraandcollaborators.2Theysug-
n
The orbital order, also called flux or orbital current
o gested the existence of a hidden incommensurate orbital
phase, has already been discussed in the context of the
c order in the heavy fermion URu2Si2 below the second
: orderphasetransition. Theorbitalorderphaseisassoci- hightemperaturesuperconductors. Thestandardtwo-leg
v t-J ladders model present a short-range orbital order,13
i ated with currents circulating around the plaquettes, as
X while an extended version has long-range orbital order
illustrated in Fig. 1. In the case of URu Si , this cur-
2 2 for some parameters.14 A recent detailed discussion of
r rents produce a very week orbital moment 0.02µ that
B
a explains the large entropy loss.2 the orbital order in the context of a Hubbard model can
be found in Ref. 15.
Very recently, neutron scattering measurements were
unable to detect the orbital order in the heavy fermion Weclosethissectionmentioningthatamodelverysim-
URu2Si2.8 Although the orbital order was not detected ilar to the KLM was used to describe the magnetism of
itisnotpossibleyettodiscarditasthehiddenorderdue URu Si . Sikkemaand collaborators16, througha mean
2 2
totheresolutionlimitationoftheexperimentsperformed. field calculation, showed that the Ising-Kondo lattice
Note that the orbital order is expected to be 50 times model with transversefield presents a weak ordered mo-
smaller than the spin order.9 ment, similar to the one observed in experiments. How-
Our goalin this work is to investigate the existence of ever,theIsing-KLMmodelwasnotabletoreproducethe
an incommensurate orbital order in the Kondo Lattice large specific heat jump.
2
II. MODEL der(LRO)isexpectedduetheRuderman-Kittel-Kasuya-
Yosida interaction, whereas for large J a paramagnetic
In order to investigate the heavy fermion systems the phase emerges. Doniach21 was the first to point out the
existenceofaquantumcriticalpoint(QCP)duethecom-
minimumingredientsthatamicroscopicmodelmustcon-
sideraretwotypes ofelectrons,the conductionelectrons petition between these two phases.
in the s-, p-, or d-, orbital as well the electrons in the Unlikeothermodels,suchasthet−J model,muchless
inner f-orbitals.17 In the literature there are two well isknownabouttheKondolatticemodel. Evenintheone
known standard models that consider these two kind of dimension version, where the ground state of the Kondo
electrons, the periodic Anderson model (PAM) and the chain is quite well known17 (see also Ref. 22). New
KLM.17 In an appropriate parameter regime (mainly (i) phases have been reported recently, such as a new ferro-
the mobility of the f electrons is very small, which is magnetic phase23 inserted into the paramagnetic phase
relevant for the heavy fermion system and (ii) that the aswellas a dimerizedphaseatquarter-filling.24 The lat-
Coulomb interaction of the electrons in the f orbitals terhasbeenquestionedrecentlybyHottaandShibata.25
is very large) Schrieffer and Wolff17,18 showed that the Those authors claim that the dimerizedphase is an arti-
KLM can be derived from the PAM. We consider in this factoftheopenboundaryconditions. Indeed,thebound-
worktheKLMwhichhaslessdegreesoffreedomperunit aryconditionisveryimportant,aswellasthenumberof
cellthanthePAManditiseasiertoexplorenumerically. sitesconsidered. InRef. 25theauthorsobserved,mainly,
TheKLMincorporateaninteractionbetweenthelocal- that with an odd number the sites the dimer state does
ized spins and the conduction electrons via exchange in- not exist. The parity of the number of sites is thus very
teractionJ. To attackthis model intwoor three dimen- relevantandanoddnumberdestroysthedimerization.26
sion by unbiasednon-perturbative numericalapproaches In quasi-one-dimensional systems, such as the N-leg
is an impossible task at the present moment. However, ladders, very few non-perturbative studies have been re-
it is possible to consider quasi-one-dimensional systems ported. Recently, quantum Monte Carlo27 (QMC) and
such as the N-leg ladders model. DMRG28 calculations of the half-filled Kondo lattice
We consider the 2-LKL with 2xL sites defined by modelinsmallclustersfoundtheexistenceofaquantum
criticalpoint (QCP)atJ ∼1.45,inagreementwith pre-
H =− (c† c +H.c.)+J S ·s vious approximated approaches29,30 (see also Ref. 31).
KM X i,σ j,σ X j j
Note that the QMC calculations were feasible only at
<i,j>,σ j
half-filling, where the famous sign problem is absent.
Moreover,theDMRGresultsoftheN-LKLathalf-filling
+J S ·S , (1) show that the spin and charge gaps are nonzero for any
AH X i j
number of legs and Kondo coupling J. These results
<i,j>
arequite differentfromthe wellknownN-legHeisenberg
wherec annihilatesaconductionelectroninsitej with ladders were the spin gap is zero for an even number of
jσ
spin projectionσ, S is a localized spin 1 operator,s = legs.32
j 2 j
1 c† σ c istheconductionelectronspindensity The phase diagram of the 2-LKL has also been ex-
o2pPerαaβtorj,aαndασβ j,βarePaulimatrices. Here<ij >denote plored numerically.33 In this case, a ferromagnetic phase
αβ
nearest-neighborsites,J >0(when the KLMisdeduced was observed only for small densities, very distinctively
from the PAM obtain J > 017) is the Kondo coupling from the phase diagram of the 1D Kondo lattice chain,
constant between the conduction electrons and the local wheretheferromagnetismispresentatallelectronicden-
moments and the hopping amplitude was set to unity to sities for large values of J. However, it is similar to
fix the energy scale. the mean field phase diagram of the 3D Kondo lattice
We also consider an interaction between the localized model.34 In this sense, the 2-LKL presents a better sig-
spins J , we choose J >0 since antiferromagnetims nature of the phases appearing in real systems than its
AH AH
hadbeenobservedinURu Si .19 Thesamemodelabove one-dimensional version. Interesting that it was also ob-
2 2
alsorepresentsthemanganiteswhenJ <0.20 Inthislat- served dimerization in the 2-LKL33 at conduction elec-
tercase,theinteractionbetweenthelocalizedspinsseems tron densities n = 1/4 and n = 1/2. As in the one-
to be important to stabilize some phases.20 This is the dimension version, the RKKY interaction explains these
motivation to also consider this interaction. Note that unusual spin structures. In fact, in some real heavy
severalotherstermsinthe Hamiltoniancouldalsobe in- fermionsystemssomeunusualspinorderstructureshave
cluded, like the Coulomb interaction of the electrons in indeed been observed.35
the conductionband, extraelectrons hopping,etc. How- Here,weconsiderelectronicdensitiesnlargerthan0.4,
ever, at the present moment, there are no evidences in- where a paramagnetic phase have been observed.33 In
dicating that such extra terms are relevant to the low particular,wefocusontheelectronicdensitiesn=1and
energyphysicsoftheheavyfermionsystems. Uptonow, n = 0.8. We choose these densities since the magnitude
it is well established that J is essential to describe the of the rung-rung current correlation is bigger for larger
magnetism observed in the heavy fermion systems. At electronic densities. We investigate the model with the
small values of J, an antiferromagnetic long-range or- DMRG technique under open boundary conditions and
3
use the finite-size algorithm for sizes up to 2×L= 120, boundary conditions it is convenient to define an aver-
keeping up to m = 1600 states per block in the final aged rung-rung current correlation in order to minimize
sweep. The discardedweightwastypicallyabout10−5− boundary effects. We have defined the averaged rung-
10−7inthefinalsweep. Wealsocross-checkedourresults rung current correlationas
with Lanczos technique for small systems.
1
C(l)= Jˆ(i)Jˆ(k) , (3)
M X D E
III. RESULTS
|i−k|=l
where M is the number of site pairs (i,k) satisfying l =
Before presenting our results, we briefly discuss the
|i−k|. Typically, M in our calculation vary from 3 to
order parameter associated with a circulating current
10.
phase. Such a phase breaks rotational, translational as
There is a true long-range orbital order if
well as time reversal symmetries. The appropriated or-
lim C(l) 6= 0. Through this criterion, we can
derparametertodetectthisphaseisthecurrentbetween l→∞
infer about the existence of the orbital order by measur-
two nearest-neighbour sites, i. e., Jˆ where the cur-
D l,jE ing C(l) at large distances. If C(l) has an exponential
rent operator between two nearest-neighbours i and j is decay, the linear-log plot shows a linear decay. On the
given by other hand, if C(l) has an power law decay, the log-log
plot present a linear decay.
We also measure the cosine transform of C(l), i. e.,
Jˆ =i (c† c −c† c ),
l,j X l,σ j,σ j,σ l,σ
σ
L
Strictly speaking, a spontaneous symmetry breaking N(q)= C(l)cos(lq),
X
only appears in the thermodynamic limit. Only in this
l=1
limit Jˆ 6= 0 in the ordered phase. The signature
D l,jE in order to infer about periodicity of the oscillatory part
of a spontaneous symmetry breaking appears in the two of C(l).
point correlation function of the operator that measures Inthe nexttwo subsectionswe investigatethese corre-
the symmetry. We utilize this fact to infer about the or- lationsforthe2-LKLathalf-fillingandclosetohalf-filing,
bitalorder. Ifacontinuoussymmetryisbroken,nolong- respectively. We did not find any evidence of long-range
range order exist at finite temperature in one and two orbital order in the ground state of the 2-LKL. Our re-
dimensions,asstatedbytheMermin-Wagner-Hohenberg sults support that the rung-rung current correlation has
theorem.36 At zero temperature a true long-range order an exponential decay at half-filling. Close to half-filling
is still possible in two dimensions, while in one dimen- our results indicate an incommensurate quasi-longrange
siononlyaquasi-long-rangeordercanmanifest,i. e.,the orbital order.
twopointcorrelationfunctiondecayalgebraic. However,
if the translational symmetry (a discrete symmetry) is
broken, even in one dimension a true long-range order A. Half-filling
may exist (a famous example is the dimerized phase of
the Majumdar-Ghosh model37,38,39 ). Since the transla-
We start presenting some results for the conduction
tional symmetry is broken in the orbital phase, a true
density n = 1. We observed, in this case, that the aver-
long-range order may occurs in the ground state of the
aged rung-rung current correlationbehaves as
2-LKL.
In order to observe any trace of orbital order in the
ground state wave function of the 2-LKL, we measure C(l)=a (−1)lexp(−l/ξ), (4)
0
the rung-rung current correlations defined as
forallvaluesofJ andJ exploredinthiswork. InFig.
AH
2(a), we present a typical example of the magnitude of
C(l,k)= Jˆ(l)Jˆ(k) , C(l) at half-filling for a system size L = 30. As we see,
D E
ourresultsindicatestronglythatC(l)hasanexponential
where Jˆ(l) is the rung current operator for the lth rung decay due to the linear decay in the linear-log plot. The
inset in Fig. 2(a) also shows that C(l) is staggered. The
given by
solid line in Fig. 2(a) correspond to a fit of Eq. 2 with
a = 0.16 and a decay length ξ = 1.43.40 We performed
0
Jˆ(l)=i (c† c −c† c ), (2) a least-squares fitting, resulting in a root mean square
X l2,σ l1,σ l1,σ l2,σ (RMS) of 0.0018 and a correlation coefficient of 0.996.
σ
We found that C(l) has a very small dependence on the
andc annihilatesaconductionelectrononrungl and numbermofstatesretainedinthetruncationprocessfor
lλ,σ
leg λ = 1,2 with spins σ =↑↓. Since we work with open J > 0.8, as can be observed in Fig. 2(a). For J <0.8 is
4
(a) (a)
0 0
10 10
10-4 J =0.0 L=30
n=1 AF
0
10-4 C(i) -6
10
i)| m=600 8 i 12 )| n=1
C( m=1000 C(i J=0.35
|10-8 fit |-12 J=0.8
J=0.8 J =0.0 L=30 10 J=1.8
AF
0 10 20 30 0 5 10 15
i i
(b) (b)
0.4 100
J=0.35 L=30
0.2
n=1
)
q -3
N(0 n=1 10
| J =0.0
)
i AF
( J =1.0
-0.2 C AF
J=0.8 J =0.0 L=30 | J =2.0
AF -6 AF
0 0.5 1 π1.5 2 10 0 10 20 30
q/ i
(c)
(c)
0
0 10
10
J=1.8 n=1 L=30
n=1
J=0.35 J =0.0
AF 10-7
-4
10 |
)
)| i J =0.0
(i C( AF
C L=30 J =1.0
|-14 AF
| L=60 10 J =2.0
10-8 AF
0 10 20 30
0 10 20 30 40 50 60 i
i
Figure 3: (Color online) The linear-log plot of |C(l)| for a
Figure 2: (Color online) (a) The linear-log plot of |C(l)| for
set of representative values of J and JAH for L=30 at half-
two distinct valueof m with L=30 at half-filling. The solid
filling. (a) |C(l)| for JAH =0 and J =0.35, 0.8 and 1.8. (b)
line in Fig. 2(a) correspond to a fit of Eq. 2 with ξ = 1.43
|C(l)| for J =0.35 and some values of JAH. (c) Same as (b)
anda0 =0.06, theRMSpercenterroris 0.18. Inset: C(l) vs butfor J =1.8.
distance with m = 1000. Only few sites are presented. The
couplingsareJ =0.8andJAH =0. (b)Thecosinetransform
N(q) of C(l) presented in Fig. 1(a) with m = 1000. (c)
The linear-log plot of |C(l)| for two distinct size, both with The signature of the sign alternation is observed
m=1000. The couplings are J =0.35 and JAH =0. through the cosine transform of C(l). In Fig. 2(b), we
show the cosine transform of C(l) present in Fig. 2(a)
with m = 1000. Clearly, we observed a peak at q = π
veryhardto get accurateresults,howeverevenfor small duethe signalternationofC(l). Notethatthe finite-size
J we believe to havecapturedthe correctqualitative be- effects are small, as can be seen in Fig. 2(c). For this
haviour. Nevertheless,we present mostof our results for reason, we restrict most of our calculus to system size
J >0.8, where the results are more accurate. 2x30 in order to save computational time.
5
(a)
0
10
0.4 J=0.8 J =1.0 n=0.8 L=30
AH
L=30 n=1
)
π m=1000
( m=1600
N
J=0.8 10-4 fit
0.2 J=1.8 |
) 0.03
C(i (i)
C0
0 2 4 J 6 8 10 |
-8
AH 10 4 i 8
1 10
i
Figure 4: (Color online) Thepeak intensity of N(q) at q=π
for J = 0.8 and J = 1.8 as a function of JAH for a system
(b)
size L=30 at half-filling.
0
10
n=0.8
OurresultsindicatethatforsmallJ,wheretheRKKY
is expectedto be dominant,the rung-rungcurrentcorre-
J=0.8 J =1.0
lations has a bigger correlation length as we see in Fig. AF
3(a). On the other hand, for large J, which favors for- -3
10
mation of singles, the correlation length is smaller. This
|
resultisexpected,sinceforJ →∞therung-rungcurrent )
i L=30
correlations must go to zero. (
C L=40
The Hamiltonian Eq. (1) with J =0 does not lead
AH |
to a long-range orbital order at half-filling, as we have -6
10
observed. Since J seems to be important to stabilise
AH
some phases for J <020 , it may be possible that it also
1 10
stabilises the orbital phase for J > 0. For these reason, i
we also investigate the effect of J in the ground state
AH
of the 2-LKL. As we see in Fig. 3(b), for small values of
J, JAH does not affect significantly C(l). On the other Figure 5: (Color online) The log-log plot of |C(l)| for the
hand, for largerJ as shownin Fig. 3(c), JAH clearly en- conductiondensityn=0.8. (a)|C(l)|vs distanceforJ =0.8,
hances the length correlation. Although JAH enhanced JAH = 1.0, and L = 30 for two distinct value of m. The
C(l), at half-filling only short-range orbital order is ob- dashed line is the fit using Eq. (1) with α1 = 2, α2 = 3,
served for several parameters investigated. a0 =0.16 and a1 =−0.33, the RMS per cent error is 6 with
a correlation coefficient of 0.998. The solid line is guide by
At half-filling,for all parametersstudied, N(q) always
eyes. (b) |C(l)| for the same parameters of Fig. 5(a) and
presents a peak at q = π. In Fig. 4, we present this
two distinct sizes, both cases with m= 1600. Inset: C(l) vs
peak intensity for J = 0.8 and J = 1.8 as function of
distancewith m=1600. Only few sites are presented.
J . As we see, the peak intensity increases with J
AH AH
and saturates for large J around ∼0.45.
AH
Ourmainconclusion,forthehalf-fillingcase,isabsence
thetruncationprocess. Althoughweobtainedresultsfor
of long-rangeorbitalorder. Note that it may be possible
afewdensitiesawayfromhalf-filling,wefocusondensity
that the inclusion of the Coulomb interaction between
n=0.8 where the magnitude of C(l) is larger. For small
the electrons in the conduction band leads the system
densities is very hard to get accurate results since the
into a phase with long-range orbital order, as occurs in
an extend t−J model.14This is under investigation in current intensity is very small. In Fig. 5(a), we present
thelog-logplotof|C(l)|atconductiondensityn=0.8for
the present moment by one of the authors.
a system size 2×30 with J =0.8 and J =1.0 for two
AH
different values of m. Since in the log-logplot we obtain
alineardecay(seethe solidline inthis figure)C(l)must
B. Close to half-filling
have a power law decay. If we use a linear-log plot our
data does not have a linear decay. As can be seen from
Away from half-filling the DMRG calculation of C(l) that Figure, it is very hard to get good accuracy even
is less stable, for this reason we consider system sizes working with m = 1600 states. Although we were not
smaller than 2x40 and keeping up to m=1600 states in able to obtain the current-current correlations at large
6
keptinthe truncationprocesssincewe alsoobtainedthe
same effect for small clusters with exact diagonalization.
0.2
J=0.8 J =1.0 In Fig. 6 we present N(q) for a representative set of
AF parameters at conduction density n = 0.8. As shown
) in that Figure, there is no peak at q = π, signaling an
q
( n=0.8 absence of staggeredrung-rung current correlations. For
N0 the conduction density n = 0.8 we observed a cusp at
q = nπ. These results indicate that close to half-filling
the 2-LKLpresentsanincommensurate quasi-long-range
orbital order.
-0.2
0 0.5 π 1
q/
IV. CONCLUSION
Figure 6: The cosine transform N(q) of C(l) presented in In this paper, we have investigatedthe possibility of a
Figure 5(a) with m=1600. two-legKondoladderpresentanorbitalorder. Inpartic-
ular,wefocusonthedensitiesn=1andn=0.8. Forthe
severalcouplingsinvestigatedwedidnotfindanytraceof
distances with a high accuracy, we believe to have cap- a true long-range orbital order, which would be relevant
tured the correct behavior, i.e., a power law decay. The to explain the large entropy loss observed in the second
large oscillations appearing in those Figures are due to order phase transition of URu2Si2 . Our data indicate
fact that some values of C(l) are very close to zero. thatthehalf-fillingcasepresentsa staggeredshort-range
Since our data ofC(l) in the log-logplotstronglysug- orbitalorder,whileclosetohalf-fillingourresultsarecon-
gest a power law decay close to half-filling (note that for sistent with an incommensurate quasi long-rangeorbital
the half-filling case the decay is exponential) we tried to order. Althoughwedidnotfindevidenceofalong-range
fit C(l) with the function orbital order in the ground state of the two-leg Kondo
ladder, we can not yet completely discard this possibil-
ity. It may occur that an extended versionof the Kondo
cos(nπl) cos(2nπl)
C (l)=a +a , (5) lattice model presents the long-range orbital order. So,
fit 0 lα1 1 lα2 we may conclude that either the orbital phase does not
exist and is not the origin of the mysterious phase tran-
where n = 0.8 is the density. The dashed curve in Fig.
sitionobservedinthe the heavyfermionURu Si orthe
5(a) corresponds to a fitting of our data with m=1600. 2 2
standard Kondo lattice Model is not able to reproduce
We were not able to reproduce precisely C(l), however
the correctorder observed in the experiments.
the general behavior is quite well described.
Note also that finite-size effect are larger away from
half-filing,aswecanseebycomparingtheFigs. 5(b)and
2(c). It is important to mention that we observed, away Acknowledgments
fromhalf-fillingandinveryfewdistancesl,thatthesign
of the averaged correlation C(l=|j-k|) does not has the The authors thank E. Miranda for useful discussions.
samesignofC(j,k)forsomepairsof(j,k)satisfyingl=|j- This workwas supportedby BrazilianagenciesFAPESP
k|. Thisdoes notseemtobe due to the numberofstates and CNPq.
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