Charge and spin excitation spectra in the one-dimensional Hubbard model with next-nearest-neighbor hopping

0.41 MB·English

by  S. Nishimoto

Charge and spin excitation spectra in the one-dimensional Hubbard model with
next-nearest-neighbor hopping
S. Nishimoto,1 T. Shirakawa,2 and Y. Ohta2,3
1Max-Planck-Institut fu¨r Physik komplexer Systeme, D-01187 Dresden, Germany
2Graduate School of Science and Technology, Chiba University, Chiba 263-8522, Japan
3Department of Physics, Chiba University, Chiba 263-8522, Japan
(Dated: February 6, 2008)
7
0
The dynamical density-matrix renormalization group technique is used to calculate spin and
0
chargeexcitationspectraintheone-dimensional(1D)Hubbardmodelatquarterfillingwithnearest-
2 ′
neighbortandnext-nearest-neighbort hoppingintegrals. Weconsideracasewheret(>0)ismuch
n smaller than t′ (> 0). We find that the spin and charge excitation spectra come from the two
a nearlyindependentt′-chainsandarebasicallythesameasthoseofthe1DHubbard(andt-J)chain
J at quarter filling. However, we find that the hopping integral t plays a crucial role in the short-

4 rangespinandchargecorrelations;i.e.,theferromagneticspincorrelationsbetweenelectronsonthe
2 neighboring sites isenhanced and simultaneously thespin-triplet pairing correlations is induced,of
which theconsequences are clearly seen in thecalculated spin and charge excitation spectra at low

] energies.
l
e
- PACSnumbers: 71.10.Pm,71.10.Fd,78.30.Jw,72.15.Nj,71.30.+h,71.45.Lr
r
t
s
.
t I. INTRODUCTION desired.
a
m As for real materials, this mechanism4 may be of pos-
Spin-tripletsuperconductivityhasbeenoneofthema- sible relevance to superconductivity observed in quasi-
-
d jorissuesinthefieldofcondensed-matterphysics. Nearly 1Dorganicconductor(TMTSF)2X[X=PF6, ClO4, etc.],
n alltheconventionalandunconventionalsuperconductors the so-called Bechgaard salts.6,7 The system exhibits a
o known to date are spin-singlet paired. The best-known rich phase diagram upon variation of the pressure and
c
example of triplet pairing is not a superconductor but a temperature. At low temperatures, the phase changes,
[
superfluid3He,whereatomicCooperpairsareformedin in the order, as the spin-Peierls insulator, antiferromag-

1 spin-tripletchannel.1 Onlyafewmaterialsofspin-triplet netic insulator, spin-density-wave (SDW) insulator, su-
v
superconductivityhavesofarbeenconfirmedinstrongly perconductivity, and paramagnetic metal, with increas-
9
correlated electron systems, which iclude ruthenium ox- ing pressure. So far, experimental evidences that the
7
5 ide Sr2RuO42 and some heavy-fermion compounds such superconducting state occurs in the triplet channel have
1 as UPt3.3 Here, some questionswill naturally arise. One been piled up,8 although not much is known on the na-
0 is whether the electron correlation can take an essential ture of the pairing. A newly synthesized copper-oxide
7 part in superconductivity carried by spin-triplet pairs. compoundPr2Ba4Cu7O15−δ9 mayalsobe arelatingsys-
0 Another is how the behavior differs from that of spin- tem. ThismaterialconsistsofboththesingleCuOchains
/
t singlet superconductivity. In this manner, research on (as in PrBa2Cu3O7) and the double CuO chains (as in
a
m spin-triplet superconductivity may offer an opportunity PrBa2Cu4O8), and those chains are separated by insu-
to expose unknown physical phenomena. latingCuO2 plains. Ithasbeenreportedthatthedouble
- chains turn into a superconducting state below T 10
d Quite recently, a new mechanism of the spin-triplet K.10 Although the signs of the hopping integralsc ∼seem
n superconductivity has been proposed in a fairly simple
o correlated electron system.4 The model consistes of two not to satisfy the ferromagnetic sign rule, the structure
c of the double CuO chains bears a certain similarity to
Hubbard chains coupled with zigzag bonds and has a
: our model.11
v unique structure of hopping integrals: sign of the hop-
i pingintegralschangesalternatelyalongthezigzagbonds The purpose of the present study is therefore to build
X
connecting two chains, while the sign along the one- up understanding of the ring-exchange superconducting
r
a dimensional (1D) chain is always negative. [A model mechanism by calculating dynamical quantities for the
where all the hopping integrals are taken to be posi- same model proposed in Ref. 4. To see the excita-
tiveisequivalentundercanonicaltransformation.] Under tions in the spin and charge degrees of freedom sepa-
this sign rule of the hopping integrals,the ring-exchange rately, we here calculate the momentum-dependent dy-
mechanism5 yields ferromagnetic spin correlations, and namical spin-spin and density-density correlation func-
accordingly, attractive interaction between electrons is tions. We use the dynamical density-matrix renormal-
derived. In Ref. 4, the argument was developed on the ization group(DDMRG) method for calculating dynam-
basisofonlythestaticpropertiessuchaspairbindingen- ical quantities,12 which is an extension of the standard
ergy, spin excitation gap, and pair correlation function, DMRG method.13 The obtained results with high res-
as well as spin correlation function. Therefore, further olutions enable us to discuss details of the fundamen-
investigations including dynamical properties have been tal properties on the spin and charge excitations. Thus,
2
we can find some interesting features in the low-energy where c† (c ) is the creation (anihilation) operator of
iσ iσ
physics of our model. an electron with spin σ (σ = , ) at site i, n = c† c
We will show that, although the spin and charge exci- isthe number operator,t′ and↑t↓arethe nearesiσt-neigihσboiσr
tation spectra are basically the same as those of the two andnext-nearest-neighborhoppingintegralsrespectively,
weakly-coupled 1D Hubbard (and t J) chains at quar- andU istheon-siteCoulombinteraction. Wechoosethe
−
ter filling, the small hopping integralbetween the chains signs of the hopping integrals t and t′ to be positive,
plays a crucial role in the short-range correlations and so that the two spins on a triangle of the lattice align
low-energy excitaions; i.e., the ferromagnetic spin corre- ferromagneticallywhenU islarge.4 Thelattice structure
lations between electrons on the neighboring sites is en- ofourmodelisshownschematicallyinFig.1(a). We call
hanced and the spin-triplet pairing correlations between the chain along the t (t′) hopping integral the t-chain
the electrons is induced, of which the consequences are (t′-chain). The dispersion relation is given by
clearly seen in the calculated spin and charge excitation
spectra at low energies. We hope that the present inves- ε =2tcoska+2t′cos2ka, (2)
k
tigation will provide deeper insight into the mechanism
of the spin-triplet superconductivity. where a is the lattice constant along the t-chain (we set
Ourpaperis organizedasfollows. InSec.II,wedefine a = 1 hereafter). We consider the case where there are
the1Dt-t′-U modelandintroducethephysicalquantities fourFermimomenta kF1and kF2(kF2 > kF1 );i.e.,
± ± | | | |
ofinterest,namely,spinandchargeexcitationspectra. In the case where there are two branches in the noninter-
Sec.III,exactsolutionofthespin(andcharge)excitation acting band dispersion [see Fig. 1(b)]. This case occurs
spectrum in the noninteractingcase is presented,and by when the hopping integrals satisfy the condition
comparingourresultwiththeexactone,weevaluatethe
performance of our DDMRG method. We then study t′ cos2(cid:2)(2 ρ)π/2(cid:3)
> − (3)
the spectra with onsite Coulomb interaction and discuss t sin2 (2 ρ)π
(cid:2) (cid:3)
−
relevance to the spin-triplet superconductivity. We close
with a summary in Sec. IV. where ρ is the band filling. In this paper, we restrict
ourselves to the case where t′ is a few times as large as
t and the system is at quarter filling, ρ = 1/2. Hence,
II. MODEL AND METHOD the model can be regardedas a double t′-chainHubbard
model weakly coupled by the t-chain.
Becausetismuchsmallerthant′,itwouldbeveryuse-
(a) ful to allow a case of t = 0 for familiarization with our
U
t results. In the limit of t 0, the system is equivalentto
′ →
thetwoindependent1DHubbardchainsatquarterfilling
t
since electrons are distributed equally to the two chains.
The noninteracting band dispersion readsε =2t′cos2k
k
and there are four Fermi momenta k = 5π/8 and
F1
± ±
k = 3π/8 in the original Brillouin zone defined for
(b) t±heF2t-cha±in. We define 2k∗ = k k = π/4 as the
F F2 − F1
‘nesting’ vector in our model at t = 0. We use this defi-
nition of the Brillouin zone throughout the paper.
We calculate the spin excitation spectrum
1 1
ε S(q,ω)= Im Ψ s+ s− Ψ , (4)
F kF1 kF2 π h 0| q Hˆ +ω E0 iη −q| 0i
− −
defined with s+ =(1/√L) eiqrc† c , and the charge
π 0 π q Pr r↑ r↓
− excitation spectrum
1 1
FIG.1: Schematicrepresentationsof(a)thelatticestructure N(q,ω)= Im Ψ n n Ψ , (5)
ofourmodeland(b)noninteractingbanddispersionatt′/t> π h 0| qHˆ +ω E iη −q| 0i
0
1. − −
defined with n = (1/√L) eiqrn . Here, Ψ and
q Prσ rσ | 0i
We consider the 1D Hubbard model defined by the E are, respectively, the wavefunction and energy of the
0
Hamiltonian ground state of the Hamiltonian Eq. (1). The DDMRG
technique is applied to calculate the excitation spectra.
H = tX(c†i+1σciσ +H.c.) We here use the open-end boundary condition (OBC)
i,σ for accurate calculation, because the system is relatively
+ t′X(c†i+2σciσ+H.c.)+UXni↑ni↓. (1) hard to deal with by the DMRG method due to large
long-range hopping integrals.13 When the OBC is used,
i,σ i
3
0:S(q)5 0:S(q)5
0
0 q (cid:25) q !(cid:25) 00 q (cid:25)
q !(cid:25)
kF1+kF2! kF1+kF2!
2kF1! 2kF1!
2(cid:25)(cid:0)2kF2! 2(cid:25)(cid:0)2kF2!
kF1(cid:0)kF2! kF1(cid:0)kF2!
0! 0!
q !0 q !0
0 1 2 3 4 5 0 1 0 1 2 3 4 5
! !
FIG. 2: Exact spin excitation spectrum 2S(q,ω) [=N(q,ω)] FIG. 3: Spin excitation spectrum 2S(q,ω) [= N(q,ω)] in
′
for t = 0.25 and t = 1 in the noninteracting case (U = 0). the noninteracting case (U = 0) calculated by the DDMRG
Broadening of the spectrum η = 0.03 is introduced. Five method. The same set of the parameter values as in Fig. 2
momentaindicatedwitharrowsintheleftsidedenotegapless is used. Right (left) panel shows the result for L = 24 and
points. Inset: Exact spin structurefactor S(q). η = 0.1 (L = 48 and η = 0.04). Inset: Spin structure factor
S(q) obtained from ω-integration of S(q,ω).
we need to use the quasimomenta k = πm/(L+1) for
integers 1 m L on a chain with L sites in order wheresmallηisintroducedtoregularizethepolesatpar-
≤ ≤
to express the momentum-dependent operators s+ and ticular frequencies ω. Note that the spectrum of charge
q
n .14 excitations at U = 0 is exactly twice as large as that
qσ
of the spin excitations, i.e., N(q,ω) = 2S(q,ω), because
N(q,ω) is just a sum over both up and down spins.
III. RESULTS In Fig. 2, we show the exact noninteracting spin ex-
citation spectrum S(q,ω) given in Eq. (6). For small
t/t′ (< 1), the spectrum contains two predominant fea-
Noninteracting spectrum
tures: (i) large-weightedstructure consisting of the dou-
ble sine curve, whose dispersions are approimately writ-
First, let us consider the noninteracting case, U = 0, ten as ω (4t′ 2t)sinq, and (ii) small-weighted con-
where the model is exactly solvable. In this case, an ∼ ±
tinuum structure at low frequencies, which arises from
excitationcorrespondstocreatingaparticle-holepairfor
excitations between different branches of the noninter-
the ground state and therefore we can obtain the exact
acting bands. A zero-energy excitation is caused by the
spectrum of spin excitations:
creationofaparticle-holepairjustatthe Fermilevelε ,
F
so that the gap closes at five momenta q = 0, k k
1 η F2− F1
S(q,ω)= lim X (=π/4), 2kF1, 2kF1, and kF1+kF2.
η→+0πLεk<εF<εk+q (ω−εk+q+εk)2+η2 Now,usingtheDDMRGmethod,weattempttorepro-
(6) duce the noninteracting spin excitation spectrum. The
4
resultisshowninFig.2. Since the noninteractingmodel
poses a nontrivial problem to the DDMRG technique, q
1
it gives us a relevant accuracy test. When carrying out )
q
the DDMRG calculation, we have to take into account (
S
the required CPU time. Usually, the DDMRG method
takes much longer CPU time than the standard DMRG
0
methodbecausethe excitedstatesmustbe obtainedand 0 q (cid:25)
(cid:25)
an asked quantity must be calculated (almost) individ-
ually for each frequency. Additionally, a required CPU
timeτ increasesrapidlywithfrequencyω andsystem
CPU
sizeLintheDDMRGcalculation,whichisestimatedap-
proximately as τ ωα (1<α<2) and as τ L
CPU CPU
∝ ∝
ifwekeepotherconditions. Hence,itwouldbeefficientto 3(cid:25)=4
take a relatively small system for obtaining an overview
of the spectrumand a largersystemfor studying the de-
tailed structure at low frequencies.
Let us then check our DDMRG result with the exact
spectrum. In the right panel of Fig. 3, we show the spin
(cid:25)=2
excitation spectrum S(q,ω) at U = 0 calculated with
the DDMRG method in a chain with L = 24 sites. The
structure of the double sine curve can be clearly seen.
However, it is hard to see dispersive structures in the
continuum spectra at low frequenciese because only dis-
crete peaks can be obtained due to the finite-size effect. (cid:25)=4
We need to take larger systems to resolve this problem
since the resolutionof spectrum canbe improvedin pro-
portiontothesystemsize. Wethereforechoosetodouble
the system size, L=48,and consider the low-energyex-
citations. The result is shown in the left panel of Fig. 3.
0
The resolution is obviously improved and we can now 0 0.5 0 1 2 3
confirm the five momenta at which the zero-energy ex-
citations occur. Moreover, we can see good agreement !
in the spin structure factor S(q) [= S(q,ω)] between
Pω
the DDMRG and exact results, as shown in the inset of FIG.4: Spinexcitation spectrum S(q,ω)at t=0.25, t′=1,
Fig.3. Thus,weareconfidentthatourDDMRGmethod and U =10 calculated by the DDMRG method. We use the
indeed enable us to study the detailed structures of the clusters L = 24 with the broadening η = 0.1 (right panel)
relatively complicated spectrum. and L=48 with η =0.04 (left panel). Inset: Spin structure
For the information, we keep m = 400 (800) density- factor S(q) obtained from ω-integration of S(q,ω).
matrixeigenstatestoobtainthespectrumforL=24(48)
sites. Note that a larger m value should be necessary to
get the true ground state and excited state s−−q|Ψ0i (or havesuggestedthat nospingapexists inthe strongcou-
n−q Ψ0 ) of the system. We therefore set m = 1200 in pling regime. On the other hand, the higher edge of the
| i
the first 4 5 DDMRG sweeps. spectrum is approximately represented as a sine curve
−
ω sinq as in the case of U = 0, which comes from
∼
the creation of the particle-hole pairs within the same
Spin excitation spectrum branch.
Let us now take a closer look at the spectrm. As far
Next, let us see how the spin excitation spectrum is as the spin degrees of freedom are concerned, the model
modified by the inclusion of the onsite Coulomb interac- Eq. (1) for large U may be mapped onto a two-chain
tion U. In Fig. 4, we show our DDMRG result for the t J model coupled with zigzag bonds. For small t/t′,
spin excitation spectrum S(q,ω) at t=0.25, t′ =1, and t−he antiferromagnetic interaction along the t′-chain, J′,
U = 10, where we use the clusters with L = 24 (right must be much larger than that along the t-chain, J, if
panel) and L=48 (left panel). we assumethat the exchangeinteractioncomes fromthe
Roughlyspeaking,theloweredgeofthespectrumcon- second-order perturbation of the hopping integrals with
sists of three sine curves with four nodes at q 0, a fixed U, i.e., J′ ( t′2) J ( t2). Consequently,
∼ ∼ ≫ ∼
k k (=π/4),π (k k )(=3π/4),andπ. The the features ofthe spectrumcanbe basically interpreted
F2 F1 F2 F1
exci−tation gap seems−to clos−e around these nodes. This as those of the 1D quarter-filled t J model.17,18,19 The
is consistent with previous theoretical studies,4,15 which nodes at the lower edge of the DD−MRG spectrum occur
5
at the momenta q = 0, 2k∗, 2π 2k∗, and π. Also, we t′-chains, which may be estimated as
F − F
can see considerable enhancement of spectral intensities
SzSz cos (π/4)R (decaying term), (8)
around q = π/4 (= kF2 kF1) in comparison with the (cid:10) i i+R(cid:11)∝ (cid:2) (cid:3)×
−
noninteracting spectrum. This indicates that the onsite
where R is an odd number. For R = 1, we obtain
Coulombinteractionenhancestheantiferromagneticcor-
SzSz = 0.00745, which indicates the presence of
relationwithaperiodof4timesthelatticeconstantalong (cid:10) i i+1(cid:11)
ferromagnetic correlation between two electrons at the
the t′-chain, which can be easily expected from the fact
neighboring sites. This result is consistent with a sce-
that the 2k∗-SDW correlation is the most dominant for
F nario of spin-triplet superconductivity where the pairing
small J in the 1D t J model at quarter filling. This re-
− of electrons occurs between the inter t′-chain nearest-
sultmay be comparedwith the 2k -SDWstate observed
F neighbor sites, which is proposed in Ref. 4.
experimentally in (TMTSF) X.16
2
We then examine the effects of a small hopping in-
tegral t, which leads to the antiferromagnetic interac- Charge excitation spectrum
tion J along the t-chain as mentioned above. From the
viewpoint of the spin degrees of freedom, magnetic frus-
Finally, we study the charge correlation function in
tration must be brought because triangular lattices are
the presence of the onsite Coulomb interaction U. In
formed of only the antiferromagnetic interactions. Al-
Fig.5,weshowthechargeexcitationspectrumN(q,ω)at
though J is much smaller than J′, we can clearly see t=0.25, t′ =1, and U =10, calculated by the DDMRG
the influence in our DDMRG spectrum; there are two
methodforthechainswithL=24(rightpanel)andwith
nodes around q = 3π/4, i.e., q 2k (> 3π/4) and
∼ F1 L=48(leftpanel). Theoutlineofthespectrum,whichis
2π −2kF2 (< 3π/4). These nodes are collected into a roughly expressedas ω 4t′sinq, seems to be similar to
single node at q = 3π/4 when t = 0. This split actually ≈
that in the noninteracting spectrum. This result reflects
signifiesatendencytoaformationofthe2k -SDWstate
F the fact that the overall dispersion is hardly affected by
alongthet-chainaswellastoacollapseofthe2k -SDW
F theonsiteCoulombinteraction. However,wenoticethat
state along the t′-chain. With increasing t/t′, the node
two distinct features emerge in the low-frequency range
at q = 2k approaches q = π/2 and the adjacent spec-
F2 of the spectrum, which are discussed below.
tral weight increases,whereas the node at q =2k goes
F1 One is the increase in the low-energy spectral weight
away from q = 3π/4 and the weight goes to zero. In
around q = π/2 in the excitation spectrum. The exci-
other wards, the hopping term t weakens the 2k -SDW
F tation gap seems to close there, so that the lower edge
oscillationalongthet′-chainsincethecompetingantifer-
of the spectrum consistes of the two sine curves with
romagnetic correlation along the t-chain is enhanced. In
three nodes at q 0, π/2, and π. We may thus have
fact, the spectral weight around q = π/4 will certainly ∼
one gapless charge mode. This result reflects a tendency
diminish as t/t′ increases.
toward the Peierls instability, i.e., the formation of the
Another noticeable feature is that the spectral weight 4k -charge-density-wave (CDW) along the t′-chain. We
F
around q = 3π/4 is obviously smaller than that around find that the spectral weight for the 2k -CDW correla-
F
q = π/4, while the spectrum should be symmetrical tionisinsteadmuchreduced. Thisfeaturewouldbemore
about q = π/2 in the case of t = 0. This result im- evident if we observe the momentum-dependent charge
pliesthepresenceofferromageticspinfluctuationsatlow structure factor N(q), which is shown in the inset of
energies. For clearer understanding, we study the spin Fig. 5. We find that N(q) takes the maximum value
structure factor S(q). As shown in the inset of Fig. 4, at q =π/2 and goes down along the practically straight
it is evident that S(q) around q = π/4 is greater than lines to q =0 and π. This dependence is consistent with
that around q =3π/4; otherwise, S(q) should be almost results in the 1D quarter-filled Hubbard chain at large
symmetrical about q =π/2. This result indicates an en- U.20; in fact, the resultis almostequivalentto the result
hancementofthe ferromagneticspincorrelationbetween for noninteracting spinless fermions at half filling, as far
theneighbouringsitesalongthet-chain. Wemayexplain as the charge degrees of freedom are concerned.
this as follows. The other is the apperance of large-weighted sharp
A real-spacebehaviorof the spin-spin correlationmay peaksaroundq =0atω 0. Thepointisasfollows;the
≈
be derived from the Fourier transform of S(q), spectral weight of the peaks around q =0 is larger than
that around q = π in the low-frequency range, and they
1 are also gathered at lower frequencies. This implies that
(cid:10)SizSjz(cid:11)= LXS(q)eiq(ri−rj), (7) the electrons tendto comein the neighboringsites along
q the t-chain, which is associated with the pairing of two
electronsbetweentheintert′-chainsites;accordingly,the
withSz =(n n )/2. Wefindthatthespincorrelation pairs tend to be arranged alternately along the t′-chain.
along tihe t′-ci↑h−aini↓is not affected so much by a small t Notethatthetwomomentaq =0andπshouldbeequiv-
value but the decay lengthof the 2k -SDW oscillationis alent when t=0, so that the present result implies that
F
slightly shortened. On the contrary, the t hopping term the small hopping integralt enhances the pairing fluctu-
plays a prominent role in spin correlation between the ations at low energies.
6
IV. SUMMARY
q 0:5) We have calculated the spin and charge excitation
q
( spectra of the two-chain zigzag-bond Hubbard model at
N
quarter filling in order to seek for consequences of the
spin-triplet pairing induced by the ring-exchange mech-
0
0 q (cid:25) anism. The model is topologically equivalent to the 1D
(cid:25) Hubbard model with nearest-neighbor t (>0) and next-
nearest-neighbor t′ (> 0) hopping integrals. We here
have considered the case at t t′. We have used the
≪
DDMRG technique to calculate the excitation spectra.
We have first demonstrated the accuracy of the
DDMRG method by reproducing the noninteracting ex-
3(cid:25)=4
act spectrum. We have suggested that, for practical
calculations, it is neccesary to adopt a relatively small
system for obtaining an overview of the spectrum and
a larger system for investigating detailed structures of
the spectrum at low frequencies because the required
(cid:25)=2
CPUtimeincreasesrapidlywithincreasingthefrequency
and/or system size. Thus, the DDMRG method enables
us to study the details of relatively complicated struc-
tures of the spectra.
Then, we have investigated how the spectra are de-
(cid:25)=4 formed when the strong onsite Coulomb interaction U
setsin. Wefindthatthespinandchargeexcitationspec-
tra are basically the same as those of the 1D Hubbard
(and t-J) chain at quarter filling; i.e., the spectra come
from the two nearly independent 1D chains where the
0 2k -SDW and 4k -CDW correlationsalong the t′-chains
F F
0 1 0 1 2 3 4
are enhanced. However, we find that the hopping inte-
gral t plays a crucial role in the short-range spin and
!
charge correlations; i.e., the ferromagnetic spin correla-
tions between electrons on the neighboring sites is en-
FIG.5: ChargeexcitationspectrumN(q,ω)calculatedbythe
hanced and the spin-triplet pairing correlations between
DDMRGmethodforthesameset oftheparametervaluesas
the electrons is induced, of which the consequences are
inFig.4. WeusetheclustersL=24withbroadeningη=0.1
clearly seen in the calculated spin and charge excitation
(right panel) and L = 48 with η = 0.04 (left panel). Inset:
spectraatlowenergies. OurDDMRGcalculationsforthe
Charge structure factor N(q) obtained from ω-integration of
N(q,ω). spin and charge excitation spectra have thus supported
the ring-exchange mechanism for spin-triplet supercon-
ductivity where the pairing of electrons occurs between
Additionally,wecanestimatetheso-calledTomonaga- the two chains.
Luttinger liquid parameter K from the derivative of
ρ
N(q) at q =0,
Acknowledgments
πdN(q)(cid:12)
K = (cid:12) , (9)
ρ 2 dq (cid:12)(cid:12)(cid:12)q=0 woWrkewtahsansukpTpo.rTteadkiimnoptaortfobryuGserfaunltsd-iisnc-uAsisdiofnosr.ScTiehnis-
whereby we find the value K 0.637. Because, in the tific Research (Nos. 18028008,18043006,and 18540338)
ρ
≈
presence of one gapless charge mode, the pairing cor- from the Ministry of Education, Science, Sports, and
relation is dominant in comparison with the 4k -CDW Culture of Japan. A part of computations was carried
F
correlation when K > 0.5,21,22 our estimated value of out at the Research Center for Computational Science,
ρ
K isconsistentwiththe occurenceofspin-tripletsuper- Okazaki Research Facilities, and the Institute for Solid
ρ
conductivity proposed in Ref. 4. State Physics, University of Tokyo.
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