Low-energy excitations of the one-dimensional half-filled SU(4) Hubbard model with an attractive on-site interaction: Density-matrix renormalization-group calculations and perturbation theory

0.13 MB·English

by  Jize Zhao

Low-energy excitations of the one-dimensional half-filled SU(4) Hubbard model with
an attractive on-site interaction: Density-matrix renormalization-group calculations
and perturbation theory
Jize Zhao and Kazuo Ueda
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
7
Xiaoqun Wang
0
Department of Physics, Renmin University of China, Beijing 100872, China
0
(Dated: February 6, 2008)
2
n We investigate low-energy excitations of the one-dimensional half-filled SU(4) Hubbard model
a withanattractiveon-siteinteractionU <0usingthedensitymatrixrenormalizationgroupmethod
J aswellasaperturbationtheory. Wefindthatthegroundstateisachargedensitywavestatewitha
8 longrangeorder. Thegroundstateiscompletelyincompressiblesincealltheexcitationsaregapful.
1 Thechargegapwhichisthesameasthefour-particleexcitationgapisanon-monotonicfunctionof
U,whilethespingapandothersincreasewithincreasing|U|andhavelinearasymptoticbehaviors.

]
l PACSnumbers: 71.10.Fd,71.10.Pm,71.30.+h,05.10.Cc
e
-
r
t
s The Hubbard model is one of the most classic mod- theless, some aspects of physical properties can be re-
.
t els for strongly correlatedelectronic systems and has at- liably explored by some analytical approaches as well as
a
tracted long-term interest since the pioneering work in numerical methods such as density matrix renormaliza-
m
1960s[1]. By taking the on-site Coulombinteractioninto tion group (DMRG)[9, 10] and Quantum Monte Carlo
-
d account, it well explains the puzzle that some materials simulations[11, 12]. Most recently, an SO(8) symmetry
n with half-filled bandare insulators. However,this model regime was proposed between 0 < U < 3t at half-filling
o whichconsideronlyasingleband,on-siteCoulombinter- by Assaraf et al[12] with using a nonperturbative renor-
c
actionandthenearest-neighborhoppingisoftenthought malizationgroupmethodandquantumMonteCarlosim-
[
being oversimplified. To account for other features be- ulation. They found that the low-energy spectrums are

2 yond the Mott physics[2], there are two kinds of natu- gapfulinthisregime. Thesimilarresultswerealsoshown
v
ral extensions: one is to incorporate the orbital degree laterbySzirmaiandSo´lyomfortheotherN >2case[13].
9
of freedom which may be called multi-band Hubbard Those studies were concentrated on the repulsive case
4
0 model[3],andtheotheristoconsiderthehoppingand/or U > 0, but for the U < 0 case few results are obtained
9 the Coulomb interaction with longer ranges. so far. Onthe other hand, it is well knownthat the one-
0 In the last several years, ultra-cold atomic experi- dimensional attractive half-filled SU(2) Hubbard model
6
ments evoke systematic studies of correlation effects in is described by a Luther-Emery liquid[14], in which the
0
the optical lattice systems where interactions are tun- charge excitation is gapless, whereas the spin excitation
/
t able through Feshbach resonance. The Hubbard model isgapful. BythehiddenSU(2)transformationtheSU(2)
a
m becomes againanappropriateone to envisagesomerele- Hubbard model with U can be mapped to the one with
vant issues with both positive and negative interactions. −U, while for the SU(4) case such a mapping does not
-
d Recently, fermionic atoms with higher spins are success- existsothatonecannotobtainanyinsightsintothelow-
n
fullytrappedintoopticallattices[4]. Thiscallsforagen- energypropertiesthroughthemapping. Inthispaper,we
o
eralizationoftheSU(2)Hubbardmodelinto theSU(N) will show that the SU(4) Hubbard model at half-filling
c
: case[5,6]. Inthispaper,westudyone-dimensionalSU(4) withthe attractiveinteractionbelongsto adifferentuni-
v
Hubbard model which is represented as versality class from the SU(2) one.
i
X Let us start with a perturbation theory for the strong
L L
ar H=−tXX(c†iσci+1σ+h.c.)+U2 X X niσniσ′ (1) ctoonuipalning(1r)egaismHe. F=orHthi+spHurp,owseh,ewreertehwerhitoeptphiengHatmeriml-
i=1 σ i=1σ6=σ′ H = −t (c† c t+h.c.u) is regarded as a perturba-
wofhtehree lta>tti0ceissiatehs,opσpainngdmσ′attrhiexsepleinmienndt,icLestthaekinnugm−be3r, taisotnthaendzePrtohiteσhoonirσ-sdieit+re1HiσnatemrailcttoinoinanHhua=s hU2igPhlyiσ6=dσe′gneniσenraiσte′
2
−1, 1 and 3. c† (c ) denotes the creation(annihilation) ground states in which each site is either fully occupied
2 2 2 iσ iσ
operator of a particle with spin σ at the site i, n = by four particles forming a SU(4) singlet, or empty. Up
iσ
c† c is the corresponding number operator and U is tothesecond-order,theeffectiveHamiltonianisgivenby
iσ iσ
the on-site interaction.
2t2 t2
This model is not exactly solvable even in one di- H(2) = P n P − P n n P, (2)
mension in contrast to the SU(2) case[7, 8]. Never- eff 3U X i 6U X i i+1
i i
2
where P is a projection operator which projects a state term can only appear after calculating to higher order.
onto the subspace spanned by the ground states of H , Therefore,theperturbationtheoryinthestrongcoupling
u
and n = n the number operator at the site i. The regime shed light on the different nature of the ground
i Pσ iσ
hoppingtermH liftsthedegeneracyofH andgivesrise states between the SU(2) and SU(4) cases.
t u
to both the energy gain of 4t2 per site at half-filling as In order to go beyond the validity range of the above
3U
denoted by the first term of (2) and an effective repul- perturbation theory for a full exploration of the low-
siveinteractionbetweenparticlesonthenearest-neighbor energyproperties,wehaveperformedsystematicDMRG
sites as denoted by the second term of (2). The second computations. Thereareactuallysomedifficulties inher-
term induces essentially a charge-density-wave (CDW) ent to this model, such as a large number of degrees of
ground state with a true long range order such that ev- freedom for each site and different edge states so that
ery other site is fully occupied with an empty site in some measures are taken necessarily to reach sufficient
between, which is consistent with the mean field result accuracy in our computations. To calculate ∆c, we have
fortheweakcouplingregion[6]. Moreover,H(2) hastwo- to use the periodic boundary condition (PBC) with the
eff
fold degenerate ground states, each of which is a SU(4) even number of sites because of multi-edge excitations
singlet. in the SU(4) singlet subspace when open boundary con-
An important question concerning the CDW ground dition (OBC) is imposed. For other gaps, we can effi-
state is whether it is metallic or insulating. To address ciently expel the corresponding edge excitations by an
this question, we need to examine charge excitations. OBC algorithm. In particular, one lattice site is added
The charge gap ∆ is the energy difference of the lowest at each step and broken into two pseudo-sites. When
c
excitation in the spin singlet channel from the ground the infinite system algorithm is conducted with the size
state as defined by ∆ = E (L,2L,0) − E (L,2L,0), of the superblock up to some odd number of sites L
c 1 0
in which E (L,N,S) stands for the the n−th excita- which are preselected, the sweeping procedure is per-
n
tion energy in a spin-S channel with L sites and N formed for those L. The necessary extrapolations for
particles. And another interesting issue is to explore the thermodynamic limit are finally made properly on
the relevance of the four-particle excitation gap ∆ to the data with these preselected sites. In this case, we
4
∆ for the SU(4) symmetry. ∆ represents the en- have to redefine the gaps correspondingly, for instance
c 4
ergy cost of adding four particles or holes into the sys- ∆4 =E0(L,2L+6,0)−E0(L,2L+2,0)−6U,where the
tems such that ∆ = 1[E (L,2L+4,0) +E (L,2L− particle-hole symmetry is explicitly taken into account.
4 2 0 0
4,0)−2E (L,2L,0)]. From H(2) , one can easily find Inthe strongcouplingregion,PBCis oftenusedtoiden-
0 eff tify the bulk values of a gap rather than edge-excitation
∆c = −83tU2 based on the fact that the motion of four energy obtained under OBC. In our computations, t is
particles from one of fully occupied sites to its neighbor set to be unit and 2000 states are kept for most cases
costs a minimal energy. Similarly, adding four particles and the maximal truncation error is the order of 10−7.
tothesystemwouldgaintheenergy6U+2×(−8t2),while
3U
addingfourholescosts−6U,thusonehas∆ =−8t2. It 0
4 3U
turns out that the four-particle excitation gap is essen-
tially the same as the charge gap and finite.
However, the situation is completely different for the
−1
SU(2) case, where the ground state is metallic. To un- DMRG
derstand this, one can write the effective Hamiltonian in δε
4t2/3U
the strong coupling limit for the SU(2) case as[15]:
−2
2t2
H(2) = P n P (3)
eff,su(2) U X iσ
iσ
t2
− UP X(niσnjσ −c+iσc+iσ¯cjσ¯cjσ)P. −3−10 −8 −6 −4 −2 0
hijiσ
U
This Hamiltonian distinguishes itself from H(2) with its
last additional term which allows a “pair hopefpfing” pro- FIG. 1: Ground state energy correction per site from H(e2f)f
(solid line) versus U in comparison with DMRGdata ((cid:13)).
cess between the two neighbor sites. Although the sec-
ond repulsive term would be in favor of forming a CDW
ground state with gapful charge excitations, the extra Figure 1 displays the ground state energy correction
“pair hopping” term eventually destabilizes this CDW per site arising from the hoping term for the thermody-
long range order resulting in gapless charge excitations. namic limit. Both the numerical results and the pertur-
On the other hand for the SU(4) case, similar hopping bation theory show this energy increases monotonically
3
as |U| increases and approaches to zero asymptotically. 1.5
When−2<U <0,theresultsoftheperturbationtheory
deviates from DMRG ones, and the deviation becomes ∆
significant for small |U|. This is reasonable because in ∆4
this region the hopping term H is no longer perturba- 1 c
t 2
−8t /3U
tive. However,for U <−2, the perturbation theory pro-
∆
vides very good results which accurately agrees with the
DMRG data. On the other hand, our DMRG calcula-
tionswithbothOBCandPBCshowthatforafiniteand 0.5
even L, the ground state is unique and a SU(4) singlet
which belongs to the irreducible representation [14][16].
In addition, slightly above the ground state there is one
accompanied SU(4) singlet excited state, whose energy 0
−10 −8 −6 −4 −2 0
difference from the ground state diminishes as L → ∞. U
Therefore,oneobtainstwo-folddegenerategroundstates
in the thermodynamic limit, which are consistent with
FIG. 2: DMRG results for charge gap (+) and four-particle
(2)
our analysis based on Heff. These degenerate CDW gap((cid:13))versusU incomparisonwithperturbationones(solid
ground states with the long range order result from the line). Error bars are smaller than size of symbols except for
translational symmetry breaking. ∆c at U =−1 as estimated with keeping different states.
In FIG. 2, we show the DMRG results on the charge
gap ∆c and the four-particle excitation gap ∆4 for the proximation. The CDW state in HF approximation can
entire range of U < 0. First of all, one can see that be achieved by simply writing the on-site interaction as
both ∆c and ∆4 are non-vanishing for all finite U < 0 niσniσ′ ≈hniσiniσ′ +niσhniσ′i−hniσihniσ′i andassum-
so that the ground state is insulating rather than metal- ing hn i=n+(−1)iδn, where n is the average number
iσ
lic in contrast to the SU(2) case. Secondly, these two of particles per site for eachspin andδn the correspond-
gaps behave non-monotonically with U. The maximum ing order parameter. At half-filling, one has n = 1 and
2
shown around U = −2 indicates a crossover region be- 0 ≤ δn ≤ 1. By further introducing a = c and
2 lσ 2lσ
tween weak and strong interaction regimes. Third, the b = c for each sublattice of the bipartite lattice,
lσ 2l+1σ
perturbation theory for the strong coupling regime pro- respectively,andtakingtheFouriertransformation,then
videscorrectlytheasymptoticbehaviorforlarge|U|limit we can write down the Hartree-Fock Hamiltonian as:
and shows a qualitative agreement with the DMRG re-
sultsinthestrongcouplingregime. Thevisibledeviation HHF = −tX(cid:0)(1+e−ik)a+kσbkσ+h.c.(cid:1)
from the DMRG results sets on at about U ≈ −5 lower kσ
than that (U ≈ −2) for the ground state energy correc- +3Uδn (a+ a −b+ b )+const. (4)
X kσ kσ kσ kσ
tion shown in Fig. 1. In the weak coupling regime, ∆
c kσ
and∆ shownbyDMRGdecreasewithincreasingU. Fi-
4
nally,whileH(2) canpredict∆ =∆ onlyinthestrong Diagonalizing this Hamiltonian, one can obtain two
eff c 4 bands for eachspin species σ with the quasi-particledis-
couplinglimit, ourDMRGcalculationsshowthatwithin
the numerical accuracy ∆c remains equal to ∆4 beyond persions wσ± = ±q∆21+4t2cos2 k2 where ∆1 = −3Uδn.
the strong coupling regime. Although it is difficult from Moreover, one has ∆ = ∆ = 2∆ in the HF ap-
c s 1
theDMRGcalculationstoobtainsufficientlyaccurate∆c proximation. These gaps can be then evaluated af-
for −1<U <0 yet, it is reasonableto conclude that ∆ ter solving the following self-consistent equation 2δn =
4
is equal to ∆c for all U <0 in the SU(4) case. ha+lσalσi − hb+lσblσi for the order parameter δn. In or-
dertocalculatethetwo-particleexcitationgap,however,
Now we turn to the other three types excitations: the
it is necessary to account for the particle-particle cor-
first one is the quasi-particle gap ∆ for adding single
1
relations into the HF approximation, which is nothing
particle or single hole to the system, the secondone spin
but the random phase approximation (RPA). For this
gap ∆ corresponding to the excitation energy in the
s
purpose, one first constructs the basis with two-particle
spin triplet channel from the ground state, and the last
excitations from the ground state of HHF as follows:
onetwo-particlegap∆ definedasenergycostwhentwo
2
particles or two holes are added to the system. While
these three gaps together with ∆ and ∆ essentially in- |Ψpp′σi=α+pσα+p′σ¯|Ψgi, |Ψgi=Yβk+σ|0i) (5)
c 4
kσ
volve all kinds of relevant excitations, they have signifi-
cantly different behaviors. Since the ground state is the whereα+ isanoperatorcreatingonequasi-particlewith
pσ
CDW state with the long range order, it is insightful to momentump andspinσ inthe w+ bandandin|Ψ i the
g
analyze those excitations in the Hartree-Fock (HF) ap- bandw− isfully filledupby quasi-particlesβ+ withthe
σ kσ
4
momenta k and spin σ. Then ∆ can be obtained by erties of the one-dimensional half-filled SU(4) Hubbard
2
diagonalizing H on the above basis (5). modelwiththeattractiveon-siteinteractionbyusingthe
DMRG method as well as the perturbation theory. We
6 found that the ground state is a CDW insulating state
3 withthelongrangeorderinwhichthetranslationalsym-
5 metryisbrokenandallkindsofexcitationsaregapfulfor
2
∆/|U| finite U < 0. Within our numerical accuracy, we found
4 1 that the four-particle excitation gap is the same as the
charge gap. While the charge gap (the four particle ex-
∆ 0 citation gap) behaves non-monotonically, the others in-
3 −10 −6 −2
creasewithincreasing|U|andhavealinear-Uasymptotic
U
behaviorwithdifferentcoefficients. Therefore,webelieve
2
that the one-dimensional attractive half-filled Hubbard
model for the SU(4) and SU(2) cases belong to differ-
1
ent universality classes. Moreover, we find that the na-
ture forthe SU(4)casecanbe further generalizedto the
0 other SU(N >2) cases[18]. At the end, it is worthwhile
−2 −1 0
to mention that since the four-particle excitation gap as
U
well as the charge gap are the smallest energy scale for
the SU(4) case with U <0, it would be very interesting
FIG. 3: DMRG results for one particle gap ∆1 (×), two- to detect four-particle process (excitations) in an ultra-
particlegap∆2((cid:3)),spingap∆s (△),four-particleexcitation cold fermionic atom system with the hyperfine spin-3/2.
gap ∆4 ((cid:13)) and charge gap ∆c(+) are shown as a function
We would like to thank Y. Yamashita and Y.Z. Zhang
of U. The results of HF-RPAfor ∆1, ∆2 and ∆s denoted by
solid, dot-dashed,and dashed lines, respectively. Insetshows for fruitful discussions. J. Zhao acknowledges J. So´lyom
∆/|U| versusU for ∆1 (×),∆2 ((cid:3)) and ∆s (△). forthehelpfulcorrespondence. X.Wangissupportedun-
der the Grants 2005CB32170Xand NSFC10425417.
AscomparedtotheDMRGresults,wefoundthatHF-
RPA can provide a qualitatively correct description for
∆ , ∆ , and ∆ . Figure 3 shows the DMRG data on
1 2 s
all five gaps as well as ∆ ,∆ , and ∆ from HF-RPA
1 2 s [1] J. Hubbard, Proc. Roy. Soc. London, Ser.A 276, 238
approximation for the region of −2 ≤ U ≤ 0 and the
(1963); 281, 401 (1964)
inset illustrates ∆ , ∆ , and ∆ by showing ratios for
1 2 s [2] N.F. Mott, Proc. Phys. Soc. London, Ser.A 62, 416
them over |U| up to |U| = 10. In contrast to ∆c and (1949); Can. J. Phys.34, 1356 (1956).
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turns out that the relation ∆ = ∆ , given by the HF [4] G. Modugno, et al. Phys.Rev.A 68, 011601(R) (2003).
c s
approximation, is invalid for general U < 0. Moreover, [5] I. Affleck and J.B. Marston, Phys. Rev. B 37, 3774
(1988); J.B. Marston and I. Affleck, Phys. Rev. B 39,
it is unclear but beyond the present approaches whether
11538 (1989).
there is a symmetry enlargement similar to the one pro-
[6] C.J. Wu, J.P. Hu and S.C. Zhang, Phys. Rev. Lett. 91,
posed for the repulsive case[12]. Nonetheless, HF-RPA
186402 (2003).
presents precise asymptotic behaviors for ∆1, ∆2 and [7] E.H.LiebandF.Y.Wu,Phys.Rev.Lett.20,1445(1968).
∆s. In the weak coupling limit, the exponential open- [8] T.C.Choy,Phys.Lett.80A,49(1980);F.D.M.Haldane,
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with taking into account the two-particle correlations.
[10] I. Peschel, X. Wang, M. Kaulke and K. Hallberg, Den-
In the strong coupling limit, one has ∆ ∼ −1.5U and
1 sity Matrix Renormalization, LNP528, Springer-Verlag,
∆ ∼ −3U from δn → 0.5, and ∆ ∼ −2U from the
s 2 1999.
HF-RPAcalculations[17]. Thecorrespondingcoefficients [11] R. Assaraf, et al. Phys.Rev.B 60, 2299 (1999).
are in good agreement with the DMRG results as can [12] R. Assaraf, et al. Phys.Rev.Lett. 93, 016407(2004).
be seen from the inset. On the other hand, the results [13] E. Szirmai and J. S´olyom, Phys. Rev. B 71, 205108
of HF-RPA deviate from the DMRG data apparently in (2005).
[14] A. Luther and V.J. Emery, Phys. Rev. Lett. 33, 589
the intermediate coupling regime, but this is quite un-
(1974).
derstandable since correlationsinvolvedin (1) cannotbe
[15] V.J. Emery,Phys. Rev.B 14, 2989 (1976).
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t u [16] e.g. Y. Yamashita, N. Shibata and K. Ueda, Phys. Rev.
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In summary, we have studied the low energy prop- [17] The asymptotic behavior of ∆1, ∆2 and ∆s can be ob-
5
tained alternatively from Hu. For instance, ∆1 is given [18] J. Zhao, K. Ueda, and X.Wang, (unpublished).
by considering that adding one hole costs energy −3U
and adding one particle gains noenergy.