Ground state phases of the Half-Filled One-Dimensional Extended Hubbard Model

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by  A. W. Sandvik

Ground state phases of the Half-Filled One-Dimensional Extended Hubbard Model
Anders W. Sandvik,1,2,3 Leon Balents,3 and David K. Campbell1,4
1Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215
2Department of Physics, ˚Abo Akademi University, Porthansgatan 3, FIN-20500 Turku, Finland
3Department of Physics, University of California, Santa Barbara, California 93106
4Department of Electrical and Computer Engineering, Boston University,
44 Cummington Street, Boston, Massachusetts 02215
4
0 (Dated: February 2, 2008)
0
Using quantum Monte Carlo simulations, results of a strong-coupling expansion, and Luttinger
2
liquid theory, we determine quantitatively the ground state phase diagram of the one-dimensional

n extendedHubbardmodelwithon-siteandnearest-neighborrepulsionsU andV. Weshowthatspin
a frustration stabilizes a bond-ordered (dimerized) state for U ≈ V/2 up to U/t ≈ 9, where t is the
J
nearest-neighborhopping. Thetransitionfrom thedimerizedstatetothestaggered charge-density-

3 wavestate for large V/U is continuousfor U <∼5.5 and first-order for higher U.
2
PACSnumbers: 71.10.Fd,71.10.Hf,71.10.Pm,71.30.+h
]
l
e
- The one-dimensional Hubbard model, which describes the accompanyingspin gapwere subsequently confirmed
r
t electrons on a tight-binding chain with single-particle usingquantumMonteCarlo(QMC)simulations[11,12].
s
. hopping matrix element t and on-site repulsion U, has The BOW phase now also has a weak-coupling theory
t
a a charge-excitation gap for any U > 0 at half-filling [1]. [13].
m
In the spin sector, the low-energy spectrum maps onto The existence of an extended BOW phase has re-
- that of the S = 1/2 Heisenberg chain; the spin coupling cently been disputed. Jeckelmann argued, on the basis
d
n J = 4t2/U for U → ∞. The spin spectrum is therefore ofdensity-matrix-renormalization-group(DMRG) calcu-
o gaplessandthespin-spincorrelationsdecaywithdistance lations, that the BOW exists only on a short segment of
c r as (−1)r/r [2]. Hence, the ground state is a quantum thefirst-orderpartoftheSDW-CDWboundary[14],i.e.,
[ critical staggered spin-density-wave (SDW). In the sim- that the transition alwaysis SDW-CDW and that BOW
1 plest extended Hubbard model, a nearest-neighbor repul- order is only induced on part of the coexistence curve.
v sion V is also included. The Hamiltonian is, in standard However, QMC calculations demonstrate the existence
2 notation and with t=1 hereafter, of BOW order well away from the phase boundary [12].
7
14 H = −t X X(c†σ,i+1cσ,i+c†σ,icσ,i+1) extAelntdheodugBhOsWeveprahlassetu[d1i0e,s1a1g,r1ee2,o1n3]t,hteheexsishtaepneceofofthains
σ=↑,↓ i
0 phase in the (U,V) plane has not yet been reliably de-
4 + UXn↑,in↓,i+V Xnini+1. (1) termined. The system sizes used in the exact diago-
0
i i nalization study [10] were too small for converging the
/
at The low-energy properties for V <∼ U/2 are similar SDW-BOW boundary (i.e., the spin gap transition) for
m to those at V = 0. For higher V the ground state U >∼4. InthepreviousQMCstudies[11,12],theempha-
- is a staggered charge-density-wave (CDW), where both sis was on verifying the presence of BOW order and the
d the charge and spin excitations are gapped. The tran- phase transitions for U ≈ 4. In this Letter, we present
n sition between the critical SDW and the long-range- the completephasediagram. Takingadvantageofrecent
o
ordered CDW has been the subject of numerous studies QMC algorithm developments—stochastic series expan-
c
: [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Until recently, it sionwith directed-loopupdates [15]in combinationwith
v
was believed that the SDW-CDW transition occurs for thequantumgeneralization[11]oftheparalleltempering
i
X all U >0 at V >∼U/2 and that it is continuous for small method [16]—we have carried out high-precision, large-
r U (<∼ 5) and first-order for larger U. However, based chain (up to L = 1024) calculations for sufficiently high
a
on a study of excitation spectra of small chains, Naka- U (≤ 12) to locate the point at which the BOW order
muraarguedthatthereisalsoabond-order-wave(BOW) vanishes. In agreement with Ref. [14], we find that the
phase [10], where the groundstate has a staggeredmod- BOW exists also above the U at which the transition to
ulation of the kinetic energy density (dimerization), in a theCDWstatebecomesfirst-order. However,long-range
narrowregionbetweenthe SDW andCDW phasesfor U BOW order exists also below this point, and hence the
smaller than the value at which the transition changes point at which the nature of the transition changes from
to first order. Previous studies [6, 7, 8, 9] had indi- continuous tofirst-order is on the BOW-CDW boundary.
cated an SDW state in this region. Nakamura’s BOW- The phase diagram we find here is qualitatively sim-
CDWboundarycoincideswiththepreviouslydetermined ilar to that obtained in a 4th order strong-coupling ex-
SDW-CDWboundary. Thepresenceofdimerizationand pansion, where the transition to the CDW state is de-
2
S(q)/q S(q)/q
ρ 1 1 σ 1 1
SDW−BOW (strong coupling)
5 SDW/BOW−CDW (strong coupling) 1.0 U=4
SDW−BOW (QMC) 0.44
BOW−CDW; continuous (QMC)
BOW−CDW; first−order (QMC) 0.8
4 SDW−CDW (QMC) 0.42 0.9 0.7
V=2.10
V3 0.40 0.8 0.60.00 0.02 0.041/L
2.14 2.16 2.18 1.8 1.9 2.0
CDW
2 1.0 U=6
0.30
SDW
1 0.9
0.25
0 0.8
0 2 4 6 8 10 0.20
3.150 3.152 3.154 3.156 2.9 3.0 3.1
U
0.20 U=8
1.0
FIG.1: QMCandstrong-couplingphasediagram. TheBOW
0.15 L=32
is located between the SDW-BOWand BOW-CDW curves. L=64
0.10 L=128
L=256
0.05 0.9 L=512
terminedbycomparingthe energiesofthe large-V CDW L=1024
state and the effective spin model including nearest- and 0.040.136 4.138 4.140 4.00 4.05 4.10
next-nearest-neighbor interactions J and J′ [8]. The V V
BOWphase correspondsto the spontaneously dimerized
FIG. 2: Long-wavelength charge (left panels) and spin (right
phaseofthe spinchain,i.e., J′/J >0.241[17]. InFig.1,
panels) structure factors vs V for U = 4 (top), 6 (middle),
wecompareourQMCphaseboundarieswiththestrong- and 8 (bottom). The system sizes are indicated in the low-
coupling result; the procedures giving the QMC bound- right panel. The U = 4 inset shows the dependence on the
aries will be discussed below. We will show that the sys- inverselattice size at V =2.10.
tem is a Luther-Emery liquid on the continuous BOW-
CDW boundary, i.e., the charge gap vanishes and the
the charge gap vanishes). In contrast, if the transition
spin gap remains open. Evidence supporting this type
is first-order, then K = 0 also on the phase boundary.
of transition was also presented in Ref. [11]. Here we ρ
Usingthe relation(3)forafinite system,anydiscontinu-
will further argue that the change to a first-order tran-
itieswillnaturallybesmoothed,andonecanonlyexpect
sition corresponds to the Luttinger charge exponent K
ρ
to observe S (q )/q developing sharp features as L is
reaching the value 1/4. ρ,σ 1 1
increased. In Fig. 2 we show results demonstrating this
We extract the SDW-BOW and BOW-CDW bound-
for several different system sizes at U =4, 6, and 8.
aries using the charge and spin exponents K and K .
ρ σ
Looking first at the chargeexponent, if K >0 on the
If there is a spin or charge gap, the corresponding ex- ρ
BOW-CDW boundary and K = 0 elsewhere, then one
ponent vanishes. Otherwise the equal-time correlation ρ
function Cρ(r) ∼ r−(Kσ+Kρ), Cσ(r) ∼ r−(Kσ−1+Kρ). If canexpectapeakdevelopinginSρ(q1)/q1 versusV. The
peakpositioncorrespondstothecriticalV,andthepeak
non-zero, the spin exponent K = 1 as a consequence
σ
height should converge to K . If the transition is first-
of spin-rotation invariance [18]. On periodic chains the ρ
order,S (q )/q shouldconvergetozeroforallV,butone
exponents can be conveniently extracted from the static ρ 1 1
can still expect some structure at the phase boundary
structure factors S (q) [19],
ρ,σ
for finite L as the nature of the ground state changes.
1 In Fig. 2, for U = 4 and 6 the development of sharp
Sρ,σ(q)= LXeiq(j−k))h(n↑j ±n↓j)(n↑k±n↓k)i, (2) peaks is apparent. For U = 4 the peak-height converges
j,k to a non-zero value, implying a continuous transition at
V ≈2.160withK ≈0.43. ForU =6theconvergenceto
in the limit q →0+: ρ
a value >0 is not clear,but the transitionpoint is given
K =S (q )/q , q =2π/L, L→∞. (3) accurately by the peak-position, which shows very little
ρ,σ ρ,σ 1 1 1
size-dependence. It has been shown previously that the
If there indeed are three successive phases, SDW-BOW- transition is first-order for U = 6 [6, 11], and the peak
CDW, as V is increased at a fixed value of U, then the shouldthereforeinfactconvergetozero. Theratherslow
spin exponent K = 1 in the SDW phase and K = 0 decay reflects the proximity to the point at which the
σ σ
everywhere else. The charge exponent K = 0 every- transitionbecomes continuous. For U =8 the peak does
ρ
where,exceptexactlyattheBOW-CDWtransitionpoint not sharpen, but instead a step develops at the critical
if this is a continuous quantum phase transition (i.e., if V. The whole curve converges to 0 as L → ∞. The
3
1.0
transition is hence strongly first-order in this case, in
U=4
agreement with previous calculations. As seen in Fig. 1, 0.45
0.9
and as observed already by Hirsch [6], the locations of
the U = 6 and 8 critical points agree very well with the 0.8 Kρ0.44
strong-coupling expansion [20].
0.7
In the SDW phase, one cannot expect to easily find
Sσ(q1)/q1 → 1 exactly, due to logarithmic corrections Kρ0.6 0.430.00 0.01 1/L0.02 0.03
that affect various quantities strongly even for very long
0.5
chains [21, 22]. However, the log-corrections are known
to vanish in the frustrated J − J′ spin chain at its 0.4
dimerization transition [23], and hence, since the SDW-
0.3
BOW transition should be of the same nature, the log-
0 1 2 3 4 5 6
corrections should vanish here as well. The transition at
U
fixed U should therefore be signaled by S (q )/q cross-
σ 1 1
ing 1 from above as V is increased. Because of the van- FIG. 3: QMC results for Luttinger charge exponent on the
ishinglog-correctionsatthetransition,thecrossingpoint BOW-CDW boundary (solid circles). The inset shows the
finite-size scaling for U =4.
with1shouldnotmovesignificantlyasLisincreased,but
the drop below 1 should become increasingly sharp, and
eventually S (q )/q should approach0 inside the BOW
σ 1 1
phase. This method was used in Ref. [11] and gave a the spin-gap curve crosses the CDW-transition curve.
slightlyhighercriticalV fortheSDW-BOWtransitionat This is slightly lower than what we find based on QMC.
U =4 than the exact diagonalization[10]. We now have Thestrong-couplingBOWextendsdowntosmallerU,V,
results fora widerrangeofcouplings. The resultsshown but clearly the 4th-order result cannot be expected to
in Fig. 2 are in accord with the above discussion for all be quantitatively accurate in this region. Nevertheless,
three U-values; S (q )/q crosses 1 at a V-point which the spin-frustration mechanismconsistently explains the
σ 1 1
does not move visibly between L=64 and L=256. For presence of an SDW-BOW transition and an extended
largerV,onecanseeasharperdropforthelargersystem BOW phase. Spin-frustration was previously cited by
sizes. The size dependence at U = 4,V = 2.1 is shown Jeckelmann [14], but, surprisingly, he used it in support
in an inset. Here the convergence to 0, i.e., the presence of a BOW of vanishing extent.
of a spin gap, is apparent. If the spin gap is small, as it Next, we consider the nature of the BOW-CDW tran-
is close to the phase boundary, the convergenceto 0 will sition. AsdiscussedinRefs.[10,13],the continuouscrit-
obviously occur only for very large systems. ical point for small U is described by a Gaussian free
Results suchas those shownin Fig.2 wereused to de- (charge) boson theory, characterized by the parameter
terminethephaseboundariesinFig.1. Asalreadynoted, Kρ. At generic values of (repulsive) U,V, the leading
the BOW aspect of the phase diagram differs from pre- “4kF” umklapp process is present, and has scaling di-
vious proposals [10, 11, 13, 14] in that the BOW-CDW mension ∆4kF = 2Kρ and is hence relevant (∆4kF < 2)
transitioncanbeeithercontinuousorfirst-order,i.e.,the for Kρ < 1. At the BOW-CDW transition, this oper-
changeoforderoccursontheBOW-CDWboundary. The ator vanishes, leading to a vanishing of the charge gap.
existence oftwospecialpoints, onewhere the transition- For consistency, no other relevant operators should be
order changesand one athigher U where the BOW van- present, which would otherwise require fine tuning to
ishes, was also suggested by Jeckelmann [14], who, how- zero, making the Gaussian theory a multicritical point.
ever, insists that the BOW does not exist for small U Themostdangerouscandidateisthe“8kF”umklapppro-
wherethetransitiontotheCDWstateiscontinuous(i.e., cess, with ∆8kF = 4∆4kF = 8Kρ, so a continuous Gaus-
his phase diagram has no continuous BOW-CDW tran- siancriticalpointispossibleonlyfor1>Kρ >1/4. Even
sition). We have carried out calculations for U down to in this range, the Gaussian theory is an unusual critical
1, and, as shown in Fig. 1, we still find a BOW phase point with non-universal behavior, e.g., the correlation
there. Most likely, in view also of weak-coupling argu- length exponent ν =1/(2−2Kρ).
ments [13], the BOW extends down to U =0+. We find Extrapolated QMC results for K on the BOW-CDW
ρ
no BOW for U >∼ 9. In the strong-coupling expansion, boundaryareshowninFig.3. The finite-sizecorrections
using the couplings J and J′ derived by van Dongen [8], appear to be of the form 1/Lα, with an U dependent
the effective spin model is gapped, i.e., J′ >0.241J [17], exponent α. At U = 4, α ≈ 1, as shown in the inset of
abovethe dashedcurve inFig. 1. The J−J′ mapping is Fig. 3. For larger U, α decreases rapidly and is difficult
notapplicablebeyondthetransitionintotheCDWstate, to determine accurately for U >∼ 5. The extrapolated
which(thesolidcurveinFig.1)waspreviouslycalculated K value at U = 5 in Fig. 3 should be regarded as an
ρ
by comparing the 4th-order CDW and J −J′ energies upperbound. AtU =6,theextrapolatedK <1/4,and
ρ
[8]. The 4th-order BOW region ends at U ≈ 7, where henceweexpectaneventualdropto0. Thisisconsistent
4
U=3 U=4 phase diagram of the extended Hubbard model at half-
0.2
0.10 Kρ=0.52 Kρ=0.43 filling. The dimerized BOW phase can be explained
by spin-frustration. The BOW-CDW transition changes
ρK−2LW0.05 0.1 Ofronmthceocnrtitinicuaolu(sU,tVo)ficrustr-voerdtheresbyesttwemeenisUaL=uth5era-nEdm5e.r5y.
D
χC liquid,withachargeexponentK decreasingfrom1asU
ρ
is increased from 0. We have argued that the minimum
0.00 0.0
K = 1/4 and that the BOW-CDW transition becomes
ρ
0.06 first-order when this value is reached.
0.1
ρK−2 0.04 This work was supported by the Academy of Finland,
L
W project No. 26175 (AWS), by the NSF under Grants
O
χB0.02 No.DMR-9985255(LB)andDMR-97-12765(DKC),and
bythe SloanandPackardfoundations(LB).The numer-
0.00 0.0
1.60 1.65 1.70 2.14 2.15 2.16 2.17 ical calculations were carried out at the CSC in Espoo,
V V
Finland, and at the NCSA in Urbana, Illinois.
FIG.4: Finite-sizescalingoftheBOWandCDWsusceptibil-
itiesforU =3(left)and4(right). Thesymbolscorrespondto
different system sizes as in Fig. 2. The dashed lines indicate
[1] E. H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445
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[6] J. E. Hirsch, Phys.Rev. Lett. 53, 2327 (1984).
to a first-order transition occurs between U =5 and 5.5.
[7] J. W. Cannon and E. Fradkin, Phys. Rev. B 41, 9435
What is the nature of the tricritical point at which
(1990); J. W. Cannon, R. T. Scalettar and E. Fradkin,
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ρ
predictsthatthecriticalK continuouslyapproaches1/4 [10] M. Nakamura, J. Phys. Soc. of Japan 68, 3123 (1999);
ρ Phys. Rev.B 61, 16377 (2000).
asthe tricriticalpointU =U isapproachedfrombelow,
t
[11] P.Sengupta,A.W.Sandvik,andD.K.Campbell,Phys.
as Kρ−1/4∼p(Ut−U)/Ut. We do not have sufficient Rev. B 65, 155113 (2002).
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[12] A.W.Sandvik,P.Sengupta,andD.K.Campbell,Phys.
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at U = 5.5 the transition should be first-order. Hence, [13] M. Tsuchiizu and A. Furusaki. Phys. Rev. Lett. 88,
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91, 089702 (2003).
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[15] O. F. Sylju˚asen and A. W. Sandvik, Phys. Rev. E 66,
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046701 (2002).
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decay as (−1)rr−Kρ [18], implying that the susceptibili- [18] J. Voit, Rep.Prog. Phys. 57, 977 (1994).
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curves for different L should intersect at the critical
[20] The U = 8, L = 256 data are not completely repro-
BOW-CDW point. Fig. 4 shows results for U = 3 and
ducible: The location of the Sρ(q1)/q1 maximum fluc-
4, using the Kρ values determined above. For U =4 the tuates slightly between runs, due to meta-stability at
expectedscalingcanbe observedevenforsmallsystems. the strongly first-order transition. The parallel temper-
For U =3 the correctionsare larger,andthe asymptotic ingscheme[11]overcomesthisproblemscompletelyonly
scalingsetsinonlyforL>∼128. Thisisclearlyduetothe for smaller L and/or smaller U. For U = 4 and 6, the
results are reproducible for all L considered.
smallerspingapatU =3,whichimpliesalongerlength-
[21] F. Woynarovich and H.-P. Eckle, J. Phys. A 20, L97
scale below which remaining spin correlations affect the
(1987).
charge and bond fluctuations.
[22] S.Eggert,I.Affleck,andM.Takahashi,Phys.Rev.Lett.
In summary, we have determined the ground state 73, 332 (1994).
5
[23] S.Eggert, Phys. Rev.B 54, R9612 (1996).