Table Of ContentEur. Phys. J. B manuscript No.
(will be inserted by the editor)
S = 3 σ = 1
Alternating-spin and Heisenberg chain with
2 2
three-body exchange interactions
N. B. Ivanov1,2, S. I. Petrova3, and J. Schnack1
6
1 1 Department of Physics, Bielefeld University,P.O. box 100131, D-33501 Bielefeld, Germany
0 2 Instituteof Solid State Physics, Bulgarian Academy of Sciences, Tzarigradsko chaussee 72, 1784 Sofia, Bulgaria
2 3 Department of Engineering Sciences and Mathematics, University of Applied Sciences, D-33619 Bielefeld, Germany
n
a Received: date/ Revised version: date
J
0 Abstract. Thepromotionofcollinearclassicalspinconfigurationsaswellastheenhancedtendencytowards
2 nearest-neighborclusteringofthequantumspinsaretypicalfeaturesofthefrustratingisotropicthree-body
exchange interactions in Heisenberg spin systems. Based on numerical density-matrix renormalization
] groupcalculations, wedemonstratethattheseextrainteractionsintheHeisenbergchainconstructedfrom
l
e alternating S = 3/2 and σ = 1 site spins can generate numerous specific quantum spin states, including
2
- some partially-polarized ferrimagnetic states as well as a doubly-degenerate non-magnetic gapped phase.
r
t In thenon-magneticregion of thephasediagram, themodel describes acrossover between thespin-1and
s
spin-2 Haldane-typestates.
.
t
a
m PACS. 75.10.Jm Quantizedspinmodels–75.40.Mg Numericalsimulationstudies–75.45.+j Macroscopic
quantum phenomenain magnetic systems
-
d
n
o 1 Introduction berg chain [7] defined by the Hamiltonian
c
[
L
1 =J1 S2n (σ2n−1+σ2n+1)
H ·
v nX=1
9 L
531 tThhreeeb-siqpuinaderxacthicanspgien-cspouinpliinntgesra(cStionSs ()S(iS·SjS)2)an+dhth.ce. +J2nX=1[(S2n·σ2n−1)(S2n·σ2n+1)+h.c.], (1)
i j i k
0 (S > 1, i = j,k; j = k) naturally·appear in· the fourth
. | i| 2 6 6 in the extremely quantum case of on-site spins S =1 and
1 order of the strong-coupling expansion of the two-orbital σ = 1. Here J = cosθ, J = sinθ (0 θ < 2π), and L
0 Hubbard model [1]. Since in this case both types of cou- denot2es the nu1mber of uni2t cells contai≤ning two different
6 plings are controlledby one and the same model parame-
spins (S >σ). The modelprovidesa simple,but realistic,
1 ter–whichisabouttwoordersofmagnitude weakerthan
exampleofaHeisenbergsystemwiththree-bodyexchange
:
v theprincipalHeisenbergcoupling–itmightbeachallenge interactions. For the class of models with σ = 1 the bi-
Xi tfeocitdseonfthifiygheexrp-eorridmeernintatellryacatcicoensssicbalnesbyesdteemfinsiwtehlyeriesotlhaeteedf-. quadratic terms (σi·Sj)2 reduce to bilinear He2isenberg
spin-spin interactions, so that Eq. (1) represents already
r Unlikethebiquadraticexchangecouplings[2],sofarthere
a the general form of the alternating-spin Heisenberg chain
is no clear evidence for effects in real systems related to
with higher-order isotropic exchange interactions.
three-bodyexchangeinteractions,althoughpossiblethree-
Inthis article,weanalyzethe quantumphasediagram
body exchange effects in some magnetic molecules [3,4]
ahnavdeinbetehnedsipscinu-s25sedH.eisenberg chain CsMnxMg1−xBr3 [5] oσf=th21e.aObuovremmotoidvaeltifoonrftohretphaisirwoofrklofcoallloswpsinfsroSm=a p32raevnid-
ously established tendency towards formation of compos-
Onthetheoreticalside,onlyrecentlysomespecificfea- ite spins from the local S and σ spins in the unit cell–an
turesofthethree-bodyexchangeinteractioninHeisenberg effect of the three-body exchange interactions in the re-
spin models in space dimensions D=1 [1,6,7,8] and D=2 gion π < θ < 3π of the classical phase diagram, which
4 4
[9,13,14] have been discussed in the literature. In partic- is characterized by a macroscopic (2L) degeneracy of the
ular, two of us (N.B.I and J.S) recently analyzed the full ground state (GS) [7]. Therefore, one may expect com-
quantum phase diagram of the alternating-spin Heisen- pletely different phase diagrams for systems with integer
2 N. B. Ivanov et al.: Alternating-spin (3,1) Heisenberg chain
2 2
and half-integer total spin (S + σ) in the unit cell, es- 2.5
pecially in the highly degenerate classical region. Based 2
s
on density-matrix renormalization group (DMRG) simu- n 2n−1 2n 2n+12n+2
o 1.5
lations,inthenextSectionweanalyzethequantumphase i
diagram of the model (1) with S = 3 and σ = 1 and dis- lat 1 CR
2 2 e
cuss different properties of the phases appearing in the r 0.5
r
interval 0 < θ < π. The last Section contains a summary co 0 Cσ
of the results. ge −0.5 C
n L
a −1
r
−−1.5
t
2 Quantum phase diagram or −2
h C
S
S−2.5
−3
20 40 60 80 100 120 140 160
FiM θ1θ2 SL θ3 θFFM θ [deg]
0 30 60 90 120 150 180 θ[deg] Fig.2.(Coloronline)Short-rangespin-spincorrelationsofthe
F(1i)gf.or1.S(=Co32loarnodnσlin=e)12Qiunatnhteumintperhvaasle0d<iagθr<amπ:oTf thheeremgoiodnesl (cid:0)SC32R2,n+21≡(cid:1)2ihc;Shn2ani=n·1vσs,2.nθ.+.(1,DiL,M.CRσG≡, OhσB2Cn−,1L=·σ224n)+.1CiL, a≡ndhσC2Rn−≡1·hSS22nni·,
θ < θ1 and θ > θF are occupied, respectively, by the N´eel
ferrimagnetic (FiM) and ferromagnetic (FM) phases, whereas
the intervals θ1 < θ < θ2 and θ3 < θ < θF are occupied by
2.1 Partially-polarized magnetic states
different types of partially-polarized magnetic states. A large
parameter region, θ2 <θ <θ3, is occupied by a non-magnetic
doubly-degenerate gapped phase (SL). The FM point θF = The establishedpartially-polarizedmagnetic states inthe
π−arctan(1)≈153.43◦ isanexactboundaryoftheFMstate,
2 intervals θ < θ < θ and θ < θ < θ do not appear in
◦ ◦ ◦ 1 2 3 F
θ1 =20.1 , θ2 =25.5 , and θ3 ≈132 . theclassicalphasediagram.ThecriticalFiMphaseinthe
first interval is identical to the partially-polarized phase
discussedfortheextremequantumcase(S,σ)=(1,1)[7]:
The generalstructure of the phase diagram,as wellas 2
Itischaracterizedbyamonotonicallydecreasingmagneti-
the accepted abbreviations for the phases, are presented
zationfromm =(S σ)=1atθ =θ downtom =0at
0 1 0
inFigure1.Mostoftheresultsinthissectionareobtained −
the phase boundary θ with the non-magnetic phase. At
2
through DMRG simulations by performing seven sweeps
thephaseboundaryθ thegapoftheAFMbranchofexci-
1
and keeping up to 500 states in the last sweep [10,11,12].
tations∆ =E(M +1) E(M )vanishesandthesystem
A 0 0
This ensures a good convergence with a discarded weight −
becomes critical. Here M =(S σ)L corresponds to the
of the order of 10−8 or better. The numerical DMRG 0 −
GS of the Lieb-Mattis FiM. Unlike the extreme quantum
analysis of the lowest energy eigenvalues E(M) in sectors
case,wherethephaseboundaryθ marksthetransitionto
2
with a fixed z component of the total spin M imply (i)
agaplesscriticalphase,hereθ isrelatedwiththevanish-
2
adoubly-degeneratenon-magneticgappedGS(SL)inthe
ing of the triplet gap ∆ of the non-magnetic phase SL.
T
intervalθ <θ <θ and(ii)anumberofspecificpartially-
2 3 Skipping further discussions on this interesting FiM criti-
polarizedmagneticstatesinthe intervalsθ <θ <θ and
1 2 calstate,we onlymentionthatsimilarpartially-polarized
θ <θ <θ . Many features of the phase diagram in Fig-
3 F (non-Lieb-Mattis-type) FiM phases have been identified
ure 1 are also encoded in the behavior of the short-range
and studied in other spin models, as well [15,16,17].
correlations (SRC) for open boundary conditions (OBC)
Now,letusturntothemagneticstatesstabilizedinthe
(see Figure 2). In particular, most of the phase bound-
interval θ <θ <θ close to the FM point θ . The exact
ary points in Figure 1 can be associatedwith pronounced 3 F F
phase boundary θ coincides with one of the instability
rearrangements of the SRC. As in the previously studied F
points of the one-magnon FM excitations and is charac-
extreme quantum case of Eq. (1) with S = 1 and σ = 1
2 terized by a complete softening of the dispersion function
[7],thebasicrearrangementsconcerntheSRCbetweenthe
in the whole Brillouin zone. As a result, one observes a
largerS spins,whereas–apartfromtheregionclosetothe
strong reconstruction of the FM state for smaller values
FM point θ – the SRC between the σ = 1 spins remain
F 2 ofθ.Asamatteroffact,forθ <θ weobserveabehavior
almost constant.1 The tendency towards spin clustering F
of the SRC which is similar to one in the extreme quan-
is revealed by different values of the spin-spin correlators
tumsystem(seeFigure4binRef.[7]).Forthisreason,we
C = σ S and C = S σ in the SL
L h 2n−1 · 2ni R h 2n · 2n+1i shall restrict our discussion mainly to the region which is
state (see Figure 2).
extremelycloseto the FMpointθ ,asitisnaturaltoex-
F
pect that the formationofspecific plateaustates depends
1 The equation for the exact FM boundary θF for arbitrary on the values of the local spins: According to the general
spins S and σ reads cosθF +σ(2S+1)sinθF =0 [7]. rule, the number of unit cells in the periodic structure q
N. B. Ivanov et al.: Alternating-spin (3,1) Heisenberg chain 3
2 2
1.42 M 6
2n+4
M
1.41 2n
5
s
n total
o M
i 1.4 n
t
a 4
z
i
et 1.39 4
n M 0
g 2n+2 1 3
a x
m 1.38 A
S S ∆
l 2n 2n+4
a 2
c 1.37
o
L S
2n+2
1.36 1 θ = 153.4o
70 80 90 100 110 120 130 140
n
Fig. 3. (Color online) On-site magnetizations Mk = hSkzi 0
(k=2n,2n+2,2n+4)andMntotal≡(M2n+M2n+2+M2n+4)/3 0 0.02 0.04 0.06 0.08
as functions of the cell index n (DMRG, θ=153.4◦, L=144, 1/L
OBC). The results indicate a periodic three-cell (q =3) mag-
neticstructure close to theFM transition point θF. TheInset Fig. 4. (Color online) Finite-size scaling of the AFM gap
shows the magnetic supercell containing six spins (i.e., three ∆A=E(M0+1)−E(M0) abovetheplateau state with mag-
unit cells). The total magnetization Mntotal in the supercell is netization m0 ≡M0/L= 35 (DMRG, OBC).
constant.
S . Thus, as a first approximation, S +σ (n =
2n 2n 2n−1
and the magnetic moment per unit cell m of the plateau 1,...,L)canbetreatedasaspin-1operatorlocatedatthe
0
states fulfill the equation q(S+σ m )=integer [18]. n th unit cell. Respectively, the low-energy sector of the
0
− −
InFigure3weshowDMRGresultsforsomelocalmag- chaincanbeanalyzedbyusingtheprojectedHamiltonian
netic moments related to the S spins at θ = 153.4◦, i.e., eff =Q† Q, where the operator Q is defined as
H H
extremelyclosetotheexactFMboundaryθ .Theresults
F L
clearly indicate a periodic magnetic structure with a pe-
Q= Q , Q = α α .
n n n n
riod of three unit cells. As required for a plateau state, | ih |
the established magnetization at this point, m = 5, ful- nY=1 αnX=0,±
0 3
fills the mentioned general rule with q = 3. The DMRG Here α (α = 0, ) are the canonical basis states of
n n
results for ∆A at θ = 153.4◦ shown in Figure 4 give fur- the co|mpiosite-spin o±perator S2n + σ2n−1 in the spin-1
ther supportforthe suggestedplateaustate since the gap subspace.IntermsoftheIsingstates Sz ,σz thebasis
| 2n 2n−1i
is very small but definitely non-zero. Unfortunately, due states α read
n
| i
tostrongfinite-sizeeffects,itisdifficulttodecideifthein-
1 1 1 1 1
dicatedstateisrealizedonlyatθ =θF,orinasmallinter- 0 n = , ,
valclosetothispoint.Further,asintheextremequantum | i √2(cid:18)(cid:12)− 2 2E−(cid:12)2 −2E(cid:19)
(cid:12) (cid:12)
case,thenearest-neighborspin-spincorrelatorCS remains = 1 √(cid:12) 3 3, 1(cid:12) 1, 1 , (2)
positive and signals a FM ordering of the spin-S subsys- |±in 2(cid:18)∓ (cid:12)± 2 ∓2E±(cid:12)± 2 ±2E(cid:19)
temintheentireintervalθ <θ <θ .Thetransitiontoa (cid:12) (cid:12)
3 F where for simplicity we(cid:12)have omitted(cid:12)the cell index n on
non-magnetic state is accompanied by an abrupt change
the right-hand side of the equations.
of the sign of the correlator C . Approaching the transi-
S Calculating the matrix elements of the operators S
tionpointθ ,the boundaryeffects inopenchainsbecome 2n
3 and σ in the basis (2), one obtains
stronger, so that by using DMRG simulations it is diffi- 2n−1
citusltpotositsitound.y the vicinity of θ3 and to fix more precisely Q†nS2nQn = 54S′n, Q†nσ2n−1Qn =−41S′n, (3)
where the effective spin-1 operators S′ are defined as fol-
2.2 The non-magnetic SL phase lows: S′z = + + , S′+ =√2(+ 0 + 0 ),
| ih |−|−ih−| | ih | | ih−|
′− ′+ †
and S = S for each unit cell. Finally, a substi-
The numericalresults presentedin Figure2 showthat for
(cid:16) (cid:17)
OBCthe non-magneticphase (SL)occupyingthe interval tution of Eqs. (3) in the expression for eff leads to the
H
θ <θ <θ is characterized by different nearest-neighbor following effective Hamiltonian
2 3
spin-spin correlations, C =C . Excluding some vicinity
of the phase boundary θ3L,6the nRumericalestimates for CL = 5J L+J L S′ S′ , (4)
arelocatedneartheeigenvalue 5 oftheoperatorσ Heff −4 1 eff n· n+1
−4 2n−1· nX=1
4 N. B. Ivanov et al.: Alternating-spin (3,1) Heisenberg chain
2 2
The dimerization effect of the three-body interaction
a) in the whole interval θ < θ < θ can be approximately
2 3
studiedbyasimpledecouplingofthe three-bodytermsin
the original Hamiltonian (1):
b)
(S σ )(S σ )+h.c.
2n 2n−1 2n 2n+1
· ·
= 2C (S σ )+2C (S σ ) 2C C .
L 2n 2n+1 R 2n 2n−1 L R
· · −
c)
SubstitutingtheaboveexpressioninEq.(1),weobtainthe
following ”mean-field” spin Hamiltonian with alternating
FM-AFM exchange bonds
d) L
= [J (S σ )+J (S σ )] E , (5)
MF AF 2n 2n−1 F 2n 2n+1 0
H · · −
Fig. 5. (Color online) Valence-bond-solid picture of the dou- nX=1
bly degenerate non-magnetic phases according to Eq. (5) in where J = J +2C J , J = J +2C J , and E =
AF 1 R 2 F 1 L 2 0
the limits |JF|≪JAF (a,b) and |JF|≫JAF (c,d). The small 2LJ2CLCR. Note that the decoupling procedure violates
blackdotsdenotespin-21 variables.Thelinesbetweentwospins the translational symmetry of the original Hamiltonian
12 denoteasinglet bond,whereas thedashedellipses andrect- (1). Since the unit cell in Eq. (5) is doubled, there is
anglesdenotesymmetrizationofthespin-21 variables.Thefirst a pair of such Hamiltonians (connected by the symme-
two(thelasttwo)VBSstatesapproximatelyrepresentground
try transformation J J ) related to both types
F AF
states of the open spin-1 (spin-2) AFM Heisenberg chain. In ←→
of dimerization functions (Ψ ) introduced above. The
L,R
theintermediateregion(|JF|≈JAF)onlyapartthecomposite decoupling procedure can |be roiughly justified by noting
cell spins form spin-2states.
that almost in the whole non-magnetic interval the val-
ues of C are close to the eigenvalue 5 of the operator
where Jeff = 156 25J2−J1 . Ssp2inn-·1σs2tLna−te1s (inseethFeiguunriet 2ce).llsTfhoirs eaapcph−ro4nxim=a1te,ly im,Lp.liIens
As a matter(cid:0)of fact, E(cid:1)q. (4) coincides with the first- thespin-1subspace,thematrixelementsofthet·h·e·e-body
order effective Hamiltonian resulting from the decoupled-
term in Eq. (1) coincide with the matrix elements of the
dσi2mn−er1.limDeitpdenefidninegdobnytthheeHsiagmnioltfoJneiffa,ntHhe0a=boJv1ePHLna=m1iSlto2n-· Hopeeisreantobrersgtrtuecrtmur−e52oJf2EPq.Ln(=51)Sca2nn·bσe2nr+ep1,rosoduthceadt.thebasic
niansupportsagappedHaldane-typephase(J >0)and
eff In approaching the phase boundary θ , the coupling
2
a partially-polarizedFiM phase (J <0). The transition
eff constant J goes to zero (see Figure 6), so that in this
point at J = 0 (i.e., θ = 21.8◦) corresponds to a com- F
eff case the decoupled-dimer limit becomes a valid approx-
pletelydimerizedGSconstructedfromindependentspin-1
imation. Up to first order in J /J , the Hamiltonian
F AF
dimers.Thispointisrelatedtothenumericallyestablished | |
isequivalenttotheprojectedspin-1Hamiltonian(4)
MF
phase boundary at θ =θ2. wHith J = 5 J (J >0). The obtained phase bound-
InFigure5(a,b)wepresentthesuggestedvalence-bond- ary (noeffw de−fin16edFas Jeff = 0) surprisingly well reproduces
F
solid (VBS) states |ΨLi and |ΨRi implementing the dis- the numerical estimate θ2 = 25.5◦. As far as the param-
cusseddimerizationfeaturesoftheGS.UnderOBC,there eter J increases with θ, it seems relevant to evaluate
F
aretwosuchstatesdependingonthepositionoftheAFM the e|ffec|t of the second-order perturbation in J /J ,
F AF
bond in the three-spin clusters σ2n−1–S2n–σ2n+1 (n = as well. However, such a perturbation does no|t l|ead to
1, ,L).Usingthe Schwingerrepresentationforanarbi- any qualitative changes of the GS because its effect is re-
···
traryspin-Soperatorwithtwotypesofcommutingbosons stricted to a small renormalization of J and to appear-
eff
(i.e., S+ =a+b, Sz =a+a b+b, where a+a+b+b=2S), ance of an irrelevant (FM) next-nearest-neighbor Heisen-
−
wthreittreenlatinedthVeBfoSrmstate |ΨLi for a periodic chain can be sbterrugcttievremtoinanEaql.yz(e5)t.hAecottuhaelrly,defocroulpalregde-rdi|mJFer| iltimisitino-f
Eq. (5) based on non-interacting (FM) spin-2 dimers and
L
using the small parameter J /J 1. Up to first
Ψ = a+ b+ b+ a+ a+ b+ b+ a+ 0 . AF | F| ≪
| Li 2n 2n+2− 2n 2n+2 2n−1 2n− 2n−1 2n | i order in JAF/JF , this gives Eq. (4), but now with the
nY=1(cid:0) (cid:1)(cid:0) (cid:1) AFM coupling|J | = 3 J and the effective spin-2 op-
eff 16 AF
Herea+a +b+b =2S (2σ)fori=2n(i=2n 1)and 0 eratorsS′n.IntermsofVBSstates,the formationoflocal
i i i i − | i spin-2statescorrespondstoanadditionalsymmetrization
isthevacuumbosonstate.Noticethatthestates Ψ and
| Li of the cell spins, as shown in Figure 5(c,d), without any
Ψ foranopenchainhavedifferentnumberof”dangling”
s|pRini-1 free bonds suggesting different degeneracy of the abrupt change in the topological structure of the singlet
2 bonds.Therefore,itmaybespeculatedthatthetransition
GSinthethermodynamiclimit.Thisfactmayexplainthe
between both dimer limits is realized through a smooth
observed automatic selection of one of both states in the
crossoverbetween both Haldane-type gapped states.2
DMRG simulations (see, e.g., Figure 2) and considerably
complicatesthe analysisofthe low-energysectorfor open 2 The alternating-bond FM-AFM Heisenberg chain (5), de-
chains. scribing a smooth transition between the spin-1 and spin-2
N. B. Ivanov et al.: Alternating-spin (3,1) Heisenberg chain 5
2 2
2
1.5 0.12
J
AF
1
0.5
0.08
Τ
0 ∆
−0.5
0.04
−1
J
F
θ
−1.5
2
0
−2 20 40 60 80 100
θ
θ
2 30 50 70 90 110
θ[deg] Fig.7.(Coloronline)Theextrapolated(uptoL=60)thriplet
gap∆T vsθ intheSLstatecalculated byDMRGunderOBC
(open circles) and PBC (filled circles). The filled triangles de-
Fig.6.(Coloronline)TheeffectiveexchangeconstantsJF and notethe quintedgap underPBC.
JAF in the Hamiltonian HMF as functions of θ.
3 Summary
InFigure7wepresentnumericalresultsforthetriplet
Wehaveestablishedthe generalstructureofthe quantum
energy gap ∆ in the discussed parameter region. The
growth of theTgap approximately up to θ 45◦ can be phase diagram of the alternating-spin S = 32 and σ = 21
related with the established increase of the≈effective ex- Heisenbergchainwithextraisotropicthree-bodyexchange
change constant J in Eq. (4). In accord with the sug- interactions. To some extent the established partially-po-
eff
gested VBS state in Figure 5(a), for OBC one observes larized FiM phases resembler the magnetic phases of the
the expectedstructureofthe lowestexcitedstatesinclud- extreme quantum chain with alternating spins S =1 and
ing a singlet GS, which is degenerate with the Kennedy σ = 21 [15], apart from the vicinity of the FM point θF
where both systems support different plateau states. On
edge triplet in the thermodynamic limit [19]. The first
the other hand, due to the clustering effect of the three-
bulk excitation, related to the Haldane gap, appears as a
body interactions, both models support completely dif-
spin-2 (quintet) state resulting from the combination of
ferent quantum phases in the non-magnetic region of the
the bulk andKennedy’s edge triplets. Onthe other hand,
for θ > 45◦ the structure of the lowest excited states be- phasediagram:thecriticalphaseinthe(1,12)modelisre-
placedbyaspecificdoubly-degeneratephase,whichcanbe
comes very complicated due to the presence of many par-
described as a Haldane-type gapped state predominantly
asitic edge excitations. Namely, as demonstrated in Fig-
composed of effective cell spins with quantum spin num-
ure5(b,c,d),thenumberoffreeedgespinsandtheirvalues
depend on (i) the type of established VBS states (Ψ or bers1or2.Itmaybeexpectedthatmostofthepredicted
L
| i effects and phases persist in higher space dimensions.
Ψ ) and (ii) the increased tendency (with θ) towards
R
| i
formationoflocalspin-2states.Forthis reason,forlarger
θ the gap ∆ is presented for periodic chains. Since the
T
Acknowledgment
increase of J is restricted to J . 2, the true spin-
f f
| | | |
2 dimer limit is not reached. Nevertheless, as far as the
This work was supported by the Deutsche Forschungsge-
pure spin-2 phase is characterized by an extremely small
meinschaft(SCHN615/20-1)andbyanexchangeprogram
energygap–∆=0.085(5)J accordingtotheDMRGresult
between Germany and Bulgaria (DAAD PPP Bulgarien
in Ref. [20]–it is reasonable to admit that the established
decrease of ∆ for θ & 45◦ is connected with a smooth 57085392 & DNTS/Germany/01/2). S. P. was partially
T
supportedbyDAAD(PPPProjectID57067781),theFSP
crossoverbetweenthe spin-1andthe spin-2 Haldane-type
AMMOattheUniversityofAppliedSciencesinBielefeld,
non-magnetic states. Finally, due to the extremely small
and the bulgarian NSF (Grant DFNI I-01/5). We thank
gap∆ andthelargenumberoflow-lyingenergystates,it
T
J. Ummethum for help with the DMRG program.
is difficult to give a precise DMRG estimate for the other
phase boundary θ and the properties of the GS close to
3
this boundary.
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