Emergent incommensurate correlations in the frustrated ferromagnetic spin-1 chains

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by  Hyeong Jun Lee

Emergent incommensurate correlations in the frustrated ferromagnetic spin-1 chains
Hyeong Jun Lee and MooYoung Choi
Department of Physics and Astronomy and Center for Theoretical Physics, Seoul National University, Seoul 08826, Korea
Gun Sang Jeon∗
Department of Physics, Ewha Womans University, Seoul 03760, Korea
(Dated: January 12, 2017)
We study the frustrated ferromagnetic spin-1 chains, where the ferromagnetic nearest-neighbor
coupling competes with the antiferromagnetic next-nearest-neighbor coupling. We use the density
matrix renormalization group to obtain the ground states. Through the analysis of spin-spin cor-
relations we identify the double Haldane phase as well as the ferromagnetic phase. It is shown
7 thattheferromagneticcouplingleadstoincommensuratecorrelationsinthedoubleHaldanephase.
1 Suchshort-rangecorrelationstransformcontinuouslyintotheferromagneticinstabilityatthetran-
0 sition to the ferromagnetic phase. We also compare the results with the spin-1/2 and classical spin
2 systems, and discuss the string orders in the system.

n
a PACSnumbers: 75.10.Jm,75.50.Ee,75.40.Mg
J

1
I. INTRODUCTION ical studies on frustrated spin chains have been per-
1
formed.5,27–30 Onegoodexampleisthespin-1chainwith

] One-dimensionalquantumspinsystemshaveattracted two types of antiferromagnetic interactions, one between
l
e much interest for the past decade. One of the reasons is NN spins and the other between next-nearest-neighbor
- (NNN) spins.28–30 The chain exhibits a discontinuous
r that quantum fluctuations play a more dominant role in
t theirgroundstatesthanforhigher-dimensionalsystems.1 phase transition between two kinds of the AKLT states;
s
they are distinguished by the patterns of singlet bonds
. Generally quantum fluctuations suppress long-range or-
at der of a many-body system in low dimensions. In the in the chain. The two phases are also characterized by
m the existence of the hidden order due to breaking of the
presence of antiferromagnetic nearest-neighbor (NN) in-
Z ×Z symmetry.
- teractionstheinteger/half-integerHeisenbergspinchains 2 2
d
were suggested to have ground states with/without exci-
n
tation gaps;2,3 this was confirmed by extensive numeri-
o Ontheotherhand, wehaveanotherkindoffrustrated
c cal studies in the spin-1/24,5 and in the spin-14,6–11 sys- system,so-calledfrustratedferromagneticone,wherethe
[ tems. The resulting ground state of the spin-1 system
NN coupling is ferromagnetic. Numerous studies have
is known to have a finite gap9 and a finite correlation
1 been performed on such frustrated ferromagnetic sys-
length.10,12 It was also verified8 that the ground state is
v tems, especially for the spin-1/2 case.5,31–38 It has been
2 connected to the Affleck-Kennedy-Lieb-Tasaki (AKLT) revealedthatthespectrumdataincupratematerialssuch
3 state,13,14 which was originally obtained in the presence
as LiCu O and LiCuVO can be explained as the prop-
8 of bilinear and biquadratic interactions. In addition to 2 2 4
erties of the frustrated ferromagnetic system with good
2 theoreticalstudies,therehavebeenextensiveexperimen-
agreement.37 Itisalsofoundthatinterestingphasetran-
0 tal studies:15–22 Some of the materials considered can
. sitions exist at zero temperature in this system. This
1 be understood as one-dimensional spin-1 systems, giv-
motivatesustoexamineafrustratedferromagneticspin-
0 ing supports for the theoretical results. The data on
1 system, of which the full understanding still lacks.
7 Ni(C H N ) NO ClO provideevidencefortheHaldane
2 8 2 2 2 4
1
gap15,16 and additional spin-1/2 degrees of freedom17,18
:
v at the singlet-bond-broken sites in the presence of impu- Inthispaper,westudyfrustratedferromagneticquan-
Xi rities. GappedexcitationswerealsoobservedinCsNiCl3 tum spin-1 systems in one dimension. We obtain the
with a correlation length smaller than the prediction ground state through the use of the density-matrix-
r
a from the theory,20–22 which is now explained by weak renormalization-group(DMRG)method,andanalyzethe
couplings between spin-1 chains. The zigzag spin chain resulting spin-spin correlation functions. It is found that
in NaV(WO4)2 manifests a spin gap in the magnetic incommensurate short-range correlations are induced in
susceptibility.23 Measurements of magnetic properties of this system. Such correlations turn out to be smoothly
ANi2V2O8 (A=Pb,Sr) have also shown that they are de- connected to the ferromagnetic phase at the transition
scribed as a Haldane gapped phase without long-range with divergent correlation length. We also discuss the
order.24–26 string order parameters in the double Haldane phase.
Frustration, which is caused by the competition of This paper is organized as follows. In Sec. II, we de-
two or more exchange interactions or the system geom- scribe the model and the method. Section III is devoted
etry, turns out to generate enormous quantum emer- to the presentation and the discussion of numerical re-
gent phenomena in exotic phases. Extensive numer- sults. Finally, a summary is given in Sec. IV.
2
(a)
II. MODEL AND METHOD
Weconsideraone-dimensionalspin-1systemwithNN
and NNN couplings. The model is described by the
Hamiltonian
(b)
H=J (cid:88)Sˆ ·Sˆ +J (cid:88)Sˆ ·Sˆ (1)
1 i i+1 2 i i+2
i i
where Sˆ ≡(Sˆx,Sˆy,Sˆz) is a vector spin-1 operator at the
i i i i
ith site. The NN and the NNN exchange couplings are
denoted by J and J , respectively. We investigate the
1 2
Hamiltonian with ferromagnetic NN couplings (J < 0) L
1
and antiferromagnetic couplings (J > 0) at zero tem-
2
perature in this work.
In the absence of the NN coupling (J = 0) the sys- FIG. 1. (Color online) Schematic representation of a spin-1
1
temreducestotwodecoupledsubchains; eachconsistsof chain in (a) modified and (b) standard open boundary con-
NNN pairs interacting with the antiferromagnetic NNN ditions. The NN coupling J1 and the NNN coupling J2 are
interaction. The individual subchain lies in the Haldane represented by solid and dashed lines, respectively. Shaded
regions denote the singlet bonds which are expected to be
phase,whichischaracterizedbyanexcitationgap,expo-
formedbetweentwohalf-spinsoftwositesinthedoubleHal-
nentially decaying spin-spin correlations, and long-range
dane phase with J <0 and J >0.
string order.10 Such a phase is called a double Haldane 1 2
phase, in which singlet bonds are formed between NNN
pairs.
the average value for the systems with size L ≥ 50 and
In the presence of the antiferromagnetic NN coupling
marktheerrorbarswhentheyarelargerthanthesizeof
(J > 0) it was revealed28,29 that the double Haldane
1 symbols. Wehavealsocheckedouttheconvergenceofthe
phase undergoes a discontinuous transition to the Hal-
data with respect to m, where m is the number of states
dane phase at J /J (cid:39) 0.744. At the transition the sys-
2 1 perblockaftertruncation,byincreasingmconsecutively.
tem exhibits a jump in the string order parameter with
The truncation errors in our calculations with m = 200
the bulk gap and the correlation length remaining finite.
are of the order of 10−5 to 10−7 for the double Haldane
The transition is attributed to the breaking of a hidden
phase, in which the main interest of this work lies. They
Z ×Z symmetry in the Haldane phase.38
2 2 are larger in the central region of the double Haldane
When the ferromagnetic NN coupling is dominant
phase due to the increase of frustration. In the region
(|J | (cid:29) J ), the ground state is expected to be a fer-
1 2 of the ferromagnetic phase the truncation errors reduce
romagnetically ordered state with total spin S = L
tot significantly below 10−15. We have performed the calcu-
with gapless spin-wave excitations.39 It is also known
lationwithmupto250,andfoundthatmostphysicalre-
thattheferromagneticphaseisrobustagainstadditional
sultsdonotdependsignificantlyonm. Somem-sensitive
competing interactions. In the intermediate region the
physical quantities such as the correlation length have
thermodynamic behavior for the S=1 chains is less clear
been presented together with the extrapolation value to
in contrast with spin-1/2 chain systems.5,31–33,35,36
m=∞, and relevant error bars have been marked when
In this work, we use the DMRG method9,10,40 with
they are larger than the size of the symbols. Henceforth
the infinite algorithm to obtain the ground state of the
we will denote the energy and the length in units of J
system. We adopt slightly modified open boundary con- 2
and of the lattice constant, respectively, throughout this
ditions, which are sketched in Fig. 1(a): the last two NN
paper.
spins interact with J . In the standard open boundary
2
conditions in Fig. 1(b) the system tends to have two free
1/2-spins at each end, one on each subchain. The anti-
III. RESULTS AND DISCUSSION
ferromagneticinteractionJ inthemodifiedopenbound-
2
ary conditions helps the two free spins in standard open
boundary conditions to form a singlet bond, leading to To examine how the system evolves as the couplings
an effective finite-size calculation in the double Haldane vary, we first calculate the spin-spin correlation function
phase. We also performed comparative computation in C (l) defined by
S
thestandardopenboundaryconditionsandfoundnosig-
nificant qualitative difference in the bulk state. C (l)≡(cid:104)Sˆ ·Sˆ (cid:105). (2)
S i i+l
We have performed calculations in chains up to L =
200. Thespin-spincorrelationsindifferentsizesturnout Inordertominimizefinite-sizeeffects,wetakesiteisuch
to collapse to a single curve in a wide range of coupling that both sites i and i+l are as far from the boundaries
parameters as will be displayed later. We estimate the as possible. Figure 2(a) shows the spin-spin correlation
uncertainty in the data by the maximum deviation from function for J =0. The ground state for J =0 can be
1 1
3
8
(a)
1
0 6
-1
)
k 4
(
1 (b) S
0 2
) -1
l( 0
S
C 1 (c) -1 -0.5 0 0.5 1
k / π
0
FIG. 3. (Color online) Spin structure factor S(k) for var-
-1
ious NN couplings J . The data for J = 0, −0.5, −1, and
1 1
−2 with various lengths L = 80 to 200 are marked by (red)
(d)
1
squares,(green)circles,(blue)triangles,and(purple)inverted
triangles. The structure factor has two peaks in the double
0 Haldanephase,whicharenotdivergentinthethermodynamic
limit. AsJ isreduced,thepeaksmoveclosertok=0andthe
1
-1 system exhibits incommensurate short-ranged correlations.
0 25 50 75 100
l Until J reaches the value J ≡ −4 the system does not
1 c
showanyabruptchangeinthecorrelations,whichimplies
that in the region of J < J < 0 the system remains
c 1
FIG. 2. (Color online) Spin-spin correlation functions as in the double Haldane phase. For J < J , the spin-
1 c
a function of the separation l between the spins at J1 = spin correlation function takes just the constant value of
0,−0.5,−1, and −2 from top to bottom. The period of the
unity, independent of l, and the total spin of the ground
oscillation is 4 (i.e. k =π/2) at J =0 and becomes incom-
1 state turns out to be S = L, signifying that the sys-
mensurate as J decreases. tot
1 temisintheferromagneticphase. Thecriticalvalueinto
the ferromagnetic phase for S = 1 is the same as that
in the case of classical spin systems5 and S = 1/2 spin
simplyunderstoodintermsofthetwocompletelydecou-
systems,31,33–36 as pointed out in Ref. 41.
pled chains, resulting in finite spin-spincorrelations only
Foraquantitativeanalysisoftheevolutionofthestates
for even-integer separations l with vanishing correlations
with J varied, we examine the spin structure factor
for odd-integer separations. Within each subchain the 1
system lies in the Haldane phase, which exhibits short- (cid:88)
S(k)≡ eiklC (l),
rangeantiferromagneticcorrelations. Suchspin-spincor- S
l
relations are in good agreement with those plotted in
Fig. 2(a). The correlations decay exponentially with the which is the Fourier transform of C (l). We have used
S
separation l, superposed by the oscillating correlations fast Fourier transform algorithms for C (l) to obtain
S
with the period of four. S(k)andplottheresultingspinstructurefactorinFig.3
We also plot the spin-spin correlations for various fer- for various values of J . For J =0 we have broad peaks
1 1
romagneticNNcouplingsJ inFig.2. WhenJ isfinite, at k = ±π/2, which reflects oscillations with period 4.
1 1
thetwosubchainsarenolongerdecoupledandfinitecor- The peak broadens due to the exponentially decaying
relations show up for odd-integer separations. Interest- correlations. As|J |increases,thepositionsoftwopeaks
1
ingly, the correlations still display oscillating behavior move towards k =0. It is also clear that the peak grad-
apparently with a single period, which is different from ually becomes narrower with the increase of J . This
1
4, the value for J = 0; it increases monotonously with resultforS(k)givesquantitativesupporttoalltheobser-
1
the increase of |J |. It is also of interest to note that the vations on the spin-spin correlation function C (l) men-
1 S
oscillating period does not always appear commensurate tioned above. The structure factor remains finite even
with the lattice period. Another conspicuous feature is whenLisincreasedindefinitely,andthesystemdoesnot
that the decay of the correlations becomes slower as the exhibit long-range spin order in this region.
ferromagnetic NN coupling becomes stronger. While the Wecandeterminethepitchanglek∗ofthespincorrela-
correlations become negligible around l ≈30 for J =0, tionsbythepositionofthemaximuminS(k)ontheside
1
we can observe clear oscillations for l (cid:38) 60 at J = −2. ofpositivek. ForJ =0,thepeakislocatedatk∗ =π/2,
1 1
4
displays rather gradual decrease even near J = 0. The
1
1 curveofk∗forthespin-1chainlocatesbetweentheclassi-
calandthespin-1/2systems,whichmaybeattributedto
the reduction of quantum fluctuations in spin-1 systems
c double Haldane in comparison with spin-1/2 systems. It would be in-
eti
n terestingtostudyvariationsofthepitchangleforhigher
π g
/ 0.5 ma spins,whichshouldreflecttheeffectsofboththechanges
*k ro Haldane in quantum fluctuations and the alternating behavior of
er 100 integer and half-integer spins.
f
IntheinsetofFig.4,weplotk∗ asafunctionofJ −J
1 c
10-1 inthelog-logscale. Nearthecriticalpointthepitchangle
10-1 100 101 displays the power-law behavior
0
-5 -4 -3 -2 -1 0 1 2 3 k∗ ∼(J −J )α. (3)
J 1 c
1
The best fit in the range −3.9 ≤ J ≤ −3.0 gives the
1
exponent α = 0.47(3). Although the best-fit value of α
FIG. 4. (Color online) The pitch angle k∗ as a function
is a bit smaller than α = 0.5 for the classical spiral
of the NN couplings J1. The pitch angles, which are deter- cl
mined by the maximum position in the structure factor, are state, the two values are consistent within numerical er-
marked by (green) solid circles. The pure NNN-AKLT state rors. Numericalerrorsintheexponentaremainlydueto
(J = 0) has k∗ = π/2 reflecting the short-ranged antiferro- the large relative errors near J . It is also notable that
1 c
magneticcorrelationineachsubchain. Theantiferromagnetic thecurveofk∗ versusJ −J isslightlyconvexupinthe
1 c
correlation becomes incommensurate at finite J1 in the dou- log-log plot, which tends to give an additional underes-
ble Haldane phase. We also plot the results of the classical timate of α in the power-law fit over a finite window of
spin system, arccos(−J /4) (dot-dashed line), and those of
1 J .
1
the spin-1/2 system from Ref. 35 (triangles) and Ref. 38 (di-
In Fig. 4 we also plot the pitch angle in the case of
amonds) for comparison. The data marked by a solid line as
J >0: the solid line represents the data from an earlier
well as the transition point to the Haldane phase are quoted 1
study29 for S = 1 and the data points denoted by solid
fromRef.29. Theinsetshowsthepitchangleinalog-logplot
for J −J >0 with J =−4 and the dashed line represents circles are obtained from our calculation. The double
1 c c
the best power-law fit to the data. The data points without Haldane phase in this region gives pitch angles in the
errorbarshaveerrorsnotlargerthanthesizeofthesymbols. range π/2 < k∗ < π.28,29 The pitch angle increases with
J andbecomescommensuratewithk∗ =π aroundJ ≈
1 1
2.67. Remarkably, the incommensurate spin correlations
persist even in the region of the Haldane phase, which
which is consistent with the NNN AKLT state, the pro- sets in at J ≈ 1.34. This is in contrast to the fact that
1
totype state in the double Haldane phase. In Fig. 4 we thepitchanglereducestok∗ =0exactlyatthetransition
plotk∗ asafunctionofJ1. As|J1|increases,k∗ decreases to the ferromagnetic phase for J1 <0.
monotonously from k∗ = π/2 for J1 = 0 to k∗ = 0 for In order to examine the correlation length, we divide
J1 = Jc, and the ground state connects smoothly with thespincorrelationfunctionCS(l)bytheoscillatingfac-
theferromagneticstateatJ1 =Jc. Thedecreasingcurve tor cos(k∗l) and estimate the correlation length ξ by fit-
corresponds to a convex-up function. ting it to the exponential decay ∼exp(−l/ξ). In Figs. 5
Such behavior is reminiscent of the spiral state which and6onecanseethatC (l)/cos(k∗l)forJ =−1and−2
S
shows up in the classical spin system. In the presence decays exponentially in a wide range of l. Figure 5 also
of NN ferromagnetic couplings and NNN antiferromag- demonstratesthatthequantitiesindifferentsizesexhibit
netic couplings, the classical spins exhibit a spiral state almost the same exponentially-decaying behavior except
for −J < J < J with the wave number given by for near the edges. The decay tends to become slower as
c 1 c
q =arccos(−J /4).5 Similar behaviors of the pitch angle the number m of the kept states is increased. In Fig. 7
1
were also reported in previous numerical studies of the we plot the correlation length ξ estimated from the best
spin-1/2chainsviathetransfer-matrixDMRGmethod35 fitasafunctionofJ forvariousvaluesofmaswellasin
1
andtheinfinitetimeevolvingblockdecimationalgorithm the limit m→∞. We have also reproduced the data for
method37,38. For comparison, we have also plotted the J >0 from an earlier work.28 For J >0 our numerical
1 1
data for classical and spin 1/2 chains with a dot-dashed results are consistent with the peak associated with the
line and empty symbols, respectively, in Fig.4. In all transition between the Haldane and the double Haldane
the three systems the pitch angle k∗ reduces with the phases. For J < 0 the correlation length is enhanced
1
increase of |J | starting from k∗ = π/2 at J = 0, and as J approaches J . We also note that there exists a
1 1 1 c
approaches k∗ = 0 continuously at the transition into small bump around J =0, which signifies stronger spin
1
the ferromagnetic phase. In the case of the spin-1/2 fluctuations near the decoupled subchains.
chains, the plateau-like region persists near J = 0 up We also examine string order and double-string order
1
toJ ≈−2. Ontheotherhand,theclassicalspinsystem forJ <0, whichcanbeprobedbytheuseofthefollow-
1 1
5
40
100 (a) (b)
) 30
*klos(10-4 gnetic double Haldane Haldane
Cl() / cS ξ 1200 ferroma
-8
10
0
0 50 100 150 0 50 100 150 200 -5 -4 -3 -2 -1 0 1 2 3
l l J
1
FIG. 5. (Color online) Semi-log plot of the spin-spin corre- FIG. 7. (Color online) Correlation length as a function of
lationfunctionC (l)dividedbycos(k∗l)forseveralvaluesof theNNcouplingJ form=100[(green)squares],150[(blue)
S 1
L and (a) J = −1; (b) J = −2. The pitch angle k∗ is de- triangles], 200 [(pink) inverted triangles], and its extrapola-
1 1
terminedbythemaximumpositionofthestructurefactorfor tion to m = ∞ [(red) circles]. As we approach the phase
each J . The data denoted by (red) squares, (green) circles, boundary to the ferromagnetic phase the correlation length
1
(blue) triangles, and (pink) inverted triangles correspond to becomes very large and is expected to diverge at J = −4.
L=50,100,150, and 200, respectively. Solid lines denote the data quoted from Ref. 28. The data
pointswithouterrorbarshaveerrorsnotlargerthanthesize
of the symbols.
100 (a) (b)
0.15
O
1
O
) 2
l
*k 100
os(10-4 0.10 10-2
c
/ 10-4
) O
l 0.05
(S 10-6
C -8 -4 -3 -2 -1 0
10
0.00
0 50 100 150 0 50 100 150 200
l l -5 -4 -3 -2 -1 0
J
1
FIG. 6. (Color online) Semi-log plot of the spin-spin cor-
relation function C (l) divided by cos(k∗l) for several values
S
of m and (a) J = −1; (b) J = −2. The pitch angle k∗ is FIG.8. (Coloronline)Stringorderparametersasafunction
1 1
determined in the same way as in Fig. 5. The data denoted of the NN coupling J1. We represent O1 and O2 by (red)
by(green)circles,(blue)triangles,and(pink)invertedtrian- solidcirclesand(green)emptycircles,respectively. Theinset
glescorrespondtom=50,100,and150,respectively,andthe displays the semi-log plot of O2. The data points without
solid lines are the best linear fits. errorbarshaveerrorsnotlargerthanthesizeofthesymbols.
ing nonlocal correlators38,42,43 The string order parameters O and O can then be de-
1 2
fined by
(cid:42)   l(cid:48)−1  (cid:43)
O1(l,l(cid:48))≡− Slzexp (cid:88) iπSjzSlz(cid:48) O1 ≡|l−lli(cid:48)m|→∞O1(l,l(cid:48)), (4)
j=l+1 O ≡ lim O (l,l(cid:48)) (5)
2 2
|l−l(cid:48)|→∞
and
in the limit of infinite separations. Figure 8 shows string
(cid:42)   l(cid:88)(cid:48)−2  (cid:43) order parameters O1 and O2 versus J1. For J1 = 0, we
O2(l,l(cid:48))≡ SlzSlz+1exp iπSjzSlz(cid:48)−1Slz(cid:48) . have nonzero O2 with vanishing O1, which is a typical
j=l+2 characteristicofthedoubleHaldanephase. Ontheother
6
00 hibits an apparent exponential decrease, which indicates
1100
the absence of a chiral phase. This also implies that the
emergence of a new phase is less probable. We presume
that the great reduction in O for −4 < J < −2 is
2 1
1100--11 caused by the large ferromagnetic fluctuations which are
)| enhancedmarkedlyinthatregionascanbeseeninFig.7.
l
(κ
C
| --22
1100
IV. SUMMARY
--33
1100 We have investigated the one-dimensional spin-1 sys-
00 2200 4400 6600 8800 110000
tem frustrated by the combination of the spin exchange
l
interactions: the ferromagnetic one between NN spins
and the antiferromagnetic one between NNN spins. Via
FIG.9. (Coloronline)Semi-logplotofthechiralitycorrela- theDMRGcalculationswehaveconfirmedexplicitlythat
tion function Cκ(l) for J1 = −3. The data denoted by (red) the ferromagnetic phase transition for S = 1 occurs at
squares, (green) circles, (blue) triangles, and (pink) inverted J = −4(≡ J ),41,44 below which the spin-spin correla-
1 c
triangles correspond to m = 50,100,150, and 200, respec-
tion function becomes constant. The robustness of J
c
tively.
suggests that quantum fluctuations due to the quantum
natureofthespindonotaffectsignificantlytheferromag-
netic phase transition; this is in sharp contrast with the
hand, in the ferromagnetic phase (J <J ) both O and
1 c 1
fact that other phase boundaries of the frustrated spin
O vanish. As J decreases from zero, O also reduces.
2 1 2
chains exhibit strong dependence on S and a variety of
It is remarkable that O is significantly reduced for
2
J (cid:46) −2 while the transition into the ferromagnetic distinct phases.27 Such interesting results may be tested
1
through experiments on ultracold atoms.45–47
phase occurs at J = −4. Such an abrupt reduction
1
maysuggestthepossibleexistenceofemergentphasebe- In the double Haldane phase, on the other hand, the
tween the double Haldane phase and the ferromagnetic system shows short-ranged antiferromagnetic spin-spin
phase. However, the detailed analysis has revealed that correlations. Such behavior remains robust in a wide
the string nonlocal correlator O (l,l(cid:48)) shows quite dis- rangeoftheNNferromagneticinteractionstrengthJ up
2 1
tinct behavior for −4 < J (cid:46) −2 from that of the ferro- to J . The pitch angle of the incommensurate spin-spin
1 c
magnetic phase where the string order is absent. In the correlationsdecreasesfromπ/2asJ approachesJ ,van-
1 c
ferromagnetic phase O (l,l(cid:48)) shows a clear exponential ishing at J = J . Similar behaviors were also reported
2 1 c
decaywith|l−l(cid:48)|,implyingthatthecorrespondingstring in earlier works on the classical and spin-1/2 systems. It
orderparameterO vanishesinthethermodynamiclimit. hasbeenrevealedthattheresultsofthespin-1systemare
2
For −4 < J (cid:46) −2, in contrast, O (l,l(cid:48)) remains finite closer to the classical results than those of the spin-1/2
1 2
although its values are very small. The semi-log plot of one. We have also discussed the behavior of string order
O , which is presented in the inset of Fig. 8, also sup- parameters in the double Haldane phase in the presence
2
ports a direct transition from the double Haldane phase of ferromagnetic NN couplings. It has been found that
to a ferromagnetic one. It demonstrates that the de- the string order parameter O undergoes a substantial
2
creasing behavior is rather consistent with the transition reduction far above J . However, detailed analysis has
c
atJ =−4althougharapiddecreaseofO startsaround suggested that a new phase is less likely to emerge in
1 2
J ≈−2. that region. It is presumed that the enhancement in the
1
Itisalsoworthwhiletoexaminewhetherachiralphase ferromagnetic fluctuations gives rise to such substantial
ispresentbetweenthedoubleHaldanephaseandthefer- reduction in O .
2
romagnetic phase. In the spin-1/2 case the chiral phase
Various extensions of our study are expected to pro-
has been reported in a close vicinity of the region,37,38
duce fruitful results in future study. The introduction
and accordingly it is a strong candidate for a new phase of the anisotropic interaction of the XXZ-type27 in our
if it emerges also in the spin-1 case. We have thus com-
model will reduce the rotational symmetry and possi-
puted the chirality correlation function defined by
bly give rise to another symmetry-broken state such as
C (l)≡(cid:104)κˆzκˆz (cid:105) (6) the chiral or the dimer-like ones. It will deepen our
κ i i+l
understanding of the roles of quantum fluctuations in
with connection with the Kosterlitz-Thouless transition. It
κˆz ≡zˆ·(Sˆ ×Sˆ )=SˆxSˆy −SˆySˆx . (7) is also worth while to investigate the effects of the uni-
i i i+1 i i+1 i i+1 axial single-ion anisotropy.48 It is expected to enrich the
Figure 9 presents C (l) for J = −3, where O is sig- physics in quantum spin-1 chains and to draw out a va-
κ 1 2
nificantly reduced. The chirality correlation function ex- riety of implications on experimental results.
7
ACKNOWLEDGMENTS
This work was supported by the National Re-
search Foundation of Korea through Grant No. 2008-
0061893(G.S.J.),GrantNo. 2013R1A1A2007959(H.J.L.
and G.S.J.), and Grant No. 2016R1D1A1A09917318
(M.Y.C.).
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