Spontaneous trimerization in a bilinear-biquadratic S=1 zig-zag chain

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by  Philippe Corboz

Spontaneous trimerization in a bilinear-biquadratic S = 1 zig-zag chain
Philippe Corboz,1 Andreas M. La¨uchli,2 Keisuke Totsuka,3 and Hirokazu Tsunetsugu4
1Institut fu¨r Theoretische Physik, ETH Zu¨rich, CH-8093 Zu¨rich, Switzerland
2Institut Romand de Recherche Num´erique en Physique des Mat´eriaux (IRRMA), CH-1015 Lausanne, Switzerland
3Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa Oiwake-Cho, Kyoto 606-8502, Japan
4Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
(Dated: February 1, 2008)
RecenttheoreticalstudiesraisedthepossibilityofarealizationofspinnematicstatesintheS =1
8 triangular lattice compound NiGa2S4. We study the bilinear-biquadratic spin 1 chain in a zig-zag
0 geometrybymeansofthedensitymatrix renormalization groupmethodandexactdiagonalization.
0 We present the phase diagram focusing on antiferromagnetic interactions. Adjacent to the known
2 Haldane-double Haldane and the extended critical phase with dominant spin nematic correlations
n we find a trimerized phase with a nonvanishing energy gap. We discuss results for different order
a parameters,energygaps,correlationfunctions,andthecentralcharge,andmakeconnectiontofield
J theoretical predictions for thephase diagram.

1
2 PACSnumbers: 75.10.Jm,75.10.Pq,75.40.Cx,75.40.Mg

] 1 3 5 7 9 11
l Introduction—Quantumspinsystemshaveprovideda
e
- verywide playgroundfor the questof novelquantum or- J1
r
ders, and the short catalog includes Haldane gap, dimer
t
s order, chiral order and others. Recently the discovery of 2 J2 4 6 8 10 12
.
t spin liquid like behavior in the spin-1 triangular magnet
a
m NiGa S [1]hasstimulatedincreasinginterestsinanother FIG.1: Zig-zaggeometryofthemodelwithJ1 (J2)thenear-
2 4
est (nextnearest) neighbor coupling strength.
type of quantum order: spin nematic order. This is the
-
d long range order of quadrupole moments of local spins,
n
in contrast to dipole moment order parameter in con-
o of bilinear and biquadratic coupling. For θ = 0 (θ =
ventional magnetic long range orders. In a spin nematic
c
π/2) the biquadratic terms (bilinear terms) vanish. For
[ ordered phase, although spins show no static moment,
θ = π/4 the Hamiltonian exhibits SU(3) spin symme-
spin rotation symmetry is spontaneously broken due to
2 try. J andJ arethe nearestandnext nearestneighbor
anisotropic spin fluctuations. For S=1 spin operators, 1 2
v
coupling strengths, respectively. The model is best visu-
5 anisotropic spin fluctuations still have uniaxial symme-
alized in a zig-zag geometry where the J bonds couple
9 try, andthis symmetry axis is called director,in analogy 1
1 to a liquid crystal. Ferro and antiferro spin nematic or- twochainsandtheJ2 bondsarelocatedalongthechains
1 (Fig. 1). We concentrate on antiferromagnetic interac-
ders were independently proposed by different groups as
7. an explanation for the unusual low-temperature proper- tions on all bonds with 0 ≤ θ ≤ π/2 and J1,J2 ≥ 0. By
0 ties of NiGa S [2, 3, 4]. In particular,the antiferrospin setting θ = 0 the model reduces to the J1 −J2 spin 1
2 4
7 chainwhich exhibits a first order transitionbetween two
nematic order is possible to match the triangular lattice
0
distinct topological orders [5]. The Haldane phase for
structure without any frustration, and the ground state
:
v is unique aside from degeneracy due to global spin rota- J2/J1 <αt ≈0.775,whichalsocontainsthe wellstudied
i spin1chainforJ =0,ischaracterizedbya finite string
X tion. Although it remains open if antiferro spin nematic 2
order parameter (SOP). The ground state is a valence
order is realized in this material, it is interesting and
r
a also important to investigate further this state and ob- bond solid (VBS) where each site consists of two spin
1/2 forming a triplet and two spin 1/2 on neighboring
tain a better understanding. To this end, we shall study
sites couple to a singlet. The transition into the double
a one-dimensional analog with the same three sublattice
Haldane (DH) phase corresponds to a decoupling of the
structure, a zig-zag chain.
Model — The Hamiltonian of the bilinear-biquadratic VBS string into two intertwined substrings, where each
string exhibits string order [5], leading to a finite double
(BLBQ) spin 1 zig-zag chain is given by
string order parameter (DSOP).
H = J1 cosθ(Si·Si+1)+sinθ(Si·Si+1)2 The line J2/J1 = 0 corresponds to the BLBQ chain
X studied, e.g., in [6]. The gapped Haldane phase (includ-
i
+ J2 cosθ(Si·Si+2)+sinθ(Si·Si+2)2. (1) ing the Affleck-Kennedy-Lieb-Tasaki (AKLT) point for
tanθ =1/3)ends at the SU(3) symmetric point θ =π/4
X
i
(exactly solvable Uimin-Lai-Sutherland model) where a
Si’sarespin1operatorsandθ parametrizesthestrength second order phase transition of Berezinskii-Kosterlitz-
2
3.5 three neighboring spins couple predominantly to a sin-
TOP
glet. The trimer ground state is threefold degenerate
1st order line
3 Trimerized andbreakstranslationalinvariance. Thisphase–includ-
2nd order line
SU(3) line ing the frustration process leading to it – is reminiscent
2.5 c=2 of the dimerized phase of the J -J spin 1/2 chain for
1 2
ED J /J ≥0.2411 [10].
2 1
2
1 Double Haldane Initially a massive trimerized phase for the spin 1
J/J21.5 Critical Heisenberg chain (J2/J1 = 0) for π/4 < θ < π/2 was
put forward[11, 12], but laterworks[6, 7, 13,14, 15, 16]
showed that the region remains massless and has domi-
1
nant nematic correlations. In our model the additional
0.5 Haldane next nearest neighbor coupling J2 allows us to stabilize
the trimerized state.
0 In previous work [17, 18] parent Hamiltonians have
0 0.1 0.2 0.3 0.4 0.5
θ/π beenconstructedusingcomplicatedfoursiteinteractions,
whichexhibitexacttrimergroundstates(inRef.[19]yet
FIG.2: (Coloronline)Thephasediagramofthespin1BLBQ adifferentkindoftrimerizedstatewasconstructed). We
zig-zag chain. The area of the black dots in the trimerized now have found a trimerized phase that is stable in a
phase scales with the magnitude of the trimer order param-
finite region of the parameter space of a much simpler
eter. We verified that the central charge is 2 in the critical
and possibly realistic spin Hamiltonian. We expect our
phase for several points (red diamonds). The green triangles
model (1) to be related to these parent Hamiltonians in
mark the phase boundary of the trimerized phase obtained
from a level spectroscopy analysis from theED data. Refs. [17] and [18] in the same spirit as the Heisenberg
spin 1 chain is related to the AKLT model.
One of the most direct indications for a trimerized
Thouless (BKT) type occurs [7]. Thus there is an expo- phaseisthe period3inthelocalbondenergies(insets of
nentially slow opening of the gap on the Haldane side. Fig.3). Thispatternisformedbecausethetwobondsbe-
For π/4 ≤ θ < π/2 one finds an extended critical phase longing to a trimer have a lower energy than the bonds
with soft modes at k = 0,±2π/3 and central charge connecting two trimers. We determine this oscillation
c=2. Thedominantcorrelationsinthisphaseawayfrom amplitude ofthe localbondenergiesinthe middle ofthe
the SU(3) point are quadrupolar (spin nematic) [6, 7]. chain for different system sizes L and extrapolate it to
Numerical simulations — The numerical results have L → ∞. In the Haldane (respectively DH) phase the
been obtained by the density matrix renormalization amplitude vanishes exponentially with L. In the critical
group (DMRG) [8, 9] method and exact diagonaliza- region the amplitude extrapolates to zero with a power
tion (ED). For the DMRG calculations we used up to law. Butinthetrimerizedphasetheextrapolationofthe
m = 2000 states with typically 6 sweeps and system oscillation amplitude clearly yields a finite value, which
sizeswithopenboundaryconditionsuptoL=300sites. we call the trimer order parameter (TOP) (see Fig. 3).
Quantitiesofinterestareextrapolatedinmandifneeded The magnitude of the TOP in the trimerized phase is
also for L→∞. Error bars are estimated from the con- proportional to the area of the black dots in Fig. 2. On
vergencebehaviorinm. WithEDweconsideredsystems
with periodic boundary conditions up to L=21.
The overall phase diagram — Figure 2 summarizes
the phase diagram of the Hamiltonian (1) obtained by 0.4
DfoMrtRhGepahnadseEsDonstimheualaxteiso:nHs.aWldaenree,ptrhoedDucHed,atnhdetrheesuelxts- ude0.3 DexMtrRapGolation S=0 S=0 S=0 S=0
tendedcriticalphasewithc=2. Allthesephasesextend plit 1
intothe plane. Interestinglytheyalltouchthe dominant Am0.2 Hi,i+10.5
0
phase in the center, a gapped trimerized phase, which 1
will be discussed below. 0.1 Hi,i+20.5
0
The Haldane - double Haldane transition [5] point ex- 0 0 10 20 30 40
0 0.005 0.01 0.015 0.02
tends as a first order line in the parameter space which
1/L
terminatesupontouchingtheboundaryofthetrimerized
phase. We confirmed the first order nature of the tran- FIG.3: (Coloronline)Extrapolationofthebondenergyoscil-
sitions by calculating the SOP and DSOP along several lationamplitudeinthemiddleofthechainleadingtoafinite
cuts for fixed θ and varying J /J . trimer order parameter in the trimerized phase (θ = 0.28π,
2 1
Trimerized phase — The most exciting feature of the J2/J1=1). Inset: The local bond energies form a pattern of
period 3 (L=48 in this example).
phase diagram is the gapped, trimerized phase, where
3
theSU(3)lineDMRGpredictsthatthetrimerizedphase 0.1
sets in at J2/J1 ≈ 0.45 and ends at ≈ 3.5. We used J2/J1=1.25
a level spectroscopy analysis of the ED data to comple- ∆ E0.05
ment the results (green triangles in Fig. 2). This tech-
DH Trimerized
nique has been successfully applied in the case of the
0
spin 1/2 chain to locate the transition point into the 50
L=90
dimerized phase [10]. ED yields a consistent result with 40 DH Trimerized L=120
L=150
DMRG for the lower boundary of the trimerized phase, ξ 30
butitsuggestsasubstantiallysmallervaluefortheupper 20
boundary. Thismismatchstemspartlyfromstrongfinite 10
0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22
size effects on the location of the level crossings (which θ/π
are much stronger than in the spin 1/2 case). We ob-
servethattheEDphaseboundaryshiftstoslightlylarger FIG.5: (Coloronline)Energygapbetweenthegroundstate
of singlet and triplet sector (upperplot) and spin correlation
J values upon taking bigger system sizes into account,
2
length(lowerplot)fordifferentsystemsizesforfixedJ2/J1 =
approaching somewhat the DMRG results. A different
1.25. The finite energy gap at the transition and the weak
source of error is the rather slow convergence of DMRG
sizedependenceofthecorrelationlengthindicateafirstorder
in m in or close to a critical phase, especially for large phasetransition. Thecorrelationlengthwasobtainedbyafit
J /J . Theimportantresult,however,whichwegetfrom to theexponential decay of the spin-spin correlations.
2 1
both methods is the finiteextent ofthe trimerizedphase
onthe J /J axis,in contrastto the spin 1/2case where
2 1
tionlengthacrossthetransitionforJ /J =1.25(Fig.5).
the dimerized phase extends to infinity. We comment on 2 1
ForJ /J =0.75wecannotcompletelyruleoutavanish-
a possible explanation in the field theory section below. 2 1
ing gap as it is rather small at the transition. Another
Figure 4 shows the finite spin gap in the trimerized
indication for the first order nature is the jump of the
phase, ∆E =E (1)−E (0) where E (Sz) is the ground
0 0 0 TOPatthe phase boundary,incontrastto the exponen-
state energy in the Sz sector. On the SU(3) line the
tial suppression of the TOP towards the critical phase.
lowest excitation is eightfold degenerate (i.e. spin and
Critical phase — We havedetermined the value of the
quadrupolar excitations are symmetry related on this
central charge numerically from the entanglement en-
line). Away from the SU(3) line the lowest excitations
tropy for different points in the critical phase (red di-
have S = 1 for θ < π/4 and S = 2 for θ > π/4.
tot tot
amonds in Fig. 2), and all the values agree with c = 2
AccordingtotheanalysisofRef.18,thenatureofexcita-
(to within 5%), corresponding to a level-1 SU(3) Wess-
tionsinthetrimerizedphasesaregapped,deconfineddo-
Zumino-Witten (WZW) model [7]. We made use of the
main walls, very similar to the deconfined spinons of the
universal scaling behavior of the entanglement entropy
spontaneouslydimerizedphaseofthe frustratedS =1/2
for conformally invariant one-dimensional quantum sys-
spin chain.
tems[20]ofthe formS(x,L)= c ln 2Lsin(πx) +const,
The BKT transition line lies on the SU(3) line except 6 π L
where S(x) is the von Neumann ent(cid:2)ropy of the(cid:3)reduced
for 0.45≤J /J ≤3.5 where it follows the right bound-
2 1 density matrix ρˆ(x) of a subsystem starting at the open
ary of the trimerized phase. Field theory predicts that
boundary of length x embedded in a system of length
the transition line separating the Haldane (respectively
L, and c is the central charge. In the critical phase
DH) phase from the trimerized phase is of first order
the spin and quadrupolar correlation functions are ex-
(see below). We confirmed this numerically from a non-
pected to decay as a power law with distance x with an
vanishingspingapandanondivergingspin-spincorrela-
exponent η =4/3 and multiplicative logarithmic correc-
tions[7]∼cos(2πx/3)(lnx)σ/xη,whichsuppressthespin
(σ = −2) and enhance the quadrupolar (σ = 2) correla-
0.08 J2/J1=1 Stot=1 0.08 θ=π/4 tions, such that the latter are dominant (Fig. 6), similar
S =2
tot to the single chain case [6].
0.06 0.06
Fieldtheoreticalconsiderations —Asisdiscussedin[7,
E
∆ 0.04 0.04 8−fold 21], the low-energy effective Hamiltonian for the model
excited (1)aroundthepointJ =0,θ =π/4isthe level-1SU(3)
2
0.02 0.02 states WZWmodelperturbedbymarginallyirrelevantcurrent-
current interactions:
0 0
0.2 0.3 0.4 0.5 1 1.5 2
θ/π J2/J1 H +vg dx JAJA+vg dx JAJA , (2)
WZW 1Z L R 2Z L R
A∈XSO(3) A∈Xothers
FIG. 4: (Color online) Energy gaps of spin excitations along
cuts in the parameter space with J2/J1 = 1 (left-hand side) wherethefirstsummationisovertheSO(3)-subsetofthe
and θ=π/4 (right-handside). SU(3)-currents JA . The choice g = g = g recovers
L,R 1 2
4
vantanddrivesacrossovertogaplesssingle-chainbehav-
ior, in contrast to the S=1/2 case, where the marginal
interaction lets the system flow to a dimerized strong
coupling state.
As a perspective we believe that the SU(N) J −J
1 2
model contains a general mechanism where the critical
staterealizedfortheJ modelflowstowardsaN-merized
1
gapped state when the ratio J /J is beyond a certain
2 1
critical value. Another interesting point is the possibil-
ity of stabilizing trimerized phases in two dimensional
FIG.6: (Coloronline)Spin(stars)andquadrupolar(crosses) lattices, possibly in a J −J BLBQ model on the tri-
1 2
correlation functions (normalized) for θ = 0.42π, J2 = 0.25. angular lattice. Furthermore in the light of the present
The two solid lines are fits to the functional form predicted
study, where we revealeddominant quadrupolar correla-
byfieldtheory. Logarithmiccorrectionstothepowerlawlead
tionsinthevicinityoftheSU(3)line,itwillbeinteresting
to dominant quadrupolar correlations.
toexplorewhetherotherspinnematicorspin-multipolar
phases[23]remaintobeuncoveredclosetoSU(N)regions
in higher spin antiferromagnets.
SU(3) symmetry and the integrable model (J = 0, θ =
2
π/4) corresponds to g <0. PC acknowledges inspiring discussions with M. Mat-
Attheleadingorder,theeffectofJ (>0)(withθ =π/4 sumoto, M. Troyer,and U. Schollw¨ock, and ETH Zurich
2
fixed)maybetakenintoaccountbyincreasingg(=g =g ) forallocationofCPUtimeontheGonzalescluster. AML
1 2
in Eq. (2). At some critical value of J , g changes its acknowledgesthesupportoftheSwissNationalFundand
2
signtopositiveandthesystemflowsintoamassivephase thanks F. Mila and K. Penc for work on related topics.
with three degenerate ground states [22]. The model (2) The ED simulations havebeen enabled by the allocation
has another massive phase characterized by the asymp- of computing time at CSCS in Manno. HT was sup-
totic trajectory g =−g , which has a unique (Haldane) portedby aGrant-in-Aidfor Scientific Research(Grants
1 2
ground state and is separated from the trimerized one No. 17071011 and No. 16540313),and also by the Next
by a first-order transition[22]. The Y-shaped boundaries Generation Super Computing Project, Nanoscience Pro-
(amongtrimerized,Haldaneandcritical)andotherquali- gram, from the MEXT of Japan.
tativefeaturesinthelowerportionofFig.2areconsistent
withwhatispredictedbythemodel(2). Theappearance
ofthegaplessphaseinthe large-J region(J /J &3.5),
2 2 1
as revealed by DMRG and ED may be understood as
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