Vector chirality and inhomogeneous magnetization in frustrated spin tubes in high magnetic fields

0.17 MB·English

by  Masahiro Sato

Vector chirality and inhomogeneous magnetization in frustrated spin tubes in high
magnetic fields
Masahiro Sato and Tˆoru Sakai
Synchrotron Radiation Research Center, Japan Atomic Energy Agency,
Sayo, Hyogo 679-5148, Japan and CREST JST, Japan
(Dated: February 6, 2008)
7
Thelow-energyphysicsofthree-legfrustratedantiferromagneticspin-Stubesinthevicinityofthe
0
upper critical field are studied. Utilizing the effective field theory based on the spin-wave approx-
0
imation, we argue that in the intermediate-interchain-coupling regime, the ground state exhibits a
2
vector chiral order or an inhomogeneous magnetization for the interchain (rung) direction and the

n low-energyexcitationsaredescribedbyaone-componentTomonaga-Luttingerliquid(TLL).Inboth
a chiralandinhomogeneousphases,theZ2 paritysymmetryalongtherungdirectionisspontaneously
J broken. Itisalsopredictedthatatwo-componentTLLappearsandallthesymmetriesarerestored

1 in the strong-rung-couplingcase.
1
PACSnumbers: 75.10.Jm,75.10.Pq,75.30.Kz,75.40.Cx,75.50.Ee

]
l
e
- I. INTRODUCTION (l = 1,2,3), J > 0 (J > 0) is the intrachain (in-
tr terchain) coupling, and⊥the PBC S~4,j = S~1,j is im-
s
. Frustrated spin systems1 have been continuously ex- posed. Focusing on the vicinity of the upper critical
t
a plored for more than five decades. Frustration is consid- field and applying an effective field theory approach,
m eredasanimportantkeywordto generateexotic,uncon- we show the possibility of two interesting long-range-
- ventional magnetic orders, disorders and excitations in- ordered states: for a certain high-magnetic-field area, a
d cluding even spin-liquid states. Actually, frustrated sys- vector chirality z = (S~ S~ )z or an inhomo-
n hVl,ji h l,j × l+1,j i
tems have provided several peculiar concepts and phe- geneous magnetization along the rung direction occurs
o
nomena so far: resonating-valence-bond picture, non- in a one-component Tomonaga-Luttinger-liquid (TLL)
c
[ collinear orders, symmetry-unrelated degeneracy, order- state. In the chiral phase, the Z2 rung-parity symme-
3 by-disorder mechanism, etc. try Slα,j ↔ Slα+1,j, by which Vlα,j changes its sign, is
In recent years, frustrated magnets containing four- spontaneously broken, while the inhomogeneous mag-
v
spin exchanges as well as standard two-spin ones have netization in another phase breaks the one-site trans-
9
been intensively studied.2 In such magnets, fascinating lational symmetry for the rung as well as the rung-
4
5 magnetic orders (nematic, chiral, dimer orders, etc.), parity one. We also predict that a two-component TLL
1 whichorderparameteris definedby productsofspinop- emerges and all the symmetries are preserved in the
1 erators,areshowntobepresent. Inasense,thesenewor- strong-rung-couplingregime. Recentlyaspintubemate-
6 ders are a naturalconsequence of the four-spinexchange rial [(CuCl tachH) Cl]Cl (Ref. 3) has been synthesized
2 3 2
0
becauseforsuchaninteraction,itispossibletoperforma anditsmagneticpropertiescouldbedescribedbyathree-
/
t mean-field approximation, SαSβSγSδ SαSβ SγSδ+ leg frustrated spin-tube model.4,5,6 This also promotes
a i j k l →h i ji k l
m SαSβ SγSδ SαSβ SγSδ . Furthermore, it is well the motivation of studying the spin tube (1).
i jh k li − h i jih k li
- knownthateffectsoffour-spinexchangesarefairlysmall Existingresultsofthemodel(1)aresummarizedhere.
d in a large number of real magnets. Thus, to discover IntheS = 1 case,thezero-fieldgroundstatesaregapped
n 2
intriguing magnetic orders within spin systems contain- and doubly degenerate with spontaneously breaking the
o
ing only two-spin exchanges could more stimulate many one-site translationalsymmetry along the chain, at least
c
: experimentalistsandwouldbe theoreticallyamorechal- when J & 0.5J.7 In addition, a semi-quantitative
v lenging issue. ground-s⊥tatephasediagramintheJ -H plane(J >0),
i
X In one dimension, as representatives of geometrically which only shows gapless and gap⊥ful regimes, ⊥is con-
r frustrated spin systems with only two-spin exchanges, structed in Ref. 8; there exists an intermediate magne-
a onecanconsiderzigzagspinchainsandthree-legantifer- tization plateau with M = Sz = 1/6. In the case of
romagnetic (AF) spin tubes, i.e., ladders with a periodic S = integer and H = 0, thhe ls,yjistem is predicted to be
boundary condition (PBC) along the interchain (rung) always gapful and to conserve all symmetries.9
direction. In this paper, we study the latter model in a
Beforeanalyzingthequantumspintube(1),todiscuss
magnetic field. The Hamiltonian is written as
its classical version is instructive. The classical ground
3 state is an umbrella structure as in Fig. 1. In this state,
= JS~ S~ +J S~ S~ HSz ,(1) symmetries of the U(1) spin rotation around the spin
H Xl=1Xj h l,j · l,j+1 ⊥ l,j · l+1,j − l,ji z axis, one-site translations, and parity transformations
along both the chain and the rung directions are allbro-
where S~ is spin-S operator on site j of the lth chain ken. Consequently, the system exhibits a finite vector
l,j
2
spin moment U(1) degeneracy εK εK
around the Sz axis band
4SJ interband
width
Κ=0 width
j+1 3 3SJ⊥
H H K=+−2π/3 lowest gap 0 k 0 k
3 −π π −π π
0 0
saturation state magnon-condensed state
j u u
H > Hc H < Hc
l=1 2
l=1 2
FIG. 2: Magnon bandsin Eq. (2).
umbrella structure j-th rung
FIG. 1: Classical ground state of the spin tube(1).
we obtain the bosonic spin-wave Hamiltonian. As ex-
pected,thebilinearpartof˜b reproducesthefree-spin-
K,j
chirality z = √3(1 H2 ). From this re- wave dispersion ǫK(k). In order to study the low-energy
hVl,ji 2 − S2(4J+3J⊥)2 andlong-distance propertiesof the spin tube, we further
sult, the vector chiral order is expected to exist even in
introduce continuous boson fields Ψ as follows:
the quantum version. However,since generally quantum q
flduescttruoaytiaonnyisorqdueirtiensgt,roitnigsinnoonntreivdiiamlewnhsieotnhearnodrtnenodtsthtoe ˜b0,j →(−1)j√a0Ψ0(x), ˜b±23π,j →(−1)j√a0Ψ±(x), (5)
chiralorderremainsand brokensymmetries are restored
wherea is the latticespacing,andx=ja . Usingthese
0 0
in the model (1).
andtakingintoaccountthemagnoninteractiontermsup
tothelowestorderofthe1/S expansion,wearriveinthe
following effective Hamiltonian,
II. EFFECTIVE THEORY
1
= dx ∂ Ψ ∂ Ψ µ ρ
Hereweconstructtheeffectivetheoryforthequantum Heff Z (cid:20)2m x †q x q− q q(cid:21)
spin tube (1) in a high magnetic field. Let us begin with q=X0,+, q
−
the fully polarized state with M = S. For the state, +g ρ2+g (ρ +ρ )2+f ρ (ρ +ρ )+f ρ ρ
0 0 1 + 0 0 + 1 +
ethxeacetnlyercgaylcduilsapteerdsiaosn of one magnon with ∆Sz = −1 is +λ0(Ψ20Ψ†+Ψ†−+h.−c.) − −
ǫK(k)=H 2S(J +J )+2SJcosk+2SJ cosK,(2) +λ1(Ψ2+Ψ†0Ψ†−+Ψ2−Ψ†0Ψ†++h.c.)+··· , (6)
− ⊥ ⊥
where K (= 0, 2π) is the wave number for the rung where ρq = Ψ†qΨq is the magnon-density field. (This
± 3 Hamiltonian can also be derived via the path-integral
and that for the chain, k, is in k < π. The lowest
| | approach.10) The first two terms correspond to the free-
bandsǫ 2π arealwaysdegenerateduetotherung-parity
symmet±ry3, transformation of which induces K K. spin-wave part, and if the chemical potential µq is posi-
→ − tive,themagnonΨ iscondensed.11Wesetµ =4SJ H
As we explaininFig. 2, whenH becomes lowerthanthe and µ = µ = qS(4J + 3J ) H so th0at Hu −and
u3SppJer),(lmowagern)oncrsitoifcatlhevallouweesHtcuba=nd4sSbJeg+in3StoJ⊥con(Hdeclns=e Hc′ are±fixed. Other parame⊥ter−s in Eq. (6) arec eval-
(are⊥fully condensed). Moreover, as H < H = 4SJ, uated as 1/mq = 2SJa20 (mq = m), g0 = 2Ja0/3,
c′ g = (4J + 3J )a /6, f = 8Ja /3, f = 4Ja /3,
magnons in the remaining band ǫ are also condensed. 1 0 0 0 1 0
0 λ =(8J 3J )a⊥/6,andλ =(16J 3J )a /12. These
Supposing that multimagnon bound states are absent va0lues wo−uld⊥be 0somewhat1changed−due⊥to0high-energy
or their excitation energies are higher than those of one-
modes, the curvature of the dispersion, higher-order in-
magnon states (this is highly expected in antiferromag-
teractions, and the hard-core property of magnons ne-
netic systems and we have numerically verified it near
glected in the spin-wave theory.
the saturation), we may describe the low-energy physics
around H Hu using one-magnon excitations. A suit-
∼ c
able method for such a description is spin-wave theory
III. LOWEST-BAND-MAGNON CONDENSED
(1/S expansion). It makes spins bosonize as
STATE
Slz,j =S−nl,j, Sl−,j =b†l,j 2S−nl,j, (3)
p Based on the effective theory (6), we investigate the
where b is the magnon annihilation operator, and
l,j spin tube near saturation. In this section, we consider
nl,j = b†l,jbl,j denotes the magnon number. Substitut- thelowest-magnon-condensedcase,whereµ>0,µ0 <0,
ing Eq. (3) in the model (1) and introducing the Fourier and max[Hl,H ] < H < Hu. For this case, the low-
c c′ c
transformation of b for the rung as energyphysicsmustbegovernedbytwocondensedfields
l,j
Ψ . The effective theory is derived by integrating out
1
b = eiKl˜b , (4) th±e massive magnon Ψ via the cumulant expansion in
l,j K,j 0
√3
X terms of the free-spin-wave part of Ψ in the partition
K=0, 2π/3 0
±
3
function. The main effect of the Ψ sector is that an It is clear that as f˜ > 0, the potential is minimized by
0 1
attractiveinteractionbetweenρ andρ originatesfrom imposing ρ =ρ . Moreover,it is found that
+ +
the second cumulant of the λ term. −As a result, the 6 −
0
coupling constant f is changed as 3
1
f f˜ =f C λ20a−02 , (7) ρ+−ρ− ∝˜b†23π,j˜b23π,j −˜b†−23π,j˜b−23π,j ∼Xl=1Vlz,j. (10)
1 1 1
→ − m|µ0|3 We thus conclude that for f˜1 > 0, a finite long-range
p vector chiral order z exists, and the rung-parity
where C is a positive dimensionless constant of O(1). hVl,ji
symmetry is spontaneously broken. For J J (i.e.,
Here, we have approximated the Matsubara Green’s µ /J 1) or J J [i.e., λ O(J⊥ )≪], f˜ < 0
faunndctiJoan0τhTτaΨre0(xsm,τa)lΨle†0r(0(,l0a)rigear)s 1th/aa0n (tzheero)cowrrheelnati|oxn| |gweh0ne|enraHll≪ybheoclodms,1e2s,1⊥c7low≫sehriletofoHr uJ,⊥f0˜∼∼iOnc(−Jre)as(ei.⊥se.a,nλd01t∼en0d)s,
length (mµ ) 1/2 [τ : imaginary time], and assumed c 1
| 0| − to be positive. Consequently, the chiral phase is present
that (mµ ) 1/2 is at most O(a ).12 For the resultant
| 0| − 0 inanintermediate-rung-couplingregime. Supposingthat
Hamiltonian [Ψ ], the Haldane’s harmonic-fluid ap-
proach (i.e., bHoe′sffoniz±ation) (Refs. 13,14,15) could be ap- ρ+ >ρ holdsin the chiralphase,we canspeculate that
the Ψ −mode constructs a massive spectrum, whereas
plicable. Using the bosonization formulas ρ (x)
{∂ρρx¯±φ±+,/wπ∂}ex1φo/±2b/Ptπai}∞nn=P−a∞n∞b=oe−si∞2onn(eiφiz2±en−d(φπH±ρ¯±−axmπ)ρ¯ei±l−txio)θn±ai,anndwoΨhfe†±rteh∼±eρ¯{p±ρ¯h±as≈+=e eTvthxihoeiseutΨsepn−+srcteuespedoanyfrcteitnhpoerRfoctevhhfi.ider8eaTs.lLoaILfrdTHiesLrLaalnssHodtausttu,ehp.et1ph0ToeNrLTtLaeLdmLiseblypyp,artrethadhemiecptecetrodeer--.
fihe±ldis (φ ,θ ). Introducing the new fields φs,a = (φ+ would be close to the univers∼al vaclue 1. The correlation
± ± ±
φ )/√2 and θs,a =(θ+ θ )/√2 further, we can repre- function of the chirality might exhibit a power decay:
se−nt the phase-field Ham±ilto−nian as z z z 2 const/j2+ at j .10
hVl,jVl,0i≈hVl,ji − ··· →∞
Letusnowdiscussthecaseoff˜ <0,whereρ¯ =ρ¯ is
H[φ,θ] = Z dxqX=s,a2vπq (cid:2)Kq(∂xθq)2+Kq−1(∂xφq)2(cid:3) irnesEtoqr.ed(8a)nydietlhdesbaosToLnLiz,awtiohnichisias1vsatirlaobnlgel.yTshtaeb+φilsizseedc−tboyr
symmetries, while the low-energy physics of the φ sec-
+g cos(2√2φ )+g cos(3√2θ )+ , (8) a
φ a θ a ··· tor depends on whether cos(2√2φa) and cos(3√2θa) are
relevant or not: the scaling dimensions of these two are
where we have assumed ρ¯ = ρ¯ = ρ¯ (see below) and
+ 2K and 9/(2K ), respectively. The Hamiltonian
dropped terms with spatially osc−illating factors ei2nπρ¯x. a a He′ff
leadstoK ( ρ¯/f˜)1/2. Therefore,whenf˜ 0andρ¯
aTnhde gthφeanthdirgdθ ctuermmuslafnotr,erxeasmpepclteivoerliyg.inaUtenfforrotmunρa+teρly−, islargeenoau∝gh,−Ka is1alwaysmuchlargerthan1 ∼1. Atthis
case,cos(3√2θ )and cos(2√2φ ) arerespectivelyhighly
the values of g cannot be evaluated quantitatively a a
φ,θ relevant and irrelevant, and then the φ sector obtains
within the present approach. In the phase-field pic- a
ture, a spin rotation around the Sz axis S+ eiγS+, a massive spectrum. If gθ > 0 (< 0), the phase field θa
the one-site translation along the chain Slα,j → Sα l,j, is pinned on lines θa = √2(2n+1)π/6 (√2nπ/3) in the
that along the rung Sα Sα , and thl,ej s→ite-pla,jr+it1y θ+-θ plane. Among these lines, only six lines intersect
transformation along thl,ej c→hainl+S1α,j Sα are, respec- the p−hysically relevant “Brillouin” zone, π < θ+ π
(tφivel(yx,+exapr)esseπdρ¯aas ,θθ± (→x+θa±)+l,jγπ→,),(φθ±l,−(xj),θθ±(x2))π/→3, astnadte−sπpo≤sseθs−s t<heπs.ixTfohlids dreesguelnteirmacpyl.ieTsothi−natvetshteigagtreo≤utnhde
an±d (φ (x0),−θ (x±))0 ±( φ (0 x−),θ (±x→)). ±F±urther- physical meaning of locking θa and the ground-state de-
more, t±he run±g-parit→y tra−nsf±orm−ation±S−α Sα may generacy,letus focusonthemagnetizationpersite. The
1,j ↔ 3,j bosonization represents it as
be realizedbyρ¯ =ρ¯ and(φ ,θ ) (φ ,θ ). Owing
+
tothesesymmetries,i−nallverte±xo±per→ators∓wit∓houtoscil-
2 4
alarteinagllofawcetdortso,oenxliystcoins[2Enq(.φ(+8)−.φT−h)e]amnodstcorse[l3env(aθn+t−nθ=−)1] hSlz,ji≈M − 3ρ¯a0Dcos(cid:16)√2θa+ 3πl(cid:17)E+··· . (11)
terms indeed appear in Eq. (8). One can see that the second term in Eq. (11) causes
The bosonization approach for He′ff evaluates the ve- a down-down-up magnetization structure in the case of
locityv asv ( f˜ρ¯/m)1/2. Therefore,iff˜ >0,then g >0,while forg <0anup-up-downstructureoccurs:
a a 1 1 θ θ
v becomesima≈gin−aryanditmeansthatthebosonization for instance, if θ is locked to zero for g < 0, Sz =
a a θ h 1,ji
isinvalid. Tounderstandthephysicalmeaningofthisin- Sz = M +δ and Sz = M 2δ [δ cos(√2θ ) ].
stability,16 we should consider the magnon-density part Wh e2,jtihus conclude thaht3a,njiinhomo−geneous∝mhagnetizaatioin
in He′ff and then define the following Ginzburg-Landau for the rung is induced by pinning θa. Obviously, the
(GL) potential: parity and translational symmetries for the rung direc-
tion are spontaneously broken in this state. Three of
= g (ρ +ρ )2+f˜ρ ρ µ(ρ +ρ ). (9) the sixfold degenerated states are indeed explained by
1 + 1 + +
F − −− −
4
this inhomogeneousdistribution. The meaningofthe re- H (A) or (B) ? BKT 4SJ+3SJ⊥
maining twofold degeneracy is unknown.18 Remarkably, 1st order transition
saturation transition (B) two-component TLL
the inhomogeneously magnetized phase is not at all ex-
M=1/2
pected from the classical tube system (see Fig. 1). We 4SJ (A) TLL with
inhomogeneous
note that this inhomogeneousdistributionmightslightly TLL with magnetization plateau
be modified if cos(3n√2θ ) with n 2 are also rele- vector chiral M=1/6
a
vant.19Fromthepredictionsofthechi≥ralorderforf˜1 >0 TLL order
andtheinhomogeneousphaseundertheconditionf˜ <0 critical region
1
and f˜ 0, the boundary f˜ = 0 is expected to be a 0.28J
first-|or1d|e∼r transition. 1 0 translational symmetry broken
When f˜/ρ¯increases so that K < 9/4, cos(3√2θ ) 0 J⊥ M=0
1 a a
−
becomes irrelevant and the low-energy physics of the φ
a
sector is described by a Gaussian model. This transi- FIG. 3: Schematic ground-state phase diagram of the S = 1
2
tion must be of a Beresinskii-Kosterlitz-Thouless(BKT) spintube(1). Theareaawayfromthesaturationisdiscussed
type.20 After the transition, the system is in a two- elsewhere (Ref. 24). Seethe Endnotes12 and 17.
component TLL phase with all symmetries enjoying. If
f˜/ρ¯ is further increased due to the growth of J or
1
t−he decrease of ρ¯, cos(2√2φ ) seems to become ⊥rele-
a the boundary between these two regime, one might ob-
vant. However, the exact results for the integrable Bose
serve a weak singularity such as a magnetization cusp.
gas21 imply that in a one-dimensional Bose system with
a short-range repulsive interaction, the TLL parameter
is not usually smaller than 1 even when the interaction
becomes extremely strong. The two-component TLL is
hence expected to continue even when J J or ρ¯ is
small (see the Endnote 17). The predicti⊥on≫of the two- V. SUMMARY AND DISCUSSIONS
componentTLL in the strong-rung-couplingregime is in
agreement with a previous study applying the strong-
We have studied the three-leg frustratedspin tube (1)
rung-coupling approach to the S = 1 tube.22
2 near the upper critical field. It has been predicted that
the vector chiral order or the inhomogeneously magne-
tized order emerges in the magnetic-field-driven TLL
IV. THREE-BAND-MAGNON CONDENSED phase in the intermediate-rung-coupling regime. It is
STATE remarkable that in these two phases, the TLL critical-
ity (massless modes) and the spontaneous breakdown
Here, we consider the case where all three kinds of of discrete parity or translational symmetries for the
magnons Ψ are condensed. This situation could be rung direction coexist. We have also shown that when
+, ,0
realizedunder−the conditionofµ>0,µ >0,Hl <H < the rung coupling becomes strong enough, the inhomo-
0 c
H , and J < 4J/3. This means that the three-band- geneous phase vanishes and instead the two-component
c′
magnon co⊥ndensed state is allowed to exist only in the TLL occurs with preserving all the symmetries.
weak-rung-couplingregime. LikeEq.(9),letusintroduce Combiningourresultsandtheexistentones,7,8 wecan
the GL potential for the present case as follows: drawthe ground-statephasediagramfortheS = 1 tube
2
as in Fig. 3. The global phase structure near the sat-
G = g0ρ20+g1(ρ++ρ−)2+f0ρ0(ρ++ρ−) uration would common to all the cases with arbitrary
+f ρ ρ µ ρ µ(ρ +ρ ). (12) S, as far as S . O(1). Although in general the spin-
1 + 0 0 +
−− − − wave approach used in this paper is not very reliable for
To find the stable magnon-density profile (ρ ,ρ ,ρ ), small-S cases, we believe that it is valid if we consider
0 +
the Hessian matrix H = [ ∂2 ] is useful. At the−lo- the regionwhere M is sufficiently closeto the saturation
i,j ∂ρi∂Gρj value: insucharegion,multimagnonscatteringprocesses
cal minimum point (ρ ,ρ¯,ρ¯) satisfying ∂ /∂ρ = 0, the
0 G j are expected to be negligible. When J is changed from
eigenvalues of H are 4J/3, C , and C ( 4J/3 <
i,j − 1 2 − +0 to + with M fixed near the sa⊥turation, the fol-
C1 <0andC2 >0). The correspondingeigenvectorsare lowing sc∞enario is expected: TLL plus chirality [first-
w(δhρe0r,eδρC+3,δ>ρ−0). ∝Th(e0,n1e,g−a1ti)v,e(e−igCe3n,v1a,l1u)e, −an4dJ/(3Ca3n,1d,1it)s, otirodner tr[BanKsTitiotrna]→nsitTioLnL] plutwsoin-choommpoogneennetouTsLmLa.g→netiza-
eigenvector indicate that the ground state takes ρ+ → →
−
ρ = 0. Moreover, a positive eigenvalue C implies the Wefinallynotethatthepredictedfirst-orderandBKT
2
ex−is6tence of the TLL. We therefore predict that the chi- transitions could not be detected by observing the mag-
ralorder(ρ =ρ )andaone-componentTLLstatestill netizationM becauseH couplesto∂ φ andρ +ρ ,but
+ x s +
remainwhen6the−systemmovesfromthe lowest-magnon- it does not directly interact φ and ρ ρ . A sp−ecific-
a +
condensed regime to all-magnon-condensed one.23,24 At heat measurement would be efficient in−the−detection.
5
Acknowledgments Education, Culture, Sports, Science and Technology of
Japan.
This work is supported by a Grant-in-Aid for Scien-
tific Research (B) (No. 17340100) from the Ministry of
1 Forexample,see Frustrated Spin Systems, editedbyH.T. ford University Press, New York,2004).
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2 For examplesee T.Hikihara, T. Momoi and X.Hu,Phys. (2003).
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N.Shannon,T.MomoiandP.Sindzingre,Phys.Rev.Lett. lower critical field, this type instability induced by the
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3 J.Schnack,H.Nojiri, P.K¨ogerler, G.J.T.CooperandL. 17 Note that in the extremely strong-rung-coupling area, the
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Prog. Theor. Phys.Suppl.159, 297 (2005). pend on coupling constants of cos(3n√2θa). However, we
7 K.KawanoandM.Takahashi,J.Phys.Soc.Jpn.66,4001 cannot quantitatively calculate them within the present
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8 D. C. Cabra, A. Honecker, and P. Pujol, Phys. Rev. Lett. 20 For example, see A. O. Gogolin, A. A. Nersesyan and A.
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2
as the situation where Ψ± magnons are fully condensed, 23 If a similar analysis using the Hessian is applied to the
but theΨ0 part is still massive. ferromagnetic- (FM-) rung case, the chiral order is pre-
12 As µ0 /J approacheszero,thespatialrangeoftheinterac- dicted to exist even in J⊥ < 0. However, the expectation
| |
tion betweenΨ+ and Ψ− becomeslarger. Forsuchacase, isdoubtfulbecausemagnonboundstateswouldbeimpor-
the bosonization approach in this paper is less valid. tant in theFM-rung regime.
13 F. D. M. Haldane, Phys. Rev.Lett. 47, 1840 (1981). 24 M. Sato, cond-mat/0612165.
14 T. Giamarchi, Quantum Physics in One Dimension (Ox-