Low-energy properties of two-leg spin-1 antiferromagnetic ladders with commensurate external fields and their extensions

0.82 MB·English

by  Masahiro Sato

Low-energy properties of two-leg spin-1 antiferromagnetic ladders with commensurate
external fields and their extensions
Masahiro Sato
Department of Physics, Tokyo Institute of Technology,
5 Oh-okayama, Meguro-ku, Tokyo 152-8550, Japan
0
0 This study addresses low-energy properties of 2-leg spin-1 ladders with antiferromagnetic (AF)
2 intrachaincouplingunderauniformorstaggered externalfieldH,andafewoftheirmodifications.
The generalization to spin-S ladders is also discussed. In thestrong AFrung (interchain)-coupling
n
J⊥ region, degenerate perturbation theory applied to spin-S ladders predicts 2S critical curves in
a
J theparameter space(J⊥,H)for thestaggered-field case, in contrast to2S finitecritical regions for
theuniform-field case. All critical areas belong to a universality with central charge c=1. On the
2
otherhand,weemployAbelianandnon-Abelianbosonizationtechniquesintheweakrung-coupling
1
region. They show that in the spin-1 ladder, a sufficiently strong uniform field engenders a c = 1

] critical state regardless of the sign of J⊥, whereas the staggered field is expected not to yield any
h singular phenomena. From the bosonization techniques, new field-theoretical expressions of string
c order parameters in thespin-1 systems are also proposed.
e
m PACSnumbers: 75.10.Jm,75.10.Pq,75.40.Cx
-
t
a
I. INTRODUCTION tools. Our main target is the simple Hamiltonian,
t
s
t. ˆ = J S~ S~ +J S~ S~ + ˆ , (1)
a Spin ladder systems have been investigated theoreti- H l,j · l,j+1 ⊥ 1,j · 2,j HZ
m Xl,j Xj
callyformorethanadecade. Severalrealmagnetscorre-
d- spondingtothemhavebeensynthesizedandobserved.1–6 whereS~l,j isthespin-1operatoronthesite(l,j);l(=1or
n A trigger of such studies may be the discovery of high- 2)isthechain-numberindexandtheintegerj runsalong
o T materials and the connection between the Hubbard eachchain. TheintrachaincouplingJ ispositive. Wewill
c
c model(one ofthe high-T models)andthe antiferromag- refer to such ladders as “AF” ladders. The interchain
c
[ netic (AF) Heisenbergmodel.7 Recently,it has beenrec- coupling J is called the rung coupling. The Zeeman
2 ognized that spin ladders themselves can provide several term ˆZ he⊥re is chosen as two types
H
v theoretically interesting phenomena: for example, quan-
9 tum critical phenomena, nontrivial magnetization pro- ˆ = H (Sz +Sz ), (2a)
9 Hu − 1,j 2,j
cesses (plateaux and cusps), the connection with field j
6 X
5 theories and integrable models, topological or exotic or- ˆ = H ( 1)j(Sz +Sz ), (2b)
ders, etc. Especially, the intensive studies have largely Hs − − 1,j 2,j
0
4 developed the physics of 2-leg spin-1 ladders. Xj
2
0 where H( 0) is the strength of external fields. The
at/ temIns,tthheeisrpriensplaodndseesrtsoyestxetmersn,allikmeagontheetircmfiealgdnsehtaicvesyres-- latter (2b)≥has a staggeredfieldalong the chain(−1)jH.
Obviously, the AF rung coupling and the staggeredfield
m ceivedtheoreticalandexperimentalattentions. Recently,
compete with each other.
inaddition to the standarduniformmagnetic field, stag-
- Regarding the case without external fields, some pre-
d gered fields, which have an alternating component along
ceding theoretical studies16–18 of the spin-1 ladder (1)
n a directionof the system, have been in the spotlight.8–15
o exist (although, as will see in Sec. III, there are also a
Actually, severalphenomena inducedby themhavebeen
c few studies dealing with ˆ ). Their results deserve to
observed, and some mechanisms generating them in real u
: H
v magnets are known.9,14 As will be discussed in the next be summarized here for our consideration in later sec-
i tions. These studies explain how the first-excitation gap
X section, uniform- or staggered-field effects in the 2-leg
spin-1 ladderwithAFintrachaincouplinghavebeenun- varies in dependence upon the rung coupling J . At the
r 2 decoupled point J = 0, the model (1) is re⊥duced to
a derstood well. In the uniform-field magnetization pro-
two spin-1 AF cha⊥ins. As known well, the chain has a
cess,a masslessphase exists between the saturatedstate
finite first-excitation gap (Haldane gap).19 Around the
and a massive spin-liquid state. A staggered field yields
decoupledpoint, the gapreductiontakes place with J
a quantum phase transition.
increasing. Namely, the gaphas a cuspstructure aro|un⊥d|
Here, the following natural question would arise: how J = 0. Far from the decoupled point, in the AF-rung
theexternalfieldsinfluencehigh-spinorN-legladdersys- si⊥de, the gap increases together with the growth of J .
tems? To answer this (partially), this study specifically Itapproachesthe gapofthe rungdimer(twospinsalon⊥g
addresses low-energy properties of 2-leg spin-1 ladders the rung) which is the strong AF rung-coupling limit of
with external fields through the use of severalanalytical the model (1). On the other hand, for the ferromagnetic
2
gap gap of rung dimer J⊥ = 0
∆
Hal
Haldane gap 2 Haldane Chain
of spin-2 chain
J⊥
0
Plaquette-Singlet Solid
FIG. 1: Schematic gap profile in the spin-1 AF ladder. The
symbol∆ denotestheHaldanegap(≃0.41J). Fordetails,
Hal
see Ref. 18.
J = oo
⊥
Rung Dimer
(FM)-rungside,evenawayfromthedecoupledpoint,the
= a + b
gapdecreasesmonotonically. It reachesthe Haldane gap
of the spin-2 AF chain with the bond J = J/2, which
′
is the strong FM rung-coupling limit. (In general, the FIG. 2: RVB pictures in the GS of the spin-1 AF ladder (1)
2-leg spin-S AF ladder with the intrachaincoupling J is without external fields. The black point denotes a spin-1
2
reduced to the spin-2S AF chain with J = J/2 in the state. The ellipse including two black points indicates the
′
strongFMrung-couplinglimit.)20,21 Thefirst-excitation- symmetrization of two spin-1 states and recovers an original
2
gap profile is summarized as Fig. 1. A recent quan- spin-1 site. The black line represents for the singlet bond,
tum Monte Carlo analysis18 quantitatively surveys the which consists of a spin-12 pair. For 0 < J⊥ < ∞, the GS is
well described by the PSS state. With increasing the rung-
ground-state (GS) properties of the model (1) without
couplingJ⊥,theweight a(b) increases (decreases) in theGS
Zeeman terms. It estimates the first-excitation gap and
wavefunction,anditapproachesthetensorproductofsinglet
the spin-spin correlationlength. Moreover,it proposes a
dimers [singlet of the spin-1pair]. For details, see Ref. 18.
new string-type parameter, which is discussed in a later
section. The authors conclude that the GS is always
massive from J = 0 to , and is characterized by
the “plaquette-s⊥inglet” solid∞(PSS) state, one of short- spin-S casethereare2S criticalregions(points)through
the uniform(staggered)-field magnetization process. In
range resonating-valence-bond (RVB) states,22 which is
addition, we propose RVB-type pictures of noncritical
explained in Fig. 2. It smoothly connects two limiting phases in the uniform-field case with ˆ . In Sec. IV,
u
states: the Haldane state (J = 0) and the rung-dimer H
we consider the weak rung-coupling region (J J )
cstaaptteur(eJ⊥a f→eat∞ur)e.ofTthheenPeSwS⊥ssttraitneg. parameter is able to and employ some field-theoretical approaches. ≫In|p⊥ar|-
ticular, to map 1D spin-1 systems onto a field theory,
In addition to the external-field effects, we will reex- we utilize a non-abelian bosonization (NAB),23–25 i.e., a
amine and revisit some of the results without external
Wess-Zumino-Novikov-Witten (WZNW) model descrip-
fields.
tion. The first two subsections are devoted to an ex-
Our analysis specifically addresses two regions: the planation of the NAB. Through it, the spin-1 ladder (1)
strong AF rung-coupling region (J J) and the weak is described by using a fermion field theory, which was
rung-coupling one (J J ). Fo⊥r≫the former (latter) originally proposed by Tsvelik.26 Subsequently, we will
region, we mainly emp≫loy| d⊥e|generate perturbation the- considerthe uniform-fieldcaseinSec.IVC. Inthis case,
ory (DPT) (field-theoretical methods). In the former theNABisconsiderablyeffectivebecauseitcantreatthe
region, we can also investigate 2-leg spin-S ladders. It uniform Zeeman term nonperturbatively.26 From conse-
contributes to a systematic understanding of spin ladder quences of the DPT in Sec. III, the NAB here, and the
systems. Furthermore,thecomparisonofourresultsand gap profile in Fig. 1, we can determine the whole GS
existing results of spin-1 ladders will help to elucidate phase diagram in Sec. IVD. The critical regime is “sim-
2
the spin-ladder systems. ply connected” and has a c=1 criticality except for the
The organization of the remainder of this paper is as decoupled point J = 0. In Sec. IVE, we devote our
follows. First, we give a brief review of 2-leg spin-1 lad- attention to spin-1⊥ladders without external fields. Sec-
2
ders in Sec. II. As mentioned previously, it is useful tionIVE1isassignedtotheevaluationofstring-typepa-
in comparing our spin-1 case with the spin-1 case and rameters in one-dimensional (1D) spin-1 systems within
2
highlighting the features of the spin-1 case. Section III our field-theoretical framework. We propose their field-
specificallyaddressesthestrongAFrung-couplingregion theoretical expressions. Using a RG analysis, we con-
(J J). For the spin-1 ladder (1), the DPT pro- sider the GS phase diagram of a spin-1 ladder extended
vid⊥es≫effective Hamiltonians and predicts that there are from the model (1) in Sec. IVE2. Field-theoretical ap-
two critical areas (points) in the magnetization process proaches used here are not sufficiently efficient for the
applying the uniform (staggered) field. We also apply staggered-field case with ˆ . A brief discussion on it is
s
H
the DPT to the 2-leg spin-S ladders, to find that in the in Sec. IVF. In Sec. V, we summarize all results and
3
briefly discuss them. The Appendix offers readers sup-
plements of field-theoretical techniques and calculations (a)
in Sec. IV.
-1 +1 -1 0 0 +1 -1 +1 0 0 -1
II. REVIEW OF SPIN-1/2 LADDERS
(b)
In this section, we review low-energyproperties of the -1 0 0 0 0 +1 -1 0 +1 -1 0 0
spin-1 AF ladder which is equivalent to the model (1),
2
in which spin-1 operators are replaced with spin-1 oper- FIG. 3: RVBpictures of theGS of thespin-1 ladder. Panels
2 2
ators. (a) and (b) are typical spin configurations in the AF-rung
For the case without external fields, a finite rung cou- and FM-rung spin-liquid phases, respectively. The gray line
pling engenders a gapped spin-liquid (no long-range or- denotesasingletbond. UpanddownarrowsrepresentSlz,j =
ders occur) GS irrespective of the sign of J . This is +21 and −21, respectively. Each number under the dashed
true because the GS of the spin-1 AF Heisen⊥berg chain loopencirclingtwositesshowsthevalueofS˜jz =S1z,j+S2z,j+1
is critical (massless) and the run2g coupling is relevant in (a), and one of Sjz =S1z,j+S2z,j in (b). Removingall sites
from the standpoint of the perturbative renormalization of S˜jz = 0 or Sjz = 0, one can see a “hidden” N´eel order
group (RG) picture.27 The spin liquid can be illustrated (+1,−1,+1,−1,···). For more details, see Refs. 30 and 32.
using the short-rangeRVB picture28–32 as inFig.3. The
figureindicates thatthe excitationgapinthe spin-liquid H
H
phase has the same order as the energy required to cut saturated phase 2
a singlet bond. In the AF-rung side, singlet bonds tend z
S = +1
to occur along both chain and rung directions. In con- 2+ (c=1 chain) j
trast, two spins on the rung tend to construct a triplet
2J
stateontheFM-rungside. Thetendencyremovessinglet
c=1 critical phase H
bonds along rungs, and allows those along the diagonal 1
direction. The FM-rung spin liquid must connect with z
c=1 S = 0
the GS of the spin-1 AF chain (J ) smoothly. j
Therefore, we call it the Haldane p⊥ha→se. −In∞Refs. 30 and 0 spin-liquid phase
32,theauthorsshowthatthesetwokindsofmassivespin 0 z J⊥
liquids [(a) and(b) inFig. 3]can be detected by the two Haldane phase S j = 0
string-type parameters,
FIG. 4: Schematic GS phase diagram of the spin-1 ladder
Ooadd =−|j−lkim|→∞DSjaeiπPkn−=1j+1SnaSkaE, (3a) with theuniform Zeeman term (2a). 2
Oeaven =−|j−lkim|→∞DS˜jaeiπPkn−=1j+1S˜naS˜kaE, (3b) aancdomumppeenrsucrraitteic-ainlcofimeldmsen(sHu1rataend(CH-IC2)). tyTpeh.e4y0–4a3reThoef
where a = x,y,z, represents the expectation value upper critical field H can be determined by calculating
h···i 2
of the GS. We defined two new operators Sja = S1a,j + the exact spin-wave excitation energy in the saturated
Sa and S˜a = Sa +Sa . Indeed, from Fig. 3, one (perfect ferromagnetic) state: H = 2J + J for AF-
2,j j 1,j 2,j+1 2
caannd coanfirm=t0h)atarOeearveenali6=zed0ianndtheOoAadFd-r=un0g(sOpioand-dliq6=uid0 rlougnigc ssihdoew,santhdaHt 2th=e u2pJpfeorrctrhieticFaMl fi-reuldngofsitdhee.⊥sTphine-1saAmFe
Oeven
(Haldane) phase. chain with J′ = J/2 (strong FM rung-coupling limit) is
For the uniform-field case, low-energy properties have 2J, which is consistent with the result H2 =2J.
become well understood, but the quantitative GS phase Wang et al. have investigated the spin-1 staggered-
2
diagramhasnotbeen constructedyetasfar aswe know. field case with the term (2b) by employing the DPT
For details, see e.g., Refs. 33–37 and references therein. and Abelian bosonization.38,39,44–50 They predict that
TheschematicGSphasediagramisexpectedasinFig.4. the competition between the AF rung coupling and the
The spin-liquid, Haldane and saturated phases corre- fieldH createsasecond-orderquantumphasetransition.
spond to massive plateau regions in the uniform-field It belongs to a Gaussian type with c = 1, and sepa-
magnetization process. The critical phase, except for rates a N´eel phase and the massive spin-liquid phase,
the decoupled line J = 0, can be regarded as an one- which continuously connects with the spin-liquid (a) in
component Tomonag⊥a-Luttinger liquid (TLL),38 which Fig.3. Furthermore,Ide,Nakamura,andSato51 recently
is identical to a conformal field theory (CFT)39 with determinedthetransitioncurvewithhighaccuracyusing
the central charge c = 1. In this area, the magneti- the level-crossing method52 and a new twisted operator
zation per rung Sz changes continuously. When the method.53 The curve starts from the origin in the space
h ji
fieldisincreased,quantumtransitionstakeplaceatlower (J ,H) because both the rung-coupling term and the
⊥
4
H H ≅ J ⊥ - 0.50 J |2,-2)
|2,-1)
E
|1,-1)
Ne’el phase J⊥ |2,0)
J⊥ 2J⊥
spin-liquid 0
H
phase -J⊥ |1,0)
H ≅ 0.60 J ⊥3 /2 / J1/2 -2J⊥ |0,0)
0
0 J⊥ |2,1)
|2,2) |1,1)
FIG. 5: Schematic GS phase diagram of the spin-1 lad-
2
der with the staggered Zeeman term (2b). In weak and
strong rung-coupling regions, the transition curve follows FIG.6: EigenstatesandeigenenergiesofHˆj withthestrength
h ≃ 0.60×J⊥3/2/J1/2 and h ≃ J⊥ −0.50×J, respectively, of the uniform field H varying. The thick dotted line corre-
where the factors (0.60 and 0.50) are determined by our nu- sponds to the GS. Each vector |··· ,···) represents a state
merical calculations.51 These results areconsistent with ana- |S,Szij.
lytical ones in Ref. 14.
H = 2J . Near one line, H = 2J , two low-lying states
staggered Zeeman term are relevant to the single chain. on the j⊥-th rung are ⊥
The GS phase diagram is given in Fig. 5.
+ 1,1 , 2,2 . (5)
1,j j 1,j j
| i ≡| i |−i ≡| i
III. STRONG RUNG-COUPLING LIMIT Low-lying states near another line are
This section employs the DPT for the strong rung- + 2,j 0,0 j, 2,j 1,1 j. (6)
| i ≡| i |−i ≡| i
coupling region: J J. Frequently in 1D systems,
theories of the stro⊥ng≫coupling limit such as the DPT The first-order calculation of the DPT is equivalent to
provide visualizations of GSs and low-lying excitations. projecting the total Hamiltonian (1) onto the subspace
This is the case in the strong rung-coupling limit of the which consists of the set of two states (5) [or (6)] over
model (1), as we discuss in this section. all rungs. Two projection operators can be defined as
Pˆ = Pˆ and Pˆ = Pˆ , where
1 j 1,j 2 j 2,j
Q Q
A. Spin-1 ladders Pˆ = + + + ,
1,j 1,j1,j 1,j1,j
| i h | |−i h−|
Pˆ = + + + . (7)
First, we investigate the spin-1 AF-rung ladder with 2,j | i2,j2,jh | |−i2,j2,jh−|
the uniformfield. The DPT35 forthis model hasalready For convenience, we define new spin-1 operators as
been performed, for example in Ref. 54. However, we 2
reproduce it here as a preparation for later discussions.
1
UnderconditionJ J,the0-thapproximationtakes Uz = + + ,
the following Hamilt⊥on≫ian: j 2(cid:20)| i1,j1,jh | − |−i1,j1,jh−|(cid:21)
ˆ0-th = ˆ , (4a) Uj+ =|+i1,j1,jh−|, Uj− =|−i1,j1,jh+|. (8)
j
H H
j Similarly,anotherspin-1 operatorT~ isdefinedbyreplac-
X 2 j
ˆ = J S~ S~ H(Sz +Sz ), (4b) ing the subscript (1,j) to (2,j) in Eq. (8). The relation
Hj ⊥ 1,j · 2,j − 1,j 2,j between these pseudo-spin operators and original spin-1
where the originalmodel is reduced to a set of two-body ones is
problems on each rung. Because the j-th rung Hamil-
tcoanniaenasHiˆlyjshoalvsetwito. Ecoancsherevigeednqstuaatnethitaiessa, S~ojn2ea-ntod-oSnjze,coonre- Pˆ1,jSlz,jPˆ1,j =−21Ujz+ 43, Pˆ1,jSl±,jPˆ1,j =(−1)lUj∓,(9a)
rneitsupdonesdeonfctehetoruansgtastpein|SS~,SzainjdwithserzecSomanpdonSeznta,rreesmpaegc-- Pˆ2,jSlz,jPˆ2,j =−21Tjz+ 41, Pˆ2,jSl±,jPˆ2,j = 2(√−31)lTj∓.(9b)
j
tively. The solution is given in Fig. 6. The GS of the
rung encounters two levelcrossingsin the magnetization We read that hSjzi=0, 1 and 2 correspond to hTjzi= 21,
process. In parameter space (J ,H), we concentrate on Tz = 1 (or Uz = 1) and Uz = 1, respectively.
vicinities of these two level-cro⊥ssing lines; H = J and hUnjdier t−he2se prehpajriation2s, we chanjiobta−in2 the following
⊥
5
effective Hamiltonian near H =2J : (α)
⊥
ˆeff Pˆ ˆPˆ = Pˆ ˆPˆ
Hu,1 ≡ 1H 1 1,iH 1,j
i,j
X (β)
= UxUx +UyUy +∆ UzUz
J1 j j+1 j j+1 1 j j+1
j
X (cid:2) (cid:3)
H Uz+const, (10)
− u,1 j { } { }
j = a + + b +
X
where = 2J, ∆ = 1 and H = 2J H + 3J.
SimilarJly1, near H =1J , w4e obtainu,1 ⊥ − 2
⊥ FIG. 7: Intermediate plateau state in the spin-1 ladder (1)
ˆeff = TxTx +TyTy +∆ TzTz with the uniform field (2a): the case with J → 0 (α) and
Hu,2 J2 j j+1 j j+1 2 j j+1 thecasewhereJ isfinite(β). Thinblackbondsmeansinglet
j
X (cid:2) (cid:3) bonds of spin-1 pair. The dotted bonds mean triplet bonds
−Hu,2 Tjz+const, (11) where both z c2omponents of two spins are +12. In the panel
j (β),grayplaquettesrepresentasuper-positionstatedrawnin
X
thelowest part of thefigure.
where = 8J, ∆ = 3 and H =J H+ J. Both
(10) anJd2 (113) are2a sp1i6n-1 XXuZ,2chain⊥w−ith a 2uniform
2
field.55 Known exact results on the spin-1 XXZ chain whereS andM are,respectively,thesumofspinmag-
2 u u
immediately lead to the following predictions. From the nitudes and magnetizations within the unit cell of the
model (10), in the region Hu,1 1(1+∆1) the sys- plateau state. According to this condition, relations (9),
| | ≤ J
tem has a c = 1 criticality, in which the uniform mag- and two effective models, the plateaustate should be in-
netization and the critical exponents of pseudo-spin cor- variantundertheone-sitetranslationalongthechainand
relation functions vary continuously with Hu,1 varying. the exchange of two chains. These allow us to illustrate
Otherwise [i.e., Hu,1 > 1(1 + ∆1)], the magnetiza- the bond picture of the intermediate plateau as Fig. 7.
| | J
tion is saturated. Similarly, from (11), another massless The state (β) connects with the state (α) smoothly by
region with c = 1 exists in Hu,2 2(1 + ∆2). By decreasing the weight b. Hereafter, we refer to bond pic-
| | ≤ J
interpreting these in the context of the original ladder tures including singlet and triplet bonds as “RVB” pic-
model in the uniform field, we predict that two c = 1 tures.
critical areas exist: J 8J H J + 11J and Next, we turn to the staggered-field case, which has
2J J H 2J +⊥4J−. 3The≤refore,≤in t⊥he str3ong AF never been discussed in previous studies. In the 0-th
run⊥g−-coup≤ling≤regio⊥n, the spin-1 AF ladder has an inter- order DPT, the even-j rung Hamiltonian has the same
mediate plateau region with Sz =1 in addition to two formasEq.(4b),buttheodd-j onehastheuniformfield
h ji
trivial plateau regions: the saturated state and the PSS pointingtotheoppositedirectiontoEq.(4b). Therefore,
state described in Fig. 2. The presence of the interme- low-lyingstatesinoddrungsmustbemodifiedasfollows:
diate plateau is contrastedwith the spin-1 case (Fig. 4).
2
From effective models (10) and (11), we also see that
all critical phenomena in the magnetization process do + 1,j=odd 1, 1 j, 1,j=odd 2, 2 j, (13a)
| i →| − i |−i →| − i
notinvolveanyspontaneoussymmetrybreakings. While + = 0,0 , 1, 1 . (13b)
2,j=odd j 2,j=odd j
| i | i |−i →| − i
effective models also predict the intermediate plateau
vanishes at the point (J ,H) = (14J,25J), where the Consequently, projection and pseudo-spin operators are
lower and upper bounda⊥ry curves H3 =3J + 11J and redefined: new pseudo-spins in odd-j sites satisfy the
H = 2J J cross each other. However,⊥this3estima- same-type relation as Eq. (9), in which −12Ujz + 34
tion is to⊥o−rough because the lowest-order DPT is prob- (−12Tjz + 41) and Uj∓ (Tj∓) are replaced with 12Ujz − 34
acabslyes.vaAlidreocnelnyt innumtheericsualffisctiuedntyl5y6,5s7treovnaglurautnegs-cthoeupvlainng- (n21ewTjzto−ol14s,)waendca−nUajl±so(c−oTnjs±t)r,ucrtesepffeeccttiivveelym. oUdseilnsgfotrhtehsee
ishing point (J ,H) (1.44J,2.7J). It claims that in staggered-field cases in a similar way deriving (10) or
the subspace fix⊥ing th≈e total magnetization, the transi- (11). In the vicinity of the line H = 2J , the effective
tion between the c = 1 region and the plateau is of a Hamiltonian is ⊥
Beresinski-Kosterliz-Thouless(BKT) type.58
Imitating the RVB picture in Fig. 2, we attempt to Hˆse,ff1 = J1 U˜jxU˜jx+1+U˜jyU˜jy+1+∆1U˜jzU˜jz+1
regardthe intermediateplateaustateasastate compris- Xj h i
innogtebtohnadtsthoef npeaairrsescto-nnteaiginhbtroirplseptibno-12ndpsaairssw. eOllnaessshinogulledt −Hs,1 (−1)jU˜jz +const, (14)
onesbecausetheplateauhas Sz =1. FromRef.59,the Xj
h ji
necessary condition for a plateau state is whereU˜x,y =( 1)jUx,y,U˜z =Uz andH =2J H
j − j j j s,1 ⊥− −
S M = an integer, (12) 3J. Similarly, near the line H = J , the effective one
u− u 2 ⊥
6
ˆeff is the same type as ˆeff, in which ( ,∆ ,U˜α) external fields (2a) or (2b). Even for the spin-S case, in
Hs,2 Hs,1 J1 1 j →
(liJk2e,t∆h2e,Tu˜njαi)foarmnd-fiHelds,1ca→sesH, tsh,2e=aboJv⊥e−twHo m−od12eJl.s hUavne- nwihainchisS~slo,jlvmabelaentsharosupgihn-sStaotpeseraSt,oSrz, tjh,einruwnhgicHhaSmriultnos-
effective staggeredfields Hs,1 and Hs,2, respectively. Be- from0 to 2S. The number of in|depenident states |S,Szij
cause such alternating terms are relevant for the spin-1 is 2S (2m+1)=(2S+1)2, and the table of Clebsch-
2 m=0
critical chain, infinitesimal values of staggered fields im- Gordan (CG) coefficients (or the Wigner-Eckart theo-
mediately yield a finite excitation gap. In other words, remP) can determine the relation between them and the
only when the field Hs,1(2) vanishes, the GS is critical. states of originaltwo spins (S~1,j,S~2,j). The eigenenergy
Therefore, there exist two critical lines (not areas) with of the state S,Sz is
j
c = 1: H = J 1J and H = 2J 3J. The former | i
(latter)critical⊥li−ne2satisfies U˜z =⊥0a−nd2 Sz = 3( 1)j J
h ji h ji 2 − ⊥S(S+1) J S(S+1) HSz, (15)
( T˜z =0 and Sz = 1( 1)j). Both lines are of a c=1 2 − ⊥ −
h ji h ji 2 −
Gaussian-typetransitionincommonwiththespin-1 case
2 where we consider the even-j rung. In a subspace with a
(Fig. 5). On the critical lines, the staggered susceptibil- fixed value of S, the GS of the rung is S,S , although
ity ( 1)j∂hSjzi diverges [see the next subsection]. Like at H = 0, all (2S+1) states are degen|eraitje. Because
− ∂H
the uniform-field case, any symmetry breakings do not the GSenergyofthe abovesubspacegoesdownwiththe
occur at the transitions. Numerical10 and analytical9,60 slope S when H is increased, the GS of the full space
worksshow that the staggeredmagnetizationin ˆeff (or has 2S−level crossings: H = J ,2J ,...,2SJ . (This
ˆeff)changescontinuouslywiththestaggeredfieHlds,v1ary- is a general result of Fig. 6.) ⊥Hen⊥ce, one re⊥adily ex-
iHngs,.2Thereby, we can confirmthat the originalstaggered pects that the spin-S case has 2S critical phenomena in
magnetization( 1)j Sz hasnoplateau,contrarytothe the uniform(staggered)-fieldmagnetization process. The
uniform-field ca−ses. h ji first-orderDPTneareachlevel-crossingline givesthe ef-
fective model, which is the same type as the spin-1 case:
Weshoulddiscusseffectsofthehigher-ordertermsofJ
we always obtain a spin-1 XXZ chain with a uniform
in the DPT, although they have already been explained 2
(staggered) field for the uniform(staggered)-field cases.
inRef. 14. Especially,letus considerwhether anymech-
Therefore, even in the spin-S cases, similar conclusions
anisms varying the properties of above c=1 criticalities
asthespin-1casescanbeavailable. Inprinciple,onecan
emerge or not, from higher-order effects. In the vicin-
obtain effective models in all vicinities of level-crossing
ity of each transition, both two low-lying states in the
lines. However,derivationsofthemodelsassociatedwith
rung and the perturbative intrachain coupling part are
lower-fieldcrossinglinesmustcalculatenumerousCGco-
invariant under spin rotations around the z axis of the
efficients(synthesesofspins). TablesIandIIdepictonly
totalspin. Therefore,theU(1)symmetrycannotbebro-
the effective models near the highest-field and second-
ken by higher-order terms. For instance, an anisotropic
highest-field lines. Nevertheless, these two will be suf-
XY exchange interaction, which brings a mass genera-
ficient to conclude that the spin-S case has 2S critical
tion, does not occur. The continuous U(1) symmetry is
regions(points) in the uniform(staggered)-fieldmagneti-
one of the characteristic natures in the c = 1 criticality.
zationprocess. Note that in these two tables,we use the
Because the original model (1) has a site-parity symme-
same symbols ( ,∆ ,H ) in the spin-1 cases
try, we can also say that bond-alternating terms, which 1(2) 1(2) u(s),1(2)
J
to represent effective models.
produce a mass too, do not appear. As long as we focus
on the strong rung-coupling cases (J J), anisotropy From these tables, we can extract the following infor-
parameter ∆ will not exceed the⊥v≫alue of the BKT mationontheGSofthespin-Scase. (i)Foruniform-field
1(2)
transition point ∆ = 1. These considerations imply cases, the critical areas around the highest-field line and
BKT
that higher-orderperturbations cannot makec=1 crit- second-highest-field line, respectively, are
icalities of transitions change, although they will modify
2SJ J H 2SJ +4SJ, (16a)
speavrearmaletneerws(bJu1t(2s)m,∆al1l(2t)e,rHmus(s()e,1.g(2.),)nselxigthntleyaraensdt ngeeingehrbaoter ⊥− ≤8S2+≤2S ⊥2
(2S 1)J − J H
interaction terms) in effective models. − ⊥− 4S 1 ≤
−
Using analytical tools, we can also evaluate several 24S2 14S+1
physical quantities around each transition. We will dis- (2S 1)J + − J, (16b)
≤ − ⊥ 4S 1
cusstheminthenextsubsectionalongwithgeneralspin- −
S cases. which determine the width ofthe criticalregionΓ in the
tables. TheplateauregionsexistoutsidetheregionsΓ,as
withthespin-1case. (ii)Thelargerthemagnitudeofspin
B. Generalization to spin-S ladders S becomes, the more the width Γ increasesas a resultof
the growth of the effective coupling . The effective
1(2)
J
Viewing results of DPTs in spin-1 cases (Figs. 4, 5, model approaches a spin-1 XY chain. Because plateaux
2 2
Refs. 14 and 34) and above spin-1 cases, we find it pos- arecharacteristicin“quantum”spinsystems,thegrowth
sible to generalize them to 2-leg spin-S AF ladders with ofΓ,orthedecreaseofplateauregions,meansthesystem
7
TABLEI:Effectivemodelsaroundthehighest-fieldlinesH =2SJ⊥. Parameters(J1,∆1,Hu,1,Hs,1)arevaluesoftheeffective
exchangecouplingconstant,theanisotropyparameter,theeffectiveuniformfieldandtheeffectivestaggered field,respectively.
The sign Γ denotes the width of the c = 1 critical region along the H-axis in the original parameter space (J⊥,H), Mu,1
represents thepossible values of hSz i in the critical region Γ, and M is the valueof |hSz i| at thetransition line H =0.
l,j s,1 l,j s,1
spin J ∆ H Γ M H M
1 1 u,1 u,1 s,1 s,1
1/2 J 1/2 J⊥−H+ 1J 3J 0→ 1 J⊥−H− 1J 1/4
2 2 2
1 2J 1/4 2J⊥−H+ 3J 5J 1 →1 2J⊥−H− 3J 3/4
2 2 2
3/2 3J 1/6 3J⊥−H+ 5J 7J 1→ 3 3J⊥−H− 5J 5/4
2 2 2
2 4J 1/8 4J⊥−H+ 7J 9J 3 →2 4J⊥−H− 7J 7/4
2 2 2
S 2SJ 1/(4S) 2SJ⊥−H+ 4S−1J (4S+1)J S− 1 →S 2SJ⊥−H− 4S−1J S− 1
2 2 2 4
TABLEII:Effectivemodelsaroundthesecond-highest-fieldlinesH =(2S−1)J⊥. Parameters(J2,∆2,Hu,2,Γ,Mu,2,Hs,2,Ms,2)
have the same roles as (J ,∆ ,H ,Γ,M ,H ,M ) in Table I, respectively. Note that the spin-1 case does not possess
1 1 u,1 u,1 s,1 s,1 2
theeffective model.
spin J ∆ H Γ M H M
2 2 u,2 u,2 s,2 s,2
1 8J 3/16 J⊥−H+ 1J 19J 0→ 1 J⊥−H− 1J 1/4
3 2 3 2 2
3/2 24J 5/48 2J⊥−H+ 3J 53J 1 →1 2J⊥−H− 3J 3/4
5 2 5 2 2
2 48J 7/96 3J⊥−H+ 5J 103J 1→ 3 3J⊥−H− 5J 5/4
7 2 7 2 2
5/2 80J 9/160 4J⊥−H+ 7J 169J 3 →2 4J⊥−H− 7J 7/4
9 2 9 2 2
S 8S4(S2S−−11)J 16S4(S2−S1−1) (2S−1)J⊥−H+ 4S2−3J 32S24−S−121S−1J S−1→S− 21 (2S−1)J⊥−H− 4S2−3J S− 34
approachestheclassicalvectorspinsystem. Ontheother andsuchmethodscanprovideaccuratepredictionsofthe
hand,inthecasefixingS,thehighest-fieldcriticalregion original spin-S ladders.
Γ is smaller than the secondhighest-field one. Therefore First we discuss the uniform-field cases. According to
we expect that the lower-field critical regions are larger. Bethe ansatz34,61,62 and Abelian bosonization,38,39,44–50
(iii) Like the spin-1 case, all intermediate plateaux must the low-energy physics of the massless area H
u,1(2)
| | ≤
vanish at a sufficiently weak rung-coupling region. (iv) (1+∆ )isgovernedbythefreebosonfieldtheory
1(2) 1(2)
J
For the staggered-field cases, the transition lines with withthespin-wave[masslessexcitation]velocityv and
1(2)
the highestfield andthe second-highestfield are,respec- thecompactificationradiusR . (Wereferthereaderto
1(2)
tively, AppendixAforanexplanationofAbelianbosonization.)
Utilizingtheirknowledge,onecanderivetheuniformsus-
H = 2SJ (4S 1)J/2, (17a) ceptibility formula9,63
⊥− −
H = (2S 1)J (4S 3)J/2. (17b)
− ⊥− − χ ∂hUjzi or ∂hTjzi = a0 , (19)
(v) All critical phenomena do not involve spontaneous 1(2) ≡ ∂Hu,1 (cid:18) ∂Hu,2(cid:19) (2π)2R12(2)v1(2)
symmetry breakings as with the spin-1 case.
where a is the lattice constant. Because the effective
Now, let us investigate the physical quantities around 0
field H varies linearly with the original field H (see
each criticality in more detail. If we represent the effec- u,1(2)
tivemodelneareachtransitionwiththe(second)highest Tables I and II), the relation χ = 1∂hSlz,ji is real-
1(2) 2 ∂H
field using the pseudo-spin operator Uα (Tα) in a simi- ized for the magnetization process varying only H,64 at
j j
larmanneroftheprecedingsubsection,theoriginalspins least in the first-orderDPT. One thus can regardall the
are projected out as follows: behavior of χ as those of the original susceptibility,
1(2)
except for the difference of the factor 1. The linear re-
Slz,j →−21Ujz+S− 41, Sl±,j →(−1)l√SUj∓, (18) lation between the effective and origin2al fields is often
used below. The Bethe ansatz can determine the radius
1 3 S(2S 1)
Slz,j →−2Tjz+S− 4, Sl±,j →(−1)l2 4S −1 Tj∓, aRn1d(2)Hand t.heEvsepleocciiatlylyv1fo(2r)Has funct=ion0,stohfeJr1a(d2)i,us∆a1n(2d)
r − u,1(2) u,1(2)
the velocity are represented analytically as
whereweconsidereven-j sites,again. Forthe staggered-
field cases, as Eqs. (13) and (14), we must redefine Ujα 1 1 1/2
(Tjα)inallodd-j rungsanddefineU˜jα (T˜jα). Therelation R1(2) = √2π 1− π arccos∆1(2) ,
(18), of course, contains Eq. (9), and provides the mag- (cid:18) (cid:19)
netic relation between the effective and original models. π 1−∆21(2)
As mentioned earlier, the low-energy properties of the v1(2) = 2 aqrccos∆ J1(2)a0. (20)
1(2)
effective models are elucidated well by severalanalytical
methods. Therefore,thecombinationoftherelation(18) Inserting Eq. (20) to Eq. (19), we see that when S goes
8
fromonehalf(one)to ,thepseudo-spinmagnetization where φ(x) [x = j a ] and A are the boson field
0 1(2)
∞ ×
curveslope[i.e.,thesusceptibility(19)]atthemidpointof and a nonuniversal constant,68–70 respectively. Here we
the(second)highest-fieldcriticalregime[i.e.,atHu,1(2) = neglect the so-called Klein factor.47–50 From Eq. (24),
0] decreases monotonically from 1 = 0.184 J 1 a finite H leads to a perturbationtermproportional
√3πJ ∼ × − s,1(2)
(0.0954 J 1) to 1 ( 1 ). Fixing S, we also find toHs,1(2)sin(φ/R1(2))fortheeffectivebosonfieldtheory,
× − 2πSJ 4πSJ and it then becomes a sine-Gordon model. The vertex
thatχ (H =0)islargerthanχ (H =0). Forexam-
1 u,1 2 u,2
pχl2e(,Hinu,2th=e0sp)i∼=n-10.0ca9s5e4,×χ1J(−H1u.,1T=he0s)e∼=im0p.l1y19a×reaJs−o1naabnlde o4pπRe1r21a(2t)oransidn(iφs/aRlw1(a2y))s hrealsevtahnets(cia.eli.n,gxd1(i2m)e<nsi2o)n. xT1(h2e)r=e-
factthatthelargerΓis,the smalleristhemagnetization fore the scaling argument9 tells us that any small fields
slope at H = 0. Subsequently, let us consider how
u,1(2) H yieldanexcitationgapm andthepseudo-spin
s,1(2) 1(2)
magnetization approaches the value of the plateau (sat-
staggeredmagnetization as:
uration). Without utilizing formula (19), studies of the
C-ICtransition40–43haveshownthatnearthesaturation, m1(2) Hs,1(2) 1/[2−x1(2)], (25a)
the magnetization behaves as ∼ | |
hU˜jzi(orhT˜jzi) ∼ (−1)j|Hs,1(2)|x1(2)/[2−x1(2)],(25b)
Uz (or Tz ) Hcr H 1/2, (21)
h ji h ji ∼ | u,1(2)− u,1(2)| where if H < 0, we take the replacement ( 1)j
s,1(2)
− →
where |Huc,r1(2)| is the critical value J1(2)(1 + ∆1(2)).65 (−1)j+1 in Eq.(25b). Substituting R1(2) in Tables I and
Hence,inthemagnetizationprocess,theoriginalmagne- II for x1(2), we find that the larger the value S becomes,
tization Sz also behaves, near each plateau, Hcr the slowerthe growthsof both the gapand the magneti-
h l,ji ∼| − zation become. In other words, the singularity of the
H 1/2, where Hcr is the critical field of each c = 1 re-
gio|n. This power law is a universal property. That is, staggered susceptibility ∂∂hHU˜sjz,1i|Hs,1→0 (or ∂∂hHT˜sjz,2i|Hs,2→0)
it is independent of the spin magnitude S and the level- decreases with S increasing. Particularly, in the limit
crossing line we choose in the two-spin problem of the S , m H and U˜z (or T˜z )
→ ∞ 1(2) ∼ | s,1(2)| |h ji| |h ji| ∼
rung. The spin-wave analysis can exactly calculate the H , which means the staggered susceptibility does
s,1(2)
| |
gap in the saturated state of each effective model. For not diverge at the “transition” line H = 0. These
s,1(2)
H > Hcr , the excitation gap is estimated as can be again interpreted as a sign of the approach to
| u,1(2)| | u,1(2)|
H Hcr .66 Again, translating this into the the classical spin system. On the other hand, fixing S,
| u,1(2)| − | u,1(2)| one observes that when the fields H = H are small
originalmodel,weseethatwhenH ismovedjustoutside s,1 s,2
eacPhercfo=rm1inreggitohne, tshame geasppignr-owwasvaesa|nHal−ysHiscirn|.6t7he satu- erensopuegchti,vmely1. aWnde t|hhUe˜rjzeif|oraereanlatircgiperattehtahnatmt2heantrdan|hsTi˜tjzioi|n,
rated state of the original spin-S ladders, we can deter- with the higher field has a stronger singularity. In the
mine the upper critical uniform field Hc. The field gives spin-1 case, we have m1 ∼ |Hs,1|0.878, m2 ∼ |Hs,2|0.903,
theboundarybetweenthesaturationwith Sz =2S and U˜z H 0.756 and T˜z H 0.806. In common
h ji |h ji| ∼ | s,1| |h ji| ∼ | s,2|
the c=1 phase just under it. The result is with the uniform-field cases, H has a linear relation
s,1(2)
with H. In order to translate all consequences into ones
Hc =4SJ+2SJ . (22) of the model (1) in the original staggered-field magneti-
⊥ zationprocess, it is sufficient to replace H and U˜z
Surprisingly, H perfectly agrees with the upper critical s,1(2) h ji
c (or T˜z ), respectively,withH and Sz ( 1)jM ,
fieldderivedfromtheeffectivemodelaroundthehighest- h ji h l,ji− − s,1(2)
where M is the staggered magnetization par site at
fieldline[seethe region(16a)]. Inotherwords,H isnot s,1(2)
c each transition line (see Tables I and II).
modified by the higher-order perturbation effects of the
Summarizing all the above results about the spin-S
DPT.FortheFM-rungside,thespin-wavetheoryimplies
cases, we can draw GS phase diagrams and the uni-
that
formandstaggeredmagnetizationsasinFig.8. Notably,
Fig. 8 is valid in the strong AF rung-coupling limit.
H =4SJ, (23)
c
Utilizing solutions of the Bethe ansatz integral equa-
where Hc does not depend upon the rung coupling J . tions,9 values of the nonuniversal constants A1(2),14,70
Equations (22) and (23) are a generalizationof the crit⊥i- etc.,wecanservemorequantitativepredictions. Weomit
cal field H in the spin-1 case (Fig. 4). them here.
2 2
Next, we shift our focus to the staggered-field cases. Finally,wespeculatetheshort-range“RVB”pictureof
When the effective staggered field H is vanishing the plateau states in the uniform-field case, without any
s,1(2)
[i.e., the GS is massless], the bosonization translates the computations. The spin-S case has 2S +1 plateau re-
pseudo-spin operator to the following boson representa- gions including two trivial plateaux: the saturated state
tion: and the H = 0 state. The guides to guess the “RVB”
pictures are the effective models, in which translational
U˜z orT˜z a0∂xφ +( 1)jA sin φ ,(24) symmetry does not break, the plateau condition (12),
j j ≈ 2πR − 1(2) R and the expectation that the bonds along the chain are
(cid:16) (cid:17) 1(2) (cid:18) 1(2)(cid:19)
9
Uniform-Field Case Staggered-Field Case
(α1) saturation (α2)
(A)
H H
...... ....
(B)
4SJ H c =4SJ+2SJ⊥ c=1 region c=1 curve
0 0
0 J⊥ 0 J⊥
FIG. 9: Expected “RVB” pictures for the plateaux with
S z +Sz (β1) S z +Sz (β2) hSzi=1 (A) and hSzi=2 (B) in the spin-3 case.
1,j 2,j 1,j 2,j j j 2
2S saturation 2S-1/2
2S-1 2S-3/2
........ ....... mappings exist:72 Haldane’s method based on a non-
3 3/2 transition point linear sigma model (NLSM)19 and the method applying
2
1 1/2 2SJ⊥ -(4S-1)J/2 a NAB.23–25 We will use the latter in this section. It is
0 0 useful for treating several additional terms, in first prin-
0 H 0 H
(2S-1)J ⊥ -(4S-3)J/2 ciple, by the perturbation theory and the RG picture.
FIG.8: Panels (α1) and(α2) are, respectively,theGS phase
diagramsintheuniform-fieldcase(2a)andthestaggered-field
one(2b). Panels (β1) and(β2) are, respectively,theuniform
A. Critical points of spin-1 AF Chain
magnetizationcurveinthecase(2a)andthestaggeredonein
thecase(2b). Inthepanel(β2),wedenotethemagnetization
of the even-j rung. The application of the NAB to 1D spin-1 sys-
tems17,26,73,74 is supported by low-energy properties of
the spin-1 Takhtajan-Babujian (TB) chain75,76 and the
subjecttotakingthesingletstateandallplateauxvanish following spin-1 bilinear-biquadratic chain:
in the sufficiently weak AF rung-coupling region. From
these,inordertobuildthe“RVB”picturefortheplateau ˆ = J S~ S~ +δ(S~ S~ )2 , (26)
with Sz =S˜inthespin-S case,weshouldperformjust Hδ j · j+1 j · j+1
the fohllojwiingtwoprocedures: (i)putting S˜tripletbonds Xj h i
tawnod(n2eSar−esSt˜-)nseiingghlbeotrborunndgsspberyasirnugnlegt,aonrdtr(iipi)le“tjobinoinndgs”, where J > 0, S~j is the spin-1 operator on the site j,
and the TB chain corresponds to δ = 1 = δ . This
nottobreakthetranslationalsymmetryalongthechain. subsection present a brief review the mo−del (26T).B77
The plaquette states of the lowest panels in Figs. 2 and
The efforts of several people78–87 since the late 1980’s
7 are available for the procedure (ii). For instance, two
plateau states with Sz =1 and Sz =2 in the spin-3 have advanced our understanding of the model (26). In
h ji h ji 2 particular,the low-energyproperties in the regionJ >0
case are described as (A) and (B) in Fig. 9, respectively.
Similarly, the plateau with Sz = 0 could be captured and δ 1 have been elucidated well. The GS phase
h ji diagr|am| ≤and the excitation gap in this region are sum-
by the set of a PSS state and a spin-liquid state (a) in
marized as in Fig. 10. At least three special points ex-
Fig. 3. Following the above speculations, one would eas-
ist aside from our target, the Heisenberg point δ = 0.
ily produce “RVB” pictures for any plateaux.
The TB chain is integrable and has massless excita-
Within the “RVB” picture, every time the GS moves
tions with the wave numbers (momenta) k = 0 and
from a plateau to the plateau just above it with H in-
π/a . The point δ = 1, called the Lai-Sutherland
creasing, one singlet bond along each rung is cut and 0 LS
(LS) model,88 is also integrable. It has massless exci-
exchanged for a triplet bond. This phenomena is rem-
tations with k = 0 and 2π/3a . The low-energy lim-
iniscent of successive transitions in the spin-S bond- ± 0
its of TB and LS points are, respectively, equal to the
alternated chain.25,45,71
level-2SU(2) WZNW model89–92 (a CFT) and the level-
1 SU(3) WZNW model. On the δ axis, these two points
arelocatedinaquantumphasetransition. TheTBpoint
IV. WEAK RUNG-COUPLING LIMIT separates the Haldane phase (δ 1) and the massive
| | ≤
dimerized phase (δ < δ ), which has twice degenerate
TB
This section describes the spin-1 ladder (1) under the GSs. Ontheotherhand,theLSpointseparatesthe Hal-
opposite condition to the last section: J J ,H. We dane and massless “trimerized” phase (δ > δ ). The
LS
start from the limit J = H = 0. There,≫th|e ⊥la|dder be- transition belongs to a generalized SU(3) BKT type.93
comes two decoupled⊥spin-1 AF Heisenberg chains. The Features of the Haldane phase are the existence of a fi-
Haldane gap of the single chain complicates the map- nite gap between the unique GS and the isolated spin-1
ping to some field theories. Nevertheless, two famous magnon mode, and a “hidden” AF order detected by a
10
massive massless three copiesofmasslessMajorana(real)fermions,as the
gap
dimerized trimerized WZNW model has c = 3/2 and the Majorana fermion
phase phase
theory, which is equivalent to a 2D critical Ising model,
incommensurability has c = 1/2. This identification allows a discussion of
δ
-1 0 1/3 1 low-energy properties of the TB chain using the c=1/2
CFT, instead of the WZNW model. In imaginary time
TB point Heisenberg AKLT Lifshitz Lai-Sutherland formalism where τ = it (t is real time), we have the
(massless) δ=δ ≅ 0.4 δ≅ 0.44 (massless) fermionic Euclidean action for the TB chain,
dis
Haldane phase
S = dτdx v[ξa∂ ξa +ξa∂ ξa], (28)
TB L z¯ L R z R
FIG. 10: GS phase diagram and the schematic gap behavior Z
in themodel (26). where ξa(z) and ξa(z¯) [a = 1,2,3] are, respectively, the
L R
leftmoverofaMajoranafermionwithweights(1,0)and
2
the right mover with (0,1). The velocity of the exci-
string order parameter: 2
tations v is the order of Ja .89,95 Here, we introduced
0
Oa = −|j−lkim|→∞DSja eiπPkn−=1j+1Sna SkaE, p∂thzeea=t“eldig21ih(ntv1d-∂cicτoen−se”air∂cexo)so,urmdainmndaetd∂e.z:¯ Iz=n=t21h(vev1τo∂+pτe+irxa,ti∂oz¯xr)=f.orvmTτha−elisirmxe-,,
ˆa , (27)
≡ hO i thefermionssatisfythefollowingequal-timeanticommu-
tationrelations: ξa(x),ξb(y) =δ δ δ(x y). Onthe
whichisthesameformasthestringparameter(3a)inthe { α β } ab αβ −
spin-1 ladder, except that Sα is a spin-1 operator. The topofthe action,there existcorrespondencesamongthe
2 j WZNW primary fields and fields of the c=1/2 CFT. In
AKLT94 point δ = 1 is noteworthy because its GS is
aklt 3 accordancewithRefs.96,74and17,Φ (z,z¯)comprises
exactlyidenticaltoavalence-bond-solidstate. Moreover, mn
a bilinear form of ξa(z) and ξa(z¯), which indeed has the
the point δ is the onset of an incommensurability: in L R
aklt weights (1,1). In addition, from Ref. 17, we have
(1>)δ >δ ,thereal-spacespincorrelationshasanin- 2 2
aklt
commensurate fluctuation, that connects smoothly with
3
the three-site period one at the LS point.
g (z,z¯) (τ ) (z,z¯), (29)
mn α mn α
Asides from points above, recent studies have de- ∝ G
α=0
scribed there are two characteristic points: δ = 0.4 X
dis ∼
(Ref. 84) and Lifshitz point δLif ∼= 0.44.81,82 In the re- where (τ0)mn = δmn and τ1,2,3 are the Pauli matrices.
gion−1<δ <δdis,themomentumofthelowestmagnon The fields Gα(z,z¯) are determined as
excitation stays at k =π/a . However, on the right side
0
of δ , it has a deviation from k =π/a , and splits into =σ σ σ , =iσ µ µ , (30)
dis 0 G0 1 2 3 Ga a a+1 a+2
twoincommensuratemomenta,whichsmoothlyreachthe
massless points +3π/2a0 and 3π/2a0 at the LS chain, where σa(z,z¯) and µa(z,z¯) are, respectively, the order
respectively. At the Lifshitz po−int, an incommensurabil- and disorder fields in the critical Ising model (c = 1/2
ity appears in the spin structure factor. CFT). Here, we use the cyclic index a+3 = a. Both
fields σ and µ have weights ( 1 , 1 ). An imaginary
a a 16 16
unit i is embedded in Eq. (30) to let the field g be a
B. Effective field theory SU(2) matrix. The SU(2) current operators Ja(z) and
L
Ja(z¯) in the WZNW model can be defined by fermions
R
as follows:
Reviewing the last paragraph, one notes that the TB
point inthe model(26)canbe adoptedas anunderlying i i
point to consider an effective field theory for the Heisen- JLa(z)=−2ǫabcξLbξLc, JRa(z¯)=−2ǫabcξRbξRc, (31)
berg chain (δ = 0). It is likely that the LS point is also
available for the field theoretical description. However, whereǫ isthetotallyantisymmetrictensorandǫ =
abc 123
as mentioned already, there are three points changing 1. ThecurrentsJa (x)satisfylevel-2SU(2)Kac-Moody
L,R
the low-energy properties between the LS and Heisen- algebra. Through the NAB,24,25,97 the spin operator in
berg points. Therefore, the LS point is not appropriate the TB chain is translated into the following sum of the
for deriving the effective theory of the Heisenberg point. uniform and staggeredparts:
This subsectionprovidesthe effectivefieldtheoryforthe
spiTnh-1ellaedvedle-r2(S1U)(i2n)oWurZnNoWtatmioond.1e7l, which describes the a1 Sja ≈ JLa+JRa +iC0(−1)jTr[(g−g†)τa]
0
TB point, has two primary fields: the 2 2 matrix field = Ja+Ja +C ( 1)jσ µ µ , (32)
g [m,n 1,2 ]with left andrightco×nformalweights L R 1 − a a+1 a+2
mn
(136,136) an∈d{the 3}×3 matrix field Φmn [m,n∈{1,2,3}] where both C0 and C1 are nonuniversal constants. On
with weights (1,1). This WZNW model is identical to the left-hand side, a = 1, 2, and 3 correspond to x, y
2 2