Two leg quantum Ising ladder: A bosonization study of the ANNNI model

6 Pages·0.13 MB·English

by  Dave Allen

Two leg quantum Ising ladder: A bosonization
study of the ANNNI model
1 D. Allen†, P. Azaria†, and P. Lecheminant‡
0
†LaboratoiredePhysiqueTh´eoriquedesLiquides,Universit´ePierreetMarie
0
Curie,4PlaceJussieu,75252Paris,France
2
‡LaboratoiredePhysiqueTh´eoriqueetMod´elisation,Universit´ede

n Cergy-Pontoise,5MailGay-Lussac,NeuvillesurOise,95301Cergy-Pontoise
a Cedex,France
J

6 Abstract.
2 Thequantum ANNNIchaininatransverse fieldisinvestigated bymeans of
thebosonizationapproachinthelimitoflargenext-nearestneighborinteraction.

1 Inthisregime,thismodelcanbeviewedasaweaklycoupledtwo-legzigzagladder
v whichenablesustoderiveitslowenergyeffectivefieldtheory. Inparticular,itis
4 shown that the effect of frustrationinthe system iscaptured bythe presence of
0 anonzeroconformalspinperturbationthat accounts forthe existence ofallthe
4 incommensuratephasesofthemodel.
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PACSnumbers: 75.10.Jm,64.70.Rh
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Two leg quantum Ising ladder: A bosonization study of the ANNNI model 2
Oneofthemoststrikingeffectoffrustrationinmagneticsystemsisthatitcanleadtoa
hugenumberofdegenerategroundstates. Asaresult,thenatureoffinitetemperature
phases may not, in contrast with ferromagnetic systems, be linked only to the sole
nature of the microscopic degrees of freedom and the dimension of space-time. Since
frustration induces strong fluctuations that involve large number of spins, the low
energy physics is determined by interferences between very different regions of phase
space. With this picture in mind one may not expect the field theoretical description
of the frustration to be an easy task. Of course there have been several attempts to
describefrustratedmagnetsbyfieldtheories. But,toourknowledgetheyweremostly
restrictedtomodelsthatdisplayedwelldefinedgroundstates,suchashelicalordering,
and the main effect of frustration was captured by an enlargement of the dimension
of the order parameter space[1]. Here our aim is to single out an operator, in the
Renormalization Group (RG) sense, that captures the effect of frustration. In this
respect, weshallconsiderthe moststudied frustratedsystemi.e. the twodimensional
ANNNI model (see Refs.[2, 3, 4] for a review). This model is characterized by a
nearest-neighborferromagnetic interaction( J <0) and a competiting next-nearest
1
−
neighbor antiferromagnetic interaction (J >0):
2
= J σ σ +J σ σ , (1)
1 i j 2 k l
H −
(Xi,j) X[k,l]
where (i,j) denotes nearest neighbor pairs and [k,l] next-nearest ones along a single
space direction. The phase diagramof the model (1) is very rich and well known and
can be summarize as follows. At zero temperature, the ground state is ferromagnetic
for J < J /2 and antiferromagnetic (actually an antiphase ) for J > J /2. At
2 1 2 1
↑↑↓↓
the special point J = J /2, the ground state is infinitely degenerate and correlation
2 1
functions areshortranged. At finite temperatures(T =0), one observesfive different
6
phasesdependingontheparameters(seefigure(1)): ferromagnetic(F),paramagnetic
commensurate (PC), paramagnetic incommensurate (PI), incommensurate critical
phase (IC, also usually called “floating phase”) and antiphase (A). Three different
kind ofphase transitionsarepresentinthis diagram: An Ising transitionbetweenthe
F and PC phases, a commensurate-incommensurate transition[5] between the A and
ICphases,andaBerezinski-Kosterlitz-Thouless(BKT)transitionseparatestheICand
PI phases. There is also a disorder line that extends down to zero temperature which
divides the PC and PI phases[6]. As a most notable feature of this phase diagram is
the existence of an incommensurate critical phase in a finite region of the parameter
space. In a seminal work,Villain and Bak [7] have proposedan approximate effective
theoryintermsoffermionsvalidinthevicinityofthedegeneratepointJ =J /2. All
2 1
their predictions have been further confirmed by numerical investigations and series
expansions. It is the purpose of the present letter to propose a complementary and
alternative low energy description of the ANNNI model valid in the large J limit.
2
Furthermore, within our approach, we exhibit a particular operator that is at the
origin of the incommensurate phases found in the phase diagram (figure (1)).
Our starting point is to map (1) into a quantum Ising model in a transverse
magneticfield. Thisisdonebyintroducingananisotropybetweenthenearestneighbor
interactioninthex-direction(J )andinthey-direction(J ). Asitiswellknown(see
1 τ
Ref. [4] for a review), the physics of the classical model (1) should be equivalent to
the one described by the following quantum Hamiltonian:
= βJ∗σz +βJ σxσx βJ σxσx , (2)
H − τ n 2 n n+2− 1 n n+1
n
X(cid:2) (cid:3)
Two leg quantum Ising ladder: A bosonization study of the ANNNI model 3
T/J
1
PC
PI
IC
F
A
0 0.5 1 J/J
2 1
Figure 1. Phasediagramoftheclassicaltwo-dimensionalANNNImodel.
where σx, σz are Pauli matrices and 2βJ∗ = lncoth(βJ ). The model (2) is nothing
n n τ τ
but the one-dimensional axial next-nearest neighbor Ising (ANNNI) model in a
transverse field. As seen in figure (2), it can also be viewed as two quantum Ising
chains labelled (1) and (2) coupled by a “zigzag” interaction with strength J and
1
thus identifies with a two-leg quantum Ising ladder with Hamiltonian:
a a a
= βJ∗σz(n+ ) βJ σx(n+ )σx(n+1+ )
H − τ a 2 − 2 a 2 a 2
n,Xa=1,2h i
βJ 1
1 σx(n+ )[σx(n) σx(n+1)]+(1 2). (3)
− 2 1 2 2 − 2 →
n
X
Notice that in order to obtain (3) we have performed an (unphysical) gauge
transformation on the ath chain (a = 1,2): σx(n + a/2) ( 1)n+aσx(n + a/2).
a → − a
As weshallnowsee,the model(3)canbe consistentlydescribedbya continuousfield
theory in the limit J <<J and J J∗.
1 2 2 ∼ τ
. . . . .
−J
2
−J . +J . . . .
1 1
−J
2
Figure 2. Inthe largeJ2 limit,theANNNImodelisbetter seenas twoweakly
coupledquantum Isingchains.
Continuum limit. We shall study the model in the vicinity of the
antiferromagnetic phase where J << J . One can take advantage that in the limit
1 2
J =0 the model is equivalent to two decoupled Ising models which are critical when
1
J = J∗. One can therefore expand the theory around the conformal invariant fixed
2 τ
pointwithsymmetryZ Z . InthecriticalregimethetwoIsingchainsaredescribed
2 2
⊗
Two leg quantum Ising ladder: A bosonization study of the ANNNI model 4
by two pairs of right and left Majorana (real) fermions ψ ,a=1,2:
a(R,L)
v
= i (ψ ∂ ψ ψ ∂ ψ ) m(ǫ +ǫ ), (4)
0 aR x aR aL x aL 1 2
H − 2 − −
a=1,2
X
ǫ =iψ ψ being the energy operatorof the ath Ising model whereas the mass gap
a aR aL
and the velocity are given by: m = 2(J∗ J )/T << 1, v = a J∗/T (a being the
τ − 2 0 τ 0
lattice spacing). In the absence of interchain interaction, the system is disordered
(respectively ordered) when m > 0 (respectively m < 0) and one has σ = 0
a
h i
(respectively σ = 0) where σ are the order operators associated with the two
a a
h i 6
Ising models. We consider now the interacting case and take the continum limit of
the second term in equation (3) in the regime J J to obtain:
1 2
≪
1/4
+g , =a (σ ∂ σ σ ∂ σ ). (5)
H≃H0 V V 0 1 x 2− 2 x 1
When m = 0 and g = J /2T = 0, the model is conformal invariant with central
1
charge c = 1 + 1 = 1. In the generic case, this fixed point is perturbed by the
2 2
thermaloperatorsǫ whichhavescalingdimension∆=1andtheoperator which
1(2)
V
has scaling dimension ∆ = 5/4. It is important to stress that both operators reflect
verydifferentphysicalbehaviors. Whiletheformeristhestandardoperatormeasuring
deviationfromcriticalityinnon-frustratedIsingmagnets,thelatterencodesthewhole
effectoffrustrationandisresponsibleofthenontrivialphasesoftheANNNImodelas
we shall see. The operator manifests itself by the fact that it is a parity symmetry
V
breaking perturbation with a conformal spin equals to 1. The effect of such non-zero
conformal spin term is non trivial since the usual irrelevant versus relevant criterion
doesnotholdforsuchanon-scalarcontribution(seeforinstancethediscussioninRef.
[8]). In fact, a similar operator also appears in the study of the S=1/2 Heisenberg
zigzag ladder[9] and has been called a “twist term”. Such contribution is difficult to
handle non-perturbatively but it has been stressed that this term may represent a
new mechanism for incommensuration in one-dimensionalsystems[9, 10]. As we shall
see, in the particular case of the ANNNI model, the effect of this operator can be
elucidated by means of the bosonization approach and accounts for the formation of
the non trivial incommensurate phases depicted in figure (1).
Bosonization and Effective Theory. We shall use the well known equivalence
betweentwocriticalIsingmodels,characterizedbyfourchiralrealfermionsψ and
1R(L)
ψ ,andafreebosontheorydescribedbythe chiralfieldsφ . Thebosonization
2R(L) R(L)
rules are (see Ref. [8] for a review):
1
(ψ +iψ ) = exp( i√4πφ ), (6)
1 2 R(L) √πa ± R(L)
0
from which it follows that is equivalent to a sine-Gordon model at β2 = 4π (free
0
H
fermionpoint). Thebosonizationofthetwistterm(5)requiresmorework. Asuitable
bosonicexpressionisobtainedbyconsideringthefollowingoperatorproductexpansion
(OPE) that stems from the fact that the order operator σ is a primary field:
a
1
(T T )(z)σ σ (w,w¯) [σ ∂σ σ ∂σ ](w,w¯), (7)
1 2 1 2 2 1 1 2
− ∼ z w −
−
where T (a = 1,2) are the energy momentum tensors in the holomorphic sector
a
(z = vτ +ix) associated with the Ising models. The fields in the left hand side of
equation (7) can be expressed in terms of the bosonic fields: T T =cos(√16πφ )
1 2 L
−
Two leg quantum Ising ladder: A bosonization study of the ANNNI model 5
and σ σ =√2sin(√πΦ) (Φ=φ +φ ). We thus deduce by performing the OPE in
1 2 L R
the bosonic theory the following representationof the twist term:
v m g√2
= (∂ Φ)2+(∂ Θ)2 cos √4πΦ i cos √πΦ sin √4πΘ ,(8)
x x
H 2 − πa − a
0 0
Θ =φ (cid:16)φ being the dua(cid:17)l field. The n(cid:16)ext ste(cid:17)p of our appr(cid:0)oach i(cid:1)s to(cid:16)map (8)(cid:17)onto
L R
−
the XXZ Heisenberg model in a magnetic field:
2πu
= J2 +J2 +4πma (JxJx+JyJy)+πg√2(Jy Jy)+2πλ JzJz, (9)
H 3 R L 0 R L R L R− L 0 R L
where u = (cid:0)5v/4, λ (cid:1)= 3v/2 and the SU(2) Kac-Moody currents are given by
0 1
Jz = ∂ φ /√2π and J+ = exp( i√8πφ )/(2πa ). The two expressions (8)
L,R x L,R L,R ± L,R 0
and (9) can be shown to be equivalent by a canonical transformation at the special
value λ = 3v/2. As a result, in an appropriate basis, the twist operator acts as a
0
magnetic field. The model (9) is more conveniently analyzed in a basis where the
magnetic field lies along the z axis. To do so, we perform a π/2 rotation around the
x-axis in the spin space so that one finally obtains after rebosonizing once again the
currents:
u∗
= (∂ Φ)2+(∂ Θ)2 g cos 8πQΦ g cos 8π/QΘ h∂ Θ, (10)
x x 1 2 x
H 2 − − −
where (cid:16) (cid:17) (cid:16)p (cid:17) (cid:16)p (cid:17)
1 ma /u
0 ∗
Q= − , u =u (1+ma /u)(1 ma /u), (11)
0 0
s1+ma0/u −
p
g = (2m λ /a )/(4πa ), and h = g π/Q. The effective field theory (10) is the
1(2) 0 0 0
±
mainresultofthisworkandallthedifferentphases(apartfromtheFandPCphases)
p
observed in the ANNNI model can be deduced from a simple analysis of it.
Phase Diagram. Consider first the high-temperature phase, i.e. when m > 0.
Since Q < 1, for sufficiently large m, the term cos( 8π/QΘ) is strongly irrelevant
and can be dropped. The remaining theory is then easy to analyse and a mass gap
p
to all excitations is generated due to the presence of the relevant cos(√8πQΦ) term.
On the other hand the operator ∂ Θ leads to incommensurate fluctuations of the
x
Θ dependent correlation functions. This phase corresponds to the incommensurate
paramagnetic phase PI. This picture is confirmed by an exact solution at the special
value Q = 1/2 where the model becomes equivalent to that of free fermions with
dispersion ǫ2(k)=v2 k g√2π/v 2+9v2/16a2. As readily seen, there is a spectral
± ± 0
gap and incommensuration develops as soon as g =0 with wavevectork∗ =√2πg/v.
(cid:0) (cid:1) 6
Similarily one can study the low temperature regime i.e. m < 0 and Q > 1.
For sufficiently large m, it is now the cos(√8πQΦ) term that is irrelevant and can
be dropped. After a duality transformation, the resulting Hamiltonian is equivalent
to the XXZ chain in a magnetic field along the z axis which is equivalent to the
bosonizedversionoftheVillain-Baktheory[11]derivedinthevicinityofthedegenerate
point J /J = 1/2, T = 0. At small g, there is a gap to all excitations with
2 1
no incommensuration: it corresponds to the A phase. As g grows, the magnetic
field increases until it reaches the gap at some critical value (g ) above which
c1
the excitations become massless. For g > g , the system displays as well an
c1
incommensuratebehaviorwithwavevectork∗ g2 g2 andoneentersthefloating
∼ − c1
phaseIC.Thenatureofthetransitionatg =g isofacommensurate-incommensurate
c1 p
type[5]. Finally, as g further increases, the cos(√8πQΦ) term eventually becomes
Two leg quantum Ising ladder: A bosonization study of the ANNNI model 6
relevant and opens a gap at a critical value g where a BKT transition to the PI
c2
phase takes place. This picture is, as above, confirmed by an exact solution at the
point Q=2 where the model (10) becomes equivalent to massive free fermions.
As seen, the previous analysis correctly reproduces the phase diagram of the
ANNNI model in the vicinity of the A phase where frustration plays its tricks. To
do so, we have assumed that m was sufficiently large to be able to neglect one of
| |
the cosine terms in equation (10). One can question the validity of this scheme when
m 0wherebothcos(√8πQΦ)andcos( 8π/QΘ)operators,being almostmarginal,
∼
compete. AdetailedanalysisoftheRGequationsassociatedwith(10)isthuscalledfor
p
but one is faced with the difficulty that the coupling constants are not small since λ
0
is of order one. One has therefore to make the hypothesis that the qualitative feature
of the RG approach does not depend on the strength of g and g and treat them as
1 2
small couplings. The RG equations have already been obtained in a different context
byGiamarchiandSchulz[12]. Itfollowsfrom[12]thattheresultswehaveobtainedfor
large m remainvalid for smallm confirming our hypothesis. However,there is room
| |
foraspin-floptransitionfromtheAphasedirectlytothePIphaseifthecos(√8πQΦ)
perturbation blows up before the ∂ Θ term closes the gap. The occurence of such
x
a transition strongly depends on the microscopic couplings of the bare Hamiltonian.
TheanalysisoftheRGequationsforλ =6u/5revealsthataspinfloptransitiondoes
0
not occur for the ANNNI model. However, care has to be payed since for this value
ofλ ,perturbationtheorystrictly doesnotapplyandthis leavesopenthe questionof
0
the existence of a Lifshitz point in the ANNNI model. Furthermore, notice that the
presenceofaLifshitzpointisagenericfeatureof(10). Thereforeonemayexpectthat
other lattice Hamiltonians displaying the same qualitative phase diagram than the
ANNNI model will exhibit such a point. In summary, we have derived a low energy
descriptionofthe ANNNI modelinthe largeJ limit wherethe systemcanbe viewed
2
as a weakly coupled two-leg zigzag ladder. This enables us to start from a conformal
invariant fixed point in the vicinity of which the continuum limit is well defined and
frustrationmanifests itselfthroughthe twist operator. This approachaccountsfor all
the incommensurate phases of the ANNNI model and in the low temperature limit
matches the Villain-Bak description near the degenerate point.
D. Allen is a CNRS post-doctoral fellow and thanks l’universit´e Cergy-Pontoise
for hospitality while this work was initiated.
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