Massive and Massless Behavior in Dimerized Spin Ladders

a r X i v :c o n d -m a t /981026

3

v

2 [c o n d -m a t .s t r -e l ] 12 F e b 1999

Massive and Massless Behavior in Dimerized Spin Ladders

D.C.Cabra 1,#and M.D.Grynberg 2

1

Physikalisches Institut der Universit¨a t Bonn,Nussallee 12,53115Bonn,Germany

2

Departamento de F′?sica,Universidad Nacional de la Plata,C.C.67,(1900)La Plata,Argentina

We investigate the conditions under which a gap vanishes in the spectrum of dimerized coupled spin-1/2chains by means of Abelian bosonization and Lanczos diagonalization techniques.Although both interchain (J ′)and dimerization (δ)couplings favor a gapful phase,it is shown that a suitable choice of these interactions yields massless spin excitations.We also discuss the in?uence of di?erent arrays of relative dimerization on the appearance of non-trivial magnetization plateaus.PACS numbers:75.10.Jm,75.60.Ej

Published in Phys.Rev.Lett.82,1768(1999).

Current developments in the ?eld of spin ladders have revealed intriguing features of low-dimensional quantum antiferromagnets (AF)[1].The appearance of plateaus in the magnetization curves of these systems has received much attention from both the theoretical and experimen-tal side [2].A necessary quantization condition for the appearance of such plateaus in generic ladders was de-rived in [2](see also [3]for the case of 1D chains and [4–7]for other particular cases).In a very recent paper [8],an unexpected phenomenon has been observed in the mag-netization curves of NH 4CuCl 3at high magnetic ?elds.The crystal structure of this material at room tempera-tures is known to be composed of double chains (2-leg lad-ders),with three di?erent nearest neighbor interactions.Thus,from previous theoretical studies of such arrange of couplings in ladders systems,plateaus at M =0and 1/2are expected to be visible at su?ciently low temper-ature.However,measurements performed in [8]at very low temperatures,clearly show two net plateaus at 1/4and 3/4,while no plateaus are observed at the theoreti-cally expected values M =0and 1/2.

As a ?rst step to reconcile some of these facts with the current understanding of ladder materials,in this work we study two important issues namely,(i)the possibility of closing the gap in a two-leg dimerized ladder by a com-bined e?ect of the dimerization and the interchain cou-pling and,(ii)the emergence of a plateau at M =1/2depending on the realization of the relative dimerization (see Figs.1(a)-(b)).Though (i)was studied on a quali-tative level using non-linear sigma model techniques [9],here we present a quantitative and more systematic treat-ment,whereas (ii)is analyzed for the ?rst time in this work.These two issues could shed light in the study of the experimental measurements such as those performed in [8]as well as in related aspects of CuGeO chain com-pounds [10],in which dimerization becomes staggered be-tween weakly coupled chains.

As is well known,the isotropic spin-1/2Heisenberg chain is already in a critical state.Thus any relevant perturbation,such as the Peierls dimerization instability [11]or a weak interchain coupling [12],[13],can dras-tically alter the nature of the ground state,whereas a massive spin gap excitation appears simultaneously in the energy spectrum.Interestingly,it was suggested that

a combined e?ect could lead to a massless regime [9].In the present work we study this issue quantitatively by means of both bosonization and numerical techniques,and ?nd that there exists indeed a ?ne-tuning of the cou-plings responsible for the appearance of massless lines.Though this is a ?ne-tuning e?ect,our result implies that there is a ?nite region of the coupling parameter space where the gap is expected to be small.Thus,when the latter becomes comparable with thermal ?uctuations,measurement attempts of zero magnetization plateaus could be smeared out even at low temperatures.

In analyzing the abelian bosonization of spin-1/2Heisenberg ladders,we follow a similar methodology de-veloped as in [2,5],[13],[14],[15],which is particu-larly suitable to elucidate the behavior of weak coupling regimes.Speci?cally,here we study two dimerized spin chains interacting through a Hamiltonian

H = a,n

J (a )n S (a )n · S (a )n +1+J ′

n

S (1)n · S (2)n ,(1)

(a =1,2),where the S

n denote spin-1/2operators.For staggered ladders,e.g.Fig.1(a),the array of coupling

exchanges are set as J (2)

n ≡J (1)n +1,and parametrized by J (1)

n =J [1+(?1)n δ]say for chain (1),whereas for the

non-staggered situation we just set J (2)

n ≡J (1)n ,as shown in Fig.1(b).To maintain pure AF and non-frustrated exchanges,the dimerization parameter is kept bounded by |δ|<1throughout the 2L spins of the ladder length with periodic boundary conditions (L even).

On general grounds [2],it is expected that gapful mag-netic excitations should appear for all magnetizations M ≡1

2

n S z (1)n

+S z (2)

n ,so as to unravel the

interplay between the above kinds of coupling arrays and applied magnetic ?elds h ,say along the z -direction.It is well known that the low-energy properties of the Heisenberg chain,(δ=0),are described by a c =1con-formal ?eld theory of a free bosonic ?eld compacti?ed at radius R =R ( M ,?)[2,4,5,14,15],for any given mag-netization M and XXZ anisotropy |?|<1(see e.g.[16]).The functional dependence of R can be obtained using the exact Bethe Ansatz solution by solving a set of integral equations obtained in [17,18]and [19](for a fuller review consult for instance Ref.[2]).The bosonized expression of the low-energy e?ective Hamiltonian for a single homogeneous chain,(δ=0),in the presence of an external magnetic ?eld h and with an XXZ anisotropy ?1

ˉH

=

d x π4πR

2

.

In the limit of both weak dimerization |δ|?1,and interchain coupling J ′/J ?1,the bosonized action reads

H int ≈λ1

x

?x φ(1)?x φ(2)+

λ2

x

cos(4k F x +

√

4π(φ(1)?φ(2)))+λ4

x

cos(

√

4πφ(a )(x )).(3)

where λi ∝J ′/J ,α(1)∝δand α(2)∝±δ,(the minus sign corresponding to Fig.1a),and the Fermi momentum k F is related to the total magnetization M via k F =(1? M )π/2.In (3)a marginal term has been neglected for the sake of simplicity since it does not change our results.

(i)Closing of the gap

At zero magnetization (i.e.k F =π/2),all terms in (3)are commensurate and relevant.Due to the λ1term,we have to ?rst diagonalize the derivative part of the Hamil-tonian which is achieved by introducing new variables φ±=(φ1±φ2)/

√1±J ′/(4Jπ2R 2).

For AF interchain coupling,(J ′>0),and in the new

basis,the λ4term is the most relevant and orders the ?φ

??eld (the dimerization terms mix the φ+and φ?variables

and will be treated as a perturbation).The αinterac-tion in (3)is then wiped out after integrating the massive

φ??eld,since it always contains a contribution from φ?.Thus we are left with the φ+?eld with a relevant inter-action given by λ2.Hence,this ?eld is in general massive so we can expect a plateau at zero magnetization.The interesting point here is that the αinteraction generates radiatively a term that can cancel the λ2perturbation,(which is the one responsible for the mass of the sym-metric boson).From this plain weak coupling analysis we see that this happens on the critical line given by

J ′

Let us now consider Eq.(3)at non-zero magnetiza-

tion M .As was referred to above,this can be read-ily accounted through the radius of compacti?cation

R( M ,?)resulting from the e?ect of the magnetic?eld h and the Fermi momentum k F=(1? M )π/2.Then, theλ2andαinteractions are incommensurate,and we have in principle a massless mode corresponding to the φ+?eld.However an interesting phenomenon can be ob-served:a plateau in the magnetization curve is expected to appear at1/2of the saturation value due to the ap-pearance of the radiative correction

γ

L

x=1

(?1)x cos(4k F(x+1/2)+

√

4πφ(1))+cos(2k F x+

√

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(1999).

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preparation.

FIG.1.Schematic view of alternating ladders with inter-

chain coupling J′and dimerization parameterδfor(a)stag-

gered and,(b)non-staggered arrays

0.5

1

1.5

2

2.5

-1-0.500.51

G

a

p

L=4

6

8

1012

FIG.2.Gap spectrum for di?erent sizes of the staggered

spin ladder for J′/J=1.The dashed lines display extrapo-

lations to the thermodynamic limit.

0.5

1

1.5

2

-1-0.500.51

J

’

/

J

FIG.3.Critical line of the staggered spin ladder in the

(δ,J′)coupling parameter space,extrapolated from?nite

samples.Solid lines are guide to the eye whereas the dashed

parabola stands for the bosonization approach.

00.250.50.7510

2

468

h/J

(a)

00.250.50.7510

2

468

h/J

(b)

FIG.4.Magnetization curves of dimerized ladders for δ=0.5and J ′/J =2using both (a)staggered and,(b)non-staggered coupling arrays.Solid,dashed and short dashed lines denote respectively magnetizations for L =12,10and 8.The thick full line denotes the expected form in the thermodynamic limit