Chain Breaks and the Susceptibility of Sr2Cu1�xPdxO3��and Other Doped Quasi-One-Dimensional Antiferromagnets

J. Sirker,1 N. Laflorencie,1 S. Fujimoto,2 S. Eggert,3 and I. Affleck1

1Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z12Department of Physics, Kyoto University, Kyoto 606-8502, Japan

3Department of Physics, University of Kaiserslautern, D-67663 Kaiserlautern, Germany(Received 5 October 2006; published 27 March 2007)

We study the magnetic susceptibility of one-dimensional S � 1=2 antiferromagnets containing non-magnetic impurities which cut the chain into finite segments. For the susceptibility of long anisotropicHeisenberg chain segments with open boundaries we derive a parameter-free result at low temperaturesusing field-theory methods and the Bethe ansatz. The analytical result is verified by comparing withquantum Monte Carlo calculations. We then show that the partitioning of the chain into finite segmentscan explain the Curie-like contribution observed in recent experiments on Sr2Cu1�xPdxO3��. Possibleadditional paramagnetic impurities seem to play only a minor role.

DOI: 10.1103/PhysRevLett.98.137205 PACS numbers: 75.10.Jm, 02.30.Ik, 75.10.Pq

Measurements of the magnetic susceptiblity, �, on anti-ferromagnetics at low temperatures, T, often reveal a(Curie) term � / 1=T. This is usually associated withparamagnetic impurities (free spins). For one-dimensionalS � 1=2 systems, however, a second mechanism existswhich leads to a contribution with somewhat similar Tdependence. Nonmagnetic impurities can cut the chaininto finite segments with essentially free boundaries. AtT=J� 1=L (J being the exchange constant) a segmentwith odd length L will lock into its doublet ground stateyielding also a 1=T contribution [1–4]. However, the be-havior is considerably more complicated at higher T. Aswe show, fitting data to a Curie form for a limited range ofT can lead to an underestimate of the impurity concentra-tion by as much as a factor of 10. In general, the different Tdependence could make it possible to distinguish this typeof impurity from paramagnetic ones providing usefulstructural information.

These results are relevant for materials like Sr2CuO3��which is known to be an almost ideal realization of thespin-1=2 Heisenberg chain with a nearest-neighbor cou-pling constant J� 2200 K and a low Neel temperatureTN � 5 K� 0:002J [5,6]. Measurements of the magneticsusceptibility have revealed a Curie contribution whichcould be dramatically reduced by annealing. The Curieterm therefore is believed to be mainly caused by excessoxygen [5,6]. Recently, susceptibility measurements havealso been reported on Sr2Cu1�xPdxO3�� with Pd serving asa nonmagnetic impurity [7].

In this Letter we show that these susceptibility data canbe explained by taking the segmentation of the chain due tothe nonmagnetic impurities into account properly withoutincluding any additional paramagnetic impurities. To thisend we derive a parameter-free analytical result for thesusceptibility of an anisotropic Heisenberg chain withfinite length and open boundary conditions (OBCs) atlow temperatures using field-theory methods. This extendsand generalizes previous results in the scaling limit [2,3]

and for the boundary susceptibility in the thermodynamiclimit [8–11]. We will verify our analytical result usingquantum Monte Carlo (QMC) calculations.

We consider an anisotropic spin-1=2 Heisenberg chainconsisting of segments with length L and OBCs

H � JXL�1

i�1

�Sxi Sxi�1 � S

yi S

yi�1 ��SziS

zi�1�: (1)

Here, � parameterizes the exchange anisotropy. We as-sume that T � TN so that interchain couplings can besafely ignored. As argued at the end of this Letter,longer-range interactions which could bridge between thechain segments can be neglected as well in the parameterregime we are interested in. For a concentration of chainbreaks p the average chain length is given by �L � 1=p� 1and the averaged susceptibility by [12]

�p � p2XL

L1� pL�L: (2)

We now calculate the susceptibility �L �hPiSzi

2i=LT in the low T large L limit using field-theory methods. We start with the Hamiltonian (1) in thescaling limit, i.e., ignoring irrelevant operators. TheHamiltonian is then equivalent to a free boson model [13]

H �v2

Z L

0dx��2 � @x�

2�: (3)

Here � is a bosonic field obeying the standard commuta-tion rule ��x;�x0� � i�x� x0 with � � v�1@t�.The velocity v is a known function of the anisotropy �.Using the mode expansion for OBCs [2]

�x; t � �R� 2�RSzxL�X1n�1

sin�nx=L��������np

�e�i�nvt=Lan � ei�nvt=Layn ; (4)

where R is the compactification radius of the bosonic field,

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the Hamiltonian (3) can also be expressed as

H ��vKL

S2z � hSz �

�vL

X1n�1

naynan � 1=2: (5)

Here an is a bosonic annihilation operator, Sz an integer(half-integer) for L even (odd), and h the magnetic field.K—the so called Luttinger parameter—is related to thecompactification radius by K � 1=2�R2 and is a knownfunction of anisotropy � as well. The susceptibility in thescaling limit is then given by

�s � �@2f

@h2

��������h�0�

1

LT

PSz S

2z exp�� �v

KLT S2z�P

Sz exp�� �vKLT S

2z�

: (6)

For LT=v! 0 and L even �s �2LT exp�� �v

KLT�, whereasfor L odd �s � 4LT�1. For LT=v! 1 the thermody-namic limit result �s � K=2�v is recovered. Correctionsto scaling occur due to irrelevant bulk and boundary op-erators. The leading bulk irrelevant operator for 0< � � 1is due to umklapp scattering yielding the following correc-tion to (3)

�H � �1

Z L

0dx cos2�=R; (7)

with �1 being the umklapp scattering amplitude. To obtain� to first order in the operator (7) for L and T finite theexpectation value hexp 2i�=Ri has to be calculated.Using the mode expansion (4) this expectation value splitsinto an Sz (zero mode) and an oscillator part. Upon usingthe cumulant theorem for bosonic modes we obtainhexp 2i�=Ri � hexp 2i�=RiSz exp�2h��iosc:=R

2

with

h��iosc: �X1l�1

sin2�lx=L�l

�1�

2

e�vl=TL � 1

�: (8)

Introducing a cutoff � for the zero temperature part in (8)and using

P1l�1 z

l=l � � ln1� z for jzj< 1 we obtainthe following correction to (6)

��1�2 ~�1

T2

��L

�2K�6K

�e��v=TL

�Z 1=2

0dy

g0y;e��v=KLT

�2K1 �y;e

��v=2TL

�22Ksin2K�y�1�e2�iye���=LH:c:��K (9)

with

g0y; q � �

PSz S

2z cos4�Szyq

S2zP

Sz qS2z

�PSz cos4�Szyq

S2z PSz S

2zq

S2z

PSz q

S2z 2

: (10)

Here�x is the Dedekind eta-function, �1u; q the elliptictheta function of the first kind [14] and ~�1 � �2K�1. ForK < 3=2 the integral in (9) is convergent, the cutoff can bedropped, and the last line becomes equal to one. An addi-

tional correction arises due to the presence of an irrelevantboundary operator. In essence, this operator changes thelength of the chain L to some effective length L� a [2].Replacing L by L� a in the exponentials in (6) andexpanding to lowest order in a yields

��2 � ��vag1exp���v=KLT�=KT2L3 (11)

with

g1q �PSz S

2zq

S2z 2

PSz q

S2z 2�

PSz S

4zq

S2zP

Sz qS2z: (12)

For K > 3=2 we isolate the cutoff dependence by subtract-ing the Taylor expand of the integrand in (9) to first non-vanishing order in y, and then take �! 0, yielding aconvergent integral. The cutoff dependent term has exactlythe same form as (11). We can therefore think of a in (11)as a parameter incorporating both contributions. In thethermodynamic limit g1e

��v=KLT ! K2T2L2=2�2v2and ��2 ! Ka=2�vL. In this limit we can compare thefield-theory result with a recent calculation of the boundarysusceptibility based on the Bethe ansatz [9,10] leading to

a � 2�1=2 sin��K=4K � 4�= cos��=4K � 4�: (13)

Because the amplitude ~�1 is known exactly as well [10,15],the obtained result is parameter-free. In Fig. 1 a compari-son between this field-theory result and QMC data for � �3=4 is shown and in the left inset the corrections to thescaling limit (6) due to the leading irrelevant bulk andboundary operator are visualized. More generally, we al-ways find excellent agreement if T=J & 0:1 and L * 10.The range of validity can be extended in principle by

0.01 0.1 1T/J

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Jχ

0 1 2LT/J

0

0.05

0.1

0.15

Jχ

0.01 0.1 1T/J

0

0.1

0.2

0.3

0.4

Jχ

scaling function [Eq. (6)]

L = 20

L = 40

L = 100

FIG. 1 (color online). QMC data for � � 3=4 and L � 99 (redsquares), L � 100 (black dots) compared to the field-theoryresult derived here (green solid lines). The blue dashed linedenotes � � �bulk � �B=L [see Eqs. (148, 149) in [10]]. Leftinset: Corrections to scaling for even length chains and � � 3=4.QMC data are represented by symbols, the field theory by thegreen lines. Right inset: Same as main figure for � � 1 with a �5:8 and �bulk, �B as obtained in [8,15].

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including terms in higher order perturbation theory in �1.However, if T=v� 1=L it is justified to separate � into abulk, �bulk, and a boundary contribution, �B, and to per-form the thermodynamic limit for each of them separately.A combination of this result [10] (blue dashed line inFig. 1) with the result derived in this letter yields anexcellent parameter-free description of the susceptibilityfor anisotropic chains of finite length up to T � J=2.

For K � 1 (� � 1) perturbation theory alone is notsufficient because umklapp scattering becomes marginal.In this case the coupling constant ~�1 has to be replaced by arenormalization group (RG) improved running couplingconstant [15,16]. ~�1 then becomes a function of the twolength scales L and v=T. For small enough energies thesmaller scale will always dominate and the running cou-pling constant becomes ~�1L; v=T � g=4 with [15]

1=g� lng=2 � ln���������2=�

pe1=4� min�L; v=T�: (14)

where is the Euler constant. ��1 in the isotropic case istherefore given by (9) with the last line dropped and ~�1

replaced by ~�1L; v=T. Furthermore, K in (6) should bereplaced by 1� gL; T=2 so that KL in the exponentialsgets replaced by L1� g=2� a=L. For this case a is notknown and we will use it as a fitting parameter. This waywe obtain excellent agreement if T=J & 0:1 and L * 10(see right inset of Fig. 1). At LT � v the corrections to thescaling formula (6) in the isotropic case due to the marginalinteraction can also be understood as follows. In this limitthe most important correction to (6) arises from the cor-rection to the excitation energy of the lowest excited stateswith Sz � 1, E! �v=L�1� gL�, [17] correspond-ing to the replacement 1=K ! 1� gL in (6). Here gLis given by (14) but with min �L; v=T� simply replacedby L.

These analytical results, supplemented by numericaldata for L & 10, can be used to calculate the averagedsusceptibility (2) at low temperatures. In addition, we havecalculated �L numerically for all L 2 �1; 100� and cer-tain additional L up to L � 1000 at various temperatures.An interpolation and extrapolation was then used to obtain�L for all lengths up to L � 5000 at a given temperature[18]. This allows us to reliably calculate �p (2) for impu-rity concentrations p * 0:2% ( �L & 500).

Experimental data for the susceptibility of the quasi one-dimensional compounds Sr2Cu1�xPdxO3�� have been an-alyzed in [5–7] by decomposing it into � � �0 � �C ��inf :. Here �0 represents a constant part due to core dia-magnetism and Van Vleck paramagnetism, �C a Curieterm, and �inf : the susceptibility (per unit length) of theinfinite S � 1=2 Heisenberg chain. However, our resultsshow that nonmagnetic impurities (chain breaks) have aquite different effect than paramagnetic impurities. In theextreme low-T limit, T � pv, chain breaks will lead to1=T behavior with a strength reduced from that whichwould arise from the same density of paramagnetic impu-

rities by the fraction of chains of odd length, 1� p=2�p � 1=2. In the opposite limit pv� T (but T still� J),chain breaks lead to a correction to the pure result of�p=�12T ln2:9J=T� corresponding to an ‘‘effective para-magnetic impurity concentration’’ of �p=�3 ln2:9J=T�which can be� p for T � J. This behavior is illustratedin Fig. 2.

It has been shown that the Curie-like contribution to thesusceptibility of Sr2CuO3�� can be significantly reducedby annealing under Ar atmosphere indicating that thiscontribution is mostly due to excess oxygen [19]. Weexpect the additional interstitial oxygen ions to be in aO2� ionization state. Each oxygen ion would then dopetwo holes into the chains. If these holes are immobile theyeffectively act like chain breaks. It has been pointed outthat the next-nearest-neighbor coupling is not that small,J2 � J=16 � 140 K [20]. Whereas we expect this irrele-vant coupling to cause only small corrections to the sus-ceptibility of an isolated chain segment the fact that it canbridge a chain break could be significant. A first orderperturbative calculation, however, shows that the correc-tion to the averaged susceptibility is given by ��p �pJ2�2

p. We can therefore ignore both interactions as longas T � maxpJ2; J? making the model discussed hereapplicable. As shown in the inset of Fig. 3 it is indeedpossible to explain the data in [5] for the unannealedsample using a 0.6% concentration of chain breaks. If ourpicture is correct the amount of excess oxygen is signifi-cantly larger than assumed in [5], 0.06%.

The Sr2Cu1�xPdxO3�� samples used in [7] were notannealed. This complicates the analysis of the susceptibil-

0.01 0.1T/J

0.001

0.01

χ p T

0.1 0.2 0.3 0.4T/J

0.02

0.04

0.06

0.08

0.1

χ p T

FIG. 2 (color online). T�p for p � 0:01, 0.03, 0.05, 0.07, 0.1from bottom to top (black dots with dashed lines being a guide tothe eye). The blue solid lines denote the value p=4�1� p=2� p reached in the limit T � pv. Inset: Samedata for T�p with subsequent curves shifted by 0.01 comparedto T�bulk � p=�12 ln2:9J=T� (red dashed lines) reached in theopposite limit.

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ity data because excess oxygen again leads to chain breaksin addition to those caused by the nonmagnetic Pd ions.The amount of excess oxygen hereby seems to vary sig-nificantly from sample to sample. This is most apparentwhen comparing the measured susceptibilities for x �0:5% and x � 1% which are almost identical apart froma small constant shift. We therefore treat the impurityconcentration p as a fitting parameter. Results where weused J � 2200 K and a gyromagnetic factor g � 2 areshown in Fig. 3 and Table I. The figure clearly showsthat the susceptibility data are indeed quite different fromwhat would be expected for paramagnetic impurities. Mostof the data are taken in the nontrivial crossover regimepv� T and are well described by our theory. Note, that thedeviations at low temperatures are expected because the

suppression of the susceptibility due to interchain and next-nearest-neighbor interactions is not taken into account inour model. For x � 0:005 and 0.01 the effective impurityconcentrations p are larger than the nominal Pd concen-tration. p� x roughly coincides with the number of chainbreaks in the ‘‘as grown’’ Sr2CuO3�� supporting our pic-ture of additional chain breaks due to excess oxygen. Forx � 0:03, however, p is smaller than the nominal Pdconcentration. Possible explanations are that for largerdoping levels some of the Pd ions go in interstitially insteadof replacing Cu ions or that the sample exhibits some sortof phase separation. Future measurements on annealedSr2Cu1�xPdxO3�� samples would be helpful to clarifysome of these issues.

The authors acknowledge valuable discussions withW. N. Hardy, K. M. Kojima, and G. A. Sawatzky. Thisresearch was supported by NSERC (J. S., N. L., I. A.), theDFG (J. S.) and the CIAR (I. A.). Numerical simulationshave been performed on the Westgrid network.

[1] J. C. Bonner and M. E. Fisher, Phys. Rev. 135, A640(1964).

[2] S. Eggert and I. Affleck, Phys. Rev. B 46, 10 866 (1992).[3] S. Wessel and S. Haas, Phys. Rev. B 61, 15262 (2000).[4] B. Schmidt et al., Eur. Phys. J. B 32, 43 (2003).[5] N. Motoyama, H. Eisaki, and S. Uchida, Phys. Rev. Lett.

76, 3212 (1996).[6] T. Ami et al., Phys. Rev. B 51, 5994 (1995).[7] K. M. Kojima et al., Phys. Rev. B 70, 094402 (2004).[8] S. Fujimoto and S. Eggert, Phys. Rev. Lett. 92, 037206

(2004).[9] M. Bortz and J. Sirker, J. Phys. A 38, 5957 (2005).

[10] J. Sirker and M. Bortz, J. Stat. Mech. (2006) P01007.[11] A. Furusaki and T. Hikihara, Phys. Rev. B 69, 094429

(2004).[12] S. Eggert, I. Affleck, and M. D. P. Horton, Phys. Rev. Lett.

89, 047202 (2002).[13] I. Affleck, in Fields, Strings and Critical Phenomena,

Proceedings of the Les Houches Summer School,Session XLIX, edited by E. Brezin and J. Zinn-Justin(Elsevier, Amsterdam, 1988), p. 563.

[14] I. Gradshteyn and I. Ryzhik, Table of Integral, Series, andProducts (Academic, New York, 2000).

[15] S. Lukyanov, Nucl. Phys. B522, 533 (1998).[16] I. Affleck, D. Gepner, H. J. Schulz, and T. Ziman, J. Phys.

A 22, 511 (1989).[17] S. Qin and I. Affleck, J. Phys. A 32, 7815 (1999).[18] See EPAPS Document No. E-PRLTAO-98-083714 for

the QMC as well as the interpolated and extrapolateddata. For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html.

[19] From annealed samples we know that the true paramag-netic impurity concentration is 0.013% at most [5] and cantherefore be neglected in the following.

[20] H. Rosner et al., Phys. Rev. B 56, 3402 (1997).

10 100T[K]

10-3

10-2

χT [

K e

mu/

mol

]

10 100T[K]10

-3

10-2

χT [

K e

mu/

mol

]

FIG. 3 (color online). Measured susceptibility T�� �0 forSr2Cu1�xPdxO3�� with x � 0:5%, 1%, 3% (crosses from bottomto top) where a constant �0 (see Table I) has been subtracted.The green dots represent the best theoretical fit with p as inTable I. Subsequent curves are shifted by 3� 10�3. The bluesolid lines represent p=41� p=2� p and the red dashedlines T�bulk � p=�12 ln2:9J=T�. Note that fits � ��0 � �inf : � ppara=4T in the regime T > 50 K would be simi-lar to the red dashed lines with effective paramagnetic impurityconcentrations ppara � p=�3 ln2:9J=T� an order of magnitudesmaller than the nominal Pd concentrations. Inset: Same for asgrown sample of Sr2CuO3�� from Ref. [5].

TABLE I. Concentration x of Pd ions in experiment comparedto impurity concentration p and constant contribution �0 yield-ing the best theoretical fit. The first line corresponds to the asgrown sample of Sr2CuO3�� from Ref. [5].

x (Exp.) p (Theory) �0 [emu/mol]

0.0 0.006 �7:42� 10�5

0.005 0.012 �7:7� 10�5

0.01 0.014 �7:5� 10�5

0.03 0.024 �7:5� 10�5

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