VOLUME 84, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 7 FEBRUARY 2000

Nuclear Spin Relaxation Rates in Two-Leg Spin Ladders

F. Naef and Xiaoqun WangInstitut Romand de Recherche Numérique en Physique des Matériaux (IRRMA), INR-Ecublens, CH-1015 Lausanne, Switzerland

(Received 9 July 1999)

Using the transfer-matrix density-matrix renormalization group method, we study the nuclear spinrelaxation rate 1�T1 in the two-leg s � 1

2 ladder as a function of the interchain �J�� and intrachain�Jk� couplings. In particular, we separate the qy � 0 and p contributions and show that the lattercontribute significantly to the copper relaxation rate 63�1�T1� in the experimentally relevant coupling andtemperature range. We compare our results to both theoretical predictions and experimental measureson ladder materials.

PACS numbers: 76.60.–k, 75.10.Jm, 75.40.Gb

Dynamical properties, while providing the most de-tailed information on the physics of low-dimensionalantiferromagnets, are also the most difficult for theoreticalpredictions. Measured in NMR experiments probing thelow-frequency spin dynamics, they continue to revealunexpected features. For instance, the recent experimentsby Imai et al. [1] on the Cu2O3 two-leg s � 1

2 laddersin La6Ca8Cu24O41 have shown the different tempera-ture T dependence of the nuclear spin relaxation rate1�T1 for the oxygen �17�1�T1�� and copper �63�1�T1��atoms. Although both exhibit activated behavior belowT � 350 K, 63�1�T1� shows a crossover to a linear Tdependence above T � 350 K. Shortly after these experi-ments were performed, Ivanov and Lee [2] proposed thatthe crossover may be understood from the momentumtransfer q � �p , p� contributions in the weakly coupledchain limit. The importance of the qy � p processes wasalso mentioned earlier by Sandvik et al. [3] in a study of63�1�T1� for SrCu2O3, where they consider the isotropicladder at T�J $ 0.2. Despite these findings, a bettertheoretical understanding of 1�T1 in isolated ladders isnecessary, as emphasized in Refs. [1,2].

Our purpose is to determine 1�T1 for the experimen-tally relevant ratio of interchain �J�� and intrachain �Jk�couplings, separating the qy � 0 and p contributions. Wewill use the transfer-matrix density-matrix renormalizationgroup (TMRG) for the evaluation of thermodynamic prop-erties [4] and local imaginary time correlations [5,6] ofone-dimensional tight-binding models. Combined with ananalytical continuation, this technique has recently provento provide reliable real frequency correlations down tolow temperatures [6]. In the low-T regime, our resultsshow a good agreement to analytical results obtained inthe weak or strong coupling limits [2,7]. At higher T , wedemonstrate the importance of the �1�T1�p contributions to63�1�T1� experiments; in particular, our result reproducesthe crossover to the linear paramagnetic regime observedin the 63�1�T1� rate.

The most recent estimates of Jk�J� from susceptibilityand neutron scattering agree for a value of about 0.5 [8,9].On the other hand, estimating Jk�J� from NMR rates, wefound that the standard ladder, together with the hyperfine

1320 0031-9007�00�84(6)�1320(4)$15.00

couplings measured in Refs. [1,10], are unable to accountfor Jk�J� � 0.5 but favors Jk�J� � 1.

The two-leg ladder is described by the antiferromagneticHeisenberg Hamiltonian

H �NX

i�1

Jk�S1,i ? S1,i11 1 S2,i ? S2,i11� 1 J�S1,i ? S2,i ,

(1)

where Sn,i denotes a s � 12 spin operator at the ith rung

and the nth chain. It is by now well established thatthe spectrum of the Hamiltonian (1) consists of an S � 0ground state, the lowest-lying excited states forming agaped S � 1, ky � p single magnon branch with mini-mum D at kx � p [7,11,12].

The issue of determining the values of Jk and J� hasraised some controversy; indeed estimates from variousauthors [1,8,10,13,14] range from J��Jk � 0.5 to 1, withan emerging consensus for about 0.5. Fitting our numeri-cal results to susceptibility [13] and Knight shift [1] ex-periments, we confirm in Fig. 1 that the best agreementis found when J��Jk � 0.5 for SrCu2O3 [15], implyingenergy scales D � 420 K and Jk � 2000 K. Neverthe-less, a reasonable fit to J��Jk � 1 is also possible in a

0 200 400 600T [K]

0

0.1

χ/gµ

B

2

Imai La6Ca8Cu24O41

Azuma SrCu2O3

0 0.1 0.2 0.3T/J||

0.00

0.05

0.10

0.15

χ/gµ

B

2

Azuma, J||=2000 KAzuma, J||=850 KImai, J||=2750 KImai, J||=1200 KJ⊥ /J||=0.5J⊥ /J||=1.0

FIG. 1. Comparison of the susceptibility and Knight shifts toTMRG results for J��Jk � 0.5 and 1. Inset: Susceptibility andKnight shift measurements compared.

© 2000 The American Physical Society

VOLUME 84, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 7 FEBRUARY 2000

slightly narrower T -range, showing that the susceptibilityis relatively insensitive to J��Jk at low T . Moreover, wepoint out in the inset that the susceptibility and Knight shiftmeasurements are not fully consistent through the entireT -range, implying material-dependent Jk values. In prin-ciple, Knight shifts are better suited for a comparison withtheoretical results, since they involve no subtraction of aCurie term and no unknown g factor [8,13].

The nuclear relaxation rate 1�T1 is given bybµ 1

T1

∂�

Xqy�0,p

ZbF�q�S�q, vN � dqx , b � 63, 17 ,

S�q, v� �1

2N

Xi,j,m,n

Z�Sz

n,i�t�Szm,jei�q?� j2i,m2n�1vt� dt .

(2)

Here, vN is the Larmor frequency and bF�q� are theappropriate hyperfine couplings. According to Ref. [1],63F�q� � A2 for 63Cu, and 17F2�q� � 4F2 cos2�qy�2� forthe rung O(2) oxygen atoms [16]. Contrary to CuO2planes, it has been argued that considering only a local hy-perfine interaction for the 63Cu nucleus in ladders [3,10]is sufficient.

To determine 1�T1, the TMRG method permits a veryprecise evaluation of the Green’s function [6]

G�t� � �Sz1,1�t�Sz

m,j , (3)

where Szn,i�t� � etHSz

n,ie2tH and the indices �m, j� [�1, 1�, �2, 1�, �1, 2�, �2, 2�� run over the four corners of aplaquette. For all calculations, we chose an imaginary timeslice e � 0.025�Jk and m � 200 states were kept to rep-resent the transfer matrix. After an analytical continua-tion using the maximum entropy method, we can resolveS�q, v� in qy � 0 or p and obtain the averages over theqx momentum transfer:

S̄0�qy , v� �1

4p

Zdqx cos2�qx�2�S�qx , qy , v� ,

S̄p�qy , v� �1

4p

Zdqx sin2�qx�2�S�qx , qy , v� .

(4)

As indicated by the notation, S̄0�qy , v� �S̄p �qy , v�� isdominated by processes with qx close to 0 �p�. For thehyperfine couplings given above, the copper and oxygenrates are fully determined by appropriate combinations ofS̄q̄x �qy , v�, q̄x � 0, p . Accordingly, we also defineµ

1T1

∂qy

� S̄0�qy , vN � 1 S̄p �qy , vN � , (5)

the dimensionless contributions to 1�T1 from the qy �0, p momentum space sectors.

For illustration, we show in Fig. 2 S̄q̄x �qy , v� for thecase J��Jk � 1 as a function of T . As the latter drops,we can see clearly the signatures of the low-lying spectrumemerging. Indeed, the main peak is due to excitationsfrom the ground state to the single magnon branch for theqy � p sector and to the 2-magnon continuum for qy �0; for instance, it appears correctly from S̄p �p, v� that

0 1 2 3 4ω/J||

0

0.2_ S_ q x

(qy,

ω)

0

0.05

0.1

0 1 2 3 4ω/J||

0

0.5

0

0.2

0.4

(0,0)

(π,0)

(0,π)

(π,π)

FIG. 2. S̄q̄x �qy , v� for J��Jk � 1. T�Jk � 1�2 (dotted line),1�3 (dashed line), 1�4 (thin line), and 1�6 (thick line).

the minimum gap approaches the T � 0 value D � 0.502.Such contributions to S̄q̄x �qy , v� become T independentfor T ! 0; however, they do not contribute to 1�T1 asthey involve frequencies v $ D ¿ vN ��3 mK�. Onthe contrary, there are thermally activated contributions to1�T1, as they involve excitations for v ! 0 [17]. As seenin Fig. 2, this limit is dominated in the low-T regime byprocesses with �q̄x , qy� � �0, 0� and �p , p�.

The first contributions involve two thermally ex-cited magnons near the minimum of the branche�kx , ky � p� � D 1 a�kx 2 p�2; according to Ref. [7],they lead toµ

1T1

∂0

~1a

e2D�T

∑0.809 2 ln

µvN

T

∂∏. (6)

Corrections to the quadratic minimum [18] change only theprefactor of the exponential term slightly for T�D & 0.3.The logarithmic divergence in Eq. (6) cannot be resolvedby the maximum entropy method; however, it does notchange the dominant exponential behavior. For instance,the factor in the square brackets changes only by about 5%from T � D�2 to D�4.

The second important contribution with qy � p in-volves scattering of single magnons with the 2-magnoncontinuum. By representing the low-lying excitations interms of free massive fermions [12] in the weakly coupledchain limit, it was proposed in Ref. [2] thatµ

1T1

∂p

~

µTD

∂e22D�T . (7)

As the scale 2D corresponds to the bottom of the 2-magnoncontinuum at k � �0, 0� independent of J��Jk [11], thisresult should hold over a wider coupling range. In fact,exact diagonalization (ED) results show the existence oflarge matrix elements j�njSz

qjmj2 between the 2-magnoncontinuum at k � �0, 0� and the single magnon branch.Such processes are characterized by a momentum trans-fer qx which rapidly shifts close to p with decreasing

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VOLUME 84, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 7 FEBRUARY 2000

J�. From the analysis of the spectrum [11], it followsthat qx�p � 0.5, 0.8, and 0.95 for J��Jk � 2, 1, and0.5. Other important qy � p processes occur between thesingle magnon branch at k � �0, p� and the continuumnear k � �p , 0� when J��Jk & 0.5. However, these havea larger activation gap of about 3.7D.

Let us first discuss our results for 1�T1 as a func-tion of J� and T . We should point out that the valuesS̄q̄x �qy , v ! 0� become very small compared to the mag-nitude of the main peak as T is lowered. Therefore, fora reliable estimate of 1�T1, it is essential (i) to work withvery precise imaginary time data and (ii) to separate thedifferent q space contributions, as the integrated weight ofthe qy � 0 and p sectors differ considerably. In Fig. 3, wepresent �1�T1�0 and �1�T1�p in the low-T regime, whereT is scaled by the spin gap D.

�1�T1�0 shows the correct behavior of Eq. (6), especiallyfor large J��Jk, when low enough T�D can be reachedto clearly identify the activate regime. This is verifiedin the inset where we have plotted a line of slope oneon a logarithmic scale. We also observe that the prefac-tor of the exponential strongly depends on J��Jk, con-sistently with the factor 1�a in Eq. (6). Indeed, as thedispersion of the single magnon branch flattens with in-creasing J�, 1�a grows and limJ�!` 1�a � ` [7,11]. Theqy � p results reveal a larger gap which can be fittedconsistently to Eq. (7) when J��Jk � 1.5, 2 (see inset).When J��Jk & 1, we find some deviations from Eq. (7).In this regime, the low-T extraction of �1�T1�p becomesmore delicate as the growing main frequency peak nearv � D (due to excitations from the ground state to the

0 0.5 1T/∆

0

0.1

0.2

(1/T

1)π

0

0.05

0.1

0.15

(1/T

1)0

1 3 5 7∆/T2.5

5

−ln(

1/T

1)

2 3∆/T2

6

−ln(

1/T

1)

qy=π

qy=0

FIG. 3. Low-T behavior of �1�T1�0 and �1�T1�p for J��Jk �5 ���, 2 ���, 1.5 �1�, 1 (�), 0.8 ���, and 0.6 ���. Insets: Thedot-dashed lines have slopes 1 (2) for qy � 0 �p�.

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single magnon branch) may bias the limit S�q, v ! 0�(see Fig. 2). On the other hand, we cannot exclude thatthe above mentioned processes with a gap �3.7D con-tribute significantly at our lowest temperatures. For thecase J��Jk � 5 (not shown in Fig. 3), �1�T1�p � 0 sincethe single magnon band and the 2-magnon continuum donot yet overlap.

A wider T range is shown in Fig. 4. �1�T1�p obtainedat T�Jk � 1 agree within 5% with those from ED of an8-rung ladder, as seen in the inset for J��Jk � 0.6. How-ever, due to finite size effects, the determination of 1�T1from ED becomes meaningless below T�Jk � 0.8.

As �1�T1�0 obeys a predominantly linear intermediate-to high-T behavior for all J� above T � 0.4Jk, �1�T1�p

exhibits a qualitative change as J��Jk is decreased. In-deed, for J��Jk # 0.6 �1�T1�p develops a maximum nearT�Jk � 0.2 becoming sharper with decreasing interchaincoupling. We believe such a feature is reminiscent ofthe weak coupling limit, as our �1�T1�p for J��Jk #

0.6 shows a good agreement to the behavior predicted inRef. [2]. There, Ivanov and Lee also argued that withincreasing T , �1�T1�p flattens to the single Heisenbergchain result [19], as we observe for J��Jk � 0.4 and 0.6above T�Jk � 0.4. In spite of the larger gap, we find that�1�T1�p dominates over �1�T1�0 in a wide intermediate-Tregime as can be verified from Figs. 3 and 4. For instance,when T�D � 1, �1�T1�p exceeds �1�T1�0 by a factor of2.5 (9) for J��Jk � 1 (0.6). In particular, for J��Jk � 1,we observe the crossing of �1�T1�p and �1�T1�0 at T�D �0.35. This consideration implies that both qy � 0 and p

contributions are relevant to copper 63�1�T1� experimentsabove T � 0.35D (typically �200 K in Cu2O3 ladders).

0 0.2 0.4 0.6 0.8 1T/J||

0

0.1

0.2

0.3

(1/T

1)π

0

0.1

0.2

(1/T

1)0

0 44ω/J||

0

0.25

TMRGED

qy=0

qy=π

T/J||=1

J⊥ /J||=0.6

FIG. 4. �1�T1�0 and �1�T1�p as a function of T�Jk forJ��Jk � 1.5 �1�, 1 ���, 0.8 ���, 0.6 ���, and 0.4 ���. Inset:Comparison of S̄�0, p, v� 1 S̄�p, p, v� from ED and TMRG.

VOLUME 84, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 7 FEBRUARY 2000

In fact, taking into account the usually omitted qy � p

processes to fit the low-T 63�1�T1� data of Ishida et al. [10]on SrCu2O3, we obtain for T between 100 and 300 K a gapD � 520 K, while considering only the form for �1�T1�0leads to D � 700 K. Hence, the discrepancy between es-timates of the gap from 63�1�T1� and susceptibility [10,13](leading to D � 420 K) is reduced.

In Fig. 5, we compare our results with experimentalmeasurements of 63�1�T1� and the rung oxygen 17�1�T1�rate in SrCu2O3 and La6Ca8Cu24O41. To convert our val-ues to the experimental unit, we considered the hyperfinecouplings given in Ref. [10] for the 63Cu and in Ref. [1]for the 17O nuclei, so that the remaining free parametersare only J� and Jk, which we have chosen according tothose obtained from the susceptibility fits in Fig. 1. Con-sidering that T scales proportionally and 1�T1 inverselyproportional to Jk, we found good overall magnitudes;however, the precise T dependence is not reproduced veryaccurately. Our results tend to indicate that J��Jk � 1,in disagreement with J��Jk � 0.5 from the susceptibil-ity. Especially, the “linear” high-T behavior of 63�1�T1�extrapolating to zero (inset) cannot be reproduced with asmaller ratio J��Jk.

These observations raise two important issues aboutladder materials. First, the hyperfine couplings measuredin Refs. [1,10] may not be sufficient for a quantitativediscussion of NMR experiments within the standard lad-der model. A possible improvement may be to considertransferred fields 63F ~ �1 1 2r cosqx�2 1 r 0 cos2�qy�2�on the copper atom, which can suppress the �p, p�-

0 200 400 600T [K]

0

50

100

150

17(1

/T1)

[1/s

]

J⊥ /J||=1, J||=1300 KJ⊥ /J||=0.6, J||=2000 KLa6Ca8Cu24O41, Imai

0 200 400 600 8000

1000

2000

63(1

/T1)

[1/s

]

La6Ca8Cu24O41, ImaiSrCu2O3, ThurberJ⊥ /J||=1, J||=1300 KJ⊥ /J||=0.6, J||=2000 K

0 8000

1500La6Ca8Cu24O41

63Cu

17O(2) rung oxygen

Imai

FIG. 5. Comparison of NMR rates in SrCu2O3 [20] andLa6Ca8Cu24O41 [1] to numerical results. Inset: Imai’s data, thehigh-T regime extrapolates to zero.

fluctuations by large factors [21] and favor a smallerratio J��Jk. This suggests that a better knowledge of thehyperfine couplings in ladder materials is needed for aquantitative interpretation of NMR data. Second, smallcorrections to the standard ladder such as interladder orfrustration couplings are known to have little effect onbulk properties such as the spin gap [22]. Therefore, theaccessible susceptibility measurements �T�D & 1� arerelatively insensitive to such corrections. On the otherhand, local dynamic properties such as NMR rates can bemore substantially affected.

To summarize, we have shown that the TMRG techniqueprovides reliable NMR rates 1�T1 in the standard ladderover a wide temperature range. In particular, we demon-strated that the qy � p contributions are crucial for under-standing the crossover observed in 63�1�T1� experiments.Finally, we point out how quantitative results can indicateinconsistencies in estimates of J��Jk from susceptibilityand NMR measurements.

We thank X. Zotos and M. Long for helpful sugges-tions, T. Imai for kindly providing us with the experimentalmeasurements, and D. C. Johnston and T. Xiang for usefulcommunications. Our work was supported by the SwissNational Foundation Grant No. 20-49486.96.

[1] T. Imai et al., Phys. Rev. Lett. 81, 220 (1998).[2] D. A. Ivanov and P. A. Lee, Phys. Rev. B 59, 4803 (1999).[3] A. W. Sandvik, E. Dagotto, and D. J. Scalapino, Phys.

Rev. B 53, R2934 (1996).[4] X. Wang and T. Xiang, Phys. Rev. B 56, 5061 (1997).[5] T. Mutou, N. Shibata, and K. Ueda, Phys. Rev. Lett. 81,

4939 (1998).[6] F. Naef et al., Phys. Rev. B 60, 359 (1999); X. Wang

et al., in Density-Matrix Renormalization, Lecture Notesin Physics Vol. 528, edited by I. Peschel et al., (Springer-Verlag, New York, 1999).

[7] M. Troyer, H. Tsunetsugu, and D. Würtz, Phys. Rev. B 50,13 515 (1994).

[8] D. C. Johnston, Phys. Rev. B 54, 13 009 (1996).[9] R. S. Eccleston et al., Phys. Rev. Lett. 81, 1702 (1998).

[10] K. Ishida et al., Phys. Rev. B 53, 2827 (1996).[11] T. Barnes and J. Riera, Phys. Rev. B 50, 6817 (1994).[12] D. G. Shelton, A. A. Nersesyan, and A. M. Tsvelik, Phys.

Rev. B 53, 8521 (1996).[13] M. Azuma, Z. Hiroi, M. Takano, K. Ishida, and Y. Kitaoka,

Phys. Rev. Lett. 73, 3463 (1994).[14] T. F. A. Müller et al., Phys. Rev. B 57, R12 655 (1998).[15] D. C. Johnston (private communication).[16] We have absorbed constants in the hyperfine couplings;

the link to Ref. [1] is given by A2 � 63g2N�m

2Bh̄�A2

a 1 A2b�,

G2 � 17g2N �m

2Bh̄�G2

a 1 G2b� when G � C, D, F.

[17] For all estimates of 1�T1, we consider S̄q̄x �qy , v ! 0�.[18] R. Melzi and P. Carretta, cond-mat /9904074.[19] S. Sachdev, Phys. Rev. B 50, 13 006 (1994).[20] K. R. Thurber et al., Phys. Rev. Lett. 84, 558 (2000).[21] For instance, r � 10.25, r 0 � 0 suppresses �1�T1�q̄x�p

over �1�T1�q̄x �p by factor of �5.[22] X. Wang, cond-mat /9803290.

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