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Correlations and fluctuations in the 2D Heisenberg antiferromagnet
Journal of Magnetism and Magnetic Materials 236 (2001) 4–5
Correlations and fluctuations in the 2D Heisenberg antiferromagnet$ H.M. Rønnowa,b,*, D.F. McMorrowb, A. Harrisonc, I.D. Youngsonc, R. Coldead, T.G. Perringd, G. Aepplie, O. Syljua( senf a
Institut Laue Langevin, 6 rue Jules Horowitz, BP 156, 38042 Grenoble Cedex 9, France b Department of Solid State Physics, Risø National Laboratory, Denmark c Department of Chemistry, University of Edinburgh, UK d ISIS, Rutherford Appleton Laboratory, UK e NEC Research, Princeton, NJ 08540, USA f NORDITA, Copenhagen, Denmark
Abstract The correlations and fluctuations in Cu(DCOO)2 4D2 O, which is a good physical realization of the 2D S ¼ 1=2 Heisenberg antiferromagnet on a square lattice, have been measured by neutron scattering measurements. The quantum fluctuations cause a non-uniform renormalization of the spin-wave dispersion with a zone boundary dispersion of 7%, while the spin wave amplitude is reduced to 5174% of its classical value. The temperature dependence of the correlation length, the spin-wave damping and the spin-wave softening agrees with theoretical predictions over a large temperature range. r 2001 Elsevier Science B.V. All rights reserved. Keywords: Two dimensional; Antiferromagnet; Correlations; Fluctuations
The last decade has witnessed considerable efforts and achievements in the understanding of the 2D Heisenberg antiferromagnet. Particular interest has been devoted to the quantum (S ¼ 1=2) system on a square lattice, due to its relation to the cuprate superconductors. Indeed, most of the experimental results have so far been obtained by neutron scattering from the cuprates La2 CuO4 and Sr2 CuCl2 O2 ; but the large coupling constant in these materials complicates the studies at temperature and energy scales of the order of JC1500 K. We have conducted a neutron scattering study of Cu(DCOO)2 4D2 O (CFTD), which is another physical realization of the 2D S ¼ 1=2 Heisenberg antiferromagnet on a square lattice (2DQHAFSL) [1]. Owing to the lower value of J ¼ 72 K, it has been possible to obtain a quite complete $
This paper is part of the ICM 2000 proceedings, published in volumes 226–230. The Publisher deeply regrets that the paper was not printed in these volumes. *Correspondence address: Department of Solid State Physics, Risø National Laboratory, Denmark. Fax: +33-476483906. E-mail address: [email protected] (H.M. R nnow).
characterization of the system over a wide range of temperatures. Though no rigorous proof exists, it is well established that the ground state of the 2DQHAFSL has long-range magnetic order, and the excitation spectrum is dominated by a doubly degenerate spin wave mode. At any finite temperature, long-range order is destroyed by thermal fluctuations, but there are still strong correlations decaying on the length scale of the correlation length x: As a consequence, the spin waves are damped and softened. This behaviour is contained P within the instantaneous structure factor SðqÞ ¼ r eiqr /SR0 Sr S Pand the dynamic structure factor Sðq; oÞ ¼ dt r eiqriot /S0 ð0ÞSr ðtÞS; which are the Fourier transforms of, respectively, the time-independent and the time-dependent spin–spin correlation functions. They are related by R SðqÞ ¼ do Sðq; oÞ: Operating the triple axis spectrometer RITA at Ris , Denmark, in an energy integrating configuration described in Ref. [2], SðqÞ has been measured for 0:2oT=Jo1:4: The dynamic structure factor Sðq; oÞ has been measured up to T=JB0:6 on the time-of-flight spectrometer HET at ISIS, UK. Representative dataset
0304-8853/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 1 1 9 4 - X
H.M. Rønnow et al. / Journal of Magnetism and Magnetic Materials 236 (2001) 4–5
5
Fig. 1. SðqÞ (top) and Sðq; oÞ (bottom) at, respectively, 16 K (left) and 36 K (middle). For SðqÞ; the lines shows resolution convoluted fits to Lorentzians, and the corresponding values of x and G are indicated. Right top: xðT Þ extracted from energy integrating neutron scattering, compared to the NLsM (solid) and HTE (dashed) predictions. Bottom: temperature dependence of GðT Þ=J and Zc ðT Þ for the data (circles), QMC (triangles) [14] and theory (solid line).
are shown in the left and middle panels of Fig. 1. The data for SðqÞ were fitted to resolution convoluted Lorentzians, extracting the correlation length xðTÞ as shown in the upper right panel of Fig. 1. Through a process, where experimental data and quantum Monte Carlo (QMC) calculations played an important role, the behaviour of xðTÞ is now well established. At low temperatures, xðTÞ was predicted by mapping the 2DQHAFSL onto the non-linear s model (NLsM) [3–5]. The high-temperature expansion (HTE) [6] and pure-quantum self-consistent harmonic approximation [7] calculations are valid for T\J=2: These limiting results are connected by QMC calculations [8,9], and the combined prediction agrees well with both previous measurements [10–12] and the current data, as illustrated by the solid (NLsM) and dashed (HTE) lines. The Sðq; oÞ data were fitted to a resolution convoluted damped harmonic oscillator (DHO) line shape with the dispersion oq and amplitude given by linear spin-wave theory modified by the constants Zw and Zc to allow for renormalzations due to quantum fluctuations in, respectively, the amplitude and the energy. The amplitude renormalization Zw ¼ 0:5170:04 is in perfect agreement with theoretical predictions [13]. For q along (1,1) the dispersion is uniformly renormalized by Zc ; but between the two zone boundary points ðp; 0Þ and ðp=2; p=2Þ there is a dispersion of 7%. This non-uniform renormalization was found in series expansion studies [13] and has recently been confirmed by QMC calculations [14]. As a function of temperature, the uniform Zc ðTÞ along (1,1) is in good agreement with the prediction Zc ðT Þ ¼ Zc0 ð1 þ 0:765 ðT=JÞ3 Þ1 [6].
Because spin waves can only propagate within a correlated region, they acquire a finite lifetime 1=G: As shown by the solid line in lower right panel of Fig. 1, the simple estimate GðT Þ ¼ vs ðT Þ=xðT Þ; where vs ðT Þ ¼ v0s Zc ðT Þ=Z 0c is the spin wave velocity, gives a remarkably good account of the data. In summary, we have presented an exhaustive characterization of the correlations and fluctuations in the 2DQHAFSL. Combining a number of theoretical methods, the behaviour of the system is now remarkably well understood. We acknowledge discussions with S. Chakravarty, S. Sachdev, M. Troyer, R. Singh and P. Verrucchi.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
S.J. Clarke, et al., Solid State Commun. 112 (1999) 561. H.M. R nnow, et al., Phys. Rev. Lett. 82 (1999) 3152. S. Chakravarty, et al., Phys. Rev. B 39 (1989) 2344. P. Hasenfratz, F. Niedermayer, Phys. Lett. B 268 (1991) 231. P. Hasenfratz, Europhys. J. B 13 (1999) 11. N. Elstner, et al., Phys. Rev. B 51 (1995) 8984. A. Cuccoli, et al., Phys. Rev. B 56 (1997) 14456. J.-K. Kim, M. Troyer, Phys. Rev. Lett. 80 (1998) 2705. B.B. Beard, et al., Phys. Rev. Lett. 80 (1998) 1742. M. Greven, et al., Z. Phys. B 96 (1995) 465. R.J. Birgeneau, et al., Phys. Rev. B 59 (1999) 13788. P. Carretta, et al., Phys. Rev. Lett. 84 (2000) 366. R.R.P. Singh, M.P. Gelfand, Phys. Rev. B 52 (1995) R15695. O.F. Sylju(asen, H.M. R nnow, J. Phys. C 12 (2000) L1.
Correlations and fluctuations in the 2D Heisenberg antiferromagnet$ H.M. Rønnowa,b,*, D.F. McMorrowb, A. Harrisonc, I.D. Youngsonc, R. Coldead, T.G. Perringd, G. Aepplie, O. Syljua( senf a
Institut Laue Langevin, 6 rue Jules Horowitz, BP 156, 38042 Grenoble Cedex 9, France b Department of Solid State Physics, Risø National Laboratory, Denmark c Department of Chemistry, University of Edinburgh, UK d ISIS, Rutherford Appleton Laboratory, UK e NEC Research, Princeton, NJ 08540, USA f NORDITA, Copenhagen, Denmark
Abstract The correlations and fluctuations in Cu(DCOO)2 4D2 O, which is a good physical realization of the 2D S ¼ 1=2 Heisenberg antiferromagnet on a square lattice, have been measured by neutron scattering measurements. The quantum fluctuations cause a non-uniform renormalization of the spin-wave dispersion with a zone boundary dispersion of 7%, while the spin wave amplitude is reduced to 5174% of its classical value. The temperature dependence of the correlation length, the spin-wave damping and the spin-wave softening agrees with theoretical predictions over a large temperature range. r 2001 Elsevier Science B.V. All rights reserved. Keywords: Two dimensional; Antiferromagnet; Correlations; Fluctuations
The last decade has witnessed considerable efforts and achievements in the understanding of the 2D Heisenberg antiferromagnet. Particular interest has been devoted to the quantum (S ¼ 1=2) system on a square lattice, due to its relation to the cuprate superconductors. Indeed, most of the experimental results have so far been obtained by neutron scattering from the cuprates La2 CuO4 and Sr2 CuCl2 O2 ; but the large coupling constant in these materials complicates the studies at temperature and energy scales of the order of JC1500 K. We have conducted a neutron scattering study of Cu(DCOO)2 4D2 O (CFTD), which is another physical realization of the 2D S ¼ 1=2 Heisenberg antiferromagnet on a square lattice (2DQHAFSL) [1]. Owing to the lower value of J ¼ 72 K, it has been possible to obtain a quite complete $
This paper is part of the ICM 2000 proceedings, published in volumes 226–230. The Publisher deeply regrets that the paper was not printed in these volumes. *Correspondence address: Department of Solid State Physics, Risø National Laboratory, Denmark. Fax: +33-476483906. E-mail address: [email protected] (H.M. R nnow).
characterization of the system over a wide range of temperatures. Though no rigorous proof exists, it is well established that the ground state of the 2DQHAFSL has long-range magnetic order, and the excitation spectrum is dominated by a doubly degenerate spin wave mode. At any finite temperature, long-range order is destroyed by thermal fluctuations, but there are still strong correlations decaying on the length scale of the correlation length x: As a consequence, the spin waves are damped and softened. This behaviour is contained P within the instantaneous structure factor SðqÞ ¼ r eiqr /SR0 Sr S Pand the dynamic structure factor Sðq; oÞ ¼ dt r eiqriot /S0 ð0ÞSr ðtÞS; which are the Fourier transforms of, respectively, the time-independent and the time-dependent spin–spin correlation functions. They are related by R SðqÞ ¼ do Sðq; oÞ: Operating the triple axis spectrometer RITA at Ris , Denmark, in an energy integrating configuration described in Ref. [2], SðqÞ has been measured for 0:2oT=Jo1:4: The dynamic structure factor Sðq; oÞ has been measured up to T=JB0:6 on the time-of-flight spectrometer HET at ISIS, UK. Representative dataset
0304-8853/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 1 1 9 4 - X
H.M. Rønnow et al. / Journal of Magnetism and Magnetic Materials 236 (2001) 4–5
5
Fig. 1. SðqÞ (top) and Sðq; oÞ (bottom) at, respectively, 16 K (left) and 36 K (middle). For SðqÞ; the lines shows resolution convoluted fits to Lorentzians, and the corresponding values of x and G are indicated. Right top: xðT Þ extracted from energy integrating neutron scattering, compared to the NLsM (solid) and HTE (dashed) predictions. Bottom: temperature dependence of GðT Þ=J and Zc ðT Þ for the data (circles), QMC (triangles) [14] and theory (solid line).
are shown in the left and middle panels of Fig. 1. The data for SðqÞ were fitted to resolution convoluted Lorentzians, extracting the correlation length xðTÞ as shown in the upper right panel of Fig. 1. Through a process, where experimental data and quantum Monte Carlo (QMC) calculations played an important role, the behaviour of xðTÞ is now well established. At low temperatures, xðTÞ was predicted by mapping the 2DQHAFSL onto the non-linear s model (NLsM) [3–5]. The high-temperature expansion (HTE) [6] and pure-quantum self-consistent harmonic approximation [7] calculations are valid for T\J=2: These limiting results are connected by QMC calculations [8,9], and the combined prediction agrees well with both previous measurements [10–12] and the current data, as illustrated by the solid (NLsM) and dashed (HTE) lines. The Sðq; oÞ data were fitted to a resolution convoluted damped harmonic oscillator (DHO) line shape with the dispersion oq and amplitude given by linear spin-wave theory modified by the constants Zw and Zc to allow for renormalzations due to quantum fluctuations in, respectively, the amplitude and the energy. The amplitude renormalization Zw ¼ 0:5170:04 is in perfect agreement with theoretical predictions [13]. For q along (1,1) the dispersion is uniformly renormalized by Zc ; but between the two zone boundary points ðp; 0Þ and ðp=2; p=2Þ there is a dispersion of 7%. This non-uniform renormalization was found in series expansion studies [13] and has recently been confirmed by QMC calculations [14]. As a function of temperature, the uniform Zc ðTÞ along (1,1) is in good agreement with the prediction Zc ðT Þ ¼ Zc0 ð1 þ 0:765 ðT=JÞ3 Þ1 [6].
Because spin waves can only propagate within a correlated region, they acquire a finite lifetime 1=G: As shown by the solid line in lower right panel of Fig. 1, the simple estimate GðT Þ ¼ vs ðT Þ=xðT Þ; where vs ðT Þ ¼ v0s Zc ðT Þ=Z 0c is the spin wave velocity, gives a remarkably good account of the data. In summary, we have presented an exhaustive characterization of the correlations and fluctuations in the 2DQHAFSL. Combining a number of theoretical methods, the behaviour of the system is now remarkably well understood. We acknowledge discussions with S. Chakravarty, S. Sachdev, M. Troyer, R. Singh and P. Verrucchi.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
S.J. Clarke, et al., Solid State Commun. 112 (1999) 561. H.M. R nnow, et al., Phys. Rev. Lett. 82 (1999) 3152. S. Chakravarty, et al., Phys. Rev. B 39 (1989) 2344. P. Hasenfratz, F. Niedermayer, Phys. Lett. B 268 (1991) 231. P. Hasenfratz, Europhys. J. B 13 (1999) 11. N. Elstner, et al., Phys. Rev. B 51 (1995) 8984. A. Cuccoli, et al., Phys. Rev. B 56 (1997) 14456. J.-K. Kim, M. Troyer, Phys. Rev. Lett. 80 (1998) 2705. B.B. Beard, et al., Phys. Rev. Lett. 80 (1998) 1742. M. Greven, et al., Z. Phys. B 96 (1995) 465. R.J. Birgeneau, et al., Phys. Rev. B 59 (1999) 13788. P. Carretta, et al., Phys. Rev. Lett. 84 (2000) 366. R.R.P. Singh, M.P. Gelfand, Phys. Rev. B 52 (1995) R15695. O.F. Sylju(asen, H.M. R nnow, J. Phys. C 12 (2000) L1.