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2 quantum spin system TlCuCl3
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) 195–196
Spin dynamics in the BEC phase of the S=1/2 quantum spin system TlCuCl3 a, Ch. Ruegg . *, N. Cavadinia, A. Furrera, K. Kr.amerb, H.-U. Gudel . b, P. Vorderwischc, K. Habichtc, H. Mutkad, A. Wildesd a
Laboratory for Neutron Scattering, ETH Zurich & Paul Scherrer Institute, 5232 Villigen PSI, Switzerland b Department of Chemistry and Biochemistry, University of Berne, 3000 Bern 9, Switzerland c BENSC, Hahn-Meitner Institute, 14109 Berlin Wannsee, Germany d Institut Laue-Langevin, P.B. 156, 38042 Grenoble Cedex 9, France
Abstract The copper salt S ¼ 1=2 TlCuCl3 is a three-dimensional quantum spin system with a singlet ground-state of dimer origin. A spin energy gap DE0:7 meV separates the ground-state from the triplet waves, which propagate along all directions in reciprocal space. Across the quantum critical point at Hc ¼ D=gmB Bose–Einstein condensation of the triplet quasi-particles has been observed. The inelastic neutron scattering spectra at H > Hc proof the coexistence of a gapless Goldstone mode, which has a linear dispersion, with renormalized quadratic Zeeman modes. r 2003 Elsevier B.V. All rights reserved. PACS: 75.10.Jm; 75.40.Gb; 78.70.Nx Keywords: Quantized spin models; Dynamic properties; Neutron inelastic scattering
Model quantum spin systems, with a proper realization of spin and interaction dimensionality, are an ideal testing ground to experimentally as well as theoretically investigate novel phenomena in solid-state physics like quantum phase transitions or the recently proposed Bose–Einstein condensation of magnetic quasi-particles [1–4]. In this regard inelastic neutron scattering is a powerful tool to study the microscopic spin correlations, also under extreme conditions as mK temperatures and high magnetic fields. The compound TlCuCl3 is such a model three-dimensional quantum spin system with a singlet ground-state of dimer origin caused by a dominant antiferromagnetic (AF) coupling between S ¼ 1=2 moments at the structurally neighbouring Cu2+ ions. Above a spin energy gap DE0:7 meV elementary triplet waves propagate with a characteristic dispersion along all directions in reciprocal space [5,6]. Compared to the isostructural sister compound KCuCl3 the interdimer *Corresponding author. Tel.: +41-56-310-31-79; fax: +4156-310-29-39. E-mail address: [email protected] (C. Ruegg). .
couplings are considerably enhanced in the Tl systems but the dominant intradimer coupling is almost unchanged [7]. This leads to an increased spectral width of the triplet band and as a consequence to a reduction of the spin energy gap. The degeneracy of the triplet waves is lifted by an external magnetic field acting as an additional and commuting Zeeman term to the Heisenberg–Hamiltonian describing the strongly interacting system. While increasing the field strength, the energy of the SZ ¼ þ1 component is linearly reduced [8]. At a critical value Hc ¼ D=gmB E6 T, the quantum critical point, the spin energy gap vanishes and a soft mode transition to a fieldinduced ordered phase has been observed [8–10]. For the case of TlCuCl3 the field-induced ordering persists also at finite temperatures and the quantum critical phase boundary is reproduced by a power law, which is consistent with a description of the ordering process by the condensation of strongly interacting triplet quasiparticles, leading to a three-dimensionally ordered Bose– Einstein condensate (BEC) [3]. The condensate is theoretically expected to support a gapless and linear Goldstone mode as the elementary excitation [4].
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.11.075
ARTICLE IN PRESS 196
C. Ruegg et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 195–196 .
Inelastic neutron scattering experiments have been performed on the V2 three-axis spectrometer FLEX (HMI, Berlin) to investigate the excitations in the BEC phase of TlCuCl3. The single crystal oriented for
scattering along the b direction has been mounted inside an Hmax ¼ 14:5 T vertical cryomagnet and optionally a mK dilution insert. Standard focussing conditions as well as radial collimations 400 –open–open (setup I, T ¼ 2 K) and 600 –600 –600 (setup II, T ¼ 50 mK) have been applied. The observed energy widths of the excitations are in agreement with expectations from Monte-Carlo simulations of the instrumental resolution. Typical constant energy scans at H ¼ 14 T > Hc around the field-induced AF Bragg point Q ¼ ð0; 4; 0Þ are presented in Fig. 1. The observed inelastic neutron intensity shows resolution limited excitations, which follow a linear dispersion [10]. At higher energy transfer, additional signal from the renormalized and quadratic dimer excitations could also be detected, Fig. 1A. Increased Instrumental resolution (setup II) as well as T ¼ 50 mK allowed to verify the linearity and commensurability of the Goldstone mode down to lower energy transfers, Fig. 1B. We therefore conclude that for strongly coupled three-dimensional S ¼ 1=2 dimer quantum spin systems above the quantum critical point Hc ; BEC of magnetic quasi-particles can be observed. The characteristic feature of the excitation spectrum inside this phase is the coexistence of two renormalized and quadratic dimer excitations with a gapless and linear Goldstone mode as predicted from theory. The fraction of the condensate depends on temperature and magnetic field-strength. Additional experiments are planed to determine the dependence of the spectrum on these parameters. Recently pressure as well as impurity induced AF ordering have been reported for TlCuCl3 [11,12]. The comparison of the nature of the excited states in these ordered phases with the field-induced BEC would additionally be of interest. We would like to thank S. Kausche, P. Smeibidl and M. Meissner for their technical support during the V2 experiments. This work was supported partially by the Swiss National Science Foundation. References
Fig. 1. Linear Goldstone mode as the elementary excitation of a magnetically ordered BEC of triplet quasi-particles. Solid lines represent a global fit to the data with Gaussian peaks above a common background. (A) Inelastic neutron intensity observed in constant energy scans around the field-induced AF Bragg point Q ¼ ð0; 4; 0Þ: The scans have a common intensity scale (right) but are vertically displace according to the corresponding energy transfer (left), setup I. At energy transfer E ¼ 2 and 2.5 meV additional signal for the gapped and renormalized dimer excitation is observed. (B) At the lowest temperature T ¼ 50 mK and increased instrumental resolution the field-induced nature, the linearity as well as the commensurability of the excitation is demonstrated down to lower energy transfer, setup II [7].
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
T.M. Rice, Science 298 (2002) 760. S. Sachdev, Science 288 (2000) 475. T. Nikuni, et al., Phys. Rev. Lett. 84 (2000) 5868. M. Matsumoto, et al., Phys. Rev. Lett. 89 (2002) 077203. N. Cavadini, et al., Phys. Rev. B 63 (2001) 172414. A. Oosawa, et al., Phys. Rev. B 65 (2002) 094426. N. Cavadini, et al., J. Phys.: Condens. Matter 12 (2000) 5463. N. Cavadini, et al., Phys. Rev. B 65 (2002) 132415. Ch. Ruegg, . et al., Appl. Phys. A 74 (2002) S840. Ch. Ruegg, . et al., Nature 423 (2003) 62. A. Oosawa, et al., Phys. Rev. B 66 (2002) 020405. H. Tanaka, et al., Physica B 329 (2003) 697.
Journal of Magnetism and Magnetic Materials 272–276 (2004) 195–196
Spin dynamics in the BEC phase of the S=1/2 quantum spin system TlCuCl3 a, Ch. Ruegg . *, N. Cavadinia, A. Furrera, K. Kr.amerb, H.-U. Gudel . b, P. Vorderwischc, K. Habichtc, H. Mutkad, A. Wildesd a
Laboratory for Neutron Scattering, ETH Zurich & Paul Scherrer Institute, 5232 Villigen PSI, Switzerland b Department of Chemistry and Biochemistry, University of Berne, 3000 Bern 9, Switzerland c BENSC, Hahn-Meitner Institute, 14109 Berlin Wannsee, Germany d Institut Laue-Langevin, P.B. 156, 38042 Grenoble Cedex 9, France
Abstract The copper salt S ¼ 1=2 TlCuCl3 is a three-dimensional quantum spin system with a singlet ground-state of dimer origin. A spin energy gap DE0:7 meV separates the ground-state from the triplet waves, which propagate along all directions in reciprocal space. Across the quantum critical point at Hc ¼ D=gmB Bose–Einstein condensation of the triplet quasi-particles has been observed. The inelastic neutron scattering spectra at H > Hc proof the coexistence of a gapless Goldstone mode, which has a linear dispersion, with renormalized quadratic Zeeman modes. r 2003 Elsevier B.V. All rights reserved. PACS: 75.10.Jm; 75.40.Gb; 78.70.Nx Keywords: Quantized spin models; Dynamic properties; Neutron inelastic scattering
Model quantum spin systems, with a proper realization of spin and interaction dimensionality, are an ideal testing ground to experimentally as well as theoretically investigate novel phenomena in solid-state physics like quantum phase transitions or the recently proposed Bose–Einstein condensation of magnetic quasi-particles [1–4]. In this regard inelastic neutron scattering is a powerful tool to study the microscopic spin correlations, also under extreme conditions as mK temperatures and high magnetic fields. The compound TlCuCl3 is such a model three-dimensional quantum spin system with a singlet ground-state of dimer origin caused by a dominant antiferromagnetic (AF) coupling between S ¼ 1=2 moments at the structurally neighbouring Cu2+ ions. Above a spin energy gap DE0:7 meV elementary triplet waves propagate with a characteristic dispersion along all directions in reciprocal space [5,6]. Compared to the isostructural sister compound KCuCl3 the interdimer *Corresponding author. Tel.: +41-56-310-31-79; fax: +4156-310-29-39. E-mail address: [email protected] (C. Ruegg). .
couplings are considerably enhanced in the Tl systems but the dominant intradimer coupling is almost unchanged [7]. This leads to an increased spectral width of the triplet band and as a consequence to a reduction of the spin energy gap. The degeneracy of the triplet waves is lifted by an external magnetic field acting as an additional and commuting Zeeman term to the Heisenberg–Hamiltonian describing the strongly interacting system. While increasing the field strength, the energy of the SZ ¼ þ1 component is linearly reduced [8]. At a critical value Hc ¼ D=gmB E6 T, the quantum critical point, the spin energy gap vanishes and a soft mode transition to a fieldinduced ordered phase has been observed [8–10]. For the case of TlCuCl3 the field-induced ordering persists also at finite temperatures and the quantum critical phase boundary is reproduced by a power law, which is consistent with a description of the ordering process by the condensation of strongly interacting triplet quasiparticles, leading to a three-dimensionally ordered Bose– Einstein condensate (BEC) [3]. The condensate is theoretically expected to support a gapless and linear Goldstone mode as the elementary excitation [4].
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.11.075
ARTICLE IN PRESS 196
C. Ruegg et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 195–196 .
Inelastic neutron scattering experiments have been performed on the V2 three-axis spectrometer FLEX (HMI, Berlin) to investigate the excitations in the BEC phase of TlCuCl3. The single crystal oriented for
scattering along the b direction has been mounted inside an Hmax ¼ 14:5 T vertical cryomagnet and optionally a mK dilution insert. Standard focussing conditions as well as radial collimations 400 –open–open (setup I, T ¼ 2 K) and 600 –600 –600 (setup II, T ¼ 50 mK) have been applied. The observed energy widths of the excitations are in agreement with expectations from Monte-Carlo simulations of the instrumental resolution. Typical constant energy scans at H ¼ 14 T > Hc around the field-induced AF Bragg point Q ¼ ð0; 4; 0Þ are presented in Fig. 1. The observed inelastic neutron intensity shows resolution limited excitations, which follow a linear dispersion [10]. At higher energy transfer, additional signal from the renormalized and quadratic dimer excitations could also be detected, Fig. 1A. Increased Instrumental resolution (setup II) as well as T ¼ 50 mK allowed to verify the linearity and commensurability of the Goldstone mode down to lower energy transfers, Fig. 1B. We therefore conclude that for strongly coupled three-dimensional S ¼ 1=2 dimer quantum spin systems above the quantum critical point Hc ; BEC of magnetic quasi-particles can be observed. The characteristic feature of the excitation spectrum inside this phase is the coexistence of two renormalized and quadratic dimer excitations with a gapless and linear Goldstone mode as predicted from theory. The fraction of the condensate depends on temperature and magnetic field-strength. Additional experiments are planed to determine the dependence of the spectrum on these parameters. Recently pressure as well as impurity induced AF ordering have been reported for TlCuCl3 [11,12]. The comparison of the nature of the excited states in these ordered phases with the field-induced BEC would additionally be of interest. We would like to thank S. Kausche, P. Smeibidl and M. Meissner for their technical support during the V2 experiments. This work was supported partially by the Swiss National Science Foundation. References
Fig. 1. Linear Goldstone mode as the elementary excitation of a magnetically ordered BEC of triplet quasi-particles. Solid lines represent a global fit to the data with Gaussian peaks above a common background. (A) Inelastic neutron intensity observed in constant energy scans around the field-induced AF Bragg point Q ¼ ð0; 4; 0Þ: The scans have a common intensity scale (right) but are vertically displace according to the corresponding energy transfer (left), setup I. At energy transfer E ¼ 2 and 2.5 meV additional signal for the gapped and renormalized dimer excitation is observed. (B) At the lowest temperature T ¼ 50 mK and increased instrumental resolution the field-induced nature, the linearity as well as the commensurability of the excitation is demonstrated down to lower energy transfer, setup II [7].
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
T.M. Rice, Science 298 (2002) 760. S. Sachdev, Science 288 (2000) 475. T. Nikuni, et al., Phys. Rev. Lett. 84 (2000) 5868. M. Matsumoto, et al., Phys. Rev. Lett. 89 (2002) 077203. N. Cavadini, et al., Phys. Rev. B 63 (2001) 172414. A. Oosawa, et al., Phys. Rev. B 65 (2002) 094426. N. Cavadini, et al., J. Phys.: Condens. Matter 12 (2000) 5463. N. Cavadini, et al., Phys. Rev. B 65 (2002) 132415. Ch. Ruegg, . et al., Appl. Phys. A 74 (2002) S840. Ch. Ruegg, . et al., Nature 423 (2003) 62. A. Oosawa, et al., Phys. Rev. B 66 (2002) 020405. H. Tanaka, et al., Physica B 329 (2003) 697.