Reply to: Comment by S.-L. Drechsler et al. (arXiv:1006.5070v1)
In a comment on arXiv:1006.5070v1, Drechsler et al. present new band-structure calculations suggesting that the frustrated ferromagnetic spin-1/2 chain LiCuVO should be described by a strong rather than weak ferromagnetic nearest-neighbor interaction, in contradiction with their previous calculations. In our reply, we show that their new results are at odds with the observed magnetic structure, that their analysis of the static susceptibility neglects important contributions, and that their criticism of the spin-wave analysis of the bound-state dispersion is unfounded. We further show that their new exact diagonalization results reinforce our conclusion on the existence of a four-spinon continuum in LiCuVO, see Enderle et al., Phys. Rev. Lett. 104 (2010) 237207.
In their Comment, Drechsler et al. Drechsler () argue that the frustrated ferromagnetic spin-1/2 chain LiCuVO should be described by a strong ferromagnetic (FM) nearest neighbor (NN) interaction , rather than a weak FM as found by an analysis of the static susceptibility and the dispersion of the bound state Enderle05 () and also from an analysis of the intensity of the continuum scattering Enderle10 (). Drechsler et al. have in Drechsler () revised their band-structure calculations of Enderle05 () and now propose two alternative sets of exchange integrals with strongly FM , which are claimed to be in [better] agreement with experiment. The set A has and meV while set B has , , and an antiferromagnetic (AFM) interchain coupling (cf. Enderle05 ()) meV.
We first note that the large AFM of set B leads to a propagation vector with a component along the axis, in contradiction with the observed magnetic structure.
Secondly, the inverse static susceptibility of Enderle05 () was reanalyzed in Drechsler () by fitting a linear relation to the high-temperature part above 500 K, with the -factor and as free parameters. Figure 1a shows the Curie-Weiss temperature from such linear fits of the same data Enderle05 () as a function of the lowest temperature used (triangles). Clearly, the cut-off at =500 K used by Drechsler et al. Drechsler () (dashed line) is fully arbitrary and their is not unique. If the susceptibility is corrected for the known constant diamagnetic and Van-Vleck susceptibilities and the -factor fixed to the value precisely known from ESR Nidda02 (), is rather independent of the fit interval (circles) and always negative, in agreement with the exchange integrals of Enderle05 (); Enderle10 (), but not with set A or B of Drechsler (). Additionally, our exchange integrals, where is weakly FM, agree with the more sophisticated analysis of the static susceptibility performed in Enderle05 (), which included high-temperature series expansion and =16-ring calculations.
Thirdly, Drechsler et al. further argue that it is not possible to find a unique set of and from the dispersion of the main peaks in inelastic neutron scattering (INS) data. Within spin-wave (SW) theory, this statement is incorrect. Figure 1b shows that the SW dispersion along the chain direction depends strongly on the choice of and . We note that the red curve shown in Fig. 1b,c of Drechsler () does not correspond to the SW description Enderle05 ().
The exact diagonalization Drechsler () reveals that the intensity maximum at energy and /4 varies little between and ( and 1.41). If we use this exact diagonalization result to estimate directly from the experimental value meV we find =3.8–4.4 meV in agreement with Enderle05 (); Enderle10 (). as large as 5 meV or even 6.5 meV can clearly be excluded. The INS data reveal considerable intensity above an upper two-spinon boundary calculated with even the largest meV. Interestingly, such intensity above is also visible in the exact diagonalization results for both weak () and strong () FM coupling Drechsler (). This intensity is in both cases much stronger than in the AFM Heisenberg chain Caux06 (). For weak FM coupling, this confirms four-spinon excitations above a two-spinon continuum, and reinforces our conclusion Enderle10 () on the existence of a four-spinon continuum in LiCuVO.
In our RPA model of two coupled Heisenberg chains Enderle10 () a significant modification of the isolated Heisenberg chain two-spinon continuum is found for FM coupling between the chains, in good agreement with the main experimental features. The fit of both and leads to a quantitatively satisfactory overall description of the observed intensity with and values close to those found in Enderle05 (). The RPA-approach appears therefore justified, at least a posteriori.
In conclusion, the parameter sets suggested in the Comment Drechsler () are at odds with several experimental properties, such as the susceptibility, the magnetic ordering vector for one of the parameter sets, and the observed dispersion of the intensity maxima Enderle05 (), while the parameters proposed in Enderle05 (); Enderle10 () lead to a consistent description of these properties.
- (1) S.-L. Drechsler et al., arXiv:1006.5070v1.
- (2) M. Enderle et al., Europhys. Lett. 70, 237 (2005).
- (3) M. Enderle et al., Phys. Rev. Lett. 104, 237207 (2010).
- (4) H.A. Krug von Nidda et al., Phys. Rev. B 65, 134445 (2002).
- (5) J.S. Caux et al., J. Stat. Mech. 2006, P12013.