Exotic criticality in the dimerized spin-1 $XXZ$ chain with single-ion anisotropy

We would like to thank the Referee for Her/His careful reading of the manuscript and the helpful comments.
In the following, we address the three points of “Weaknesses” and three points of “Requested changes” raised by the Referee.
(See also reply to Report 1 for further modifications in the revised manuscript.)

Weaknesses 1:
Our usage of “Gaussian model” for the free (compactified) relativistic boson in 1+1 dimensions is standard in the field, see e.g. the textbook by Gogolin, Nersesyan and Tsvelik on bosonization.

Weaknesses 2:
As usual field theoretic RG is defined in the vicinity of a critical point. In our case this is the Takhtajan-Babujian (TB) point and we may describe any (small) perturbation away from it as we know the exact operator content at the critical point. So the applicability of our field theory description in the vicinity of the TB point is guaranteed, irrespective of which perturbations (dimerization, biquadratic interaction etc.) we apply. The applicability of the same field theory description, with appropriately adjusted coupling constants at the cutoff scale, is as we clearly state an assumption. This assumption is verified a posteriori by comparing the field theory descriptions to DMRG computations. As we noted in our manuscript, using Schulz's bosonization prescription leads to equivalent results.

Weaknesses 3:
The steep drop of the susceptibility suggests that this Ni-compound lies at (or close to) the Gaussian or Ising transition point, where the system becomes gapless. In the case of the Gaussian transition with central charge $c=1$, the behavior of the susceptibility might be similar to the spin-1/2 Heisenberg chain as suggested by the Referee. Note that the logarithmic correction in the spin-1/2 Heisenberg chain becomes only visible in the very low-temperature regime, see, e.g., Eur. Phys. J. B 5, 677 (1998).

In Fig.5(b) we have demonstrated that the experimental data of Ref.[19] can be reproduced by the modern numerical technique (infinite TEBD) with the same parameter set $\delta=0.25$ as used in Ref. [19] (note that the very small single-ion anisotropy $D/J=0.02$ was also reported in Ref.[19]). Figure 5(a) shows the phase diagram in the $D/J$-$\Delta$ plane, where the corresponding parameter set is denoted by the star symbol. If the Ni-compound would have been described by a larger value of $D/J$, it might be located on the $c=1$ line (“plus” symbol), however, the steep drop of the susceptibility in the low-temperature regime doesn’t show up [compare the dashed line in Fig.5(b)]. Here a sharp drop of the susceptibility will appear at extremely low temperatures at most, just as for the spin-1/2 Heisenberg chain. Therefore we are led to the conclusion that a reasonably well fit of the experimental data indicates that the Ni compound is parametrized close to the Ising transition point.


Requested changes 1:
We added a sentence with respect to $\langle\sigma^3\rangle$ in the caption of Fig.1:
“$\langle \sigma^3\rangle$ denotes third Ising order parameter, determining the Ising QPT between the D-H or D-LD phase and the D-AFM phase.”

Requested changes 2:
We have modified the corresponding sentence in the revised version.

Requested changes 3:
We have modified the sentence as follows (added “where”):
In the regimes where the mass scale associated with $S_3$ is much smaller (larger) than the one associated with $S_B$ and “where" $S_{\rm int}$ can be treated as a perturbation, we may integrate out the bosonic (fermionic) degrees of freedom, see e.g. Ref. [33].

We would like to thank the Referee for Her/His careful reading of the manuscript and the helpful comments.

In the revised version we comment on the accuracy level of our numerical scheme, discussing the discarded weights in the beginning of Section 2 and the precision of the tricritical point in the caption of Fig.1 (actually the error bar now is significant smaller than the symbol size). Estimating the error bar of the tricritical point we corrected the given value of the tricritical point somewhat compared to the previous version.

Moreover, we added a new appendix (Appendix B) in order to demonstrate the high accuracy of the transition points obtained by the infinite DMRG (iDMRG) technique. In our model, all three phases are gapped, and the system becomes critical only on the continuous transition lines. Therefore the correlation length can be used to pinpoint the transition point by determining the correlation length for various $\Delta$ at fixed $D/J$ as shown in Fig.6. Thereby a finite-size scaling is, however, not necessary since we use the iDMRG technique, which provides physical quantities directly in the thermodynamic limit.

Differently from our model, in the half-filled extended Hubbard model with nearest-neighbor Coulomb interaction the spin degrees of freedom are gapless in the spin-density-wave (SDW) phase, which makes it difficult to determine the phase boundaries between SDW and the so-called bond-order-wave (BOW) phases. Hence the correlation length obtained by iDMRG cannot be used to determine, e.g., the SDW-BOW transition point.

The phase diagram for $\delta=0.25$ (Fig.5) can be produced in the same way. In the absence of the single-ion anisotropy $D$, the D-H phase disappears at about $\delta\simeq 0.26$ from the ground-state phase diagram in the $\delta$-$\Delta$ plane, see Ref. [10,11]. Therefore, our observation of a D-H phase in the $D/J$-$\Delta$ plane for $\delta=0.25$ corroborates previous results.

In addition, we added an explanation for the topological order parameter in the presence of dimerization shortly after Eq. (48). See also reply to Report 2 for further modifications in the revised manuscript.