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Open XXZ chain and boundary modes at zero temperature
by Sebastian Grijalva, Jacopo De Nardis, Veronique Terras
- Published as SciPost Phys. 7, 023 (2019)
|As Contributors:||Jacopo De Nardis · Véronique Terras|
|Submitted by:||De Nardis, Jacopo|
|Submitted to:||SciPost Physics|
|Subject area:||Condensed Matter Physics - Theory|
We study the open XXZ spin chain in the anti-ferromagnetic regime and for generic longitudinal magnetic fields at the two boundaries. We discuss the ground state via the Bethe ansatz and we show that, for a chain of even length L and in a regime where both boundary magnetic fields are equal and bounded by a critical field, the spectrum is gapped and the ground state is doubly degenerate up to exponentially small corrections in L. We connect this degeneracy to the presence of a boundary root, namely an excitation localized at one of the two boundaries. We compute the local magnetization at the left edge of the chain and we show that, due to the existence of a boundary root, this depends also on the value of the field at the opposite edge, even in the half-infinite chain limit. Moreover we give an exact expression for the large time limit of the spin autocorrelation at the boundary, which we explicitly compute in terms of the form factor between the two quasi-degenerate ground states. This, as we show, turns out to be equal to the contribution of the boundary root to the local magnetization. We finally discuss the case of chains of odd length.
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Published as SciPost Phys. 7, 023 (2019)
Author comments upon resubmission
-"The authors added a complete new discussion about the odd length chain. I think this is a great improvement to the paper. Quite interestingly, footnote 9 explains that the thermodynamic limit should be taken with a boundary magnetic field which changes sign between odd and even length chains. Would the authors agree that this is a consequence of the anti-ferromagnetic nature of the chain? Can this be seen perhaps in the Ising limit?"
We have added an extra comment in footnote 9 stressing that indeed this is due to the antiferromagnetic nature of the chain and that it can be also easily understood in the Ising limit.
-"The authors are explaining very clearly that the odd length case is different, because there is a level crossing even at finite
L. On the other hand, in the even length case there is a level avoidance, and the numerics is shown on the right of Fig. 3. However, from this figure it seems to me that in the L to infinity
limit we would indeed get a level crossing, because clearly the gap has to smoothen out, it becomes exponentially small. I think that this should be mentioned: based on the
L to infinity values the odd and even length cases are not so much different, given that the magnetic field is changed alternatingly. This is also a nice physical picture, that we don't really get different behaviour for an infinitely long chain, apart from the issue of the anti-ferromagnetic ordering."
We believe we have substantially explained the different nature of odd or even L. The even L case displays a degeneracy that closes exponentially with the system size while in the L odd case the degeneracy is exact due to Z_2 symmetry. We have stressed that we indeed recover the same behaviour in the thermodynamic limit for the quantities we have computed. This is also commented in the conclusion. We do not see reasons to stress this more.
-"Fig 3. is very informative and useful. Nevertheless I would like to see a finite volume data of the GS boundary magnetization too. This is presented in Fig 15, but there the finite volume data is only used to confirm the infinite volume predictions. Or at least it seems to me that this is the role. I think that it would be much more pedagogical if some finite volume data for
would be added already around Fig 3. It would be nice to see data from two
values, perhaps the
and some smaller value. Such that we could see the closing of the gap like on the right of Fig 3 now, and the appearance of the discontinuity like on Fig 15. I stress again that these figures are already quite good and informative, but the understanding would be much better if they would be presented close to each other, with two
L values, with continuous curves. Also, perhaps these figures could be presented earlier. Now Section 1. has Figs. 1 and 2, and they are very informative. But why not present the other data here, before the discussion of the complicated Bethe Ansatz solution? It is always useful to have the simple physical picture in mind, before going to the technical details."
We have added a comment on the relation between the closing of the gap and the discontinuity in the spontaneous magnetisation in the caption of figure 11, where the boundary magnetization for L even is numerically plotted at finite size versus its analytical value in the thermodynamic limit. This comment refers directly to Fig. 3 so that the reader can more easily make the connection. We believe that our manuscript has already many figures and all the aspects are covered, we do not see a reason to add extra figures. The boundary magnetisation is plotted in figure 11 and figure 15 and we believe this figures are enough for our publication. We moreover think that the position of the figures is adequate in the text to illustrate our analytical results. We hope that the referee will understand that further modifications will go against the format we wish to have for our manuscript.