SciPost Submission Page
Open XXZ chain and boundary modes at zero temperature
by Sebastian Grijalva, Jacopo De Nardis, Veronique Terras
This is not the current version.
|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.
Author comments upon resubmission
We have indeed implemented the two requested changes suggested in report 2. Concerning the many insightful comments of report 1, they deserve some more explanations.
- The sentence « the boundary magnetization becomes a discontinuous function of the other field at h− = h+, point at which the boundary root jumps from one edge of the chain to the other » was supposed to be an image describing the significant change in the mathematical description of the ground state (defined as the lowest energy state in finite volume) when crossing this point: the boundary root changes localization in its mathematical description as a solution of the Bethe equation (from being exponentially close to a zero associated to the boundary parameter at one end of the chain, it becomes exponentially close to the zero associated to the boundary parameter at the other end of the chain, as it becomes clear from our study of the ground state in section 4, and this also corresponds to a change of localization of the corresponding contribution to the wave function), and this change of localization is in fact responsible for the discontinuity that we observe here. This is not really due to a crossing of level since the degeneration is not exact at h_-=h_+ for L even (so there is no crossing of level for L finite in this case, see Figure 3 that we have slightly modified to make this point more clear). This change of localization of the boundary root with respect to the two zeroes of the boundary factor corresponds of course to a rearrangement of all the Bethe roots, but the latter is continuous when crossing this point (since the two zeroes coincide at this point). Nevertheless, since it seems that this sentence can be misunderstood, we have slightly modified it, as well as a few sentences in the paragraph before, so as to make it more precise.
- "It is difficult to imagine that any measurement would detect the discontinuity investigated by the authors ».
We are not expert in experimental measurements and, of course, since the discontinuity that we computed for the boundary magnetization may be smoothened by the temperature, it may not be easy to measure it experimentally (and we never pretended it should be). However, it seems to us that, in order to cool down the system to one of its ground state, it is sufficient to choose one of the magnetic fields at the edges to be larger (or smaller) than the other and then cool down the chain. This way the system is gapped, there is no degeneracy and the ground state 1 (or 2) has a finite energy difference with the other. We agree that this may be difficult to do in practice when h_- ~ h_+ (the temperature should be kept smaller than |h_- - h_+|) but in principle the protocol is clear.
- Concerning the discussion about finite temperature, we hope it is clear that we do not consider the temperature case in the present paper. This is indeed an interesting problem that we mention as a possible further development in the conclusion. In particular, our study of the boundary magnetization in the ground state is performed strictly at zero temperature. Therefore, we do not pretend to correct anything about references [50,51] (which are now [53,54] in the new version): indeed, these references should in fact not appear in the footnote on page 28, and we thank the referee for pointing out this misprint, we have corrected this. We mentioned these references only by completeness since the same quantity was computed there in the temperature case. In particular, in  (now ), the zero-temperature limit of the boundary magnetization has been recovered (in the case of a zero magnetic field at the other end).
- Concerning the question of whether the thermodynamic and the zero-temperature limits commute or not, it is difficult to answer this question without having investigated the temperature case. Nevertheless, the authors of  (now ) have investigated the zero-temperature limit of the expression they have obtained for the boundary magnetization at finite-temperature by taking the thermodynamic limit first, and they have recovered the result (including the discontinuity) that had been previously obtained at zero-temperature for the case of a zero-boundary magnetic field at infinity. This seems to indicate that, at least in the case considered in , these limits do commute.
- « What happens here when we change only one of the fields, and how does it relate to the size of the system, and the order of limits ? »
We thank the referee for this very good question. What we have found is that the form factor of the sigma_1^z operator between the two ground states decays exponentially in system size whenever the two fields are chosen to be different. Therefore in order to have a finite t->infty limit for the auto-correlation one should take first h_- -> h+ and then the thermodynamic limit. We have commented this in the manuscript, and added the computation of the form factor for different fields.
- We focused on the even length case because it is the only case where there is an interesting quasi (and not exact) degeneracy at h_-=h_+. In the odd length case, there is no degeneracy at h_-=h_+ unless both are zero. There is however an exact degeneracy for h_-= - h_+, which is also present for finite L and can be seen as a consequence of the spin flip symmetry of the chain, and the two degenerate ground Bethe states are in this case in different magnetization sectors. It is obvious that in this case the discontinuity of the boundary magnetization is due to a crossing of levels of these two ground Bethe states with different magnetization, and therefore this case seemed to us a priori not so interesting from the physical point of view. It is nevertheless interesting to compare what happens in the even and odd length cases, since the appearance of the discontinuity is due to very different microscopic mechanisms. Hence, as suggested by the referee, we have added a section devoted to the odd L case, in which we redo the computation in this case and explain the differences with respect to the even L case. We have also commented this case in introduction and in conclusion.
- Finally, we thank the referee for pointing out the misprint in formula (1.8). Indeed in the bulk case, the spontaneous staggered magnetisation can be observed either by applying a small staggered field or a small field on one of the sites (or at the boundary). We have nevertheless preferred to remove the mention to this quantity, since finally we do not find it so significant for our study. We have preferred to add instead a small comment about the odd L case that we now also study in our paper.
Submission & Refereeing History
Reports on this Submission
Anonymous Report 1 on 2019-6-4 Invited Report
The authors addressed my questions, and I think that the manuscript has improved considerably. Nevertheless I have some further questions. I am sure that the article can be published soon, once these remaining questions are settled.
-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?
-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 $L\to\infty$ 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 $L\to\infty$ 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.
-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 $\sigma_1^z$ would be added already around Fig 3. It would be nice to see data from two $L$ values, perhaps the $L=12$ 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.
1. Discuss a bit more the nature of the TDL, and the alternating boundary field.
2. Discuss a bit more clearly that the level avoidance becomes a level crossing eventually. The physical insights should be added to the Conclusions too.
3. Reposition some of the figures somehow, and add a bit more numerical data, for better visibility and understanding.
4. A typo: page 2 has ''two decoupled Majoranas fermions''