Open XXZ chain and boundary modes at zero temperature

Submission summary

 As Contributors: Jacopo De Nardis · Sebastian Grijalva · Véronique Terras Arxiv Link: https://arxiv.org/abs/1901.10932v2 (pdf) Date submitted: 2019-02-19 01:00 Submitted by: De Nardis, Jacopo Submitted to: SciPost Physics Discipline: Physics Subject area: Condensed Matter Physics - Theory Approach: Theoretical

Abstract

We study the open XXZ spin chain in the regime Delta>1 and for generic longitudinal magnetic fields at the two boundaries. We discuss the ground state via the Bethe ansatz and we show that, 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 the length L of the chain. 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.

Ontology / Topics

See full Ontology or Topics database.

Current status:
Has been resubmitted

Submission & Refereeing History

Resubmission 1901.10932v4 on 8 July 2019
Resubmission 1901.10932v3 on 27 May 2019

Submission 1901.10932v2 on 19 February 2019

Reports on this Submission

Anonymous Report 2 on 2019-3-15 Invited Report

• Cite as: Anonymous, Report on arXiv:1901.10932v2, delivered 2019-03-15, doi: 10.21468/SciPost.Report.872

Strengths

The paper is mathematically robust and clear, and considers a non trivial problem. They obtained an unexpected result. In a special region of the model, the boundary magnetization depends on the magnetic field applied in one end of the chain. This is a surprise since the quantum chain is gaped.

Weaknesses

(opitional) Perhaps more physical discussions of the resuls would be desired

Report

The authors study the XXZ open quantum chain in the antiferromagnetic regime,
in the presence of magnetic fields coupled with the spins at the boundaries
of the lattice. Using the quantum inverse scattering method with boundaries
they were able to connect the expected double degeneracy of the ground state
(since the model has an antiferromagnetic order) with the existence of
a Bethe complex root (boundary root), related to an excitation localized at the
boundaries of the chain. As a result of their calculation they show that
the boundary magnetization at one end of the lattice (in the bulk limit)
depends on the value of the magnetic field at the other end. This is a
surprise.

The paper in my opinion is interesting from both, mathematical and physical
perspectives, and I recommend its acceptance for publication.

Requested changes

1)The authors should state in the abstract and several places that the model are considered in the anti ferromagnetic regime ($\Delta >1$ is not enough for that).

2) The integrability of the model were obtained in their Re.[21], this should be mentioned in starting
Section 2.

• validity: high
• significance: high
• originality: high
• clarity: high
• formatting: good
• grammar: good

Anonymous Report 1 on 2019-2-21 Invited Report

• Cite as: Anonymous, Report on arXiv:1901.10932v2, delivered 2019-02-21, doi: 10.21468/SciPost.Report.839

Strengths

1. New results.
2. Exact treatment.
3. Good presentation.
4. Correcting earlier mistakes in the literature.

Weaknesses

1. Insufficient physical explanations.
2. Treatment of earlier literature is a bit rushed.

Report

This paper deals with the boundary XXZ chain with diagonal boundary field. It discusses the nature of the ground state at even length and it has very interesting results. In certain cases it corrects mistakes made in some papers in the literature, including a classic paper [15].

Nevertheless I have certain comments about the presentation. The main issue is about the meaning of the results and establishing the connection to experiments and to earlier results in the literature.

This work deals with the ground state energy, and ground state boundary magnetization. Also, it discusses the boundary spin autocorrelation, which is important for non-equilibrium problems. The work defines the ground state as it should be: the lowest energy state in any finite volume. Nevertheless it discusses cases when $h_1 = h_L$ and there will be two degenerate states in the TDL, with an exponentially decaying (with volume) gap. Let us denote the true finite volume ground states by $GS_1$, and the second state as $GS_2$. The work deals with discontinuities in certain quantities, for example in the boundary magnetization. It is defined as
$<GS_1 | \sigma_{1,z} | GS_1 >$
It is noted that this quantity is discontinuous, and it depends also on the value of the magnetic field on the other side of the chain.

The statements are correct and the formulas are mathematically precise. Nevertheless the physical interpretation is not correct in my opinion. The authors explain that the reason for the discontinuities is that when we keep $h_1$ fixed and change $h_L$ to cross $h_1$, the boundary excitation jumps'' from one end to the other end. Now on a big chain there is clearly no instantenous jump and rearrangement. This discontinuity is an artifact of the mathematical definitions, but it has no immediate physical meaning. Its origin is that $GS_1$ and $GS_2$ are exchanged when we change one of the magnetic fields to cross the other one. There is a physical consequence though: the spin-autocorrelation, which is measurable, but any sudden jump in $\sigma_{1,z}$ as we change the magnetic field at the other end is not measurable.

Also, it is quite questionable what is relevant for experiments. When we cool a system down, it is hard to imagine that we would actually get to the $GS_1$ mean values in any situation. Instead, lowering the temperature would result in the average

$\frac{1}{2}(<GS_1 | \sigma_{1,z} | GS_1 > + <GS_2 | \sigma_{1,z} | GS_2 > )$

And this average is a continuous function of any of the magnetic fields. It is difficult to imagine that any measurement would detect the discontinuity investigated by the authors. Only the effect in the autocorrelation is easily detected.

The connection to the finite T results of [50,51] is a little bit rushed, but it should be certainly discussed. It is claimed that [50,51] can be valid only when there is free boundary condition at the other end. On the other hand, the authors explain in the Conclusion, that they don't know yet what happens at finite T.

From the results of this manuscript it seems to me that the authors might be correct that [50,51] (and possibly other works too) were wrong with regard to the boundary magnetization in the ground state, in the regimes discussed here. However, [50,51] take the $T\to 0$ limit of the finite $T$ formulas, which were obtained already in the $L\to\infty$ limit. It seems to me that the two limits $T\to 0$ and $L\to\infty$ do not commute in this case. [50,51] does the TDL first and $T\to 0$ afterwards, and the present manuscript does $T\to 0$ first (by selecting the true ground state in finite volume) and doing the TDL afterwards.

I think that this difference should be discussed. Also, it should be discussed which one might be more relevant for experimental situations.

The Conclusion leaves open the discussion of the finite $T$ case. I doubt that any such phenomenon would survive at finite $T$, but it is certainly worthwhile to investigate.

Also, it would be important to discuss the order of limits for the autocorrelation function. The hard statement is about the $t\to\infty$ limit in the cases when $h_1=h_L$. 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?

One more problem with the present work is that it discusses only the even lenghts. Given the amount of mathematical precision in the definitions, it is a bit strange to focus only on even lengths. How would the situation look like at odd lengths? Again, at any finite $T$ we would expect that the length does not matter, but it is known that for the ground states it does matter. I am not so sure what should be suggested here: asking for a major revision to do the odd length case is perhaps too much. Nevertheless it should be mentioned and at least a little bit discussed.

Requested changes

1. Discuss the odd length case.
2. Discuss the physical meaning of the results.
3. Discuss the exchange of limits with regards to the finite $T$ case. (or try to convince me if I am wrong)
4. Discuss the order of limits with respect to the autocorrelation function.
5. Formula (1.8) can not be correct in this form. If the chain is periodic, and there is only a global field, then nothing distinguished the odd sites and even sites, so the formula could be $(-1)^j$ but also $(-1)^{j+1}$. This should be somehow corrected, perhaps with a staggered field going to zero, or in any other way.
6. There are a few small misprints, the text should be carefully read once more.

• validity: top
• significance: high
• originality: good
• clarity: top
• formatting: perfect
• grammar: perfect