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The Folded Spin-1/2 XXZ Model: I. Diagonalisation, Jamming, and Ground State Properties
by Lenart Zadnik, Maurizio Fagotti
This is not the current version.
|As Contributors:||Maurizio Fagotti · Lenart Zadnik|
|Date submitted:||2020-09-22 10:38|
|Submitted by:||Zadnik, Lenart|
|Submitted to:||SciPost Physics|
We study an effective Hamiltonian generating time evolution of states on intermediate time scales in the strong-coupling limit of the spin-1/2 XXZ model. To leading order, it describes an integrable model with local interactions. We solve it completely by means of a coordinate Bethe Ansatz that manifestly breaks the translational symmetry. We demonstrate the existence of exponentially many jammed states and estimate their stability under the leading correction to the effective Hamiltonian. Some ground state properties of the model are discussed.
Submission & Refereeing History
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Reports on this Submission
Report 3 by Paul Fendley on 2020-12-30 Invited Report
- Cite as: Paul Fendley, Report on arXiv:scipost_202009_00018v1, delivered 2020-12-30, doi: 10.21468/SciPost.Report.2346
The authors first generalize an old procedure from MacDonald et al for doing a systematic expansion of a Hamiltonian around a limit of a single large coupling, doing what is often called a Schrieffer-Wolf transformation. They then analyze the particular case of the XXZ chain expanded around large $\Delta$. They find the striking result that the resulting zeroth order Hamiltonian is much simpler than generic models solvable by the coordinate Bethe ansatz -- the momenta can be determined exactly, as the bare scattering matrices cancel in the Bethe equations.
I think this result is very interesting and should be published in SciPost. My main comment is that both parts of the paper are closely related to earlier results, and very possibly some of the work done previously could be of use in simplifying and/or extending the authors' analysis.
I take the two parts in turn. For the first part, the procedure of expanding Hamiltonians as described in section 2 of this paper was developed in great depth by Abanin, de Roeck, Huveneers and Ho (arXiv:1509.05386, published in CMP). ADHH prove (as in rigorous) that such an expansion exists up to corrections almost exponentially small in the large parameter ($1/\kappa\,$ in this paper). The authors don't discuss the validity of their expansion at all, and so should mention this. In the integrable XXZ case described in depth here, I would guess the procedure will work to all orders, but that is not guaranteed.
Also, thinking about the expansion the way ADHH do probably will lead to some useful intution. ADHH think of the large term in the Hamiltonian as an symmetry (approximate if the series does not converge). In the case that the authors study, this symmetry amounts to domain-wall number conservation. Also note that the expansion for the Ising chain is given explicitly earlier in my paper with Else, Kemp and Nayak (arXiv:1704.08703, published in PRX), where it is also explained how in essence that this expansion was how Onsager solved the Ising model.
While ADHH explicitly construct the unitary transformation order by order, their expression is unwieldy, and so while useful for their proof, it is not so useful in understanding the resulting physics. So the authors' explicit analysis of the XXZ chain here is still well worth doing. In fact, even though the zeroth order Hamiltonian was already solved by Bariev, their method exhibits a feature not seen here before -- the Bethe equations can be solved explicitly, giving in essence a free fermion spectrum, except with unusual degeneracies.
This brings me to the second observation of my report, concerning the zeroth order approximation to XXZ that the authors devote the bulk of their paper to. A Hamiltonian with essentially the same behavior was analysed in depth in a paper I wrote with Schoutens a while ago (arXiv:cond-mat/0612270, published in JSTAT). Very possibly the authors here could get some useful intuition from these results. Moreover, since the model Zadnik and Fagotti study is simpler (the only interaction is that fermions hop only if the intermediate site is empty), possibly something could be learned about the earlier model as well. In the old model, the degeneracies were interpreted as arising from Cooper pairs, and the remaining "free" quasiparticles were understood as exclusons in the sense defined by Haldane. I'd guess a similar nice interpretation is possible here, and may simplify the analysis, which gets a little techinical. In addition, comparing the enhanced symmetry algebras giving the degeneracies (there it is a generalization of supersymmetry) possibly will lead to some useful intution and/or progress.
Anonymous Report 1 on 2020-10-17 Invited Report
- Cite as: Anonymous, Report on arXiv:scipost_202009_00018v1, delivered 2020-10-17, doi: 10.21468/SciPost.Report.2091
1-It is a clear paper that introduces a procedure to study models in a restricted context of large ones.
2-The paper is interesting and clear.
In this paper the authors provides a prescription to produce effective
Hamiltonians (folded Hamiltonian) that acts on a smaller Hilbert space
("restricted space") as compared with the original one. This folded
Hamiltonian is obtained by appropriate strong-coupling limit.
The main application in the paper is an XXZ-like chain (Eq. 27), where
pairs of nearest neighbors spins (up and down) interchange positions
only if the pair has as neighbors two equal spins. They made an detailed
study of the models's charge conservations, and obtained, that the ground state
belongs to the constrained sector where two up spins are not allowed. Since
restricted to this sector the Hamiltonian (27) is equivalent to the
constrained XXZ Hamiltonian solved in Ref., their solution recovers that
of this mentioned reference.
In fact, it seems that the several Hilbert space sectors associated to (27) can be mapped
to the Hilbert spaces of a constrained model that generalizes the one in Ref..
In this generalization we have a mixture of of clusters of l up spins
("molecule of size l"). We have a set of n cluster of up spins (c_1,...,c_n),
and on each of these clusters we have a number of s_i=,1,2,...,l_i (size l_i)
up spins. This general model, in the sectors where all the particles have
the same size t (t=1,2,3,...) reproduces the result of , being a special sector
of the Hamiltonian (27), in the case where t=2. This general Hamiltonian was introduced in order
to describe the asymmetric diffusion of a mixture of particles with arbitrary size.
It was solved by the coordinate Bethe ansatz in Alcaraz and Bariev, Phys. Rev. E (33) (1999),
and also by a Matrix-Product ansatz in Alcaraz and Lazo, Braz. J. Phys. (33) (2003).
An more general case related to the constrained spin-1 XXZ chains was also solved
in Alcaraz and Bariev Braz. J. Phys. (30) 655 (2000).
I think the paper is interesting and contains interesting results, and
present a way to see quantum chains in restricted Hilbert spaces.
For this reason it should be published, but it would be interesting if the authors
made a contact with the above mentioned considerations.