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The Folded Spin1/2 XXZ Model: I. Diagonalisation, Jamming, and Ground State Properties
by Lenart Zadnik, Maurizio Fagotti
Submission summary
As Contributors:  Maurizio Fagotti · Lenart Zadnik 
Preprint link:  scipost_202009_00018v1 
Date submitted:  20200922 10:38 
Submitted by:  Zadnik, Lenart 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We study an effective Hamiltonian generating time evolution of states on intermediate time scales in the strongcoupling limit of the spin1/2 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.
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Submission & Refereeing History
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Anonymous Report 1 on 20201017 Invited Report
Strengths
1It is a clear paper that introduces a procedure to study models in a restricted context of large ones.
2The paper is interesting and clear.
Report
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 strongcoupling limit.
The main application in the paper is an XXZlike 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.[20], 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.[20].
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 [20], 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 MatrixProduct ansatz in Alcaraz and Lazo, Braz. J. Phys. (33) (2003).
An more general case related to the constrained spin1 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.
(in reply to Report 1 on 20201017)
Dear Referee,
Thank you for the review of our paper and positive feedback. We would like to clarify few points that
might have not been stressed enough in the submitted manuscript.
1.) The projection of the folded Hamiltonian (27) onto the subspace with forbidden occupation of two
neighbouring sites is equivalent to the analogous projection of the XX Hamiltonian, defined in Ref. [20].
The corresponding projector indeed commutes with the folded Hamiltonian and defines a sector containing
the ground state of the folded Hamiltonian. The situation is however a bit different with regard to the projectors
onto subspaces of states, in which two spins that are less than $r >1$ sites apart cannot simultaneously point
downwards. Upon additional checks we have observed that the projections of the folded Hamiltonian and of
the XX model onto such sectors still coincide, but the projectors themselves no longer commute with the folded
Hamiltonian (27). Thus, the particles with higher radii discussed in Ref. [20] do not coincide with any configuration
of our folded Hamiltonian, i.e., they do not define the sectors of Hamiltonian (27) in the submitted manuscript.
We have incorporated this into the discussion in Section 3.1, around Eqs (32) and (33)  see the newest version of
the manuscript, accessible on the arXiv: https://arxiv.org/abs/2009.04995
2.) Related to the previous observation, there is a point that, arguably, we didn’t stress enough: the Bethe Ansatz
that we presented produces $2^L$ eigenstates that, in turn, span the entire Hilbert space (including the kernel of
the Hamiltonian, which, in our case, includes a set of jammed states). This goes beyond the standard Ising limit of the
XXZ Bethe Ansatz, discussed in Refs [18,19], or the Bethe Ansatz used in Ref. [20] to diagonalise the sector with
prohibited nearestneighbour occupation. To the best of our knowledge, the structure that we pointed out was not
investigated before, and we think that it provides a general framework that is suitable, if not even necessary, for
investigations into quench dynamics in the strong coupling limit. Appropriate comments have been inserted in the
introduction to Section 5.
3.) We have recently realised that the more general folded XYZ model was solved by Bariev using the nested Bethe
Ansatz method – it now goes under the name of Bariev model. Bariev’s solution does not rely on the additional
symmetries present at the XXZ point. The main difference between his and our Ansatz is in the quantum numbers:
in our case only one set of quantum numbers is necessary for the characterisation of the eigenstates; we somehow
circumvent the nested structure. We have added some comments and references into the text: see the last paragraphs
after Eqs (41) and (93) in the newest version of the manuscript.
4.) Finally, we would like to remark that the point of discussing the largeanisotropy limit of the Heisenberg
model lies in the fact that it provides access to the dynamics on intermediatetime scales, one of the goals
being the study of prerelaxation phenomena.
We hope that the new version of the manuscript is clearer.
The Authors