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The Folded Spin-1/2 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: 2020-09-22 10:38
Submitted by: Zadnik, Lenart
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approach: Theoretical

Abstract

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.

Current status:
Editor-in-charge assigned


Submission & Refereeing History


Reports on this Submission

Anonymous Report 1 on 2020-10-17 Invited Report

Strengths

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.

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 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.[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 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.

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

Author Lenart Zadnik on 2020-11-17
(in reply to Report 1 on 2020-10-17)
Category:
remark
answer to question

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 nearest-neighbour 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 large-anisotropy limit of the Heisenberg
model lies in the fact that it provides access to the dynamics on intermediate-time scales, one of the goals
being the study of pre-relaxation phenomena.

We hope that the new version of the manuscript is clearer.

The Authors

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