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Necklace Ansatz for strongly repulsive spin mixtures on a ring
by Gianni Aupetit-Diallo, Giovanni Pecci , Artem Volosniev, Mathias Albert, Anna Minguzzi, Patrizia Vignolo
Submission summary
| Authors (as registered SciPost users): | Anna Minguzzi · Patrizia Vignolo · Artem Volosniev |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202407_00048v1 (pdf) |
| Date submitted: | 2024-07-26 14:13 |
| Submitted by: | Vignolo, Patrizia |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We propose an alternative to the Bethe Ansatz method for strongly-interacting fermionic (or bosonic) mixtures on a ring. Starting from the knowledge of the solution for single-component non-interacting fermions (or strongly-interacting bosons), we explicitly impose periodic condition on the amplitudes of the spin configurations. This reduces drastically the number of independent complex amplitudes that we determine by constrained diagonalization of an effective Hamiltonian. This procedure allows us to obtain a complete basis for the exact low-energy many-body solutions for mixtures with a large number of particles, both for $SU(\kappa)$ and symmetry-breaking systems.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report
In the work ``Necklace Ansatz for strongly repulsive spin mixtures on a ring''
the authors propose an ansatz which reminds of perturbation theory
and apply it to a strong (infinite) repulsive regime for SU(2) and SU(3) cases.
This strongly reminds of the situation of Ogata and Shiba which exactly describe the situation for the SU(2)
in this limit. There is also a work which builds on top of their work precisely on SU(N) extensions.
Both works are not at all mentioned in this manuscript since these are basic examples
where the authors could test the strength of the method proposed.
Here there is a contact matrix that has to be elaborated for each single case
and which seems the interaction transported into the new basis of the problem which is
given by necklaces and snippets.
It is shown in few examples that choosing this basis is a notable reduction in states to be considered
and this would be more pronounced for larger system sizes.
The end-product, however, is a system of largely overdetermined equations for the parameters which describe the infinitely system.
And it does not help to just state that it seems just to be overdetermined and referring to some gauge invariance.
It is there and the authors apparently do not have a proper method to deal with this problem.
Seeing the slightly small system sizes given as examples (up to SU(3) 3 particles) one is tempted to
a) see what is written in the work on the extension of Ogata and Shiba and compare with that work.
b) ask the authors where they think to arrive with their ansatz in terms of number of sites, particles, kappa?
This work is interesting for this infinitely interacting regime but elsewhere not applicable.
Therefore it is a rather niche application than a work of broader interest and should appear elsewhere.
Consequently, it should not be published in SciPost Physics.
Remarks and questions:
The article (beginning with the title) speaks of boson-fermi mixtures to be addressed, however I don't see where this is done.
At the moment every regarded system is just of purely bosons or of purely fermions,
and no mixture of the both has been considered.
In the abstract there is a ``repulsive'' missing, I guess.
For an even number of particles there is no k=0 value, right?
Equation (8) there is a symbol R: is it permutations as usual?
The authors speak of ``relative'' integers: what are they?
The different sums in (13),(14) should be marked otherwise; now it seems that two parameters areto be summed over.
The Young-tableaus are sometimes flipped in certain tables; it should be stated
a) that they appear flipped and hence the axix of (anti)-symmetrization are exchanged
b) the ``more exotic'' tableaus correspond to a different representation of the symmetric group S_n
c) standard young tableaus do not show the singlets in the states; for example the young tableau for SU(2)
would be composed only of the first (of two: anti-symmetrization) row.
I have a problem with formula (19) and values for \gamma^{(2)} in that it is not clear to me
whether the summation starts in 0 or 1, or wherever it starts; I cannot reproduce the values of the various Young tableaus in either case.
I have looked into Ref. [62] and could not get an answer to my question.
According to Table 1 it could seem that it is the spin/casimir operator, but it is probably not like that.
Where (in which chapter or equation) in the books Refs. [60],[61] is this found?
Please give with books also at least the corresponding chapter as reference.
Recommendation
Reject
Strengths
1. New approach.
2. Interesting physical system.
3. Highly non-trivial many-body quantum effects.
Weaknesses
1. Very technical.
2. At the end the method requires heavy numerical calculations.
Report
The authors develop an interesting ansatz to solve the problem of a mixture of strongly-interacting fermions in a ring. The physical systems is surely relevant for the current experiments in low-dimensional quantum gases.
Their "necklace ansatz" seems somehow better with respect to the familiar Bethe ansatz. In the case of a large number of fermions one has to face highly numerical calculations which, however, involve an Hilbert space with a reduced dimension.
The paper is surely interesting and accurately written. However, I do not see exciting new physical results. Maybe Sci Post Physics Core could be a more appropriate destination for this paper.
Requested changes
None.
Recommendation
Accept in alternative Journal (see Report)