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Exactly solvable quantum few-body systems associated with the symmetries of the three-dimensional and four-dimensional icosahedra
by T. Scoquart, J. J. Seaward, S. G. Jackson, M. Olshanii
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Submission summary
| Authors (as registered SciPost users): | Maxim Olshanii |
| Submission information | |
|---|---|
| Preprint Link: | http://arxiv.org/abs/1608.04402v2 (pdf) |
| Date submitted: | 2016-09-27 02:00 |
| Submitted by: | Olshanii, Maxim |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The purpose of this article is to demonstrate that non-crystallographic reflection groups can be used to build new solvable quantum particle systems. We explicitly construct a one-parametric family of solvable four-body systems on a line, related to the symmetry of a regular icosahedron: in two distinct limiting cases the system is constrained to a half-line. We repeat the program for a 600-cell, a four-dimensional generalization of the regular three-dimensional icosahedron.
List of changes
1. Added a reference to Heckman and Opdam's "Yang’s system of particles and Hecke algebras."
2. Added a second paragraph to the concluding section about extending the results to finite-strength delta-potentials.
3. Minor grammatical, spelling and word-choice changes.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 2) on 2016-10-3 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1608.04402v2, delivered 2016-10-03, doi: 10.21468/SciPost.Report.26
Strengths
1. Finding specific mass ratios in the few-body hard-core models with particle-particle contact interaction for which one obtains H_3 and H_4 type Lieb-Liniger models with infinite delta interaction strenghts by a change of coordinates.
2. Obtaining an explicit wave expansion of the solutions of the associated spectral problem.
Weaknesses
1. Identification of Lieb-Liniger models associated to exceptional reflection groups and few-body models
can only be done for infinite delta-interaction strengths (hard-core particles). A precise analysis what prevents extension to finite delta-interaction strengths is missing.
2. The ideas in this paper have been worked out before for reflection group of type F_4 in ref. [4] of the paper under review. The paper is a rather straightforward exercise to adjust the techniques to reflection
groups of type H_3 and H_4.
Report
The Lieb-Liniger Bose gas on the line is a famous integrable one-dimensional many body system naturally attached to the symmetric group. It admits a generalisation as integrable system in which the role of the symmetric group is taken over by a finite reflection group. The particle-particle contact interaction of the Lieb-Liniger Bose gas is replaced by delta-interactions at the root hyperplanes of the reflection group.
For classical Weyl groups there still is a reasonable interpretation as a one-dimensional many body system. For other types, in case of infinite delta-interaction strengths, the model can sometimes admit an interpretation as a hard-core few-body model on the line with distinguishable particles.
The paper under review gives an example of this phenomenon.
The idea that one-dimensional particles with different masses and with particle-particle contact interaction can be related by a (nonorthogonal) change of coordinates to a model describing equal particles but with delta-potential interactions along "nonphysical" hyperplanes goes back to McGuire in 1963 (it is reference [7] in the paper under review). The main point of the paper is to show that if the masses of the particles are exactly such that the hyperplanes x_i=x_{i+1} turn into the simple root hyperplanes for some nonclassical finite reflection group, then the model is equivalent to the associated generalised Lieb-Liniger model with infinite delta-interaction strengths. In ref. [4] of the paper under review this was already worked out in case of the finite reflection group of type F_4 and its affine version. In the present paper the same techniques are used for the noncrystallographic reflection groups H_3 and H_4.
Requested changes
1. See item 1 at weaknesses. A thorough analysis of what goes wrong for finite interaction strengths should be added.
Small comments:
2. plains --> planes (several times).
3. Section 2: Formula for e_{COM}\cdot\mathbf{z} is wrong (only so for \mu=M).
4. "m_{i-1}-m_i and m_i-m_{i+1} planes" need explanation.
5. I do not understand why giving an arctan-formula for the angle, instead of the (more standard) way of expressing the angle using arccos (which does not make the mass-dependence more difficult).
6. Page 3: Ref. [15] should be [16] I suppose.
Author: Maxim Olshanii on 2016-10-09 [id 59]
(in reply to Report 1 on 2016-10-03)We thank the referee for their careful reading and insightful critique.
>> 1. See item 1 at weaknesses. A thorough analysis of what goes wrong for finite
>>interaction strengths should be added.
We added an explanatory paragraph in the conclusion section.
>> Small comments:
>> 2. plains --> planes (several times).
Corrected
>> 3. Section 2: Formula for e_{COM}\cdot\mathbf{z} is wrong (only so for \mu=M).
Corrected
>> 4. "m_{i-1}-m_i and m_i-m_{i+1} planes" need explanation.
Corrected
>> 5. I do not understand why giving an arctan-formula for the angle, instead of the (more >> standard) way of expressing the angle using arccos (which does not make the mass-
>> dependence more difficult).
The convention is consistent with the well accepted convention used in the
foundational McGuire article and the rest of our work in this topic.
>> 6. Page 3: Ref. [15] should be [16] I suppose.
Corrected