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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

This is not the latest submitted version.

This Submission thread is now published as SciPost Phys. 1, 005 (2016)

Submission summary

As Contributors: Maxim Olshanii
Arxiv Link: http://arxiv.org/abs/1608.04402v2 (pdf)
Date submitted: 2016-09-27 02:00
Submitted by: Olshanii, Maxim
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • Quantum Physics
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.

Ontology / Topics

See full Ontology or Topics database.

Exactly solvable models Icosahedra Reflection groups
Current status:
Has been resubmitted


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.

Submission & Refereeing History

Published as SciPost Phys. 1, 005 (2016)

Resubmission 1608.04402v4 on 20 October 2016

Reports on this Submission

Anonymous Report 1 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.

  • validity: good
  • significance: good
  • originality: ok
  • clarity: ok
  • formatting: good
  • grammar: reasonable

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

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