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Exactly solvable quantum fewbody systems associated with the symmetries of the threedimensional and fourdimensional 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/1608.04402v1 (pdf) 
Date submitted:  20160817 02:00 
Submitted by:  Olshanii, Maxim 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The purpose of this article is to demonstrate that noncrystallographic reflection groups can be used to build new solvable quantum particle systems. We explicitly construct a oneparametric family of solvable fourbody systems on a line, related to the symmetry of a regular icosahedron: in two distinct limiting cases the system is constrained to a halfline. We repeat the program for a 600cell, a fourdimensional generalization of the regular threedimensional icosahedron.
Ontology / Topics
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Submission & Refereeing History
Published as SciPost Phys. 1, 005 (2016)
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Reports on this Submission
Anonymous Report 1 on 2016913 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1608.04402v1, delivered 20160913, doi: 10.21468/SciPost.Report.15
Strengths
1 An new way of looking at Guadin's system associated with the finite Coxeter groups H_3, H_4 in the *impenetrable* regime, namely as fewbody particle systems with different masses.
Weaknesses
1 Only hardcore particles are considered. It is a natural question whether Gaudin's system [1] associated with H_3 and H_4 in the *interacting* regime can also be interpreted as a fewbody system.
Report
Gaudin observed in a landmark paper that the quantum boson system of $n$ particles with deltapotential interaction admits natural generalisation in terms of finite Coxeter groups [1]. Furthermore, the generalised systems from [1] also admits generalisations in terms of affine Weyl groups [1,2, 8,9,10,11,12,13].
For classical Coxeter groups these systems have natural particle interpretations. For example, the original nparticle model with pairwise interaction corresponds to the permutation group $S_n$ (on the line) or affine $S_n$ (on the circle). The other classical Coxeter groups appear when restricting the particles to a halfline or interval, or by distributing them symmetrically around the origin.
Recently the last two authors showed that the system associated with the affine Weyl group $\tilde F_4$ in the impenetrable regime (i.e. when the interaction goes to +infinity) can be mathematically identified with a system of four onedimensional hardcore particles with mass ratios 6, 2, 1, and 3 in a hardwall box.
The preprint under review is a continuation of the above idea, but now applied to the finite noncrystallographic Coxeter groups of type H_3 and H_4. The paper can be summarised as follows: In the impenetrable regime Gaudin's systems [1] associated with H_3 and H_4 can be mathematically identified with a system of four (for H_3) or five (for H_4) onedimensional hardcore particles on a line, if the ratios of the masses m_j/m_2 are certain explicit rational functions of a dimensionless parameter \xi. For very special values of $\xi$ the system reduces to a system of three (for H_3) or four (for H_4) onedimensional hardcore particles on a halfline.
Requested changes
1 In the paper "Yang's system of particles and Hecke algebras", Heckman and Opdam relate the spectral problem for the system in [1] to the representation theory of the graded Hecke algebra of the corresponding Coxeter group. Furthermore, they also prove the crucial Plancherel formula (in the repulsive and attractive regime). This important paper should be cited.
2 Typo: Page 6, figure 3: morion > motion
3 Discuss possible extension to interacting case (sea Weakness)
Author: Maxim Olshanii on 20160913 [id 55]
(in reply to Report 1 on 20160913)First of all, we thank the referee for a truly thorough reading of our manuscript and for s/his suggestions. We are currently
preparing a new version to be posted at the arXiv
Changes:
(1) We will add the suggested reference, it is indeed very relevant;
(2) We will correct the typo;
(3) We will add a note on finite interaction extensions to the Outlook section.
Concerning (3), a comment is in order:
Prior to our $\tilde{F}_{4}$ paper, even among the four classic reflection groups, only two were producing \emph{local twobody} interaction
implementations, and only one of them unconditionally.
(*) $A_{N1}$($\tilde{A}_{N1}$) corresponds to the relative motion of $N$ $\delta$interacting particles on a line(ring), with any value of the
interaction strength allowed;
(*) $B_{N}=C_{N}$($\tilde{C}_{N}$) corresponds to $N$ $\delta$interacting particles on a halfline(box) bounded by one(two)
$\delta$barriers of an \emph{infinite} strength.
Already for $B_{N}=C_{N}$, one can see containing the requirement for having only local twobody interactions is. If one replaces an
infinite barrier by a finite one, integrability will still be preserved. However, in that case, one would need to allow some particles to venture
on the other side of the barrier. And this is where they will nonlocal (albeit still twobody) interactions: a particle a distance $l$ from the
barrier, will interact with another particle also at a distance $l$ from the barrier but \emph{on the other side} of it. Formally, these nonlocal
interactions are present even in the case of an infinite barrier: but particles can be contained indefinitely on one side of the barrier, and in this
case, the nonphysical interactions do not lead to any measurable consequences.
For other reflection groups, the situation is even more severe, and we currently believe that no finitestrengh
(at least for some pairs of particles) particle implementations of reflection groups, with local twobody interactions, exists,
besides the $A_{N1}$ and $C_{N}$. In the paper, we give a simple counting argument for why the $H_{3}$ group can not
produce a physically sound finitestrengthinteracting system.