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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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Authors (as registered SciPost users): Maxim Olshanii
Submission information
Preprint Link:  (pdf)
Date submitted: 2016-08-17 02:00
Submitted by: Olshanii, Maxim
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Quantum Physics
Approach: Theoretical


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.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 1 on 2016-9-13 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1608.04402v1, delivered 2016-09-13, doi: 10.21468/SciPost.Report.15


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 few-body particle systems with different masses.


1- Only hard-core 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 few-body system.


Gaudin observed in a landmark paper that the quantum boson system of $n$ particles with delta-potential 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 n-particle 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 half-line 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 one-dimensional hard-core particles with mass ratios 6, 2, 1, and 3 in a hard-wall box.

The preprint under review is a continuation of the above idea, but now applied to the finite non-crystallographic 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) one-dimensional hard-core 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) one-dimensional hard-core particles on a half-line.

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)

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

Author:  Maxim Olshanii  on 2016-09-13  [id 55]

(in reply to Report 1 on 2016-09-13)
answer to question

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

(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 two-body} interaction
implementations, and only one of them unconditionally.

(*) $A_{N-1}$($\tilde{A}_{N-1}$) 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 half-line(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 two-body 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 two-body) 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 non-physical interactions do not lead to any measurable consequences.

For other reflection groups, the situation is even more severe, and we currently believe that no finite-strengh
(at least for some pairs of particles) particle implementations of reflection groups, with local two-body interactions, exists,
besides the $A_{N-1}$ and $C_{N}$. In the paper, we give a simple counting argument for why the $H_{3}$ group can not
produce a physically sound finite-strength-interacting system.

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