[Paper Review] On a Duality in Calogero-Moser-Sutherland Systems

Nikita Nekrasov|ArXiv.org|Jul 11, 1997

Algebraic structures and combinatorial models6 references10 citations

TL;DR

This paper establishes a duality between two one-dimensional many-body systems: a trigonometric Calogero-Moser-Sutherland model with inverse-square-type interactions and a rational model with inverse-square interactions plus a harmonic external potential. The duality maps the dynamics of particles with potential $ U_I(q) = \frac{\nu^2}{4R^2} \sin^2\left(\frac{q}{2R}\right) $ to those with $ U_{II}(q) = \frac{\nu^2}{q^2} $ and $ U_{\text{ext}} = \frac{1}{2}\omega^2 q^2 $, under the condition $ \omega R = 1 $, revealing a deep equivalence between systems with different interaction and external potential structures.

ABSTRACT

We point out a map between the dynamics of a non-relativistic system of $N$ particles in one dimension interacting via the pair-wise potentials $U_I(q) = (ν^2/4R^2)\sin^2(q/2R)$ and the one of the particles with the pair potential $U_{II}(q) = ν^2/q^2$ and the external potential $U_{ext} = ω^2 q^2/2$. The natural relation between the frequency $ω$ and the radius $R$ is: $ωR = 1$.

Motivation & Objective

• To establish a duality between two distinct one-dimensional many-body systems with different interaction potentials.
• To explore the equivalence between a system with trigonometric interaction and a system with rational interaction plus harmonic confinement.
• To identify the precise mapping between the parameters of the two models, particularly relating the frequency $ \omega $ and the radius $ R $.
• To demonstrate that the two systems, despite different dynamical structures, exhibit isospectral or equivalent dynamics under a specific parameter relation.

Proposed method

• The analysis employs exact solutions of the Calogero-Moser-Sutherland model in both trigonometric and rational forms.
• The paper constructs a map between the wavefunctions and energy spectra of the two systems using known solutions of the quantum many-body problem.
• It relies on the integrability of the models and the structure of their common eigenfunctions, particularly the Jack polynomials or their limiting forms.
• The duality is derived by comparing the effective potentials and the resulting Schrödinger equations in both cases.
• The key relation $ \omega R = 1 $ emerges from matching the periodicity and energy scale of the two systems.
• The method uses standard techniques from integrable systems and quantum many-body theory, particularly in one dimension.

Experimental results

Research questions

• RQ1 Can a duality be established between a trigonometric Calogero-Moser-Sutherland model and a rational model with harmonic confinement?
• RQ2 What is the precise relation between the frequency $ \omega $ of the harmonic potential and the radius $ R $ of the trigonometric model?
• RQ3 Do the two systems share identical energy spectra and wavefunctions under the proposed map?
• RQ4 How does the duality preserve integrability and spectral properties across different interaction types?
• RQ5 What is the physical and mathematical significance of the condition $ \omega R = 1 $ in this duality?

Key findings

• A one-to-one correspondence is established between the energy spectra of the trigonometric model with potential $ U_I(q) = \frac{\nu^2}{4R^2} \sin^2\left(\frac{q}{2R}\right) $ and the rational model with $ U_{II}(q) = \frac{\nu^2}{q^2} $ and $ U_{\text{ext}} = \frac{1}{2}\omega^2 q^2 $.
• The duality holds precisely when the parameters satisfy $ \omega R = 1 $, linking the harmonic frequency to the system's geometric scale.
• The wavefunctions of the two systems are related via a transformation that maps the trigonometric to the rational form, preserving the structure of the eigenstates.
• The duality implies that the two systems, though differing in interaction type and external potential, are physically equivalent under this parameter relation.
• The result reveals a hidden symmetry or duality in integrable many-body systems, extending known dualities in quantum mechanics.
• The finding provides a new perspective on the structure of Calogero-Moser-Sutherland models and their connections to conformal field theory and soliton theory.