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arXiv:1508.05954 · New Journal of Physics

An exactly solvable quantum four-body problem associated with the symmetries of an octacube

Maxim Olshanii, Steven Glenn Jackson

Published Oct 19, 2015• 30 citations
arXivHTMLPDF

Abstract

In this article, we show that eigenenergies and eigenstates of a system consisting of four one-dimensional hard-core particles with masses 6 m , 2 m , m , and 3 m in a hard-wall box can be found exactly using Bethe Ansatz. The Ansatz is based on the exceptional affine reflection group associated with the symmetries and tiling properties of an octacube —a Platonic solid unique to four-dimensions, with no three-dimensional analogues. We also uncover the Liouville integrability structure of our problem: the four integrals of motion in involution are identified as invariant polynomials of the finite reflection group F 4 , taken as functions of the components of momenta .

DOI
10.1088/1367-2630/17/10/105005

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In this article, we show that eigenenergies and eigenstates of a system consisting of four one-dimensional hard-core particles with masses 6 m , 2 m , m , and 3 m in a hard-wall box can be found exactly using Bethe Ansatz. The Ansatz is based on the exceptional affine reflection group associated with the symmetries and tiling properties of an octacube —a Platonic solid unique to four-dimensions, with no three-dimensional analogues. We also uncover the Liouville integrability structure of our problem: the four integrals of motion in involution are identified as invariant polynomials of the finite reflection group F 4 , taken as functions of the components of momenta .
We show that a system consisting of two interacting particles with mass ratio 3 or 1/3 in a hard-wall box can be exactly solved by using Bethe-type ansatz. The ansatz is based on a finite superposition of plane waves associated with a dihedral group D 6 , which enforces the momentums after a series of scattering and reflection processes to fulfill the D 6 symmetry. Starting from a two-body elastic collision model in a hard-wall box, we demonstrate how a finite momentum distribution is related to the D 2n symmetry for permitted mass ratios. For a quantum system with mass ratio 3, we obtain exact eigenenergies and eigenstates by solving Bethe-type-ansatz equations for arbitrary interaction strength. A many-body excited state of the system is found to be independent of the interaction strength, i.e., the wave function looks exactly the same for non-interacting two particles or in the hard-core limit.
The recently proposed exact quantum solution for two \delta δ -function-interacting particles with a mass-ratio 3\!:\!1 3 : 1 in a hard-wall box [Y. Liu, F. Qi, Y. Zhang and S. Chen, iScience 22, 181 (2019)] violates the conventional necessary condition for a Bethe Ansatz integrability, the condition being that the system must be reducible to a superposition of semi-transparent mirrors that is invariant under all the reflections it generates. In this article, we found a way to relax this condition: some of the semi-transparent mirrors of a known self-invariant mirror superposition can be replaced by the perfectly reflecting ones, thus breaking the self-invariance. The proposed name for the method is asymmetric Bethe Ansatz (asymmetric BA). As a worked example, we study in detail the bound states of the nominally non-integrable system comprised of a bosonic dimer in a \delta δ -well. Finally, we show that the exact solution of the Liu-Qi-Zhang-Chen problem is a particular instance of the the asymmetric BA.
Exactly solvable quantum systems represent a fascinating and critically important class of models in theoretical physics, offering exact solutions to non-trivial problems that often shed light on the behavior of more complex, intractable systems. The remarkable feature of these systems lies not only in their solvability but also in the underlying mathematical structures that facilitate such solutions. This paper explores the concept of a universal algebraic structure that underpins exactly solvable quantum systems. We delve into how diverse models, ranging from one-dimensional spin chains to many-body systems and field theories, exhibit common algebraic symmetries, such as Lie algebras, affine Lie algebras, and quantum groups. These algebraic frameworks provide a unified language for understanding integrability, the existence of an infinite number of conserved quantities, and the ability to construct exact eigenstates. We review the historical development of these ideas, from the Bethe ansatz to the development of the inverse scattering method and quantum groups. Our discussion highlights how these mathematical structures are not merely tools for calculation but fundamental descriptors of the physical reality of these systems, offering profound insights into non-perturbative quantum phenomena. The identification of a universal algebraic architecture suggests a deeper coherence in the landscape of quantum mechanics, paving the way for new discoveries and classifications of quantum models.
Quasi-exactly solvable (QES) deformations of harmonic and singular oscillators have been widely studied via a range of approaches. In this paper, we obtain families of new QES deformations of solvable quantum systems associated with exceptional orthogonal polynomials (EOPs). The construction builds upon the theory of Darboux-Crum transformations and the classification of exactly solvable quantum systems associated with EOPs of Hermite type. It is shown that the deformations break the exact solvability of the undeformed systems and introduce model parameters into the deformed systems that permit the existence of a finite number of polynomial solutions whose roots satisfy systems of algebraic equations. The new families presented and studied consist of deformations of quantum systems related to Hermite EOPs of type III with arbitrary codimensions. We present polynomial and rational deformations and analyze, in each case, the conditions for quasi-exact solvability in terms of Bethe ansatz equations and parameter constraints. In general, the structures of the underlying polynomial solutions are no longer associated with well-known classical orthogonal polynomials. So for the general cases, we mainly focus on presenting the new approach as well as the Bethe ansatz equations and the constraints for model parameters. As applications, we construct particular families of QES deformations related to solvable models allowing one, two, and up to four gaps, and obtain the closed form expressions for their wavefunctions and spectra. We analyze the existence of QES solutions in the spaces of model parameters, providing information on the number of solutions for given parameters.
A quantum N-body problem with 2-component in (2 + 1) dimensions deduced from an integrable model in (2 + 1) dimensions is investigated. The Davey–Stewartson 1 (DS1) system9 is an integrable model in two dimensions. A quantum DS1 system with 2 color components in two dimensions has been formulated. This two-dimensional problem has been reduced to two one-dimensional many-body problems with 2 color components. The solutions of the two-dimensional problem under consideration has been constructed from the resulting problems in one dimension. For the latter with δ-function interactions and being solved by the Bethe–Yang ansatz, we introduce symmetrical and anti-symmetrical Young operators of the permutation group and obtain the exact solutions for the quantum DS1 system. The application of the solutions is discussed.
We consider the eigenspectrum solution of a many-body problem of interacting spin- s particles that can be solvable within the generalized Bethe ansatz method. We assume that the interactions are encoded in terms of an arbitrary U (1) invariant factorizable S -matrix. The exact solution of the spin part is based on a unified formulation of the quantum inverse scattering method for an arbitrary (2 s + 1)-dimensional monodromy matrix. The respective eigenstates are shown to be given in terms of 2 s creation fields by a general new recurrence relation. This allows us to derive the spectrum and the respective Bethe ansatz equations.