Extension of the class of integrable dynamical systems connected with semisimple Lie algebras

Inozemtsev, V. I.; Meshcheryakov, D. V.

doi:10.1007/bf00398546

Cited by 42 publications

(38 citation statements)

References 12 publications

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“…where This model is a complexified version of the BC-type Inozemtsev model [27] and belongs to the family of generalizations of the Calogero model. It is integrable on the classical level for time-independent parameters.…”

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confidence: 99%

Non-equilibrium dynamics of Gaudin models

Barmettler¹,

Fioretto²,

Gritsev³

2013

EPL

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-In classical mechanics the theory of non-linear dynamics provides a detailed framework for the distinction between near-integrable and chaotic systems. Quite in opposition, in quantum many-body theory no generic microscopic principle at the origin of complex dynamics is known. Here we show that the non-equilibrium dynamics of homogeneous Gaudin models can be fully described by underlying classical Hamiltonian equations of motion. The original Gaudin system remains fully quantum and thus cannot exhibit chaos, but the underlying classical description can be analyzed using the powerful tools of the classical theory of motion. We specifically apply this strategy to the Tavis-Cummings model for quantum photons interacting with an ensemble of two-level systems. We show that scattering in the classical phase space can drive the quantum model close to thermal equilibrium. Interestingly, this happens in the fully quantum regime, where physical observables do not show any dynamic chaotic behavior.Many aspects of the transition from regular dynamics of an integrable system to erratic behavior of a complex system are understood in classical mechanics. On the one hand, there is the Kolmogorov-Arnold-Moser (KAM) theorem [1], that proves the stability of weakly perturbed integrable systems. On the other hand, a variety of mechanisms leading to chaos and eventually to the ergodic exploration of phase space have been found (see, e.g., [2]). For quantum systems, the main paradigms for the description of quantum chaotic phenomena are quasiclassical [3] and random matrix [3,4] theories. Moreover, in a number of specific model studies thermalization processes have been observed (e.g., [5]). However, there remains an important conceptual gap between regular and complex behaviors. In this letter we investigate a non-trivial integrable quantum system without going to the quasiclassical limit and gain microscopic insight into the emergence of irregularity when breaking integrability by driving an internal parameter.A good starting point to approach regular dynamics of non-trivial quantum systems are Bethe ansatz (BA) integrable models, which possess a complete set of integrals of motion. The exact solutions of time-independent BA many-body solvable systems played a crucial role in the understanding of various fundamental phenomena and concepts in physics. Famous examples are the solutions for the Ising model, the Heisenberg spin chain, the onedimensional Hubbard model or the Lieb-Liniger gas [6,7]. Also certain aspects of quantum chromodynamics can be described by the integrable quantum spin chain with complex spin [8]. However, the non-equilibrium dynamics of these models are rich [9][10][11][12][13][14][15][16][17][18] and much more difficult to be calculated within the BA than the static properties. Formulating a theory of integrability breaking for timedependent problems is thus not only a conceptual, but also a technical challenge.In this letter we restrict ourself to a certain class of Gaudin-type models for which we can...

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confidence: 99%

Non-equilibrium dynamics of Gaudin models

Barmettler¹,

Fioretto²,

Gritsev³

2013

EPL

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“…This is the rational A type Inozemtsev model [51,52,53]. Inozemtsev models are known as a family of deformed CS models which preserve the classical integrability.…”

Section: A Case Imentioning

confidence: 99%

Corrigendum to “sl(M+1) construction of quasi-solvable quantum M-body systems” [Ann. Phys. 309 (2004) 239–280]

Tanaka¹

2005

Annals of Physics

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We propose a systematic method to construct quasi-solvable quantum many-body systems having permutation symmetry. By the introduction of elementary symmetric polynomials and suitable choice of a solvable sector, the algebraic structure of sl(M +1) naturally emerges. The procedure to solve the canonical-form condition for the two-body problem is presented in detail. It is shown that the resulting two-body quasi-solvable model can be uniquely generalized to the M -body system for arbitrary M under the consideration of the GL(2, K) symmetry. An intimate relation between quantum solvability and supersymmetry is found. With the aid of the GL(2, K) symmetry, we classify the obtained quasi-solvable quantum many-body systems. It turns out that there are essentially five inequivalent models of Inozemtsev type. Furthermore, we discuss the possibility of including M -body (M ≥ 3) interaction terms without destroying the quasi-solvability.

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“…After completion of the present work, we came across [23] which discusses the integrability of spin BC r model with harmonic confining potential, or "spin Inozemtsev model" [4] in terms of the Dunkl operator formalism [20].…”

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confidence: 99%

Universal Lax pairs for spin Calogero–Moser models and spin exchange models

Inozemtsev¹,

Sasaki²

2001

J. Phys. A: Math. Gen.

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For any root system ∆ and an irreducible representation R of the reflection (Weyl) group G ∆ generated by ∆, a spin Calogero-Moser model can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member µ of R, to be called a "site", we associate a vector space V µ whose element is called a "spin". Its dynamical variables are the canonical coordinates {q j , p j } of a particle in R r , (r = rank of ∆), and spin exchange operators {P ρ } (ρ ∈ ∆) which exchange the spins at the sites µ and s ρ (µ). Here s ρ is the reflection generated by ρ. For each ∆ and R a spin exchange model can be defined. The Hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only. It is obtained by "freezing" the canonical variables at the equilibrium point of the corresponding classical CalogeroMoser model. For ∆ = A r and R = vector representation it reduces to the well-known Haldane-Shastry model. Universal Lax pair operators for both spin Calogero-Moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for degenerate potentials.

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Extension of the class of integrable dynamical systems connected with semisimple Lie algebras

Cited by 42 publications

References 12 publications

Non-equilibrium dynamics of Gaudin models

Non-equilibrium dynamics of Gaudin models

Corrigendum to “sl(M+1) construction of quasi-solvable quantum M-body systems” [Ann. Phys. 309 (2004) 239–280]

Universal Lax pairs for spin Calogero–Moser models and spin exchange models

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