On quantum integrable models related to nonlinear quantum optics. An algebraic Bethe ansatz approach

1989 ◽  
Vol 30 (8) ◽  
pp. 1739-1743 ◽  
Author(s):  
Branislav Jurčo
Keyword(s):  
Quantum Optics ◽  
Bethe Ansatz ◽  
Integrable Models ◽  
Algebraic Bethe Ansatz ◽  
Nonlinear Quantum ◽  
Quantum Integrable Models ◽  
Ansatz Approach

scholarly journals Introduction to the nested algebraic Bethe ansatz

2020 ◽  
Author(s):  
Nikita Slavnov
Keyword(s):  
Bethe Ansatz ◽  
Trace Formula ◽  
Integrable Models ◽  
Explicit Formulas ◽  
Algebraic Bethe Ansatz

We give a detailed description of the nested algebraic Bethe ansatz. We consider integrable models with a \mathfrak{gl}_3𝔤𝔩3-invariant RR-matrix as the basic example, however, we also describe possible generalizations. We give recursions and explicit formulas for the Bethe vectors. We also give a representation for the Bethe vectors in the form of a trace formula.

Quantum Tavis–Cummings Model

2019 ◽  
pp. 474-488
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Bethe Ansatz ◽  
Transfer Matrix ◽  
Strong Interaction ◽  
Building Blocks ◽  
Vertex Model ◽  
Algebraic Bethe Ansatz ◽  
Quantum Matter ◽  
Ansatz Approach

This chapter extends the algebraic Bethe ansatz to the quantum Tavis–Cummings model, an N atom generalization of the Jaynes–Cummings model to describe the strong interaction between light and quantum matter. In the case of the quantum Tavis–Cum- mings model there is no underlying vertex model to suggest the constituent building blocks of the algebraic Bethe ansatz approach, e.g.like the L-matrix or ultimately the transfer matrix. The algebraic Bethe ansatz is then first applied to the Tavis–Cummings Hamiltonian with an added Stark term using a conjecture for the transfer matrix. The original Tavis–Cummings model and its algebraic Bethe ansatz are obtained in the limit of vanishing Stark term, which requires considerable care.

scholarly journals QUASI-EXACTLY SOLVABLE DEFORMATIONS OF GAUDIN MODELS AND "QUASI-GAUDIN ALGEBRAS"

1998 ◽  
Vol 13 (04) ◽  
pp. 281-292 ◽  
Author(s):  
A. G. USHVERIDZE
Keyword(s):  
Bethe Ansatz ◽  
Complete Integrability ◽  
Integrable Models ◽  
Exact Solvability ◽  
New Class ◽  
Algebraic Bethe Ansatz ◽  
Exactly Solvable ◽  
Algebraic Deformation ◽  
Bethe Ansatz Solution ◽  
Gaudin Models

A new class of completely integrable models is constructed. These models are deformations of the famous integrable and exactly solvable Gaudin models. In contrast with the latter, they are quasi-exactly solvable, i.e. admit the algebraic Bethe ansatz solution only for some limited parts of the spectrum. An underlying algebra responsible for both the phenomena of complete integrability and quasi-exact solvability is constructed. We call it "quasi-Gaudin algebra" and demonstrate that it is a special non-Lie-algebraic deformation of the ordinary Gaudin algebra.

scholarly journals Nested Algebraic Bethe Ansatz in integrable models: recent results

2018 ◽  
Author(s):  
Stanislav Pakuliak ◽  
Eric Ragoucy ◽  
Nikita Slavnov
Keyword(s):  
Bethe Ansatz ◽  
General Expression ◽  
Monodromy Matrix ◽  
Form Factors ◽  
Integrable Models ◽  
Quantum Deformation ◽  
Algebraic Bethe Ansatz ◽  
Composite Models ◽  
Scalar Products

We review the recent results we have obtained in the framework of algebraic Bethe ansatz based on algebras and superalgebras of rank greater than 1 or on their quantum deformation. We present different expressions (explicit, recursive or using the current realization of the algebra) for the Bethe vectors. Then, we provide a general expression (as sum over partitions) for their scalar products. For some particular cases (in the case of gl(3)gl(3) or its quantum deformation, or of gl(2|1)gl(2|1)), we provide determinant expressions for the scalar products. We also compute the form factors of the monodromy matrix entries, and give some general methods to relate them. A coproduct formula for Bethe vectors allows to get the form factors of composite models.

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