Lattice Nonlinear Schrödinger Model: History, Applications and Open Problems

The lattice nonlinear Schrödinger model has applications in many fields. In this paper, we briefly review the basic concepts of the model. Special emphasis is given to the algebraic content of quantum integrable models. Quantum lattice nonlinear Schrödinger is equivalent to XXX spin chain with negative spin. It can be solved by the quantum inverse scattering method (algebraic Bethe ansatz). For the spin s = -1 case, XXX spin chain appears as an effective theory of Quantum Chromodynamics. The Yang-Yang equation describes quantum thermodynamics of lattice nonlinear Schrödinger model. We also provide the conformal field theory description and entanglement entropy time evolution of the model. Symmetry, correlations, and other applications are mentioned.