Converting Lattices into Networks: The Heisenberg Model and Its Generalizations with Long-Range Interactions

Ternary generalization of Heisenberg's Algebra

A concise study of ternary and cubic algebras with $Z_3$ grading is presented. We discuss some underlying ideas leading to the conclusion that the discrete symmetry group of permutations of three objects, $S_3$, and its abelian subgroup $Z_3$ may play an important role in quantum physics. We show then how most of important algebras with $Z_2$ grading can be generalized with ternary composition laws combined with a $Z_3$ grading. We investigate in particular a ternary, $Z_3$-graded generalization of the Heisenberg algebra. It turns out that introducing a non-trivial cubic root of unity, $j = e^…

Integrable generalizations of Schrodinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces

A moving frame formulation of non-stretching geometric curve flows in Euclidean space is used to derive a 1+1 dimensional hierarchy of integrable SO(3)-invariant vector models containing the Heisenberg ferromagnetic spin model as well as a model given by a spin-vector version of the mKdV equation. These models describe a geometric realization of the NLS hierarchy of soliton equations whose bi-Hamiltonian structure is shown to be encoded in the Frenet equations of the moving frame. This derivation yields an explicit bi-Hamiltonian structure, recursion operator, and constants of motion for each …

Inhomogeneous Heisenberg Spin Chain and Quantum Vortex Filament as Non-Holonomically Deformed NLS Systems

Through the Hasimoto map, various dynamical systems can be mapped to different integrodifferential generalizations of Nonlinear Schrodinger (NLS) family of equations some of which are known to be integrable. Two such continuum limits, corresponding to the inhomogeneous XXX Heisenberg spin chain [Balakrishnan, J. Phys. C 15, L1305 (1982)] and that of a thin vortex filament moving in a superfluid with drag [Shivamoggi, Eur. Phys. J. B 86, 275 (2013) 86; Van Gorder, Phys. Rev. E 91, 053201 (2015)], are shown to be particular non-holonomic deformations (NHDs) of the standard NLS system involving g…

Generalized Theta Functions. I

Generalizations of classical theta functions are proposed that include any even number of analytic parameters for which conditions of quasi-periodicity are fulfilled and that are representations of extended Heisenberg group. Differential equations for generalized theta functions and finite non-unitary representations of extended Heisenberg group are presented so as other properties and possible applications are pointed out such as a projective embedding of tori by means of generalized theta functions.

基于 Heisenberg-Weyl 群的积分量子化

Integral quantization based on the Heisenberg-Weyl group

本文提出一种基于 Heisenberg-Weyl 群相干态的相对论性积分量子化框架，利用正定算符值测度处理 Minkowski 时空中的无自旋粒子运动，并证明其在一维谐振子量子化中能恢复标准 Hamiltonian。

On Soliton Interactions for a Hierarchy of Generalized Heisenberg Ferromagnetic Models on SU(3)/S(U(1) $\times$ U(2)) Symmetric Space

We consider an integrable hierarchy of nonlinear evolution equations (NLEE) related to linear bundle Lax operator L. The Lax representation is Z2 \times Z2 reduced and is naturally associated with the symmetric space SU(3)/S(U(1) \times U(2)). The simplest nontrivial equation in the hierarchy is a generalization of Heisenberg ferromagnetic model. We construct the N-soliton solutions for an arbitrary member of the hierarchy by using the Zakharov-Shabat dressing method with an appropriately chosen dressing factor. Two types of soliton solutions: quadruplet and doublet solitons are found. The one…