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arXiv:nlin/0604073 · Journal of Physics A Mathematical and Theoretical

Exchange operator formalism forN-body spin models with near-neighbours interactions

Alberto Enciso B, Federico Finkel, Artemio González-López, Miguel A. Rodrı́guez

Published Feb 6, 2007• 8 citations
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Abstract

We present a detailed analysis of the spin models with near-neighbours interactions constructed in our previous paper (Enciso et al 2005 Phys. Lett. B 605 214) by a suitable generalization of the exchange operator formalism. We provide a complete description of a certain flag of finite-dimensional spaces of spin functions preserved by the Hamiltonian of each model. By explicitly diagonalizing the Hamiltonian in the latter spaces, we compute several infinite families of eigenfunctions of the above models in closed form in terms of generalized Laguerre and Jacobi polynomials.

DOI
10.1088/1751-8113/40/8/012

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We present a detailed analysis of the spin models with near-neighbours interactions constructed in our previous paper (Enciso et al 2005 Phys. Lett. B 605 214) by a suitable generalization of the exchange operator formalism. We provide a complete description of a certain flag of finite-dimensional spaces of spin functions preserved by the Hamiltonian of each model. By explicitly diagonalizing the Hamiltonian in the latter spaces, we compute several infinite families of eigenfunctions of the above models in closed form in terms of generalized Laguerre and Jacobi polynomials.
Several generalizations of one-dimensional and two-dimensional exactly solvable low-dimensional quantum spin-1/2 models, some of which contain off-diagonal terms of the exchange tensor, are proposed. It is shown that off-diagonal terms of the exchange (symmetric or nonsymmetric with respect to the exchange of spins) yield renormalization of the diagonal ones and phase shifts. The latter can be moved to eigenfunctions of the models and do not affect eigenvalues for open geometries of the lattices. For closed geometries of the lattices the phase shifts manifest themselves in the finite size corrections. The advantage of the proposed models is in their simplicity and in possible realizations of the models in a number of applications, e.g., for the description of the wide variety of correlated electron systems, ultracold atoms, and in the theory of topological quantum computation.
In this short review paper the detailed analysis of six two-dimensional quantum {\it superintegrable} systems in flat space is presented. It includes the Smorodinsky-Winternitz potentials I-II (the Holt potential), the Fokas-Lagerstrom model, the 3-body Calogero and Wolfes (equivalently, G2G_2G2​ rational, or I6I_6I6​) models, and the Tremblay-Turbiner-Winternitz (TTW) system with integer index kkk. It is shown that all of them are exactly-solvable, thus, confirming the Montreal conjecture (2001); they admit algebraic forms for the Hamiltonian and both integrals (all three can be written as differential operators with polynomial coefficients without a constant term), they have polynomial eigenfunctions with the invariants of the discrete symmetry group of invariance taken as variables, they have hidden (Lie) algebraic structure g(k)g^{(k)}g(k) with various kkk, and they possess a (finite order) polynomial algebras of integrals. Each model is characterized by infinitely-many finite-dimensional invariant subspaces, which form the infinite flag. Each subspace coincides with the finite-dimensional representation space of the algebra g(k)g^{(k)}g(k) for a certain kkk. In all presented cases the algebra of integrals is a 4-generated (H,I1,I2,I12≡[I1,I2])(H, I_1, I_2, I_{12}\equiv[I_1, I_2])(H,I1​,I2​,I12​≡[I1​,I2​]) infinite-dimensional algebra of ordered monomials of degrees 2,3,4,5, which is a subalgebra of the universal enveloping algebra of the hidden algebra.
Exchange operator formalism describes many-body integrable systems using phase-space variables involving an exchange operator that acts on any pair of particles. We establish an equivalence between models described by exchange operator formalism and the complete infinite family of parent Hamiltonians describing quantum many-body models with ground states of Jastrow form. This makes it possible to identify the invariants of motion for any model in the family and establish its integrability, even in the presence of an external potential. Using this construction we establish the integrability of the long-range Lieb-Liniger model, describing bosons in a harmonic trap and subject to contact and Coulomb interactions in one dimension. We further identify a variety of models exemplifying the integrability of Hamiltonians in this family.