Topological Basis Realization Associated With Non-Hermitian Heisenberg XXZ Model

Temperley-Lieb(TL)algebra plays an important role in quantum integrable model and knot theory.As a tool for analyzing various related lattice models,it first appeared in statistical mechanics.In the subsequent development,it has important connections with topological quantum field theory,statistical physics,quantum teleportation,entanglement swapping and quantum computation.In statistical physics,such algebra has been used to study one-dimensional and two-dimensional statistical models.Heisenberg XXZ spin chain model is the simplest spin chain model.It has very important research value and has been widely used in simulating quantum computers.The high-dimensional Heisenberg XXZ model has attracted more and more attention.However,in the process of solving high-dimensional spin chains,the solution process of the high dimension of Hilbert space often complicated.With the introduction of the theory of topological basis,this problem has been solved very well.In fact,the properties of the total space of the system can be reflected by studying the properties of the topological subspace.It has been shown that a kind of spin chains is closely related to TL algebras.Topological base space is the action space of TL algebra and braid algebra.Based on this point of view,some scholars have constructed the spin realization of the topological bases of Hermitian Heisenberg XXX model and XXZ model.Based on the existing research,this paper mainly studies the realization of the non-Hermite Heisenberg XXZ model based on the matrix representation of TL algebra and Birman-Murakami-Wenzl(BMW)algebra.Particularly,the four-qubit Heisenberg XXZ model is investigated in detail.The results show that when the parameter is complex and || = 1,the corresponding model is a non-Hermitian Heisenberg XXZ model.The non-Hermitian Heisenberg XXZ model has not only quantum group symmetry but also PT symmetry.Its topological space is composed of topological bases related to TL algebra and BMW algebra.When the system consisted of spin-(1?2)quasiparticles,the orthogonal normalized topological basis is constructed by the generators of TL algebra.When the system consisted of spin-1 quasiparticles,the BMW algebraic generators are used.Then the properties of the topological basis are studied.The results show that the topological subspace is equivalent to the q-deformed spin single subspace and the ground state for the antiferromagnetic Heisenberg XXZ spin chain is one of the topological basis state.On this basis,the TL algebra,BMW algebra and system Hamiltonian are reduced by using the constructed topological basis.