Yangian Algebra Realization Of Spin-1 And Its Application In Spin Chain Model

The Yang-Baxter equation proposed by C.N.Yang and R.J.Baxter in the 1960s and 1970s plays a crucial role in the in-depth study of quantum integrable model problems.RTT relation is one of the most critical basic relations in the study of the relevant theory of the Yang-Baxter equation.It not only specifies the commutative relation of operators,but also provides a set of conserved quantities,and on this basis judges the quantum integrability of the relevant model.Therefore,RTT relation is at the core of quantum scattering theory.In 1985,the Soviet mathematician Delinfeld proposed the theory of quantum groups with the help of RTT relation.Quantum groups include quantum algebra and Yangian,which are important developments in the theory of YangBaxter equation and provide a powerful mathematical tool for the study of quantum integrable models.Yangian algebras contain very rich physical connotations.After decades of development,people have made considerable progress in various physical implementations of Yangian algebras and the study of fully integrable quantum models.Yangian algebras have two values.One is that Yangian describes greater symmetry than Lie algebras,and the other is that Yangian can achieve transitions of quantum states beyond Lie algebras.Yangian has conducted in-depth and systematic research on the implementation of spin 1/2,while the high spin implementation of Yangian operators and their role in the high spin chain model need to be further studied.The case of high spin is different from the case of spin 1/2.When the spin is 1/2,the Pauli matrix constitutes a concrete representation of Lie algebra,and the Pauli matrix and identity matrix constitute the complete basis vector of the spin 1/2 operator space.Therefore,the Yangian algebra generator can be represented using a local spin operator.When the spin is 1,the local spin operator is no longer complete,and it is not possible to use the spin operator to represent the generators of Yangian algebras,which poses difficulties in studying the implementation of high spin Yangian algebras.Through this research,we hope to find suitable research methods for the construction of high spin Yangian algebras,and study their transitions in the Eigenstates of the Heisenberg XXX model with the help of high spin Yangian implementations.The model studied in this thesis is the Heisenberg XXX model for spin 1.We extend it to the case of spin 1 based on the known Heisenberg spin chain model for spin 1/2.Firstly,the Heisenberg XXX spin chain model of spin 1 is constructed from the RTT relation,and the spin representation of the Hamiltonian of the Heisenberg XXX model of spin 1 is obtained.Then,because the commutation relation between the T(1)and T(2)operators in the global transfer matrix expansion is the basis for the representation of higher order commutation relation,the so(3)algebra is not complete for the case of higher order T(n)(n≥2)with spin 1.Therefore,in order to represent the higher order T(n).we need to use the Gell-Mann matrix to represent the component operator of T(n),and then obtain a Yangian algebra implementation with spin 1.Based on the expansion of the RTT relation,the commutative relation between the component operators of T(1)and T(n).the commutative relation between the component operators of T(2)itself,and the representation of the component operators of T(3)are obtained,and the results are compared.Finally,we use Yangian’s spin representation to construct the transition operator for the XXX spin chain model.