Thermodynamics Of One-Dimensional Anisotropic Heisenberg Spin Chain Models

U(1)symmetry is one of the most important symmetries in physics,which corresponds to the law of conservation of the number of particles.In physics,there are many exact solvable models without U(1)symmetry,such as XYZ spin chain with odd lattice points,anti-periodic anisotropic spin chain and quantum spin chain with off-diagonal boundary field.The exact solution of these models plays a decisive role in understanding the system without U(1)sym-metry.In recent years,the off-diagonal Bethe Ansatz method was proposed,which can be used to solve the eigenvalue of a large class of integrable models without U(1)symmetry.Based on the exact solution,the thermodynamic limit and boundary energy of the models have attracted great attention.On the other hand,the solvable Buck-Sukumar(BS)model is widely used in quantum optics and is a basic model to describe the interaction between field and matter.It becomes an interesting research direction to generalize this model by q-boson theory.In this paper,we mainly study the anisotropic XXZ model with topological boundary conditions and q-deformed BS model.Since the anti-periodic XXZ spin chain model was solved in 2013,we obtain the thermo-dynamic limit,boundary energy and the first excitation energy of the anti-periodic XXZ model for the first time.In chapter 3,we introduce a method to systematically solve the thermody-namic limit of the model without U(1)symmetry.Firstly,we obtain the exact solution of the model through the off-diagonal Bethe Ansatz method,that is,the eigenvalue of the correspond-ing transfer matrix can be parameterized by the inhomogeneous T-Q relation.Secondly,the contribution of inhomogeneous term to ground state energy,momentum and higher conserved charge is studied by numerical simulation.When the contribution of the inhomogeneous term to the system tends to zero with the increase of the size,it indicates that the inhomogeneous term in the inhomogeneous T-Q relation can be erased and reduced into the usual T-Q relation.The reduced Bethe Ansatz equation can be obtained from the reduced T-Q relation,which can still be used to describe the anti-periodic XXZ model in the limit of thermodynamic-s.Finally,the thermodynamic limit of the model can be solved through the traditional method,and the boundary energy and the first excitation energy of the model can be analyzed.The method we use is also applicable to integrable models which are exactly solved by off-diagonal Bethe Ansatz method.The BS model is a basic model in quantum optics.We construct an exactly solvable q-deformed BS model,which is obtained by replacing the boson algebra in the BS model with the q-deformation algebra.The extended BS model needs to introduce deformation parame-ters q and Bargmann parameter s.By a q-deformed Holstein-Primakoff transformation,the q-deformed BS model can be described by the algebra su_q(1,1)⊕su(2).In order to study the physical significance of the q-deformed boson field,the atomic inversion and von Neumann entropy evolution over time are studied with the coherent state of q-deformed Glauber as the initial state.