The Jordan-Wigner Transformation And Its Applications

In the thesis we firstly introduce one-dimensional spin-1/2 Jordan-Wigner transformation and the Holstein-Primakoff transformation that are used widely in the quantum many-body theory. Secondly, another form of one-dimensional spin-1/2 Jordan-Wigner transformation is discussed. Then, one-dimensional spin-3/2 Jordan-Wigner transformation is introduced. At last, some models in which one-dimensional spin-1/2 Jordan-Wigner transformation has been applied are presented.Chapter!: Firstfy,one-dimensional spin-1/2 Jordan-Wigner transformation is introduced detailedly, which maps spins onto spinfess fermions. Next,we discuss the statistical relations of spin operators by means of this mapping. Moreover, from these relations we can notice that the S=l/2 spin operators partly resemble Fermi operators, and as well as, they partly resemble Bose operators. Finally, the Holstein-Primakoff transformation is mentioned simply.Chapter2: An important tool in studying one-dimensional spin lattice models is the Kramers-Wannier duality transformation. Based upon this transformation, we introduce another form of one-dimensional spin-1/2 Jordan-Wigner transformation that maps the PauH spin operators onto the real Majorana lattice fermions. Next, its application in one-dimensional quantum Ising spin chain model is displayed. We also discuss the statistical relations of the Majorana lattice fermions and the Majorana Fermi fields respectively.Chapter3: New Jordan-Wigner-rype transformation between spin-3/2 and fermions of two types is shown. By using the double-fennion representation, we discuss the statistical relations of the S=3/2 spin operator. Then, we introduce the double-fermion representation of one-dimensional isotropic XY spin-3/2 chain model.Chapter4: The applications that one-dimensional spin-1/2 Jordan-Wigner tnmsfermation is used for some spin chain models are displayed.