Exact Results Of A Class Of Quantum Ising Model With Single-ion Anisotropy

A class of quantum Ising model with single-ion anisotropies is solved exactly, including spin-1quantum Ising model with single-ion anisotropy and mixed spin-1/2and spin-1quantum Ising model with both longitudinal and transverse single-ion anisotropies, the bond can be uniformly or alternatingly changed. This model belongs to few rigorously solved models in condensed physics, shows abundant physics picture and quantum behavior of low dimensional quantum spin systems, provides theoretical evidence for synthesizing and researching on the mixed spin magnetism materials, and offer new thinking to searching exact results for other mixed spin model or high dimensional quantum spin models. The main researches are as follows:1. Exact solutions of one dimensional spin-1quantum Ising modelIn this paper we study the spin-1quantum Ising chain with single-ion anisotropies, based on the generally properties of this model, combining analytic methods such as Jordan-Wigner transformation, the spin-1Ising uniform chain and dimerized chain can be rigorously solved. The ground state energy, excitation energy spectra can be obtained under these two cases. By using the partition function, the thermodynamic quantities such as the internal energy, the entropy, the specific heat and so on can be exactly obtained by numerical calculation. The ground state phase diagram is also given. The results show that the system exhibits a series of quantum phase transitions depending on the dimerization strength of the crystal fields, while the quantum critical points are determined exactly.2. Exact results of a mixed spin-1/2and spin-1quantum Ising uniform chain with both longitudinal and transverse single-ion anisotropiesThere is a conservation quantity hidden in this model that is hole number, here, the Hamiltonian acting on a state with some m=0sites, we refer to these as "holes". The hole number is a conserved quantity which is independent of the parameters in the model. The general framework to solve the model is by dividing the total Hilbert space of the original model into a number of subspaces labeled by the number of holes. Though solving each subsystem, properties of the whole system can be determined by that of subsystems.As for the ground state always lie in the none hole subspace, the mixed spin-1/2and spin-1Ising uniform chain with both longitudinal and transverse single-ion anisotropies is solved exactly by means of a mapping to the spin-1/2Ising chain with the alternating transverse fields and the Jordan-Wigner transformation. The analytic expressions of the ground state energy, the quasi-particles’spectra, the minimal energy gap, the transverse magnetization, the static transverse susceptibility, the nearest neighboring longitude spin-spin-correlation function, and the phase diagram of the ground state are also given. The results show that when longitudinal single-ion anisotropy is positive and for any finite value of transverse single-ion anisotropy, there is no quantum critical point and the ground state is always lie in a spin ordered phase disregard of the boundary condition in the present system.3. Exact results of an alternating-bond mixed spin-1/2and spin-1quantum Ising chain with both longitudinal and transverse single-ion anisotropiesUsing the same method as in the uniform bond case, we can obtain the ground state properties of this model. The most striking finding to emerge here is that the alternating bond just quantitatively changes the ground state properties, and the system also has no quantum disordered phase when longitudinal single-ion anisotropy is positive and no mater the exchange interaction is alternating changed or uniform. Unlike a dimerized spin-1Ising chain or alternating spin-1/2Ising chain, in which the regularly alternating bonds and fields will lead to the appearance of additional quantum phase transitions.4. Thermodynamic properties of the mixed spin-1/2and spin-1quantum Ising chain with both longitudinal and transverse single-ion anisotropiesWe study a recursion formula derived for the partition function based on the ground state properties have known. The thermodynamic quantities such as the free energy, the internal energy, the entropy, and the specific heat are calculated from the partition function, which can be evaluated recursively from an original no-hole system. Based on the evidence that the specific heat shows a sharp peak in the low-temperature region where the hole excitations proliferate, we conclude that the hole excitations can enhance the thermodynamic fluctuations.To our knowledge the mixed spin-1/2and spin-1quantum Ising chain with both longitudinal and transverse single-ion anisotropies is solved exactly for the first time. As for the hole number is a good quantum number, this model is solved exactly by means of a mapping to the spin-1/2Ising chain with transverse fields and the Jordan-Wigner transformation. The ground state properties such as ground state energy, excitation energy spectra, phase diagram of the ground state and so on can be obtained. Based on the ground state properties, by utilizing a recursion formula derived for the partition function, the thermodynamic properties such as the free energy, the internal energy, the entropy and the specific heat also can be exactly obtained.