Quantum Phase Transition in a Transverse Ising Chain with Regularly Varying Parameters * the One-dimensional Spin

Using rigorous analytical analysis and exact numerical data for the spin-1 2 transverse Ising chain we discuss the effects of regular alternation of the Hamiltonian parameters on the quantum phase transition inherent in the model.1 2 Ising model in a transverse field (the transverse Ising chain) defined by the Hamiltonian H = n 2Is x n s x n + n Ωs z n (1) is known to be the simplest system exhibiting a quantum (zero-temperature) phase transition driven by the transverse field [1]. Most of the performed studies for this model use the exact eigenvalues and eigenfunctions of its Hamiltonian (1) that makes the problem amenable for rigorous analysis [1, 2]. It is generally known that the critical value of the transverse field is Ω c = |I| (and Ω c = −|I|). The longitudinal (Ising) magnetization per site m x = 1 N n s x n is the order parameter of the system. |m x | varies from 1 2 (for Ω = 0) to 0 (for Ω ≥ Ω c) according to |m x | = 1 2 1 − Ω Ωc 2 1 8. The quantum phase transition at Ω c is equivalent to the thermal phase transition of the square-lattice Ising model. After the understanding of the properties of the basic model was achieved the models with various modifications were introduced and the effects of introduced changes on the quantum phase transition were discussed. Among numerous works in this field one may mention