Phase Transition Of One Dimension Spins Interacting With Phonons Of Spin-lattice System

For systems with short-range interactions the various physical ensemble such as the canonical ensemble and the microcanonical one are equivalent in the infinite volume limit. The description of phase transitions in the canonical ensemble and the microcanonical formalism has gained growing interest in recent years. And people found that for systems with long-range interactions the various physical ensemble such as the canonical ensemble and the microcanonical one are unequivalent.The main purpose of this paper is to demonstrate the difference of the microcanonical and the canonical ensemble in the infinite system for one-dimension Ising model when an infinite range interaction in the form of dimerizing lattice distortion is incluced.First, we introduce the study about phase transitions of an Ising model with both short-range and long-range interactions without external electric field as an example. The model gives the different phase transitions temperature T0due to different nearest neighbor coupling K in different statistical ensembles. It is demonstrated that the model system will not have any phase transitions when the different nearest neighbor coupling K<-0.5. We will compare the K-T phase diagrams of the microcanonical ensemble with the one of the canonical ensemble.In the canonical ensemble the large K transition is continuous down to the tricritical point CTP where it becomes first order.In the microcanonical ensemble the continuous transition coincides with the canonical one at large K. It persists at lower K down to the tricritical point MTP where it turns first order with a branching of the transition line. The region between these two lines is not accessible.Second, we will study the main work of this text about one-dimension Ising model with lattice distortion interaction based on the example model. The lattice distortion is the order parameter and there is phase transition where the uniform chain is the high-temperature phase, the distorted chain is the low-temperature phase chain. We will compare the phase transitions in different statistical ensembles,In the canonical ensemble,we study the phase transition by sovling derivatives of the free energy per spin and we find there is no phase transition when ω>0.25(ω is the frequency of the dimerizing distortion). Tere is a second-order continuous phase transition when0<ω<0.219, up to the tricritical point CTP (ω=0.219, T=1.519) and it becomes first order when0.219<ω<0.25.In the microcanonical,we study the phase transition by sovling derivatives of the entropy per spin and we find there is no phase transition when ω>0.25(ω is the frequency of the dimerizing distortion). Tere is a second-order continuous phase transition when0<ω<0.222, up to the tricritical point MTP (ω=0.222, T=1.439) and it becomes first order when0.222<ω<0.25.It is different in the first-order phase transition but equivalent in the second-order phase transition of the diagram in various physical ensemble, also the tricritical point does not coincide with each other.