The Yang-lee Zero Theory In Non-equilibrium Phase Transitions And Quantum Phase Transitions In The Applications

The Yang-Lee theory of equilibrium phase transitions is generalized to study the non-equilibrium phase transitions of an urn model for the separation of sand and the quantum phase transitions of one-dimensional anisotropic XY quantum spin chain in a transverse field.Firstly, we briefly review the Yang-Lee theory which provides an important qualitative and quantitative tool in the study of critical phenomena. Yang and Lee proposed to investigate the distribution of partition function zeros in the complex plane, i.e., the field or fugacity were treated as complex variables. In the thermodynamic limit, the zeros accumulate nearly the transition point z₀ in the real axis and the density of zeros near z₀ determines the order of the phase transition. We also present the applicability of the theory to an Ising model and a lattice gas.Secondly, a second-order phase transition in a non-equilibrium urn model for the separation of sand is studied. Dynamical indicators of the transition such as the order parameter and its fluctuations are exhibited. The stationary probability distribution in each phase has also been calculated. We focus on the application of the Yang-Lee theory in the model. The normalization factor plays the role of an effective partition function, which can be expressed as a polynomial of the effective fugacity z. Numerical calculations show that in the thermodynamic limit the zeros of the effective partition function are located on the unit circle in the complex z plane. In the complex plane of the actual control parameter, certain roots converge to the transition point of the model. Thus, it is further evidence for the application of the Lee-Yang theory in a wider class of nonequilibrium systems.Finally, we apply the theory to study the quantum phase transitions of one-dimensional anisotropic XY quantum spin chain in a transverse field at zero temperature. In complex field plane, real and positive roots of the partition function mark the type and location of transition. For uniform chain, there is one critical point. For periodic and quasiperiodic chain, there is more than one phase transition point at some parameter region because of the competition between the spin cluster and anisotropy. Our results are in good agreement with that obtained by employing the transfer matrix and numerical method. Therefore, our work opens new perspectives for the application of Lee-Yang theory.