Dynamical properties of classical and quantum spin systems

Abstract

The Kibble-Zurek mechanism (KZM) was originally proposed to describe the evolution and "freezing" of defects in the early universe, but later it was generalized to study other quantum and classical systems driven by a varying parameter. The basic idea behind the KZM is that, as long as the changing rate (velocity) of the parameter is below a certain critical velocity, 𝑣_crit, the system will remain adiabatic (for isolated quantum systems) or quasi-static (for classical systems with a heat bath). The nonequilibrium finite-size scaling (FSS) method based on KZM has been exploited systematically. Through applying the scaling hypothesis, we can extract the critical exponents and study the dynamic properties of the system.

In the first few chapters of this dissertation, we discuss the applications of KZM in several classical systems: first, we study the dynamics of 2D and 3D Ising model under a varying temperature as well as a varying magnetic field. Secondly, we examine the classical ℤ₂ gauge model, in which we show that KZM also works for topological phase transitions. Moreover, we also investigate the dynamics of other models with topological ordering only at T=0, where KZM cannot be applied. Lastly, we explore the 2D Ising spin glass with bimodal and gaussian couplings. With bimodal couplings, we find dual time scales associated with the order parameter and the energy correspondingly, while in the gaussian case one unique time scale is involved.

The systems mentioned above are all classical and the dynamics are approached through simulated annealing (SA), in which thermal fluctuations drives systems to explore the energy landscape in finding the ground state. In the last chapter, we explore the efficiency of Quantum Annealing (QA) on a fully-connected spin glass (or Sherington-Kirkpatrick model) with a transverse field. QA is the counterpart of SA, where quantum fluctuations drive the system toward the ground state when the quantum terms are reduced. QA is currently widely explored as a paradigm for quantum computing to solve optimization problems. Here we compare the scaling of the dynamics (with system size) of the fully-connected spin glass through QA versus SA.