Dissecting Quantum Phase Transition in the Transverse Ising Model

Abstract

Despite the fact that a complete theoretical description of critical phenomena in connection with phase transitions has been well-established through the renormalization group theory, the microscopic nature of the phase transitions remains to be understood in a satisfactory way. For example, how does the interaction between individuals drive a system from one phase to another as a specific parameter varies, and how do the individuals respond to changes in this parameter during the process? Here we take the well-studied quantum phase transition (QPT) in the one-dimensional transverse Ising model (TIM) as an example to exhibit such a microscopic process. We first introduce $2L$ collective structures,referred to as patterns, for the TIM with $L$ ferromagnetically interacting spins, and then analyze the contributions of these patterns to the system's states, e.g., the ground state, the first excited state, and so on, from which the analogue of the QPT process between the disordered phase in the weakly coupling regime and the ferromagnetic phase in the strongly coupling regime is clearly identified around the interaction strength $J_c =1$. We systematically explore this process for small lattice sizes of $L=6, 8, 10, 12$, whose ground state energies are identical to those obtained by direct numerical exact diagonalization. Increasing the system size up to $L=128$, the actual QPT point located at $J_c = 1$ in the thermodynamical limit is gradually approached. Our results show that the pattern picture is not only able to provide a microscopic process of phase transitions, but also of practical interest in analyzing analogues of QPT in diverse quantum simulation platforms.