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.

Figures (11)

• Figure 1: The patterns and their relative phases for $L=6$ obtained by the first diagonalization, marked by the single-body operators $\hat{A}_n = \sum_{i=1}^L \left[u_{n,2i-1} (i\hat{\sigma}^y_i) + u_{n,2i}\hat{\sigma}^z_i\right]$ with $(\pm,\pm)$ denoting the signs of $(u_{n,2i-1},u_{n,2i})$. All patterns are divided into two groups marked by the red and blue frames with $\lambda_n < 0$ and $\lambda_n > 0$, respectively. The characteristic difference between these two groups is that for the pattern with $\lambda_n < 0$ (red frame) the operators within the sites are in-phase, but for the pattern with $\lambda_n > 0$ (blue frame) they are out-of-phase. For these two groups of patterns, different patterns are distinguished by the phases between site $i$ and its nearest neighbor sites, which form domains if they are in-phase for $\hat{\sigma}^z_i$, otherwise, kinks if they are out-of-phase for $\hat{\sigma}^z_i$. Here it should be mentioned that the eigenvectors $(u_{n,2i-1},u_{n,2i})$ are free of a total phase factor $e^{i\pi}$ but their relative phases remain fixed.
• Figure 2: The patterns and their relative phases for $L=8$. Other information is consistent with Fig. \ref{['fig1']}.
• Figure 3: The patterns and their relative phases for $L=10$. Other information is consistent with Fig. \ref{['fig1']}.
• Figure 4: The patterns and their relative phases for $L=12$. Other information is consistent with Fig. \ref{['fig1']}.
• Figure 5: The eigenenergies $\lambda_n$ of the patterns as functions of the Ising interacting strength $J$ for $L =6, 8, 10, 12$. The patterns are divided into two groups, which have positive and negative eigenenergies, respectively, satisfying with $\lambda_n = - \lambda_{2L - n +1}$ ($n = 1, 2, \cdots, 2L$). There are still degenerate for the patterns such as $\lambda_{2,3}$, $\lambda_{4,5}$, $\cdots$, and their positive eigenenergy counterparts. For simplicity, we only give the legend of (a), from bottom to top (or from the red line to wine line) corresponding pattern $\lambda_1$, $\lambda_{2,3}$, $\lambda_{4,5}$, $\lambda_6$, $\lambda_{7}$, $\lambda_{8,9}$, $\lambda_{10,11}$, $\lambda_{12}$. The legend of (b): from bottom to top corresponding pattern $\lambda_1$, $\lambda_{2,3}$, $\lambda_{4,5}$, $\lambda_{6,7}$, $\lambda_8$, $\lambda_{9}$, $\lambda_{10,11}$, $\lambda_{12,13}$, $\lambda_{14,15}$, $\lambda_{16}$. The legend of (c): from bottom to top corresponding pattern $\lambda_1$, $\lambda_{2,3}$, $\lambda_{4,5}$, $\lambda_{6,7}$, $\lambda_{8,9}$, $\lambda_{10}$, $\lambda_{11}$, $\lambda_{12,13}$, $\lambda_{14,15}$, $\lambda_{16,17}$, $\lambda_{18,19}$, $\lambda_{20}$. The legend of (d): from bottom to top corresponding pattern $\lambda_1$, $\lambda_{2,3}$, $\lambda_{4,5}$, $\lambda_{6,7}$, $\lambda_{8,9}$, $\lambda_{10,11}$, $\lambda_{12}$, $\lambda_{13}$, $\lambda_{14,15}$, $\lambda_{16,17}$, $\lambda_{18,19}$, $\lambda_{20,21}$, $\lambda_{22,23}$, $\lambda_{24}$.