Nonequilibrium-relaxation approach to quantum phase transitions: Nontrivial critical relaxation in cluster-update quantum Monte Carlo

Although the nonequilibrium-relaxation (NER) method has been widely used in Monte Carlo studies on phase transitions in classical spin systems, such studies have been quite limited in quantum phase transitions. The reason is that the relaxation process based on cluster-update quantum Monte Carlo (QMC) algorithms, which are now standards in Monte Carlo studies on quantum systems, has been considered “too fast” for such analyses. Recently, the present authors revealed that the NER process in classical spin systems based on cluster-update algorithms is characterized by stretched-exponential critical relaxation, rather than conventional power-law relaxation in local-update algorithms. In the present article, we show that this is also the case in quantum phase transitions analyzed with the cluster-update QMC. As the simplest example of isotropic quantum spin models that exhibit quantum phase transitions, we investigate the Néel-dimer quantum phase transition in the two-dimensional $S=1/2$ columnar-dimerized antiferromagnetic Heisenberg model with the continuous-time loop algorithm, and we confirm stretched-exponential critical relaxation consistent with the three-dimensional classical Heisenberg model in the Swendsen-Wang algorithm.

Figure 1

Schematic figure of the columnar dimer pattern. Broad red lines represent the dimerized bonds.

Figure 2

Simulation-time dependence of the absolute value of the Néel order at $\delta =0.90947$ for $L=\beta =56$ (green stars), 80 (pink triangles), 112 (blue squares), and 160 (red circles). Error bars are much smaller than symbols in this scale.

Figure 3

Nonequilibrium-to-equilibrium scaling plot of the absolute value of the Néel order at $\delta =0.90947$ (circles) for $L=\beta =56$ (green symbols), 80 (pink symbols), 112 (blue symbols), and 160 (red symbols) with $\beta /\nu =0.514\left(1\right)$, $\sigma =0.502\left(8\right)$, and $c_m=0.423\left(14\right)$. The data at $\delta =0.90952$ (squares) and 0.909 42 (triangles) for the same sizes are scaled with $\beta /\nu =0.516\left(1\right)$, $\sigma =0.501\left(9\right)$, $c_m=0.427\left(18\right)$ and $\beta /\nu =0.512\left(1\right)$, $\sigma =0.504\left(6\right)$, $c_m=0.421\left(10\right)$, respectively. Error bars are much smaller than symbols in this scale. The data except for $\delta =0.90947$ are shifted upward or downward for clear visualization.

Figure 4

Nonequilibrium-to-equilibrium scaling plot of the absolute value of the Néel order at $\delta =0.90947$ in the vicinity of equilibrium for $\beta /\nu =0.5125$ (diamonds), 0.5130 (squares), 0.5135 (downside-marked circles), 0.5140 (circles), 0.5145 (upside-marked circles), 0.5150 (triangles), and 0.5155 (inverse triangles) with the same colors for $L=\beta =56$, 80, 112, and 160 as those in Fig. 3 using $\sigma =0.502$ and $c_m=0.423$ used in Fig. 3 at the same $\delta$. The data except for $\beta /\nu =0.5140$ are shifted upward or downward for clear visualization.

Figure 5

Nonequilibrium-to-equilibrium scaling plot of the absolute value of the Néel order in the vicinity of equilibrium at $\delta =0.90954$ (diamonds with $\beta /\nu =0.517$), 0.909 52 (squares with $\beta /\nu =0.516$), 0.909 47 (circles with $\beta /\nu =0.514$), 0.909 42 (triangles with $\beta /\nu =0.512$), and 0.909 40 (inverse triangles with $\beta /\nu =0.511$) for $L=\beta =56$ (green symbols), 80 (pink symbols), 112 (blue symbols), and 160 (red symbols) using $\sigma =0.502$ and $c_m=0.423$ used in Fig. 3 at $\delta =0.90947$, for all the data in this figure. The data except for $\delta =0.90947$ are shifted upward or downward for clear visualization.

Figure 6

Nonequilibrium-to-equilibrium scaling plot of the staggered susceptibility at $\delta =0.90952$ (squares), 0.909 47 (circles), and 0.909 42 (triangles) for $L=\beta =56$ (green symbols), 80 (pink symbols), 112 (blue symbols), and 160 (red symbols) using $\sigma =0.501$ in Eq. (10), with $\gamma /\nu =1.9703\left(8\right)$ and $c_{\chi }=1.7480\left(24\right)$, $\gamma /\nu =1.9730\left(8\right)$ and $c_{\chi }=1.7484\left(26\right)$, $\gamma /\nu =1.9752\left(8\right)$ and $c_{\chi }=1.7488\left(26\right)$, respectively. Error bars are much smaller than symbols in this scale. The data except for $\delta =0.90947$ are shifted upward or downward for clear visualization.