Monte Carlo Renormalization Flows in the Space of Relevant and Irrelevant Operators: Application to Three-Dimensional Clock Models

We study renormalization group flows in a space of observables computed by Monte Carlo simulations. As an example, we consider three-dimensional clock models, i.e., the $XY$ spin model perturbed by a $Z_q$ symmetric anisotropy field. For $q=4$, 5, 6, a scaling function with two relevant arguments describes all stages of the complex renormalization flow at the critical point and in the ordered phase, including the crossover from the U(1) Nambu-Goldstone fixed point to the ultimate $Z_q$ symmetry-breaking fixed point. We expect our method to be useful in the context of quantum-critical points with inherent dangerously irrelevant operators that cannot be tuned away microscopically but whose renormalization flows can be analyzed as we do here for the clock models.

Figure 1

MC RG flows for $q=6$. Each set of connected dots represents a fixed $T$ and sizes $L=2,3,4,\dots$. The sets for the highest and lowest $T$ and $T=T_c$ are shown with bigger dots in black, red, and blue, respectively. The inset shows detailed flows in the critical region.

Figure 2

Log-log plot of the critical angular order parameter ${\varphi }_q$ versus the linear system size $L$ for several $q$ and $h$ values. The fitting lines correspond to the power-law form ${\varphi }_q\propto L^{-|y_q|}$ and the resulting exponents are summarized in Table 1.

Figure 3

(a) Distance $d_1(L)$ to the $XY$ fixed point. Black and blue solid circles correspond to $T=2.193$ and $T=2.201$, respectively, and open circles show temperatures in between. The minimums (red circles) were obtained by polynomial fits. (b) Power-law behaviors in $t$ of the minimum distance $D_1$ and corresponding size $L_1$ [red dots in (a)].

Figure 4

(a) Distance of the curves in Fig. 1 to the $x$ axis. The black and blue solid circles correspond to $T=2.06$ and 2.14, respectively, and the open circles are for equally spaced $T$. (b) The minimums (red dots) in (a) exhibit scaling in $t$ of the minimum distance $D_2$ and the corresponding size $L_2$.

Figure 5

(a) ${\varphi }_6$ versus $L$ for temperatures from $T=1.85$ (blue circles) and 1.95 (black solid circles). The crossing points with three horizontal lines at 0.5,0.55, and 0.6 are analyzed in (b) with a joint power law with a common exponent.

Figure 6

(a) The distance $d_3(L)$ to the NG fixed point for temperatures between $T=2.00$ (black solid circles) and 2.05 (blue circles) (b) Power-law behaviors in $t$ of the minimum-distance quantities $D_3$ and $L_3$.