Moving from continuous to discrete symmetry in the 2D XY model

We study the effects of discretization on the U(1) symmetric XY model in two dimensions using the higher order tensor renormalization group approach. Regarding the $Z_N$ symmetric clock models as specific discretizations of the XY model, we compare those discretizations to ones from truncations of the tensor network formulation of the XY model based on a character expansion and focus on the differences in their phase structure at low temperatures. We also divide the tensor network formulations into core and interaction tensors and show that the core tensor has the dominant influence on the phase structure. Lastly, we examine a perturbed form of the XY model that continuously interpolates between the XY and clock models. We examine the behavior of the additional phase transition caused by the perturbation as the magnitude of perturbation is taken to zero. We find that this additional transition has a nonzero critical temperature as the perturbation vanishes, suggesting that even small perturbations can have a significant effect on the phase structure of the theory.

Figure 1

Specific heat $C_V$ versus temperature $T$ for the truncated XY model with varying initial bond dimension $D$.

Figure 2

Specific heat $C_V$ versus temperature $T$ for the $Z_N$ clock model with varying $N$.

Figure 3

Magnetization per site $⟨m⟩$ versus temperature $T$ for the truncated XY model with magnetic field $h={10}^{-4}$ and varying initial bond dimension $D$.

Figure 4

Magnetization per site $⟨m⟩$ versus temperature $T$ for the $Z_N$ clock model with magnetic field $h={10}^{-4}$ and varying $N$.

Figure 5

Cross-derivative $-{\partial }_T⟨m⟩$ versus temperature $T$ for the truncated XY model with $h={10}^{-4}$ and varying $D$.

Figure 6

Cross-derivative $-{\partial }_T⟨m⟩$ versus temperature $T$ for the $Z_N$ clock models with $h={10}^{-4}$ and varying $N$.

Figure 7

Specific heat $C_V$ versus temperature $T$ for different combinations of core tensor and interaction matrix from the truncated XY model at $D=5$ and the $Z_5$ clock model. The combination with the XY interaction matrix and $Z_5$ (periodic) core (squares) shows two peaks, similar to the $Z_5$ model (triangles), while the $Z_5$ interaction and $XY$ core (diamonds) shows no sign of a second phase transition.

Figure 8

Specific heat $C_V$ versus temperature $T$ for a mixed model with the XY interaction matrix with a $Z_5$ core tensor generalized to different numbers of initial states $D$. For $D=5$, this corresponds to the same mixed model in Fig. 7 (squares). For large $D$, this approaches the $Z_5$ clock model result (diamonds). The $XY$ model (circles) is also shown for comparison.

Figure 9

Magnetization per site $⟨m⟩$ versus temperature $T$ for the perturbed XY model with magnetic field $h={10}^{-5}$ and varying $h_5$.

Figure 10

Cross-derivative $-{\partial }_T⟨m⟩$ versus temperature, $T$, for the perturbed XY model with magnetic field $h={10}^{-5}$ and varying $h_5$.

Figure 11

Lower peak temperature $T_{p1}$ versus magnetic field $h$ for the perturbed XY model with varying $h_5$.

Figure 12

Lower peak temperature $T_{p1}$ versus $h_5$ for the perturbed XY model with varying $h$.

Figure 13

Lower peak temperature extrapolated to $h_5=0$ ($T_{p1}^{*}$ in the text) versus magnetic field $h$ for the perturbed XY model. The smooth line is a fit to the power-law form in Eq. (14). The smallest $h$ value also has a result for $D_{\mathrm{cut}}=101$, which is consistent with the $D_{\mathrm{cut}}=91$ results.

Figure 14

Lower peak height $S_{p1}$ versus magnetic field $h$ for the perturbed XY model for varying $h_5$.

Figure 15

Lower peak temperature $T_{\mathrm{peak}}$ versus magnetic field $h$ for the perturbed XY model at $h_5=0.2$ for varying $D_{\mathrm{cut}}$.

Figure 16

Lower peak height versus magnetic field $h$ for the perturbed XY model at $h_5=0.2$ for varying $D_{\mathrm{cut}}$.