Shadow of complex fixed point: Approximate conformality of $Q>4$ Potts model

We study the famous example of weakly first-order phase transitions in the $\left(1+1\right)$-dimensional quantum $Q$-state Potts model with $Q>4$. We numerically show that these weakly first-order transitions have approximately conformal invariance. Specifically, we find that entanglement entropy on considerably large system sizes fits perfectly with the universal scaling law of this quantity in conformal field theories (CFTs). This supports that the weakly first-order transitions are proximate to complex fixed points, which are described by recent conjectured complex CFTs. Moreover, the central charge extracted from this fitting is close to the real part of the complex central charge of these complex CFTs. We also study the conformal towers and the drifting behaviors of these conformal data (e.g., central charge and scaling dimensions).

Figure 1

(a) A bipartition of a periodic Potts chain. (b) The critical point (CP) and tricritical point (TCP) when $Q<4$. $\lambda$ is irrelevant at the critical point and relevant at the tricritical point. (c) FPs at different $Q$.

Figure 2

Scaling of the EE ($S$) at the transition point of the $Q$-state quantum Potts model for different system sizes $L$. We fit the data with the function in Eq. (3). The fitted central charges are listed in Table 1.

Figure 3

The central charge at the transition point for $Q=3,5,6,7$ as a function of system size is fitted with the formula in Eq. (6). As a comparison, it is also fitted with the usual polynomial of $1/L$.

Figure 4

(a) Schematic illustration of tensor network calculation of a thermal partition function. The tensors denoted as circles are identified due to the periodic boundary condition along $\beta$. (b) Conformal tower of the 5-state quantum Potts model. Crosses (red: primaries; black: descendants) are theoretical values while lines are numerical results. The spectrum is separated into sectors with different conformal spin ($s$). (c) The drift of the ratio of scaling dimensions ${\mathrm{\Delta }}_{\varepsilon }/{\mathrm{\Delta }}_{\sigma }$.

Figure 5

The drift of central charge at $Q=4$ is fitted with the drift formula Eq. (6) and the polynomial of $1/L$.

Figure 6

Conformal tower of the critical $Q=3$ ($\beta =40$) and $Q=4$ ($\beta =50$) Potts models. Red (primary) and black (descendant) crosses are theoretical values while lines are numerical results. The spectrum is separated into sectors with different symmetry charge ($q$).

Figure 7

Sound velocity of the $Q=3$ ($20\le \beta \le 40$ and $\chi =300$), $Q=4$ ($20\le \beta \le 50$ and $\chi =500$), $Q=5$ ($25\le \beta \le 50$ and $\chi =400$), and $Q=6$ ($25\le \beta \le 50$ and $\chi =300$) Potts models at the transition point.

Figure 8

(a) Conformal tower of the $Q=6$ Potts model at the transition point. Red (primary) and black (descendant) crosses are theoretical values while lines are numerical results. The spectrum is separated into sectors with different symmetry charge ($q$). (b) The drift behavior of ${\mathrm{\Delta }}_{\varepsilon }/{\mathrm{\Delta }}_{\sigma }$.