Reclaiming the Lost Conformality in a non-Hermitian Quantum 5-state Potts Model

Abstract

Conformal symmetry, emerging at critical points, can be lost when renormalization group fixed points collide. Recently, it was proposed that after collisions, real fixed points transition into the complex plane, becoming complex fixed points described by complex conformal field theories (CFTs). Although this idea is compelling, directly demonstrating such complex conformal fixed points in microscopic models remains highly demanding. Furthermore, these concrete models are instrumental in unraveling the mysteries of complex CFTs and illuminating a variety of intriguing physical problems, including weakly first-order transitions in statistical mechanics and the conformal window of gauge theories. In this work, we have successfully addressed this complex challenge for the (1+1)-dimensional quantum $5$-state Potts model, whose phase transition has long been known to be weakly first-order. By adding an additional non-Hermitian interaction, we successfully identify two conjugate critical points located in the complex parameter space, where the lost conformality is restored in a complex manner. Specifically, we unambiguously demonstrate the radial quantization of the complex CFTs and compute the operator spectrum, as well as new operator product expansion coefficients that were previously unknown.

Figures (9)

• Figure 1: (a) The complex CFT scenario for the Q-state Potts model: The phase transitions are second-order for $Q\le 4$, while weakly first-order for $Q>4$ because the location of the critical branches move into the complex parameter plane. (b) Illustration of phase transitions of the non-Hermitian 5-state Potts model $H_{\text{NH-Potts}}(J,h,\lambda)$ on the self-dual plane ($J/h=1$): Two complex fixed points located at $\lambda_c\neq0$ (green and cyan squares), which are determined via conformal perturbation calculation (see main text). The transition point of original 5-state Potts model is marked by the red dot.
• Figure 2: Finite-size scaling of the low-lying excitation energy gaps (distinguished by colors) at the complex critical point, where (a) real and (b) imaginary part of energy gaps identically scales to zero.
• Figure 3: Conformal multiplet for 11 low-lying Virasoro primary operators: real part of scaling dimension Re$(\Delta)$ versus Lorentz spin $s$. Different symbols and colors label different conformal towers. The spectrum is calibrated by setting the scaling dimension of energy momentum tensor $\Delta_T=2$. The dots are results from the 5-state Potts model and short lines are prediction from the analytical continuation of the Coulomb Gas partition function Gorbenko2018b. The translucent arrows denote Virasoro generators connecting different states.
• Figure S1: The distribution of optimized coupling strength $g_{\epsilon'}$ by scanning $\lambda$ on the self-duality plane $J=h=1$. The subfigures are for system size $L=8,9,10,11,12$. The minimum of $|g_{\epsilon'}|$ is shown by the cyan dot, which gives the critical point $(J_c=1,h=1,\lambda_c=0.079+0.060i)$ of microscopic Hamiltonian $H_{\text{NH-Potts}}$.
• Figure S2: Scaling of the real (left) and imaginary (right) part of the operator dimensions. The round circles are numerical results evaluated at different total system size $L$ with different operators distinguished by colors. The dashed lines are fitted from \ref{['eq:ope_dim_scaling']}. The diamonds denote prediction from complex CFT Gorbenko2018b.