Spiral renormalization group flow and universal entanglement spectrum of the non-Hermitian 5-state Potts model

TL;DR

Problem: characterize the complex fixed point and walking RG flow of the non-Hermitian deformed 5-state Potts model on the lattice. Approach: employ tensor-network methods (DMRG/QPA with MPS) to simulate up to $L \approx 64$, and perform finite-size scaling and entanglement-spectrum analysis via the entanglement Hamiltonian. Contributions: obtain a refined fixed point $\lambda_c = 0.0788 + 0.0603i$, observe spiral flow of the CCFT perturbation $g_{\varepsilon'}$, and reveal a boundary CCFT spectrum consistent with the free-free CBC towers through the lattice entanglement spectrum. Significance: demonstrates emergent CCFT-like conformal invariance in non-Hermitian critical phenomena and showcases tensor-network methods as effective tools for probing weakly first-order transitions and CCFT data, with avenues toward infinite-size TN techniques.

Abstract

The quantum $5$-state Potts model is known to possess a perturbative description using complex conformal field theory (CCFT), the analytic continuation of ``theory space" to a complex plane. To study the corresponding complex fixed point on the lattice, the model must be deformed by an additional non-Hermitian term due to its complex coefficient $λ$. Although the variational principle breaks down in this case, we demonstrate that tensor network algorithms are still capable of simulating these non-Hermitian theories. We access system sizes up to $L = 28$, which enable the observation of the theoretically predicted spiral flow of the running couplings. Moreover, we reconstruct the full boundary CCFT spectrum through the entanglement Hamiltonian encoded in the ground state. Our work demonstrates how tensor networks are the correct approach to capturing the approximate conformal invariance of weakly first-order phase transitions.

Figures (10)

• Figure 1: The flow of the running coupling parameter $g_{\varepsilon'}$ acquired through the cost function fitting Tang2024supplementary (a) for $\mathcal{\overline{C}}$ and (b) for $\mathcal{C}$ around their respective fixed points for system sizes $L=16-24$. The perturbation parameter at $\lambda_c$ is denoted by a blue dot, while red arrows show the walking behavior of values on the real axis.
• Figure 2: The $1/L^2$ scaling of the difference between the theoretical and estimated scaling dimensions Eq. \ref{['eq:Deltadeviation']} at $\lambda_{c} = 0.0788 + 0.0603i$ for (a) the real part and (b) the imaginary part.
• Figure 3: The entanglement spectrum of the ground state in the (a) zero sector and (b) a charged sector of $\mathbb{Z}_5$ for the deformed 5-state Potts Hamiltonian at $\lambda_c = 0.0788 + 0.0603i$ for system sizes $L = 12 - 64$ and periodic boundary conditions. The dashed lines and red numbers are the theoretical energy levels and their degeneracies, respectively Tang:2025bju. We fit the data to their theoretical values for the first two energy levels of the zero sector.
• Figure S1: The drift of the critical value due to subleading finite-size corrections on the $g_{\varepsilon'}$ minimum of the cost function, Eq. \ref{['eq:costfunction']}, for system sizes $L =10 -24$. We note that the minimum at $L = 18$ and $L = 19$ overlap. We perform a fit of $y = ax^2 +b$ with $x = 1/L$ for all data points and for $L = 17-24$. Their zero points $b$ are marked by a cross and yield an estimate for the critical point, denoted $\lambda_{CF}$. The estimate $\lambda_c = 0.0788 + 0.0603i$ is also included.
• Figure S2: The energy gaps $E_{\sigma} - E_0$ collapsing to $\frac{v \Delta_{\sigma}}{2\pi}$ at the fixed points $\lambda_c = 0.0788 \pm 0.0603i$ when multiplying with different system sizes.