Boundary Criticality of Complex Conformal Field Theory: A Case Study in the Non-Hermitian 5-State Potts Model

TL;DR

This work numerically investigates boundary criticality in a non-Hermitian 5-state Potts model that exhibits complex conformal invariance in the bulk. By leveraging blob and two-boundary Temperley–Lieb algebras and analyzing the annulus partition functions, the authors identify free, fixed, and mixed boundary conditions and reveal conformal tower structures in the boundary spectrum, including complex scaling dimensions. The results demonstrate that conformal boundary fixed points persist in the complex CFT regime and expose dualities between boundary conditions via Kramers–Wannier transformations, offering a framework to extend BCFT concepts to non-Hermitian, irrational settings. The findings provide a groundwork for understanding boundary critical phenomena in complex CFTs and motivate future explorations of boundary RG flows, logarithmic BCFT aspects, and algebraic structures at $Q>4$.

Abstract

Conformal fields with boundaries give rise to rich critical phenomena that can reveal information about the underlying conformality. While most existing studies focus on Hermitian systems, here we explore boundary critical phenomena in a non-Hermitian quantum 5-state Potts model which exhibits complex conformality in the bulk. We identify free, fixed and mixed conformal boundary conditions and observe the conformal tower structure of energy spectra, supporting the emergence of conformal boundary criticality. We also studied the duality relation between different conformal boundary conditions under the Kramers-Wannier transformation. These findings should facilitate a comprehensive understanding for complex CFTs and stimulate further exploration on the boundary critical phenomena within non-Hermitian strongly-correlated systems.

Figures (7)

• Figure 1: Kac index within $0\leq r \leq 3$ and $-4\leq s \leq 4$ for bulk Potts CFT with generic $Q$Grans-Samuelsson2020Jacobsen2022. The representations for $S_Q$ singlet operators correspond to the direct product of two irreducible Verma module $V_{1,n} \otimes \bar{V}_{1,n}$ ($n \in \mathbb{N}^*$), marked by blue circles. The $S_Q$ vector fields are also diagonal with Kac index ($0,m+1/2$) ($m \in \mathbb{N}$), marked by green squares, while each module is non-degenerate in this situation since $r=0$. The primary fields with $S_Q$ higher representation correspond to logarithmic operator pairs combining Verma modules with opposite $s$: $V_{r,n/r} \otimes \bar{V}_{r,-n/r}$ or $V_{r,-n/r} \otimes \bar{V}_{r,n/r}$ ($r,n \in \mathbb{N}$ and $r \geq 2$).
• Figure 2: Analytic continuation of the blob-boundary parameter $Q_s(r)$ for $Q=5$. The curve shows $Q_s=\sqrt{Q}\,\sinh[(r+1)\lambda]/\sinh(r\lambda)$ with $\lambda=\mathrm{arccosh}(\sqrt5/2)$. The special integer points $r=1,2,-2$ correspond respectively to the free $(Q_s=5)$, mixed $(Q_s=4)$, and fixed $(Q_s=1)$ boundary conditions. For $r\ge3$, the boundary fugacity $Q_s$ takes non-integer values between $3$ and $4$, approaching $Q_s(+\infty)\approx3.618$ as $r\to+\infty$. The dashed blue lines indicate the asymptotic limits $Q_s(\pm\infty)$.
• Figure 3: Finite-size scaling of raw energy gaps with free-free boundary condition. We numerically compute several low-lying energy gaps with total system size $L=6,7,\cdots,11$ and fit them through $(E_n-E_0) \propto A/L+B/L^3$. Vanishing energy gaps in the thermodynamic limit signals the boundary criticality. Different colors label states belonging to different conformal multiplets, which will be elucidated in detail below. The extracted speed of light $v=\frac{A}{2\pi} \simeq 2.7205-0.6906i$ is close to the periodic case where $v_{\text{PBC}} \simeq 2.8810-0.7091i$ has been numerically determined in Tang2024.
• Figure 4: Finite-size scaling of rescaled energy gaps with free-free boundary condition. At each size, we rescale the whole spectrum by setting the first descendant of the identity operator to be $h_{L_{-2} \mathbb{I} }=2$. Then an extrapolation is performed through $h(L)=h(\infty)+C/L^2$. Up to Re$(h) \leq 5$ (here $h$ labels the scaling dimension of the boundary field), we classified three conformal multiplets according to their degeneracy and conformal tower structure. The $h_{1,1}$ multiplet labels the conformal family of identity operators, whose lowest field corresponds to the ground state in this case. The $h_{3,1}$ multiplet denotes the most relevant $S_5$ 4-dimensional standard representation operators, corresponding to the scaling dimension of boundary magnetization. Lastly, the representation of the field $h_{5,1}$ under $S_5$ decomposes as a direct sum of irreducible representations with 11-fold degeneracy.
• Figure 5: Real parts of boundary conformal spectrum of free-free boundary condition. The solid symbols label numerical results after an extrapolation (see main text), while the dashed lines marked theoretical values from each identified Verma module. Different multiplets are distinguished through their ground states degenercay, originating from different $S_Q$ representations. The imaginary parts belonging to the same conformal multiplet are almost the same, see Tab.\ref{['tab:free-free']}.