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arXiv:2502.02001 · Physical review. B./Physical review. B

Complex entanglement entropy for complex conformal field theory

Haruki Shimizu, Kohei Kawabata

Published Jul 17, 2025• 7 citations
arXivHTMLPDF

Abstract

Conformal field theory (CFT) conventionally possesses positive central charges and describes continuous phase transitions in quantum many-body systems. Recently, CFT with complex-valued central charges has been proposed to relate to the discontinuous phase transitions of the Potts models. Here, the authors demonstrate that the scaling of complex-valued entanglement entropy characterizes complex CFT by applying the density matrix renormalization group method to the non-Hermitian five-state Potts model. This study suggests a way to investigate the entanglement properties of complex CFT.

DOI
10.1103/n578-ljd5

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Conformal field theory (CFT) conventionally possesses positive central charges and describes continuous phase transitions in quantum many-body systems. Recently, CFT with complex-valued central charges has been proposed to relate to the discontinuous phase transitions of the Potts models. Here, the authors demonstrate that the scaling of complex-valued entanglement entropy characterizes complex CFT by applying the density matrix renormalization group method to the non-Hermitian five-state Potts model. This study suggests a way to investigate the entanglement properties of complex CFT.
We demonstrate three types of transformations that establish connections between Hermitian and non-Hermitian quantum systems at criticality, which can be described by conformal field theories (CFTs). For the transformation preserving both the energy and the entanglement spectra, the corresponding central charges obtained from the logarithmic scaling of the entanglement entropy are identical for both Hermitian and non-Hermitian systems. The second transformation, while preserving the energy spectrum, does not perserve the entanglement spectrum. This leads to different entanglement entropy scalings and results in different central charges for the two types of systems. We demonstrate this transformation using the dilation method applied to the free fermion case. Through this method, we show that a non-Hermitian system with central charge c = -4 c = − 4 can be mapped to a Hermitian system with central charge c = 2 c = 2 . Lastly, we investigate the Galois conjugation in the Fibonacci model with the parameter \phi \to - 1/\phi ϕ → − 1 / ϕ , in which the transformation does not preserve both energy and entanglement spectra. We demonstrate the Fibonacci model and its Galois conjugation relate the tricritical Ising model/3-state Potts model and the Lee-Yang model with negative central charges from the scaling property of the entanglement entropy.
We study the famous example of weakly first-order phase transitions in the (1+1)\left(1+1\right)(1+1)-dimensional quantum QQQ-state Potts model with Q>4Q>4Q>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).
Quantum entanglement is one essential element to characterize many-body quantum systems. However, the entanglement measures are mostly discussed in Hermitian systems. Here, we propose a natural extension of entanglement and Rényi entropies to non-Hermitian quantum systems. There have been other proposals for the computation of these quantities, which are distinct from what is proposed in the current paper. We demonstrate the proposed entanglement quantities which are referred to as generic entanglement and Rényi entropies. These quantities capture the desired entanglement properties in non-Hermitian critical systems, where the low-energy properties are governed by the non-unitary conformal field theories (CFTs). We find excellent agreement between the numerical extrapolation of the negative central charges from the generic entanglement/Rényi entropy and the non-unitary CFT prediction. Furthermore, we apply the generic entanglement/Rényi entropy to symmetry-protected topological phases with non-Hermitian perturbations. We find the generic n-th Rényi entropy captures the expected entanglement property, whereas the traditional Rényi entropy can exhibit unnatural singularities due to its improper definition.
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The bipartite entanglement entropy of a segment of length lll in 1+11+11+1-dimensional conformal field theories (CFT) follows the formula $S=\frac{c}{3}\mathrm{ln}l+\ensuremath{\gamma}$, where ccc is the central charge of the CFT and $\ensuremath{\gamma}$ is a cutoff-dependent constant which diverges in the absence of an ultraviolet cutoff. According to this formula, systems with larger central charges have more bipartite entanglement entropy. Using quantum Fisher information (QFI), we argue that systems with bigger central charges not only have larger bipartite entanglement entropy, but also have more multipartite entanglement content. In particular, we argue that since a system with a smaller smallest scaling dimension has a larger QFI, the multipartite entanglement content of a CFT is dependent on the value of the smallest scaling dimension present in the spectrum of the system. We show that our argument seems to be consistent with some of the existing results regarding the von Neumann entropy, negativity, and localizable entanglement in 1+11+11+1 dimensions. Furthermore, we also argue that the QFI decays under renormalization group flow between two unitary CFTs. Finally, we also comment on nonconformal but scale-invariant systems.
Here we show that the Rényi entanglement entropy of a region of large size ℓ in a one-dimensional critical model whose ground state breaks conformal invariance (such as in those described by non-unitary conformal field theories), behaves as , where is the effective central charge, c (which may be negative) is the central charge of the conformal field theory and is the lowest holomorphic conformal dimension in the theory. We also obtain results for models with boundaries, and with a large but finite correlation length, and we show that if the lowest conformal eigenspace is logarithmic ( with N nilpotent), then there is an additional term proportional to . These results generalize the well known expressions for unitary models. We provide a general proof, and report on numerical evidence for a non-unitary spin chain and an analytical computation using the corner transfer matrix method for a non-unitary lattice model. We use a new algebraic technique for studying the branching that arises within the replica approach, and find a new expression for the entanglement entropy in terms of correlation functions of twist fields for non-unitary models.