Extending finite-size tensor-network-flow extraction of conformal data beyond the Ising and three-state clock benchmarks: complex models, scheme generality, and non-unitary CFTs

Chan and Chen (Phys. Rev. Research 2026) introduced a framework that extracts conformal data — central charge, scaling dimensions, and conformal spins — from transfer-matrix spectra of tensor-network approximations of critical two-dimensional classical partition functions, without requiring a controlled critical fixed-point tensor. The method identifies a self-consistent finite-size window via integer quantization of the conformal spin and a crossover scale that separates finite-size scaling from finite-entanglement scaling, and it was validated only on two unitary minimal-model benchmarks (critical Ising, $c=1/2$, and the three-state clock model, $c=4/5$, both described by unitary minimal models, the discrete family of exactly solvable two-dimensional conformal field theories) with three coarse-graining schemes: the higher-order tensor renormalization group (HOTRG), the periodic transfer-matrix renormalization group (PTMRG), and the core-tensor renormalization group (CTRG). The authors pose as open directions: applications to more complex classical models, systematic comparisons across a wider class of tensor-network renormalization schemes, and extensions to settings where fixed-point-tensor approaches are difficult to control, such as non-unitary conformal field theories or models with intricate operator spectra. The decisive stress test is the non-unitary extension, where the ground-state-identity assumption behind the central-charge estimator is explicitly tied to unitarity and the $c/3$ coefficient of the entanglement scaling is assumed rather than derived.

Background

At a continuous phase transition, a two-dimensional classical lattice model is described in the continuum limit by a conformal field theory (CFT), a scale-invariant theory whose universal content is encoded in the conformal data: the central charge $c$, the scaling dimensions $\Delta_i$ of the scaling operators, and the conformal spins $S_i$ (eigenvalues of the rotation generator, $S=h-\bar{h}$ in terms of left and right conformal weights). A standard way to access this data on the lattice is the transfer matrix, the operator whose repeated application builds the partition function of a strip; for a strip of width $L_x$ with periodic boundary conditions, Cardy's finite-size scaling theory relates the transfer-matrix levels to the conformal data through $E_i(L_x)/L_x = f_\infty + \frac{2\pi}{L_x^2}\left(\Delta_i-\frac{c}{12}\right)+\dots$, where $f_\infty$ is the bulk free-energy density (Cardy 1984; Cardy 1986; Blöte, Cardy, and Nightingale 1986).

Tensor-network methods make this program practical at large sizes: the partition function is written as a contraction of local tensors and coarse-grained iteratively, starting with the tensor renormalization group (TRG) of Levin and Nave, with higher-order TRG (HOTRG, based on higher-order singular value decompositions) and variants such as the core-tensor renormalization group (CTRG) and the periodic transfer-matrix renormalization group (PTMRG; Fedorovich et al. 2025) among its descendants. The bond dimension $D$ controls the truncation error of the network. The most direct route to conformal data is to iterate the coarse graining to a critical fixed-point tensor and read the data off its structure (Gu and Wen 2009; Hauru et al. 2016; reviewed by Guo and Wei 2024), or to extract scaling dimensions from the eigenvalues of the linearized coarse-graining map (Lyu, Xu, and Kawashima 2021); these strategies are technically delicate because of corner double-line artifacts and gauge redundancies, and a rigorous general definition of such fixed points is an active subject (Kennedy and Rychkov 2022).

The source paper (Chan and Chen 2026) culminates a series of works (Huang, Chan, Kao, and Chen 2023 on the Ising model; Maiti, Chan, and Chen 2025 on the three-state clock model) that developed an alternative which stays entirely at finite size: approximate the finite-width transfer matrix and the shift operator (the lattice counterpart of a rotation, whose eigenvalues give the conformal spin) by coarse-grained tensor networks (the transfer matrix of a strip of width $L_x$ is assembled by stacking $n$ copies of a coarse-grained building-block tensor, where $n$ is the stacking number), and follow the flow of the finite-size scaling dimensions $X_k^Q(L_x)$ and spins $S_k^Q(L_x)$. The levels are organized by their charge $Q$ under the model's discrete internal spin-rotation symmetry (the $\mathbb{Z}_2$ symmetry enforced for the Ising model and the $\mathbb{Z}_3$ symmetry for the three-state clock model; this internal-symmetry label is independent of the stacking number $n$) into nearly degenerate groups, which are then resolved into spin subgroups. Three ingredients make the method self-contained:

1. A self-consistent window: while $S_k^Q(L_x)$ stays close to the nearest integer, the data are trusted; once the deviation exceeds a threshold, the data are discarded.
2. A crossover scale $L_x^*$: beyond it, the finite bond dimension acts as a relevant perturbation (Ueda and Oshikawa 2023) that drives the flow into a finite-entanglement regime where the levels collapse — a phenomenon previously observed in boundary TRG (Iino, Morita, and Kawashima 2019). $L_x^*$ is located as the size at which the absolute value of the derivative of the relevant estimator with respect to $l=\ln L_x$ (for example the spin-resolved subgroup average $\bar X_S^Q(L_x)$ or the estimator $c_{E_0}$) is minimal, and conformal data are evaluated there; the crossover scale depends on the observable, the coarse-graining scheme, the bond dimension $D$, and the stacking number $n$.
3. Two central-charge estimators: $c_{E_0}$ from the ground-state scaling $E_0(L_x)/L_x = f_\infty - \frac{\pi c}{6 L_x^2}+\dots$ (the $\Delta_{\mathbf{I}}=0$ case of the Cardy relation above), and $c_S = 3\frac{dS_E}{d\ln L_x}$ from the bipartite entanglement entropy $S_E(L_x,L_x)$ of the dominant transfer-matrix eigenvector of a strip of width $2L_x$ (interpreted as a quantum state), bipartitioned into two halves of width $L_x$, which at criticality is expected to scale as $S_E = \frac{c}{3}\ln\frac{2L_x}{\pi}+\alpha$.

The framework was benchmarked with HOTRG (best, relative errors $2.2\times10^{-6}$ to $1.1\times10^{-3}$ for Ising scaling dimensions up to $\Delta\approx7.1$, and $1.8\times10^{-5}$ to $1.2\times10^{-2}$ up to $\Delta\approx4.8$ for the clock model), PTMRG, and CTRG. One structural assumption is explicitly tied to unitarity in the paper: the ground state corresponds to the identity operator with $\Delta_{\mathbf{I}}=0$ (stated for unitary CFTs, and load-bearing for $c_{E_0}$). The $c/3$ coefficient of the logarithmic entropy scaling is instead posited at criticality without derivation and without an explicit unitarity restriction. In non-unitary CFTs — theories with negative-norm states, such as the Yang-Lee edge singularity ($c=-22/5$) realized by the Ising model at imaginary magnetic field, or logarithmic conformal field theories (theories whose correlation functions contain logarithms alongside powers), such as percolation ($c=0$) — the lowest scaling dimension $\Delta_{\min}$ is negative, the identity is not the lowest state, and both the finite-size free energy and the entanglement scaling are governed by an effective central charge $c_{\rm eff}=c-12\Delta_{\min}$ rather than $c$; lattice realizations moreover involve complex Boltzmann weights and non-Hermitian transfer matrices. Models with denser operator spectra, such as the four-state Potts model ($c=1$, with marginal operators and logarithmic corrections) or the tricritical Ising model ($c=7/10$), stress the group-resolution step: the clock-model benchmark already showed errors roughly ten times larger than Ising and a narrower usable window. This is the context in which the source poses its open directions.

Problem Statement

Establish the generality of the finite-size tensor-network-flow framework of Chan and Chen for extracting conformal data (central charge, scaling dimensions, conformal spins) from critical two-dimensional classical lattice models beyond the two unitary benchmarks on which it has been demonstrated. The source poses three directions for future work, which bear on whether the framework is a general method rather than a technique tuned to the Ising and three-state clock models: (1) apply it to more complex critical classical models, in particular models with denser operator spectra and marginal operators (for example the four-state Potts model at $c=1$ with its logarithmic corrections, or the tricritical Ising model at $c=7/10$); (2) establish systematic consistency across a wider class of tensor-network renormalization schemes than the three tested; and (3) extend it to settings where fixed-point-tensor approaches are difficult to control, above all non-unitary conformal field theories (such as the Yang-Lee edge singularity realized by the Ising model at imaginary magnetic field, or logarithmic CFTs like percolation) and models with intricate operator spectra. Direction (3) is the decisive stress test, because the framework's components have untested or unitary-dependent status there: the self-consistency window discards data once the conformal spin deviates from the nearest integer, a criterion whose applicability to non-unitary content or to operators with non-integer conformal spin is untested; the ground-state estimator $c_{E_0}$ assumes the identity is the lowest state, and whether it tracks $c$ or the effective central charge $c_{\rm eff}=c-12\Delta_{\min}$ in a non-unitary CFT is not established; the entropy-based estimator's logarithmic coefficient likewise needs reexamination; and complex Boltzmann weights make the transfer matrix non-Hermitian. All three directions are required for a complete resolution: for direction (1) it suffices to treat the four-state Potts model and the tricritical Ising model, for direction (2) it suffices to demonstrate consistency for at least one renormalization scheme beyond the three tested, and direction (3) requires treating the named non-unitary models, the Yang-Lee edge singularity and a logarithmic CFT such as percolation. The open question is whether the framework's components — the integer-spin self-consistency window, the crossover-scale criterion $L_x^*$, and the two central-charge estimators $c_{E_0}$ and $c_S$ — survive these extensions and, where they fail, what must replace them.

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