Parent Hamiltonian and Gossamer-Tail Control for Chiral Projected Entangled-Pair States: Bulk Gaps of the Z3 Kitaev Chiral Spin Liquid

Projected entangled-pair states (PEPS) reproduce chiral spin liquid ground states but generically carry tiny-amplitude, slowly decaying 'gossamer' correlation tails that raise doubts about their bulk spectra. Chen, Tan, Capponi, Poilblanc, Ye and Mei [Phys. Rev. B 113, 085130 (2026)] showed within a tangent-space variational principle that tails decaying faster than $r^{-2}$ do not force spurious gapless modes, and computed the first bulk excitation gaps of the $Z_2$ (benchmark) and $Z_3$ Kitaev chiral spin liquids. The open items it leaves — the future-work suggestions in its conclusions plus one stated limitation of its $Z_3$ results — comprise three linked questions: (i) construct and characterize the exact parent Hamiltonian of the chiral PEPS, for which the no-go theorem (rigorous for Gaussian fermionic PEPS) leaves open whether it can be both strictly local and gapped, so the explicit form, magnitude scaling, and physical consequences of any required small-amplitude long-range interactions are undetermined; (ii) determine the asymptotic functional form of the long-distance correlation tails of the $Z_3$ Kitaev chiral spin liquid, which finite-$\chi$ effects at the current $D=6$, $\chi=108$ state of the art prevent resolving; (iii) develop finite-$\chi$/finite-$D$ scaling diagnostics that distinguish negligible gossamer tails from physically genuine critical correlations in chiral phases.

Background

Projected entangled-pair states (PEPS) represent two-dimensional lattice wave functions by placing a tensor on each site and contracting virtual bonds between neighbors; the bond dimension $D$ controls the variational accuracy, and the ansatz naturally satisfies the area law of entanglement entropy (entanglement scaling with the boundary of a region rather than its volume). In the infinite, translation-invariant version (iPEPS), contractions are performed approximately by the corner transfer matrix renormalization group (CTMRG), whose environment bond dimension $\chi$ is a second control parameter. For non-chiral gapped phases the method is quantitatively controlled: observables converge with $D$ and $\chi$, and near critical points the finite correlation length $\xi(D)$ induced by finite entanglement obeys a finite-correlation-length scaling theory analogous to finite-size scaling (Corboz et al., Phys. Rev. X 8, 031031 (2018); Rader and Läuchli, Phys. Rev. X 8, 031030 (2018)).

Chiral spin liquids (CSLs) are lattice analogs of fractional quantum Hall states: gapped, topologically ordered (ordered by long-range entanglement and anyonic quasiparticles rather than by local symmetry breaking), and breaking time-reversal symmetry, so that they carry protected chiral edge modes. The paradigm is Kitaev's honeycomb model (Ann. Phys. 321, 2 (2006)): spin-$\frac{1}{2}$ sites with bond-dependent Ising couplings, exactly solvable by Majorana fermionization (a representation of the spins in terms of Majorana fermions, particles that are their own antiparticles), with a three-spin chiral term $\kappa$ that drives a chiral Ising topological order; its exact excitation spectrum is known (Lahtinen et al., Ann. Phys. 323, 2286 (2008)). Barkeshli, Jiang, Thomale and Qi (Phys. Rev. Lett. 114, 026401 (2015)) introduced $Z_N$ generalizations; the $Z_3$ model has three-state clock variables, an exact $Z_3$ 1-form symmetry (a generalized symmetry whose charged excitations live on loops, implemented by commuting plaquette loop operators $W_p$), and a chiral ground state established by cylinder DMRG (density matrix renormalization group, a quasi-one-dimensional variational method applied to long cylinders; Chen et al., Phys. Rev. B 109, 155161 (2024)) and by fusion-category analysis (Eck and Fendley, SciPost Phys. 18, 170 (2025)); it is not exactly solvable, and the nature of its energy gap in the two-dimensional thermodynamic limit was unresolved before the source paper.

PEPS variational states for chiral phases have a structural problem, visible since Wahl, Tu, Schuch and Cirac (Phys. Rev. Lett. 111, 236805 (2013)) and Poilblanc (Phys. Rev. B 96, 121118 (2017)) and confirmed in all later optimizations (Hasik et al., Phys. Rev. Lett. 129, 177201 (2022); Niu et al., Phys. Rev. B 109, L081107 (2024); Weerda and Rizzi, Phys. Rev. B 109, L241117 (2024)): their connected correlation functions decay exponentially only up to a crossover scale, beyond which a tiny-amplitude power-law 'gossamer' tail persists and is pushed outward but never eliminated as $D$ grows. A parent Hamiltonian is a Hamiltonian for which the given state is an exact ground state; for injective PEPS (those whose site tensors map virtual boundary states one-to-one onto physical states) it can be constructed locally and is frustration-free (every interaction term attains its minimum on the state) and gapped, but chiral PEPS are non-injective and the construction fails. The obstruction is a theorem in the Gaussian sector only: Dubail and Read (Phys. Rev. B 92, 205307 (2015)) proved that any strictly short-range quadratic parent Hamiltonian of a Gaussian fermionic PEPS carrying nonzero Chern number (the integer topological invariant of the occupied single-particle bands, which forces chiral edge modes) is gapless, and explicitly stated that for the fractional (interacting) analogs obtained by Gutzwiller projection (removing configurations that violate a fixed particle number per site from a Gaussian fermionic state) gaplessness of the parent is not known. For the interacting 1-form-symmetric chiral PEPS of the source paper, the expectation — not a proven theorem — is that the exact parent must contain small-amplitude long-range interactions.

The source paper (Chen et al., Phys. Rev. B 113, 085130 (2026)) addresses whether gossamer tails preclude trustworthy bulk gaps. Using 1-form-symmetric PEPS (Tan et al., Commun. Phys. 2026, DOI 10.1038/s42005-026-02729-x) and a variational principle for excited states in the tangent space of the PEPS manifold (the linear space of infinitesimal deformations of the tensors) within a local mode approximation, they argue that correlations decaying faster than $r^{-2}$ keep the structure factor (the zero-momentum sum of the excited-state norm kernel) finite, so the tails do not force gapless modes. They validate on the $Z_2$ chiral Kitaev model ($D=4$, $\chi=80$; vison gap $0.52J$ versus the exact benchmark; ground-state energy within $10^{-4}E$), and give the first bulk-gap determination for the $Z_3$ model ($D=6$, $\chi=108$; ground-state energy $-2.9600J$ per unit cell versus exact-diagonalization $-3.013J$/$-3.016J$ on 18-/24-site tori), classifying single-site excitations into 81 vison configurations (visons: the gauge-flux excitations, labeled by plaquette-operator eigenvalues) grouped into 15 classes. The residual limitations they state are the open items: the exact parent Hamiltonian is unknown; finite-$\chi$ effects in the $Z_3$ model are large enough to prevent determining the long-distance tail's functional form; and the standard finite-correlation-length scaling frameworks may not separate gossamer tails from genuine critical correlations in chiral phases.

Problem Statement

The common objective is to make the finite-bond-dimension chiral PEPS representation of gapped chiral topological phases quantitatively trustworthy: to identify which Hamiltonian the optimized chiral PEPS is actually the ground state of, and to determine whether its tiny-amplitude algebraic ('gossamer') correlation tails are representation artifacts or physical. Throughout, PEPS denotes two-dimensional tensor-network wave functions built from site tensors contracted along virtual bonds, with bond dimension $D$ controlling the variational accuracy and environment dimension $\chi$ controlling the approximate contraction; the parent Hamiltonian of a state is a Hamiltonian for which the state is an exact ground state. Three jointly supporting deliverables capture the open items left by Chen et al., Phys. Rev. B 113, 085130 (2026):

1. Parent Hamiltonian. Construct explicitly and characterize the exact parent Hamiltonian of the 1-form-symmetric chiral PEPS (PEPS invariant under the commuting loop-plaquette operators of a 1-form symmetry) optimized for the $Z_2$ and $Z_3$ Kitaev models with chiral terms (honeycomb spin models whose three-spin chiral terms produce $\mathbb{Z}_2$ and $\mathbb{Z}_3$ gauge-structure topological orders; the $Z_2$ model is exactly solvable, the $Z_3$ model is not). By the no-go theorem (proven for Gaussian fermionic PEPS: any strictly short-range quadratic parent of a Chern-carrying Gaussian PEPS is gapless; expected but unproven for the interacting case), the parent is expected to deviate from the Kitaev Hamiltonians by long-range interactions with negligible amplitudes, but a strictly local gapped parent is not excluded. Determine whether the exact parent Hamiltonian can be simultaneously strictly local and gapped, and if it cannot, characterize the required long-range terms, their amplitude and decay scaling with distance, and their physical consequences.

2. Long-distance tail asymptotics of the $Z_3$ Kitaev chiral spin liquid. Determine the asymptotic functional form and decay exponent of the long-distance correlation tails of the $Z_3$ model, which finite-$\chi$ effects at the current $D=6$, $\chi=108$ state of the art prevent resolving.

3. Scaling diagnostics for chiral PEPS. Develop a refinement of the standard finite-correlation-length scaling frameworks (Corboz et al. 2018; Rader and Läuchli 2018) — the analog of finite-size scaling in which the finite correlation length induced by finite $D$ replaces the system size — that, for direct chiral PEPS, distinguishes negligible gossamer tails from physically relevant critical correlations, so that PEPS studies of chiral phases and of transitions involving chiral phases can decide whether an observed algebraic tail signals criticality or a representation artifact.

These three results jointly answer the source question: the parent Hamiltonian fixes what the computed gaps are gaps of, deliverable 2 settles the one concrete unresolved instance, and deliverable 3 supplies the criterion transferable to future chiral-PEPS studies.

References

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