Whether projected entangled-pair states (PEPS) of finite bond dimension can describe interacting chiral topological phases in two spatial dimensions - phases with nonzero chiral central charge c−c_- - is a foundational open question of the tensor-network formalism, posed explicitly by Arildsen, Chen, Schuch, and Ludwig (Phys. Rev. B 110, 235147 (2024)) while developing entanglement-spectrum chirality diagnostics for an SU(3) chiral spin liquid PEPS. For non-interacting fermions the question is settled, but adversely: chiral free-fermion PEPS exist, and a no-go theorem states that any strictly local quadratic parent Hamiltonian for them is gapless. For intrinsically interacting phases the question remains undecided as of September 2026: Gutzwiller-projected chiral PEPS exhibit the expected edge and entanglement-spectrum signatures and increasingly convincing evidence for gapped bulk excitations, yet every known example retains tiny-amplitude power-law correlation tails (gossamer tails), no exact gapped local parent Hamiltonian is known, and no no-go theorem beyond the free case exists. The problem asks for a resolution in either direction: a certified construction of an intrinsically interacting chiral PEPS with a local gapped parent Hamiltonian and certified nonzero chiral central charge, or a proof that no such PEPS exists.

Background

Chiral topological phases in two spatial dimensions are gapped quantum phases that break time-reversal symmetry and carry gapless, topologically protected edge modes described by a chiral conformal field theory (CFT, a 1+1-dimensional scale-invariant quantum field theory). Their canonical representatives are the fractional quantum Hall states and chiral spin liquids. Chirality is quantified by the chiral central charge c−c_-, whose exact value fixes, for instance, the thermal Hall response carried by the edge; given only the anyon content, c−c_- is determined modulo 8, since stacking an E8 phase shifts it by 8.

Projected entangled-pair states (PEPS) are variational wavefunctions for lattice systems built by contracting a network of identical local tensors whose legs of bond dimension DD are entangled in pairs; every PEPS satisfies an entanglement area law by construction, and PEPS with the appropriate virtual symmetry structure represent non-chiral ('doubled') topological order exactly, for instance string-net fixed points (exactly solvable lattice models whose topological order pairs every chiral sector with its time-reversed partner, and is hence non-chiral). Whether the ansatz class can reach beyond doubled topological order, to genuinely chiral phases, is a long-standing foundational question for the two-dimensional tensor-network program.

For non-interacting fermions the question is settled, with a twist. Wahl, Tu, Schuch, and Cirac (2013) and Dubail and Read (2015) constructed exact chiral PEPS for free-fermion topological phases (Chern insulators and p+ipp+ip superconductors, both characterized by a nonzero Chern number - an integer topological invariant of the occupied single-particle bands) and proved a 'no-go theorem': any strictly local Hamiltonian quadratic in the fermion operators that has such a state as its ground state is necessarily gapless, so a gapped parent Hamiltonian, if it exists, must be long-ranged. Within the free class, chiral PEPS are therefore never exact ground states of local gapped Hamiltonians.

The interacting case is approached through Gutzwiller projection, that is, enforcing one fermion per site on a partially filled chiral band, which yields a spin liquid inheriting the band's anyon and edge structure. Projected chiral PEPS (Yang et al. 2015; Poilblanc et al. 2015; Chen et al. 2020 and 2021, the first of these being the SU(3)1_1 chiral spin liquid on the square lattice analyzed by Arildsen et al. (2024); the subscript 1 denotes the level of the chiral SU(3) topological field theory) exhibit the characteristic Li-Haldane structure of the entanglement spectrum: the spectrum of the entanglement Hamiltonian −log⁡ρA-\log\rho_A of a bipartition mirrors the chiral edge CFT, with a universal counting of low-lying levels at fixed momentum (Li and Haldane, 2008). Two obstructions keep this from settling representability. First, 'apparently chiral' non-chiral PEPS that preserve only the product RT\mathcal{RT} of reflection and time reversal can reproduce the same Li-Haldane counting in some sectors (Kurečić et al. 2019; Arildsen et al. 2023); Arildsen et al. (2024) answered this diagnostic worry by proving that in a genuinely chiral SU(3)1_1 entanglement spectrum conjugate SU(3) irreducible representations are exactly degenerate (the operators that would split them are excluded by symmetry), providing a sharp chirality diagnostic. Second, every numerically studied chiral PEPS develops tiny-amplitude power-law 'gossamer' correlation tails (Hasik et al. 2022), which in the free-fermion case signal gaplessness of local parents; recent variational work argues that tails decaying faster than r−2r^{-2} do not preclude a bulk gap and finds evidence for gapped chiral bulk excitations (Chen et al. 2026), while leaving the exact parent Hamiltonian - expected to contain small long-range terms - undetermined. Chirality certification from the bulk state is in principle available through the modular commutator, an entanglement quantity that determines c−c_- (Kim et al. 2022; Zou et al. 2022).

Within this landscape, Arildsen et al. (2024) pose the fundamental question that this problem records: whether PEPS can generally represent interacting (2+1)-dimensional chiral topological phases. As of September 2026 the literature treats the question as open and controversial: no interacting chiral PEPS with a proven gapped local parent is known, no no-go theorem beyond the free-fermion case exists, and the newest contributions engineer workarounds - for example embedding the chiral system together with a time-reversed auxiliary partner to obtain a non-chiral enlarged PEPS (Niu and Huang, 2026) - rather than resolving the pure-state question.

Problem Statement

Decide whether projected entangled-pair states of finite bond dimension can represent intrinsically interacting (2+1)-dimensional chiral topological phases as ground states of local, gapped Hamiltonians. 'Intrinsically interacting' excludes the free-fermion (Gaussian PEPS) case, which is already settled by the no-go theorem of Wahl et al. (2013) and Dubail and Read (2015). The question admits two decisive forms, either of which resolves it.

(A) Construction: exhibit, on a two-dimensional lattice, a finite-bond-dimension PEPS ∣ψ⟩|\psi\rangle together with a local Hamiltonian HH (finitely many, uniformly bounded-range interaction terms) such that: (i) HH has a spectral gap in the thermodynamic limit; (ii) ∣ψ⟩|\psi\rangle is provably a ground state of HH (unique up to topological degeneracy on closed manifolds); (iii) the phase has nonzero chiral central charge, c−≠0c_-\neq 0, certified either exactly (for instance via topological data or entanglement-spectrum structure such as exact conjugate-representation degeneracy in SU(N)-symmetric states) or via a chirality measure such as the modular commutator with rigorous convergence in the thermodynamic limit; and (iv) the state is not reducible to a Gaussian/free-fermion PEPS.

(B) No-go theorem: prove that no such PEPS exists - that any finite-bond-dimension PEPS that is the ground state of a local, gapped two-dimensional Hamiltonian has vanishing chiral central charge - thereby extending the free-fermion no-go theorem to interacting states, with the class of admissible Hamiltonians and PEPS precisely delimited.

Forms (A) and (B) are each sufficient to resolve the task; the construction-or-no-go dichotomy above is the required decisive core. A characterization of which interacting chiral phases admit (or cannot admit) such representations, or a proof that all interacting chiral phases admit such representations, is a strictly stronger and optional result answering the question in its strongest 'generally represent' form. All statements are to be understood in the thermodynamic limit, since any fixed finite lattice system can be approximated by PEPS arbitrarily well. Numerical variational evidence alone - which currently exists on both sides of the issue (gossamer tails suggesting obstruction, variational excitation gaps suggesting realizability) - does not constitute an answer.

References

  1. M. J. Arildsen, J.-Y. Chen, N. Schuch, and A. W. W. Ludwig, 'Entanglement spectrum as a diagnostic of chirality of topological spin liquids: Analysis of SU(3) projected entangled pair states', Phys. Rev. B 110, 235147 (2024), DOI: 10.1103/PhysRevB.110.235147, arXiv:2305.13240, https://arxiv.org/abs/2305.13240 (the arXiv preprint version carries the shorter title 'Analysis of an SU(3) PEPS')
  2. T. B. Wahl, H.-H. Tu, N. Schuch, and J. I. Cirac, 'Projected entangled-pair states can describe chiral topological states', Phys. Rev. Lett. 111, 236805 (2013), DOI: 10.1103/PhysRevLett.111.236805, arXiv:1308.0316, https://arxiv.org/abs/1308.0316
  3. J. Dubail and N. Read, 'Tensor network trial states for chiral topological phases in two dimensions and a no-go theorem in any dimension', Phys. Rev. B 92, 205307 (2015), DOI: 10.1103/PhysRevB.92.205307, arXiv:1307.7726, https://arxiv.org/abs/1307.7726
  4. S. Yang, T. B. Wahl, H.-H. Tu, N. Schuch, and J. I. Cirac, 'Chiral projected entangled-pair state with topological order', Phys. Rev. Lett. 114, 106803 (2015), DOI: 10.1103/PhysRevLett.114.106803, arXiv:1411.6618, https://arxiv.org/abs/1411.6618
  5. D. Poilblanc, J. I. Cirac, and N. Schuch, 'Chiral topological spin liquids with projected entangled pair states', Phys. Rev. B 91, 224431 (2015), DOI: 10.1103/PhysRevB.91.224431, https://doi.org/10.1103/PhysRevB.91.224431
  6. J.-Y. Chen, S. Capponi, A. Wietek, M. Mambrini, N. Schuch, and D. Poilblanc, 'SU(3)1_1 Chiral Spin Liquid on the Square Lattice: A View from Symmetric Projected Entangled Pair States', Phys. Rev. Lett. 125, 017201 (2020), DOI: 10.1103/PhysRevLett.125.017201, arXiv:1912.13393, https://arxiv.org/abs/1912.13393
  7. J.-Y. Chen, J.-W. Li, P. Nataf, S. Capponi, M. Mambrini, K. Totsuka, H.-H. Tu, A. Weichselbaum, J. von Delft, and D. Poilblanc, 'Abelian SU(N)1_1 chiral spin liquids on the square lattice', Phys. Rev. B 104, 235104 (2021), DOI: 10.1103/PhysRevB.104.235104, arXiv:2106.02115, https://arxiv.org/abs/2106.02115
  8. I. Kurečić, L. Vanderstraeten, and N. Schuch, 'Gapped SU(3) spin liquid with Z3\mathbb{Z}_3 topological order', Phys. Rev. B 99, 045116 (2019), DOI: 10.1103/PhysRevB.99.045116, arXiv:1805.11628, https://arxiv.org/abs/1805.11628
  9. M. J. Arildsen, N. Schuch, and A. W. W. Ludwig, 'Entanglement spectra of nonchiral topological (2+1)-dimensional phases with strong time-reversal symmetry breaking, Li-Haldane state counting, and PEPS', Phys. Rev. B 108, 245150 (2023), DOI: 10.1103/PhysRevB.108.245150, arXiv:2207.03246, https://arxiv.org/abs/2207.03246
  10. J. Hasik, M. Van Damme, D. Poilblanc, and L. Vanderstraeten, 'Simulating Chiral Spin Liquids with Projected Entangled-Pair States', Phys. Rev. Lett. 129, 177201 (2022), DOI: 10.1103/PhysRevLett.129.177201, arXiv:2201.07758, https://arxiv.org/abs/2201.07758
  11. S. Budaraju, D. Poilblanc, and S. Niu, 'Simulating chiral spin liquids with fermionic projected entangled pair states', Phys. Rev. B 110, 064402 (2024), DOI: 10.1103/PhysRevB.110.064402, arXiv:2403.01243, https://arxiv.org/abs/2403.01243
  12. J.-Y. Chen, Y. Tan, S. Capponi, D. Poilblanc, F. Ye, and J.-W. Mei, 'Simulating bulk gap in chiral projected entangled pair states', Phys. Rev. B 113, 085130 (2026), DOI: 10.1103/g8cb-fz29, arXiv:2502.20142, https://arxiv.org/abs/2502.20142
  13. H. Chen, T. Neupert, and J. Hasik, 'Simulating Fermionic Fractional Chern Insulators with Infinite Projected Entangled-Pair States', Phys. Rev. Lett. 137, 046502 (2026), DOI: 10.1103/c9dr-bpn4, arXiv:2512.20697, https://arxiv.org/abs/2512.20697
  14. R. Fan, Y. Wu, Y. Bao, and Z. Dai, 'No-go theorems for sequential preparation of two-dimensional chiral states via channel-state correspondence', arXiv:2511.19612 (2025), https://arxiv.org/abs/2511.19612
  15. S. Niu and R.-Z. Huang, 'Accessing Gapped Chiral Phase with Auxiliary-Assisted PEPS', arXiv:2608.15732 (2026), https://arxiv.org/abs/2608.15732
  16. I. H. Kim, B. Shi, K. Kato, and V. V. Albert, 'Chiral Central Charge from a Single Bulk Wave Function', Phys. Rev. Lett. 128, 176402 (2022), DOI: 10.1103/PhysRevLett.128.176402, arXiv:2110.06932, https://arxiv.org/abs/2110.06932
  17. Y. Zou, B. Shi, J. Sorce, I. T. Lim, and I. H. Kim, 'Modular Commutators in Conformal Field Theory', Phys. Rev. Lett. 129, 260402 (2022), DOI: 10.1103/PhysRevLett.129.260402, arXiv:2206.00027, https://arxiv.org/abs/2206.00027
  18. H. Li and F. D. M. Haldane, 'Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States', Phys. Rev. Lett. 101, 010504 (2008), DOI: 10.1103/PhysRevLett.101.010504, arXiv:0805.0332, https://arxiv.org/abs/0805.0332
  19. Y. Xu, S. Capponi, J.-Y. Chen, L. Vanderstraeten, J. Hasik, A. H. Nevidomskyy, M. Mambrini, K. Penc, and D. Poilblanc, 'Phase diagram of the chiral SU(3) antiferromagnet on the kagome lattice', Phys. Rev. B 108, 195153 (2023), DOI: 10.1103/PhysRevB.108.195153, arXiv:2306.16192, https://arxiv.org/abs/2306.16192
  20. S. Srinidhi, S. C. L. Srivastava, and J. N. Bandyopadhyay, 'Entanglement signatures of gapless topological phases in a p-wave superconductor', Phys. Rev. B 113, 064509 (2026), DOI: 10.1103/s23b-sgpp, arXiv:2509.23300, https://arxiv.org/abs/2509.23300
  21. G. Scarpa, A. Molnár, Y. Ge, J. J. García-Ripoll, N. Schuch, D. Pérez-García, and S. Iblisdir, 'Projected Entangled Pair States: Fundamental Analytical and Numerical Limitations', Phys. Rev. Lett. 125, 210504 (2020), DOI: 10.1103/PhysRevLett.125.210504, https://doi.org/10.1103/PhysRevLett.125.210504