Numerical corner transfer matrices (CTMs) and their higher-dimensional generalizations encode universal data of infinite two-dimensional lattice systems in their spectra: conformal-field-theory characters, boundary central charges, phase-transition locations, and — for chiral topological projected entangled-pair states (PEPS) — the entanglement spectrum of half the system, whose degeneracies reproduce the Virasoro-tower counting of the gapless edge theory. Huang, Wei, and Orús [Phys. Rev. B 95, 195170 (2017)] identified one structural limitation of this corner framework: unlike cylinder-based calculations, where the entanglement spectrum resolves into one-dimensional bands versus momentum, the corner-derived eigenvalues of the half-plane reduced density matrix carry no momentum quantum number. They conjectured that appropriate translation operators acting on the CTMs could restore momentum labels and left the construction open. The open problem is to construct such translation operators — operators acting on the corner and edge environment tensors that implement lattice translations parallel to the entanglement cut and commute with the half-plane reduced density matrix — thereby assigning momentum labels to CT-derived entanglement eigenvalues, and to demonstrate on benchmark models (the Ising PEPS band structure and chiral topological PEPS with SU(2)kSU(2)_k edges) that the construction reproduces the momentum-resolved entanglement spectra of cylinder calculations directly in the thermodynamic limit.

Background

Consider a two-dimensional (2D) quantum lattice system in its ground state, divided by a straight cut into two half-planes. The entanglement spectrum is the set of eigenvalues of the reduced density matrix ρ\rho of one half; by the Li–Haldane correspondence it often mirrors the spectrum of a physical boundary of the same system, so for topological phases it reproduces the edge theory. A projected entangled-pair state (PEPS) is a tensor-network wavefunction in which each lattice site carries a tensor whose virtual indices are contracted along the lattice bonds; for an infinite PEPS (iPEPS) the network is contracted numerically with the corner transfer matrix renormalization group (CTMRG), which represents the environment around the bipartition by corner transfer matrices (a CTM is the matrix associated with one quadrant of a two-dimensional tensor network, related historically to exact solutions of classical lattice models) together with half-row and half-column edge tensors. The eigenvalues of the CTMs are directly related to those of ρ\rho for a half-plane, so the entanglement spectrum is available without any finite-size direction. Huang, Wei, and Orús (2017) showed that these corner spectra encode universal data directly in the thermodynamic limit: corner energies realize the correspondence between dd-dimensional quantum and (d+1)(d+1)-dimensional classical models, corner spectra locate phase transitions (topological or not) without needing observables, and for chiral topological PEPS — interacting spin-liquid wavefunctions with gapless chiral edges — the corner-derived entanglement spectrum of half the system reproduces the degeneracy counting of the vacuum Virasoro tower (the organized levels of the edge conformal field theory) of the SU(2)kSU(2)_k Wess–Zumino–Witten (WZW) edge theory for k=1,2k=1,2. On an infinitely long cylinder of finite circumference, by contrast, translation around the cylinder commutes with the bipartition, so the entanglement spectrum resolves into one-dimensional bands versus momentum, and per-momentum level counting is the standard diagnostic of chiral topological order in tensor-network calculations. The corner framework works directly on the infinite plane, which has no finite direction, and its eigenvalues carry no momentum label; the spectrum can be matched to conformal-field-theory degeneracies but not assigned to momentum sectors. Huang, Wei, and Orús pointed out this limitation explicitly, proposed that 'appropriate translation operators' acting on CTMs might restore momentum labeling, and left the construction to future work. Since the half-plane reduced density matrix of a translation-invariant system commutes with lattice translations along the cut, its eigenvalues do admit momentum labels in principle; the open question is how to realize this labeling inside the truncated corner-tensor formalism, where the translation operator must act on a finite bond-dimensional boundary space that the standard truncation isometries do not treat symmetrically.

Problem Statement

Within the corner-transfer-matrix / corner-tensor framework for translation-invariant infinite two-dimensional systems — both 2D classical partition functions and 2D quantum ground states written as PEPS, including the double-layer norm networks (the tensor network obtained by contracting the bra and ket copies of the state, which computes ⟨ψ∣ψ⟩\langle\psi|\psi\rangle) and the single-layer quantum-state renormalization scheme of Huang–Wei–Orús — construct operators that implement lattice translations parallel to the entanglement cut, act on the corner and edge environment tensors, and commute with the reduced density matrix of a half-plane, so that each eigenvalue of the CT-derived entanglement spectrum of a half-plane can be assigned a momentum quantum number. This general framework is the setting in which the construction is to be formulated; the two benchmark models below define the required level of demonstration, and resolving the task consists in the operator construction together with demonstrations on both. The construction must make precise in what sense momentum labels exist in the thermodynamic limit, where the half-plane entanglement spectrum is continuous and the discreteness of the corner-tensor spectrum is induced by the finite corner bond dimension χ\chi. The construction must be validated on those two benchmark models: (a) a system such as the 2D Ising PEPS, whose low-lying half-plane entanglement spectrum is known to form one-dimensional bands versus momentum; and (b) a chiral topological PEPS with an SU(2)kSU(2)_k WZW edge, where the momentum-resolved level counting in each momentum sector should match the conformal-field-theory Virasoro-tower counting obtained from finite-circumference cylinder calculations. The common objective is to reproduce, directly in the thermodynamic-limit corner framework, the momentum-resolved entanglement spectra that cylinder geometries provide — the missing structural element of the corner-spectrum diagnostic program identified by Huang, Wei, and Orús (2017).

References

  1. Ching-Yu Huang, Tzu-Chieh Wei, Roman Orús, 'Holographic encoding of universality in corner spectra', Phys. Rev. B 95, 195170 (2017). DOI: 10.1103/PhysRevB.95.195170; arXiv:1702.01598; https://arxiv.org/abs/1702.01598
  2. Hui Li, 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; https://doi.org/10.1103/PhysRevLett.101.010504
  3. Tomotoshi Nishino, Kouichi Okunishi, 'Corner Transfer Matrix Renormalization Group Method', J. Phys. Soc. Jpn. 65, 891 (1996). DOI: 10.1143/JPSJ.65.891; https://doi.org/10.1143/JPSJ.65.891
  4. Román Orús, Guifré Vidal, 'Simulation of two-dimensional quantum systems on an infinite lattice revisited: Corner transfer matrix for tensor contraction', Phys. Rev. B 80, 094403 (2009). DOI: 10.1103/PhysRevB.80.094403; arXiv:0905.3225; https://arxiv.org/abs/0905.3225
  5. J. Ignacio Cirac, Didier Poilblanc, Norbert Schuch, Frank Verstraete, 'Entanglement spectrum and boundary theories with projected entangled-pair states', Phys. Rev. B 83, 245134 (2011). DOI: 10.1103/PhysRevB.83.245134; https://doi.org/10.1103/PhysRevB.83.245134
  6. Didier Poilblanc, Norbert Schuch, David Pérez-García, J. Ignacio Cirac, 'Topological and entanglement properties of resonating valence bond wave functions', Phys. Rev. B 86, 014404 (2012). DOI: 10.1103/PhysRevB.86.014404; arXiv:1202.0947; https://arxiv.org/abs/1202.0947
  7. Didier Poilblanc, Norbert Schuch, Ian Affleck, 'SU(2)_1 chiral edge modes of a critical spin liquid', Phys. Rev. B 93, 174414 (2016). DOI: 10.1103/PhysRevB.93.174414; https://doi.org/10.1103/PhysRevB.93.174414
  8. Matthieu Mambrini, Román Orús, Didier Poilblanc, 'Systematic construction of spin liquids on the square lattice from tensor networks with SU(2) symmetry', Phys. Rev. B 94, 205124 (2016). DOI: 10.1103/PhysRevB.94.205124; https://doi.org/10.1103/PhysRevB.94.205124
  9. S. N. Saadatmand, I. P. McCulloch, 'Detection and characterization of symmetry-broken long-range orders in the spin-1/2 triangular Heisenberg model', Phys. Rev. B 96, 075117 (2017). DOI: 10.1103/PhysRevB.96.075117; arXiv:1704.03418; https://arxiv.org/abs/1704.03418
  10. Augustine Kshetrimayum, Hendrik Weimer, Román Orús, 'A simple tensor network algorithm for two-dimensional steady states', Nat. Commun. 8, 1291 (2017). DOI: 10.1038/s41467-017-01511-6; arXiv:1612.00656; https://arxiv.org/abs/1612.00656
  11. Nicholas Pomata, Ching-Yu Huang, Tzu-Chieh Wei, 'Phase transitions of a two-dimensional deformed Affleck-Kennedy-Lieb-Tasaki model', Phys. Rev. B 98, 014432 (2018). DOI: 10.1103/PhysRevB.98.014432; arXiv:1711.00036; https://arxiv.org/abs/1711.00036
  12. Ji-Yao Chen, Laurens Vanderstraeten, Sylvain Capponi, Didier Poilblanc, 'Non-Abelian chiral spin liquid in a quantum antiferromagnet revealed by an iPEPS study', Phys. Rev. B 98, 184409 (2018). DOI: 10.1103/PhysRevB.98.184409; arXiv:1807.04385; https://arxiv.org/abs/1807.04385
  13. Adil A. Gangat, Ying-Jer Kao, 'Phase boundary location with information-theoretic entropy in tensor renormalization group flows', Phys. Rev. B 100, 094430 (2019). DOI: 10.1103/PhysRevB.100.094430; arXiv:1901.08193; https://arxiv.org/abs/1901.08193
  14. Sara Murciano, Giuseppe Di Giulio, Pasquale Calabrese, 'Symmetry resolved entanglement in gapped integrable systems: a corner transfer matrix approach', SciPost Phys. 8, 046 (2020). DOI: 10.21468/SciPostPhys.8.3.046; arXiv:1911.09588; https://arxiv.org/abs/1911.09588
  15. Philipp Schmoll, Augustine Kshetrimayum, Jens Eisert, Román Orús, Matteo Rizzi, 'The classical two-dimensional Heisenberg model revisited: An SU(2)-symmetric tensor network study', SciPost Phys. 11, 098 (2021). DOI: 10.21468/SciPostPhys.11.5.098; arXiv:2106.06310; https://arxiv.org/abs/2106.06310
  16. Ravi Teja Ponnaganti, Matthieu Mambrini, Didier Poilblanc, 'Tensor network variational optimizations for real-time dynamics: Application to the time-evolution of spin liquids', SciPost Phys. 15, 158 (2023). DOI: 10.21468/SciPostPhys.15.4.158; arXiv:2304.13184; https://arxiv.org/abs/2304.13184
  17. Mark J. Arildsen, Ji-Yao Chen, Norbert Schuch, Andreas 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
  18. I. V. Lukin, A. G. Sotnikov, 'Corner transfer matrix renormalization group approach in the zoo of Archimedean lattices', Phys. Rev. E 109, 045305 (2024). DOI: 10.1103/PhysRevE.109.045305; arXiv:2401.07274; https://arxiv.org/abs/2401.07274
  19. Illia Lukin, Andrii Sotnikov, 'Single-layer tensor network approach for three-dimensional quantum systems', Phys. Rev. B 110, 064422 (2024). DOI: 10.1103/PhysRevB.110.064422; arXiv:2405.01489; https://arxiv.org/abs/2405.01489
  20. Valeriia Bilokon, Elvira Bilokon, Illya Lukin, Andrii Sotnikov, Denys I. Bondar, 'Dispersion relations in two- and three-dimensional quantum systems', Phys. Rev. Research 8, L012063 (2026). DOI: 10.1103/rnr1-bbjw; arXiv:2509.15483; https://arxiv.org/abs/2509.15483
  21. Hao Chen, Titus Neupert, Juraj 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