Natural momentum labeling of corner-tensor entanglement spectra for infinite two-dimensional quantum systems

Huang, Wei, and Orús [Phys. Rev. B 95, 195170 (2017)] showed that corner transfer matrices (CTMs) and corner tensors (CTs) computed in standard tensor-network algorithms holographically encode universal bulk information, including the entanglement spectrum of half of an infinite two-dimensional quantum system, with the degeneracy structure of the $SU(2)_k$ Wess-Zumino-Witten edge theories of chiral topological PEPS emerging for $k=1,2$. A basic limitation they identified is that this corner-based entanglement spectrum carries no momentum quantum numbers: the half-infinite-plane construction provides no natural labeling of the eigenvalues by momentum along the entanglement boundary. They proposed that appropriate translation operators defined directly on the CTMs could supply such a labeling and announced the idea for future work, but it has not been developed. Subsequent work reached momentum-resolved entanglement spectra of infinite two-dimensional tensor-network states only by bypassing the corner construction: either by wrapping the boundary tensors on finite-width cylinders, which quantizes the momentum, requires a boundary-condition choice, and needs width extrapolation, or by variationally reconstructing a one-dimensional boundary entanglement Hamiltonian whose quasiparticle excitations carry momentum — leaving the corner spectrum itself without a momentum labeling. The open problem is to construct a momentum-resolved corner entanglement spectrum directly in the infinite half-plane geometry and validate it against known momentum-resolved band and counting structures.

Background

Ground states of two-dimensional quantum lattice systems are efficiently represented by tensor networks. A projected entangled pair state (PEPS) assigns a tensor to each lattice site, with virtual indices connecting neighboring sites, and physical indices carrying the local Hilbert space; an infinite, translation-invariant PEPS (iPEPS) describes the thermodynamic limit directly. Evaluating observables requires contracting an infinite network, which is done approximately by iterative algorithms. The workhorse is the corner transfer matrix renormalization group (CTMRG) method of Nishino and Okunishi, adapted to quantum states by Orús and Vidal: the contraction is approximated by four corner transfer matrices (CTMs) — the environment tensors of the four quadrants of the plane — together with edge tensors, all truncated to a finite bond dimension $\chi$.

Beyond being a numerical device, the CTM spectrum itself carries physics. Via the imaginary-time path-integral mapping between a $d$-dimensional quantum partition function and a $(d+1)$-dimensional classical one, the CTM spectrum of the classical network equals the entanglement spectrum of the dual quantum state: the eigenvalues of the reduced density matrix $\rho_A$ of half the system, conventionally written as levels of the entanglement Hamiltonian $H_{\mathrm{ent}}=-\log\rho_A$ (Li and Haldane). Huang, Wei, and Orús (HWO) systematically explored this holographic encoding. They showed that corner spectra and the corner entropy $S=-\mathrm{tr}\,\rho_{\mathrm{corner}}\log\rho_{\mathrm{corner}}$ (built from the four corner matrices) pinpoint quantum phase transitions, topological or not, without computing observables; that corner data confirm the $d$-dimensional-quantum versus $(d+1)$-dimensional-classical correspondence for several universality classes; and — via a quantum-state renormalization scheme built on corner tensors — that the entanglement spectrum of half of an infinite two-dimensional system can be extracted from corner data. For chiral topological PEPS (built from the tensors of Poilblanc, Cirac, and Schuch), the corner spectra reproduce the degeneracy pattern of the $SU(2)_k$ Wess-Zumino-Witten (WZW) conformal field theories describing the gapless chiral edges for $k=1,2$, consistent with the bulk-edge correspondence between a topological phase and its boundary theory.

One piece of information is missing. In cylinder-geometry computations (the standard route, then and since), the entanglement cut of a cylinder of finite circumference $W$ is a one-dimensional system, so the entanglement levels fall into momentum sectors $K=2\pi n/W$; this momentum resolution is what reveals, for example, the chiral branch structure of a topological edge mode, level by level, following the tower structure (the Virasoro tower) of the edge conformal field theory. In contrast, the half-infinite-plane corner construction of HWO yields entanglement eigenvalues $\omega_\alpha$ that are naturally indexed only by magnitude. HWO observe that the low-lying spectrum of, e.g., the disordered Ising PEPS of Rispler, Duivenvoorden, and Schuch forms one-dimensional bands versus momentum, and that the finite $\chi$ induces an effective boundary length that discretizes the momenta sampled — but which band position each corner eigenvalue corresponds to is not determined. In their conclusions they state that the construction "does not provide a 'natural' way of labelling the different eigenvalues in terms of a momenta quantum number", propose that "this may be possible by defining appropriate translation operators on CTMs", and announce this idea for future work.

After 2017 the practical need for momentum-resolved entanglement spectra of infinite two-dimensional tensor-network states was met by bypasses rather than by the corner construction itself. The main route returned to the cylinder geometry, following the alternative already sketched in HWO's own conclusions (wrapping the boundary tensors around a finite width): Chen, Vanderstraeten, Capponi, and Poilblanc computed momentum-resolved entanglement spectra of a non-Abelian chiral spin-liquid iPEPS by contracting the CTMRG boundary tensors on finite-circumference rings, reproducing the $SU(2)_2$ edge counting. The same work also extracted a momentum-resolved boundary spectrum in the infinite geometry by interpreting the fixed point of the PEPS transfer matrix as a boundary entanglement Hamiltonian and diagonalizing it through a variational one-dimensional uniform matrix-product state with a quasiparticle excitation ansatz. Chen, Neupert, and Hasik (2025) extended the cylinder route to fermionic fractional Chern insulators using a compression scheme for the boundary matrix-product operators. The cylinder constructions quantize the boundary momentum by the cylinder width, require a periodic-versus-antiperiodic boundary-condition choice, and require width extrapolation, while the boundary-Hamiltonian route replaces the corner spectrum by a separate variational object rather than labeling it. The half-plane corner spectrum itself remains momentum-unlabeled, and the correspondence between corner eigenvalues and momentum sectors has not been established in the literature.

Problem Statement

For a translation-invariant infinite two-dimensional quantum state in PEPS form (the setting of the source), construct a momentum-resolved version of the corner-tensor entanglement spectrum in the half-infinite-plane geometry, as proposed by Huang, Wei, and Orús: define appropriate translation operators acting on the corner transfer matrices — or an equivalent momentum-labeling structure on the CTM fixed-point environment — such that the eigenvalues of the corner-based entanglement spectrum of half the plane are labeled by a momentum quantum number along the entanglement boundary. An admissible answer must: (i) make the construction precise and well-defined for the infinite system, with momenta taking continuous values on the infinite boundary rather than being quantized by a finite cylinder width; (ii) demonstrate it on the source's benchmark settings, namely the disordered Ising PEPS whose low entanglement spectrum is known to form one-dimensional bands versus momentum, and the chiral topological PEPS with $SU(2)_1$ and $SU(2)_2$ gapless edges, whose low-lying degeneracies follow the WZW Virasoro-tower counting with its characteristic chiral momentum dependence; and (iii) establish consistency with momentum-resolved spectra obtained by wrapping the boundary tensors on finite-width cylinders, including how the corner eigenvalues distribute among momentum sectors as the CTM bond dimension $\chi$ and the cylinder circumference grow.

References

1. C.-Y. Huang, T.-C. Wei, and R. 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. T. Nishino and K. 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
3. R. Orús and G. Vidal, "Simulation of two-dimensional quantum systems on an infinite lattice revisited: Corner transfer matrix for tensor contraction and renormalization", Phys. Rev. B 80, 094403 (2009). DOI: 10.1103/PhysRevB.80.094403, https://doi.org/10.1103/PhysRevB.80.094403
4. J. I. Cirac, D. Poilblanc, N. Schuch, and F. 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
5. 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, https://doi.org/10.1103/PhysRevLett.101.010504
6. 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
7. D. Poilblanc, N. Schuch, and I. 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. M. Mambrini, R. Orús, and D. 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. M. Rispler, K. Duivenvoorden, and N. Schuch, "Long-range order and symmetry breaking in projected entangled-pair state models", Phys. Rev. B 92, 155133 (2015). DOI: 10.1103/PhysRevB.92.155133, https://doi.org/10.1103/PhysRevB.92.155133
10. J.-Y. Chen, L. Vanderstraeten, S. Capponi, and D. 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
11. H. Chen, T. Neupert, and J. Hasik, "Simulating Fermionic Fractional Chern Insulators with Infinite Projected Entangled-Pair States", arXiv:2512.20697 (2025), https://arxiv.org/abs/2512.20697
12. N. Pomata, C.-Y. Huang, and T.-C. 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 (the arXiv version carries the shortened title "Phase transitions of a 2D deformed-AKLT model"), https://arxiv.org/abs/1711.00036
13. A. Kshetrimayum, H. Weimer, and R. 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
14. A. A. Gangat and Y.-J. 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
15. P. Schmoll, A. Kshetrimayum, J. Eisert, R. Orús, and M. 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. I. V. Lukin and 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