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This paper claims that a U(1)-symmetric fermionic iPEPS with bond dimension D≥7, optimized for the Haldane–Hubbard model at ν=1/3, faithfully represents the fractional Chern insulator phase, as shown by the (1,1,2,3,5) entanglement-spectrum

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2026-08-03 14:21 UTC pith:NUO7OK2R

load-bearing objection First credible iPEPS simulation of a fermionic chiral topological phase; the D≥7 FCI evidence is solid, but the 'critical bond dimension' is a heuristic supported by a finite-size ED gap proxy that the paper never validates. the 2 major comments →

arxiv 2512.20697 v3 pith:NUO7OK2R submitted 2025-12-23 cond-mat.str-el quant-ph

Simulating fermionic fractional Chern insulators with infinite projected entangled-pair states

Hao Chen , Titus Neupert , Juraj Hasik This is my paper
classification cond-mat.str-el quant-ph
keywords fractional Chern insulatoriPEPSfermionic topological orderentanglement spectrumchiral edge modesLaughlin stateHaldane-Hubbard modeltensor networks
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tries to establish that infinite projected entangled-pair states (iPEPS), a tensor network ansatz defined directly in the thermodynamic limit, can faithfully represent a fermionic fractional Chern insulator (FCI) once the bond dimension D reaches 7. Working with a U(1)-symmetric fermionic iPEPS on the Haldane–Hubbard model at 1/3 filling, the authors find a critical bond dimension: states with D≤6 fall into variational basins with charge modulations and no FCI signatures, while D≥7 states reproduce the universal fingerprints of the topological phase. These fingerprints are the (1,1,2,3,5) momentum-resolved entanglement-spectrum counting of a chiral U(1) boson edge, a pair-correlation function matching the ν=1/3 Laughlin state, and variational energies below the first-excitation gap computed by exact diagonalization on small tori. If correct, this makes iPEPS a working tool for chiral fermionic topological order in two dimensions, complementing methods limited to small clusters or finite-width cylinders.

Core claim

The central discovery is that a finite-bond-dimension fermionic iPEPS can faithfully represent a fractional Chern insulator, despite the no-go theorem that forbids any exact finite-correlation-length iPEPS representation of a gapped chiral phase. For the specific model studied, the authors find D_min=7: below this bond dimension the optimized variational states are energetically above the first-excitation gap and show charge inhomogeneity; at and above D=7 the states settle into the FCI phase. The faithfulness of the D≥7 states is established by three independent signatures: the bulk equal-time single-particle Green's function decays with a short correlation length followed by a small 'gossa

What carries the argument

The key machinery is the U(1)-symmetric fermionic iPEPS built from three independent rank-5 tensors arranged in a 3×3 unit cell, with fermionic statistics encoded by swap gates and the fractional filling ν=1/3 enforced exactly by attaching an auxiliary charge leg to one tensor. Optimization uses fixed-point automatic differentiation of the CTMRG environment, which computes gradients directly at the fixed point and avoids storing the iteration history of the slowly converging CTMRG. For the entanglement spectrum, the load-bearing identity is spec(ρ_L)=spec(σ_L^T σ_R), which expresses the spectrum of the reduced density matrix of a cylinder half in terms of left and right leading eigenvectors

Load-bearing premise

The D_min=7 threshold is anchored by comparing iPEPS energies to the first-excitation gap from exact diagonalization on tori of up to 24 unit cells; if that small-cluster gap drifts with system size, the boundary between 'wrong basin' and 'faithful phase' could shift.

What would settle it

Compute the first-excitation gap of the same Haldane–Hubbard model on larger tori (36 or 48 unit cells, i.e., 12 or 16 fermions at ν=1/3) or on wider iDMRG cylinders: if the gap falls below the D=6 iPEPS energy, or if a D≥10 iPEPS state loses the (1,1,2,3,5) entanglement-spectrum counting, the central claim would need revision.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • iPEPS with D≥7 becomes a variational method for fermionic chiral topological phases directly in the thermodynamic limit, without cluster-size or cylinder-width constraints.
  • The (1,1,2,3,5) entanglement-spectrum counting serves as a sharp fingerprint of the ν=1/3 Laughlin-type FCI in iPEPS, stable under APBC/PBC and across bond dimensions.
  • The MPO unit-cell compression scheme enables entanglement-spectrum studies of large iPEPS unit cells, with the low-lying spectrum already converged at modest environment and cutoff dimensions.
  • The methodology extends to other filling fractions, lattice geometries, and potentially to hierarchy and non-Abelian FCI states, as the authors state.
  • The existence of a minimal bond dimension for faithful representation provides a practical criterion: below D_min, variational energies above the excitation gap signal a wrong variational basin.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the D_min threshold behavior holds generally, it could serve as a diagnostic 'topological basin' criterion: a variational manifold that fails to reach below the gap may systematically miss chiral topological order, and checking energy versus gap could be a cheap pre-screening step in other FCI or chiral spin liquid models.
  • The single-magnetic-length Laughlin match suggests that the pair-correlation function of lattice FCI states could be used as a quantitative estimator of the effective magnetic length, which may help bridge lattice simulations and moiré materials experiments.
  • A natural extension the paper leaves implicit is computing neutral and charged excitations on top of the D≥7 ground state to extract the bulk gap directly in the thermodynamic limit; this would replace the small-torus ED gap as the primary benchmark for D_min.
  • The HOSVD-based MPO unit-cell compression may transfer to other tensor-network boundary-spectrum calculations where multi-site unit cells make direct diagonalization infeasible.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The manuscript reports variational infinite projected entangled-pair-state (iPEPS) simulations of a spinless-fermion Haldane-Hubbard model on the honeycomb lattice at filling ν=1/3, using U(1)-symmetric fermionic tensors with bond dimensions up to D=9, optimized by automatic differentiation with fixed-point CTMRG. The central claims are (i) a critical bond dimension D_min=7 above which the ansatz faithfully represents the fractional Chern insulator (FCI) phase; (ii) the D≥7 states are identified as FCI by the momentum-resolved entanglement spectrum with chiral U(1) boson counting (1,1,2,3,5), the pair-correlation function g(x) matching the ν=1/3 Laughlin state, and variational energies below the first excited level from exact diagonalization; and (iii) a new MPO unit-cell compression scheme that makes entanglement-spectrum calculations for large unit cells feasible and well converged at modest cutoffs.

Significance. If correct, this is the first demonstration that iPEPS can variationally capture a fermionic chiral topological phase directly in the thermodynamic limit, extending earlier bosonic chiral spin liquid and bosonic FQH work. The D≥7 entanglement-spectrum fingerprint is robust across environment bond dimensions χ, compression cutoffs D_c, boundary conditions APBC/PBC, and bond dimensions 7–9, which is strong internal evidence. The compression method for MPO unit cells is simple and reusable, and the use of the open-source YASTN library supports reproducibility. The comparison of g(x) to the continuum Laughlin curve with a single magnetic length is suggestive. The main caveat is that the specific D_min=7 threshold rests on a finite-size ED comparison that is not fully documented.

major comments (2)
  1. [Results, Fig. 2(a) inset] The critical-bond-dimension claim D_min=7 is based on comparing the thermodynamic-limit iPEPS energy density e0(D) with ED results on tori up to 24 unit cells, but no finite-size scaling of the ED gap is presented. The text is also dimensionally ambiguous: e0 is an energy density, while the ED 'first excitation gap' is a total energy. The authors should specify the ED energy density E1(N)/N for each cluster and show that it is stable with N. Without this, the D=6 point e0=-0.855254 may lie below the true thermodynamic first-excited energy, and the threshold is not established. Please provide the ED finite-size data and extrapolation, or rephrase the claim as an observed variational crossover supported by charge variance and the absence of ES signatures.
  2. [Results, critical-bond-dimension paragraph] The paper interprets the D<7 energies as evidence that the variational manifolds of D≤6 do not contain the FCI state. However, an upper-bound energy above the ED first-excited level is equally consistent with the optimizer being trapped in a local minimum within an expressive manifold. No information is given about optimization convergence for D≤6, such as number of restarts, dependence on initial tensors, or gradient norms. Since the 'minimal bond dimension' language refers to representational power, please add convergence checks for D=4–6, or rephrase the conclusion as a statement about the optimized states rather than about the ansatz's capacity.
minor comments (4)
  1. [Appendix A] Please clarify the charge-attachment counting. If q0=1 is attached to each occurrence of the A1 tensor in the 3×3 unit cell, then total Q0=3 and ν=3/9=1/3, consistent with the main text. If only one tensor carries q0, the filling would be 1/9. State this explicitly.
  2. [Results, energy comparison] Replace 'first excitation gap' by 'first-excited energy density' or define the comparison in the same units throughout, to avoid the density-versus-total-energy ambiguity.
  3. [Fig. 1(c,d) and Fig. 1(d) discussion] The magnetic length ℓ_B/a=1.63 is used for both the IQH and Laughlin continuum curves. State explicitly that this is a fitted parameter for the comparison, not an input derived from the lattice model.
  4. [Fig. 5 caption] The momentum-resolved ES is shown only for the U(1) charge sector n=0; state the particle-number convention (electrons per cylinder circumference) in the caption.

Circularity Check

0 steps flagged

No significant circularity: the FCI identification is benchmarked against external ED and chiral-boson entanglement-spectrum data.

full rationale

The derivation chain is not circular. The iPEPS tensors are optimized by minimizing the energy density e0 of the Haldane–Hubbard Hamiltonian (Eq. 1) with a U(1)-symmetric ansatz at fixed filling; the optimization loss contains no entanglement-spectrum or pair-correlation target. The (1,1,2,3,5) ES counting is therefore an emergent diagnostic computed after optimization from the CTMRG environment via Eq. (5), and its agreement with the chiral U(1) boson edge of the ν=1/3 Laughlin state is an external benchmark. The D≥7 claim is supported by variational upper bounds on e0 compared with ED first-excitation energies on tori up to 24 unit cells; ED is an independent method, not an input to the ansatz. The only free-parameter comparison is the magnetic length ℓ_B/a≈1.63 used for the continuum g(x) overlay, but this is a plotting scale for a side-by-side comparison, not a fitted quantity renamed as a prediction, and it does not enter the ES or the D_min conclusion. Self-citations ([40], [55]) are background statements about persistence of FCI with longer-range interactions and about the YASTN library; neither is load-bearing for the central result. The finite-size ED gap proxy is a potential correctness/robustness caveat, not a circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The central claim rests on established tensor-network machinery (U(1) block-sparse fermionic iPEPS, CTMRG, fixed-point AD) and on two physics postulates imported from prior literature: the bulk–edge correspondence for reading the ES, and the small-torus ED gap as a phase proxy. No new physical entity is introduced; the single matched scale is ℓ_B/a≈1.63. The ansatz construction (3×3 unit cell, auxiliary charge leg, per-D sector splits) introduces variational hyperparameters chosen by hand.

free parameters (2)
  • magnetic length ℓ_B/a for continuum comparison = ≈1.63
    One scale used to overlay continuum Laughlin (ν=1/3) and IQH (ν=1) g(x) onto the lattice iPEPS data (Fig. 1c,d). The value is not derived in the paper (a naive flux-density mapping for a C=1 band gives ≈0.64), so the stated agreement is partly a comparison at a matched scale.
  • per-bond-dimension U(1) sector splits (D_-1, D_0, D_1) = Table I: D=4:(1,2,1), D=5:(1,2,2), D=6:(2,2,2), D=7:(2,3,2), D=8:(2,3,3), D=9:(3,3,3)
    Chosen by hand; the D≥7 threshold and the quality of the ES could in principle depend on how the total bond dimension is distributed among charge sectors. This is an ansatz hyperparameter rather than a fitted physical constant.
axioms (5)
  • domain assumption Gapped chiral phases cannot be exactly represented by finite-correlation-length iPEPS (Dubail–Read no-go theorem)
    Used in the Green's-function paragraph to interpret the long-range 'gossamer' tail as a finite-D artifact rather than physical gaplessness; the paper assumes the tail does not spoil the ES and bulk diagnostics.
  • domain assumption The entanglement spectrum of a half-cylinder bipartition of a gapped 2D state reproduces the physical chiral edge spectrum (Li–Haldane / Cirac–Poilblanc–Schuch–Verstraete)
    The whole identification of the FCI phase rests on reading (1,1,2,3,5) from the ES (Eq. 5 and Fig. 5); if the bulk–edge correspondence fails for this finite-D iPEPS, the fingerprint is not a phase proof.
  • domain assumption The first-excitation gap of ED on tori up to 24 unit cells is a faithful proxy for the thermodynamic-limit gap of the same FCI model
    Underpins the D_min=7 claim: iPEPS energies for D≤6 are 'above the first excitation gap', so those manifolds are judged not to contain the FCI ground state.
  • domain assumption The FCI ground state lies in the variational manifold of a U(1)-symmetric 3×3-unit-cell iPEPS with an auxiliary charge-injection leg
    The ansatz fixes filling exactly by construction and assumes no symmetry breaking beyond the 3×3 cell; a larger unit cell or symmetry relaxation could in principle change the variational landscape.
  • standard math Swap-gate encoding correctly implements fermionic statistics in the U(1) block-sparse tensor network
    Standard, well-tested encoding (Refs 27-28) but correctness is assumed rather than re-derived; it underlies the sign structure of all observables and the ES.

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Cite this review

Pith. "Pith review of Simulating fermionic fractional Chern insulators with infinite projected entangled-pair states." pith.science (2026). https://pith.science/paper/NUO7OK2R

@misc{pith2026251220697,
  author       = {Pith},
  title        = {Pith review of: Simulating fermionic fractional Chern insulators with infinite projected entangled-pair states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NUO7OK2R}},
  note         = {Machine review of arXiv:2512.20697}
}
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read the original abstract

Infinite projected entangled-pair states (iPEPS) provide a powerful variational framework for two-dimensional quantum matter and have been widely used to capture bosonic topological order, including chiral spin liquids. Here we extend this approach to \emph{fermionic} topological order by variationally optimizing $U(1)$-symmetric fermionic iPEPS for a fractional Chern insulator (FCI), with bond dimensions up to $D=9$. We find evidence for a critical bond dimension, above which the ansatz faithfully represents the FCI phase. The FCI state is characterized using bulk observables, including the equal-time single-particle Green's function and the pair-correlation function, as well as the momentum-resolved edge entanglement spectrum. To enable entanglement-spectrum calculations for large iPEPS unit cells, we introduce a compression scheme and show that the low-lying part of the spectrum is already well converged at relatively small cutoff dimensions.

Figures

Figures reproduced from arXiv: 2512.20697 by Hao Chen, Juraj Hasik, Titus Neupert.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Distinct hopping processes of the Haldane model [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Equal-time single-particle Green’s function [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Left/right dominant eigenvectors [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Momentum-resolved ES of the [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. CTMRG routine. (a) Double-layer tensors used to [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Action of the gauge-fixed map [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Momentum-resolved ES of the [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗

discussion (0)

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Gauge-covariant projected entangled paired states for interacting systems in a magnetic field

    quant-ph 2026-04 unverdicted novelty 7.0

    A gauge-covariant PEPS ansatz with virtual flux tensors ensures translation-invariant physical expectation values for 2D interacting systems in a magnetic field, allowing gauge-independent simulations without enlarged...

  2. Fast two-dimensional tensor-network contraction via subspace iteration

    cond-mat.str-el 2026-07 conditional novelty 6.5

    A new CTMRG variant, SI-CTMRG, substitutes QR-based subspace iteration for the dominant large SVD, shifting cost to tensor contractions and enabling state-of-the-art iPEPS calculations on a single H100 GPU.

  3. Implicit differentiation of tensor network algorithms

    quant-ph 2026-07 conditional novelty 6.0

    PEPS energy gradients can be computed by implicit differentiation of characteristic equations for the contraction environment, avoiding unstable subroutine backpropagation and reducing asymptotic cost.

Reference graph

Works this paper leans on

56 extracted references · 1 linked inside Pith · cited by 3 Pith papers

  1. [1]

    Neupert, L

    T. Neupert, L. Santos, C. Chamon, and C. Mudry, Frac- tional quantum hall states at zero magnetic field, Phys. Rev. Lett.106, 236804 (2011)

    2011

  2. [2]

    Regnault and B

    N. Regnault and B. A. Bernevig, Fractional chern insu- lator, Phys. Rev. X1, 021014 (2011)

    2011

  3. [3]

    Sheng, Z.-C

    D. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Fractional quantum hall effect in the absence of landau levels, Na- ture communications2, 389 (2011)

    2011

  4. [4]

    K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, Nearly flatbands with nontrivial topology, Phys. Rev. Lett.106, 236803 (2011)

    2011

  5. [5]

    Tang, J.-W

    E. Tang, J.-W. Mei, and X.-G. Wen, High-temperature fractional quantum hall states, Phys. Rev. Lett.106, 236802 (2011)

    2011

  6. [6]

    J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtzmann, Y. Zhang, F. Fan, T. Taniguchi, K. Watanabe,et al., Signatures of fractional quantum anomalous hall states in twisted mote2, Nature622, 63 (2023)

    2023

  7. [7]

    H. Park, J. Cai, E. Anderson, Y. Zhang, J. Zhu, X. Liu, 6 C. Wang, W. Holtzmann, C. Hu, Z. Liu,et al., Observa- tion of fractionally quantized anomalous hall effect, Na- ture622, 74 (2023)

    2023

  8. [8]

    Y. Zeng, Z. Xia, K. Kang, J. Zhu, P. Kn¨ uppel, C. Vaswani, K. Watanabe, T. Taniguchi, K. F. Mak, and J. Shan, Thermodynamic evidence of fractional chern in- sulator in moir´ e mote2, Nature622, 69 (2023)

    2023

  9. [9]

    Z. Lu, T. Han, Y. Yao, A. P. Reddy, J. Yang, J. Seo, K. Watanabe, T. Taniguchi, L. Fu, and L. Ju, Fractional quantum anomalous hall effect in multilayer graphene, Nature626, 759 (2024)

    2024

  10. [10]

    E. J. Bergholtz and Z. Liu, Topological flat band models and fractional chern insulators, International Journal of Modern Physics B27, 1330017 (2013)

    2013

  11. [11]

    A. G. Grushin, J. Motruk, M. P. Zaletel, and F. Poll- mann, Characterization and stability of a fermionicν= 1/3 fractional chern insulator, Phys. Rev. B91, 035136 (2015)

    2015

  12. [12]

    Y.-C. He, F. Grusdt, A. Kaufman, M. Greiner, and A. Vishwanath, Realizing and adiabatically preparing bosonic integer and fractional quantum hall states in op- tical lattices, Phys. Rev. B96, 201103 (2017)

    2017

  13. [13]

    Dubail and N

    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. B92, 205307 (2015)

    2015

  14. [14]

    Poilblanc, Investigation of the chiral antiferromagnetic heisenberg model using projected entangled pair states, Phys

    D. Poilblanc, Investigation of the chiral antiferromagnetic heisenberg model using projected entangled pair states, Phys. Rev. B96, 121118 (2017)

    2017

  15. [15]

    Chen, J.-W

    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 chiral spin liquids on the square lattice, Phys. Rev. B104, 235104 (2021)

    2021

  16. [16]

    Hasik, M

    J. Hasik, M. Van Damme, D. Poilblanc, and L. Vander- straeten, Simulating chiral spin liquids with projected entangled-pair states, Phys. Rev. Lett.129, 177201 (2022)

    2022

  17. [17]

    S. Niu, J. Hasik, J.-Y. Chen, and D. Poilblanc, Chiral spin liquids on the kagome lattice with projected entan- gled simplex states, Phys. Rev. B106, 245119 (2022)

    2022

  18. [18]

    Budaraju, D

    S. Budaraju, D. Poilblanc, and S. Niu, Simulating chi- ral spin liquids with fermionic projected entangled pair states, Phys. Rev. B110, 064402 (2024)

    2024

  19. [19]

    J.-Y. Chen, Y. Tan, S. Capponi, D. Poilblanc, F. Ye, and J.-W. Mei, Simulating bulk gap in chiral projected entangled-pair states, arXiv preprint arXiv:2502.20142 (2025)

    Pith/arXiv arXiv 2025

  20. [20]

    E. L. Weerda and M. Rizzi, Fractional quantum hall states with variational projected entangled-pair states: A study of the bosonic harper-hofstadter model, Phys. Rev. B109, L241117 (2024)

    2024

  21. [21]

    J.-Y. Chen, L. Vanderstraeten, S. Capponi, and D. Poil- blanc, Non-abelian chiral spin liquid in a quantum an- tiferromagnet revealed by an ipeps study, Phys. Rev. B 98, 184409 (2018)

    2018

  22. [22]

    Niu, J.-W

    S. Niu, J.-W. Li, J.-Y. Chen, and D. Poilblanc, Chiral spin liquids with projected gaussian fermionic entangled pair states, Phys. Rev. B109, L081107 (2024)

    2024

  23. [23]

    I. P. McCulloch, From density-matrix renormalization group to matrix product states, Journal of Statistical Me- chanics: Theory and Experiment2007, P10014 (2007)

    2007

  24. [24]

    Singh, R

    S. Singh, R. N. C. Pfeifer, and G. Vidal, Tensor net- work decompositions in the presence of a global symme- try, Phys. Rev. A82, 050301 (2010)

    2010

  25. [25]

    Bauer, P

    B. Bauer, P. Corboz, R. Or´ us, and M. Troyer, Implement- ing global abelian symmetries in projected entangled-pair state algorithms, Phys. Rev. B83, 125106 (2011)

    2011

  26. [26]

    Singh, R

    S. Singh, R. N. C. Pfeifer, and G. Vidal, Tensor network states and algorithms in the presence of a globalU(1) symmetry, Phys. Rev. B83, 115125 (2011)

    2011

  27. [27]

    Corboz and G

    P. Corboz and G. Vidal, Fermionic multiscale entangle- ment renormalization ansatz, Phys. Rev. B80, 165129 (2009)

    2009

  28. [28]

    Corboz, R

    P. Corboz, R. Or´ us, B. Bauer, and G. Vidal, Simula- tion of strongly correlated fermions in two spatial di- mensions with fermionic projected entangled-pair states, Phys. Rev. B81, 165104 (2010)

    2010

  29. [29]

    Liao, J.-G

    H.-J. Liao, J.-G. Liu, L. Wang, and T. Xiang, Differ- entiable programming tensor networks, Phys. Rev. X9, 031041 (2019)

    2019

  30. [31]

    Naumann, E

    J. Naumann, E. L. Weerda, M. Rizzi, J. Eisert, and P. Schmoll, An introduction to infinite projected entangled-pair state methods for variational ground state simulations using automatic differentiation, SciPost Physics Lecture Notes , 086 (2024)

    2024

  31. [32]

    Francuz, N

    A. Francuz, N. Schuch, and B. Vanhecke, Stable and effi- cient differentiation of tensor network algorithms, Phys. Rev. Res.7, 013237 (2025)

    2025

  32. [33]

    Nishino and K

    T. Nishino and K. Okunishi, Corner transfer matrix renormalization group method, J. Phys. Soc. Jpn.65, 891 (1996)

    1996

  33. [34]

    Nishino and K

    T. Nishino and K. Okunishi, Corner transfer matrix algo- rithm for classical renormalization group, Journal of the Physical Society of Japan66, 3040 (1997)

    1997

  34. [35]

    Or´ us and G

    R. Or´ us and G. Vidal, Simulation of two-dimensional quantum systems on an infinite lattice revisited: Cor- ner transfer matrix for tensor contraction, Phys. Rev. B 80, 094403 (2009)

    2009

  35. [36]

    Corboz, T

    P. Corboz, T. M. Rice, and M. Troyer, Competing states in thet-jmodel: Uniformd-wave state versus stripe state, Phys. Rev. Lett.113, 046402 (2014)

    2014

  36. [37]

    M. T. Fishman, L. Vanderstraeten, V. Zauner-Stauber, J. Haegeman, and F. Verstraete, Faster methods for con- tracting infinite two-dimensional tensor networks, Phys. Rev. B98, 235148 (2018)

    2018

  37. [38]

    Fulsebakke, M

    J. Fulsebakke, M. Fremling, N. Moran, and J. K. Slinger- land, Parametrization and thermodynamic scaling of pair correlation functions for the fractional quantum Hall ef- fect, SciPost Phys.14, 149 (2023)

    2023

  38. [39]

    F. D. M. Haldane, Model for a quantum hall effect with- out landau levels: Condensed-matter realization of the ”parity anomaly”, Phys. Rev. Lett.61, 2015 (1988)

    2015

  39. [40]

    H. Chen, D. Mao, A. K. Dagnino, G. Wagner, M. H. Fis- cher, J. Hasik, E.-A. Kim, and T. Neupert, How quantum fluctuations freeze a classical liquid and then melt it into a topological one, Phys. Rev. Res.7, L042021 (2025)

    2025

  40. [41]

    C. V. Kraus, N. Schuch, F. Verstraete, and J. I. Cirac, Fermionic projected entangled pair states, Physical Re- view A—Atomic, Molecular, and Optical Physics81, 052338 (2010)

    2010

  41. [42]

    Mortier, L

    Q. Mortier, L. Devos, L. Burgelman, B. Vanhecke, N. Bultinck, F. Verstraete, J. Haegeman, and L. Van- derstraeten, Fermionic tensor network methods, SciPost Physics18, 012 (2025)

    2025

  42. [43]

    Y. Gao, H. Zhai, J. Gray, R. Peng, G. Park, W.-Y. Liu, 7 E. F. Kjønstad, and G. K.-L. Chan, Fermionic tensor network contraction for arbitrary geometries, Phys. Rev. Res.7, 023193 (2025)

    2025

  43. [44]

    Nocedal and S

    J. Nocedal and S. J. Wright,Numerical Optimization, 2nd ed., Springer Series in Operations Research and Fi- nancial Engineering (Springer, New York, NY, 2006)

    2006

  44. [45]

    Ponsioen, F

    B. Ponsioen, F. F. Assaad, and P. Corboz, Automatic differentiation applied to excitations with projected en- tangled pair states, SciPost Phys.12, 006 (2022)

    2022

  45. [46]

    S. M. Girvin, Anomalous quantum hall effect and two- dimensional classical plasmas: Analytic approximations for correlation functions and ground-state energies, Phys. Rev. B30, 558 (1984)

    1984

  46. [47]

    S. M. Girvin, A. H. MacDonald, and P. M. Platzman, Magneto-roton theory of collective excitations in the frac- tional quantum hall effect, Phys. Rev. B33, 2481 (1986)

    1986

  47. [48]

    Li and F

    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)

    2008

  48. [49]

    X.-L. Qi, H. Katsura, and A. W. W. Ludwig, General relationship between the entanglement spectrum and the edge state spectrum of topological quantum states, Phys. Rev. Lett.108, 196402 (2012)

    2012

  49. [50]

    J. I. Cirac, D. Poilblanc, N. Schuch, and F. Verstraete, Entanglement spectrum and boundary theories with pro- jected entangled-pair states, Phys. Rev. B83, 245134 (2011)

    2011

  50. [51]

    Z. Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang, and T. Xiang, Coarse-graining renormalization by higher- order singular value decomposition, Phys. Rev. B86, 045139 (2012)

    2012

  51. [52]

    Vanderstraeten, J

    L. Vanderstraeten, J. Haegeman, and F. Verstraete, Sim- ulating excitation spectra with projected entangled-pair states, Phys. Rev. B99, 165121 (2019)

    2019

  52. [53]

    Ponsioen and P

    B. Ponsioen and P. Corboz, Excitations with projected entangled pair states using the corner transfer matrix method, Phys. Rev. B101, 195109 (2020)

    2020

  53. [54]

    W.-L. Tu, L. Vanderstraeten, N. Schuch, H.-Y. Lee, N. Kawashima, and J.-Y. Chen, Generating function for projected entangled-pair states, PRX Quantum5, 010335 (2024)

    2024

  54. [55]

    M. M. Rams, G. W´ ojtowicz, A. Sinha, and J. Hasik, YASTN: Yet another symmetric tensor networks; A Python library for Abelian symmetric tensor network cal- culations, SciPost Phys. Codebases , 52 (2025)

    2025

  55. [56]

    Bruognolo, J.-W

    B. Bruognolo, J.-W. Li, J. von Delft, and A. Weichsel- baum, A beginner’s guide to non-abelian ipeps for cor- related fermions, SciPost Physics Lecture Notes , 025 (2021)

    2021

  56. [57]

    ∼” replaced by “=

    P. Corboz, R. Or´ us, B. Bauer, and G. Vidal, Simula- tion of strongly correlated fermions in two spatial di- mensions with fermionic projected entangled-pair states, Phys. Rev. B81, 165104 (2010). END MA TTER Appendix A: iPEPS ansatz details—We imposeU(1)- symmetry on iPEPS tensors to reduce computational cost and to fix the fermion filling. The vector s...

    2010