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Hole doping a chiral topological phase on the infinite triangular lattice yields a uniform superconducting state with a finite pairing order parameter.

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2026-08-03 14:07 UTC pith:JY35PKSZ

load-bearing objection Strong iPESS study of the Hofstadter-Hubbard model with a credible parent-phase picture, but the thermodynamic-limit SC claim needs the missing extrapolation details and a normal-state control before I'd call it established. the 3 major comments →

arxiv 2512.21503 v3 pith:JY35PKSZ submitted 2025-12-25 cond-mat.str-el

Thermodynamic-Limit Evidence for Chiral Superconductivity Induced by Doping Chiral Topological Phases

Sen Niu , D. N. Sheng , Yang Peng This is my paper
classification cond-mat.str-el
keywords chiral superconductivitychiral spin liquidHofstadter-Hubbard modeltensor networkinfinite projected entangled simplex stateMott transitionpairing order parameterthermodynamic limit
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 establishes thermodynamic-limit evidence for chiral superconductivity by simulating the triangular Hofstadter-Hubbard model with a fermionic infinite projected entangled simplex state (iPESS) approach. At half filling, it maps the undoped phases: a weak-coupling Chern insulator, an intermediate-coupling chiral spin liquid, and a large-coupling Néel state, with a Mott transition at U≈11.5. Upon hole doping, it finds a uniform superconducting state with a finite complex pairing amplitude whose linear 1/D extrapolation remains nonzero in the infinite-bond-dimension limit. The pairing phase winds anticlockwise (+1) over a broad U–δ region and clockwise (−1) in a small pocket near the Mott criticality. This converts prior finite-cylinder evidence of pairing correlations into direct evidence of superconducting long-range order in the thermodynamic limit.

Core claim

The central claim is that chiral superconductivity is a genuine ground state of the doped Hofstadter-Hubbard model on an infinite triangular lattice, not an artifact of finite system size. The authors show that the nearest-neighbor singlet pairing amplitude |Δ| is finite over a wide doping range 0<δ≤0.15, increases with doping, and survives the 1/D extrapolation to infinite bond dimension. The pairing phases are nearly uniform, with a +1 winding in the broad regime and a −1 winding near the Mott criticality, both breaking time-reversal symmetry and consistent with the parent chiral topology. The doped state also retains a finite scalar spin chirality and an entanglement spectrum that preserv

What carries the argument

The fermionic iPESS ansatz: a tensor network on the infinite triangular lattice built from rank-3 simplex tensors on up-triangles and rank-4 site tensors, with Z2 fermion parity and SU(2) spin symmetry, optimized by gradient descent with automatic differentiation and full-update imaginary-time evolution. It directly evaluates order parameters in the grand canonical ensemble, bypassing finite-width cylinders. The ansatz's ability to represent chiral topological states relies on the 'gossamer tail' phenomenon: spurious power-law correlation tails caused by the tensor-network no-go theorem for chiral states, which the authors find are confined to long distances and weaken with increasing bond d

Load-bearing premise

The finite bond dimension (up to D=12) of the iPESS ansatz, together with the known no-go obstruction for chiral topological states in tensor networks, still faithfully represents the true ground state because the spurious power-law correlation tails are benign and vanish in the infinite-bond-dimension limit.

What would settle it

Compute the nearest-neighbor singlet pairing amplitude |Δ| at bond dimensions D=14,16,20 with the same optimization scheme and extrapolate 1/D to zero; if the extrapolated value vanishes or changes sign, the claimed thermodynamic-limit superconducting order is an artifact of the ansatz's finite bond dimension.

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

If this is right

  • Chiral superconductivity emerges from purely repulsive Hubbard interactions in the thermodynamic limit, confirming that finite-cylinder pairing correlations from earlier studies correspond to genuine long-range order.
  • The pairing order parameter provides a concrete signature—uniform +1 phase winding across nearest-neighbor bonds—that can be searched for in moiré bilayer experiments.
  • The near-Mott-criticality pocket of −1 winding predicts a doping-tunable transition in pairing symmetry, potentially observable as a change in the sign of the Hall response or thermal transport.
  • The stability of the intermediate-U chiral spin liquid over a wide parameter window sharpens the distinction between this model and the standard triangular-lattice Hubbard model.

Where Pith is reading between the lines

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

  • If the gossamer-tail interpretation is correct, the same iPESS approach could be used to extract the topological Chern number of the superconducting state directly in the thermodynamic limit, a quantity not yet computed in this work.
  • The two winding sectors (+1 and −1) may represent distinct topological superconducting phases; a direct calculation of the thermal Hall conductance or edge-state count would test this.
  • The use of SU(2)-symmetric tensors may bias the variational search toward singlet pairing; an unbiased check with spin-anisotropic tensors could reveal whether triplet or mixed-pairing states are competitive.
  • The method's success suggests it can be transferred to other chiral topological models (e.g., kagome or square-lattice Hofstadter) to probe doping-induced superconductivity without finite-size extrapolation.

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

3 major / 4 minor

Summary. The paper develops a fermionic infinite projected entangled simplex state (iPESS) method and applies it to the triangular Hofstadter-Hubbard model at π/2 flux. At half filling, the authors identify a Chern insulator at weak U, a chiral spin liquid at intermediate U, and a 120° Néel phase at large U, locating transitions at U_c1 ≈ 11.5 and U_c2 ≈ 22.5 from charge/chirality derivative signatures and finite-correlation-length scaling. Upon hole doping δ ≲ 0.15, they report a uniform chiral superconducting state with a finite complex nearest-neighbor pairing amplitude whose phase winding is +1 in a broad U–δ region and −1 in a small pocket near the Mott criticality. The central thermodynamic-limit claim rests on a linear 1/D extrapolation of |Δ| shown only schematically in Fig. 5(d), with the raw finite-D data, fit residuals, and error bars deferred to the Supplemental Material.

Significance. If the central claim survives scrutiny, this would be a notable advance: direct thermodynamic-limit evidence for chiral superconductivity in a purely repulsive Hubbard-type model, and a demonstration that fermionic iPESS can access both chiral topological order and superconducting order. The paper also provides several concrete benchmarks that support the reliability of the parent-phase description: the SU(2)_1 CFT counting in the CSL entanglement spectrum (Fig. 2(d)), the convergence of variational energies against infinite DMRG (Fig. 2(a)–(b)), and sharp derivative signatures at U_c1. These strengths are real and should be credited. However, the superconducting claim currently rests on an extrapolation whose raw data are not shown and on a variational calculation without a normal-state control, so the significance is conditional on resolving those points.

major comments (3)
  1. [Chiral superconductivity at finite hole doping; Fig. 5(d)] The paper's key thermodynamic-limit claim is that |Δ| remains finite as D → ∞, but Fig. 5(d) does not show the discrete D values, the number of points in the extrapolation, the linear-fit residuals, or error bars. The accessible D range is small (D ≤ 12), and the text only states that a '1/D linearly extrapolated' value is used. The raw finite-D data and fits for each doping point must be displayed, at least in the Supplemental Material, so the reader can judge whether the extrapolation is stable and whether the quoted nonzero intercept is robust or is an artifact of a single or two dominant points. Without this, the central claim is not verifiable from the manuscript.
  2. [Chiral superconductivity at finite hole doping; missing normal-state control] The iPESS simulations are performed in the grand canonical ensemble using SU(2)-symmetric tensors without fixing particle number. This means the ansatz explicitly allows a nonzero pairing condensate even if the true ground state is a normal metal, because the finite-D tensor network cannot faithfully represent a gapless or weakly correlated state and may develop a spurious local pairing amplitude. The paper does not report a control calculation in which particle number is conserved (or a U(1)-symmetric iPESS) at the same δ and D, nor does it compare variational energies of a normal state and the pairing state. Such a control is essential to rule out that the finite |Δ| and its 1/D intercept are a variational artifact of the chiral-PEPS no-go obstruction, especially since the parent-phase benchmarks are all at half filling and do not validate the doped region.
  3. [Chiral superconductivity at finite hole doping; role of chiral-PEPS no-go and gossamer tails] The manuscript acknowledges the known obstruction for chiral topological states in tensor networks (Refs. [38,40]) and the resulting power-law 'gossamer tails', citing prior work that these tails weaken with increasing D in the parent CSL. However, for the doped superconducting state, the paper only states that 'universal gossamer tails' appear in correlations; it does not show how these tails behave with D in the doped regime, nor does it demonstrate that the pairing amplitude |Δ| is not tied to the tail amplitude. The Z2 4×4 check mentioned in the text is reported to exclude CDW and magnetic order, but it does not exclude a normal metal. A quantitative comparison of |Δ| with the spin/charge correlation tail amplitude as D increases, or an energy comparison against a normal-state ansatz, is needed to separate a physical pairing condensate from a finite-bond-dimension artifact.
minor comments (4)
  1. [Abstract and text] The phrase 'a anti-clockwise' should be 'an anticlockwise' or 'a counterclockwise'.
  2. [Fig. 5(d)] The caption and text do not specify the values of U shown in the right panel beyond 'U=9', and the axes are not labeled in the figure as reproduced. Please add explicit axis labels and legend entries for the spin chirality and Δ data.
  3. [Fig. 1(d)] The phase diagram appears to show three panels (insulating δ=0, doped δ>0, and a zoomed Néel panel) but the caption does not explain the color scale or the meaning of the boundary lines. Clarify.
  4. [General] Several references to the Supplemental Material are made (e.g., 'Z2 symmetric 4×4 iPESS simulations', CTMRG environment details, ES algorithm) without giving a clear summary of what data are contained there. Since the SM is not available for review, please ensure the main text at least states the key numerical parameters (D values, unit-cell sizes, CTMRG χ) for the central extrapolation.

Circularity Check

0 steps flagged

No significant circularity: the SC order parameter and phase windings are unconstrained outputs of variational iPESS optimization, not inputs or fitted predictions.

full rationale

The paper's central claim is a numerical variational measurement rather than a derivation: an iPESS wavefunction is optimized for the grand-canonical triangular Hofstadter-Hubbard Hamiltonian, and the complex nearest-neighbor pairing amplitude is read from the optimized tensor. No parameter is fitted to the target pairing pattern, and no anomalous pairing term is added to H; the +1 and -1 winding patterns are emergent outputs of the optimization, including an unfitted pocket of opposite winding near U_c1. The 1/D extrapolation of |Delta| is an empirical finite-bond-dimension extrapolation; it does not make |Delta| finite by construction, and the underlying finite-D values are independent variational data. The half-filling identification of CI/CSL is benchmarked against iDMRG energies and CFT entanglement-spectrum counting, so it does not rest solely on the self-cited 'gossamer tail' literature (Refs. [43-45], two involving the first author). Those citations supply context and a validity heuristic, but the paper's own energy/ES/correlation data are the operative evidence. Remaining issues - accessible D<=12, absence of a particle-number-conserving normal-state control, and qualitative doped ES - are methodological/correctness risks, not circular reductions.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

The model, gauge, and flux are taken from Ref. [10]; the central claim does not invent constituents. What the paper imports is (i) the faithfulness of iPESS for chiral topological states (partly self-cited), (ii) the empirical scaling forms used to locate the phase boundaries, and (iii) the linear 1/D extrapolation that carries the thermodynamic-limit claim. None of these is a fitted constant that predetermines the SC result; they set the interpretive frame.

free parameters (4)
  • Uc1 ≈ 11.5 (CI–CSL Mott critical point) = ≈ 11.5
    Read off from the minimum of n̄(µ), the derivative singularity of double occupancy, and the scalar chirality derivative vs. U (Fig. 3). It is a phase-boundary extraction that structures the SC claim ('near Mott criticality') but is not tuned to force SC.
  • Uc2 ≈ 22.5 (CSL–Néel critical point) = ≈ 22.5
    From finite-correlation-length scaling fit M(ξ) = M(∞) + b/ξ (Fig. 4(c)); quoted as 'roughly consistent' with the effective J–Jχ model.
  • Scaling exponents in Eq. (2): E ∝ 1/ξ³, M ∝ 1/ξ = exponents 3 and 1
    Empirical scaling hypothesis taken from Ref. [80]; the choice of exponents determines M(∞) and thus Uc2.
  • Linear-in-1/D extrapolant for |Δ| = |Δ|(∞) > 0 at U = 9
    The claim that |Δ| remains finite in the thermodynamic limit assumes a linear 1/D convergence; the D-range, number of points, and fit quality are not stated in the main text (Fig. 5(d)).
axioms (6)
  • domain assumption Finite-D iPESS faithfully represents gapped chiral topological states despite the no-go obstruction; gossamer tails are benign artifacts that weaken with D
    Invoked throughout the CI/CSL section (Fig. 2(e)-(f)); inherited from Refs. [43–45], of which [44,45] are the first author's own prior work. If false, the parent-state identification and the doped SC claim both fail.
  • domain assumption Optimized iPESS energy approaching iDMRG energies implies the variational state approximates the true ground state in the relevant sectors
    Used in Fig. 2(a)-(b) to validate the ansatz; standard tensor-network practice but not proven for chiral states with gossamer tails.
  • standard math Li–Haldane correspondence: low-lying entanglement spectrum identifies bulk topological order
    Used to classify the U = 8 state as spinful CI and U = 14 state as Kalmeyer–Laughlin CSL (Fig. 2(c)-(d)), via Ref. [69].
  • domain assumption Gauge transformation to the C6-symmetric imaginary gauge preserves the physical state and the pairing-winding classification
    Used before Fig. 5 to compare iPESS pairing patterns with DMRG; details deferred to SM [67]. Since pairing phases are gauge-dependent, the +1/−1 winding claim depends on this gauge choice.
  • domain assumption SU(2)-symmetric grand-canonical ansatz (no fixed particle number, no spin-symmetry breaking) does not bias the competing-order search
    The 2×2 doped-state ansatz forbids magnetic order by construction; the no-magnetism claim is deferred to a Z2 4×4 check in SM [67].
  • domain assumption Effective J–Jχ spin-model estimate locates Uc2 consistently with the numerics
    Used to validate Uc2 ≈ 22.5 against Refs. [10,13,74,75].

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read the original abstract

The emergence of superconductivity from doping strongly correlated chiral topological phases in purely repulsive two-dimensional fermionic systems is a problem of broad and fundamental interest. However, existing numerical evidence has been limited to finite-size studies, and direct thermodynamic-limit evidence for superconducting long-range order has remained lacking. Here we provide such evidence for chiral superconductivity in the triangular Hofstadter-Hubbard model by advancing a simplex tensor-network approach that simultaneously captures superconducting long-range order and chiral topological order in the presence of intrinsic charge fluctuations, a capability that has remained challenging for previous two-dimensional approaches. We show that a broad intermediate-$U$ chiral spin liquid is separated from the weak-$U$ Chern insulator by a Mott transition, together forming undoped parent chiral topological states. Upon hole doping, we identify a uniform chiral superconducting state in the infinite system, characterized by a finite complex pairing order parameter. The pairing field exhibits an almost universal phase winding over a broad interaction-doping regime, with a distinct pocket of opposite winding near the Mott criticality. In addition, the entanglement spectrum retains the chiral structure of the parent topological phases while developing additional low-energy branches upon doping. These results establish that chiral superconductivity emerges robustly from doped chiral topological phases.

Figures

Figures reproduced from arXiv: 2512.21503 by D. N. Sheng, Sen Niu, Yang Peng.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗

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