REVIEW 2 major objections 5 minor 75 references
Interacting Electronic Topology of Nonlocal Crystals
T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read The paper claims that one-dimensional electrons with spinful time-reversal, charge conservation, translation, and inversion symmetry can host a topological phase only when interactions are infinite-range, with ground-state inversion…
desk verdict Exact HK-type model gives a genuinely new nonlocal crystalline phase with I(L)=-1 and an odd pump, but the 'impossible locally' half rests on the completeness of the cited MAL classification rather than a proof in this paper. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the Hatsugai-Kohmoto (HK) interaction, a momentum-space-diagonal interaction that is exactly solvable and becomes infinite-range in real space (the paper's 'nonlocal crystal'). The model has two spinful orbitals, an inversion-even s orbital and an inversion-odd p orbital, at the 1a Wyckoff position, and the interaction is expressed through two time-reversal-invariant two-particle operators: \hat C_1^\dagger(k), an inversion-odd singlet, and \hat C_2^\dagger(k), an inversion-even pair. At each momentum k the ground state is \psi_1(k)\hat C_1^\dagger(k)+\psi_2(k)\hat C_2^\dagger(k) applied to vacuum, with coefficients \psi_1=i\cos(\varphi(k)/2), \psi_2=\sin(\varphi(k)/2), and \tan\varphi(k)=(V_2/V_1)\tan k. The essential identity is that the many-body inversion eigenvalue factorizes as I(L)=\xi_{k=0}\xi_{k=\pi} (for even L) or \xi_{k=0} (for odd L), and the interaction pins \xi_{k=0}=-1 and \xi_{k=\pi}=+1, giving I(L)=-1. Two derived identities carry the physical content: the polarization formula P=(i/2\pi)\log(\xi_{k=\pi}/\xi_{k=0}) quantizes P to 1/2, and the pump parity relation (-1)^Q=\xi(0,0)\xi(\pi,0)\xi(0,T/2)\xi(\pi,T/2) forces odd Q.
What would settle it
Enumerate or numerically search local, gapped, inversion- and TRS-symmetric 1D class AII Hamiltonians at half filling (for instance, exact diagonalization of short-range Hubbard-type models up to L = 12–16) and test whether any ground state has I(L) = -1 for all even L while the many-body gap stays open. If such a local model exists — or if the classification of Refs. [19,20,37,38] is shown to miss a sector — the paper's central exclusion fails. Conversely, a proof that every local gapped state with these symmetries has I(L) = +1 or (\pm1)^L would confirm the paper's claim.
Extended reading notes
Core claim
The central claim is that a 1D fermionic system in class AII with inversion and translation symmetry can be in a gapped topological phase whose many-body inversion eigenvalue is I(L) = -1 for every system size, and that this phase exists only if interactions have infinite range. The authors realize it with a Hatsugai-Kohmoto interaction that is diagonal in momentum space; at each k the ground state is a two-particle state built from an inversion-odd singlet operator and an inversion-even pair operator. At k = 0 the ground state carries inversion eigenvalue -1 and at k = π it carries +1, and because the many-body state is a tensor product over momenta, the total eigenvalue is I(L) = -1 for both even and odd L. This value is forbidden for band insulators by Kramers pairing and for local interacting insulators by the real-space/Mott atomic limit classification, which allows only I(L) = +1 or I(L) = (\pm1)^L. The nontrivial eigenvalue implies a polarization P = 1/2 and, through a parity relation on the pump cycle, an odd pumped charge Q = 1; the pump cycle Hamiltonian is unitarily equivalent to a Chern insulator with Chern number one. The authors further show that cutting off the interaction range at any finite length destroys the energy gap, so the phase is intrinsic to nonlocal crystals.
Load-bearing premise
The paper's claim that I(L) = -1 is impossible for local systems depends on the completeness of the real-space/Mott-atomic-limit classification of 1D class AII crystalline fermionic SPTs: if some local gapped inversion- and TRS-symmetric Hamiltonian outside that classification had a size-independent inversion eigenvalue of -1, the conclusion that this phase is intrinsically nonlocal would collapse.
Editorial extensions
If this is right
- The classification table of 1D crystalline FSPTs in class AII acquires a new sector, I(L)=-1, that is realizable only when the Hamiltonian is nonlocal; any complete classification must track interaction range or locality as a resource.
- An adiabatic pump in a nonlocal crystal can move an odd number of electrons per cycle (Q=1), whereas noninteracting and local interacting pumps in the same symmetry class are restricted to even Q; odd pumping is a direct experimental signature.
- Nonlocal crystals keep a meaningful notion of dimension through translation symmetry, so infinite-range interactions do not collapse topology to a zero-dimensional problem.
- Cutting the interaction range at any finite length closes the gap, so the phase cannot be adiabatically deformed to a local model without a phase transition.
- In two dimensions the same construction cannot produce I(L_x,L_y)=-1 because time-reversal symmetry forces a zero Chern number, but weak stacks of 1D nonlocal chains give I=(-1)^{L_x} or I=(-1)^{L_y}, extending the framework beyond 1D.
Reading between the lines
- If the exclusion holds, interaction range becomes a classification resource in the same spirit as symmetry: analogous size-independent eigenvalue patterns might be engineered for other point-group symmetries (mirror, rotation) and in other symmetry classes, producing nonlocal-only phases beyond class AII.
- The odd charge pump suggests a concrete experimental test in platforms with all-to-all or very long-range couplings—trapped ions, Rydberg arrays, or synthetic momentum dimensions—where the momentum-space HK interaction can be realized and the pumped charge measured per cycle.
- The paper does not analyze stability under small local perturbations; a natural next step is to add weak local hopping/interactions to the exact ground state and check whether I(L)=-1 survives with an open gap, which would determine whether the phase is truly robust or only a fine-tuned construction.
- The relation I(L)=\xi_0\xi_\pi and the polarization formula P=(i/2\pi)\log(\xi_\pi/\xi_0) offer a general diagnostic: any nonlocal crystal whose ground state factorizes over momenta can be scanned for nontrivial inversion eigenvalues, turning symmetry eigenvalue tables into a search tool for nonlocal topology.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies 1D fermionic systems with spinful time-reversal symmetry, U(1) charge conservation, lattice translation, and inversion symmetry, and asks what topological phases are possible when the Hamiltonian is a 'nonlocal crystal,' i.e., it has translation symmetry but couplings of arbitrary range. The authors construct an exactly solvable Hatsugai-Kohmoto-type model whose ground state is a momentum-space product state with inversion eigenvalue ξ(k=0)=-1 and ξ(k=π)=+1, giving a system-size-independent ground-state inversion eigenvalue I(L)=-1. They argue that this value is impossible for noninteracting systems (where I(L)=+1) and for local interacting systems in class AII with inversion (where the allowed patterns are I(L)=+1 or I(L)=(-1)^L), relying on the real-space/Mott-atomic-limit classification of crystalline FSPTs. They then compute the Resta polarization P=1/2, construct an adiabatic charge pump with pumped charge Q=1, and show numerically that truncating the interaction to finite range closes the many-body gap, which they take as corroboration that the phase is intrinsically nonlocal.
Significance. If the central claim holds, the paper establishes a genuinely new phenomenon: a topological phase that exists only because interactions have infinite range, with concrete invariants (inversion eigenvalue, polarization, pumped charge) that are absent from the local FSPT classification. The strengths of the paper are its exact solvability: the ground state, excitation gap, inversion eigenvalues, polarization, and Chern number are derived explicitly from the Hamiltonian in Eqs. (9) and (17), and the response predictions (P=1/2, Q=1) are falsifiable. The main fragility is that the negative claim 'not present for local systems' is imported from an external classification rather than proved or stated as a precise theorem in the manuscript. The paper is a useful and potentially important contribution, but the logical status of its headline claim should be clarified.
major comments (2)
- [Interacting local phases (paragraph beginning 'In 1D, there are no nontrivial FSPTs with the symmetries of class AII…] The central negative claim of the paper, namely that I(L)=-1 cannot occur for any gapped local Hamiltonian with these symmetries, is not proved in the manuscript. The text asserts that the real-space/MAL classification 'exhausts all possible topological sectors under symmetry constraints' and then concludes that a state with I(L)=-1 'cannot arise as the ground state of any local Hamiltonian.' This is a completeness theorem about the classification of interacting local FSPTs in 1D class AII with inversion, and it is load-bearing for the paper's main result and for the '–' entries in Table II. Please either (i) state the precise theorem being imported, including its hypotheses (unique gapped ground state, finite-range interactions, translation and inversion symmetry, spinful TRS, U(1) conservation, half filling), with a specific citation to the exact result in Refs. [19,20,34–38], or (ii) provide a self-contained proof in the Supplemental Material that every such local ground state has I(L)∈{+1,(-1)^L}. As written, the completeness assertion is too strong to be taken on faith, and the conditional nature of this step should be acknowledged.
- [Locality (main text) and SM Sec. IV] The finite-range truncation analysis in Fig. 2(b) and SM Sec. IV is presented as 'corroboration' that the phase is intrinsically nonlocal, but it should not be read as a proof of the impossibility of a local realization. Truncating the HK interaction to a cutoff Λ and multiplying the Hamiltonian by L produces a different local model, and the observed closing of the gap in that specific model does not rule out a different local Hamiltonian with I(L)=-1. The exclusion of local realizations must come from the classification theorem discussed in the previous comment. The authors should make this logical separation explicit, both in the 'Locality' section and in the abstract, so that the truncation result is understood as an illustrative diagnostic rather than as evidence with the same logical status as the classification.
minor comments (5)
- [SM Eq. (III.14)] Equation (III.14) in the Supplemental Material reads 'ξ(π,T/2)ξ(π,T/2)' in the last factor; this appears to be a typo for 'ξ(0,T/2)ξ(π,T/2).' The main-text Eq. (18) has the correct expression, but the SM version should be corrected to avoid confusion about the parity relation.
- [Footnote 50] Footnote 50 contains the duplicated phrase 'for details on details on'; please fix this typo.
- [SM Eq. (I.17)] In Eq. (I.17), the expression 'eiθ(k) ˆI 2' is ambiguous as typeset; the intended factor for the k/−k pair should be written with explicit parentheses, e.g., 'eiθ(k) ˆI 2 |ψ(k)⟩' or an equivalent unambiguous form.
- [Ground state inversion eigenvalue (main text)] The paper defines I(L) with respect to inversion centered at x=0, but for a periodic chain there is also an inversion center at x=La/2. The discussion of which I(L) values are allowed for local systems depends on the parity of L through the number of fixed points at these two centers. A one-sentence clarification of this convention would make the 'system-size-independent versus alternating' distinction easier to follow.
- [Stability of the nonlocal phase] The paper calls the model a 'topological phase' but does not explicitly argue stability under small symmetric perturbations that preserve the nonlocal structure. For a nondegenerate gapped ground state, the discrete inversion eigenvalue is locally constant, and the model has a finite gap for V1,V2>0, so this is a standard continuity argument; adding one sentence would close the gap between 'exactly solvable model' and 'phase.'
Circularity Check
No significant circularity: the exact nonlocal model is solved directly, and the only external load-bearing input is the cited completeness of the 1D local classification, which is a correctness assumption rather than a circular reduction.
full rationale
The central derivation is a forward exact calculation. The Hamiltonian in Eq. (9) is explicitly momentum-diagonal; its ground state is obtained by direct diagonalization in each k-sector (Eq. (13) and SM Sec. I.B), and the inversion eigenvalues xi(0) = -1 and xi(pi) = +1 are computed from the stated transformation rules (Eq. (12)), giving I(L) = -1 through the generic formula Eq. (8). This is not a definition of the target result: the model is constructed with definite orbital inversion characters, and the ground state is then solved, not assumed. The polarization P = 1/2 follows from the independent inversion-eigenvalue formula for the Resta polarization (SM Eq. (II.23)), and the pumped charge Q = 1 is evaluated as a Chern number of the explicit two-band family (SM Eq. (III.12)), with Eq. (18) serving as a consistency relation rather than an input. No parameter is fitted, and no predicted quantity is set equal to a fitted or defined input. The cited self-references [37,38] support the existence of Mott atomic limits realizing specified inversion eigenvalues, but the paper also provides the explicit local wavefunctions (Eqs. (3)-(5)), and the exhaustiveness of the real-space/MAL classification is attributed primarily to independent works [19,20,34-36]. The genuinely fragile step is the completeness assumption behind the sentence 'Other patterns, such as I(L) = -1, are excluded by the real-space classification, which exhausts all possible topological sectors under symmetry constraints': if a local gapped state outside the MAL enumeration existed with a size-independent I(L) = -1, the 'nonlocal-only' conclusion would fail. That is a correctness or completeness risk, not a circular reduction by the paper's own equations, and the locality-truncation study in Fig. 2(b) is suggestive numerical evidence rather than a proof. Overall the derivation is self-contained against its exact solvable model and the identified external-input dependence is not a circularity.
Assumptions & free parameters
assumptions (6)
- domain assumption Completeness of the real-space/MAL classification of 1D class AII crystalline FSPTs with inversion (all gapped local ground states are adiabatically connected to a Mott atomic limit with inversion eigenvalue (+1)^L or (-1)^L).
- domain assumption No nontrivial 1D FSPTs in class AII without additional lattice symmetries.
- domain assumption Many-body polarization from the twist operator P = Im log<Psi|z|Psi>/2pi and its inversion-eigenvalue relation (Eq. II.23) remain valid for nonlocal interactions.
- standard math For HK-type models, the ground state is a product over momenta and the even fermion parity per momentum sector removes sign ambiguities when reordering sectors.
- domain assumption The many-body Chern number computed on the (k,tau) torus equals the charge pumped over the adiabatic cycle, and Eq. (18) links its parity to inversion eigenvalues.
- ad hoc to paper Truncating the HK interaction to a finite range and multiplying the Hamiltonian by L is a meaningful locality diagnostic; a closing gap under this rescaling proves the phase is intrinsically nonlocal.
Cite this review
Pith. "Pith review of Interacting Electronic Topology of Nonlocal Crystals." pith.science (2026). https://pith.science/paper/IHJOVNBG
@misc{pith2026250610071,
author = {Pith},
title = {Pith review of: Interacting Electronic Topology of Nonlocal Crystals},
year = {2026},
howpublished = {\url{https://pith.science/paper/IHJOVNBG}},
note = {Machine review of arXiv:2506.10071}
}
read the original abstract
Nonlocal crystals are systems with translational symmetry but arbitrary range couplings or interactions between degrees of freedom. We argue that the notion of topology in such systems does not collapse to that in zero dimensions, as one may naively expect in view of the infinite interaction range. At the same time, we show that the range of available topological phases can be enriched in comparison to the case with local interactions. This is demonstrated by constructing an example of a fermionic symmetry-protected phase in one dimension in symmetry class AII with inversion symmetry, using a Hatsugai-Kohmoto-type model. The new phase exists only in a nonlocal crystal with electron-electron interactions and can be identified from symmetry eigenvalues. We construct an associated topological charge pump as a physical manifestation of its topology.
Figures
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Note that the prefactor ( L − 1)/2 originates from the fermionic anticommutation relation does not appear since for each momentum the Hilbert space is spanned by two- particle basis states in contrast with the band insula- tor [60]
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Interacting Electronic Topology of Nonlocal Crystals
Note that TRS preserves the sign of τ . If one instead con- siders a choice of τ such that TRS that flips its sign, i. e., ˆT ˆH(ϕ, τ) ˆT −1 = ˆH(−ϕ, −τ ), the pumped charge always vanishes Q = 0. 8 Supplemental Material for “Interacting Electronic Topology of Nonlocal Crystal...
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Case (ii): Interacting local case ( Q ∈ 2Z) 15 E
Pumped charge Q ∈ 2Z 15 D. Case (ii): Interacting local case ( Q ∈ 2Z) 15 E. Case (iii): Interacting nonlocal case 15
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Locality of the HK-type interaction in real space 16 V
Pumped charge Q ∈ Z 16 IV. Locality of the HK-type interaction in real space 16 V. I(Lx, Ly) = +1 due to the Chern number constraint for 2D nonlocal crystals 17 A. Ground state inversion eigenvalue from inversion-symmetric momenta 17 B. Parity of the Chern number from inversio...
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Model We start by considering the noninteracting case. In order to realize a nonzero pumped charge, we treat the Hamiltonian as a two-parameter family h(k, τ), and design the two-dimensional model with a nonzero Chern number that satisfies the symmetry constraint in Eq. (III.5...
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At both τ = 0 and τ = T /2, the Hamiltonian in Eq
Pumped charge Q ∈ 2Z We consider the pumped charge for the noninteracting case. At both τ = 0 and τ = T /2, the Hamiltonian in Eq. (III.6) describes an inversion and TRS band insulator with total charge polarization P = 0. At τ = 0, the individual polarizations of the Kramers ...
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[74]
In this case, an odd charge can be pumped at the end of the adiabatic cycle
Model We now consider the nonlocal crystalline model discussed in the main text. In this case, an odd charge can be pumped at the end of the adiabatic cycle. To demonstrate this, we introduce a parameter-dependent Hamiltonian ˆH(k, τ) such that at τ = 0 the Hamiltonian is give...
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[75]
Pumped charge Q ∈ Z We calculate the pumped charge for the nonlocal crystal case. The Hamiltonian ˆH(k, τ) takes the matrix form ˆH(k, τ) = 1 − cos θ(τ ) − cos k sin θ(τ ) − i sink sin θ(τ ) + i sink −(1 − cos θ(τ ) − cos k) = (sin θ(τ ))σx + (sin k)σy + (1 − cos θ(τ ) − cos k...
Reviewed August 7, 2026 · model on record in the stance chip above.
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