REVIEW 3 major objections 4 minor 94 references
Rigorous Test for Quantum Integrability and Nonintegrability
T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash
Pith's one-line read This paper proves that for a wide class of quantum spin chains, checking that no 3-local operator has a commutator of length at most 2 with the Hamiltonian guarantees there are no k-local conserved quantities for any 3 ≤ k ≤ N/2.
desk verdict A genuine reduction of k-local nonintegrability to a 3-local condition, with several new applications, but the load-bearing proof of the key lemmas is not fully checkable as written. 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 workhorse is a string-diagram calculus for commutators of local operators, together with the linear maps ι_k: B^(k-1)_<= → B^(k)_<= defined by X^(k-1) ↦ Σ_i [X^(k-1)_i, H^(2)_{i+k-2}]. These maps, which resemble the boost operator used in exactly solvable models, show that the space of k-local operators whose commutator with H has length at most k is isomorphic to the corresponding space at k-1. The critical identity is Lemma 7 (a diagrammatic doubling relation for nested commutators), and the technical core is the decomposition in Appendix B of the quantities F^(k)_i and G^(k)_i into left/right pieces with coefficients (k-2)/(k-1); this exact bookkeeping is what makes the peeling argument work for all k up to N/2. When condition (6) holds, this implies that any k-local conserved quantity descends to a 3-local quantity with len([Q,H])≤2.
What would settle it
A direct symbolic computation for a small chain (for example, S=1/2, N=6, with the XXZ interaction plus uniform field) that finds a 4-local conserved quantity while solving Eq. (78) shows no solution would disprove Theorem 1. Alternatively, testing Lemma 13 for k=5 on any concrete nearest-neighbor Hamiltonian—constructing A^(4)_L, A^(4)_R and verifying the asserted 3/4 coefficients—would settle whether the core peeling step is valid.
Extended reading notes
Core claim
The core claim, stated as Theorem 1, is: let H be a quantum spin chain with nearest-neighbor interactions and on-site potentials satisfying the injectivity condition (4) and the simplicity condition (6) that all 2-local operators with len([X,H])≤2 are proportional to the interaction H^(2). If there is no 3-local quantity Q with len([Q,H])≤2, then the system has no k-local conserved quantity for any 3≤k≤N/2. Equivalently, any k-local conserved quantity in that range would force the existence of a 3-local Q with len([Q,H])≤2. The proof works by showing linear isomorphisms B^(k)_<,b ≅ B^(k-1)_<,b that peel local operators one site at a time, so any local conserved quantity of large support descends to a 3-local one. This yields the first rigorous nonintegrability proofs for models such as the spin-1/2 Heisenberg chain with non-uniform field, the spin-1/2 XYZ model on the triangular lattice, and the general spin XYZ chain, and gives a partial affirmative answer to the conjecture that 3-local conserved quantities govern integrability.
Load-bearing premise
The whole reduction depends on the diagrammatic bookkeeping in Appendix B: the claim that the complicated commutators F^(k)_i and G^(k)_i can be split into left and right pieces carrying exactly the weights (k-2)/(k-1), which the paper itself acknowledges was found to have a proof gap.
Editorial extensions
If this is right
- Nonintegrability proofs for any Hamiltonian satisfying conditions (4) and (6) reduce to checking a single 3-local condition, eliminating the case-by-case, k-by-k analysis from earlier works.
- For translationally invariant chains, the test becomes algorithmic and independent of system size: it amounts to solving two finite linear systems, Eq. (77) and Eq. (78), whose dimension is fixed by the local Hilbert space.
- The theorem supplies the first rigorous nonintegrability proofs for the spin-1/2 Heisenberg chain with a non-uniform field, the spin-1/2 XYZ model on the triangular lattice, and the general spin XYZ chain.
- The existence of a 3-local Q with len([Q,H])≤2 is proved to be necessary for integrability in this class, partially resolving the longstanding conjecture that integrability requires a 3-local conserved quantity.
- The same peeling argument extends to spectrum generating algebras: if no 3-local operator satisfies [Q,H]=EQ with nonzero E up to length-2 remainders, then no k-local operator does, ruling out exact towers of quantum many-body scar states of that form.
Reading between the lines
- A natural extension is to turn the finite linear problems in Eqs. (77) and (78) into a certified numerical routine that proves nonintegrability for generic spin chains exactly over the rationals, making the property machine-checkable.
- If the (k-2)/(k-1) coefficients in Appendix B reflect a more general locality-renormalization flow, similar peeling maps may exist for medium-range interactions, where the same conjecture has been posed and the paper's method does not directly apply.
- The paper's criterion is the absence of a 3-local Q with len([Q,H])≤2, which is weaker than a genuine conserved 3-local operator; this suggests that 'almost conserved' 3-local operators are the true obstruction and could sharpen numerical integrability diagnostics based on level statistics.
- One could construct explicit one-parameter families of Hamiltonians satisfying (4) and (6) where the minimal len([Q,H]) for a 3-local Q is exactly 3; Theorem 1 predicts no local conserved quantities for k≥3, a prediction checkable by exact diagonalization at moderate N.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a rigorous test for the absence of local conserved quantities in quantum spin chains with nearest-neighbor interactions and on-site potentials. Under an injectivity assumption (4) and the assumption that the only 2-local operators with commutator length at most 2 are proportional to the interaction term (assumption (6)), Theorem 1 reduces the absence of k-local conserved quantities for all 3≤k≤N/2 to the absence of a 3-local operator Q with len([Q,H])≤2. The proof is built on a string-diagram calculus and on linear isomorphisms between the spaces B^(k)_<,b (Theorem 3, proved in Appendix B). The paper applies the theorem to several models, including the XXZ chain with a non-uniform magnetic field, the XYZ model on the triangular lattice, and the general spin-S XYZ chain, and it proposes an N-independent algorithm for translationally invariant chains. A separate result on spectrum generating algebras is also given.
Significance. If the proof is completed, this is a substantial contribution: it unifies and simplifies earlier nonintegrability proofs, provides new examples, and gives an algorithm whose cost is independent of system size. The main theorem is a genuine mathematical reduction, not a fit to data, and no parameters are fitted. However, the current version is not fully self-contained at load-bearing points: the diagrammatic lemmas in Appendix B, the structural membership claims in Section V that are used in Appendix C, and the basis of A_2 in Appendix C all require correction or detailed proof. These issues prevent the main results from being certified as written, although they appear to be local and potentially fixable.
major comments (3)
- [Appendix B, Lemmas 13 and 14 (Eqs. (B3)–(B6))] The proof of Theorem 3, and hence of Theorem 1, depends on the decomposition of F_i^(k) and G_i^(k) into left/right parts with the exact coefficient (k−2)/(k−1). The proof given in Appendix B states that Eqs. (B16)–(B19) and (B27)–(B30) imply Eqs. (B3)–(B6), but no intermediate algebra is displayed. The Acknowledgments state that 'Yuuya Chiba ... pointed out the gap in the proof,' yet the manuscript does not say which proof contained the gap or how the gap was resolved. Since the entire chain of isomorphisms (13) and the reduction in Theorem 1 pass through these lemmas, the current text does not certify the main theorem. Please provide a complete verification of Eqs. (B4) and (B6), or explicitly identify and prove the corrected statement.
- [Section V, last paragraph, and Appendix C] The text asserts that if H^(2)_1∈A⊗2, then a solution Q^(2)_{[2],1} of Eq. (77) satisfies Q^(2)_{[2],1}∈A⊗2, and that a solution of Eq. (78) satisfies Q^(2)_{[3],1}∈A_2⊗A_2. These membership claims are used in Appendix C to reduce the spin-S XYZ chain to the S=1/2 computation: Eq. (C3) presumes Q^(2)_{[2],1}∈A⊗A, and the basis of A_2 is used to rule out solutions of Eq. (78). No proof of either claim is given; 'by the injectivity and Eq. (77)' is not by itself an argument, because injectivity controls only commutators with one-sided operators and does not force solutions of a generalized eigenvalue problem into a prescribed subspace. Please supply proofs of both claims, or state the additional structural assumptions under which they hold.
- [Appendix C, Eq. (C7)] The operators labeled Ŝαα := (Ŝα)^2 − 2I are claimed to form a basis of A_2 for all S≥1. For S>1 these operators are not traceless: for S=3/2, Tr((S^x)^2−2I) = −3, whereas every element of A_2, as defined in Eq. (79) with H^(1)=0, is traceless. Thus the stated basis is not a basis of A_2. The subsequent coefficient comparison using Ŝxx_3 is therefore not justified as written. The argument should be rephrased with the correct traceless basis, e.g., (S^α)^2 − Tr((S^α)^2)I/d.
minor comments (4)
- [Section V, Eqs. (77) and (78)] The two sides of the eigenvalue equations are typeset in a way that makes them look identical; please clarify the diagram notation (e.g., by explicitly writing the operator order in the [ , ] box) so the reader can identify the actual eigenvalue problem.
- [Acknowledgments] The Acknowledgments mention 'the gap in the proof' without locating it or explaining how it was addressed; please add a note in the relevant appendix describing what the gap was and how the current version resolves it.
- [Throughout] The range 3≤k≤N/2 should be stated as 3≤k≤⌊N/2⌋ to avoid ambiguity when N is odd.
- [Section III A, Eq. (26)] The coefficient c_{zyy,i−1} is used before its subscript convention is defined; please spell out the index shifts in Eqs. (23)–(26).
Circularity Check
No circular derivation found: Theorem 1 is a self-contained reduction of k-local conservation to a 3-local condition; the sole self-citation (Ref. [24]) is non-load-bearing, and the acknowledged proof gap (Chiba) and unproved Sec. V assertion are correctness risks, not circularity.
full rationale
The central derivation is self-contained and non-circular: Theorem 1 reduces the absence of k-local conserved quantities (3 ≤ k ≤ N/2) to the strictly weaker 3-local condition that there is no Q with len([Q,H]) ≤ 2, and its proof runs through Theorem 2's explicit boost isomorphism ι_k and Theorem 3's isomorphism chain (13) with an explicitly constructed map and inverse; no parameter is fitted and no predicted quantity defines an input. The examples in Section III and Appendix C verify injectivity (4), assumption (6), and the 3-local condition by direct computation, so the applications are not renamed prior results. The lone self-citation, Ref. [24] (Hokkyo, Yamaguchi, Chiba), appears in the historical survey ('spin-1 chains [23, 24]') and in the statement that previously proven nonintegrable models satisfy the theorem's sufficient condition; it is not load-bearing for Theorems 1-3, whose proofs are given in the paper, and Appendix C proves the spin-S XYZ claim explicitly, so this citation does not raise the circularity score. Two flagged risks are gaps in verification, not circular steps: first, the author acknowledges that 'Yuuya Chiba ... pointed out the gap in the proof' (ACKNOWLEDGMENTS), and the proof of Lemma 13 and Lemma 14 in Appendix B, which supplies the (k-2)/(k-1) coefficients in Eqs. (B4) and (B6) that the isomorphism chain requires, is a weight-counting sketch whose decisive verification ('By using Eqs. (B16), (B17), (B18), and (B19), we can show ...') is asserted rather than displayed; if the factor were wrong, Eqs. (B10)-(B12) and hence Theorem 3 and Theorem 1 would collapse. Second, the last paragraph of Section V asserts without proof that a solution of Eq. (77) lies in A⊗2 when H^(2)_1 ∈ A⊗2 and that a solution of Eq. (78) lies in A⊗2_2, and Appendix C's Proposition 15 depends on this assertion. Neither risk is an equivalence-by-construction or a fit, so the appropriate circularity score is low rather than high.
Assumptions & free parameters
assumptions (4)
- domain assumption Injectivity condition (Eq. 4): [I⊗X, H^(2)_i]=0 implies X=0 and [Y⊗I, H^(2)_i]=0 implies Y=0 for all i.
- domain assumption B^(2)_<= = C H^(2) (assumption (6)): the only 2-local operators whose commutator with H has length at most 2 are proportional to the interaction term.
- ad hoc to paper The diagrammatic string calculus and the decompositions in Lemma 13 and Lemma 14 (Appendix B) correctly implement the isomorphisms of Theorem 3.
- ad hoc to paper In the last paragraph of Section V, the claim that for H^(2)_1 in A⊗A, the solution Q^(2) of Eq. (77) lies in A⊗A follows from injectivity and Eq. (77).
Cite this review
Pith. "Pith review of Rigorous Test for Quantum Integrability and Nonintegrability." pith.science (2026). https://pith.science/paper/LMJ2PR7H
@misc{pith2026250118400,
author = {Pith},
title = {Pith review of: Rigorous Test for Quantum Integrability and Nonintegrability},
year = {2026},
howpublished = {\url{https://pith.science/paper/LMJ2PR7H}},
note = {Machine review of arXiv:2501.18400}
}
abstract
The integrability of a quantum many-body system, which is characterized by the presence or absence of local conserved quantities, drastically impacts the dynamics of isolated systems, including thermalization. Nevertheless, a rigorous and comprehensive method for determining integrability or nonintegrability has remained elusive. In this paper, we address this challenge by introducing rigorously provable tests for integrability and nonintegrability of quantum spin systems with finite-range interactions. Our results significantly simplify existing proofs of nonintegrability, such as those for the $S=1/2$ Heisenberg chain with nearest-and next-nearest-neighbor interactions, the $S=1$ bilinear-biquadratic chain and the $S=1/2$ XYZ model in two or higher dimensions. Moreover, our results also yield the first proof of nonintegrability for models such as the $S=1/2$ Heisenberg chain with a non-uniform magnetic field, the $S=1/2$ XYZ model on the triangular lattice, and the general spin XYZ model. This work also offers a partial resolution to the long-standing conjecture that integrability is governed by the existence of local conserved quantities with small support. Our framework ensures that the nonintegrability of one-dimensional spin systems with translational symmetry can be verified algorithmically, independently of system size.
Figures
Reference graph
Works this paper leans on
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[2]
, ˆH]) ≤ 2 (16) holds for some ˆQ(2)
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[3]
(15) Here, εαβγ denotes the Levi-Civita symbol
:= i 2 ι2( ˆH (2)) = NX i=1 X α,β,γ ∈{x,y,z} JαJγεαβγ ˆσα i ˆσβ i+1 ˆσγ i+2. (15) Here, εαβγ denotes the Levi-Civita symbol. Next, we determine B(3) < . In the following, we show that this set is trivial, that is, B(3) < ⊂ B(2) if a magnetic field is non-uniform. To prove this, we derive a contra- diction by assuming that len([ ˆQ(3)
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[1]
Our goal in this step is to show that ˆX = 0
Injectivity of ˆH (2) i Consider an operator ˆX ∈ B{i}×ΛM 0 such that [ ˆX ⊗ I, ˆH (2) i ] = 0. Our goal in this step is to show that ˆX = 0. We expand ˆX and [ ˆX ⊗ I, ˆH (2) i ] by the Pauli basis: ˆX = X α∈{0,x,y,z}ΛM α̸=0 qα O j∈ΛM ˆσαj (i,j) =: X α qα ˆσα i , (31) [ ˆX ⊗ I, ˆH (2) i ] = X α β∈{x,y,z} j′∈ΛM rα,β;j′ ˆσα i ˆσβ (i+1,j′), (32) where ˆσ0 :...
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[4]
This operator can be ex- panded in terms of the Pauli operators as ˆQ(2)
∈ B(2). This operator can be ex- panded in terms of the Pauli operators as ˆQ(2)
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[5]
(17) Now we expand [ ˆQ(3)
= NX i=1 X α,β∈{x,y,z} qαβ,i ˆσα i ˆσβ i+1. (17) Now we expand [ ˆQ(3)
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[6]
The as- sumption (16) is equivalent to cαβγ,i = 0 for all α, β, γ and i ∈ ΛN
, ˆH] = 2i NX i=1 X α,β,γ ∈{x,y,z} cαβγ,i ˆσα i ˆσβ i+1 ˆσγ i+2 + (a 2-local quantity), (18) where cαβγ,i is a function of Jα′ and qα′β′,i. The as- sumption (16) is equivalent to cαβγ,i = 0 for all α, β, γ and i ∈ ΛN . First, we calculate cxxz,i. This coefficient comes from the following three commutators: [JzJx ˆσx i ˆσy i+1 ˆσz i+2, hi+1 ˆσz i+1], (19) ...
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[7]
First, we note that ˆX ∈ B(2) ≤ is equivalent to [ ˆXi ⊗ I, I⊗ ˆH (2) i+1] = [ ˆH (2) i ⊗ I, I⊗ ˆXi+1] (37) for all i ∈ ΛN
Confirmation of Assumption (6) In this step, we confirm the assumption (6), or more specifically, we show that ˆX ∝ ˆH (2) for ˆX ∈ B(2) ≤ . First, we note that ˆX ∈ B(2) ≤ is equivalent to [ ˆXi ⊗ I, I⊗ ˆH (2) i+1] = [ ˆH (2) i ⊗ I, I⊗ ˆXi+1] (37) for all i ∈ ΛN . From this equation and the injectivity, we know [58] that ˆXi consists of two-body operator...
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[8]
Thus, Thm
Absence of 3-local quantity ˆQ[3] satisfying len([ ˆQ[3], ˆH]) ≤ 2 Up to this point, we have confirmed that the Hamil- tonian (27) satisfies the assumption (4) and (6). Thus, Thm. 1 can be applied, and it suffices to show that there is no 3-local quantity ˆQ such that len([ ˆQ, ˆH]) ≤ 2 holds. By the fact (6) and Thm. 2, a 3-local quantity ˆQ[3] sat- isfy...
Show all 94 references
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[9]
(41) with some q ∈ C and 2-local quantity ˆQ(2)
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[10]
We expand this operator ˆQ(2)
Our goal is to show that q = 0. We expand this operator ˆQ(2)
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[11]
basis- independent
by the Pauli basis: ˆQ(2) [3],i = X α,β∈{0,x,y,z}ΛM q(2) αβ,i ˆσα i ˆσβ i+1. (42) We focus on the following ( α, β): αj = z (j = 1) x (j = 2) 0 ( j ̸= 1, 2) , (43) βj = ( y (j = 1) 0 ( j ̸= 1) . (44) The coefficient of ˆσx (1,1) ˆσy (2,1) ˆσx (2,2) ˆσy (3,1) in [ ˆQ[3]...
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[12]
= PN i=1 ˆQ(2) [2],i is an element of B(2) ≤ . C. Isomorphisms between B(k) <,b’s As pointed out at the end of Subsec. II B, Thm. 1 fol- lows from Thm. 3. Therefore, we conclude this section by proving Thm. 3. Here, we shall prove only the case of B(3) <,b ∼= B(4) <,b to expla...
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[13]
7→ q ˆQ(3) b + ˆQ(2)
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[14]
∈ B(3) <,b, (59) where q ∈ C and ˆQ(k) b is defined by Eq. (11). Proof. First we construct a map from B(4) <,b to B(3) <,b. For ˆQ(3)
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[15]
∈ B(3), the condition q ˆQ(4) b + ˆQ(3)
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[16]
Here, [ , ˆX] : B0 ∋ ˆY 7→ [ ˆY ,ˆX] ∈ B0 is the commutator with ˆX ∈ B0
∈ B(4) <,b is equiva- lent to the following equalities: − H (2) i+2 [ , ] Q(3) [4],i + H (2) i [ , ] Q(3) [4],i+1 =q H (2) i+2 [ , ] H (2) i+1 [ , ] H (2) i α(H (2) i ) + H (2) i+2 [ , ] H (2) i+1 [ , ] H (2) i α(H (2) i+1) + H (2) i+2 [ , ] H (2) i+1 [ , ] H (2) i...
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[17]
∈ B(3) <,b, where we define ˆQ(2)
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[18]
Therefore, we have constructed a linear transformation i34 : B(4) <,b → B(3) <,b
as 2 3 PN i=1 ˆQ′(2) i . Therefore, we have constructed a linear transformation i34 : B(4) <,b → B(3) <,b. The injectivity of this map is confirmed from the injectiv- ity (50). Conversely, by using the above equalities (e.g., Eq. (65)), we can construct a mapping i43 : B(3) < ...
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[19]
∈ B(3), the con- dition q ˆQ(4) b + ˆQ(3)
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[20]
(60)) − Eq ˆQ(4) b,i (73) for all i ∈ ΛN
∈ B(4) <,E is equivalent to the following 10 equalities: − H (2) i+2 [ , ] Q(3) [4],i + H (2) i [ , ] Q(3) [4],i+1 = (the RHS of Eq. (60)) − Eq ˆQ(4) b,i (73) for all i ∈ ΛN . As in the proof of the previous subsection, Eq. (73) is equivalent to Eq. (64), where ˆQ′(3) L,i is r...
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[21]
(70)) − 2 3 Eq ˆQ(3) b,i (75) for all i ∈ ΛN , which is equivalent to q ˆQ(3) b + ˆQ(2)
∈ B(2) sat- isfying − Q(2) [3],i [ , ] H (2) i+1 + Q(2) [3],i+1 [ , ] H (2) i = (the RHS of Eq. (70)) − 2 3 Eq ˆQ(3) b,i (75) for all i ∈ ΛN , which is equivalent to q ˆQ(3) b + ˆQ(2)
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[22]
On the other hand, for any E ̸= 0, ˆQ(3)
∈ B(3) <, 2 3 E . On the other hand, for any E ̸= 0, ˆQ(3)
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[23]
Therefore, we have B(4) <,E ⊂ B(3) for every E ∈R \ {0}, which indicates that there is no 4-local quantity satisfying Eq
∈ B(3) <, 2 3 E implies q = 0 by the assumption. Therefore, we have B(4) <,E ⊂ B(3) for every E ∈R \ {0}, which indicates that there is no 4-local quantity satisfying Eq. (71). With a slight modification of the proof of Thm. 3 for the general k case (Appx. B), we can show the ...
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[24]
We note that B(2,p) ≤ = {0} except for at most (d2 − 1)2 numbers of (N -independent) momentum p due to the injectivity (50)
∈ B(2,p) sat- isfies τ ( ˆQ(2) [2],i+1) = eip ˆQ(2) [2],i. We note that B(2,p) ≤ = {0} except for at most (d2 − 1)2 numbers of (N -independent) momentum p due to the injectivity (50). If B(2,p=0) ≤ = C ˆH (2), we can move on to the next step. Next, we check whether B(3) < in t...
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[25]
For each α ∈ S, ˆQ(k) [k],α + ˆQ(k−1) [k],α ∈ B(k) < ( ˆHα) holds for some (k −1)-local operator ˆQ(k−1) [k],α on {α}× ΛN
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For each (α, i) ∈ S ×ΛN and S0 ⊋ {α}, the follow- ing holds: [[ ˆQ(2) [2],α;i, ˆH (1) S0;i+1], ˆH (2) α;i+1] + [ ˆH (2) α;i , [ ˆQ(2) [2],α;i+1, ˆH (1) S0;i+1]] = −(k − 2)[[ ˆQ(2) [2],α;i, ˆH (2) α;i+1], ˆH (1) S0;i+1]. (85)
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(86) Here, ˆQ(k) [k],α and ˆQ(2) [2],α’s are defined through Eq
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