NOTE-1534: Yang & Abanin — Integrals of motion as slow modes

Contribution

It shows that the slowest eigenoperators of a weakly dissipative many-body Lindbladian are, in the models studied, the small-sized integrals of motion (IOMs) of the underlying Hamiltonian, exact and approximate, and gives the mechanism: noise punishes size, and chaotic dynamics grows everything except what it conserves. That turns the Lindbladian’s low-lying spectrum into a tool for finding conservation laws, and into a proposed measure of how chaotic a model is (§III D). The authors place it “within the research theme of identifying symmetries using superoperators” (p. 8).

Key insight

Two facts combine (§II, pp. 2–3). Local noise makes an operator’s norm decay at a rate proportional to the number of sites it acts on: under depolarizing noise, d‖O‖/dt = −γS(O)‖O‖ (eq. 5). And under chaotic unitary dynamics an operator’s size grows, typically ballistically, S(O(t)) ∼ v_B t, so its norm falls superexponentially, ∼ e^{−γv_B t²/2} (p. 2). A conserved quantity does not grow, so it decays only as e^{−γSt}. Decompose any operator as O = O∥ + O⊥, its projection onto the IOMs and the rest: O⊥ dies fast, and “O(t) ≈ O∥(t). It is in this sense that we call IOMs O∥ the slow modes” (p. 3). Without noise nothing decays and the ranking does not exist; without operator growth the conserved quantities would not stand out.

Assumptions

• Markovian, local noise. Heisenberg-picture Lindbladian L†[O] = i[H, O] + Σⱼ γⱼ(Lⱼ†OLⱼ − ½{Lⱼ†Lⱼ, O}) (eq. 1), with local jump operators, “justified provided noise at different sites is uncorrelated, and a Markovian approximation can be applied” (p. 1).
• Depolarizing noise by default: jump operators Xᵢ, Yᵢ, Zᵢ, each at rate γ/4, so D[P] = −γS(P)P for a Pauli string P (p. 2). Dephasing–decay noise is checked in Appendix B.
• Weak noise: γ = 10⁻³ in the main numerics; γ = 0.1 in Figs. 6(b,c), 8, 9.
• Small systems: exact diagonalization of the superoperator, L ≤ 9, periodic boundaries, zero-momentum sector (p. 3; Appendix C argues no slow modes live elsewhere).
• Size, not locality. “Small-sized” means few non-identity sites per Pauli string, weighted (eq. 6). Bi-local sums like Σ_{i<j}ZᵢZⱼ count as small, and products of small-sized operators stay small (p. 2), so powers of H count as IOMs.
• Ballistic operator growth for the perturbative argument (eq. 21), i.e. fully chaotic behaviour of the complement; relaxed in §IV’s second comment to “some form of operator growth” (p. 8).

Key results

• First-order correspondence (§III, p. 3). If [H, Q] = 0, there is an eigenoperator V ≈ Q with λ ≈ (Q|D[Q])/‖Q‖² = −γS(Q). Conversely, a Hermitian eigenoperator with real λ has S = |Re λ|/γ, and ‖[H, V]‖² = γ²(⟨S²⟩_V − ⟨S⟩²_V) (Appendix A, eq. A7), so a low-lying eigenoperator is at least an approximate IOM, with ‖[H, V]‖ = O(1) × γ for “reasonable” size distributions.
• Mixed-field Ising, J = 1, hx = 1.1, hz = 0.7, L = 7 (Fig. 2a). The model is thermalizing with no local IOM but H; the first three eigenoperators are combinations of I, H, H².
• Transverse-field Ising (Fig. 2b,c, L = 7 and 9). The first five eigenoperators are combinations of the Majorana-bilinear IOMs Aₘ, Bₘ (eqs. 9–11; derived in full in Appendix E).
• Heisenberg chains, L = 9 (Fig. 3; Appendix G). In the XXX model the known IOMs and their products “account for the first 25 eigenoperators” (p. 5). In XXZ the listed IOMs are not enough: at Δ = 0.1 the spin current J = XY − YX appears, a remnant of the free-fermion XX point; at Δ = 1.5 an approximate S²ₓ + S²ᵧ, the SU(2) Casimir approximately conserved near Δ = 1; at Δ = 10 a domain-wall swap operator (Fig. 10a). The quasi-local IOM of XXX is not seen, perhaps because it is larger than size 3 (p. 19).
• Approximate conservation laws found (§III D, Fig. 4). For MFIM at hx = hz = 0.4, L = 8, the second and third eigenoperators are not powers of H but combinations of H and Wurtz and Polkovnikov’s dressed particle number N (Appendix F, eq. F2).
• A chaoticity measure (Fig. 4c). R_Σ = Σ_{i=1}^{3}(1 − Rᵢ), the part of the first three eigenoperators outside span{I, H, H²}. MFIM is fully chaotic for hx > 1, hz ≠ 0, and not fully chaotic over large regions with 0 < hx < 1, where no exact local IOM but H is known. The measure counts prethermal approximate symmetries that “the traditional measure” would call chaotic (p. 6).
• Perturbation theory (§IV, eqs. 13–22). Write L†[Q] = −γ_Q Q + ε₁J₁, with ε₁ ∼ γ for an exact IOM and ε₁ ∼ ‖δH‖ for an approximate one. The eigenoperator V = Q + V⊥ has ‖V⊥‖ ≲ |ε₁|∫₀^∞ e^{γ_V t}‖J₁(t)‖dt (eq. 20). Under unitary dynamics the integral diverges; with dissipation and ballistic growth, ‖J₁(t)‖ ∼ e^{−γv_B t²/2} (v_B t < L) or e^{−γLt} (eq. 21), so ‖V⊥‖ ≲ ε₁/E_g^< with E_g^< ∼ min(γL, √(v_Bγ)) (eq. 22), which does not shrink with L and is ≫ γ for small γ and large L. Hence “we reasonably expect the perturbation theory to hold as long as ε ≲ γ ≪ 1” (p. 8). Fig. 5 checks the convergence of the integral numerically for MFIM with hz = 0.05, L = 9.
• The authors’ own caveat on the gap (p. 8). Modulated energy densities H_k = Σₓ e^{ikx}hₓ have O(k) commutators with H, so in the Mori limit lim_{γ→0} lim_{L→∞} there is “a continuum of slow modes”, and “the Mori gap must be defined in sectors that are orthogonal to all hydrodynamic modes, and might not exist at all”.
• Weak integrability breaking (p. 8, eq. 23). For E_g^< < ε < E_g^>, Q is an approximate eigenoperator, and to lowest order its decay rate is −ε₁ε₂*∫₀^∞(J₂|J₁(t))dt, “consistent with” Fermi’s golden rule; E_g^> stays finite as γ → 0.
• Noise strength reorders the slow modes (Appendix B, Fig. 6). An approximate IOM decays at roughly −λ = Aε² + Bγ, an exact one at a rate linear in γ. At γ = 10⁻³ in MFIM (hx = 1.5, hz = 0.1) V₃ ≈ H², V₄ ≈ A₁; at γ = 0.1, V₃ ≈ A₁ and V₄ ≈ H² + B₁, and the chaoticity diagram gains a non-chaotic region near hz = 0. The authors’ gloss: “a noisy model with noise rate γ has similar physics with a noiseless model if we focus on a time scale t ≪ 1/γ … a model that looks chaotic on a long time scale may look less chaotic at shorter times” (p. 13).
• Noise channel (Appendix B, Fig. 7). With dephasing–decay noise the correspondence holds “albeit to some degree with a less perfect correspondence”; D is no longer Hermitian and the eigenoperators are no longer orthogonal at first order (V₁ ∥ V₂, V₄ ∥ V₅ in Fig. 7b). An exception: in XXX/XXZ chains with that noise, the projector onto the fully polarized state is a slow mode although its size scales as L, because both H and the noise leave that state alone (p. 14).
• Finite size (Appendix D, Fig. 9). The bulk of the spectrum moves to larger |Re λ| as L grows while the slow modes do not; IOMs larger than a threshold proportional to L, “around 3 to 4” here, are drowned in the bulk (p. 15).

Claims

id	claim	strength	support
C1	Under local noise an operator’s decay rate is proportional to its size; exactly γS(P) for a Pauli string under depolarizing noise	strong	eqs. 2, 5, exact; bounds for general local noise, eq. B1
C2	Each exact small-sized IOM has an eigenoperator near it with λ ≈ −γS(Q), and a low-lying Hermitian eigenoperator is an approximate IOM with ‖[H, V]‖ = O(γ)	strong (first statement, first order) / moderate (second, needs a “reasonable” size distribution)	§III argument; Appendix A
C3	In the Ising and Heisenberg chains studied, the slowest eigenoperators are linear combinations of known exact and approximate small IOMs	strong at the sizes studied	exact diagonalization, L ≤ 9, Figs. 2, 3, 10
C4	The low-lying spectrum finds approximate conservation laws not supplied in advance	moderate	XXZ examples and MFIM’s dressed N (Figs. 3, 4, 10a); each was identified by comparison with a candidate the authors then wrote down
C5	The correspondence holds at large L, because the controlling gap is E_g^< ∼ min(γL, √(v_Bγ)), not the level spacing	weak to moderate	perturbative argument assuming ballistic operator growth (eqs. 20–22); numerical check of the integral at L = 9 (Fig. 5); qualified by the authors’ hydrodynamic-mode caveat
C6	R_Σ measures departure from full chaos, including approximate symmetries	weak	proposed; one phase diagram at L ≤ 9; noise-dependent (Fig. 6c)
C7	Stronger noise favours approximate IOMs over exact larger ones	moderate	the Aε² + Bγ form from eq. 18, and one model at two noise strengths (Fig. 6)
C8	The correspondence does not depend much on the noise channel	weak to moderate	heuristic rotation-averaging argument and one alternative channel (Fig. 7), with a named exception

Concepts

• Integral of motion (IOM) — an operator commuting with H; an approximate IOM is one whose commutator with H has small norm (p. 1).
• Size S(O) — the weighted mean number of non-identity sites in O’s Pauli strings (eq. 6). Small-sized is wider than local.
• Slow mode / low-lying eigenoperator — an eigenoperator of L† with decay rate −Re λ = O(1) × γ, the factor not scaling with L (p. 3).
• Operator growth — the spreading of S(O(t)) under unitary dynamics, ballistic at butterfly velocity v_B in chaotic systems (p. 2).
• Mori gap — the gap of a weakly dissipative many-body Lindbladian in the Mori limit lim_{γ→0} lim_{L→∞}, finite where the level spacing is exponentially small (refs. 24, 25; p. 7).
• Frobenius norm — ‖O‖ = √((O|O)), with (A|B) = tr(A†B)/D; the only norm used (eq. 4).

Connections

• Mi et al. 2022 (ref. 13): Majorana zero modes found as the slowly decaying Pauli strings of a noisy superconducting chain; the experiment that suggested the method (p. 1).
• Mori 2024; Jacoby, Huse and Gopalakrishnan 2025 (refs. 24, 25): the Lindbladian gap of chaotic channels at infinitesimal noise, read as the inverse thermalization time. The record holds neither.
• Wurtz and Polkovnikov (ref. 69): the dressed particle number used as MFIM’s approximate conservation law (Appendix F).
• Lian; Chen and Lian; Moudgalya and Motrunich (refs. 81–83): other superoperator routes to conserved quantities, from entanglement Hamiltonians and as ground states of local superoperators. The record holds none of them.
• Within the record. THEORY-158 (LIT-773) states, for classical Markov generators, when an observable is exactly conserved. This paper’s quantities are conserved by the Hamiltonian part only, and under the full generator are slow, not conserved. Zurek’s einselection (NOTE-1229) has a related shape: what the environment leaves alone persists. Here the environment acts on every site alike, and what persists is what the system’s own dynamics keeps small.

Bearing on the record

• CLAIM-478: a mechanism of slowness its list lacks in this form. The claim’s “Not one mechanism” names near-decomposability, slaving, metastability, conservation and transport, and renormalization (A13), with broken symmetry (Toner–Tu, NOTE-1228) and dissipative coupling (CLAIM-481) added since. This paper’s route is conservation, but the separation is made by two things together: weak local noise, which punishes size, and the system’s own chaos, which grows everything it does not conserve. The fast side’s rate is set by the chaos: the gap is of order √(v_Bγ), so the ratio to the slow rates (∼ γ) grows like √(v_B/γ) as the noise weakens (eq. 22). The H_k family is A13’s transport branch arising in the same setting (p. 8). As for “separation makes a constraint”: after O⊥ has decayed, O(t) ≈ O∥(t) (Fig. 1d), so over the window between the two rates the conserved sector is all that remains of an observable. That supports the first half. It does not bear on defeater (a), since an exact conservation law is the limit of slowness, not a constraint that works without it, or on the second half.
• CLAIM-331: slowness as a mark, relative to an environment and a measure of size. Without noise every eigenoperator of i[H, ·] has zero decay rate and slowness ranks nothing; the ranking exists only once a weak environment and a notion of size, fixed by the tensor-product split into sites, are given. And the ranking moves with the environment’s strength: at γ = 0.1 an approximate IOM overtakes the exact H² (Appendix B, Fig. 6). The authors’ “a model that looks chaotic on a long time scale may look less chaotic at shorter times” is the claim’s point that slowness marks and does not define, in a case where what is marked depends on the observation window.
• CLAIM-394: a non-reversible generator where slowest-decaying and eigen-directions nearly coincide, and where they come apart. i[H, ·] is anti-Hermitian and depolarizing D Hermitian, so L† is not self-adjoint, yet the small IOM Q and the eigenoperator V near it agree to ‖V⊥‖ ≲ ε/E_g^<, small when ε ≲ γ ≪ 1 (§IV). With dephasing–decay noise D is not Hermitian either, and the eigenoperators lose orthogonality (Fig. 7b). CLAIM-394’s linear Gaussian setting is not this one, so this is a parallel, not a test.
• THEORY-158 and THEORY-160: complementary, as above; nothing in them is changed.
• No instruction for machine-learning practice.
• Two THEORYs filed: THEORY-1323 (Active), what the diagonalization showed at the sizes studied; THEORY-1343 (Proposed), the large-L claim.

Limitations

• Small systems. Every spectrum is at L ≤ 9, and IOMs of size above about 3–4 cannot be seen (Appendix D). The large-L picture is argued, not computed.
• Identification is by comparison. Each slow mode is matched to a candidate IOM by overlap; the “new” approximate IOMs were recognized against operators the authors then wrote down (J, S²ₓ + S²ᵧ, N, H_D.W.). The method finds the subspace, not the name.
• The gap argument assumes ballistic operator growth, i.e. a chaotic complement, and the authors’ own hydrodynamic caveat may remove the gap altogether (p. 8). In integrable models, where many tests are run, the growth assumption is not the obvious one; the paper does not say how the argument covers them beyond the second comment of §IV.
• The chaoticity measure depends on γ (Fig. 6c) and on the choice of three eigenoperators.
• Depolarizing noise is the main case, chosen for “mathematical appeal” and “not expected to characterize the actual dissipation in most realistic systems” (p. 14).
• The experimental protocol (shadow tomography of O(t), p. 9) is left to future work. The acknowledgements say ChatGPT was used to improve the manuscript.

Open questions

• Does the correspondence hold at large L in chaotic models where the IOM is not an exact eigenoperator (H in MFIM)? Tensor-network diagonalization or long-time evolution of the Lindbladian, which the authors propose (pp. 8–9), would say (THEORY-1343).
• Is there a Mori gap at all once hydrodynamic modes are counted (p. 8)?
• Can the low-lying spectrum name an approximate conservation law nobody has written down, rather than confirm one?