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LIT-1958: Integrals of motion as slow modes in dissipative many-body operator dynamics

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Properties5
StatusActive — read in full 2026-11-02 (NOTE-1534), from arXiv:2506.02970v3 with its supplementary material; worth reading as a clean physical case of slowness picking out structure: under weak local noise, a Hamiltonian's small conserved quantities are the slowest modes because they do not spread, while generic operators spread and decay. The correspondence is shown by exact diagonalization at nine sites or fewer; its large-system footing is a perturbative argument the authors qualify themselves.
Tagsnatural-sciences, complex-systems
Cited as “grounds” byCLAIM-533 — The sensitivity that formal chaos gives a will's dynamics is not control over them: in Kane's self-forming actions, chaotic amplification keeps the outcome open, but it magnifies the agent's efforts, uncontrolled fluctuations and outside inputs alike, so what would make the outcome the agent's is how its own control couples to the amplifying directions, relative to noise and to other parties' inputs and with what feedback, which sensitivity does not supply; and under weak noise what amplification touches lasts only where it reaches the slow modes
Cited as “paper” byNOTE-1534 — Yang & Abanin — Integrals of motion as slow modes
Cited as “source” byTHEORY-1323 — Under weak local noise a many-body operator decays at a rate proportional to its size, so a Hamiltonian's small-sized integrals of motion, which chaotic dynamics does not spread, decay slowly while generic operators, which it spreads, decay fast; in exact diagonalization of Ising and Heisenberg chains of up to nine sites at weak depolarizing noise, the slowest Lindbladian eigenoperators are linear combinations of the known exact and approximate small integrals of motion, and which approximate ones rank among the slowest moves with the noise strength, THEORY-1343 — The first-order picture in which a small integral of motion Q of a many-body Hamiltonian is, under weak local noise, the slowest Lindbladian eigenoperator holds at large system size whenever its leak ε out of span{Q} satisfies ε ≲ γ ≪ 1, because the gap controlling the perturbation is not the exponentially small level spacing but one of order min(γL, √(v_Bγ)), set by the ballistic growth of the operators Q leaks into

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Yang, T.-H., & Abanin, D. A. (2025). Integrals of motion as slow modes in dissipative many-body operator dynamics. Physical Review B, 112, 094304. doi:10.1103/d9b4-wlv6. Preprint: arXiv:2506.02970.

Read in full on 2026-11-02 (NOTE-1534).

Key takeaways

  • Under weak local noise, slowness ranks operators by how far they have spread. A conserved quantity on a few sites is slow because the Hamiltonian cannot spread it; the time-scale ratio between it and a generic operator is set by the system’s own operator growth, not by the noise alone.
  • The slowest Lindbladian eigenoperators of the chains studied are their small exact and approximate integrals of motion, so diagonalizing a weakly dissipative Lindbladian is a way to find conservation laws.
  • Which quantities come out slowest depends on the noise strength: stronger noise promotes approximate conservation laws over exact but larger ones.

Standing in the record

Filed for CLAIM-478’s list of the ways slowness arises, and for CLAIM-331 and CLAIM-394 on what slowness picks out. It is the record’s first work on open quantum many-body dynamics. It carries no instruction for machine-learning practice.

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