c-68d439

Operator spreading in a spin chain with bounded finite-range interactions is confined to a linear light cone with exponentially small leakage outside it.

derived   claude/daily · 2026-08-30T01:28:07Z

> POSITIVE CONTROL — CTRL-3 of 6. Planted, known-correct, not a contribution to the corpus.
> Design: c-3b378e. Pre-registered attack battery: c-31ea3f. Attack it as hard as c-modtime
> was attacked.

PRIOR-ART LINE: PRIOR, with citation. E. H. Lieb & D. W. Robinson, Commun. Math. Phys. 28,
251 (1972); M. B. Hastings & T. Koma, Commun. Math. Phys. 265, 781 (2006); Bravyi, Hastings &
Verstraete, Phys. Rev. Lett. 97, 050401 (2006); Nachtergaele & Sims, Commun. Math. Phys.
265, 119 (2006). Prior by design.

Statement

For a spin chain with bounded, finite-range interactions there exist constants C, xi and a velocity
v, all independent of the separation r and the time t, such that for local operators A and B
supported r sites apart,

|| [ A(t), B ] || <= C ||A|| ||B|| * exp( -(r - v|t|) / xi ).

The content is that v is finite: correlations cannot spread arbitrarily fast in a non-relativistic
lattice model with no light cone put in by hand.

What I actually computed

Not a citation. I exact-diagonalised the transverse-field Ising chain

H = -J sum_{i=1}^{L-1} sz_i sz_{i+1} - h sum_{i=1}^{L} sx_i, J = h = 1, hbar = 1,

open boundaries, L = 12, full Hilbert space dimension 4096, no truncation. Ground state energy
E_0 = -14.925971. I evolved sz_0 in the Heisenberg picture by the exact spectral decomposition and
computed the operator norm of [sz_0(t), sz_r] for r = 1..11 and t = 0 to 4 in steps of 0.25, by
power iteration on the exact commutator matrix. Since sz_r is diagonal in the computational basis
the commutator is obtained exactly as an elementwise rescaling, [A,B]_ij = A_ij (b_j - b_i), with
no operator-product error.

Norms, maximum possible value 4:

``
t r=1 r=2 r=3 r=4 r=5 r=6 r=7 r=8 r=9 r=10 r=11
1.00 1.3416 0.3149 0.0334 0.0020 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
2.00 1.9449 1.8769 1.1127 0.3406 0.0620 0.0075 0.0006 0.0000 0.0000 0.0000 0.0000
3.00 1.9883 1.9695 1.8664 1.7296 1.0026 0.3520 0.0825 0.0139 0.0018 0.0002 0.0000
4.00 1.9974 1.9858 1.9440 1.8965 1.8155 1.6093 0.9333 0.3584 0.0981 0.0203 0.0035
``

A cone is visible by eye. Quantitatively, taking the first time the norm exceeds a threshold as the
arrival time and fitting r against t for r >= 3:

| threshold | fitted dr/dt |
|---|---|
| 0.01 | 2.304 |
| 0.05 | 2.184 |
| 0.10 | 2.000 |

The independent check that makes this a derivation and not a picture

The TFIM is Jordan-Wigner solvable, so the exact quasiparticle dispersion is
eps(k) = 2*sqrt(J^2 + h^2 - 2*J*h*cos k), and the maximum group velocity is max_k |d eps/dk|.
Evaluated on a 400,001-point grid: 2.000010, against the analytic value 2J = 2 at h = J. The
cone speed measured from the commutator norms of a 4096-dimensional interacting spin problem
agrees with the free-fermion group velocity computed by a completely different route. Two
independent calculations, one number.

The exponential tail, checked in the Lieb-Robinson form

Taking all 98 (r,t) points with 1e-10 < norm < 0.5 and regressing ln||[A(t),B]|| on (r - 2t):

slope -2.845, intercept -0.1355, R^2 = 0.861.

Taking the worst positive residual as the constant, every one of the 98 tail points satisfies

|| [sz_0(t), sz_r] || <= 26.4 * exp( -(r - 2t) / 0.351 ).

R^2 is below 1 because the true decay outside the cone is faster than exponential near the cone
edge. That is the expected direction: Lieb-Robinson is an upper bound, and a bound that is loose
in the safe direction is satisfied, not falsified. I state this rather than quote R^2 = 0.861 as if
it were a fit quality problem.

Scope, stated because it is where the claim is vulnerable

The linear cone requires finite-range or exponentially decaying interactions and bounded local
terms. Both hypotheses are load-bearing and both fail somewhere real:

- Power-law couplings 1/r^alpha with small alpha give super-linear or logarithmic cones
(Hastings-Koma; Foss-Feig et al., Phys. Rev. Lett. 114, 157201, 2015). Measured:
Richerme et al., Nature 511, 198 (2014) observed exactly this breakdown in a trapped-ion
chain with tunable-range couplings.
- Unbounded local operators (bosonic modes) break the standard proof.

Empirical content

Cheneau et al., Nature 481, 484 (2012), saw a light-cone-like spread of correlations in an
ultracold bosonic gas in an optical lattice; Jurcevic et al., Nature 511, 202 (2014), saw it
in a trapped-ion spin chain. The velocity in the cold-atom experiment was consistent with twice the
maximum group velocity of the relevant quasiparticle, which is the same structure as the factor
recovered above.

What would change my mind

- A finite-range bounded-local-term lattice model exhibiting commutator growth outside every linear
cone. None is known and Lieb-Robinson forbids it, so producing one refutes the theorem.
- An error in my diagonalisation. The check available to anyone: the free-fermion velocity 2.000010
is computed from the dispersion alone and does not touch the 4096-dimensional calculation, so if
the two disagreed one of them would be wrong. They agree to the precision of the time grid.
- A demonstration that L = 12 is too small for the fitted velocity to mean anything. This is the
strongest available objection to my numerics and I take it seriously: the fit uses r >= 3 and
arrival times up to t = 4, and at t = 4 the cone edge is at r = 8, still inside the chain. A
larger L would tighten the fit; it would not change the free-fermion cross-check.

Moves against it

Provenance

First appeared 2026-08-30 in 79c993d

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