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arxiv: 2606.30635 · v1 · pith:343ALWDUnew · submitted 2026-06-29 · 🪐 quant-ph · cond-mat.dis-nn· cond-mat.stat-mech· hep-th

Provable random-matrix spectral ramp in a static, geometrically local Hamiltonian

Matteo Ippoliti This is my paper

Pith reviewed 2026-06-30 05:41 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.dis-nncond-mat.stat-mechhep-th
keywords spectral form factorquantum chaosdual-unitary circuitsFeynman-Kitaev clockrandom matrix theorygeometrically local HamiltoniansFloquet systems
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0 comments X

The pith

Static geometrically local Hamiltonians exhibit a random-matrix spectral ramp via embedded Floquet spectra.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the connected spectral form factor of certain static, geometrically local Hamiltonians displays the linear ramp of random matrix theory. This follows from embedding the quasienergy spectrum of a dual-unitary Floquet circuit into the static Hamiltonian spectrum through a variant of the Feynman-Kitaev clock construction, with the ramp inherited inside a symmetry sector. A sympathetic reader would care because earlier rigorous proofs of the ramp applied only to periodically driven systems, and the result extends the signature of quantum chaos to time-independent many-body Hamiltonians with finite local dimension.

Core claim

By embedding the Floquet quasienergy spectrum of a dual-unitary circuit into the energy spectrum of a static local Hamiltonian using a variant of the Feynman-Kitaev clock construction, the connected spectral form factor of the static Hamiltonian inherits the BKP ramp within a symmetry sector. This is the first proof of a spectral ramp in a time-independent, geometrically local many-body system with finite local Hilbert space dimension.

What carries the argument

A variant of the Feynman-Kitaev clock construction that embeds the Floquet quasienergy spectrum of dual-unitary circuits into the energy spectrum of a static local Hamiltonian so that the connected SFF inherits the ramp.

If this is right

  • The connected SFF inherits the BKP ramp from the dual-unitary Floquet circuit.
  • The ramp appears in a symmetry sector of the static Hamiltonian.
  • The construction applies to geometrically local Hamiltonians with finite local Hilbert space dimension.
  • This yields the first rigorous spectral ramp for any time-independent geometrically local many-body system of this type.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar clock embeddings could be tested on non-dual-unitary circuits to see if the ramp persists beyond the BKP class.
  • The result suggests explicit constructions for checking whether other static local systems without periodic driving also obey random-matrix statistics.
  • One could ask whether the symmetry-sector restriction can be lifted while preserving the ramp.

Load-bearing premise

The connected SFF of the static Hamiltonian inherits the BKP ramp from the embedded Floquet quasienergy spectrum within the chosen symmetry sector.

What would settle it

A direct numerical or analytic computation of the connected SFF for an explicit example Hamiltonian constructed this way that lacks a linear ramp regime would falsify the inheritance.

Figures

Figures reproduced from arXiv: 2606.30635 by Matteo Ippoliti.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of the periodic Feynman-Kitaev clock [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Kernel [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Unitary equivalence between staircase and brick [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
read the original abstract

Quantum chaos is commonly associated with the emergence of random-matrix statistics in the spectra of quantum systems. A useful diagnostic is provided by the spectral form factor (SFF), which for random matrix ensembles displays a universal linear-growth regime (`ramp'). In the last decade, a landmark result by Bertini, Kos and Prosen (BKP) identified for the first time a class of geometrically local quantum dynamics of finite-dimensional particles where the SFF provably exhibits a random-matrix ramp: periodically driven (Floquet) qudit chains whose evolution is described by `dual-unitary' circuits. Here, building on the BKP result and on a recently proposed variant of the Feynman-Kitaev clock construction, we obtain a spectral ramp in a class of static, geometrically local Hamiltonians. Our strategy is to embed the Floquet quasienergy spectrum of a dual-unitary circuit into the energy spectrum of a static local Hamiltonian, and to prove that the latter's connected SFF inherits the BKP ramp within a symmetry sector. This is to our knowledge the first proof of a spectral ramp in a time-independent, geometrically local many-body system with finite local Hilbert space dimension.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper claims to obtain a provable random-matrix spectral ramp (linear growth regime of the connected spectral form factor) in a class of static, geometrically local Hamiltonians with finite local Hilbert-space dimension. The strategy embeds the quasienergy spectrum of dual-unitary Floquet circuits (known from BKP to exhibit the ramp) into the energy spectrum of a time-independent local Hamiltonian via a variant of the Feynman-Kitaev clock construction; the connected SFF of the static Hamiltonian is then shown to inherit the BKP ramp inside one symmetry sector. This is presented as the first such rigorous result for time-independent geometrically local many-body systems.

Significance. If the embedding and exact inheritance of the ramp are established without extraneous spectral contributions, the result would constitute the first rigorous proof of a random-matrix ramp in a static, geometrically local quantum many-body Hamiltonian with finite local dimension. It directly extends the BKP Floquet result to the time-independent setting while preserving geometric locality and finite local dimension, providing a concrete bridge between periodically driven and static chaotic systems.

major comments (1)
  1. [Symmetry sector restriction (clock embedding construction)] The central claim requires that the connected SFF of the static Hamiltonian exactly inherits the BKP ramp from the dual-unitary Floquet spectrum inside one symmetry sector. The variant Feynman-Kitaev construction enlarges the Hilbert space with clock degrees of freedom to embed the quasienergies as eigenvalues of a static local H. Even after restriction to the symmetry sector, it is unclear whether the sector contains precisely the original quasienergy set (with identical multiplicities) or admits additional levels arising from clock superpositions, boundary terms, or incomplete decoupling. Any extra eigenvalues would modify the two-point energy correlations that define the connected SFF, preventing exact inheritance of the linear ramp.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We address the major comment below.

read point-by-point responses

  1. Referee: [Symmetry sector restriction (clock embedding construction)] The central claim requires that the connected SFF of the static Hamiltonian exactly inherits the BKP ramp from the dual-unitary Floquet spectrum inside one symmetry sector. The variant Feynman-Kitaev construction enlarges the Hilbert space with clock degrees of freedom to embed the quasienergies as eigenvalues of a static local H. Even after restriction to the symmetry sector, it is unclear whether the sector contains precisely the original quasienergy set (with identical multiplicities) or admits additional levels arising from clock superpositions, boundary terms, or incomplete decoupling. Any extra eigenvalues would modify the two-point energy correlations that define the connected SFF, preventing exact inheritance of the linear ramp.

    Authors: We appreciate the referee drawing attention to this subtlety in the embedding. In our variant of the Feynman-Kitaev construction, the symmetry sector is defined via the eigenspaces of a conserved clock operator whose eigenvalues label the discrete time steps of the original Floquet evolution. Within this sector the Hamiltonian reduces exactly to the dual-unitary Floquet operator (up to a global energy shift), so the spectrum consists solely of the quasienergies with their original multiplicities. States involving clock superpositions or boundary-induced mixing lie in orthogonal symmetry sectors and are excluded by construction; the locality of the Hamiltonian ensures no coupling between sectors. This spectral equivalence is established rigorously in Section 3.2 and Appendix B of the manuscript. If the presentation leaves room for ambiguity we are prepared to expand the discussion of the sector projection in a revision. revision: partial

Circularity Check

0 steps flagged

Minor self-citation on clock variant; central claim independent via external BKP result

full rationale

The derivation explicitly invokes the external BKP result on dual-unitary Floquet SFF ramps and embeds via a variant Feynman-Kitaev clock construction to obtain the static Hamiltonian spectrum. The connected SFF inheritance within the symmetry sector is presented as a proof step rather than a definitional equivalence or fitted prediction. No load-bearing step reduces by construction to self-citation or input data; the result remains self-contained against the cited external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the BKP theorem for dual-unitary circuits and the properties of the Feynman-Kitaev clock construction; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • domain assumption The BKP result holds for dual-unitary Floquet circuits
    Invoked as the source of the ramp that is embedded into the static Hamiltonian.
  • domain assumption The clock construction variant maps the Floquet spectrum into the static energy spectrum while preserving the connected SFF within the symmetry sector
    Central mapping step whose validity is required for inheritance of the ramp.

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    2022