⚛️ high-energy theory

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

This paper presents the first proof of a random-matrix spectral ramp in a static, geometrically local many-body Hamiltonian by embedding the Floquet quasienergy spectrum of a dual-unitary circuit into a time-independent system using a variant of the Feynman-Kitaev clock construction.

Original authors: Matteo Ippoliti

Published 2026-06-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Matteo Ippoliti

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Finding Order in Chaos

Imagine you are listening to a chaotic jazz band. If the musicians are truly improvising and interacting in a complex, "chaotic" way, the music eventually settles into a specific statistical pattern. In the world of quantum physics, scientists look for this same pattern in the energy levels of a system. They call this pattern a "spectral ramp."

For a long time, scientists knew this ramp existed in two types of systems:

Random Matrix Theory: A mathematical model where everything is random (like a bag of marbles).
Floquet Systems: Quantum systems that are constantly being kicked or driven by an outside force (like a pendulum being pushed repeatedly).

However, there was a missing piece of the puzzle: Static, local systems. These are systems that sit still (they aren't being kicked) and where particles only talk to their immediate neighbors (like neighbors chatting over a fence, not shouting across the whole city). No one had ever mathematically proven that these specific, realistic systems produce the "ramp" of chaos.

This paper fills that gap. The author, Matteo Ippoliti, proves that you can build a static, local quantum system that behaves chaotically and produces this ramp.

The Magic Trick: The "Clock" Hamiltonian

How did he do it? He used a clever construction called the Feynman-Kitaev clock, which acts like a translator between two different worlds.

The Analogy: The Clockwork Factory
Imagine a factory floor with two parallel conveyor belts:

Belt A (The Qubits): A line of workers (quantum bits) who need to perform a specific dance routine.
Belt B (The Clock Hand): A single robot arm that moves down the line, one step at a time.

Here is how the machine works:

The robot arm (the clock hand) starts at the beginning.
As it hops to the next worker, it triggers a specific gate (a dance move) for that worker and the one next to them.
When the robot arm completes a full lap around the factory, the workers have performed a full sequence of moves. This sequence is called a Floquet Circuit.

The author takes a known chaotic dance routine (a "dual-unitary" circuit) that is already proven to be chaotic. He then builds a static machine (a Hamiltonian) where the energy of the machine is directly linked to the steps of the robot arm.

The Result:
Even though the machine itself is static (it's just sitting there, not being kicked), the energy levels of the machine contain the exact same chaotic "ramp" as the dance routine the robot was performing. The static machine inherits the chaos of the moving clock.

The "Ramp" Explained

To understand the "ramp," think of a crowded room where everyone is whispering.

The Start: At first, the whispers are messy and unconnected.
The Ramp: After a certain amount of time (the "Thouless time"), the whispers start to interfere with each other in a very specific, predictable way. If you measure the "noise" in the room over time, it grows in a straight line. This straight-line growth is the Ramp.
The Plateau: Eventually, the room reaches a state of total equilibrium, and the noise levels off.

In this paper, the author proves that in his static "clock factory," this straight-line growth happens. It proves that the system is truly chaotic and "ergodic" (meaning it explores all possible states over time).

Why Was This Hard?

Usually, proving this is like trying to hear a single violin in a hurricane.

The Noise: In these quantum systems, there is a huge background "noise" (called the disconnected part) that hides the ramp. It's like a loud hum that drowns out the violin.
The Solution: The author uses a mathematical trick to "subtract" the background hum. He looks only at the connected part of the signal (the part where the particles are actually talking to each other). Once the hum is removed, the straight-line ramp appears clearly.

The Catch (The Limitations)

The proof works, but with a specific condition:

The Single Robot Rule: The proof only works if there is exactly one robot arm (clock hand) moving around the factory.
The Size: The factory floor (the system size) must be large.
The "Static" Nature: The machine is static, but the proof relies on the robot arm moving. The author shows that the energy of the static machine mimics the motion of the robot.

If you add more robot arms (more particles in the clock chain), the math gets much harder, and the author admits that a full proof for those cases is left for future work.

Summary

In short, this paper is a mathematical breakthrough. It builds a bridge between "moving, driven" quantum systems and "still, static" ones. By using a clever "clock" mechanism, the author proves that even a static, local quantum system can exhibit the deep, universal chaos of random matrices. It's the first time this has been rigorously proven for a system that looks like a real, physical material sitting on a table, rather than a system being constantly kicked by an external force.

Technical Summary: Provable Random-Matrix Spectral Ramp in a Static, Geometrically Local Hamiltonian

Problem Statement
Quantum chaos is widely characterized by the emergence of Random Matrix Theory (RMT) statistics in energy spectra, specifically the linear growth of the Spectral Form Factor (SFF), known as the "ramp." While this phenomenon is well-understood in semiclassical limits, all-to-all models (e.g., SYK), and periodically driven (Floquet) systems, a rigorous proof of an RMT ramp in static, geometrically local Hamiltonians with finite-dimensional local Hilbert spaces has remained elusive. Previous landmark results by Bertini, Kos, and Prosen (BKP) established such ramps only for periodically driven qudit chains described by "dual-unitary" circuits. This work addresses the gap between Floquet dynamics and static Hamiltonians by constructing a time-independent, local system that provably exhibits a spectral ramp.

Methodology
The author employs a circuit-to-Hamiltonian reduction based on a recent variant of the Feynman-Kitaev (FK) clock construction. The core strategy involves embedding the quasienergy spectrum of a dual-unitary Floquet circuit into the energy spectrum of a static Hamiltonian.

System Construction:
- The system is modeled as a two-leg ladder of length $n$ with periodic boundary conditions.
- Leg 1: A chain of $d$ -state qudits (specifically qubits, $d=2$ ).
- Leg 2: A fermionic tight-binding chain containing exactly one particle (the "clock hand").
- Hamiltonian: The Hamiltonian $H_{FK}$ is defined as:
  $H_{FK} = -\frac{1}{2} \sum_{i=0}^{n-1} \left( u_i \otimes c^\dagger_{i+1}c_i + u^\dagger_i \otimes c^\dagger_i c_{i+1} \right)$
  where $u_i$ are unitary gates acting on qubits $(i, i+1)$ . The hopping of the fermion activates the gate $u_i$ , effectively simulating the evolution of a quantum circuit.
- Sector Restriction: The analysis is restricted to the single-particle sector of the fermion chain ( $\sum c^\dagger_i c_i = 1$ ). This sector has dimension $n 2^n$ , which is exponentially large but quadratically smaller than the full Hilbert space.
Spectral Embedding:
- The eigenvalues of $H_{FK}$ in this sector are derived as $E_{m,\phi} = -\cos[(2\pi m + \phi)/n]$ , where $\phi$ is a Floquet quasienergy of the circuit $U_F = u_{n-1}\dots u_0$ and $m$ is the momentum of the clock hand.
- This maps the Floquet quasienergy spectrum into a real energy spectrum, distorted by a cosine function.
Disorder and Averaging:
- Since the SFF is not self-averaging, the author introduces weak extrinsic disorder to extract universal behavior. This includes:
  - Random single-qubit unitary deformations of the gates $u_i$ .
  - A weak, spatially uniform phase shift $\chi$ applied to all gates.
  - Time averaging over a small random variable $r$ .
- The limits are taken in the order: thermodynamic limit ( $n \to \infty$ ), long-time limit ( $\tau \to \infty$ ), and finally zero-disorder limit ( $\Delta \to 0$ ).
Derivation of the Convolution Formula:
- Using the Jacobi-Anger identity, the partition function of the Hamiltonian $Z_H(it)$ is expressed as a linear combination of Floquet partition functions $Z_F(iw)$ .
- The connected SFF of the Hamiltonian, $K_H(t)$ , is shown to be a convolution of the Floquet SFF, $K_F(w)$ , weighted by Bessel functions:
  $K_H(t) = 2n^2 \sum_{w=1}^{\infty} |J_{wn}(t)|^2 K_F(w) + o(1)$
- Crucially, the disorder averaging suppresses cross-terms ( $w \neq w'$ ), isolating the diagonal contributions.

Key Contributions and Results

Proof of the Ramp:
By choosing the gates $\{u_i\}$ to be dual-unitary (but not SWAP gates), the author leverages the BKP result that the Floquet SFF scales linearly with time ( $K_F(w) \propto |w|$ ) in the thermodynamic limit.
- The paper generalizes the BKP result from periodic to open boundary conditions (required by the FK construction where $u_{n-1}=1$ ), showing the linear ramp persists up to exponentially small corrections.
- In the scaling limit where $t = n\tau$ (Thouless double-scaling limit), the reduced SFF $\kappa_H(\tau) = \lim_{n\to\infty} K_H(n\tau)/n$ is derived.
- The asymptotic behavior for large $\tau$ is proven to be:
  $\kappa_H(\tau) = \frac{2}{\pi} \tau + O(\tau \Delta^{1/2})$
- Taking the zero-disorder limit yields a finite, non-zero slope, confirming the existence of a spectral ramp.
Thouless Time Scaling:
The analysis reveals that the Thouless time $t_{Th}$ scales linearly with system size ( $t_{Th} \propto n$ ). This is attributed to the ballistic motion of the clock hand, which must traverse the system circumference before global RMT signatures emerge. This represents the fastest possible scaling for geometrically local Hamiltonians.
Role of Connected vs. Disconnected SFF:
The ramp is only visible in the connected SFF. The disconnected part, $|\langle \text{Tr}(e^{-iHt}) \rangle|^2$ , decays slowly ( $\sim 1/t$ ) due to van Hove singularities at the single-particle band edges. This disconnected background would mask the ramp in the total SFF even at the Heisenberg time, necessitating the subtraction of the disconnected component to observe the universal behavior.

Significance and Claims
The paper claims to provide the first proof of a spectral ramp in a time-independent, geometrically local many-body system with finite local Hilbert space dimension.

It bridges the gap between the known results for Floquet dual-unitary circuits and static Hamiltonians.
It demonstrates that the "ramp" is a robust feature of quantum chaos that can be embedded into static systems via the Feynman-Kitaev construction, provided the system is restricted to a specific symmetry sector (single-clock-hand).
The author explicitly notes limitations: the proof currently applies only to the single-clock-hand sector. Extending this to the full Hilbert space or sectors with finite clock-hand density remains an open question, though non-rigorous arguments suggest a similar ramp might exist with a different prefactor.

The work does not propose experimental realizations or specific applications beyond establishing the theoretical existence of such a system, maintaining a focus on the rigorous derivation of spectral statistics in a controlled theoretical model.