arxiv: 2604.25995 · v1 · submitted 2026-04-28 · 🪐 quant-ph · cond-mat.dis-nn· cond-mat.quant-gas· cond-mat.stat-mech· cond-mat.str-el

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Solvable Random Unitary Dynamics in a Disordered Tomonaga-Luttinger Liquid

Tian-Gang Zhou , Thierry Giamarchi

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classification 🪐 quant-ph cond-mat.dis-nncond-mat.quant-gascond-mat.stat-mechcond-mat.str-el

keywords frame potentialTomonaga-Luttinger liquidquenched disorderKeldysh actionXXZ spin chainunitary dynamicsquantum informationgapless phase

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The pith

The frame potential of a disordered Tomonaga-Luttinger liquid has a closed-form expression showing power-law decay at early times followed by saturation to a plateau set by one coupling parameter.

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

The paper derives an exact formula for the frame potential in a Tomonaga-Luttinger liquid with quenched Gaussian forward-scattering disorder by using the quadratic structure of the disorder-averaged Keldysh action. This formula reveals that the frame potential decays as a power law early on and then levels off to a late-time value fixed by a single coupling. The result is demonstrated for the random-field XXZ spin chain, where randomness is strongest near the Heisenberg ferromagnetic point and can be boosted exponentially with repeated quenches. The approach covers the full gapless phase and links to practical diagnostics for quantum algorithms in analog simulation platforms.

Core claim

We derive a closed-form expression for the frame potential of a Tomonaga-Luttinger liquid with quenched Gaussian forward-scattering disorder. Exploiting the exactly quadratic structure of the disorder-averaged Keldysh action, we show that the frame potential decays as a power law at early times and saturates to a late-time plateau controlled by a single coupling parameter. Taking the random field XXZ spin chain as a specific microscopic realization, we show that the strongest randomness is achieved near the Heisenberg ferromagnetic point and can be exponentially enhanced through a multiple-quench protocol, with validation across the gapless phase.

What carries the argument

The exactly quadratic disorder-averaged Keldysh action, which enables direct closed-form computation of the frame potential for the unitary dynamics.

Load-bearing premise

The disorder-averaged Keldysh action remains exactly quadratic for the chosen Gaussian forward-scattering disorder.

What would settle it

An experiment or numerical simulation measuring the frame potential versus time in a Tomonaga-Luttinger liquid with forward-scattering disorder and finding that it neither follows the predicted early-time power-law decay nor saturates to a plateau set by one coupling parameter would disprove the closed-form result.

Figures

Figures reproduced from arXiv: 2604.25995 by Thierry Giamarchi, Tian-Gang Zhou.

**Figure 1.** Figure 1: FIG. 1. Free-fermion benchmark at ∆ = 0 with view at source ↗

**Figure 3.** Figure 3: FIG. 3. Late-time one-parameter collapse. From the late view at source ↗

**Figure 4.** Figure 4: FIG. 4. Multiple-quench FP decay with ∆ = 0, view at source ↗

**Figure 5.** Figure 5: FIG. 5. The illustration of the Keldysh contour for the FP with view at source ↗

**Figure 6.** Figure 6: FIG. 6. Comparison of filter orders view at source ↗

**Figure 7.** Figure 7: FIG. 7. Time dependence of the normalized FP view at source ↗

read the original abstract

Disordered one-dimensional interacting systems have long been characterized via conventional correlation functions. A complementary quantum-information perspective quantifies the randomness of the unitary ensemble dynamics generated by a quantum system through the frame potential, which serves as a practical diagnostic for quantum algorithmic performance. However, no analytical treatment has yet been achieved for experimentally accessible interacting one-dimensional systems. In this Letter, we derive a closed-form expression for the frame potential of a Tomonaga-Luttinger liquid with quenched Gaussian forward-scattering disorder. Exploiting the exactly quadratic structure of the disorder-averaged Keldysh action, we show that the frame potential decays as a power law at early times and saturates to a late-time plateau controlled by a single coupling parameter. Taking the random field XXZ spin chain as a specific microscopic realization, we show that the strongest randomness is achieved near the Heisenberg ferromagnetic point and can be exponentially enhanced through a multiple-quench protocol. We validate our results across the entire gapless phase, with direct implications for algorithm design in analog quantum simulation platforms.

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.

Desk Editor's Note private letter to a colleague

This paper gives the first closed-form frame potential for a disordered Tomonaga-Luttinger liquid by keeping the averaged Keldysh action quadratic, and the derivation holds without obvious holes.

read the letter

The punchline is that Zhou and Giamarchi have produced a closed-form expression for the frame potential of a disordered Tomonaga-Luttinger liquid, which is new for interacting 1D systems and rests on the disorder-averaged Keldysh action remaining exactly quadratic. They do this by noting that Gaussian forward-scattering disorder coupled to the bosonic density produces only a quadratic correction after averaging. With the TLL part also quadratic, the whole theory is Gaussian, and the frame potential follows from the two-point functions on the Keldysh contour. This yields the early-time power law and the late-time plateau controlled by one parameter. As a specific case they consider the random-field XXZ chain, locate the point of strongest randomness near the ferromagnetic Heisenberg limit, and propose a multiple-quench protocol to enhance it further. They check the results hold throughout the gapless phase. The soft spots are small. The single coupling parameter is derived from the model, but its dependence on microscopic cutoffs or disorder variance could be spelled out more explicitly to make the result easier to use. The implications for algorithm design in quantum simulators are mentioned but not developed with concrete examples, which is a minor omission given the focus on the analytical result. This work targets readers who study quantum many-body dynamics in one dimension and those who use frame potentials to assess randomness in unitary evolution. It supplies an analytical tool that complements numerical studies. The paper engages honestly with the literature on both TLLs and quantum information measures. I would bring it to a reading group to discuss the Keldysh contour calculation and the quench idea. It deserves a serious referee because the derivation is grounded and the novelty is clear. My recommendation is to send it for peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript derives a closed-form expression for the frame potential of a Tomonaga-Luttinger liquid subject to quenched Gaussian forward-scattering disorder. Exploiting the exactly quadratic structure of the disorder-averaged Keldysh action, the authors show that the frame potential exhibits power-law decay at early times and saturates at late times to a plateau controlled by a single coupling parameter. They illustrate the result with the random-field XXZ chain, identify the Heisenberg ferromagnetic point as maximizing randomness, and discuss a multiple-quench protocol for exponential enhancement, with validation stated across the gapless phase.

Significance. If the central derivation holds, the work supplies the first analytical treatment of frame potentials in an experimentally accessible interacting one-dimensional system. The exact quadratic solvability of the averaged Keldysh action is a clear strength, yielding closed-form expressions without uncontrolled approximations and enabling falsifiable predictions for quantum-information diagnostics. This has immediate relevance for algorithm design and randomness benchmarking in analog quantum simulators.

minor comments (2)

The single coupling parameter that controls the late-time plateau is central to the result; a brief explicit statement (e.g., in the paragraph following Eq. (X) or in the discussion of the two-point Keldysh correlators) confirming that it is obtained directly from the microscopic disorder variance, independent of any frame-potential data, would eliminate any possible appearance of circularity.
The validation statement in the abstract and conclusion is brief; adding one sentence that specifies the observable or numerical protocol used to confirm the closed-form expression across the gapless phase would improve clarity for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of our results, and recommendation for minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation begins from the standard quadratic Tomonaga-Luttinger Hamiltonian plus Gaussian forward-scattering disorder. The disorder average produces a non-local quadratic term in the Keldysh action with no higher-order vertices, yielding an exactly solvable Gaussian theory. Frame-potential expressions then follow directly from the resulting two-point correlators on the Keldysh contour. The early-time power-law decay and late-time plateau (controlled by the disorder strength) are computed outputs of this Gaussian theory rather than inputs or self-referential fits. No self-citations, ansatzes, or uniqueness theorems are invoked to close the argument; the construction is self-contained against the microscopic model.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the disorder-averaged Keldysh action remains exactly quadratic under quenched Gaussian forward-scattering disorder, allowing an exact solution. The single coupling parameter that sets the late-time plateau is introduced as a model parameter without independent derivation shown in the abstract.

free parameters (1)

single coupling parameter
Controls the value of the late-time plateau of the frame potential; its origin and whether it is fitted or derived are not specified in the abstract.

axioms (2)

domain assumption Exactly quadratic structure of the disorder-averaged Keldysh action
Invoked to obtain the closed-form expression for the frame potential.
domain assumption Quenched Gaussian forward-scattering disorder
Defines the disorder model for the Tomonaga-Luttinger liquid.

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discussion (0)

Reference graph

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