Ruelle-Pollicott resonances of diffusive U(1)-invariant qubit circuits

In their manuscript, the authors study so-called Ruelle-Pollicott (RP) resonances in 1D quantum circuit models composed of gates that preserve a U(1) conserved quantity, such as total particle number. Such RP resonances have their origin in the study of classical chaotic systems, where they connect the exponential decay of correlations in time with properties of the propagator of probability distributions on phase space, which are closely connected with its response when adding weak dissipation. Such RP resonances have recently experiences a renewed interest in the context of quantum many-body physics, where one can connect their behavior with the problem of operator spreading (the Heisenberg evolution of local observables) that has received much attention in the last decade.

The present paper adds to this line of research by focusing on systems with a continuous conserved quantity and asking how the diffusive transport of this quantity manifests in the structure of RP resonances. In particular, the paper focuses on discrete time dynamics defined by a quantum circuit model, which has one conserved U(1) charge (and no energy conservation). To define RP resonances, the authors consider the effective dynamics obtained when restricting to extensive observables whose densities are sufficiently local, below some cut-off r. By projecting down to this subspace of observables, they obtain a non-unitary evolution whose non-unit eigenvalues constitute the RP spectrum. To further uncover the effect of diffusion, they resolve the RP spectrum by momentum. The leading eigenvalue at momentum k is directly related to the diffusion constant D, decaying exponentially with D*k^2. The authors verify this scaling and compare the value of D obtained from the RP resonances with the value obtained from a different method (evolving a domain wall state), finding excellent agreement, which constitutes one of the paper's main results. The authors further argue that below this leading RP resonance, there should be a continuous spectrum of further resonances, having to do with the conservation of higher powers of the conserved charge, subleading hydrydynamic tails, and other more subtle effects associated to diffusive transport. They also show numerical evidence that the above scaling of the leading resonance with k breaks down far from k=0 due to a cross-over with a different resonance that resembles the structure observed in systems with no conservation laws.

I think the paper is well written and it is a welcome contribution to the field. I recommend publication in SciPost once the authors have addressed the following minor questions and concerns:

• I was a bit confused by the proper order of limits in Eq. (11). In a finite system, taking r->\inifty should restore unitarity, so that all eigenvalues have unit modulus. I'm assuming the authors imagine an infinite system, but in that case, the spectrum might be continuous (as they indeed argue happens in the systems they study) so that the approximation using only the leading eigenvalue need not be valid. Could they comment on this issue?
• In Eq. (19), in the second formula, is there a subtraction of the t=0 value missing?
• In Section 4.1, the authors give a heuristic argument for the size of the subleading RP resonances coming from powers of the conserved charge. Could this be turned into a rigorous bound, using the locality of the circuit?
• How is Eq. (24) consistent with diffusion (since the diffusion constant itself is related to the current-current autocorrelation)? Is this an issue of the proper order of limits?
• The authors mention that Ref. 47 predicted a power-law decay for current-current correlations. What is the power and how does it relate to their numerical results in Fig. 9?
• The RP resonance approach to hydrodynamics appears similar to ideas that appeared under other names in recent years, particularly PRL 131, 220403 by Ogunnaike et al and PRX Quantum 5, 040330 by Moudgalya and Motrunich. I think these papers should be mentioned somewhere in the manuscript.
• A few typos: "Infinitely dimensional", "unit call" and "does not conserve only"

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1- Clear connection between Ruelle-Pollicott resonances and hydrodynamics. 2- The presented results are expected to be applicable to a wide range of U(1)-invariant circuits.

1- The study of transport by truncating the dynamics to operators with a finite local support size is closely related to the dissipation-assisted operator evolution (DAOE) approach to quantum hydrodynamics [see, e.g. Phys. Rev. B 105, 075131 (2022) and related works]. I would have expected to see these works referenced, and I think the current manuscript would be strengthened by discussing this connection and possibly outlining how the DAOE truncation (based on operator length, i.e. number of nontrivial Pauli operators) would modify the conclusions.

2- In the summary of results and in Section 2.3, it is mentioned that the quasi-momentum dependence of the leading PR resonance is in general non-trivial, even in systems without conservation laws. Here it would be useful to discuss the connection with Ref. [22], one of the authors' previous works.

3- I'm not sure how meaningful it is to include data that is not yet converged without showing any indicator of convergence, as in Fig. 6. It is mentioned in the text that many of the presented eigenvalues appear to be converging to 1, but it would be useful to show some data to support this claim.

4-It's not obvious to me how Eq. (20) follows from the above argument. Could the authors clarify? From Fig. 7 it's also not clear to me that this argument should return the correct scaling for higher orders of M.

Publish (easily meets expectations and criteria for this Journal; among top 50%)

1. Novel extension to conserved charges: The authors present a new method for extending the Ruelle–Pollicott resonance framework to systems with conserved charges.

2. Clarity of presentation: The paper is written in a clear and accessible manner, making complex concepts understandable to a broad audience.

3. Benchmarking against existing methods: The authors compare their approach with established techniques, such as truncation via an additional Lindbladian dissipation and direct Time-Evolving Block Decimation (TEBD) evolution, carefully discussing the nuances, strengths, and weaknesses of each.

4. Complementary momentum-space perspective: While the method does not yet show a clear advantage over state-of-the-art approaches, it offers a complementary view by enabling direct examination of correlation functions in momentum space—something not as straightforward in existing algorithms (e.g., TEBD).

5. In-depth spectral analysis: The authors provide a detailed discussion on the possible existence of subleading isolated RP resonances and on corrections to the asymptotic decay of correlation functions.

6. General applicability to strongly interacting systems: The framework offers a generic tool for studying strongly interacting Floquet circuits, opening the way to future generalizations for Hamiltonian systems and different symmetry classes.

1. No clear improvement in transport coefficient extraction: The proposed method does not demonstrate a clear advantage in determining transport dynamical exponents or the diffusion constant, producing results comparable to those from existing techniques.

2. Limited new physical insights: Although the approach is novel and original, it does not substantially advance our understanding of chaotic systems with U(1) conservation laws.

The paper clearly meets two of the four specific SciPost criteria:

1. Providing a novel and synergetic link between different research areas.

2. Opening a new pathway in an existing or entirely new research direction, with clear potential for multi-pronged follow-up work.

It also clearly satisfies the general acceptance criteria.

I therefore recommend the paper for publication.

As a few minor suggestions:

1. The naming “cases” might be misleading, as it could be interpreted as referring to different models or physical regimes. Since these instead correspond to different samples of Haar-random unitaries, it may be clearer to rename them as “instances” or “samples”.

2. The reference to k≃2 at the top of page 10 is potentially confusing, as in the corresponding graph k is expressed in units of π. Referring instead to k≃0.6π would make it consistent.

Publish (easily meets expectations and criteria for this Journal; among top 50%)