Numerical Study of Disordered Noninteracting Chains Coupled to a Local Lindblad Bath

The manuscript by Berger and collaborators studies the physics of Anderson localisation in one dimensional chains (with random or quasiperiodic disorder) in presence of a dissipative coupling to a bath at one edge of the chain. The motivation for this work comes from the Many-Body-Localization transition and its possible instability due to thermal avalanches.

The Authors take advantage of the non-interacting nature of the problem and use the formalism of third quantization to solve their models and compute the Lindbladian gap. The scaling of this quantity with system size, for different disorder strengths, allows to obtain information on the stability of the localised phase. The main result of this work is that the Lindbladian gap, related to the inverse relaxation time, scales to zero exponentially for large chains. In addition, the Authors point out that for both models under scrutiny (and in particular for the case of quenched random disorder) the finite size effects are strong enough to prevent obtaining a conclusive picture. Nevertheless, the numerical evidence suggests an exponential decay the critical disorder for the avalanche instability is overestimated. The Authors conclude their work by providing a simple toy model to understand the origin of the effective decoupling of the bath from the system, at the origin of the exponential scaling of the gap.

I think the content of this manuscript is interesting and worth to be published in a suitable journal. However the manuscript does not contain in my opinion sufficient material and results to justify publication in SciPost Physics, based on its acceptance criteria which require either a major advancement/breakthrough or opening of a new research direction. In its current form I would be happy to recommend the paper for Scipost Phys. Core.

Accept in alternative Journal (see Report)

This paper studies whether localization persists in an noninteracting disordered spin chain whose edge is coupled to local baths, which may trigger avalanche instability and lead to thermalization. The key insight is that the scaling behavior of the Lindbladian gap could be used as a diagnostic of ergodicity/localization. In particular, this paper numerically computed the Lindbladian gap for the Anderson and Anbry-Andre-Harper chain (with the left boundary coupled to spin raising and lowering operators) for various system sizes and disorder strengths. This paper examines the spectral gap rescaled by $2^L$ ($L$ is the system size) and finds qualitatively different scaling behaviors for disorder strengths larger and smaller than a critical threshold. This finding is corroborated by the behavior of percent of eigenstates with non-vanishing weight on the first site for different disorder strengths. Finite size effects were also explored. This paper reports strong finite size effects for intermediate system sizes and shows that the critical disorder threshold approaches the expected value from above for longer chains.
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I have a few major comments and questions:
1. The motivation/main result of this work is not entirely clear. As the authors have also noted, thermal avalanches can only arise in interacting chains. So what exactly does simulating non-interacting chains do? Surely, much larger system sizes will be accessible numerically than the interacting case, but how does the extrapolated disorder threshold relate to the "actual" disorder threshold in interacting chains? The paper only suggests that the extrapolated value is larger than the actual critical disorder in closed chains. Can we say more about this relation? For example, can we claim the extrapolated value approaches the actual value (from above) in the thermodynamic limit? Or is it just an upper bound (if so, why?) The physics isn't really clear. Perhaps the authors could shed some light on this.

2. The mapping between a dissipative spin chain and a non-Hermitian two-leg ladder has been rather well-established (and more generally between a dissipative lattice model and its non-hermitian bilayer counterpart; see e.g. PRB 99 174303, PRB 109,085115). Since the system is a non-interacting chain of spin-halves, would it be possible to map it to a non-Hermitian problem quadratic in fermion operators and even solve it exactly? While this is a numerical study, writing down the problem in the fermion language could add more completeness to the work.

3. The motivation of Section 4 is not clear. The main point is that when there is decoupling in either real or energy space, certain degrees of freedom get a divergent lifetime/ vanishing Lindbladian gap, and this mechanism applies to the dissipative disordered chains at hand (some states can be completely decoupled from the first site and therefore the bath). However, this argument is rather straight-forward, well-known, and intuitive. A few paragraphs might have done the job of establishing this connection. Could the authors motivate/argue why a study of the toy model is needed/ should be kept in the paper?

4. Figures:
i. using legends instead of colormaps might have been sufficient for all the plots.
ii. Figure 2(c) and similarly 5(c): the authors should consider using a plot marker other than the stars (lowest curve in 2c, $\lambda = 1.7$). The stars cover the error bars almost completely.
iii. Figure 3: using log-scale for the y axis might work better. Is there a particular reason why a linear scale was chosen instead?
iv. Figure 6: it's not clear why there are still error bars. Is the plotted quantity already the standard deviation across all disorder realizations?
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In addition, I have a few minor comments/suggestions:
1. In the second paragraph of Sec 2.2.1, is there a reason to assume/claim the Lindbladian gap scales as the thermalization rate?

2. On page 4 under Eq. 6: is there a way to quantify "weak" vs. "strong" disorder? (a comparison of W against some other energy scale?)

3. The authors might consider going beyond simply citing third quantization and including more details on how the numerics were performed.

4. In computing the overlap of single-particle eigenstates with the first site, the paper chooses $10^{-14}$ as the error threshold. Why was the machine precision used? Naively, the threshold for determining whether there is significant overlap should depend on the system size as $1/L$. Would the claims in the paper remain valid with a system-size dependent threshold? Or might $10^{-14}$ suffice? Perhaps the authors can help clarify this point.

5. Typos:
i. Last paragraph of Sec 2.1.1: The sentence "If in a finite chain..." needs a bit of rewording for better clarity.
ii. Right above Eq. 8, the golden ratio should be $(\sqrt{5} + 1 )/2$ instead of $-$.
iii. At bottom of page 9: an extra space between $W$ and $(\lambda)$.
iv. Eq. 17: consider using a dummy index other than $i$. There is already an imaginary $i$ in the phase term $e^{i \phi/3}$.
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Overall, I think the paper studies an interesting question, is largely valid, and is honestly and cleanly written. However, a few major things aren't exactly clear, and the authors are politely recommended to address the comments above.

See comments above.

Ask for minor revision