Prethermalization in coupled one-dimensional quantum gases

1- good numerical results to support the proposed prethermalisation
2- nicely written
3- important subject

1- explanations in terms of extra non-trivial conserved charge unclear or incorrect

In this work, the authors study the evolution of two Lieb-Liniger models coupled by density-density coupling in the limit of small characteristic coupling momentum and with different states / tubes being coupled. They use the Boltzmann equation formalism. In this limit, only certain processes (called (1,1)-processes) are involved, which, it is claimed, do not lead to thermalisation. Convincing numerical results are provided which show that this is indeed the case. Thus, there is a pre-thermalisation plateau. The authors also provide an explanation in terms of additional, non-trivial conserved quantities that are supposedly admitted by the coupled model.

I think the overall results are good and interesting. The subject of thermalisation / prethermalisation due to integrability breaking is certainly of high interest currently, and the setup of coupled Lieb-Liniger model is very relevant. Admittedly this is a small extension / adaptation of some previous works, however it does unveil an interesting phenomenon.

But I do not understand the explanations in terms of the non-trivial conserved quantities. It seems to me that the explanations are incorrect, and that the authors do not actually explain the phenomenon in terms of such non-trivial local conserved quantities. I agree that there are objects in the Boltzmann equations that constrain the (1,1)-dynamics, but I don't think these have been correctly identified with extra local conserved quantities (or it may be that I have misunderstood something).

Thus, I would accept the paper for publication only after this has been clarified or re-written in order to be correct.

Precise comments:

After eq 16 say that "Q_0 is defined below" or the like for more readability

After eq 33: perhaps also say that we need the operation of dressing to be invertible to conclude that we must have vanishing of Q_0.

Just before eq 34: say that this is indeed a necessary condition as all other factors in eq 29 are strictly positive (on the set eqs 36,37).

Page 12 "energy momentum-energy conservation laws"

Equation 58 is valid to what order in 1/c? First order only? All orders? Please clarify.

Solution to eq 59: it is stronger than that, as what we want is to find $f_1$ and $f_2$ that are *independent of the state* (as in the value of conserved quantities on the state, eq 49, the functions $f_i(\lambda)$ do not depend on the state). The state-independent $f_i$ must satisfy eq 59 for all states. It is very different from eq 56, where $\epsilon_i$ are determined by (characterise) the state. The problem appears to be very different. The special cases consider, where dressing disappear, are ok, but the full set of solutions will be different it seems.

Pages 14-16: The language and discussion of conserved charges and GGEs is confusing and, it seems, the discussion is not entirely correct.

Eq 62 is not a conserved charge, it is the average value of the conserved charge in the state characterised by $\rho_{p,i}(\lambda)$ (GGE of the unperturbed system). Eq 64 is unclear: what is the time $T$? What is the specific dynamics? It in fact does not make sense, as the average value of the conserved charge should only depend on the state, and should not have explicit time dependence (there is no time, there is just a state). Do the authors want to take $T\to\infty$? Is the idea that $\bar I_{i,j}$ is then a function of the state, specifically the time-averaged value of $I_{i,j}(t)$ starting form that state? Please clarify this. If so, then indeed $\mathcal I_j$ is a function of the state only, and this would make more sense.

However, even then, the result would be a very nonlinear function of $\rho_{p,i}(\lambda)$. Then, of course, there is little guarantee that this is the average of a local operator.

On this point, the authors said that locality of $\mathcal I_j$ would be clear. But then, in order for this to be so, one should interpret eq 62 more "literally" as an operator equation, not just an equation between average. Just an equation between averages in GGEs (of the unperturbed system) would not guarantee that $\mathcal I_j$ is local as an operator; there are many nonlocal operators that have the same expectation values as local operators in all GGEs. Then this should be made clear, perhaps with hatted symbols, and in this case $\hat I_{ij}$ is not given by eq 63 (which is just its average).

But then, this would be a problem, as the definition of the operator $\mathcal I_j$ would involve explicitly not only the charges $I_{i,j}$, but also denominators that depend explicitly on the state. We cannot define intrinsically a charge with coefficients that depend on the state (we can take linear combinations of conserved quantities with state-dependent coefficients, e.g. in subtracting the average times the identity operator, but then this is not intrinsic). Thus, eq 62 as an operator equation does not make sense.

I conclude (if the limit $T\to\infty$ is implied) that the authors define rather complicated state-dependent functions, eq 62, which they would like to interpret as averages of local conserved quantities in the state, but there is no indication that these correspond to local conserved quantities.

Other questions are: how does one minimise the fluctuations (sentence after eq 65)? Why does on do so? How is the condition of non-zero norm of the operator translate to eq 66? Given that only averages of charges in states were defined, how does one get to an operator norm condition?

I agree nevertheless that the (1,1)-induced dynamics may be constrained, and that functions of the state that are invariant under this dynamics indeed constrain it and may stop it from thermalising. But I am not convinced that the discussion here explains this in terms of any nontrivial local conserved quantities of the low-$k_r$ perturbation.

Eq 68: I don't understand this. Naively this seems to be try from eq 67 with eq 62. However, in eq 62 each denominator seems to be a function of both $\rho_{p,1}$ and $\rho_{p,2}$, because the time evolution involves coupling between them (unless I am missing something); and also perhaps the coefficients because of the minimisation of fluctuations.

Then, I am rather lost on the rest of that section. In particular, in eq 70 there is $f_{ij}(\lambda)$, the one-particle contribution of nontrivial conserved quantities. But if the source term of the pseudo energy has this form, then the average of the nontrivial conserved quantities should be linear in $\rho_{p,i}$. But I concluded above that this couldn't be the case from the definition eq 62. So this seems to be inconsistent.

see above

The authors study a system of two one-dimensional quantum bosonic gases, coupled through a density-density non-local interaction term. Each one-dimensional system is described by an integrable Lieb-Liniger model. However, the interaction between the two systems break integrability. The authors are interested in the late-time regime, when the system is initialized in a non-equilibrium state. While at infinite-times one expects thermal behavior to emerge, the authors study the case of weak integrability breaking and aim at characterizing the pre-thermal regime emerging at large but finite times.

This is a fully theoretical work, using a hydrodynamic framework based on the Boltzmann collision integral approach. Overall, I think the draft is very well written and proceeds in a very logical way. I could follow all the sections, and I believe the style of the presentation makes the material superficially accessible even by non-experts.

Having said this, I am a bit hesitant to recommend publication in Scipost Physics, as I believe Scipost Physics Core would be more appropriate. The main reason is that in this work there is no independent numerical result supporting the analytical predictions (by "independent" I mean some ab-initio numerical simulation, using for instance tensor-network techniques). Of course, I understand that producing numerics for this model might be very challenging or even impossible. Still, without an independent numerical test it is difficult to quantitatively estimate the impact of the approximations made in the hydrodynamic setting. Accordingly, in my opinion the main value of this work relies in its technical/mathematical aspects. They are certainly impressive, but perhaps of interest for a more specialized audience.

I also have a curiosity: it seems to me that the model is integrable in the limit opposite to that studied by the authors, namely when A(x)=\delta(x) (after regularization). In this case, it seems to me the model can be mapped onto a Yang-Gaudin Hamiltonian. Is this true? Do the authors expect that this observation can be used to study the dynamics in the opposite limit of short-range interactions?

In conclusions, I would recommend publication of the draft in Scipost Physics Core.