Entanglement spreading in non-equilibrium integrable systems

Dear Editor,

I thank the referees for the prompt and very positive reports. Both referees ask for some very minor adjustments that I took in consideration as detailed below.

Best Regards,

Pasquale Calabrese

Answers to referee 2

1 -Page 2: von Neumann is misspelled: Von instead of von.

A: Thanks, fixed.

2- Page 4, 5 lines below Eq 7: double "the"

A: Fixed

3- References to numerical tests of ETH are perhaps a bit selective, in particular, since no recent papers are cited. I cannot make a specific recommendation about what to cite, but there are many other people who contributed significantly (e.g., Lea Santos, Peter Prelovsek, Lev Vidmar, Robin Steinigeweg, Jochen Gemmer, and others). This is not central to the review's main topic, though.

A: I added the more recent references 38-44. This should be enough.

4- Page 5: is the question of completeness of sets of conserved operators settled for typical integrable models, such as e.g., the spin-1/2 Heisenberg chain? I am not aware of a mathematical proof, in particular, given the newly discovered quasi-local charges. See the sentence at the end of the 1st paragraph.

A: This is beyond the goal of these notes. The issue has been settled in Ref. [61] (and some other papers by Balazs Pozsgay) for model with a simple Bethe ansatz. For nested Bethe ansatz systems, to the best of my knowledge, there is no proof yet, but the completeness of local+quasilocal charges is likely true.

5- Page 5: Is "Neither" perhaps a typo and should read "Noether"?

A: Fixed

6- Page 8, 2nd paragraph: missing "of" in "pairs quasiparticles"

A: Fixed

7- In Sec. 6.4.7, one could add a reference to Phys. Rev. B 90, 075144 (2014)

A: Done

8- The author could (optionally) add an outlook onto open technical and conceptual questions.

A: Thank you for the suggestions. Few open questions are spread out in Section 6.4, now 7 (e.g. Renyi entropies, negativity, etc). I found that discussing even more open problems is beyond the scope of the lecture notes and would lead to a real review.

Answers to Referee 1

1) Abstract: I would strongly suggest mentioning the word "quasiparticle picture" in the abstract, as this topic takes most of the discussions in these lecture notes.

A: Thanks for the suggestion. Done.

2) Page 2, "despite the fact that the dynamics governing the evolution is unitary and the initial state is pure": It would be useful to explain why is this surprising, i.e. that unitary evolution should lead to deterministic dynamics, where one can rewind time and read off the initial non-equilibrium state of the system. "Thermalisation" implies, however, that such rewinding should not be possible.

I added the sentence "Such relaxation is, at first, surprising because it creates a tension between the reversibility of the unitary dynamics and irreversibility of statistical mechanics."

3) Page 3, "Let us consider a spatial bipartition of the system": More motivation is needed here, specifically why are we interested in bipartition and local operators. One can, for example, mention that in experiments we can usually only access local observables. This then naturally implies that the question "do we see thermalisation in quantum systems?" should be reformulated to "do local measurements see thermalisation?".

A: I liked the suggestion and added, after Eq. (4) the sentence: "This line of thoughts naturally leads to the conclusion that the question Can a close quantum system reach a stationary states?'' should be reformulated asDo local observables attain stationary values?''."

4) Page 5, "only integrals of motion with some locality or extensivity properties must be included in the GGE": At this point, it would be useful to give a simple example of what integrals of motion can enter GGE, as a direct comparison to the example of projectors on the eigenstates, which cannot enter GGE.

A: I added the sentence "For examples, the energy and a conserved particle number must enter the GGE, while the projectors on the eigenstates should not."

5) Page 6, Eq. 15: It would be useful to mention that the natural logarithm used in the definition of entropies is sometimes changed to log_2, or even, log_10. log_2 is often used in the context of quantum information, where the natural unit of entropy is the maximum entropy of a spin-half singlet state.

A: After Eq. (14) I added "(hereafter $\log$ is the natural logarithm)"

6) Page 6: "Then, for integer n >= 2, they are the only quantities that are measurable in cold-atom and ion-trap experiments": I would suggest mentioning here that the n=2 Renyi entropy is directly related to purity, a quantity of interest for quantum information science.

A: I added the sentence: "(${\rm Tr} \rho_A^2$ is usually referred as purity in quantum information literature)"

7) Figure 2: For panel (b), please indicate what is the \Delta used.

A: Added \Delta=2 in the caption

8) Page 9, "(the anisotropic Heisenberg model)": Since in the figures and later in the lecture notes, the name "XXZ chain" is used, I would say "(the anisotropic Heisenberg model, also known as the XXZ chain)".

A: Done

9) Page 10, "The interpretation of Eq. (25) is obvious": For inexperienced students, it may not be so obvious, so it would be good to mention that the form of H(n) can be derived easily considering a 2x2 diagonal density matrix with n and (1-n) on the diagonal, corresponding to occupied and unoccupied modes.

A: I added a longer explanation after Eq. (25).

10) Page 11, "Jordan-Wigner and Bogoliubov transformations": I would strongly suggest putting references here. Even better, one could add a reference to a step-by-step derivation from Eq. 29 to Eq. 30.

A: I added a reference to the book of Sachdev, where one can found all details of the transformations leading to Eq. (30).

11) Page 12, "Intuitively, a n-string solution corresponds to a bound state of n elementary particles with n = 1.": I am puzzled by the clause "with n=1". Surely, the definition of a generic n-string solution should not include this clause?

A: Here I just meant that the elementary particles are those with n=1, I rephrased to be clearer.

12) Page 14: I would strongly suggest promoting the whole subchapter 6.4 to chapter 7, as it serves as a conclusion for these lecture notes and an outlook for the future.

A: Thank you. Indeed, already in the first writing I was hesitating a lot if doing it or not. Now it is done.

13) Page 17, Chapters 6.4.5 and 6.4.6: Although the author briefly mentions the topic of random quantum circuits, it would be good to stress here that there has been extraordinary interest in these systems in the recent years. They have been used to probe entanglement dynamics in various scenarios, e.g. competition between unitary evolution and measurements, which can lead to extensive or sub-extensive entanglement behaviours, a topic which also ties into Ch. 6.4.6 on open systems and effects of the environment.

A: At the end of Sec 7.5, I added the sentence "Random unitary circuits have been used to probe the entanglement dynamics in many different circumstances, providing a large number of new insightful results for chaotic models. Their discussion is however far beyond the scope of these lecture notes".

14) Minor spelling/interpunction: * "Neither theorem" -> "Noether's theorem" (page 5) * add a period to the end of the sentence "Here we just summarise the main ingredients we need and then move back to the entanglement dynamics" (page 12) * "a n-string solution" -> "an n-string solution" (page 12) * add a period to the end of the paragraph in chapter 6.4.6 (page 18)

A: done

15) Minor editing of equations, as it is sometimes difficult to read: * cos x -> cos(x), sin x -> sin(x) in Eqs. 16-18 * space after the variables of integration in Eqs. 20-22, 34, 37, 42

A: Done

The list of changes is fully given in the reply to referee