The quantum Ising chain for beginners

1- Very detailed lectures notes.
2- Reasonably self-contained.

1- Tendency to focus too much on technicalities.
2- The section on entanglement could be improved.

The lecture notes 'The quantum Ising chain for beginners' are an introduction to the various set of free fermionic techniques which can be used to study the one-dimensional quantum Ising model. The focus is on explicit computations (spectrum, eigenstates, overlaps, time-dependence, entanglement), of specific but nevertheless useful quantities, which are explained in considerable detail.

Overall I find these lectures to be pedagogical, already accessible to a master-level student. The authors often 'go the extra mile', and explain all the various hidden subtleties related to boundary conditions, fermionic parity, coherent states, etc, which are sometimes neglected in other lectures.

There are also downsides to the way things are presented. Reading these notes sometimes feels like browsing through a list of specific calculations. Those are carried out without discussing motivations, why they are possible, what is important or not. Such explanations would be useful to the targeted audience, especially if they need at some point to be able to compute something slightly different from what is presented.

Here is a list of general comments, that elaborate further on this criticism. Provided those are addressed, I would be happy to recommend publication in Scipost lecture notes.

a) The introduction is too short. It would be nice to explain some of the logic of these lectures here, together with some broader 'take home messages', which would be useful to the targeted audience. For example, a common feature shared by all sections is that to compute essentially anything in the quantum Ising chain with L sites, the only thing you need to be able to do is diagonalise a matrix of size $2L$, resulting in a massive simplification.

b) I also concur with the first referee regarding the first paragraph, which, even though it was probably not the author's intentions, sounds like trying to make excuses for not providing a proper bibliography.

c) There are too many footnotes, not even taking into account footnotes in problems and info boxes. While those can be a nice complement to the text, they should be used sparingly. There are several places where those could be put in the main text, or simply removed. For example, the fact that there are 5 long footnotes on page 19 is a sign that something went very wrong with the presentation in this particular place.

d) Specific notations are used to distinguish between the various types of operators and matrices. This is a good idea, but explaining those to the nonexpert reader would be even better. For example bold capital letters are used for square matrices of size $L$, while the hat is reserved for fermionic operators, Hamiltonians, of size $2^L$. I do not fully understand the mathbb notation, however. In some case it is just a $2L\times 2L$ matrix with blocks in bold capital letters, but the hat mathbb are used to emphasize different parity sectors, for example in section 3.

e) Section 9 and especially 10 are subpar compared to the previous ones, see comments below. While computations say of the energy spectrum does not really require discussing motivations, it is clearly necessary for the entanglement entropy.

Below is a list of more specific suggestions:

1) section 1, page 3. It would be useful to explicitly state somewhere that for a collection of L spins, the dimension of the Hilbert space is 2^L. In footnote 1, remove the 'one the same site', since these are just 2 by 2 Pauli matrices, there is no notion of 'same site' yet. 'typical of fermionic commutation rules' reads awkwardly, since there is the anti-commutator.

2) Page 5. 'Notice that $K_j=K_j^\dagger=K_j^{-1}$ and $K_j^2=1$'. It was already mentioned two lines above that $K_j$ is a sign, so the reader should be able to figure this out by themselves.

3) Footnote 4 does not fit well with the spirit of these lectures, which deals almost exclusively with finite chains. It can be safely removed.

4) In section 2 and 3, there is some ambiguity in what is meant by transverse field Ising model. The text is reasonably clear in stating that J^y=0 in (18) in this case, but then discussions later on do not always make this assumption, even though section 2 and 3 are named 'Ising model'. Perhaps give a different name to (18).

5) Page 10. It seems to me the most important point is that $\{c_k,c_{k'}^\dagger\}=\delta_{kk'}$, which is implicitly used later on. The notation in (36), (37) are somewhat nonstandard, and advantageously replaced by $\displaystyle{\sum_{k\in \mathcal{K}_p}}$ as is done on the next page. Remove footnote 7, or put back in the main text. In footnote 9, replace the first and by a comma.

6) Page 11, before (42). $\hat{H}_k$ 'lives in the 4-dimensional space' because all terms with different |k| in the Hamiltonian commute, which is related to the previous point. Also, technically $H_k$ are still matrices of size $2^L \times 2^L$, they just happen to act nontrivially only on this four-dimensional subspace.

7) Page 13. Figure 2 is more stylish, but figure 3 provides the same information and is much more easy to visualize.

8) Page 15. First line in section 3.1. 'in Eq. (59)'. Together with (50), which is nonnegative. Just before (69), the discussion of the energy spliting is not very clear. The reader 'naively expects' (69) to hold, and they would be perfectly right, since this equation deals only with the extensive part of the energy. Subleading corrections, exponential or not, are irrelevant in (69).

9) At the beginning of section 3.2. 'In this section, we will focus on the Ising case'. But section 3 is named 'Uniform Ising model', and the same for section 3.1.

10) Page 20. Footnote 20 can be safely put back in the main text.

11) While discussing the 'Nambu formalism', it would be nice to mention how one can recover the results of the previous sections, simply because the matrices $\mathbf{A}$ and $\mathbf{B}$ have a simple translationally invariant structure.

12) Page 21. Similar abuse of notation as before, with concatenation of two vectors ${\bf u}$ and ${\bf v}$ to make a larger vector.

13) Page 22, footnote 25. 'Similarly, one must have [...]'. This does not bring additional information, since a left inverse is also a right inverse for square matrices.

14) Page 23, before (108). Replace 'The $2^L$ Fock states' by 'The $2^L$ eigenstates'.

15) Page 24. $2\epsilon_\mu \equiv \epsilon_k$ looks like a clash of notations.

16) Page 25. 'More in detail' reads awkwardly.

17) Page 25, 26. Make explicitly the point that in the absence of disorder, eigenstates are extended.

18) The discussion in section 5.3 is pretty good I think. 'a theorem of linear algebra' becomes a consequence of the Schur decomposition in a later section.

19) Page 41. The notation $1(0)$ makes the formulas harder to read. It would be better to replace by $\mathbf{Z}_{\alpha}$ with $\alpha=0,1$, or something similar.

20) Discussion after (200). I understand the point, but the assumption of invertibility is actually not a problem since everything is finite-dimensional. Invertible matrices are dense in the space of all matrices, so the Onishi formula holds as a limit when $U$ is not invertible. A mathematician would even be very happy with this. Same comment in the absence of pairings.

21) The discussion in section 9 ends quite abruptly. It would be perhaps nicer to mention that these exact formulas are the starting point of many subsequent large distance results --also related to the 2d classical Ising model-- and give a few extra references. Also, footnote a in problem 6 can be put in the text.

22) Page 50. The notation $\{l\}$ for subsystem is non standard and confusing, since $\{l\}$ is usually understood precisely as the set containing only $l$. And it gets worse with 'for instance, $\{l\}=1,\ldots,l$', where a standard notation for the rhs would be $\{1,\ldots,l\}$. Also, it is not clear whether the implication is that $\{l\}$ necessarily contains $l$ elements, consecutive or not.

23) Page 50, 'For a system described by a quantum state $|\psi(t)\rangle$'. It is strange to use notations implying time-dependence without ever mentioning it, since the calculation of the entanglement entropy has in principle nothing to do with time. You just need a state where Wick's theorem holds. Of course, this is also the case when time-evolving any such state with a quadratic fermions Hamiltonian. So time could in principle be removed from this whole section, resulting in lighter notations. Perhaps the authors want to make some kind of point that any state where Wick's theorem holds can be seen in this way, or just consider time-evolved BCS states, in which case they need to explain what they want to discuss, and what assumptions they make.

24) Page 52, 'its state $|\psi(t)\rangle$ is a gaussian state'. I do not know what is the state of the quantum Ising chain.

25) Page 53. There is a section 10.1, but no section 10.2. In step 2, 'to a sub-chain $\{l\}=1,\ldots,l$, say'. This assumption is not innocent at all. For a chain with PBC, it is in fact very important that the 'subchain' be in one connected piece. Otherwise the calculation explained in this section breaks down due to issues with Jordan-Wigner strings. That point should be made clearly, otherwise the 'beginner' reader might think this holds as well for subsystem $1,3,5,8$, which it does not. Also, subsystem is possibly clearer than subchain.

26) Page 53. Footnote 53 is not very clear, since much more than normalization is lost for unitary matrices, and the Nambu matrix is not unitary.

27) The derivation presented here is fine, but other methods should at least be mentioned as well, also in the slightly broader context of entanglement computations in free bosonic or fermionic systems, dating back to Bombelli et al, Srednicki, etc. This is perhaps a matter of taste, but I find the (closely related) derivation of Peschel to be easier to understand, especially for non experts. It just explains why the reduced density matrix (RDM) is the exponential of a quadratic fermion Hamiltonian, similar to Step 1. In fact, the explicit expression for the RDM follows from (277) in the text.

28) Problem 7. Give at least a few references for the logarithmic divergence, including the paper by Holzhey, Larsen, Wilczek.

See report

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see report

The lecture notes "The quantum Ising chain for beginners" by G. B. Mbeng, A. Russomanno, G. E. Santoro,
constitute an introduction to various techniques allowing one to study one dimensional quantum Ising chains.
The lectures are comprehensible starting from a master level student and are written in a almost uniformly pedagogical way.
Most of the presented results come along with detailed derivation, what makes the notes essentially self contained. I can only regret that
the authors did not provide more sections relative to the explicit computation of correlation functions in the models they consider.
Still, it is more of a personal regret than a criticism on my part. To summarise, I believe that it is an interesting piece of work.

However, the only thing which does not allow me to recommend, so far, these for publication is the following sentence that appears in the
introduction: "We, unfortunately, do no justice to the immense literature where concepts and techniques were first introduced or derived, and even less so to the
many papers where physical applications are presented: this would require too much effort for our limited goals."
While such a statement would, in principle, be OK for a master lecture given on the blackboard, I do not believe that it is possible to do so
in a set of lecture notes to be published in a journal. It is not just the question of making justice to the various predecessors
that worked and pioneered the analysis related to this models. It is also the issue of giving the students the right impression about the
amount of work that was carried on these models so as to eventually get to such a nice and neat description. Finally, the references may also be of quite some
use to the researched that would need to remind themselves of this or that point of history of the model.

Therefore, for the time being, I do not recommend the lecture notes for publication, and kindly ask the authors to complement these with a detailed set of
references. If this is done, then I would not see any more obstruction for publication.

Below, I provide a short list of the missprints I have spotted:

\begin{itemize}


\item Page 9 and several instances that follow. The authors speak of the number of Eigenvalues as a way to count dimensionality. I would suggest to rather speak of the
number of Eigenvalues counted with multiplicities.

\item Eq (33). I would suggest to use a slightly different notation for the Fourier transforms of the fermion operators. For instance, to change the letter c typography.
That should lead to more clarity.


\item Discussion below (69). Taken the overall level of detail given in the lecture notes, I would suggest to provide more explicit details on the content of the paragraph.

\item I think it is rather Hermitian than Hermitean


\item "Unfortunately, the computer will produce eigenvectors associated with the degenerate zero-energy eigenvalues which do
not have the structure alluded at in Eq" I think that this statement strongly depends on the routine adopted by the computer. Maybe some more
explanation of the origin of this problem would be interesting.

\item Discussion around (114). Taken the overall level of detail given in the lecture notes, I would suggest to provide more explicit details on the derivation of (114). Here, the authors
could recall the central limit theorem and the convergence of averages and could explain a bit more the main steps.


\item I could not identify the definition of $\tau_L^{*}$ in the footnote of exercise 5 on page 40

\item "using" on page 49 is probably a typo

\item page 53, Section 10, paragraph 2 $\rho_{L} \hookrightarrow \rho_{l}$

\item $(t)$ is missing in the definition of $\mathbb{\Lambda}$, in the elements $\lambda_k$ as well as in the line below of (270).




\end{itemize}

see report