最近打算投个american journal of physics（个人心目中非常非常好的一个杂志，上面无数经典文章，远比nature，science上的时髦文章强），于是翻出2009年投ajp时获得的审稿意见。当时是一正一负，编辑于是找了第三个，这个偏负（somewhat reluctantly），于是拒稿。转投european journal of physics（美国版的欧洲对应）后顺利发表。  

不得不说，三个审稿意见还是非常认真的（虽然我不赞同第二个审稿人的意见）。刘全慧老师讲，这个期刊审稿很严，确实！我投pra和prl，只有一次获得比这个更认真的审稿意见。

最终发表的文章：  Exact diagonalization: the Bose-Hubbard model as an example

这个文章后来反响还可以，到目前被下载了2500多次，google显示被引用45次。

根据樊博友的建议，最后附上代码

第一个审稿人意见：

The theory of exact diagonalization for quantum many body systems is relatively simple, but the technique relies on a few special 'tricks' to be practical. Most importantly is the use of iterative algorithms, especially Lanczos, which explot the sparse nature of the Hamiltonian matrix for computational efficiency. These are not just time saving steps, but efficient use of computer memory is even more important since, unlike most other algorithms in computational physics, is is memory not computer CPU time which is the critical limiting factor. The (not well known) tricks for selecting the appropriate sectors of Hilbert space and efficiantly packing these into a single computer array are therefore essential to make the scheme practical.

This paper presents these techniques in a clear and straightforward manner, which is easily accessible by students with a background of quantum physics (up to second quantization) and a basic knowledge of computer programming (either in MATLAB or FORTRAN or similar languages). The steps for counting the states in Hilbert space, and efficiently mapping these states to a computer array are well explained in detail. This includes the use of symmetries of the model to reduce the effective matrix size. This part is presented in an intuitive manner which does not assume a deep understanding of group representation theory.

The only part of the paper which I felt could have benefitted from a bit more detail is the section on the Lanczos algorithm itself. It is stated that the algorithm converges well for extremal eigenvalues, but the algorithm itself is simply described as a matter of making a suitable subroutine call. There are some issues in the choice of initial state for the Lanczos iteration, and the idea of a Krylov sequence of states (|psi>, H|psi>, H^2|psi> etc) which might give the students a better understanding of why Lanczos is so successful. Without giving any of these details the students are left with only two choices, either using the ARPACK package in Fortran or the 'eigs' command in Matlab. Perhaps a bit more information about the Lanczos method could help students understand what these commands do, or allow them to write their own

subroutines for this part (a topic omitted in other pedagogical guides to key algorithms, such as 'Numerical recipes').  On the other hand there are plenty of references where the Lanczos algorithm and its convergence properties are described in detail, and so an interested student can always fill in these gaps by a bit more background reading.

The use of the Bose-Hubbard model is nice as the physical example, because bos models have issues of state counting which do not occur in simpler spin or fermi Hubbard models. The model also has interesting properties such as superfluidity which can be examined using the techniques presented (Fig 2).

In summary the paper is well written and should be published in AJP. Perhaps the authors should be given the option to add more on the Lanczos algorithm itself, as described above, but I do not require this if the authors feel that this is unnecessary.

=====================

第二个审稿人意见： AJP MS23144-1 R2's report.pdf

This paper describes a procedure for finding the ground state eigenenergy and eigenvector of the Bose-Hubbard model. It basically consists of three steps: (1) finding a suitable basis set in terms of

occupation numbers; (2) generating the Hamiltonian matrix (which turns out to be sparse); and (3) feeding the matrix to a blackbox code (MATLAB, in this case) for the answer.

The authors gave considerable details on a way to produce the basis vectors by using a lexicographic order, as well as to generate the H-matrix by labeling the basis vectors with unique tags and sorting. These techniques, while interesting, are neither new or uncommon. For example, one explicitly or implicitly uses the lexicographic order all the time in thermodynamics. While it is technically true that they are more “efficient” as the authors claim, a larger point is being missed. Efficiency in setting up the Hmatrix is peripheral, and of limited consequence. Actually implementing an efficient matrix solver is more important, which the present paper does not address. Instead, it just uses packaged MATLAB routine. Incidentally, any variant of the celebrated Lanczos algorithm would do the job also.

In terms of physics, clearly, only small particle numbers (N) and sites (M) on the order of 15 or so could be solved numerically with the method outlined in the paper. As such, results from this limited

model are not a good indicator of real systems. For instance, the condensate fraction fc (Fig. 2a) does not approach zero in the limit of large U/J.

Essentially, the paper describes a way to generate and solve a sparse matrix to find the ground state to machine accuracy. Methods dealing with sparse matrices are abundant, more so if used only as blackboxes as in the present paper. The paper is limited and not of general interest. In many systems of interest at undergraduate level, Hilbert space representation is more common than the second quantization method in terms of the basis vectors. The chosen model in the paper naturally uses the latter, but the method is not applicable if Hilbert space is involved.

The paper is not particularly motivated. It lacks innovation and contains few new ideas in computational physics. Given its limited educational value, it is not suitable for publication in AJP.

=====================

第三个审稿人意见：

The paper discusses exact numerical diagonalization techniques for solving the Bose-Hubbard problem to make it a problem that is “readily accessible to a senior undergraduate student.” The two referee reports make different assessments, one (A) recommending it and the other (B) suggesting rejection.

I can see why there might be a disagreement among referees. I find the discussion to be of high quality, relatively clear, and interesting to me as a reader. However, I am nevertheless somewhat on the fence about whether it should be published.

Referee A finds the material covered on setting up the Hamiltonian matrix to be well explained and to be accessible to students. Referee B finds these points “neither new or uncommon” and are “of limited consequence” and is “not particularly motivated.”  Both referees suggest that not enough detail is given concerning the actual diagonalization method.

To me the fundamental question is whether the material will of interest and be used by a sufficient number of readers to devote valuable AJP pages in its publication. Referee B suggests the paper “is limited and not of general use.” I find the material of interest to me, but not as a teacher of undergraduates, but rather as a researcher in Bose systems. But surely if I wanted to use the methods, I would be able to find all of them in the research literature as referee B suggests. Even with more detail on the diagonalization of sparse matrices, I doubt these methods would be used by any undergraduate instructors. The second quantization, the group theory, and the whole complication of the physics and the method are too advanced and would put off any instructor from assigning such a project even as an Honors project---unless perhaps the instructor already used the methods and could provide much of it ready-made.

Thus I have to come down on the side of Referee B and suggest, somewhat reluctantly, that the paper be rejected for publication.  

========================================================

欧洲物理杂志（EJP）的意见

Title: Exact diagonalization: the Bose-Hubbard model as an example

Authors: Mr J M Zhang et al

The manuscript "Exact diagonalization: the Bose-Hubbard model as

an example" by Mr J M Zhang et al. does not primarily focus on

new results but on an introduction in exact diagonalization by

means of the Lanczos algorithm. I think it is of good quality,

and if EJP likes to publish methodological papers I think it

could be published after minor changes.

My criticism concerns two things:

1. On page 2, first paragraph, a strong statement is made that

complete diagonalization is often neither possible nor

necessary in many cases. Here I would demand a more thorough

statement. If one wants to describe the full thermodynamics

of an electronic system both as function of temperature and

e.g. magnetic field a full diagonalization is

indispensable. Whether one can or cannot do it due to

dimensional restrictions is an important but different

question. The field of molecular magnetism is one example

where complete diagonalization is desired and even for larger

systems achieved by heavy use of symmetries, see e.g.

R. Schnalle, J. Schnack, Numerically exact and approximate

determination of energy eigenvalues for antiferromagnetic

molecules using irreducible tensor operators and general

point-group symmetries, Phys. Rev. B 79 (2009) 104419

I would be happy if the authors would rephrase their statements

somewhat.

2. The hashing technique of Lin is widely used in the

community. Nevertheless, one can find an analytical mapping

of basis states on natural numbers and back, see

J. Schnack, P. Hage, H.-J. Schmidt, Efficient implementation

of the Lanczos method for magnetic systems,

J. Comput. Phys. 227 (2008) 4512-4517

This works very fast. Since the authors might be able to

access this journal I would like to send a pdf copy for their

convenience.

文章代码：Untitled2.m  search.m  

需要的时候，可以把子程序search   改成这个   search.m

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