The complete scientific publication portal
Managed by professional scientists
For open, global and perpetual access to science
|As Contributors:||Balázs Pozsgay|
|Submitted by:||Pozsgay, Balázs|
|Submitted to:||SciPost Physics|
|Subject area:||Quantum Physics|
We derive contour integral formulas for the real space propagator of the spin-$\tfrac12$ XXZ chain. The exact results are valid in any finite volume with periodic boundary conditions, and for any value of the anisotropy parameter. The integrals are on fixed contours, that are independent of the Bethe Ansatz solution of the model and the string hypothesis. The propagator is obtained through a lattice path integral, which is evaluated exactly utilizing the so-called $F$-basis in the mirror (or quantum) channel. The final expression is similar to the Yudson representation of the infinite volume propagator, with the volume entering as a parameter. The contour integrals involve an amplitude describing the particular propagation process; this amplitude is similar to (but not identical with) the Bethe Ansatz wave function. An important feature is that it depends on two sets of coordinates (initial and final), and it is manifestly periodic for an arbitrary set of rapidities.
1- The paper presents an original way to compute the propagator by means of an ingenious use of the F-basis in the mirror channel.
2- The paper is clearly written and computations are given in detail (at least in the cases of 1 or 2 particles which are explicitly treated here).
1- The proof of the main result (formula for the multi-particle propagator) in not given in the paper. The authors write that they have "constructed a general proof of this result, which will be presented elsewhere due to its length and technical nature". In fact, the computations become already so cumbersome in the two-particle case that you may wonder whether it is really tractable in the general case, and you have to believe the authors when they say they indeed did it.
2- The authors seem to be unaware of the existing literature concerning dynamical correlation functions that would be especially relevant for the subject of their paper. In particular, they seem to be unaware of the following reference:
N. Kitanine, J. M. Maillet, N. A. Slavnov, V. Terras, "Dynamical correlation functions of the XXZ spin-1/2 chain", Nucl.Phys. B729 (2005) 558-580, arXiv:hep-th/0407108.
in which a not so different approach was developed for the computation of a slightly different object (a time-dependent correlation function), leading to a very similar result in its form. In particular, the result of the present paper can be easily predicted in the context of the aforementioned reference.
The aim of the paper is the computation of the propagator of the finite XXZ spin-1/2 chain, namely of the matrix elements of the time evolution operator in the local spin basis of the model.
To perform this computation, the authors use a Trotter approximation of the time evolution operator as a product of transfer matrices. The resulting matrix elements are then written as a trace of products of elements of the so-called quantum transfer matrix, and evaluated in the F-basis, a basis in which the explicit expression of these quantum transfer matrix elements take very simple form.
I find this paper interesting due to the ingenious use of the F-basis in the framework of the quantum transfer matrix. I have however two main concerns.
The first one is that I was quite disappointed that the authors did not present the general proof of their main result in the paper. In fact, they only proved explicitly the formula in the cases with one or two particles only. The computations in the two particle case are already quite involved, so that it seems not so obvious that it is still manageable in the general case. The authors pretend they have done it, and I really would have liked to see the proof here.
The second one is that there already exists in the literature a paper, not even cited here:
N. Kitanine, J. M. Maillet, N. A. Slavnov, V. Terras, "Dynamical correlation functions of the XXZ spin-1/2 chain", Nucl.Phys. B729 (2005) 558-580, arXiv:hep-th/0407108,
in which the propagator of the finite spin chain was summed up in the form of a multiple integral of the same form (i.e. on the same contour, also with the Bethe equations in the denominator), although in a slightly different context (for the study of time-dependent correlation functions). The authors of the aforementioned reference showed moreover that it is possible to obtain this result in two different ways: from a direct computation using a Trotter decomposition (see formulas (1.6), (4.1) and (4.2) there, which are exactly equivalent to formula (2.35) of the present paper), or from the insertion of a complete set of eigenstates (with deformation by a twist so as to effectively ensure the completeness), an option which is discussed by the authors of the present paper, but ruled out as too complicated. In fact, if one uses the latter option (following the procedure described in section 5 of the above 2005 paper), one obtains straightforwardly a multiple integral representation of the propagator that should be equivalent to the result presented here. The numerous discussions in the present paper of whether one can or not obtain the result by a sum over the Bethe roots clearly show that the authors are not aware of this previous work on the subject.
1- Since the authors claim they already constructed the general proof of their result by their method, I would have liked to see it in the present paper. I understand that it may be a bit long but, as anyway they want to publish this proof, I think this should be done here, at least in a sketchy way. In fact, since the result in itself is not a surprise (considering that one can obtain the multiple integral representation directly by insertion of a complete set of states, following the scheme presented in the 2005 reference I mentioned above), the main interest of the present paper is the inventive method presented here. It should therefore be shown that it is manageable in the general case as well, and not only for 1 or 2 particles. The authors may use appendices for this if they do not want to make the paper too heavy.
2- The authors should include the aforementioned reference to the 2005 paper of N. Kitanine et al., and modify accordingly their discussion, for instance before section 2.1, around formula (2.35), in section 4.4, in section 5.3, in section 6 and of course in introduction and conclusion. The result obtained by the two methods should of course coincide.
Other minor changes:
3- It would be nice to have explicit references in the first paragraph of introduction, concerning "the study of the state functions and correlation functions in the ground state or at finite temperature", concerning "the study of out-of-equilibrium situations" (even if in that case the references are presented later), and concerning "experimental advances that make it possible to measure the dynamical properties of isolated quantum systems".
4- Still in the introduction, on page 2, I think that the sentence "It was already argued that even time dependent local correlators could be computed within the QTM, somewhat analogous to the determination of finite temperature correlators" would deserve, among other references to the computation of correlation functions within QTM, a reference to the work of Sakai:
K. Sakai, "Dynamical correlation functions of the XXZ model at finite temperature", J. Phys. A 40:7523-7542 (2007), arXiv:cond-mat/0703319.
5- Just before formula (2.19), the authors should also cite
M. Gaudin, B. M. McCoy, and T. T. Wu, "Normalization sum for the Bethe's hypothesis wave functions of the Heisenberg-Ising chain" Phys. Rev. D 23 (1981).
6- There is an unfinished sentence at the end of section 4 that should be completed.