A beginner's guide to non-abelian iPEPS for correlated fermions

This review provides a detailed introduction to infinite projected entangled-pair states (iPEPS) for fermionic systems with a non-abelian symmetry. It is explained in detail how the symmetry and fermions can be implemented on the level of the PEPS tensors. The involved technical frameworks are not new and have been previously discussed in other papers, however, here they are presented in a unified way. The most important ingredients of the iPEPS algorithms based on imaginary time evolution are also discussed. As example applications, the authors present initial results for two different multi-band Hubbard models.

The paper is clearly structured and written in a pedagogical way, making it a valuable reference especially for non-experts. The example applications are interesting (although a full study will require more calculations as the authors point out), providing evidence of the validity of the approach and showing the impressive bond dimensions that can be reached by exploiting the non-abelian symmetry.

Thus, I can recommend this paper to be published in SciPost Physics. I only have minor comments, listed below, which the authors may want to address when revising their paper.

List of comments

1) Eq.6: is there a particular motivation to use a ket notation (|\sigma^y_x>) on the physical leg, rather than just putting a physical index? After Eq (4) the physical index \sigma^x_y is introduced, thus why not using the physical index instead, which is more common, rather than a ket? That notation may be confusing (also calling the tensor in Eq.6 a rank-4 tensor, rather than a rank-5 tensor may be confusing). The same comment holds also for other figures with labels on the physical leg (Eq(7), Eq(11), …)

2) In Sec 3.5 it would be good to also mention the latest developments in using automatic differentiation to perform a gradient-based energy optimization, see H.-J. Liao, J.-G. Liu, L. Wang, and T. Xiang, Phys. Rev. X 9, 031041 (2019).

3) In the study of the spinful two-band Hubbard model, it would indeed be interesting to study larger unit cell sizes (as the authors point out). But I agree that this is probably beyond the scope of the present work.

4) Figure 11(a) and (c): on this scale it is difficult to see which energy is lower. Maybe it would be better to show the difference between the two (or subtracting the mean value), in order to better see which of the two states is lower in energy and by how much.

5) Minor typos:
- on page 40, 2nd line: a verb is missing after "10^9 individual contractions"
- page 41, line 4: "But whem" -> "But when"
- page 42, in the middle: "(a) Either" -> "(a) either"
- page 46, second paragraph, after Qspace [30]: "fter" -> "After"

1- The manuscript has a pretty clear scope and structure: it provides an (almost) self-contained solid guide for PEPS-beginners, while providing some concrete examples concerning relatively uncharted systems. The latter part makes it visible, why the whole effort is worth.

2- The technical part on non-Abelian symmetries complements other existing sources in terms of a different approach to the same problem, though it would have been great to see a deeper discussion in this respect (see “Weaknesses”).

3- This review collects a lot of useful tricks, which one should otherwise search for between the lines of an increasingly large body of literature — thus offering a precious service to the beginners.

1- According to the SciPost criteria, I would rate the manuscript not to contain enough new results to be published in the standard section, but rather to perfectly fit (up to minor changes) in the Lecture Notes one. The final Sec. 7 about the Hubbard model(s) offers an interesting collection of concrete and physically relevant examples, but does not convey a clear (possibly new) take-home message about the physics of those model(s). This is perfectly fine in a Lecture Note, but sub-optimal in a standard research paper.

2- The relative weight between the more standard PEPS formalism and the more peculiar one about non-Abelian symmetries is not well balanced to my eyes. Clearly, the latter is explained in full detail in Ref.[30], but the same applies even more for the whole of (i)PEPS general formalism in Sections 3 and 4/5. Given the title and declared aim of this manuscript, I would have expected a pedestrian summary of the QSpace approach here, in order to keep the whole (i) truly self-contained and (ii) at the beginner’s level as the rest.

The Manuscript offers an ordered review of (i)PEPS techniques, highlighting the most relevant ones for the implementation of an algorithm from scratch, thus sparing considerable efforts to the beginners and newcomers to the field.

Particular emphasis is given, in the intentions at least, to the incorporation of fermionic statistics and non-Abelian symmetries, both crucial ingredients when aiming to tackle condensed matter paradigms like the multi-band Hubbard model, where a lot remains to be explored. In this respect, some concrete examples of calculations are provided in the final section, thus motivating the interested reader to struggle with the algorithm implementation to explore new physics.

Certainly, after the first thirty self-contained didactical pages on general aspects of (i)PEPS, the Reader would expect to find the same level of user-friendly description for the ingredients that set the Authors’ approach apart from other ones in the literature, namely the QSpace approach to non-Abelian symmetries. Unfortunately, however, apart a short comment about being a fully alternative approach to Ref. [48], not much more is provided to the reader, except for addressing to the extensive Ref.~[30]. At least a sketchy recapitulation of the QSpace idea and its implementation details would be highly desirable to keep the review self-contained and homogeneous in style, especially given the promise in the title.

Given the overall spirit of the Manuscript, I would rate it definitely as suitable for the Lecture Notes section of SciPost more than as Regular Article. A few more punctual remarks — anyway not as crucial as the criticism / suggestion above about QSpace — are listed below, with the aim of improving the clarity of the presentation.

1) At page 2, the claim “Computational costs scaling as $D^\alpha$ can thus potentially be reduced by a factor of (D/D^*)^\alpha” should actually be made a bit milder by taking into account the growing number of smaller objects to be contracted, as correctly discussed later in Sec. 6.1.3. E.g., for a matrix-matrix multiplication it should read $(D/D^*)^{\alpha-1}$ since there are order of $(D^*/D)$ small blocks to be multiplied with each other. Anyway, nothing to say about the fact that the gain is going to be quite spectacular :-)

2) At page 4, the statement “In other words, the computational cost to simulate a low-entangled state using a PEPS scales only polynomially with system size.” is misleading at this stage, since it somehow suggests a perfect analogy with the efficiency of MPS algorithms. However, as correctly discussed in Sec. 3.3, the contraction of a PEPS “represents an exponentially hard problem” and one should resort to “a variety of approximate schemes to deal with this issue.”

3) At the bottom of page 11, more quantitative indications and/or criteria to judge the convergence would be helpful for beginners. In particular, more quantitative substance to expressions like “typically multiple times”, “a few full steps”, “can significantly increase” would be of great help for the beginner trying to assess the performance of its own simulations. Same applies later on in Sec. 3.5.2 when discussing the convergence in terms of the imaginary time steps.

4) At the end of Sec. 3.4, when discussing the convergence in the environment bond dimension $\chi$, I would have expected a small discussion about the extrapolation criteria based on the discarded probability introduced in this specific context by P. Corboz.

5) At the end of Sec. 3.5.3, a sketchy description of the other approaches (e.g., fast-full update, but not only) might be useful, maybe under a further subsection “Alternative Approaches”. One could spare the necessary space, if needed, by dropping some little redundancies in the previous pages (e.g., Fig. 2).

6) I am not really sure why Sec. 4 and 5 about fermionic statistics should be two separate ones, and not merged into a single one :-)

7) The notation of Eq.(71) is sub-optimal: dropping the dependence of the Clebsch-Gordan coefficients on the symmetry charges (q,q’,q”) might implicitly suggest a full decoupling of the factors A and C, which is instead not there.

8) I was quite surprised not to find any discussion on how the Authors’ approach deals with non-trivial outer multiplicities, being them coming from SU(2) tensors with rank larger than three or form the symmetry group(s) themselves (like for SU(3)). In the end, they understandably claim that this is what “sets our work apart from that of other iPEPS practitioners”, but they do not provide the reader with any detail, even sketchy, about it. Especially for the beginners, who are declared target of the manuscript, that would be of vital importance, without requiring them to dig into the 75 pages of Ref.[30] at first round of reading. Same applies to the compact notation for the SU(N) labels :-)

9) About the actual data on Hubbard model(s), l wonder whether the extrapolation in $(D^*)^2$ has some deep meaning behind, or is that simply a numerical observation at this stage (I would be perfectly fine with it, if stated). Same applies to the weak (absent?) scaling with $\chi^*$ in panel 8(d): by the way, what is the strange bump in the $D^*=4$ series there? and why is $D^*=5$ not displayed?

9b) Are site occupation and bond energy extrapolated in a similar way before plotting them in Figs. 8-9? If yes, why is then not the same procedure applied to the singlet-pairing amplitudes in Fig. 10?

10) How comes that, while the SU(N) symmetries are directly incorporated in the Tensor Network Ansatz, the conservation of total particle number is not included, and the Authors resort to tuning the average filling via a chemical potential? This might be useful to be explained to the Reader.

11) In Fig. 9, orienting the $\mu$ and $\delta$ axis the same way (i.e., so that left-to-right the population grows / decreases) would probably help the reader a bit.