Hamiltonian Truncation Effective Theory

Dear editor and referees,

We thank the referees for the exceptionally detailed and helpful feedback. We apologize for the delayed response, but it took us some time to digest and properly respond to all of the comments and suggestions in the referee report. We have made extensive revisions to the paper in a good-faith effort to incorporate the insights of the referees and address their comments. We hope that the present form of the paper is acceptable for publication.

Below we provide responses to the referee comments point by point, using the same numbering as in the referee reports. A complete list of changes in the paper is listed separately at the end.

Sincerely,
Timothy Cohen, Kara Farnsworth, Rachel Houtz, Markus A. Luty

Referee 1:

1. All three referees commented that non-Hermiticity may not be a fundamental feature of the effective Hamiltonian. We will therefore address the comments of all three referees on this question here.

First, we would like to emphasize that we have defined the effective Hamiltonian by matching. This definition is then used as the basis for a systematic expansion in powers of 1/Emax. For example, it is not the same as the `exact Hamiltonian' used in previous references, which depends on the eigenvalue that is being computed. There may well be other definitions that also work, possibly defining a Hermitian effective Hamiltonian with the same spectrum. We think this is an interesting question, and we do not know what is possible. In this paper, we have focused on investigating the definition that we have proposed. The calculational and numerical checks that we have performed work as expected, and encourage us to continue to pursue this direction, which we plan to do in future work.

To address the concerns of the referees regarding the non-Hermiticity of the effective Hamiltonian, we have removed statements throughout the paper that suggested that non-Hermiticity is a fundamental feature of the effective Hamiltonian. We have also added a very preliminary discussion of some alternative definitions that may give a Hermitian effective Hamiltonian on pp. 11-12 and appendix B.

Referee 1 makes the proposal that one can define a Hermitian effective Hamiltonian by matching the Hermitian operator T +T^\dagger, rather than T, as we do in this work. This is an interesting suggestion, but unfortunately it does not work. We checked that at O(V^3) the effective Hamiltonian defined this way is ill-defined due to IR divergences. This is a manifestation of the failure of separation of scales. We have added a comment
on this to the discussion of non-Hermiticity on pp. 11-12.

Referee 1 states that matching T +T^\dagger gives the Schrieffer-Wolff effective Hamiltonian. We feel that the connection to the Schrieffer-Wolff effective Hamiltonian is rather distant from the main topic of the paper, and we have added only a brief mention of it in the paper. Briefly, here is our understanding of this relation. The result of performing the Schrieffer-Wolff similarity transform gives the O(V^2) contribution to the effective Hamiltonian

<f|H_2|i> = 1/2 \Sum_\alpha <f|V|\alpha><\alpha|V|i>(1/E_{i\alpha} + 1/E_{f\alpha}). (1)

Note that the sum is over all states $\alpha$. For this to be a useful effective Hamiltonian, one assumes that the low-energy states are separated by a large gap \Delta E from the high-energy states, and the unperturbed Hamiltonian H_0 is diagonalized on the low-lying states. With these assumptions, the sum only gets contributions over the high energy states $\alpha$, and the effective Hamiltonian gives an expansion in powers of 1/(\Delta E). Matching T +T^\dagger at O(V^2) gives the same equation as Eq. (1), but the sum over $\alpha$ is over high energy states whether or not H_0 is diagonal in the low-energy subspace. The 1/(\Delta E) expansion of the Schrieffer-Wolff effective Hamiltonian is very different from the 1/Emax expansion we are performing. We do not feel that explaining this in detail is appropriate for this paper, and we hope the referee will agree that the changes we have already made in the paper are adequate.

Referee 2 points out that we may be able to define a Hermitian effective Hamiltonian using a similarity transformation. We thank the referee for this suggestion, which we believe is interesting to pursue. We have commented on this in the main text, and discussed a very preliminary attempt to find such a similarity transformation in appendix B.

Referees 1 and 2 also point out that in the `local approximation' considered so far, the non-Hermiticity (and non-locality) do not appear, and therefore these aspects of our effective Hamiltonian are not tested by the calculations we have performed. We agree, and this was stated in the original version of this paper. We have added several additional statements to emphasize this point further. We are currently performing higher-order calculations where these features make their appearance, and we plan to publish them in a follow-up paper. We believe that the present results are sufficient for publication, since we have demonstrated 1/Emax^3 numerical errors from the first correction term in a parameter-free systematic expansion.

2. Referee 1 points out a lack of clarity regarding general power counting arguments in section 2. In section 2, we are considering a perturbative expansion in powers of V . This expansion is later used as the basis for the expansion in powers of 1/Emax in section 6, but this expansion depends on the model. We have attempted to make this two-step logic clearer by modifying both sections 2 and 6.

3. Referee 1 states that separation of scales is expected in the `local approximation' used in the text. We agree with the referee that it is essential to check separation of scales beyond the local approximation, and we are planning to do so in follow-up work. We have made a number of changes throughout the text to emphasize that the formalism needs to be tested beyond the local approximation. However, we would like to push back a bit on the idea that separation of scales is automatic in the local approximation. Matching T +T^\dagger (as suggested by the referee) gives IR divergences that result from vanishing energy denominators (Eq. (2.25) in
footnote 4 in the paper), and this is a violation of separation of scales that would be present even in the local approximation at O(V^3).

4. Referee 1 points out that non-locality and non-Hermiticity need not be connected in our effective Hamiltonian. We agree, and have added clarification of this point at the end of section 6.1. As already discussed in point 1 above, we have also de-emphasized the non-Hermiticity of our effective Hamiltonian throughout the paper.

5. Referee 1 suggests that we compute physical quantities such as critical exponents at the critical point. The focus of this paper is on the convergence of the 1/Emax expansion, while an accurate extraction of critical exponents requires a careful treatment of finite volume effects along with the truncation error. We request that we be allowed to leave this for future work.

6. Referee 1 points out that our Eq. (8.3) is incorrect. We thank the referee for catching this, and we have corrected it.

Referee 2:

Referee 2 states that their comments are not required to be addressed for publication, but we have made a number of changes to the paper in response to their comments.

1. Referee 2 requests extra details clarifying the exact procedure used in the numerics. We have added Eqs. (7.5)-(7.6) and the surrounding discussion to summarize our final results and explicitly present the Hamiltonian matrix we diagonalized.

Referee 2 also asks about the noise in our plots. We have added additional clarifying remarks near Eq. (5.26) to the effect that the Euler-Maclaurin approximation was not used for numerical results, and therefore does not explain this noise. We do not currently understand the origin of this noise. It is not large enough to be important for the results in this paper, although it will be important when we go to next order in the expansion, and we will address it then.

2. Referee 2 suggests that we clarify the relationship between our method and his previous work, which also make use of the local approximation, but does not obtain 1/Emax^3 errors. We agree that this would be interesting, but we believe it is more important for us to understand our method better by going to higher orders in our expansion. We do not know how to give a simple formula for the correction as in the referee's
work because the contribution from the states near the cutoff appears to be quite complicated, and not amenable to a simple approximation.

3. See point 1 for Referee 1.

4. Referee 2 asks about the physical meaning of the transition matrix T. This is a great question, and we do not have a good answer. Since we only use it for matching, it does not even have to have a physical meaning, it just has to define a good effective Hamiltonian. We added a comment on p. 9 pointing out that T suffers from an IR
divergence even in finite volume. This IR divergence cancels in the matching (separation of scales again), so this does not invalidate our method, but it may explain why T by itself is not useful.

5. The referee suggests we explicitly say that the diagonal elements of Eq. (2.23) agree with what is obtained by matching the spectrum. We have added this in the paragraph after Eq. (2.23). (The fact that this was not included in the original draft was an oversight on our part, we certainly did perform this crucial check!)

6. Referee 2 states that while the matching contributions are small, the contributions to T from the low-energy and high-energy theories are large. This is because T gets contributions from all scales, and the cancelation comes because the strongly-coupled IR region cancels in the matching. This cancelation is a manifestation of the separation of scales that is an essential (but standard) feature of EFT matching. We have added references to general effective field theory methodology where this is explained.

7. The referee asks whether we have considered how our results would change if we attempted to match the effective Hamiltonian for states near the cutoff. We have not done so. The standard EFT methodology is to expand all quantities around zero momentum and energy, corresponding to a derivative expansion of the effective theory. The fact that this works as expected in the calculations we have performed so far
motivates us to continue in this direction.

8. See point 1 above.

9. See point 1 above.

Referee 3:

Referee 3 did not provide numbered points, but asked us to address 3 questions (paraphrasing): Why is T the right quantity to match? Is there another quantity that could be matched to define a Hermitian effective Hamiltonian? Do the numerical results depend on the precise definition of the matching?

We believe that we have addressed all these questions with the changes discussed in point 1 of Referee 1.

Referee 3 also suggests that we more clearly summarize our main results. We have addressed this in point 1 of Referee 2.

List of changes to the draft:

Since we made many changes, we felt it would be helpful to provide a complete list of the substantive ones here to make it easy for the referees and editor to follow them.

  p. 1: Changed wording in the abstract.
  p. 5: Modified the bullet on the non-Hermiticity
  p. 5: Added a bullet emphasizing the 1/Emax expansion
  p. 6: Added a mention of the new Appendix B
  p. 6: Added footnote 2
  pp. 8-9: Added clarifying discussion to the beginning of Section 2.2
  pp. 11-12: Added a few paragraphs at the end of Section 2.3 to explain what is in the new Appendix B
  p. 20: Added clarifying remarks to the end of the second paragraph at the beginning of Section 5
  p. 23: Added clarifying remarks above Eq (5.12)
  p. 24: Added clarifying comments around Eq (5.19)
  p. 26: Added clarifying discussion below Eq (5.26)
  pp. 26-27: Expanded the first paragraph of Section 5.5
  p. 27: Shortened what is now footnote 10
  p. 30: Removed the last paragraph before Section 6.1
  p. 30: Heavily edited the first paragraph of Section 6.1
  p. 31: Added a discussion of non-locality in the paragraph below Eq (6.6)
  p. 34: Added clarifying equations to the beginning of Section 7.2, Eqs (7.5) and (7.6)
  p. 35: Added a reference to the python numerical package `scipy.sparse.linalg'
  p. 42: Added Eq (8.3) and related comments right below it
  pp. 50-51: Added Appendix B on similarity transformations to make the effective Hamiltonian Hermitian