Lightcone Hamiltonian for Ising Field Theory I: T < T_c

The paper develops an effective Lightcone Hamiltonian for Ising Field Theory in the low-temperature phase (m<0).
The authors verify the approach using integrability results, as well as broader applications including bound-state form factors and stress-tensor spectral density.

The paper is interesting, features multiple self-validation crosschecks, and therefore I recommend its publication.
However, the clarity of the exposition could be improved, and I have a number of questions that should be addressed as well as few comments.

- Towards end of page 2, the authors mention “our prior paper”. A citation is needed there.
Similarly in page 3, “As we argued in our previous paper”. Which one? It shouldn’t be left to the author to figure out which paper are referring to.
- Equation (1.5) contains no explanation at all for the first term.
- Is time ordering missing in 2.2?
- The discussions arguments around equation (2.5): is this the reason why the assumption Delta < 1 is needed? What is the effect of subleading terms on the OPE? It is never explained at which step the assumption Delta < 1 is needed.
- Similarly to the previous point, what is the order of the corrections missing in (2.9)?
- Sections 2.2, 2.3 and the crosschecks of the appendix are useful.
- The work set units such that the finite volume R=1. Nevertheless it will add clarity to include the limit of integration in the ‘key’ equations (1.5) and (1.2). In the derivation following equation 2.1, perhaps it can be stated that the authors omit limits of integration.
- Section 3 is very interesting. The comparison and agreement with integrability results is remarkable!
The work though uses input from integrability by inputing the vev of sigma in the LC diagonalization. It is unlcear to me how good will be the comparison if
<\sigma> was computed numerically, instead of using integrability input. My question is however partially addressed in the next section, when turning on “m”.
- The comparison with TFFSA/TCSA is very interesting. It is remarkable that the LC isolated the correct degrees of freedom to reproduce the observables computed here. It is correct that the authors did not include Heff corrections in the TFFSA/TCSA comparison? It would be interesting to compare against TFFSA with similar level of sophistication regarding Heff and <signa>.
- Would it be interesting to extract phase-shifts? Via the asymptotic bethe ansatz it should be quite straightforward from the authors finite volume data. In the integrability limit it is a nice observable to crosscheck.

Publish (meets expectations and criteria for this Journal)

Hamiltonian Truncation is a first-principles, numerical approach for solving interacting quantum theories nonperturbatively. The authors develop a technique in this family called Lightcone Conformal Truncation, which is applicable to QFTs defined as conformal field theories (CFTs) with relevant deformations. In this approach, the lightcone Hamiltonian (generator of translations in lightcone time $t+x$) for the QFT is constructed as an explicit matrix of finite size, that acts on a basis of states built using operators of the undeformed CFT with low scaling dimension from one chiral sector.

The authors seek to extend this method to theories where the operator deforming the CFT can acquire a vacuum expectation value (VEV). The new results of the paper include a compact, analytic expression for the extra term that must be added to the effective lightcone Hamiltonian in this case (given in equation 2.9), and numerical demonstrations of their effective Hamiltonian for the 2d Ising Field Theory. In particular, they compute the low-lying spectrum and matrix elements of the stress energy tensor and compare them with exact results from integrability (along a slice of the model parameter space), and with earlier results from equal-time Hamiltonian Truncation.

The numerical results were obtained using significantly smaller bases of states than were needed in the equal-time HT results shown for comparison. This alone provides a clear demonstration of the advantages of their method. The authors also helpfully provide Mathematica software for building the effective Hamiltonian along with their paper, lowering the barrier to entry for others to apply their framework. However, a key question for future work is whether their approach can be readily generalized. The effective Hamiltonian in equation 2.9 was derived under the assumption of a single relevant deformation with scaling dimension $\Delta<1$. If a similarly compact expression cannot be found without imposing these conditions, the applicability of the method may be significantly restricted.

Overall, this paper meets the journal’s requirement for opening a new pathway in an existing research direction, with clear potential for follow-up work, at least in a particular class of 2d QFTs. I would request only a few minor revisions to clarify the presentation:

1. The equation 2.9 is derived from equation 1.2 within the paper, but the derivation of the start point, 1.2, is only presented in references. I think that including some more information from the earlier works would clarify the logical flow. Ref [4] suggested that Eq. 1.2 is equivalent to the “Schur’s complement formulation” of the effective Hamiltonian (also given in Eq. C.4 in the current paper) at all orders in perturbation theory - and not just up to O(g^4). Including this information explicitly in the introduction would be helpful, I think. Also, it seemed as if 1.2 was derived originally in 1803.10793, which was not in the reference list (perhaps intentionally). Either way, including more detail on which principles were being used to derive 1.2, as well as more precise guidance as to where to find the clearest, most fundamental derivation of 1.2 from the literature, might be useful.

2. Following on from the previous point: On page 3, a reference should be given following the statement “As we argued in our previous paper…” in the block of text following equation 1.2.

3. What power counting rule is being employed in writing down the effective Hamiltonian? Effective Field Theories must be organised as expansions in small parameters, where each parameter is some low energy scale divided by a high energy scale. Could the authors clarify the power counting scheme being used in Eq. 1.2? It would be helpful to specify the small parameters involved and how many orders are retained in the expansion.

4. What contour is $x$ being integrated over in equation 1.3? Also, is it a lightcone $x^-$ or the periodic spatial coordinate $x$ of the cylinder?

5. Why can we not study the other half of the phase diagram (where $T>T_c$) with the current approach? Is it due to needing states from the fermionic chiral sector (as well as the bosonic) to describe this phase, or is it because the epsilon operator gets a VEV? A further comment about how to generalise to this case would clarify this.

6. When comparing data for the spectrum obtained with lightcone conformal truncation and equal time Hamiltonian Truncation, is the idea that the lightcone data is automatically in the infinite volume limit, whereas the equal time data should be extrapolated to the infinite volume limit first before making the comparison? It might be useful to include the volumes used in the data taken from Ref [8] and mention this in the discussion of Figure 5. Also, what units are the masses on the y-axis of Figure 5 measured in?

Ask for minor revision