Systematic Improvement of Hamiltonian Truncation Effective Theory

1- The paper makes progress on a central problem in Hamiltonian truncation, the problem of constructing effective Hamiltonians that take into account the contribution from states above the truncation and therefore lead to improved convergence in the method.

2- It takes a proposal for a procedure for computing such effective Hamiltonians and checks this proposal in detail at higher orders than previously studied, and considers new effects that arise at this order.

3- It has detailed numeric checks of proposed scalings of corrections, and precise results for the spectral and critical coupling of the critical point of the theory.

4- It discusses the theoretical framework for understanding these errors in the language of effective field theory and power-counting, and makes clear predictions for the form of the corrections

1- The discussion of the power-counting argument is surprisingly brief given its importance in the paper. There are several subtleties that naively seem like they could undermine the argument which I would have liked to see more discussion of, see the report.

2- At various points, "numerical noise" is invoked to explain the absence of certain scaling behavior predicted by the authors' power-counting rule, but in the current state of the paper it is not clear why "numerical noise" cannot just be invoked to explain any failure of the scaling predictions. Moreover in some cases where the fits look good, the range and number of data points are small and it is not clear how precisely the scaling is truly being measured.

1) Since the power-counting argument in (2.13) is central to the paper, it would be good if it could be made tighter. The effective Hamiltonian is expanded order by order in lambda, but Hamiltonian truncation is a numeric method for strong coupling and it is not clear to me why a perturbative expansion is trustworthy. My guess is that the authors are assuming that even if lambda is large, the expansion parameter is (lambda/Emax^2) and that this is small for large enough Emax.

a) One reason to worry about this expansion is that there are operators within the low-energy subspace that are close to the truncation and in perturbation theory, these will have small energy differences in the denominator and thus these terms will not be suppressed by powers of Emax. This subtlety has been an important issue for other prescriptions for the effective Hamiltonian, and could be an issue here as well.

b) One might also worry that the coefficients of (lambda/Emax^2)^n in (2.13) grow rapidly with $n$, especially at large volume at the critical point where there is an emergent small energy scale (the gap) which for small enough gap requires many high energy states (in the absence of an effective Hamiltonian), and therefore when these high energy states are integrated out, one is integrating out many states whose nonperturbative energy is much smaller than their naive energy in perturbation theory.

c) Moreover, in (2.13), why does each appearance of $H_0$ have to come with a power of $E_{max}$ in the denominator? My guess is that this is an appeal to dimensional analysis, but $H_0$ is a matrix with many entries and therefore there are a large number of dimensionless ratios that can be made from its elements.

d) I also did not understand how the authors can impose their constraint (2.12). For single particle states, $\omega_i$ and $E_i$ are the same, and more generally for any $E_i$ one can take states where there are one or two particles with energies $\omega_i$ with an $O(1)$ fraction of the total energy of the incoming state. Additionally, for any truncation $E_{max}$, there are initial states within the truncated space that lie just below the truncation, and so have $E_i \approx E_{max}$, not $E_i \ll E_{max}$.

2) I did not understand why the authors meant by their expansion $\Theta(E_f+X) \approx \Theta(X)+E_f \delta(X)$. They say that this should be understood as a distribution, but that would mean the expansion is valid when integrated against smooth test functions, which is clearly not true -- simply integrate both sides of the equation against a function $g(X)$ that has support in the region $-E_f < X< 0$, so that the LHS is nonzero but the RHS is exactly zero. I suspect what they mean by $\Theta(E_f+X) \approx \Theta(X)+E_f \delta(X)$ is that they are defining a new function $\delta(X) \equiv (\Theta(E_f+X)-\Theta(X))/E_f$.

3) In (2.35a) they introduce the notation $\delta_{12,56}$ which I think is shorthand for $\delta_{p_1+p_2,p_5+p_6}$ but I did not see this defined (the closest I could find is $\delta_{1+2} \equiv \delta_{p_1+p_2}$).

4) I thought the comment about $k_{UV}=1000$ as a Wilsonian UV cutoff on page 14 was confusing, since $k_{UV}$ plays no role in the rest of the paper -- my understanding is that all the sums over $k$ converge, so they can take the sums over $k$ to run up to infinity, and numerically evaluating the sum up to $k<= 1000$ is a good approximation.

5) I would have liked to see comparisons of the fits to $1/E_{max}^4$ vs $1/E_{max}^a$ for some $a\ne 4$ in order to make it clearer how well the scaling is being established. E.g. can $1/E_{max}^3.5$ be ruled out with the data?

I also would have liked to see a more careful treatment of the "numerical noise". I am not even completely sure what is meant by "numerical noise" -- is this just an issue of diagonalizing matrices numerically, which could be addressed simply by running the numerics at higher precision, or it is a finite $E_max$ effect and therefore "noise" only in some abstract sense ? If it is the former, then the clearest resolution would be if the authors could run their numerics at various levels of precision and see that the "numerical noise" varies with some level of error that they can then extract. If it is the latter, then it would be good if the authors could say more precisely in what sense it is "noise".

Ask for minor revision