Hamiltonian Truncation Effective Theory

The paper offers a new view on improving the Hamiltonian truncation method.

The authors propose that the effective Hamiltonian, as a matrix, must reproduce an observable 'related to the S-matrix'. They propose to compute this observable perturbatively with the real Hamiltonian and use this to determine the effective Hamiltonian.

I think the matching of an entire matrix of observables is an interesting proposition. (The choice of the off-diagonal terms in the counterterms was also discussed in section 4.2 of 1803.05798, where greater emphasis was put on Hermiticity.) The authors however do not explain why their particular observable is the right one to match? Would there not be a better one, perhaps one that avoids a non-Hermitian Hamiltonian?

The paper then proceeds by computing, in the example of phi^4 theory in two dimensions, the effective Hamiltonian so defined to second order in perturbation theory. The authors choose to then restrict themselves to a so-called 'local approximation' where only coupling renormalizations remain.

In numerical experiments it is observed that the cutoff dependence of the estimations is improved from E_max^{-2} to E_max^{-3}. The authors claim this is 'a strong indication that our method is working as expected'.

I am puzzled by such a strong claim. The local approximation to leading order in E_max seems to me entirely equivalent to the improvement computable by standard perturbation theory. What part of the numerical results relies on matching an entire matrix of observables rather than just the energies themselves?

The paper is in my opinion clearly written; my only stylistic comment would be that a punchline equation is sometimes missing. In particular, the effective Hamiltonian 'described above' (on page 30) needs to be cobbled together by the reader from several equations scattered throughout the previous section.

I would like to see the above questions answered.

This paper aims to improve convergence of the Hamiltonian Truncation, by adding to the truncated Hamiltonian a series of judiciously chosen correction terms. The idea itself is not new, and some proposals exist in the literature. The paper proposes a new method to do it, rooted in Effective Field Theory. The proposal identifies a new quantity - called transition matrix - which should be computed in the fundamental and in the truncated theory and matched. The matching condition determines which correction terms one should add in the truncated theory.

The paper is interesting and generally well written although omitting some details. I would like to list several questions, comments, and suggestions which occurred to me when reading it. These fall in three categories: the need to explain better conceptual foundations of their method; the need to document better their computations to allow reproducibility; the need to localize better the difference from the previous work by Rychkov and Vitale. I suggest, although I do not insist, that the authors add clarifications concerning at least some of these points. The framework and the results are promising. I wish the authors good luck in pushing their program to correction terms of higher order, and to higher dimensions.

1) Numerical section 7 could benefit from extra details clarifying the meaning of "improved" results. Full improved Hamiltonian could be described in detail. At least it should be said if the matching corrections are used for states i,f with energies all the way up to the cutoff (since the previous sections compute them under the constraint E_i,E_f<<E_max).

Elsewhere in the text, it is mentioned that sums involved in the evaluating matching corrections are evaluated sometimes analytically (in 1/E_max expansion), sometimes numerically. Summarizing the procedure in section 7 could be helpful, to allow reproducibility.

2) Their matching corrections, at the order they аrе working, are very similar, although hopefully not identical, to the correction terms used by Rychkov and Vitale in "Hamiltonian Truncation Study of the φ^4 Theory in Two Dimensions", http://arxiv.org/abs/1412.3460. Although derived in that paper via an OPE argument, Eq. (3.31) can be shown to arise from local approximation to the sums very similar to the sums used in the paper under review (appendix E in http://arxiv.org/abs/1706.09929). The 1/E_max expansion of matching corrections could shed light on why correction terms of Rychkov and Vitale do not improve correction to O(1/Emax^3), while the matching corrections in the paper do so. Currently, the comparison to the prior work by Rychkov and Vitale in section 8.3 does not explain this issue. One thing to look at is \mu_{442} in (3.31) of Eq. (3.31) of Rychkov and Vitale which contains log(Lambda/m), while all matching corrections in the paper under review are argued to be purely high-energy associated and should presumably be free of such logs. A precise formula analogous to (3.31) in Rychkov and Vitale could therefore be welcome. It is derivable by methods of appendix E in http://arxiv.org/abs/1706.09929.

3) The logic on p.8 concerning the phrase "We therefore need additional relations to fully determine effective Hamiltonian" is unclear. It looks like a good effective Hamiltonian is any matrix which has the same low-energy spectrum as the full Hamiltonian. Why is it bad that section 2.1 does not determine it fully? Why can't we e.g. put all elements which are not fixed by section 2.1 to zero, or to anything else of a given order in V? Why would this be a worse definition than the one the authors come up with in Section 2.2? Put another way, is the freedom accorded by Section 2.1 perhaps just a gauge freedom, and Section 2.2 a particular way to fix that freedom, or is there something more special about the choice of Section 2.2?

3') A related remark. The authors stress that their effective Hamiltonian is not Hermitian. Could this be cured? Hamiltonians H and H' = G^{-1} H G have the same spectrum. For G unitary, H' is Hermitian if and only if H is so. However for a general invertible G we may have H non-Hermitian and H' Hermitian. It might therefore very well be possible to transform their Hamiltonian to an equivalent Hermitian one. Such a transformation would be advantageous for numerics, since Hermitian matrices are faster to diagonalize. Do the authors see any particular reason why such a transformation may not exist, or should not be used?

4) What is the physical meaning of the transition matrix introduced in section 2.2 ? Is it observable? Quantum Mechanics textbooks contain similar quantities, but not exactly this one. In Quantum Mechanics, for adiabatically turning on interactions, transition is only possible between eigenstates having the same energy, while the transition matrix in the paper has no such constraint. So it looks like their transition matrix is something else from what it is in Quantum Mechanics. They say that it is related to S-matrix, but also S-matrix conserves energy... Given that all possible quantities of quantum-mechanical origin have been presumably considered before, it would be great to identify the previous uses of such "transition matrices between states of different energies", and cite them, to orient the reader. If to the authors' knowledge it's the first time such a quantity is used in the long history of quantum mechanics, this is remarkable and should also be mentioned.

5) It would seem required by the logic of the paper that the effective Hamiltonian satisfying the conditions from section 2.2 satisfies also those from section 2.1, but this fact is neither stated nor proven.

6) Logically, when one performs perturbative matching of the fundamental theory on the effective theory, one does it for observables which are computable in both theories, and at an energy scale where both theories are perturbative. If this condition is not satisfied the matching does not deserve to be called perturbative. The transition matrix that the authors propose to match does not seem to satisfy these requirements. The transition matrix itself does not seem to have a good perturbative expansion nor in fundamental nor in the truncated theory. It's only when one subtracts the two large terms one gets a small correction term. It would be nice to have a discussion to why this is allowed, and perhaps include references to other similar uses of Effective Field Theory, not fully legitimate at first appearance but giving correct results.

7) The status of the condition E_i,E_f<<E_max in the matching correction calculations is not fully clear. They use this condition to simplify those calculations since in this limit a local approximation can be derived. Note that if the theory is really perturbative at E_max, then by using exact (not approximated) correction terms for all the states up to E_max the results should only improve. This would be an important check on their scheme. For the previous scheme of Rychkov and Vitale such a check, done by Elias-Miro, Montull, Riembeau http://arxiv.org/abs/1512.05746, showed that in fact full correction terms gave worse results than approximate local ones. The troublesome states for which perturbativity was violated were identified; they corresponded to states with many particles. If the authors have an opinion on whether the same issue would affect their scheme, it would be nice to add it. (See the discussion below Eq. (2.14) in Elias Miro, Rychkov, Vitale http://arxiv.org/abs/1706.09929)

8) (Related to comment 1) It would be nice to comment why correction terms computed under the condition E_i,E_f<<E_max are subsequently used for all the states up to the energy E_max in the subsequent numerical computation. According to Eq. (6.7) with its H_0/E_max dependence of C_na, correct matching corrections for E_i,E_f~E_max may differ at order 1 from the ones they used.

9) Concerning the use of Euler-Maclaurin formula: k_min in (6.4) is given by the integer part of the r.h.s. of (6.3). This means that the l.h.s. of (6.5) has some step-discontinuities in E_max. There are limitations in approximating such step-discontinuous functions with a series in inverse powers. I wonder if the authors have any comment on this and how it is related to the "scatter" in their numerical plots?

The authors introduce a new strategy for performing Hamiltonian Truncation calculations.
The idea consists in constructing the truncated Hamiltonian Heff by matching a transition amplitude T.
Heff is uniquely determined by requiring that the computation of the transition amplitude T in the finite low energy Hilbert space
equals the computation of T in the fundamental theory.
The authors test this idea with phi4 theory in two dimensions, and numerically obtain a nice looking spectrum at strong coupling.

This work opens a new pathway for Hamiltonian Truncation calculations. It is a very interesting paper with original results and novel analyses.
Therefore I will recommend its publication after the authors address the observations below, and clarifications I would like to ask:

1) The Heff introduced in this paper is non-hermitian simply because the authors are matching a non-hermitian operator.
Hermitian matrices have nice properties of reality of the eigenvalues and orthogonality of the eigenstates. Therefore I believe it would be nicer to work with Hermitian effective Hamiltonian.
This could be fixed by matching a hermitian version of the author’s T operator, i.e. 1/2(<f|T |i > + h.c.).

Interestingly, matching the operator 1/2(<f|T |i > + h.c.) leads to a Heff which coincides with the so called Schrieffer-Wolff (SW) effective Hamiltonian.
The SW is a correct effective Hamiltonian by construction, because it is obtained by performing a canonical transformation to block-diagonalize the fundamental Hamiltonian (this construction is applied to a UV renormalized fundamental theory).
Therefore the SW derivation justifies the procedure of matching the operator 1/2(<f|T |i > + h.c.) as a route for arriving to the SW effective Hamiltonian.
This reasoning leads one to ask whether the Heff of the authors (retrieved by matching <f|T |i > instead of 1/2<f|T |i > + h.c. ) is exactly correct, or only approximately?
What I am pointing out is a matter of principle, whose consequences may not be detectable by computing the spectrum of the two dimensional phi4 theory with second order Heff.

2) Equations 2.24 are the correct solution to the matching equation at O(V^3).
However, towards the end of section 2, the authors claim that the correct matching equations require the extra step of expanding equations 2.24 in powers of 1/Emax.
Should one keep only the leading 1/Emax correction in 2.24, or keeping the leading and subleading is also fine?
Is one allowed to incorporate the whole series in 1/Emax, and if so, why expanding in powers of 1/Emax in the first place?
The justification “in accordance with standard effective field theory methodology” is unclear because the problem at hand departs from standard quantum field theory in many interesting ways.

3) The authors also show that their calculations obey various expectations of EFT regarding separations of scales.
In particular, a definition is given in the introduction: the coefficients of the Heff should be dominated by states close to the cutoff.
This is so because Heff is obtained from a renormalized (i.e. finite) fundamental theory. Fine.
However the authors take this expectation one step further in the main text, and show that the separation of scales to O(V^2) is obeyed in a local manner.
Namely, it turns out that leading UV/IR overlaping regions in momentum space (of the coefficients of the local, un-normal ordered, operators in Heff) vanish.
This is a manifestation of the locality of Heff at this order of the calculation: states containing soft momentum modes can have energy close to the cutoff, however these states do not dominate the contribution to the coefficients of Heff.

4) Section 6 contains a power-counting discussion. Reading through it, one has the impression that non-locality and non-hermicity are somewhat intertwined. However, as I emphasised above this does not need to be the case.

5) The authors carefully estimate the error of their approximation in Heff at second order.
This allows them to show that raw spectrum converges as 1/Emax^2 while leading order second order as 1/Emax^3.
The focus of the numerical analysis is on convergence. However, with such a good control on the convergence, it would had been interesting to compute universal data at the critical point.
Those computations could be compared across different methods and works.

6) Equation 8.3 is wrong. There the authors are referring to the exact eigenvalue equation 2.3, therefore the denominator of the exact “Delta H_exact” should be Epsilon-H0-V instead of Epsilon-H0.
References [5,6] did two essential approximations: 1) substituted the denominator of Delta H_exact by Epsilon-H0 ; 2) approximated V.1/(Epsilon-H0).V by local operators.