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Hamiltonian Truncation Crafted for UV-divergent QFTs
by Olivier Delouche, Joan Elias Miro, James Ingoldby
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Olivier Delouche · Joan Elias Miro |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202403_00021v1 (pdf) |
| Date accepted: | 2024-03-26 |
| Date submitted: | 2024-03-14 14:19 |
| Submitted by: | Elias Miro, Joan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We develop the theory of Hamiltonian Truncation (HT) to systematically study RG flows that require the renormalization of coupling constants. This is a necessary step towards making HT a fully general method for QFT calculations. We apply this theory to a number of QFTs defined as relevant deformations of $d=1+1$ CFTs. We investigated three examples of increasing complexity: the deformed Ising, Tricritical-Ising, and non-unitary minimal model $M(3,7)$. The first two examples provide a crosscheck of our methodologies against well established characteristics of these theories. The $M(3,7)$ CFT deformed by its $Z_2$-even operators shows an intricate phase diagram that we clarify. At a boundary of this phase diagram we show that this theory flows, in the IR, to the $M(3,5)$ CFT.
Author comments upon resubmission
Dear Editor,
We thank the referees for their useful comments.
We now submit to you an updated version of our manuscript incorporating changes in response to the numbered requests made by the referees.
Below we go through each of the requests in turn, describing the corresponding edits we have made to our paper:
Report 1:
1.- We added text to clarify where dependence on the regulator parameter epsilon enters on page 6, as part of step 1 of our procedure for defining the truncated Hamiltonian of a renormalised QFT.
2.- We added a comment about the log dependence of the ground state energy on page 11. In our numerical results we recover the RlogR contribution, as we should.
3.- This is correct and the alternative approach was taken in Ref. [20].
4.- That is correct, g3 is a renormalised coupling. We have added a comment stating this at the end of section 6.1 and after Eq. (6.8).
5.- Drawing energy radius plots in the deformed M(3,7) model as the referee suggests is something that could in principle be done in future, although we do not do it here. Careful tuning of the alpha parameter is required in this case however (unlike in the flow from the tricritical model). We added a paragraph about this point on page 28.
6.- We expanded the paragraph that begins “The phase transition boundary …” on page 25, to describe in more detail what the HT calculations of the spectrum show around the second phase boundary.
7.- Following Eq 2.2 we added a sentence clarifying our notation: $\Delta_i$ may refer to primary, descendant operators or both depending on the context.
8,9.- Typos fixed, thanks for bringing them to our attention.
Report 2:
1.- On page 9, we provided more details about how we are computing the matrix elements we use, and added a reference to the paper suggested by the referee (Horvath et al. in Computer Physics Communications 277: 108376). We also expanded the footnote at the bottom of the page to include more information about how we are removing null states from our truncated basis.
2.- We included a more prominent reference to the important work of Zamolodchikov, after equation (5.10). We also simplified our explanation of the derivation of (5.10), which only required computing expectation values of the energy-momentum tensor squared $T_{\mu\nu}^2$ in Ising CFT energy eigenstates.
Report 3:
1.- We changed the motivation/ overview of our procedure that we give in the introduction by reordering a couple of paragraphs and adding some further details. The changes can be found on page 3, contained between the paragraphs “Considerable progress has been made..” and “In Ref.~[12], a general procedure…”.
2.- We added further comments to address this question in the paragraph that begins at the very bottom of page 28.
3.- We increased the sizes of the labels in many of the figures to improve readability.
We caught a $2\pi$ typo missing on (5.9) and (6.13) that does not affect the numerical results and formulas presented later on the text.
We are grateful to the editor and referees for their careful consideration of our paper. We hope that our replies to the referees and changes to the manuscript will make it fully acceptable for publication in SciPost.
Best wishes,
Joan, James and Olivier
We thank the referees for their useful comments.
We now submit to you an updated version of our manuscript incorporating changes in response to the numbered requests made by the referees.
Below we go through each of the requests in turn, describing the corresponding edits we have made to our paper:
Report 1:
1.- We added text to clarify where dependence on the regulator parameter epsilon enters on page 6, as part of step 1 of our procedure for defining the truncated Hamiltonian of a renormalised QFT.
2.- We added a comment about the log dependence of the ground state energy on page 11. In our numerical results we recover the RlogR contribution, as we should.
3.- This is correct and the alternative approach was taken in Ref. [20].
4.- That is correct, g3 is a renormalised coupling. We have added a comment stating this at the end of section 6.1 and after Eq. (6.8).
5.- Drawing energy radius plots in the deformed M(3,7) model as the referee suggests is something that could in principle be done in future, although we do not do it here. Careful tuning of the alpha parameter is required in this case however (unlike in the flow from the tricritical model). We added a paragraph about this point on page 28.
6.- We expanded the paragraph that begins “The phase transition boundary …” on page 25, to describe in more detail what the HT calculations of the spectrum show around the second phase boundary.
7.- Following Eq 2.2 we added a sentence clarifying our notation: $\Delta_i$ may refer to primary, descendant operators or both depending on the context.
8,9.- Typos fixed, thanks for bringing them to our attention.
Report 2:
1.- On page 9, we provided more details about how we are computing the matrix elements we use, and added a reference to the paper suggested by the referee (Horvath et al. in Computer Physics Communications 277: 108376). We also expanded the footnote at the bottom of the page to include more information about how we are removing null states from our truncated basis.
2.- We included a more prominent reference to the important work of Zamolodchikov, after equation (5.10). We also simplified our explanation of the derivation of (5.10), which only required computing expectation values of the energy-momentum tensor squared $T_{\mu\nu}^2$ in Ising CFT energy eigenstates.
Report 3:
1.- We changed the motivation/ overview of our procedure that we give in the introduction by reordering a couple of paragraphs and adding some further details. The changes can be found on page 3, contained between the paragraphs “Considerable progress has been made..” and “In Ref.~[12], a general procedure…”.
2.- We added further comments to address this question in the paragraph that begins at the very bottom of page 28.
3.- We increased the sizes of the labels in many of the figures to improve readability.
We caught a $2\pi$ typo missing on (5.9) and (6.13) that does not affect the numerical results and formulas presented later on the text.
We are grateful to the editor and referees for their careful consideration of our paper. We hope that our replies to the referees and changes to the manuscript will make it fully acceptable for publication in SciPost.
Best wishes,
Joan, James and Olivier
Published as SciPost Phys. 16, 105 (2024)