Gaussian state approximation of quantum many-body scars

The authors propose a new type of fermionic trial wavefunction approximating the scar states of the PXP model. It suggests that after a Jordan Wigner transformation these scar wavefunctions are reasonably well approximated by a Gaussian structure. This may reveal a certain property which was hitherto not pointed out.

1) the fermionic quadratic Hamiltonian is different for every scar wavefunction. It is thus unclear whether there is any parent Hamiltonian that underlies the structure of all scar wavefunctions.

2) The Jordan Wigner transformation produces a non-local fermionic Hamiltonian. Logically it is not very clear why it is a good thing to make such a step.The scar states of that non-local fermionic Hamiltonian are then approximated by local quadratic BdG Hamiltonians.

3) The quadratic approximations have reasonable, but not extraordinarily good overlap with the scar wavefunctions from exact diagonalization. It is unclear to what extent these trial functions are better or conceptually more satisfactory than bosonic wavefunctions that have been proposed in the literature.

The authors have clarified significantly their approach (even though several details should still be made clearer, as detailed below). Now it is possible to follow what has been done and what the results of the study imply. Below we list a few points that need to be addressed and taken into consideration.

• As the authors explain now, they apply a Jordan Wigner transformation to the Hamiltonian, to make it fermionic. If they indeed used the standard transformation of Eqs. (2-4) the resulting fermionic Hamiltonian is, however, non-local, each term containing an uncompensated tail stemming from the single s_x operator. Given that there seems to be no reason to prefer open over periodic boundary conditions.
It is important to state both facts explicitly. Also, please comment on the origin of the far off-diagonal entries in the optimal coupling matrix, and how that could come about if indeed open boundary conditions were used (see next point).

• For each scar state the authors now consider quadratic fermionic Hamiltonians. They take their ground state (which is Gaussian) and construct the inversion-symmetric cat state of Eq. (8) (which is almost surely not Gaussian, as should be stated if indeed true). The overlap of these cat states with a considered scar state is the maximized by adjusting the Hamiltonian. The authors claim that the optimal Hamiltonians are local. However, Fig. 3 shows very strong matrix elements far off the diagonal. This would be natural to expect if periodic boundary conditions had been used, but is very unnatural for open boundary conditions, and seems to directly contradict the claim of locality of the fermionic Hamiltonian.
This issue has to be clarified and thoroughly discussed, possibly together with revisiting the Jordan Wigner transformation.

• The main aim of this work was to elucidate the origin of many-body scars. However, it remains unclear what insight has been gained. In the introduction, the authors write: “Indications for a Gaussian structure of quantum many-body scars would suggest that the PXP model is close to certain quadratic parent Hamiltonians, hinting on the origin of quantum many-body scars”. However, the meaning of “certain parent Hamiltonians” is unclear. Namely, as is stated in the conclusion, each individual scar state has been approximated by the ground state of a different quadratic Hamiltonian.
To justify the notion of a “parent Hamiltonian”, the authors should at least offer a speculation regarding the relation among the different quadratic Hamiltonians associated with all the scar states (are they close to each other in some sense? Or are they all representatives of some sub-class of Hamiltonians)?

• Judging from the numerical quality of the overlap, the fermionic non-interacting trial wavefunctions presented here do not seem as good as the wavefunctions constructed from (non-interacting) bosonic magnon excitations above certain reference states in the literature. Insofar it is not clear whether any statement as to a hidden fermionic (as opposed to bosonic) nature of QMBS can be made.
A statement about this consideration would help the reader situating this approach within the landscape of others.

• One detail in the text: On p. 3, 2nd column, the meaning of the following sentences is unclear: “One could alternatively choose A and B such that the ground state of Hˆ is given by the product state that has the highest overlap with the quantum many-body scar under consideration.” Do the authors mean the initial choice of A and B? They seem to exactly describe what was done anyhow, and it is thus unclear what the alternative is supposed to be.
Please clarify.
• There are still many typos and English mistakes that remain to be corrected.

In conclusion: It has become much clearer what was done in this work. With the above clarifications, the reader will have enough information to form a well-informed opinion about this fermionic approach to PXP.

With these changes the paper might just make the bar for publication in SciPost.
However, we still feel that none of the expected acceptance criteria is met.
At best the manuscript approaches criterion 4: “Provide a novel and synergetic link between different research areas.”
This may hold if one interprets different approaches to the PXP model as different research areas. The analysis is novel. Whether the link is synergetic is not so clear, though.

1) State that the JW transformation yields a non-local Hamiltonian, and that it makes no difference whether one consideres open or periodic boundary conditions. In fact periodic boundary conditions would be more natural, as translation symmetry would not be broken.

2) Please comment on the origin of the far off-diagonal entries in the optimal coupling matrix, and how that could come about if indeed open boundary conditions were used. This issue has to be clarified and thoroughly discussed, possibly together with revisiting the Jordan Wigner transformation.

3) To justify the notion of a “parent Hamiltonian”, the authors should at least offer a speculation regarding the relation among the different quadratic Hamiltonians associated with all the scar states (are they close to each other in some sense? Or are they all representatives of some sub-class of Hamiltonians)?

4) Comment on the quality of the approximation by fermionic wavefunctions, as compared to that with bosonic trial wavefunctions.