Gaussian state approximation of quantum many-body scars

Let us thank the referee again for providing us a positive and constructive report. Please find our reply to each of the points below.

The referee writes:

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.

Our response:

The primary aim of our work is to show that individual quantum many-body scars can be described surprisingly well in terms of Gaussian states. For such states, the number of parameters scales polynomially with system size, instead of exponentially (as for generic quantum many-body states). We believe that such an observation is non-trivial. Our work should be seen as a first step in exploring the posssibility that a unified description in terms of Gaussian states for all quantum many-body scars can be obtained. We have commented on this more explicitly in the revised version.

The referee writes:

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.

Our response:

While the fermionic version of the PXP Hamiltonian is indeed non-local, we do not use it in any way in the manuscript, since we work directly with its eigenstates. Since the quantum many-body scars are known to have decaying spatial correlations, and sub-volume entanglement entropy it is actually natural that the optimal quadratic Hamiltonian is local. With this said, we do not constrain the approximating quadratic Hamiltonian, such that the outcome of the optimization could have been non-local. We have clarified this issue in the text.

The referee writes:

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.

Our response:

Our approach is conceptually different from what has been done in previous works (please also see our previous reply). Our goal is not to achieve the most accurate description of quantum many-body scars. Instead, we aim to find out whether quantum many-body scars can be well described in terms of Gaussian states. We have elaborated more extensively on this in the revised version.

The referee writes:

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 $\sigma^x$-operator. Given that there seems to be no reason to prefer open over periodic boundary conditions. It is important to state both facts explicitly.

Our response:

We thank the referee for this comment, which allowed us to considerably improve our manuscript. We have changed the calculation to periodic boundary conditions, which allowed us to obtain remarkable overlaps also with scars close to the center of the band, compared to the previous open boundary conditions calclations. We have added a comment about the non-locality of the Jordan-Wigner transformed Hamiltonian to the text.

The referee writes:

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).

Our response:

We thank the referee for making this observation. In the revised version, we focus periodic instead of open boundary conditions, for which this issue does not appear. Switching from open to periodic boundary conditions does not qualitatively change our findings or conclusions. In fact, we find that overlaps are typically higher compared to what we observed before. In particular, the overlaps improved significantly for the quantum many-body scars near the center of the spectrum.

The referee writes:

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.

Our response:

The symmetrized cat state are indeed not Gaussian. We have mentioned this explicitly in the revised version of the manuscript. Let us mention that we perform optimizations for non-symmetrized (Gaussian) states in appendix A.

The referee writes:

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.

Our response:

In the revised version, we focus on periodic instead of open boundary conditions. For periodic boundary conditions, the issue does not appear.

The referee writes:

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)?

Our response:

We thank the referee for pointing out this omission. As elaborated above, our main result is the obervation that quantum many-body scars can be well described in terms of Gaussian states, the implications on the structure of models hosting quantum many-body scars are merely speculations that can provide starting points for future investigations. We have stressed this more explicitly in the revised version, and expanded the discussion on how different parent Hamiltonians could possible be related in the conclusions and outlook section.

The referee writes:

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.

Our response:

We have implemented this suggestion in the revised version. Specifically, we have added a comment similar to the one made by the referee in the conclusions and outlook section.

The referee writes:

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 $\hat{H}$ 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.

Our response:

The referee is correct regarding the first point. In the revised version, we have replaced “choose $A$ and $B$" by “choose the initial guesses for $A$ and $B$". Please let us stress that this is not the procedure adapted in the manuscript: we take $A$ and $B$ such that the ground state of $\hat{H}$ is given by the $\mathbb{Z}_2$ state. Although for some quantum many-body scars the highest-overlapping basis state is the $\mathbb{Z}_2$ state, this is not the case in general.

The referee writes:

There are still many typos and English mistakes that remain to be correct

Our response:

We have carefully checked the manuscript, and corrected all misprints that we found.