Quantum state preparation of topological chiral spin liquids via Floquet engineering

Dear Editor,

We would like to resubmit a new version of the draft and enclose below the response to the 1st referee. We believe that the changes made to comply with the comments of the 1st referee have improved the paper significantly. We are awaiting for the report of the second Referee who can then benefit from the improved version.

Sincerely yours,

M. Mambrini and D. Poilblanc

We thank Prof. Marin Bukov very much for is insightful reading of the
paper. We have made some revisions and added some references following
his comments. We believe it has improved the quality and clarity of the
paper. Below we answer point by point and describe the changes made.

## Section 2

***Referee*** Eq(3); it would be helpful if the authors define
explicitly the permutation operator in terms of the spin degrees of
freedom (I now see that you sort of have it in Eq(24) but an explicit
reference will be useful).

***Authors*** A sentence has been added to make the definition of
$P_{ijkl}$ explicit. The expression in terms of the spin degrees of
freedom has now been included in the main text.

***Referee*** Sec 2.3, "Note that such a monochromatic drive may not be
easy to implement on experimental cold atom platforms": I think there
is no difficulty with the dephased monochromatic drive itself; the
spin-spin interactions would be the real challenge, see e.g.,
https://www.nature.com/articles/s41567-020-0949-y

***Authors*** We thank the Referee for mentioning the reference. We have
rephrased the short paragraph in Sec. 2.3 on the experimental
implementation, taking into account the Referee comment.

***Referee*** Sec 2.4: before Eq (6), when the authors introduce
stroboscopic times, there appear integers p, n, times t, t0, etc,;
what's written is not wrong, but I think it can be simplified by
removing the unnecessary info for the benefit of the readers.

***Authors*** Following the comment we have simplified the formulation
(removing $t_0$, etc\...).

***Referee*** after Eq(7), "It is interesting that the next order of
the expansion is $O(1/\omega^3)$ so that we expect HF0 to capture
accurately the stroboscopic motion.": this is not obvious to me; can
the authors argue why there are no $1/\omega^2$ terms here?

***Authors*** Eq. (42) of Bukov et al. suggests that, **in the absence
of a time independent part $H_0=0$**, the expansion involves only nested
commutators containing the same number of $H_1$ and $H_{-1}$ Fourier
components of the Hamiltonian, i.e. odd powers of $1/\omega$. We have
added a note on this.

***Referee*** If I'm not mistaken, you might be able to compute the
Floquet Hamiltonian to Eq(6) alone exactly, as follows:

1. Let $L_z$ be the generator of continuous rotations about the
out-of-lattice-plane axis, i.e., the operator that rotates $H_x$
into $H_y$: $\exp(-i\pi/2L_z) H_x \exp(+i\pi/2 L_z )= H_y$; I think,
$\exp(-i\pi / 2 L_z)$ is related to the permutation operator
$P_{i,j,k,l}$ but we need the infinitesimal rotation generator. Not
sure how crazily non-local (in terms of the spin operators) $L_z$
is, though.

2. consider now a rotation about the angle $\alpha = \omega t$: in a
co-rotating frame defined by the rotation
$V(t)=\exp(-i \omega t L_z)$, the circular drive will freeze and
point, let's say, along $H_x$.

3. taking into account the Galilean term due to the frame
transformation, in the co-rotating frame the Hamiltonian will read
as $H'= \omega L_z+H_x$ (no liability regarding the correctness of
signs).

This business is very similar to the two-level system in a circular
drive, just the operators involved are quite a bit more complicated. To
relate $H'$ to HF, you may need to do extra work, especially if the
spectrum of $L_z$ is not commensurate, see Sec 2.3 of Ref [39]. In the
language of $H'$, to make the chiral structure manifest, you'd have to
consider the "symmetric gauge", see discussion in Sec 3.3 of Ref
[39].

***Authors*** We thank the Referee for his proposal. However, we believe
that, away from the high-frequency limit, the Floquet Hamiltonian
becomes more and more non local (including all sorts of chiral terms on
larger and larger closed loops). It would be very surprising that
tractable expressions could be obtained generically. In any case, we
postpone this calculation to future studies.

## Section 3

***Referee*** Sec 3.2.1: "Since the effective Floquet dynamics is
expected to be very accurate, one can use it to investigate the physics
in the $t\rightarrow \infty$ limit": this is dangerous b/c the
approximate effective Hamiltonian computed from the Magnus expansion
does not capture heating effects. The system is expected to heat up to
an infinite temperature state in the $t\rightarrow \infty$ limit for any
finite $\kappa>0$, see e.g., Ref [33]. However, what's physically
relevant in practice is the prethermal plateau regime which occurs at
finite time (and whose lifetime can be controlled by increasing the
drive frequency).

***Authors*** We agree with the referee and have reformulated the text.
We have added a comment on the role of the finite system size.

## Appendix A

***Referee*** "PEPS offer an extremely efficient variational scheme to
address local Hamiltonians": it's not clear which features/properties
are being addressed

***Authors*** Beside providing a very good variational energy, PEPS can
encode the topological nature of the state and the physics of the edge
mode. Note and Refs. [75,76] added.

***Referee*** it's worthwhile to mention in one sentence the drawbacks
of PEPS as well, or the regime of applicability

***Authors*** PEPS fail to describe the rapid increase of entanglement
entropy (e.g. in the case of a quench, see Ref. [77]) but should still
be relevant in the case of an adiabatic ramp.

## Appendix C

***Referee*** "The unitary operator $\exp(-iK(t))$ corresponds
physically to a change of basis": time-dependent unitaries correspond
rather to a change of reference frame; when evaluated at some fixed
$t_0$, one can think of them as giving kicks to the state (i.e.,
changing the basis). In this sense, Floquet's theorem is a statement
about the existence of a reference frame, where the dynamics is governed
by a static Hamiltonian $H_F$ at all times (not only stroboscopically).

***Authors*** We have changed "change of basis" into "change of
reference frame (via a change of basis)".

## Figure 2

***Referee*** a) I find it a bit confusing that the micromotion in inset
(c) seems to oscillate around the black-squares curve; however the two
curves are w.r.t. different x-axes (black squares go from times
$\sim$`<!-- -->`{=html}25 to 40, while grey micromotion curve goes from
times 30.1 to 30.5). It might be better to show inset (c) as a separate
panel of the figure to avoid this confusion.

***Authors*** We have modified the figure to avoid the confusion
mentioned by the Referee.

***Referee*** b) regarding errors caused by Trotterization: I think it
should be possible to evolve the state using a Runge-Kutta solver on a
$4\times4$ patch.

***Authors*** We have no expertise with such a technique but we have
full control on the error introduced by the Trotterization scheme. By
varying the Trotter step we can estimate the minimum number of steps per
period to reach a very good accuracy, typically better than $10^{-4}$.

## Figure 3

***Referee*** is $k$ the same integer as $p$? If yes, better use $p$.

***Authors*** We are a bit confused by the comment. Probably the Referee
has mistaken $\kappa$ (defined in Eq. (11)) for $k$ ?

## References

***Referee*** non-abelian anyons have recently been observed on
Honeywell's trapped ion quantum computer:
https://www.nature.com/articles/s41586-023-06934-4

***Authors*** We thank the Referee for this interesting paper which we
have cited in the introduction.

***Referee*** the authors might be interested in a recent preprint,
where we discuss various aspects of state preparation under strong
Floquet drives, and in particular how to speed up adiabatic ramps:
https://arxiv.org/abs/2310.02728

***Authors*** We thank the Referee for pointing out this interesting
preprint. We have cited it in the final discussion.

## Typos

***Referee***

* abstract: remarquable $\longrightarrow$ remarkable

* after Eq(1) [+ other instances]: "significantly smaller that the
smallest": that $\longrightarrow$ than

* before Eq(2): Krylov's $\longrightarrow$ Krylov

* Eq(2): summation index m missing in sum subscript

* Sec 2.2, first paragraph, "It can be view as": view
$\longrightarrow$ viewed

* Sec 3.1, penultimate paragraph, "p fixed": p $\longrightarrow$ p

* Sec 3.2, first paragraph, "is suddenly switch on", switch
$\longrightarrow$ switched

* Sec 3.4.2, last paragraph, "As shown in Figs. 7": Figs
$\longrightarrow$ Fig

* Sec 4.3: "We argue that our goal cannot be realized by branching
the drive suddenly": branching $\longrightarrow$ quenching

* Sec 4.3: "the minimum ramp time to reach a good fidelity should
qualitatively scales like N": scales $\longrightarrow$ scale

***Authors*** We thank the Referee for pointing out the typos which have
all been correcte\]d.

## Added references

Note that Refs. [19], [23] and [74] have been added.