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Gaussian state approximation of quantum many-body scars

by Wouter Buijsman, Yevgeny Bar Lev

This is not the latest submitted version.

Submission summary

Authors (as registered SciPost users): Yevgeny Bar Lev · Wouter Buijsman
Submission information
Preprint Link: scipost_202307_00041v1  (pdf)
Date submitted: 2023-07-31 09:37
Submitted by: Buijsman, Wouter
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational

Abstract

Quantum many-body scars are atypical, highly nonthermal eigenstates of kinetically constrained systems embedded in a sea of thermal eigenstates. These special eigenstates are characterized, for example, by a bipartite entanglement entropy that scales as most logarithmically with subsystem size. We use numerical optimization techniques to investigate if quantum many-body scars of the experimentally relevant PXP model are well approximated by Gaussian states. These states are described by a number of parameters that scales quadratically with system size, thereby having a much lower complexity than generic quantum many-body states. We find that this is a good description for the quantum many-body scars away from the center of the spectrum.

Current status:
Has been resubmitted



Reports on this Submission

Anonymous Report 2 on 2023-10-16 (Invited Report)

Strengths

1. This is an interesting question to explore.

2. The results are presented clearly and are interesting.

Weaknesses

1. The analysis seems shallow, and many immediate questions and issues are not addressed.

2. Does not provide adequate review of the algorithm and methodology used.

Report

This manuscript contains results on some properties of Quantum Many-Body Scars (QMBS) of the PXP model. The authors explore if these QMBS eigenstates can be approximated by the ground state of some non-interacting Bogoliubov-deGennes (BdG) fermion model, which is not necessarily local or translation-invariant. They optimize the parameters in the fermion model to maximize its overlap with the QMBS, and they use a previously developed algorithm for this optimization. They show that the overlap is large for QMBS at the edge of the spectrum of the PXP model, it is lesser in the middle of the spectrum. Further, for the QMBS, the optimized model has some locality that emerges even though it is not assumed from the start.

Given that ground states of non-interacting fermion states have low-entanglement and so do QMBS eigenstates of the PXP model, trying to explore if there is any connection between them is an interesting question. However, I think the current version of the manuscript does not present a thorough enough exploration of this question to warrant immediate publication in SciPost. Here are some questions that the authors could consider exploring.

1. The authors present the obtained results for the QMBS as-is, without any exploration on how surprising their results actually are. For example, is it clear that the overlap of the QMBS with optimized Gaussian states is significantly different from that of other eigenstates in the spectrum? If not, is there some properties of the optimized Hamiltonian that change for non-QMBS?

2. The ground state of the PXP model and its QMBS close to the edges of the spectrum are known to be well-approximated by Matrix Product States: https://arxiv.org/abs/1903.10517. This property itself makes it plausible that the QMBS states can admit several approximate parent Hamiltonians. Is this property related to the optimized non-interacting Hamiltonians that the authors discover for the QMBS close to the edge of the spectrum? For example, do the exact MPS studied in the above paper admit such non-interacting parent Hamiltonians?

3. The algorithm for optimization is used as a ``black box" and no details or motivations are presented. For example, what is the role of the initial states $|\psi_{init}\rangle$ that dictates the initial guesses for the matrix A and B? Do answers change for different choices of this initial state?

4. Another question that the authors may consider exploring if computationally feasible: Since the ground state of the free-fermion ground state does not naturally contain the Rydberg blockade constraint, do the overlaps with the QMBS change if the many-body wavefunctions are projected and normalized within the constrained subspace?

Requested changes

1. The results on QMBS eigenstates should be contrasted with results for non-QMBS eigenstates. Perhaps the same algorithms can be applied to non-QMBS, and properties that distinguish the QMBS can be pointed out clearly.

2. Potential connections to previous work addressing similar questions (https://arxiv.org/abs/1903.10517) should be discussed clearly.

3. Would be good to have an overview of important details of the algorithm used.

  • validity: high
  • significance: good
  • originality: ok
  • clarity: good
  • formatting: excellent
  • grammar: excellent

Author:  Wouter Buijsman  on 2024-01-08  [id 4226]

(in reply to Report 2 on 2023-10-16)

Let us thank the Referee for providing us with useful and constructive remarks. Please find our replies below.

The Referee writes:

The authors present the obtained results for the quantum many-body scars as-is, without any exploration on how surprising their results actually are. For example, is it clear that the overlap of the quantum many-body scars with optimized Gaussian states is significantly different from that of other eigenstates in the spectrum? If not, is there some properties of the optimized Hamiltonian that change for non-quantum many-body scars?

Our response: We have implemented this suggestion in the revised version. Our newly added results show that our method works significantly better for quantum many-body scars compared to the thermal eigenstates.

The Referee writes:

The ground state of the PXP model and its quantum many-body scars close to the edges of the spectrum are known to be well-approximated by Matrix Product States (arXiv:1903.10517). This property itself makes it plausible that the quantum many-body scar states can admit several approximate parent Hamiltonians. Is this property related to the optimized non-interacting Hamiltonians that the authors discover for the quantum many-body scars close to the edge of the spectrum? For example, do the exact MPS studied in the above paper admit such non-interacting parent Hamiltonians?

Our response: We thank the Referee for this interesting question. Indeed, since the ground state of the PXP model has a large overlap with the matrix product state constructed in the reference and our Gaussian states, it follows that the constructed matrix product state can also be well-approximated by a Gaussian state, and admits a non-interacting parent Hamiltonian.

The Referee writes:

The algorithm for optimization is used as a ``black box" and no details or motivations are presented. For example, what is the role of the initial states $|\psi_\text{init} \rangle$ that dictates the initial guesses for the matrix $A$ and $B$? Do answers change for different choices of this initial state?

Our response: In the revised version, we have expanded our discussion on the choice and details of the optimization procedure, as well as the sensitivity of the results on the initial state.

The Referee writes:

Another question that the authors may consider exploring if computationally feasible: Since the ground state of the free-fermion ground state does not naturally contain the Rydberg blockade constraint, do the overlaps with the quantum many-body scars change if the many-body wavefunctions are projected and normalized within the constrained subspace?

Our response: The Referee is right that the non-interacting Hamiltonian is constructed in the full Hilbert space and therefore not all of its eigenstates can lie within the projected space. However, this does not rule out that its ground state (or even more eigenstates) can lie within this space, similarly to the Néel state. In fact, given the high overlap of the Gaussian state with the ground state, even without the projection, we know that this is the case. We have added a comment on this in the revised version.

The Referee writes:

The results on quantum many-body scars eigenstates should be contrasted with results for non-quantum many-body scar eigenstates. Perhaps the same algorithms can be applied to non-quantum many-body scars, and properties that distinguish the quantum many-body scars can be pointed out clearly.

Our response: We have implemented this suggestion in the revised version.

The Referee writes:

Potential connections to previous work addressing similar questions (arXiv:1903.10517) should be discussed clearly.

Our response: We thank the Referee for pointing out this work. We have implemented this suggestion in the revised version (please also see our response to the first point of Referee 1). We have added a reference to the reference in the revised version.

The Referee writes:

It would be good to have an overview of important details of the algorithm used.

Our response: We have implemented this suggestion in the revised version.

Anonymous Report 1 on 2023-8-19 (Invited Report)

Strengths

Motivated by the empirically known low bipartite entanglement entropy of atypical eigenstates of the PXP model, the authors try to approximate these states by ground state wavefunctions of a quadratic, fermionic Bogoliubov-de Gennes (BdG)-like Hamiltonian.
The authors optimize the overlap between the scar states and the ansatz wavefunctions by tuning the coefficients of the BdG Hamiltonian, in particular the hopping matrix. This numerical study furnishes the following two main results:
1. the trial wavefunctions have appreciable overlap with the exact eigenstates, whereby it is higher for the scar states at the edges of the spectrum;
2. the optimal hopping matrix seems to be local, which gives back a low entanglement of the scar states.

Weaknesses

Unfortunately, it remains unclear what one should take home from this analysis. It seems that the trial wavefunctions are rather tedious to obtain, and contain many more parameters than the wavefunctions of Ref. [25-27]. The authors do not benchmark their approach with the existing approaches: Are their wavefunctions more or less accurate, and does the gained accuracy (if any) justify the computational expense to optimize the wavefunctions? What is the physical content conveyed by the ansatz (if any)? What more does one learn as compared to what one learns from the wavefunctions proposed and discussed in the literature?

At a technical level, the approximation is not well described: A bosonic wavefunction is approximated by a fermionic wavefunction. While the Hilbert spaces are of equal dimension, there are many ways to map states with the same site occupations from fermions to bosons. It should be specified which mapping (ordering of fermionic operators) was chosen and why. It seems very likely that different choices of orderings would yield different results. Is there an optimal choice?

Report

Unfortunately, neither of the 4 possible acceptance criteria is met.

Requested changes

1. Considering that the ground state of PXP, as a ground state of a gapped one-dimensional Hamiltonian, is expected to be low-entangled, property (i) seems not so surprising, and is in fact shared with several other trial wavefunctions in the literature.
Please comment.

2. To what extent the optimal hopping is really short range is hard to tell, since the authors do not provide data for different sizes. The resulting hopping range is still a quarter to a third of the system size. How does this range evolve with system size? Does it seem to saturate? It probably should, since otherwise the state would hardly have a low entanglement entropy, unlike the state to be approximated. Thus, the short range of hoppings seems to be rather a sanity check of the approximation than an actual result that bears physical content.
Since the original PXP model has open boundary and thus is not translation invariant, one should not expect translation invariant hoppings to emerge of course. However, at least in the bulk one would expect translation invariance to be recovered, as seems to happen, looking at Fig.3; however the authors claim otherwise. This should be elaborated on more if the authors claim absence of translation invariance. If they believe it is there: why should this happen, and what is its significance?

3. The first sentences of the abstract and in the second paragraph are rather misleading, as they give the impression that quantum many-body scars are only found in kinetically constrained models. However, to our knowledge there are more models with many body scars in unconstrained models (Hubbard models, XY models, Heisenberg models etc...) than in kinetically constrained models. Please reformulate.

4. Can the authors motivate their choice of open boundary conditions (even though periodic boundary conditions would significantly reduce the Hilbert space)? Presumably this is due to a problem for the mapping between fermionic and bosonic wavefunctions. If so, that should be spelt out.

5. It is unclear what is meant by "the PXP model is time-reversal symmetric" before Eq.(6). Is its meaning that the Hamiltonian has real matrix elements in the sz basis (but not standard TR-invariance of spin systems)?

6. On page 3 it is stated that many-body scars distinguish themselves from other types of non-ergodic many-body states by their overlap with Neel states. This is a rather confusing statement. Usually one considers all non-ergodic many-body states in a given model as scar states.

  • validity: low
  • significance: poor
  • originality: low
  • clarity: ok
  • formatting: reasonable
  • grammar: acceptable

Author:  Wouter Buijsman  on 2024-01-08  [id 4225]

(in reply to Report 1 on 2023-08-19)

We would like to thank the Referee for their careful reading of the manuscript, as well as their useful comments and suggestions. Please find below our replies.

The Referee writes:

Unfortunately, it remains unclear what one should take home from this analysis. It seems that the trial wavefunctions are rather tedious to obtain, and contain many more parameters than the wavefunctions of Ref. [25-27]. The authors do not benchmark their approach with the existing approaches: Are their wavefunctions more or less accurate, and does the gained accuracy (if any) justify the computational expense to optimize the wavefunctions? What is the physical content conveyed by the ansatz (if any)? What more does one learn as compared to what one learns from the wavefunctions proposed and discussed in the literature?

Our response: The goal of our work is to show that the quantum many-body scars of the PXP model can be well-approximated by Gaussian states. We believe that this considerably advances the understanding of the origin of quantum many-body scars, since one could argue that the PXP model might be in the vicinity of certain quadratic parent Hamiltonians. We did not intend to develop an efficient method that minimizes the number of optimization parameters, or provides a better overlap compared to existing methods. Our motivation is thus different from other works, for example focusing on matrix product states, where in some cases exact matrix product state constructions of quantum many-body scars can be found. We have clarified our motivation in the Abstract and Introduction of the revised version.

The Referee writes:

At a technical level, the approximation is not well described: A bosonic wavefunction is approximated by a fermionic wavefunction. While the Hilbert spaces are of equal dimension, there are many ways to map states with the same site occupations from fermions to bosons. It should be specified which mapping (ordering of fermionic operators) was chosen and why. It seems very likely that different choices of orderings would yield different results. Is there an optimal choice?

Our response: We thank the Referee for pointing this out. In the revised version, we have added more details on the mapping of the PXP model into fermionic form. It is an open question what is the optimal mapping.

The Referee writes:

Considering that the ground state of PXP, as a ground state of a gapped one-dimensional Hamiltonian, is expected to be low-entangled, property (i) seems not so surprising, and is in fact shared with several other trial wavefunctions in the literature. Please comment.

(i) The trial wavefunctions have appreciable overlap with the exact eigenstates, whereby it is higher for the scar states at the edges of the spectrum.

Our response The main point of our work is that Gaussian states can provide surprisingly good descriptions of the quantum many-body scars of the PXP model, compared to any other eigenstates of the same model. This is surprising, since the model is interacting, and the approximation works amazingly well also in the middle of the spectrum. We would like to stress that the fact that the ground state of the PXP model is low entangled does not guarantee it is well approximated by a Gaussian state. For example, a cat state is a superposition of two Gaussian states. It has area-law entanglement, but a maximal overlap of $1/2$ with a Gaussian state. The fact that our approach works better for the ground state is a side point.

While as we mention in the Introduction, the fact that the quantum many-body scars have low entanglement motivated us to conduct this study, there is no a-priori reason to believe that Gaussian states would well-approximate the quantum many-body scars (or the ground state). We have commented on this in the revised version.

The Referee writes:

To what extent the optimal hopping is really short range is hard to tell, since the authors do not provide data for different sizes. The resulting hopping range is still a quarter to a third of the system size. How does this range evolve with system size? Does it seem to saturate? It probably should, since otherwise the state would hardly have a low entanglement entropy, unlike the state to be approximated. Thus, the short range of hoppings seems to be rather a sanity check of the approximation than an actual result that bears physical content.

Our response: The Referee is right that once we show that the Guassian states provides good descriptions of the quantum many-body scars, the fact that the matrices $A$ and $B$ have large elements only close to the diagonal is merely a sanity check. To be slightly more quantative, we bring here (see the Figure in the attachment) a plot of the cut throught the anti-diagonal of the matrices depicted in Fig. 3. The decay of the hopping when moving away from the diagonal is visible. The main physical content, which should be infered from Fig. 3 is (a) the pairing matrix is not-negligible, and (b) translation invariance is broken, even in the bulk (more about this below).

The Referee writes:

Since the original PXP model has open boundary and thus is not translation invariant, one should not expect translation invariant hoppings to emerge of course. However, at least in the bulk one would expect translation invariance to be recovered, as seems to happen, looking at Fig. 3; however the authors claim otherwise. This should be elaborated on more if the authors claim absence of translation invariance. If they believe it is there: why should this happen, and what is its significance?

Our response: We thank the Referee for pointing this out. The claim on the absence of translational invariance refers to the checkerboard-pattern of the matrices $A$ and $B$ that suggests a translational invariance for translations over two sites, instead of ``full'' translational invariance. This does not appear to be surprising given the fact that our trail wavefunctions [Eq. (5)] break translational invariance of the matrices $A$ and $B$. We found that enforcing translation invariance in the trial wavefunctions yields considerably lower overlaps. This provides a hint that the exact quantum many-body scars are superpositions of states invariant under translations by two sites. We have added an elaboration on this in the revised version.

The Referee writes:

The first sentences of the abstract and in the second paragraph are rather misleading, as they give the impression that quantum many-body scars are only found in kinetically constrained models. However, to our knowledge there are more models with many body scars in unconstrained models (Hubbard models, XY models, Heisenberg models etc...) than in kinetically constrained models. Please reformulate.

Our response: We thank the Referee for pointing this out. We have corrected this in the revised version.

The Referee writes:

Can the authors motivate their choice of open boundary conditions (even though periodic boundary conditions would significantly reduce the Hilbert space)? Presumably this is due to a problem for the mapping between fermionic and bosonic wavefunctions. If so, that should be spelt out.

Our response: We use open boundary conditions to avoid the extra boundary term, which appears after mapping the spins into fermions using a Jordan-Wigner transformation.

The Referee writes:

It is unclear what is meant by "the PXP model is time-reversal symmetric" before Eq. (6). Is its meaning that the Hamiltonian has real matrix elements in the $S_z$-basis (but not standard translational-invariance of spin systems)?

Our response: Yes, practically this means that the PXP Hamiltonian is real-valued in the $S_z$-basis. We have elucidated this in the revised version.

The Referee writes:

On page 3 it is stated that many-body scars distinguish themselves from other types of non-ergodic many-body states by their overlap with N\' eel states. This is a rather confusing statement. Usually one considers all non-ergodic many-body states in a given model as scar states.

Our response: Although certain works (Ref. [46], for example) classify all non-ergodic eigenstates as quantum many-body scars, this is in our view not a standard convention. See Ref. [8], Sec. VI. C for a recent overview of the debate on this issue. We have elaborated on our precise definition of quantum many-body scars in the revised version.

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