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Weak Ergodicity Breaking in NonHermitian Manybody Systems
by Qianqian Chen, Shuai A. Chen, Zheng Zhu
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Submission summary
Authors (as registered SciPost users):  Qianqian Chen · Zheng Zhu 
Submission information  

Preprint Link:  scipost_202209_00035v1 (pdf) 
Date submitted:  20220918 08:24 
Submitted by:  Chen, Qianqian 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
The recent discovery of persistent revivals in the Rydbergatom quantum simulator has revealed a weakly ergodicitybreaking mechanism dubbed quantum manybody scars, which are a set of nonthermal states embedded in otherwise thermal spectra. Until now, such a mechanism has been mainly studied in Hermitian systems. Here, we establish the nonHermitian quantum manybody scars and systematically characterize their nature from dynamic revivals, entanglement entropy, physical observables, and energy level statistics. Notably, we find the nonHermitian quantum manybody scars exhibit significantly enhanced coherent revival dynamics when approaching the exceptional point. The signatures of nonHermitian scars switch from the realenergy axis to the imaginaryenergy axis after a realtocomplex spectrum transition driven by increasing nonHermiticity, where an exceptional point and a quantum tricritical point emerge simultaneously. We further examine the stability of nonHermitian quantum manybody scars against external fields, reveal the nonHermitian quantum criticality and eventually set up the whole phase diagram. The possible connection to the open quantum manybody systems is also explored. Our findings offer insights for realizing longlived coherent states in nonHermitian manybody systems.
Current status:
Reports on this Submission
Anonymous Report 2 on 20221127 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202209_00035v1, delivered 20221127, doi: 10.21468/SciPost.Report.6205
Strengths
1) In principle, this is an interesting model to explore.
2) Manuscript is wellwritten, presentation is clear.
Weaknesses
1) Trying to make a connection between properties of excited states such as quantum scars, level statistics to the ground state phases seems to be fundamentally misguided.
2) Does not provide an adequate review of the basic concepts in nonHermitian systems.
3) Lacks direct experimental motivation. The jump operators that connect this to the Lindblad master equation are highly artificial.
Report
This manuscript contains results on the extension of celebrated quantum manybody scars of the PXP model to an appropriate nonHermitian generalization of the PXP Hamiltonian. They use standard exact diagonalization to study the properties of this generalized model in the presence of nonHermiticity and real magnetic field. First, they study various diagnostics in the "ground state" of the model, and they map out the "phase diagram". Subsequently, in each of the ground state phases, they study the fate of quantum manybody scars and revivals, which are known to exist in the PXP model, and they claim that the scars persist in some phases whereas they appear to disappear in some others. In addition, they study the real and complex energy level statistics in each of these phases and find a variety of results.
In principle, exploring the fate of scars under nonHermiticity is an interesting direction for exploration, given that a community interested in the physics of nonHermitian Hamiltonians has developed. However, I think there are some fundamental problems in the manuscript, here are some of my major remarks:
1) The way the discussion of scars is presented seems to imply that scarring and ground state properties are correlated, i.e., the ground state phases are also somehow dynamical phases for the special initial state. This is a strong claim, since the definition of a phase implicitly assumes some robustness under perturbations, and other known examples of quantum scars do not seem to have such robustness. However, none of the data the authors present support such a claim. To claim that quantum scars survive in a given phase, it is necessary to show that there a ``sudden" transition of the quantum scar properties at the phase boundaries shown in Fig. 1. Observing a particular behavior for a fixed system size at some isolated points in the phase diagram, as shown in Figs. 3 and 4, is not enough to claim that the behaviour of scars is the same throughout the same phase. Even if the scars exhibit a weak crossover from one behavior to the other at the phase boundaries, it should simply be called a crossover and not a transition unless there is a specific order parameter and scaling collapse in the thermodynamic limit.
2) The above comment also extends to level statistics. The authors seem to claim that the level statistics is correlated with the ground state phases, as shown in Fig. 5. Again, observing a particular behavior at one point in the phase is not sufficient to make the connection between the ground state phase and level statistics. If the authors would really like to claim that, perhaps they can show the level statistics ratios everywhere in the phase diagram of Fig.~1, perhaps in the form of a heat map? To emphasize how strong the claim is  the authors seem to suggest that the equilibrium phases of matter this model are also nonequilibrium phases of matter, which is mostly unheard of in quantum manybody systems.
3) In probing the ground state physics, the authors assume that several standard concepts in Hermitian equilibrium physics straightforwardly extend to nonHermitian ground states. First, it is not obvious how ground states are defined in nonHermitian systems given that the complex spectra, and the authors don't seem to state this clearly anywhere. Second, it is not obvious how and why concepts such as "central charge", "fidelity susceptibility" should generalize  many of the references that the authors point to seem to be those of standard Hermitian physics.
For these reasons, I do not recommend the publication of the current version of the manuscript in SciPost.
Requested changes
1) The authors should not attempt to make the connection between ground state phase properties and the excited state properties of quantum manybody scars and level statistics without sufficient evidence. If they wish to study both properties, they can reorganize the paper into two disconnected sections  one focusing on ground states and the other focusing on excited states.
2) The definition of the ground state in nonHermitian systems needs to be clarified somewhere. It would also be good to briefly review terminologies from the study of nonHermitian Hamiltonians, including simple concepts such as "overlaps", "biorthogonality", "exceptional points", different types of entanglement entropy, etc. Further, when one discusses standard concepts such as "critical exponents", "central charge", "susceptibility", etc., it would help to clarify what these mean and how they are connected to their Hermitian counterparts, which readers would be familiar with.
3) In the introduction, the authors have a sentence "However, whether weak ergodicity breaking exists in the presence of nonHermiticity is elusive both theoretically and experimentally." The theoretical aspect is not true  there are papers that explicitly point out exact quantum manybody scars in nonHermitian Hamiltonians, e.g., https://arxiv.org/pdf/2106.10300.pdf.
Anonymous Report 1 on 20221024 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202209_00035v1, delivered 20221024, doi: 10.21468/SciPost.Report.5976
Strengths
1 thorough
2 easy to read
Weaknesses
1not clear how to realize jump operators
Report
The authors study a nonHermitian variant of the PXP model. They find that the weak ergodicity breaking (or more precisely longlived oscillations) only exists in the region of the phase diagram that has real energies for the scarred states, which makes sense. The phase diagram is determined numerically in line with most studies on the PXP model.
The authors are further very thorough and study the spectral statistics, fidelity from special initial states, a few different prototypical observables (known from the Hermitian case to exist longtime oscillations), entanglement, etc.
In my opinion, importantly, the authors manage to connect their nonHermitian Hamiltonian to a Lindblad master equation, which renders the setup more physical (though it remains unclear how to realize the required jump operators experimentally).
Why their model has real energies for the scars and why in that region remains unclear. I cannot find an intuitive picture, nor some sort of analytical argument.
Requested changes
1 Comment on possible experimental realizations of jump operators
2 The reason for the weak ergodicity breaking in region (I) could possible be understood through Supp Th 3 of Ref. [98] that connects scars in Lindblad master equations to purely imaginary eigenvalues of the Liouvillian and with pure state eigenmodes of the Liouvillian. I suggest that the authors maybe check this.
Author: Qianqian Chen on 20230228 [id 3412]
(in reply to Report 1 on 20221024)
Response to Anonymous Report 1 on 20221024 (Invited Report)
We would like to thank the referee for reviewing our manuscript. The constructive suggestions of the referee are of great help for us to revise our manuscript. We have accepted all of the “request changes” and revised the manuscript accordingly. We sincerely hope that these improvements and the appended pointtopoint responses could resolve the referee’s concerns in a satisfactory manner.
Requested changes:
1 Comment on possible experimental realizations of jump operators
Our reply: The jump operators we constructed based on a perturbative analysis are complicated and it may be hard to realize such jump operators in experiments.
In our manuscript, what we identified is a condition for the jump operators that correspond to our nonHermitian model, and there should be many other choices for such jump operators. Based on a perturbative analysis, we gave one of the examples of such jump operators so as to perform the numerical calculation. Future construction of experimentally realizable jump operators is fundamentally important to deepen our understanding of the QMBS in an open quantum system. We have highlighted these points in Section 3.5, the appendix, and the section “Summary and Discussion” of the revised manuscript.
2 The reason for the weak ergodicity breaking in region (I) could possibly be understood through Supp Th 3 of Ref. [98] that connects scars in Lindblad master equations to purely imaginary eigenvalues of the Liouvillian and with pure state eigenmode of the Liouvillian. I suggest that the authors maybe check this.
Our reply: We thank the referee for such constructive suggestion and for drawing our attention to Supp Th 3 of Ref. [98], which we believe is very significant for the mechanism of the weak ergodicity breaking (or more precisely, nonstationarity) in open quantum systems. However, for our model, we have found that the QMBS states are not the pure state eigenmodes of the Liouvillian we constructed. One may design other forms of Lindblad operators that satisfy the theorem, which is beyond the scope of the current work. Indeed, in both the Hermitian PXP model and the nonHermitian PXP model we studied, the QMBS states are not exactly commensurately spaced in energy, which may lead to relaxation. In addition, the mechanisms causing scars in the PXP model are only approximately understood, usually with some deformations, while here we did not consider those deformations in our model. Moreover, in our manuscript, we applied the noncommutating Hermitian Lindblad operators to every site, which may also lead to relaxation in open quantum systems. However, the phenomenon that a specific initial product state exhibits much longer time periodic revivals than other initial states is clearly seen for our constructed Liouvillian.
We thank the referee for pointing out this important theorem. We will investigate the theorem in detail for future studies of QMBS related to nonHermiticity, and we have highlighted these points in the section “Summary and Discussion” of the revised manuscript.
List of changes:
 We have commented on the jump operators we constructed in relation to experimental realizations in Section 3.5, and emphasized experimentally realizable Lindblad operators in the section “Summary and Discussion”.
 We have highlighted the research direction on the eigenmodes of the Liouvillian with purely imaginary eigenvalues from the perspective of the nonHermitian QMBS.
Author: Qianqian Chen on 20230228 [id 3411]
(in reply to Report 2 on 20221127)Response to Anonymous Report 2 on 20221127 (Invited Report)
## Report
Referee’s comment:
In principle, exploring the fate of scars under nonHermiticity is an interesting direction for exploration, given that a community interested in the physics of nonHermitian Hamiltonians has developed. However, I think there are some fundamental problems in the manuscript, here are some of my major remarks:
1) The way the discussion of scars is presented seems to imply that scarring and ground state properties are correlated, i.e., the ground state phases are also somehow dynamical phases for the special initial state. This is a strong claim, since the definition of a phase implicitly assumes some robustness under perturbations, and other known examples of quantum scars do not seem to have such robustness. However, none of the data the authors present support such a claim. To claim that quantum scars survive in a given phase, it is necessary to show that there a sudden" transition of the quantum scar properties at the phase boundaries shown in Fig. 1. Observing a particular behavior for a fixed system size at some isolated points in the phase diagram, as shown in Figs. 3 and 4, is not enough to claim that the behaviour of scars is the same throughout the same phase. Even if the scars exhibit a weak crossover from one behavior to the other at the phase boundaries, it should simply be called a crossover and not a transition unless there is a specific order parameter and scaling collapse in the thermodynamic limit. 2) The above comment also extends to level statistics. The authors seem to claim that the level statistics is correlated with the ground state phases, as shown in Fig. 5. Again, observing a particular behavior at one point in the phase is not sufficient to make the connection between the ground state phase and level statistics. If the authors would really like to claim that, perhaps they can show the level statistics ratios everywhere in the phase diagram of Fig.~1, perhaps in the form of a heat map? To emphasize how strong the claim is  the authors seem to suggest that the equilibrium phases of matter this model are also nonequilibrium phases of matter, which is mostly unheard of in quantum manybody systems.
Referee’s comment: 3) In probing the ground state physics, the authors assume that several standard concepts in Hermitian equilibrium physics straightforwardly extend to nonHermitian ground states. First, it is not obvious how ground states are defined in nonHermitian systems given that the complex spectra, and the authors don't seem to state this clearly anywhere.
Referee’s comment: Second, it is not obvious how and why concepts such as "central charge", "fidelity susceptibility" should generalize  many of the references that the authors point to seem to be those of standard Hermitian physics.
$$\rho_A=\mathrm{Tr}_\bar A\rho,$$ $$ \mathcal F(\lambda ,\delta \lambda)=\langle \psi_L(\lambda)\psi_R(\lambda+\delta\lambda)\rangle\langle \psi_L(\lambda+ \delta\lambda)\psi_R(\lambda)\rangle $$ $$ \chi≈\frac{1\mathcal F}{\delta\lambda^2} $$Requested changes
Referee’s comment: 1) The authors should not attempt to make the connection between ground state phase properties and the excited state properties of quantum manybody scars and level statistics without sufficient evidence. If they wish to study both properties, they can reorganize the paper into two disconnected sections  one focusing on ground states and the other focusing on excited states.
Referee’s comment: 2) The definition of the ground state in nonHermitian systems needs to be clarified somewhere. It would also be good to briefly review terminologies from the study of nonHermitian Hamiltonians, including simple concepts such as "overlaps", "biorthogonality", "exceptional points", different types of entanglement entropy, etc. Further, when one discusses standard concepts such as "critical exponents", "central charge", "susceptibility", etc., it would help to clarify what these mean and how they are connected to their Hermitian counterparts, which readers would be familiar with.
Referee’s comment: 3) In the introduction, the authors have a sentence "However, whether weak ergodicity breaking exists in the presence of nonHermiticity is elusive both theoretically and experimentally." The theoretical aspect is not true  there are papers that explicitly point out exact quantum manybody scars in nonHermitian Hamiltonians, e.g., https://arxiv.org/pdf/2106.10300.pdf.
List of changes: