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Rigorous bounds on dynamical response functions and timetranslation symmetry breaking
by Marko Medenjak, Tomaz Prosen, Lenart Zadnik
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Submission summary
As Contributors:  Marko Medenjak · Lenart Zadnik 
Arxiv Link:  https://arxiv.org/abs/2003.01035v1 (pdf) 
Date submitted:  20200303 01:00 
Submitted by:  Medenjak, Marko 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
Dynamical response functions are standard tools for probing local physics near the equilibrium. They provide information about relaxation properties after the equilibrium state is weakly perturbed. In this paper we focus on systems which break the assumption of thermalization by exhibiting persistent temporal oscillations. We provide rigorous bounds on the Fourier components of dynamical response functions in terms of extensive or local dynamical symmetries, i.e. extensive or local operators with periodic time dependence. Additionally, we discuss the effects of spatially inhomogeneous dynamical symmetries. The bounds are explicitly implemented on the example of an interacting Floquet system, specifically in the integrable Trotterization of the Heisenberg XXZ model.
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Reports on this Submission
Anonymous Report 2 on 2020517 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2003.01035v1, delivered 20200516, doi: 10.21468/SciPost.Report.1692
Strengths
This is an interesting contribution to the literature on dynamics of integrable systems. It builds on some recent ideas about persistent oscillations in integrable spin chains, and works out the details in a very clear way. I do not see any weaknesses, and think the paper should be published as is.
Report
I think this paper offers a very clear discussion of dynamical response, and especially its relation to dynamical symmetries in integrable spin chains (building on some previous work concerning "time crystals" and other persistent oscillations in these systems). I found the discussion very clear and have no significant comments.
Anonymous Report 1 on 2020510 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2003.01035v1, delivered 20200510, doi: 10.21468/SciPost.Report.1675
Strengths
1 the paper is clearly written;
2 the breaking of time translation symmetry is an extremely timely and interesting topic;
3 the mathematical aspects are rigorous although remain accessible to physicicst in the field;
4 the bounds are checked against numerics and are shown to be saturated;
Weaknesses
1 it is unclear if the discussion is fully general or it is focused on the behavior of specific integrable cases
2 the relevant point of the stability under perturbation is only mentioned in the conclusion
Report
This work considers the behavior of response functions, obtained by perturbing a system initially prepared at its thermal equilibrium (or in a stationary state). The approached followed retraces the construction of the celebrated Mazur's bound for dynamical correlation functions. Instead of considering conserved quantities of the system (i.e. operators commuting with the timeevolution generator), here the focus is on dynamical symmetries which generalise static ones: operators which are are eigenvectors for the adjoint action of the Hamiltonian. The authors are able to derive exact bounds which ensure that in the presence of dynamical symmetries and nontrivial overlap with relevant operators, one will observe persistent oscillations as measured by the absence of decay of the dynamical response functions.
Overall, the paper is dealing with a very timely topic which is the breaking of timetranslation symmetry and the existence of time crystals. Additionally, the paper is wellwritten and contains rigorous results and a mathematical framework to study theoretically the occurrence of this phenomenon. I also appreciate the presence of a comparison with numerical simulations.
I recommend the publication on Scipost; I have a few questions and comments, whose answer could improve the readability and the clarity of the manuscript.
Requested changes
1 at the end of the introduction, the authors comment about the relation between timetranslation symmetry breaking and the existence of extensive dynamical symmetries. Is it true in general that all forms of timetranslation symmetry breakings do require the existence of dynamical symmetries as given in Eq. 12?
2In Eq. (1), I could not understand if the $\rho$'s stand for the normalized density matrix or the unnormalized one $\rho(\beta) = e^{\beta H}$. Please clarify.
3 Although this is wellknown in the literature, can you please add some comments (or a precise reference) about why Eq. (2) is a welldefined scalar product ? In particular, since it is used a lot in the rest of the paper, why $\langle A, A\rangle \geq 0$?
4 Why in Eq. (5) you only consider the diagonal case where both $A$ and $B$ have the same $k$ component? Would it be relevant to treat the offdiagonal case?
5 In section 3, I think it would help a lot to have a couple of explicit examples of local / extensive dynamical symmetries, beyond their mathematical definitions.
6 An important point that I see is: what could be the physical origin of a relation like Eq. (9)? I understand that it can come out from explicit contruction of integrable models (as shown in Sec. 5), but beyond this is there a physical context which would lead to this "dynamical symmetries"? For conserved quantities, we know that they result from the Noether's theorem and a physical (or gauge) symmetry. What about dynamical symmetries discussed here? It would be important to know if they can come out in physical systems because of general principles.
7 Can you clarify what is the origin of the conjecture in Eq. 33?
Author: Marko Medenjak on 20200604 [id 845]
(in reply to Report 1 on 20200510)
We thank the referee for their positive report and in particular for the thorough and constructive evaluation, as well as questions and comments, which we address below.
"1 at the end of the introduction, the authors comment about the relation between timetranslation symmetry breaking and the existence of extensive dynamical symmetries. Is it true in general that all forms of timetranslation symmetry breakings do require the existence of dynamical symmetries as given in Eq. 12?"
There exists a set of systems with quantum scarred states where TTS breaking occurs only for special initial condition, which does not require the existence of dynamical symmetries. However, if TTS breaking occurs with respect to some stationary ensemble it implies the existence of dynamical symmetries that are extensive with respect to that ensemble.
"2 In Eq. (1), I could not understand if the ρ's stand for the normalized density matrix or the unnormalized one ρ(β)=exp(−βH). Please clarify."
ρ stands for the unnormalized density matrix, as introduced in the inline equation before Eq. (1). Instead the averages of observables are normalized, as described immediately after Eq. (1). We have now explicitly stated in the manuscript that this is an unnormalized state.
"3 Although this is wellknown in the literature, can you please add some comments (or a precise reference) about why Eq. (2) is a welldefined scalar product ? In particular, since it is used a lot in the rest of the paper, why ⟨A,A⟩≥0?"
The reference has been added.
"4 Why in Eq. (5) you only consider the diagonal case where both A and B have the same k component? Would it be relevant to treat the offdiagonal case?"
We assumed that the Hamiltonian H is translationally invariant, implying that the offdiagonal elements are 0. We added the comment to the manuscript.
"5 In section 3, I think it would help a lot to have a couple of explicit examples of local / extensive dynamical symmetries, beyond their mathematical definitions."
We have added a simple example concerning the spin precession, as well as a more general comment on the dynamical symmetries that arise from a possible nonabelian structure of local conservation laws.
"6 An important point that I see is: what could be the physical origin of a relation like Eq. (9)? I understand that it can come out from explicit contruction of integrable models (as shown in Sec. 5), but beyond this is there a physical context which would lead to this "dynamical symmetries"? For conserved quantities, we know that they result from the Noether's theorem and a physical (or gauge) symmetry. What about dynamical symmetries discussed here? It would be important to know if they can come out in physical systems because of general principles."
They can arise as a consequence of nonabelian conservation laws that form a closed algebra. For instance, assume that the system possesses conserved quantities X and Y, [H,X]=0, [H,Y]=0, satisfying [X,Y]=Y. Then Y will be a dynamical symmetry for the Hamiltonian H'=H+X. The discussion has been added as another example in section 3.
"7 Can you clarify what is the origin of the conjecture in Eq. 33?"
The conjecture is guessed similarly as in Ref. [35], by considering the action of the auxiliary transfer matrix, given in Eq. (61) of the Appendix. Due to Proposition 1 (Appendix) the operatorial sum in Eq. (60) converges. One then needs to compute its matrix element using a fifthorder recurrence relation determined by the elements of the auxiliary transfer matrix (see Refs. [7], [41] and [35]). The solution can be guessed so as to reproduce the various limits in which solutions exist and can be proven (for example the continuoustime limit in Ref. [41]). Then, the result is compared to the numerical evaluation of the overlap. The word "numerically", as well as a short footnote have been added.
We would like to thank the referees again, for their examination of the manuscript.
Author: Marko Medenjak on 20200604 [id 846]
(in reply to Report 2 on 20200517)We thank the referee for a very positive response.