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Quantum echo dynamics in the Sherrington-Kirkpatrick model

by Silvia Pappalardi, Anatoli Polkovnikov, Alessandro Silva

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Submission summary

Authors (as registered SciPost users): Silvia Pappalardi · Anatoli Polkovnikov · Alessandro Silva
Submission information
Preprint Link: https://arxiv.org/abs/1910.04769v2  (pdf)
Date submitted: 2020-02-17 01:00
Submitted by: Pappalardi, Silvia
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • Quantum Physics
Approach: Theoretical

Abstract

Understanding the footprints of chaos in quantum-many-body systems has been under debate for a long time. In this work, we study the echo dynamics of the Sherrington-Kirkpatrick (SK) model with transverse field under effective time reversal, by investigating numerically its quantum and semiclassical dynamics. We explore how chaotic many-body quantum physics can lead to exponential divergence of the echo of observables and we show that it is a result of three requirements: i) the collective nature of the observable, ii) a properly chosen initial state}and iii) the existence of a well-defined semi-classical (large-$N$) limit. Under these conditions, the echo grows exponentially up to the Ehrenfest time, which scales logarithmically with the number of spins $N$. In this regime, the echo is well described by the semiclassical (truncated Wigner) approximation. We also discuss a short-range version of the SK model, where the Ehrenfest time does not depend on $N$ and the quantum echo shows only polynomial growth. Our findings provide new insights on scrambling and echo dynamics and how to observe it experimentally.

Current status:
Has been resubmitted


Author comments upon resubmission

Dear Editor,

thank you for handling our submission.

We are pleased to thank the referees, whose insightful comments and observations helped us further improve and clarify our work. In this resubmission, we believe to have addressed all the point raised by the Referees. We append below a detailed response to the reports and the list of changes.

Yours sincerely,

Silvia Pappalardi, Anatoli Polkovnikov and Alessandro Silva

Reply to referee report I:

We thank the referee for her/his thoughtful reading on the manuscript. Here we reply to the corresponding points of their report, specifying the changes included in the resubmitted manuscript.

1) We thank the referee for the question since our previous draft lacked clarity. Since a similar issue has been raised also by the Referee 2, we have added few lines in the introduction, improved the discussion in Section 2 and included a new small appendix with the details of the classical limit of the echo observables.

We now explain in details the relation between $\mu(t)$ and OTOCs. First of all, let us specify the general meaning that one can associate to OTOC. Even if in recent literature the name OTOC is usually related to correlators as $\langle \hat A \hat B(t) \hat A \hat B(t) \rangle$, "out of time-ordered correlators" OTOC can be referred to all the time-dependent correlation functions which have an unusual time-ordering. Namely, they describe correlation functions which cannot be computed with conventional techniques that assume casual time evolution and therefore need to be evaluated via extended Keldysh contours, like the one developed in [Aleiner, Faoro, Ioffe, Annals of Physics 2016]. In this sense, the echo observable $\mu(t)$ can be refereed as OTOC, since it contains terms as $\langle \hat B(t) \hat A \hat B(t) \rangle$, see Eq.(4).

In the second place, in the semi-classical limit, every kind of OTOC contains quadratic terms in the derivatives of the initial conditions, as was already known by Larkin and Ovchinnikov. This is true also for the echo $\mu(t)$. This statement can be shown by the use of time-dependent {Bopp operators}, a standard formalism of phase-space methods which allows computing time-dependent correlation functions, by representing quantum operators as phase-space variables. This is extensively discussed in the Appendix of [Schmitt, Sels, Kehrein and Polkovnikov, PRB 2019] for a semi-classical limit of the SYK. In order to be as self-consistent as possible, we have added a new Appendix to the manuscript, with the derivation of the semi-classical limit of $\mu(t)$, based on Bopp operators. We show for bosonic and spin operators, that the semi-classical echo observable contains quadratic terms in the derivatives of the classical trajectory to respect to the initial conditions, hence it encodes the Lyapunov exponent, exactly as the square commutator. In what follows, we report the core of the proof.

Let us start by introducing the Bopp formalism. The Wigner-Weyl quantization is intrinsically connected with the symmetric Bopp representation of the quantum operators, which allows to map operators to functions of phase space variables. In particular, the bosonic creation and annihilation operators in the Bopp representation read

\[ \hat a^\dagger \to \alpha^\ast-{1\over 2} {\partial\over \partial\alpha},\quad \hat a\to\alpha+{1\over 2} {\partial\over \partial \alpha^\ast}. \quad \quad (1) \]

Then the Weyl symbol of, for example, the number operator is obtained by simply writing it in the Bopp representation $n^w=(\hat a^\dagger \hat a)^w=\left (\alpha^\ast-{1\over 2} {\partial\over \partial\alpha}\right)\alpha=\alpha^\ast\alpha-{1\over 2}$. Semi-classical expectation values are obtained by averaging the Weyl symbol of the operator with the Wigner function corresponding the the quantum state (see also the new Section 5 of the manuscript). Interestingly the Bopp representation immediately allows one to compute non-equal correlation functions e.g.

\[ \left (\hat a^\dagger(t_1) \hat a(t_2) \right )_w = \alpha^\ast (t_1) \alpha (t_2)-{1\over 2}{\partial \alpha(t_2)\over\partial \alpha(t_1)} \ ,\quad \quad (2) \]

where the derivative to respect to $\alpha(t_1)$ represents the non-equal time response. One can also write the Bopp operators in a more compact form where the creation and annihilation operators (and similarly the momentum and the coordinate operators) map to the corresponding phase space variables plus half of the Poisson bracket. For more complicated operators, like non-linear bosonic variables or spin operators, this simple interpretation is lost as generally higher order derivatives emerge. In order to derive the semi-classical limit of OTOC at order $\hbar^2$, it is enough to keep at most the second-order expansion in $\hbar$ of the Bopp operator. In particular, for a generic time-dependent operator $\hat B(t)$, the Bopp representation can be written as

\[ \hat B(t) \to B_t + \hbar D^{(1)}_{B_t} + \hbar^2 D^{(2)}_{B_t} \ ,\quad \quad (3) \]

where $B_t$ is the Weyl symbol of the operator $\hat B$ evaluated at time $t$, the linear order is given by half of the Poisson brackets
$D_{B_t}^{(1)}=i/2${$B_t, \cdot$}, and $D^{(2)}_{B_t}$ contains the second-order derivatives and its explicit form depends on the operator $\hat B(t)$. These formulae can be used in constructing the Weyl symbols for various time-dependent expectation values see [Polkovnikov Ann.of.Phys. (2010)] and, in particular, to compute out-of-time ordered correlators.

To do so, we consider the Bopp representation of $\hat B(t)$ [cf. Eq.(3)] and the corresponding one for $ \hat A(0) \to A_0 + \hbar D_{A_0}^{(1)}+ \hbar^2 D_{A_0}^{(2)} $.

To compute the semi-classical limit the echo observable and the square commutator, we evaluate the Weyl symbol of several correlation functions, e.g.

\[ \left( \hat B(t) \, \hat A \, \hat B(t) \, \right)_w = (B_t + D^{(1)}_{B_t} + D_{B_t}^{(2)})\, (A_0 + D^{(1)}_{A_0} + D_{A_0}^{(2)})\, B_t \ , \]

and we simplify the resulting expressions. After a tedious calculation, the Weyl symbol of the echo observable $\mu(t) = -1/2 \langle [\hat B(t), [\hat B(t), \hat A]]\rangle $ reads

\[ \left ( [ \hat B(t), [\hat B(t), \hat A(0)]\, ]\right )_w = \hbar^2\left [3 \left(D_{B_t}^{(1)} \right )^2 A_0 - D_{B_t}^{(2)}\, B_t A_0 + D_{A_0}^{(2)} B_t^2 + A_0 \, D_{B_t}^{(2)} B_t \right ]\ , \quad \quad (4) \]

while for the square commutator $c(t) = - \langle [ \hat B(t), \hat A(0)] \, ]^2\rangle$ one finds

\[ -\left ( [ \hat B(t), \hat A(0)]\, ]^2 \right )_w = - 4 \hbar^2\, \left ( D_{A_0}^{(1)} B_t \right )^2 = \hbar^2\, \{ A_0, B_t\}^2 \ . \quad \quad (5) \]

It is well known that the classical limit of the square commutator [cf. Eq.(5)] encodes the square of the derivatives of the classical trajectory to respect to the initial conditions (see Refs.[48-50] in the manuscript). This means that, whenever the classical limit is chaotic, $c(t)$ is expected to grow exponentially, with a rate given by twice the largest Lyapunov exponent. Let us consider a simple example by choosing $\hat A(0) = \hat a^2(0)$ and $\hat B(t) = \hat a^{\dagger}(t)$, for which

\[ \begin{align*} B_t& =\alpha_t^\ast\ , \quad D^{(1)}_{B_t} = - \frac 1{2} \, \frac{\partial }{\hbar \partial \alpha_t} \ , \quad D^{(2)}_{B_t}=0 \\ A_0& =\alpha^2(0)\ , \quad D^{(1)}_{A_0} = {\alpha(0)}\frac{\partial}{ \hbar\partial \alpha(0)^{\ast}}\ , \quad D^{(2)}_{A_0}=\frac 1{4} \frac{\partial^2}{\hbar^2 \partial \alpha(0)^{\ast\, 2}} \ . \end{align*} \]

It is then easy to show that Eq.(5) simply gives $ c(t) \to -4 \alpha^2(0)\left( \frac{ \partial \alpha^\ast(t)}{ \partial \alpha^{\ast}(0)} \right)^2 $. We now show that the same result applies to the semi-classical limit of the echo observable (4), for instance considering the previous example. Substituting the Bopp representation for $\hat A(0) = \hat a^2(0)$ and $\hat B(t) = \hat a^{\dagger}(t)$ into Eq.(4), and using the chain rule for the second-order derivatives, one gets

\[ \left ( [ \hat a^{\dagger}(t), [\hat a^{\dagger}(t), \hat a^2(0)]\, ]\right )_w = \frac \hbar2 \left [ 3 \left ( \frac{\partial\alpha(0)}{\partial \alpha(t)}\right )^2 + \left ( \frac{\partial\alpha^*(t)}{\partial \alpha^*(0)}\right )^2 + 3 \alpha(0)\frac{\partial^2 \alpha(0)}{\partial \alpha^2(t)} + \alpha^\ast(t)\frac {\partial^2 \alpha^{\ast}(t)}{\partial \alpha^{\ast\, 2}(0)} \right ] \ , \]

which, exactly as the square commutator, is dominated by the square of the derivatives of the classical trajectory to respect to the initial conditions. In the new Appendix B, we discuss also the Bopp representation of spin variables $\hat A=\hat B = \hat S^z$ and we show that, also in this case, the semiclassical echo observable is proportional to the square of the derivatives of classical spin trajectory $S^z_t$ to respect to the initial conditions $S_0^{x,y,z}$.

2) We thank the referee for the question and we acknowledge that the previous version of the manuscript was intended to readers who already had familiarity with the method. First of all, to improve the explanation of the TWA, we have added in the new version a broader introduction of the method, considering as phase-space example the coherent state representation. We explained how to use it to compute time-dependent expectation values and multi-time correlation functions. Furthermore, since TWA can be derived as the saddle point approximation of the Keldysh path-integral, we have added a new Appendix in which we sketch the steps of the proof. In the second place, we better discuss the justification of TWA for the SK model comes from the validity of this saddle point approximation due to the mean-field nature of the fully-interacting SK model. We made an effort also to justify this both in the appendix and in the section devoted to the TWA on the SK.

3) We thank the referee for this question since a large part of our results is based on the use of TWA. We have added the following discussion in the new version of the manuscript. As we have argued in the paper, the quantum exponential growth is restricted in the time interval ${t^* \leq t \leq t_{\text{Ehr}}\sim\log N}$. Therefore, in order to fully appreciate it numerically, one would need to simulate very big system sizes, being $\log N$ a very slow function of its argument. Since exact diagonalization is limited to very small system sizes, e.g. $[\log(8):\log(20)] \sim [0.9: 1.3]$, a different numerical approach is needed to reproduce the behaviour for large $N$. As we show numerically in Section 5.1 and now justify in Appendix B, for the model under analysis TWA meets this need, reproducing the exact quantum dynamics for big $N$ before the Ehrenfest time $t_{\text{Ehr}}$. Furthermore, TWA is more accurate in extracting the Lyapunov exponent $\Lambda$ for the additional two reasons: 1) TWA does not know about the Ehrenfest time (a fully quantum time-scale) and its exponential growth lasts for many decades. 2) In TWA $\Lambda$ becomes independent on the system size even for relatively small $N$, allowing a precise estimate. Fig.5 is used to show the usefulness of this approach. The ED data grow exponentially only at short times where they coincide with TWA (before $t\sim1.5$), see also Fig.4 (i). While the TWA data continue to grow - with the same rate - for a few decades allowing to accurately extract $\Lambda$. To summarize this discussion the TWA for the echo indeed breaks down at relatively short times unless $N$ is huge. But it breaks down in a smart way predicting what quantum dynamics would look like of $N$ becomes exponentially large. While this result seems to be paradoxical it is correct and not incidental. By our arguments, it should apply to any large N model, which has a diverging Ehrenfest time. This loosely follows from the fact that the main role of $N$ in dynamics is to set the value of $\hbar$, other corrections due to finite $N$ are small an very quickly disappear as $N$ becomes moderately large, of the order of 10. So the semiclassical-classical TWA dynamics effectively extrapolate $\hbar\to 0$ and is very efficient if we are interested in this limit.

4) Reply to the minor comments:

  • We have added the citations.

  • We have added further references.

  • We thank the referee for pointing out a mistake in the notations. Generally, if one assumes that the time-evolution of a quantum state with an hamiltonian $\hat H$ is $e^{-i \hat H t}\ket{\psi_0}$, then the Heisenberg representation for the operator $\hat A$ is $\hat A(t) = e^{i \hat H t} A e^{-i \hat Ht}$. Anyhow, right after Eq.(1), it was stated that \emph{``$\hat B(t) = e^{ -i\hat H t} \hat B \, e^{ i\hat H t}$ is the perturbing operator in the Heisenberg representation with respect to the forward Hamiltonian $\hat H$''}, which is therefore not correct. We correct this by first performing the evolution with the backward hamiltonian $-H$, then we consider the small rotation and then evolve with the reversed hamiltonian $\hat H.$

  • We explained in what sense Eq.(2) contains the OTOC, see reply to the point (1).

  • We added references.

  • We have added justification for the use of TWA for the SK model in a new Appendix B and made an effort to make the section more clear.

  • We removed the `and hence the otoc' since it is not necessary for this part.

Reply to referee report II:

We thank the referee for her/his interesting comments. Here we reply to the corresponding points of their report, specifying the changes included in the resubmitted manuscript.

1) We thank the referee for its comment. Since a similar issue has been raised also by the Referee 1, we have added few lines in the introduction, improved the discussion in Section 2 and included a new small appendix with the details of the classical limit of the echo observables.

First of all, let us specify the general meaning that one can associate to OTOC. Even if in recent literature the name OTOC is usually related to correlators as $\langle \hat A \hat B(t) \hat A \hat B(t) \rangle$, "out of time-ordered correlators" OTOC can be referred to all the time-dependent correlation functions which have an unusual time-ordering. Namely, they describe correlation functions which cannot be computed with conventional techniques that assume casual time evolution and therefore need to be evaluated via extended Keldysh contours, like the one developed in [Aleiner, Faoro, Ioffe, Annals of Physics 2016]. In this sense, the echo observable $\mu(t)$ can be refereed as OTOC, since it contains terms as $\langle \hat B(t) \hat A \hat B(t) \rangle$, see Eq.(4). In the second place, in the semi-classical limit, every kind of OTOC contains quadratic terms in the derivatives of the initial conditions, as was already known by Larkin and Ovchinnikov. This is true also for the echo $\mu(t)$. This statement can be shown by the use of time-dependent {Bopp operators}, a standard formalism of phase-space methods which allows computing time-dependent correlation functions, by representing quantum operators as phase-space variables. This is extensively discussed in the Appendix of [Schmitt, Sels, Kehrein and Polkovnikov, PRB 2019] for a semi-classical limit of the SYK. Anyhow, in order to be as self-consistent as possible, we have added a new Appendix to the manuscript, with the derivation of the semi-classical limit of $\mu(t)$, based on Bopp operators. We show for bosonic and spin operators, that the semi-classical echo observable contains quadratic terms in the derivatives of the classical trajectory to respect to the initial conditions, hence it encodes the Lyapunov exponent, exactly as the square commutator. For further details on this derivation, we refer to the response (1) to the Report 1, where we have reported the main steps of the proof.

In conclusion, we acknowledge that the square commutator $c(t) = - \langle [\hat B(t), \hat A]^2 \rangle$ contains a term $\langle B(t)A^2 B(t)\rangle$, which is missed by $\mu(t)$. For this reason, we added a sentence in Section 2. Since we find this very interesting, we would like to ask if the Referee could kindly point us the reference where "the terms that make up for their difference have been shown to be crucial in obtaining the exponential growth of the former as the difference between parametrically larger components".

2) We thank the referee for this comment since this point is indeed a non-trivial statement and it is exactly one of the purposes of the paper. If we believe that there are no other initial time scales, our claim can be re-stated as follows: If there exist a time-interval $t^{\ast}\leq t \leq t_{\text{Ehr}}$, where $t_{\text{Ehr}}$ diverges in the thermodynamic limit, then in the same interval the $\mu(t)$ should be semi-classical. Therefore it behaves like the square of the derivatives to respect to the initial conditions, see also the Refs.[16-18] of the manuscript and the new Appendix A. When the latter grows exponentially with a rate given by the classical (generalized) Lyapunov exponent, then - in that time interval - also the quantum $\mu(t)$ should grow exponentially with the same rate. On the other hand, when the system does not have a semi-classical analogue, i.e. $t_{\text{Ehr}}\sim t^{\ast}$ does not depend on the system size, there is no time-regime where the quantum $\mu(t)$ would behave semi-classically. Therefore there is no reason in principle to have exponential growth. This is typically the case of short-range interacting lattice systems, which do not have a classical analogue and for which it has been proved that the square-commutator grows at most exponentially fast [Kukulian et al.]. This statement is firmly supported by our numerics on long-range (semi-classical) and short-range (fully-quantum) models.

3) We thank the Referee for this comment since our previous version did not provide the underlying argument to our statements. We have added a discussion and a new appendix in order to better justify our claims. The justification can be summarized as follows. TWA can be proved as the saddle point of the Keldysh action in a path-integral formulation of the time-evolutions. Within this framework, representation of initial conditions through the Wigner function is exact, while the approximation occurs at the level of the time-evolution, which is solved by the saddle-point approximation expanding in small quantum fluctuations. Even if the present work deals with spins $1/2$, this all-to-all interacting model has a mean-field character and the large $N$-limit ensures the validity of the saddle point approximation, as confirmed by the numerics. More formally (see Appendix B), one can estimate the corrections to the saddle-point solution, which are found to scale as $1/N$.

4) We thank the referee for highlighting this point, since the presence of non-local interactions, shared by the SYK and SK models, is indeed crucial to have a semi-classical limit and hence to observe the exponential growth of the OTOC. Even if in the previous version we already pointed out that the models share several analogies, we have now added a sentence to signal that the SYK and SK differ in many respects. For example, they have different equilibrium phase-diagrams, which inevitably lead to differences in the dynamics. This is particularly evident at low temperatures, where the SK model possesses a glassy phase below a critical transverse field, i.e. see [Andreanov and Muller, PRL 2012]. On the other hand, such a phase-transition is absent in the SYK model. Anyhow, as pointed out by the Referee, the two models share the feature of having non-local all-to-all random interaction. Indeed, a semi-classical analysis was performed by one of us on the SYK using the TWA on fermionic bilinears, see [Schmitt, Sels, Kehrein and Polkovnikov, PRB 2019], where the authors found similar results concerning the exponential growth of the echo. In that paper, the role of the short-range interactions was not addressed, and the analysis presented in the present work shows how the non-locality of the interactions affects the classical limit (even on a system of spins $1/2$) and therefore the exponential growth. Also in that paper, there was no analysis of the observables and the echo operator and the choice, which satisfies the criteria we derived here was rather coincidental. We believe that finding these conditions is another important new result of the present work.

5) We changed the figure's captions and added the ordinate axis where missing. We made an effort to improve the presentation, correcting misspellings and changing some words.

List of changes

- We re-wrote the introduction in a more cohesive manner.
- We added a small comparison between the SK and SYK models.
- We added a new appendix containing the derivation of the semi-classical limit of the echo observable and the square commutator.
- We added a new appendix containing the derivation of TWA and its validity for the SK model.
- We re-wrote the introduction to TWA in Section 5.
- We changed the caption of the pictures and added ordinate labels when missing. In Fig.4, we added an exponential function to guide the reader's eyes to the thermodynamic limit.
- We improved the discussion of the numerical findings in Section 6.
- Updated bibliography.


Reports on this Submission

Anonymous Report 1 on 2020-2-28 (Invited Report)

Strengths

1-timely subject (OTOCs)
2-cutting edge analytics and semiclassics
3-Relevant results about time-scales and possibility to observe exponetial growth of the OTOC.

Weaknesses

1-OTOC considered is not the most widespread quantity

Report

The authors have answered the concerns of the referees rather carefully and I think the manuscript in the present form is a significant contribution to the subject of characterizing chaos in complex quantum systems. I recommend publication.

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

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Comments

Silvia Pappalardi  on 2020-02-17  [id 740]

Category:
correction

We would like to correct two typos present in the Reply to the report I:

  • In question 1), the 8-th formula from the top of the reply reads:
\[ \left ( [ \hat a^{\dagger}(t), [\hat a^{\dagger}(t), \hat a^2(0)]\, ]\right )_w = 3 \left ( \frac{\partial\alpha(0)}{\partial \alpha(t)}\right )^2 + \left ( \frac{\partial\alpha^*(t)}{\partial \alpha^*(0)}\right )^2 + 3 \alpha(0)\frac{\partial^2 \alpha(0)}{\partial \alpha^2(t)} + \alpha^\ast(t)\frac {\partial^2 \alpha^{\ast}(t)}{\partial \alpha^{\ast\, 2}(0)} \ . \]
  • In question 4), the third point to the reply to the minor remarks reads: "We thank the referee for pointing out a mistake in the notations. Generally, if one assumes that the time-evolution of a quantum state with an hamiltonian $\hat H$ is $e^{-i \hat H t}|\psi_0\rangle$, then..."

Yours sincerly, Silvia Pappalardi, Anatoli Polkovnikov and Alessandro Silva