Beyond Fermi's golden rule with the statistical Jacobi approximation

We thank Prof. Khaymovich for his detailed report. We have incorporated most of his comments and suggested changes into the revised manuscript, and believe that they help improve the paper. Those suggested changes which we have not included, we will directly address below.

We address Prof. Khaymovich's specific comments in the order he made them. We will number our responses as he did, beginning with 0 (with 8 sub-points) and ending with 19.

0.1: Systems with anomalous relaxation, other than exponential decay. All the systems we know of with this property (including those listed by Prof. Khaymovich) are in the sparse regime. The Frobenius norm of a row or column in the Hamiltonian for these models is dominated by a vanishing fraction of the elements. Our current manuscript only concerns the dense regime, but we expect that the analysis of our previous paper, Ref. [39] (now Ref. [42] in the revised version), should apply in some parameter regime of those models. (Although, in the context of (generalized) Rosenzweig-Porter models, the ratio of the Frobenius norm of the off diagonal to that of the diagonal can also vanish. We have not considered this scenario in either this paper or Ref. [39].) The remark on page 19 concerned a different scenario, where the model remains in the dense regime, but the spectral function of the perturbation diverges at small frequencies. We have not analyzed this case, nor the case where the spectral function vanishes at small frequencies, so we do not comment on these scenarios in detail. We have included a comment summarizing the above discussion to section 2.2, and included the recommended references.

0.2: Applicability to other integrable models. Our analysis should apply to any model which begins in the dense regime of the SJA. While exceptions exist, our understanding is that most integrability-breaking perturbations of integrable models tend to be in the dense regime. (One important exception is nearly localized systems, for which local perturbations are sparse.) Our analysis will always predict asymptotic decay of the fidelity, though there can be more complicated dynamics at short times. Revivals should require some phase coherence in the evolution, which we expect to be washed out at large system sizes in the dense regime. There is likely a broad array of behavior which can occur for integrability preserving perturbations. In this case, our analysis could only apply to an initial microcanonical state in the quench, not actual eigenstates.

0.3: Comparison with Lea Santos. Much of the analysis of Santos et al. is focused on long times, beyond the cutoff time $t^*$. For instance, the correlation hole is a feature of the turnover from the intermediate time decay of the fidelity to the late time saturation. Our analysis does not capture any feature at or beyond $t^*$, but we believe it should describe the system well up to that cutoff. The discussion of generic features of the fidelity in the introduction should correctly represent the conclusions of Prof. Santos's work, and already includes citations to her papers on the topic. In the revised manuscript, we more explicitly point out that the "dip and ramp" feature near $t^*$ corresponds to the correlation hole.

0.4: Comparison with the Wigner-Weisskopf approximation. As we understand it, the Wigner-Weisskopf approximation refers to a particular analysis of fidelity decay where a kind of Markovian approximation is made, which allows simple differential equations for the log-fidelity to be derived. The analysis we have referred to as "TDPT" reproduces the same result as the Wigner-Weisskopf approximation as the leading order of perturbation theory for the log-fidelity. Explicitly, the real part of Eq. (56) in the Monthus reference cited by Prof. Khaymovich should be compared to Sec. 4.3.1 in our manuscript. Thus, the comparison of the SJA to the Wigner-Weisskopf approximation is the same as its comparison to TDPT. Qualitative features are reproduced by TDPT/Wigner-Weisskopf, but the SJA shows quantitative improvement for larger perturbations. In the Monthus reference, careful attention is paid to the competition between the broadening of the LDOS compared to the level spacing. No such competition exists in the models we consider, where the level spacing is asymptotically much smaller than the LDOS width, justifying the continuum approximation for the LDOS. As such, the unusual features of fidelity decay that Monthus uncovers with the Wigner-Weisskopf approximation do not appear in our analysis.

0.5: Validity conditions for the assumptions in Table 1. We have changed Table 1 to include a column stating the interpretation of each approximation. We hope that this provides more context for when each approximation can be expected to hold. As stated in the caption, all approximations should be valid for a large many-body system which satisfied the eigenstate thermalization hypothesis, and at times before the cutoff time $t^*$.

0.6: Saturation point for the fidelity. One might attempt to place the saturation point of the fidelity by comparing the log-fidelity to the diagonal entropy of the initial state in the Hamiltonian eigenbasis. This is essentially what we have done, with an additional assumption of ergodicity to pass from the diagonal entropy to the usual thermodynamic entropy at the specified energy density. Even in the ergodic case, this prediction for the saturation time is not precise. Roughly, we are estimating the peak height of the LDOS (saturated fidelity) from its width (entropy). Still, this should give a good heuristic. We have added this discussion to a footnote on page 20 of the revised manuscript.

0.7: Comparison to Dyson Brownian motion. The comparison of Jacobi to Dyson Brownian motion is currently unexplored. One may observe that the Jacobi algorithm causes levels connected by a matrix element $w$ to repel with a kick which is, for $w\ll \omega$, given by $w^2/\omega$, as is the case for Dyson Brownian motion. However, this is the extent to which we understand the two processes are related. As such, we are not in a position to address questions related to explicit computations that one could perform. We have revised this point in the discussion to make it clearer that this relation is conjectural. Making a more quantitative connection between the dynamics on the diagonal induced by the Jacobi algorithm and the dynamics of the eigenvalues in Dyson Brownian motion is an open direction for research.

0.8: Comparison to Ref. [25]. We have included a brief intermediate comment about strong disorder renormalization and its relation to the SJA.

1: The slope of the magenta curve in Fig. 1b matches the black curve at late times. This is coincidence. The same data is plotted for longer times in Fig. 10. While we don't understand the late time behavior of the data well, it is not consistent with either SJA or TDPT. We also note that each of these methods aims to find the full log fidelity curve, not just the slope.

2: Scaling of $\rho_{\mathrm{dec}}$. The definition of $\rho_{\mathrm{dec}}$ is stated in Eq. (15). Equation (16) may be regarded as a normalization condition, though it is a consequence of the definition. Note that $\rho_{\mathrm{dec}}$ is a number density that counts all rotations performed on any state in the Hilbert space, and so naturally grows when the Hilbert space dimension is increased. Perhaps the confusing point is that the natural normalization of $\rho_{\mathrm{dec}}$ is in terms of its second moment, not its integral (zeroth moment). In fact, we expect that the integral of $\rho_{\mathrm{dec}}$ can diverge even at fixed $N$. That is, the Jacobi algorithm might never bring the Hamiltonian to an exactly diagonal form in a finite number of rotations. All that is guaranteed is that the Frobenius norm of the diagonal decreases rapidly.

3: Missing factor of two in Eq. (13). We took the square root of both sides in Eq. (13), which is why the exponent is factor of two smaller.

4: Definition of $\eta$. We are using the same definition as in the previous section. We have included the relevant equation immediately after Eq. (23).

5: Slowly varying assumption. We expect this assumption to be valid whenever we are in the dense regime. This is precisely saying that many rotations are necessary to appreciably change the Hamiltonian, and so any individual rotation typically does little.

6: Cross terms. We now state that we need the average of the cross terms to be small in comparison to the diagonal terms, rather than that they average to zero.

7.:Approximation in Eq. (52). We have corrected the error term in Eq. (52). The next terms are on the order of $w^3 t^2$ and $w^3 \tau^2$. There is an important order of limits where we take w to zero quickly enough in comparison to the increase in t and tau for these terms to be negligible.

8: Error plots in Fig. 7. We maintain that the absolute difference plots in Fig. 8 are meaningful. Note that the Jacobi autocorrelator passes through zero, which makes the relative error poorly behaved. The Jacobi spectral function is positive, but the bare spectral function is extremely small in the tails, which also causes the relative error to be large. This large relative error does not have anything to do with the log-fidelity, which does not depend much on the behavior of the Jacobi spectral function in the large omega tails. It may help to observe that all axes in Fig. 7 are plotted in terms of dimensionless ratios. None of the plots may be freely scaled by a change of units or increasing system size, making the smallness or largeness of the absolute difference compared to 1 a meaningful assessment of the change in the spectral function and autocorrelator as $J/\sigma_\omega$ is increased.

9: Replacing the sum over $\sin^2(\eta/2)$. We have expanded on the discussion in that paragraph. The particular replacement we made was essentially a small angle expansion of $\sin^2(\eta/2)$.

10: Difference between $C_{\mathrm{Jac}}$ and $C_V$. If it is available, one should always use $C_{\mathrm{Jac}}$. $C_V$ may be considered to be the leading order term in an expansion of $C_{\mathrm{Jac}}$ in the perturbation strength $J/\sigma_\omega$. The paragraph Prof. Khaymovich mentions seems to have not been updated from a previous version of the paper. We use $C_{\mathrm{Jac}}$ in both subsections. This has now been corrected.

11: The random matrix model. The perturbing off-diagonal matrix in this model is given by the ETH ansatz for the matrix elements of a local operator in the eigenbasis of a thermalizing Hamiltonian. We are not aware of any in-depth study on the properties of this model from the random matrix perspective, but we have made a comment on some readily apparent features.

12: System sizes in Fig. 8. We have stopped at $N=2^{13}$ as this is already sufficient to observe the asymptotic behavior we seek to describe. There is no fundamental impediment to going to larger sizes, only numerical effort. Also, the many-body numerics are performed in a fixed symmetry sector, so they do not actually use $2^{16}$ states.

13: Missing factor of 2. This has been corrected.

14: Damped oscillations. The invisibility of the damped oscillations in Fig. 8 may be due more to the scale of the plot. We think that the oscillations are visible in panel (b), though they are slightly disguised by the overall decay of the log-fidelity curve. We have included an inset in panel (a) which shows the damped oscillations. One may also explicitly see the oscillatory term in the second line of what is now Eq. (81) in the revised manuscript.

15: Physical origin of discrepancy between SJA and TDPT. The SJA prediction for the log-fidelity may be thought of as a resummation of a particular series in perturbation theory. As such, the physical origin of the difference between SJA and (first order) TDPT would be the occurrence of multiple virtual transitions dressing the eigenstates.

16: Calculating $C_{\mathrm{Jac}}$ needs the eigenvalues. This is not true. The $E_{a_n}$ terms appearing in Eq. (79) are the expectation value of the Hamiltonian in the Jacobi basis at step n. Only when n becomes very large (roughly $O(N^2)$) will this be a good approximation to the actual eigenvalue. Further, $C_{\mathrm{Jac}}$ is not sensitive to this precise value, but only to the density of states (properly, the LDOS). This is much coarser information, and one could in principle obtain it without doing complete diagonalization. It is true that, in running the Jacobi algorithm numerically exactly, we are essentially performing an exact diagonalization calculation to compute $C_{\mathrm{Jac}}$. It is an open avenue of research to predict $C_{\mathrm{Jac}}$ from a microscopic definition of the model, and we are currently pursuing this. We have also clarified these points with an additional paragraph in the discussion. Lastly, we note that, in the random matrix model, the SJA prediction seems to converge to the infinite $N$ limit much faster than the exact calculation. There is thus a significant saving in system size. The complexity of the Jaocbi algorithm as a numerical technique for exact diagonalization is discussed in Sec. 2.1. It is $O(N^3)$, similar to most other simple diagonalization algorithms.

17: Footnote on page 22. The footnote has been brought into the main text.

18: System size. The system size $L$ is a parameter of the model, not a fitting parameter. It cannot be freely chosen so as to optimize the accuracy of the SJA prediction. One must calculate $C_{\mathrm{Jac}}$ for the system size being studied, and then compare to the exact dynamics in that same system size. We expect that the log fidelity density will have a finite thermodynamic limit at fixed $t$, so for large enough $L$ the system size no longer matters. But several of the models we studied are still far from this regime at the system sizes we could simulate.

19: No difference between SJA and TDPT in Fig. 9. Indeed, SJA and TDPT are very similar in Fig. 9. There is, of course, a regime in which TDPT works very well. In this regime, the SJA must necessarily reproduce TDPT in order to be correct. The purpose of Fig. 9 is to show that this is the case. SJA does as well as TDPT when TDPT works well, but also continues to work when TDPT fails (Figs. 8, 10).

We thank the Referee for their report and suggested changes. We believe that all of their comments have been addressed in the revised manuscript, and that the Referee should now find the paper appropriate for publication.

We respond to the Referee's specific comments below.

1: The error term in Eq. (3). We use $O(J/\sigma_E)$ in Eq. (3) to denote the scaling of the error at fixed $t$. A more complete statement of the error would be $[\log P_0(t)] = ... + O(J/\sigma_E) f(\sigma_\omega t)$, where$ f(x)$ is finite for $J/\sigma_E \to 0$, behaves as $f(x) \sim f''(0) x^2/2$ for short times, and grows linearly in $x$ for large $x$. This leading correction can in principle be calculated using the framework in Appendix A. We have included enough of this calculation in the new appendix A.4 to verify the statements we have made here, and summarize this information in the new Sec. 4.3.5.

2: Dimensions in Eq. (64). The error term has been corrected. It is meant to be $O(J^4/\sigma_\omega^4)$, as one should expect from the next order of TDPT. The time dependent correction here is again of the form $(J^4/\sigma_\omega^4) g(\sigma_\omega t)$, where $g(x)$ is a dimensionless function which is quadratic for small $x$, linear for large $x$, and finite for $J/\sigma_\omega \to 0$.

3: "$\sim$" symbol in Eq. (69). We are using "$\sim$" in the sense defined by, for instance, this article. That is, Eq. (69) is correct for $t \to 0$, even up to constant factors. The error would be given by the next order in a short time expansion of $C_{\mathrm{Jac}}^+(\tau)$. This is $-(J^2 t^4/12) \partial_\tau^2 C_{\mathrm{Jac}}^+(E_0, 0)$, and we have included this comment in the revised manuscript. We also corrected an error in this equation. The $\mathrm{sgn}(t)|\tau|$ term should have been $\mathrm{sgn}(t)\tau$, with no absolute values, as in the previous equation.