Comment on “Chaotic-Integrable Transition
in the Sachdev-Ye-Kitaev Model”

In their Letter [1], the authors studied a variant of the
Sachdev-Ye-Kitaev model with quadratic and quartic inter-
actions (also known as a mass-deformed SYK). They
claimed that the quantum Lyapunov exponent λL would
vanish below some critical temperature. In this Comment,
we show that this is not the case. We calculate λL exactly in
the perturbative regime where the temperature T and the
quartic coupling J are much smaller than the quadratic
coupling κ. At leading order in T/κ and J/κ, we find

$$
\frac{\lambda_L}{\kappa} = \frac{3T^2 J^2}{\kappa^4}. \qquad (1)
$$

Therefore, this model has no λL = 0 phase at small but
nonzero T, J.

To derive Eq. (1), we improve upon the standard
approach followed by the Letter’s authors and analyze
the leading eigenvalue problem of the ladder kernel, Eq. (9)
of the Letter:

$$
K = K_2 K_1, \quad (K_1 f)(t) = [\kappa^2 + 3J^2 G_{lr}^2(t)] f(t), \quad (2a)
$$

$$
(K_2 f)(\omega) = \left| G_R \left( \omega + i \frac{\lambda_L}{2} \right) \right|^2 f(\omega), \quad (2b)
$$

where $G_R$ and $G_{lr}$ are retarded and Wightman Green
functions of a fermion, respectively. Recall also that $\lambda_L$
is found by imposing that the largest eigenvalue of $K$ is 1.
We shall work perturbatively in $J/\kappa$ and $T/\kappa$ up to the
leading order. Such a perturbative expansion is controlled
since the quartic term is irrelevant in the low-$T$ limit where
the model is a Fermi liquid. Our analysis thus differs from
the Letter, which considered a perturbative expansion with
respect to the relevant term.

To start, we compute $G_{lr}$ and $G_R$ to an adequate order.
For $G_{lr}$, the conformal solution of the $J=0$ limit [2]
suffices:

$$
G_{lr}(t)^2 = \frac{T^2}{\kappa^2} \operatorname{sech}(t\pi T)^2. \qquad (3)
$$

For $|G_R|^2$, the $J=0$ conformal solution $G_R(\omega) = -\mathbf{i}/\kappa$ [2]
is not enough, and we shall incorporate the leading
correction. That comes from quasiparticle decay encoded
in the imaginary part of the self-energy $\Sigma_4(\tau) := J^2 G(\tau)^3$ in

<figure><img src="image_1.png" /><figcaption>FIG. 1. Temperature dependence of the Lyapunov exponent λL for various interaction strengths (markers) compared to the prediction, Eq. (1) (solid lines).</figcaption></figure>

Euclidean time. Continuing to real time, we find that the
retarded propagator Σ<sup>R</sup><sub>4</sub> satisfies

$$
-\mathrm{Im}\Sigma_4^R(\omega) = \frac{J^2 T^2}{2\kappa^3} + \frac{J^2 \omega^2}{2\pi^2 \kappa^3} \quad (4)
$$

at leading order in $J/\kappa$. Together with the Schwinger-Dyson equation $G_R(\omega)^{-1} = \omega - \kappa^2 G_R(\omega) - \Sigma_{4,R}(\omega)$, Eq. (4) implies that

$$
\kappa^2 \left| G_R \left( \omega + i \frac{\lambda_L}{2} \right) \right|^2 = 1 - \frac{\lambda_L}{2\kappa} - \frac{J^2 T^2}{2\kappa^4} - \frac{J^2}{2\pi^2 \kappa^4} \omega^2, \quad (5)
$$

where we also omitted higher orders of λL as we anticipated
it to be small.

With Eqs. (3) and (5), the kernel, Eq. (2), reduces to

$$
K = 1 - \frac{\lambda_L}{2\kappa} - \frac{J^2 T^2}{2\kappa^4} - \frac{J^2 T^2}{\kappa^4} \left[ -\frac{1}{2\pi^2 T^2} \partial_t^2 - 3\operatorname{sech}(t\pi T)^2 \right]. \quad (6)
$$

Upon rescaling $s = t\pi T$ in the parenthesis, we recognize the
1D Schrödinger Hamiltonian with a Pöschl-Teller potential:
$-\frac{1}{2}\partial_s^2 - 3\operatorname{sech}(s)^2$. Its ground state energy is $E_0 = -2$ [3].
Thus, the largest eigenvalue of $K$ equals $1 - \lambda_L/(2\kappa) +
3J^2T^2/(2\kappa^4)$. Imposing it to be 1 leads to Eq. (1).

We verified our prediction numerically in the large-N
limit by directly computing the leading eigenvalue of the
kernel, Eq. (2), where the Green functions are (nonpertur-
bative) numerical solutions of the real-time Schwinger-
Dyson equations [2]. We find a nice agreement in the
perturbative regime (see Fig. 1). Observe, however, that
λL/κ can become very small and could be mistaken for 0
due to numerical artifacts. Finally, we note that Refs. [4–6]
observed λL ~ T² > 0 in other large-N models of disor-
dered Fermi liquid.

<footnote>Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article's title, journal citation,
and DOI.</footnote>
We thank Jiabao Yang and Ehud Altman for helpful discussions. We acknowledge support from the U.S. DOE, Award No. DE-SC0019380, and the Gordon and Betty Moore Foundation’s EPIC initiative, Grant No. GBMF4545 (X. C.).

Jaewon Kim<sup>1</sup> and Xiangyu Cao<sup>2</sup>

<sup>1</sup>Department of Physics, University of California
Berkeley, California 94720, USA

<sup>2</sup>Laboratoire de Physique de l'Ecole Normale Supérieure
ENS, Université PSL, CNRS
Sorbonne Université
Université de Paris, 75005 Paris, France

Received 11 April 2020; revised 11 August 2020;
accepted 29 January 2021; published 11 March 2021

DOI: 10.1103/PhysRevLett.126.109101

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[2] J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D **94**, 106002 (2016).

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[4] S. Banerjee and E. Altman, Solvable model for a dynamical quantum phase transition from fast to slow scrambling, Phys. Rev. B **95**, 134302 (2017).

[5] H. Guo, Y. Gu, and S. Sachdev, Transport and chaos in lattice Sachdev-Ye-Kitaev models, Phys. Rev. B **100**, 045140 (2019).

[6] N. Sorokhaibam, Phase transition and chaos in charged SYK model, J. High Energy Phys. 07 (2020) 055.