Dissipation-enhanced scrambling in the SYK model coupled to a lossy cavity

Abstract: We study the Yukawa-Sachdev-Ye-Kitaev model, a disordered model of $N$ Majorana fermions and $R=γN$ bosons in which the bosons are linearly coupled to independent realizations of SYK $p$-body interactions, in the presence of dissipation, modeled by a Lindblad master equation. Motivated by recent proposals for implementing SYK models in quantum simulators, we focus on bosonic leakage at rate $κ$. Initializing the system in the steady state, we analyze the late-time fermionic relaxation rate and the Lyapunov exponent, solving the large-$N$ theory both numerically and for $p$ large, finding a rich landscape of dynamical behaviors. Most notably, the Lyapunov exponent remains positive for every value of $κ$ and, for $p&gt;2$, can even grow as $κ$ increases. The QED case $p=2$, which lies between the fully chaotic regime $p&gt;2$ and the integrable case $p=1$, exhibits special features. We also identify a critical value of the boson-to-fermion ratio $γ_c \approx 2/p<sup>2$ separating distinct dynamical regimes.

• The paper investigates how disordered dissipative dynamics from boson loss in cavity quantum-electrodynamics settings affect scrambling behaviour in the Yukawa–Sachdev–Ye–Kitaev model.
• The observed enhancement of Lyapunov exponent known as scrambling for p>2 matches unexpected positivity for $\gamma>2 / p^2$, showing that dissipation-induced scrambling can be controlled for chaotic quantum dynamics
• This phenomenon is attributed to an attractive contribution from bath quantum fluctuations

Overview and motivation

This paper by Pelliconi, Lapierre, and Ryu studies the Yukawa–Sachdev–Ye–Kitaev (YSYK) model — $N$ Majorana fermions linearly coupled to $R=\gamma N$ bosonic modes through disordered $p$-body interactions — subject to Markovian boson loss at rate $\kappa$, described by Lindblad jump operators $L_\mu=\sqrt{\kappa}\,a_\mu$ (2608.19310). The setting is motivated by cavity-QED proposals for simulating SYK physics, in which dispersive bosons mediate the effective fermionic interactions while unavoidably leaking out of the apparatus. The central question is how such dissipation affects two chaotic observables: the late-time fermionic relaxation rate $\Gamma$ and the Lyapunov exponent $\lambda$ extracted from OTOCs.

The main result contradicts the prevailing expectation from prior work on dissipative SYK models, where sufficiently strong dissipation drives $\lambda$ negative. Here the Lyapunov exponent remains positive for every value of $\kappa$, and for $p>2$ it can be enhanced at intermediate dissipation. The authors trace this to quantum fluctuations of the dissipative bath, encoded in the bosonic Keldysh propagator, which enter the ladder kernel as an attractive contact term.

Setup: dissipative YSYK on the Schwinger–Keldysh contour

The Hamiltonian couples $R=\gamma N$0 harmonic oscillators of frequency $R=\gamma N$1 to Majorana fermions via random Yukawa vertices $R=\gamma N$2 with variance $R=\gamma N$3. The dynamics is governed by a Lindblad master equation with photon-loss jump operators. The disorder-averaged real-time partition function is formulated as a bilocal collective-field theory on a two-fold Schwinger–Keldysh contour, with fields $R=\gamma N$4 and $R=\gamma N$5 and self-energies $R=\gamma N$6, $R=\gamma N$7. The saddle-point (Schwinger–Dyson) equations are melonic and close self-consistently; the bosonic retarded propagator carries the loss rate explicitly through poles at $R=\gamma N$8.

The steady state is computed perturbatively in $R=\gamma N$9: at leading order the bosons sit in their vacuum while the fermions are maximally mixed (infinite temperature), with an off-diagonal correction of order $p$0. The analysis is carried out in the auxiliary boson limit $p$1, where integrating out the bosons would generate an effective SYK$p$2 interaction; crucially, however, the bosons are retained explicitly because their dissipative fluctuations are the focus.

Late-time relaxation rate

Solving the large-$p$3 Schwinger–Dyson equations reduces the problem to a Liouville equation for $p$4, defined through $p$5, supplemented by an attractive contact term whose sign is fixed by complete positivity of the Lindbladian evolution. The solution yields

$p$6

with two physically interpretable energy scales:

Scale	Expression	$p$7-dependence	$p$8-dependence
Effective coupling $p$9	$\kappa$0	monotonically decreasing, $\kappa$1 at large $\kappa$2	$\kappa$3
Purcell rate $\kappa$4	$\kappa$5	non-monotonic, peaked at $\kappa$6, $\kappa$7	linear

Both scales are familiar from quantum optics: adiabatic elimination of a dissipatively coupled qubit–cavity system produces precisely these forms, corresponding respectively to virtual-photon-mediated coherent exchange and Purcell decay. The distinct $\kappa$8 scalings have a simple origin: independent disorder realizations add incoherently in variance for $\kappa$9, whereas decay channels add linearly for $L_\mu=\sqrt{\kappa}\,a_\mu$0.

Expanding $L_\mu=\sqrt{\kappa}\,a_\mu$1 at small $L_\mu=\sqrt{\kappa}\,a_\mu$2 shows that its monotonicity is controlled entirely by $L_\mu=\sqrt{\kappa}\,a_\mu$3: the relaxation rate is monotonic for $L_\mu=\sqrt{\kappa}\,a_\mu$4 and non-monotonic for $L_\mu=\sqrt{\kappa}\,a_\mu$5, defining a critical ratio $L_\mu=\sqrt{\kappa}\,a_\mu$6. Numerical solution of the full Schwinger–Dyson equations at $L_\mu=\sqrt{\kappa}\,a_\mu$7 (with damping parameter $L_\mu=\sqrt{\kappa}\,a_\mu$8) agrees with the large-$L_\mu=\sqrt{\kappa}\,a_\mu$9 formula to within roughly 10% even though the formula was derived at $\Gamma$0 — a notable robustness, though one should note the comparison is made at parameters chosen so the auxiliary approximation holds.

Dissipative Lyapunov exponent

The OTOC is represented on a four-fold Keldysh contour, and its connected $\Gamma$1 piece satisfies a Bethe–Salpeter equation with a ladder kernel comprising a bosonic kernel (a propagating boson connecting the rails) and a fermionic kernel (purely fermionic rungs with non-local retarded bosonic rails). At late times the eigenvalue condition $\Gamma$2 maps, via the exponential ansatz $\Gamma$3, onto a Schrödinger problem with a Pöschl–Teller potential plus an attractive $\Gamma$4-function contact term proportional to $\Gamma$5, where $\Gamma$6.

The bound-state energy gives the central analytic result:

$\Gamma$7

Three regimes follow:

• Small dissipation: $\Gamma$8 equals the unitary YSYK value at $\Gamma$9 and is suppressed linearly in $\lambda$0, consistent with the general expectation that weak dissipation inhibits scrambling.
• Large dissipation ($\lambda$1): for $\lambda$2, $\lambda$3, remaining positive but vanishing as $\lambda$4. For $\lambda$5, $\lambda$6, with the leading coefficient independent of $\lambda$7 due to a cancellation between the $\lambda$8 and $\lambda$9 scalings.
• Intermediate dissipation ($\lambda$0): for $\lambda$1 and $\lambda$2, $\lambda$3 develops a maximum at $\lambda$4, where $\lambda$5. This is the dissipation-enhanced scrambling regime: bath quantum fluctuations, entering as the attractive contact term, outweigh the suppression of $\lambda$6.

Two structural observations deserve emphasis. First, the critical ratio $\lambda$7 coincides, including the $\lambda$8 prefactor, with the threshold separating monotonic from non-monotonic behavior of the relaxation rate — remarkable given that $\lambda$9 and $\kappa$0 are distinct observables. Second, the formula correctly returns a non-scrambling answer for the integrable case $\kappa$1, although strictly the four-point function vanishes identically there since the model is quadratic.

Numerically, the authors solve the ladder-kernel eigenvalue problem by power iteration on $\kappa$2 in frequency space, tuning $\kappa$3 until the largest eigenvalue reaches unity. For $\kappa$4 the fit $\kappa$5 agrees with the large-$\kappa$6 coefficient $\kappa$7 to about 10%. For $\kappa$8 at fixed $\kappa$9, the numerics confirm the predicted enhancement at intermediate $p>2$0 with excellent agreement against the analytic curve.

Limitations and open questions

Several caveats qualify these results. The analytic formulas are controlled only in the strict large-$p>2$1 auxiliary limit; the derivation of the Pöschl–Teller problem involves approximations (e.g., inconsistent treatment of factors $p>2$2) that are mutually valid only at $p>2$3, and finite-$p>2$4 validity is established empirically by comparison with numerics rather than systematically. The numerical checks use parameters chosen so the auxiliary approximation applies, leaving the genuinely non-auxiliary regime ($p>2$5) unexplored. Only boson leakage is treated; other experimentally relevant channels such as incoherent photon scattering involve disordered jump operators that are difficult to incorporate in the path-integral formalism. The special status of $p>2$6 — where the adiabatically eliminated jump operator is quadratic in the fermions, versus genuinely many-body for $p>2$7 — lacks a deeper explanation, particularly since no general framework connects operator growth to the Lyapunov exponent in non-unitary systems. Finally, the coincidence of $p>2$8 across both observables suggests a possible interpretation in terms of double-scaled variables (the critical boson number can be written $p>2$9), but a double-scaled formulation of dissipative YSYK remains to be constructed.

Conclusion

This paper provides an analytically tractable example of a dissipative many-body chaotic system in which the dissipative degrees of freedom also mediate the interactions. Its principal finding is that there is no universal dissipation threshold beyond which scrambling is destroyed: the Lyapunov exponent stays positive at all $R=\gamma N$00, and for $R=\gamma N$01 with $R=\gamma N$02 it is enhanced by bath quantum fluctuations at intermediate loss rates. This establishes a mechanism for "engineered scrambling" controlled by cavity dissipation, directly relevant to cQED implementations of SYK models, and identifies $R=\gamma N$03 as a sharp organizing scale shared by relaxation and scrambling diagnostics.