Special role of p=2 in non-unitary operator growth

Investigate the special role of the quadratic interaction order p=2 in operator growth and its relationship to the Lyapunov exponent in non-unitary Yukawa–Sachdev–Ye–Kitaev dynamics.

Background

The paper finds that the p=2 Yukawa–SYK model behaves qualitatively differently from models with p>2: it remains scrambling at all leakage strengths, but its Lyapunov exponent decreases monotonically with dissipation rather than exhibiting an enhancement regime. The authors attribute this distinction to the effective jump operator generated by adiabatic elimination, which is quadratic in the fermions for p=2 but genuinely many-body for p>2.

Although operator growth and the Lyapunov exponent are related in unitary dynamics, the paper notes that a general framework establishing this connection for non-unitary systems is still lacking. A deeper analysis of the p=2 case is therefore identified as an unresolved problem that could clarify how the structure of dissipative jump operators governs scrambling.

References

Understanding the special role of $p=2$ more deeply remains an interesting open problem.

— Dissipation-enhanced scrambling in the SYK model coupled to a lossy cavity  (2608.19310 - Pelliconi et al., 19 Aug 2026) in Section Discussion, subsection “The peculiar case of p = 2”

An interesting question concerns whether the enhancement of scrambling by turning on a Markovian bath, described in the paper by a larger Lyapunov exponent, occurs in other systems and impacts other observables such as the operator size growth [34, 35]. We wish to address this question in the near future.

— From integrability to many-body quantum chaos through a Markovian bath  (2608.27897 - Liu et al., 28 Aug 2026) in Conclusions and discussions, p. 5