An upper bound on Lyapunov exponent of a thermal many body quantum system has been conjectured recently. In this work, we attempt to achieve a physical understanding of what prevents a system from violating this bound. We suggest the answer is that super-maximal chaos leads to instability. Our suggestion is based on findings in a SYK lattice model, with a tuneable parameter, which the Lyapunov spectrum depends upon. In the stable regime of this parameter, along with incoherent metallic phase, the system exhibits another novel phase, where transport is controlled neither by quasi-particles nor by diffusion. At the phase transition diffusion coefficient, butterfly velocity and Thouless time diverges.
Abstract An upper bound on Lyapunov exponent of a thermal many body quantum system has been conjectured recently. It would be useful to develop a physical understanding of what prevents a system from violating this bound. In this article we suggest violation of this bound leads to instability. Our suggestion is based on careful scrutiny of the arguments that established the bound. We also present a physical system where this phenomenon is manifested. In the stable regime, along with incoherent metallic phase, the system exhibits another novel phase, where transport is controlled neither by quasi-particles nor by diffusion. At the phase transition, diffusion coefficient, butterfly velocity and Thouless time diverges.
We compute the transport and chaos properties of lattices of quantum Sachdev-Ye-Kitaev islands coupled by single fermion hopping, and with the islands coupled to a large number of local, low energy phonons. We find two distinct regimes of linear-in-temperature
(T) resistivity, and describe the crossover between them. When the electron-phonon coupling is weak, we obtain the ``incoherent metal'' regime, where there is near-maximal chaos with front propagation at a butterfly velocity
vB, and the associated diffusivity ${D}_{\mathrm{chaos}}={v}_{B}^{2}/(2\ensuremath{\pi}T)$ closely tracks the energy diffusivity. On the other hand, when the electron-phonon coupling is strong, and the linear resistivity is largely due to near-elastic scattering of electrons off nearly free phonons, we find that the chaos is far from maximal and spreads diffusively. We also describe the crossovers to low
T regimes where the electronic quasiparticles are well defined.
A remarkable feature of chaos in many-body quantum systems is the existence of a bound on the quantum Lyapunov exponent. An important question is to understand what is special about maximally chaotic systems which saturate this bound. Here we provide further evidence for the `hydrodynamic' origin of chaos in such systems, and discuss hallmarks of maximally chaotic systems. We first provide evidence that a hydrodynamic effective field theory of chaos we previously proposed should be understood as a theory of maximally chaotic systems. We then emphasize and make explicit a signature of maximal chaos which was only implicit in prior literature, namely the suppression of exponential growth in commutator squares of generic few-body operators. We provide a general argument for this suppression within our chaos effective field theory, and illustrate it using SYK models and holographic systems. We speculate that this suppression indicates that the nature of operator scrambling in maximally chaotic systems is fundamentally different to scrambling in non-maximally chaotic systems. We also discuss a simplest scenario for the existence of a maximally chaotic regime at sufficiently large distances even for non-maximally chaotic systems.
Abstract A remarkable feature of chaos in many-body quantum systems is the existence of a bound on the quantum Lyapunov exponent. An important question is to understand what is special about maximally chaotic systems which saturate this bound. Here we provide further evidence for the ‘hydrodynamic’ origin of chaos in such systems, and discuss hallmarks of maximally chaotic systems. We first provide evidence that a hydrodynamic effective field theory of chaos we previously proposed should be understood as a theory of maximally chaotic systems. We then emphasize and make explicit a signature of maximal chaos which was only implicit in prior literature, namely the suppression of exponential growth in commutator squares of generic few-body operators. We provide a general argument for this suppression within our chaos effective field theory, and illustrate it using SYK models and holographic systems. We speculate that this suppression indicates that the nature of operator scrambling in maximally chaotic systems is fundamentally different to scrambling in non-maximally chaotic systems. We also discuss a simplest scenario for the existence of a maximally chaotic regime at sufficiently large distances even for non-maximally chaotic systems.
Chaos in classical systems can be characterized by Lyapunov exponents that measure the exponential divergence of nearby trajectories, but directly extending this framework to quantum mechanics has been a persistent challenge. The wavelike nature of quantum states and the non-commutative geometry of quantum phase space obstruct a straightforward generalization of classical chaos theory. Here we develop a rigorous approach to quantum chaos by leveraging quantum optimal transport theory, which provides the missing geometric foundation for measuring distances between extended quantum distributions. We define quantum Lyapunov exponents that naturally avoid divergences encountered when naïvely generalizing classical exponents, and show that in the semiclassical limit they recover the classical global expansion rate, and hence the usual maximal Lyapunov exponent when they coincide. Our framework provides a tight connection between the divergence of classical trajectories and semiclassical phase space evolution, and additionally clarifies the role of out-of-time-order correlators as diagnostics of quantum chaos. These results establish quantum optimal transport as a unifying mathematical foundation for quantum chaos theory, providing new tools to characterize dynamical behavior across the full range of quantum dynamics from simple few-body models to complex many-body systems.
We calculate the scrambling rate
λL and the butterfly velocity
vB associated with the growth of quantum chaos for a solvable large-
N electron-phonon system. We study a temperature regime in which the electrical resistivity of this system exceeds the Mott-Ioffe-Regel limit and increases linearly with temperature - a sign that there are no long-lived charged quasiparticles - although the phonons remain well-defined quasiparticles. The long-lived phonons determine
λL, rendering it parametrically smaller than the theoretical upper-bound
λL≪λmax=2πT/ℏ. Significantly, the chaos properties seem to be intrinsic -
λL and
vB are the same for electronic and phononic operators. We consider two models - one in which the phonons are dispersive, and one in which they are dispersionless. In either case, we find that
λL is proportional to the inverse phonon lifetime, and
vB is proportional to the effective phonon velocity. The thermal and chaos diffusion constants,
DE and
DL≡vB2/λL, are always comparable,
DE∼DL. In the dispersive phonon case, the charge diffusion constant
DC satisfies
DL≫DC, while in the dispersionless case
DL≪DC.