Abstract It has been conjectured that the speed of thermalization is universally constrained by a Planckian timescale, τ Pl = ℏ /( k B T ), determined solely by the reduced Planck constant ℏ, Boltzmann’s constant ( k B ) and the temperature ( T ). However, a general and model-independent derivation of this limit has remained elusive. Here we provide a general proof of the emergence of Planckian thermalization based on quantum information geometry and quantum metrology. We formulate thermalization as a process that prepares states close to the corresponding thermal ensemble for a set of distinct Hamiltonians, and show that quantum mechanics imposes a universal lower bound τ ≥ τ Pl /2 on the thermalization time τ at finite temperature. In the low-temperature regime, the spectral gap Δ between the ground and first excited states replaces the temperature as the relevant energy scale and determines the lower bound on the thermalization time, in close connection with the quantum adiabatic theorem. These bounds, rooted in Hamiltonian estimation, establish operational limits on thermalization governed exclusively by fundamental constants and the intrinsic energy scale.
We study quantum many-body systems with a global U(1) conservation law, focusing on a theory of N interacting fermions with charge conservation, or N interacting spins with one conserved component of total spin. We define an effective operator size at finite chemical potential through suitably regularized out-of-time-ordered correlation functions. The growth rate of this density-dependent operator size vanishes algebraically with charge density; hence we obtain new bounds on Lyapunov exponents and butterfly velocities in charged systems at a given density, which are parametrically stronger than any Lieb-Robinson bound. We argue that the density dependence of our bound on the Lyapunov exponent is saturated in the charged Sachdev-Ye-Kitaev model. We also study random automaton quantum circuits and Brownian Sachdev-Ye-Kitaev models, each of which exhibit a different density dependence for the Lyapunov exponent, and explain the discrepancy. We propose that our results are a cartoon for understanding Planckian-limited energy-conserving dynamics at finite temperature.
The local equilibration time
τeq of quantum many-body systems is conjectured to be bounded below by the Planckian time
ℏ/T. We formalize this conjecture by defining
τeq as the time scale at which a hydrodynamic description emerges for conserved densities. Drawing on analytic properties of real time thermal correlators, we establish a rigorous lower bound
τeq≥αℏ/T on the onset of hydrodynamic behavior in a `regulated' thermal two-point function. The dimensionless coefficient
α depends only on dimensionality and the type of hydrodynamic or diffusive behavior that emerges, and is independent of the thermalization mechanism or other microscopic details. This bound applies universally to local quantum many-body systems, with or without a quasiparticle description, including in the presence of inelastic scattering.
We introduce a simplified version of Connes-Narnhofer-Thirring's quantum dynamical entropy for quantum systems. It quantifies the amount of information gained about the initial condition from continuously monitoring an observable. A nonzero entropy growth rate can be obtained by monitoring the thermal fluctuation of an extensive observable in a generic many-body system, away from classical or large N limits. We explicitly compute the entropy rate in the thermodynamic and long-time limit in terms of the two-point correlation functions. We conjecture a universal Planckian bound for the entropy rate. Related results on the purification rate are also obtained.
Motivated by the Planckian thermalization rate
Γ∼T observed in the two-body interacting SYK model, we study thermalization in the
q/2-body case. Previous studies of such systems have established the notion of instantaneous thermalization to leading order in
1/q. It was conjectured that the thermalization may still be Planckian but with a divergent rate
Γ∼Tq, explaining the ``instantaneous'' part. For an analytically and numerically tractable system, we calculate the effective temperatures after a quench at time
t=0 and indeed find a Planckian rate, albeit with the unexpected decaying behavior $Γ\sim T q^{-1} $. This is contrasted with the behavior of the causal Green's function
G(t1,t2), which instantly acquires a thermal form in the two-time plane block
t1,t2>0.
The resulting picture is that instant thermalization is a meaningful concept only for this block, while the off-diagonal blocks
t1⋅t2<0 are inherently non-thermal at any
q. Since the effective temperature is obtained from correlations spanning all blocks, it inherits a finite rate set by the off-diagonal. We illustrate this directly by studying the non-thermal correlations' time dependence.
The observation of Planckian scattering, often inferred from Drude fits in strongly correlated metals, raises the question of how to extract an intrinsic timescale from measurable quantities in a model-independent way. We address this by focusing on a ratio (
B) of spectral moments of the dissipative part of the optical conductivity and prove a rigorous upper bound on
B in terms of the Planckian rate. The bound emerges from the analytic structure of thermally weighted response functions of the current operator. Crucially, the bounded quantity is directly accessible via optical spectroscopy and computable from imaginary-time correlators in quantum Monte Carlo simulations, without any need for analytic continuation. We evaluate
B for simplified examples of both Drude and non-Drude forms of the optical conductivity with a single scattering rate in various asymptotic regimes, and find that
B lies far below the saturation value. These findings demonstrate that Planckian bounds can arise from fundamental constraints on equilibrium dynamics, pointing toward a possibly universal structure governing transport in correlated quantum matter.