Pole skipping away from maximal chaos

Pole skipping is a recently discovered subtle effect in the thermal energy density retarded two point function at a special point in the complex (ω, p) planes. We propose that pole skipping is determined by the stress tensor contribution to many-body chaos, and the special point is at (ω, p)p.s. = $$ i{\lambda}^{(T)}\left(1,1/{u}_B^{(T)}\right) $$ i λ T 1 1 / u B T , where λ(T) = 2π/β and $$ {u}_B^{(T)} $$ u B T are the stress tensor contributions to the Lyapunov exponent and the butterfly velocity respectively. While this proposal is consistent with previous studies conducted for maximally chaotic theories, where the stress tensor dominates chaos, it clarifies that one cannot use pole skipping to extract the Lyapunov exponent of a theory, which obeys λ ≤ λ(T). On the other hand, in a large class of strongly coupled but non-maximally chaotic theories $$ {u}_B^{(T)} $$ u B T is the true butterfly velocity and we conjecture that uB ≤ $$ {u}_B^{(T)} $$ u B T is a universal bound. While it remains a challenge to explain pole skipping in a general framework, we provide a stringent test of our proposal in the large-q limit of the SYK chain, where we determine λ, uB, and the energy density two point function in closed form for all values of the coupling, interpolating between the free and maximally chaotic limits. Since such an explicit expression for a thermal correlator is one of a kind, we take the opportunity to analyze many of its properties: the coupling dependence of the diffusion constant, the dispersion relations of poles, and the convergence properties of all order hydrodynamics.

Chaos exponents of SYK traversable wormholes

In this paper we study the chaos exponent, the exponential growth rate of the out-of-time-ordered four point functions, in a two coupled SYK models which exhibits a first order phase transition between the high temperature black hole phase and the low temperature gapped phase interpreted as a traversable wormhole. We see that as the temperature decreases the chaos exponent exhibits a discontinuous fall-off from the value of order the universal bound 2π/β at the critical temperature of the phase transition, which is consistent with the expected relation between black holes and strong chaos. Interestingly, the chaos exponent is small but non-zero even in the wormhole phase. This is surprising but consistent with the observation on the decay rate of the two point function [1], and we found the chaos exponent and the decay rate indeed obey the same temperature dependence in this regime. We also studied the chaos exponent of a closely related model with single SYK term, and found that the chaos exponent of this model is always greater than that of the two coupled model in the entire parameter space.

Binary (k, k)-Designs

We introduce and investigate binary (k,k)-designs, a special case of T-designs. Our combinatorial interpretation relates (k,k)-designs to the binary orthogonal arrays. We derive a general linear programming bound and propose as a consequence a universal bound on the minimum possible cardinality of (k,k)-designs for fixed k and n. Designs which attain our bound are investigated.

Phonon heat transport in cavity-mediated optomechanical nanoresonators

The understanding of heat transport in nonequilibrium thermodynamics is an important research frontier, which is crucial for implementing novel thermodynamic devices, such as heat engines and refrigerators. The convection, conduction, and radiation are the well-known basic ways to transfer thermal energy. Here, we demonstrate a different mechanism of phonon heat transport between two spatially separated nanomechanical resonators coupled by the cavity-enhanced long-range interactions. The single trajectory for thermalization and non-equilibrium dynamics is monitored in real-time. In the strong coupling regime, the instant heat flux spontaneously oscillates back and forth in the nonequilibrium steady states. The universal bound on the precision of nonequilibrium steady-state heat flux, i.e. the thermodynamic uncertainty relation, is verified in such a temperature gradient driven far-off equilibrium system. Our results give more insight into the heat transfer with nanomechanical oscillators, and provide a playground for testing fundamental theories in non-equilibrium thermodynamics.

Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain

We prove a two-term Weyl-type asymptotic formula for sums of eigenvalues of the Dirichlet Laplacian in a bounded open set with Lipschitz boundary. Moreover, in the case of a convex domain we obtain a universal bound which correctly reproduces the first two terms in the asymptotics.