Spatio-temporal spread of perturbations in power-law models at low temperatures: Exact results for OTOC

Abstract

We present exact results for the classical version of the Out-of-Time-Order Commutator (OTOC) for a family of power-law models consisting of $N$ particles in one dimension and confined by an external harmonic potential. These particles are interacting via power-law interaction of the form $\propto {\sum }_{\begin{array}{c}i,j=1(i\ne j)\end{array}}^N|x_i-x_j{|}^{-k}$ $\forall$ $k>1$ where $x_i$ is the position of the $i^{\text{th}}$ particle. We present numerical results for the OTOC for finite $N$ at low temperatures and short enough times so that the system is well approximated by the linearized dynamics around the many body ground state. In the large-$N$ limit, we compute the ground-state dispersion relation in the absence of external harmonic potential exactly and use it to arrive at analytical results for OTOC. We find excellent agreement between our analytical results and the numerics. We further obtain analytical results in the limit where only linear and leading nonlinear (in momentum) terms in the dispersion relation are included. The resulting OTOC is in agreement with numerics in the vicinity of the edge of the "light cone". We find remarkably distinct features in OTOC below and above $k=3$ in terms of going from non-Airy behaviour ($1<k<3$) to an Airy universality class ($k>3$). We present certain additional rich features for the case $k=2$ that stem from the underlying integrability of the Calogero-Moser model. We present a field theory approach that also assists in understanding certain aspects of OTOC such as the sound speed. Our findings are a step forward towards a more general understanding of the spatio-temporal spread of perturbations in long-range interacting systems.

14 pages, 4 figures (including supplementary material)

References in corpus (6)

• Bose-Einstein condensation of chromium
• Lyapunov Exponent and Out-of-Time-Ordered Correlator's Growth Rate in a Chaotic System
• Physics and Mathematics of Calogero particles
• Generalized hydrodynamics in strongly interacting 1D Bose gases
• Hydrodynamics of cold atomic gases in the limit of weak nonlinearity, dispersion and dissipation
• Classical many-body chaos with and without quasiparticles

Cited by in corpus (11)

• Robustness of Kardar-Parisi-Zhang scaling in a classical integrable spin chain with broken integrability
• Dynamical fluctuations in the Riesz gas
• Out of equilibrium dynamics of repulsive ranked diffusions: the expanding crystal
• Dynamics and transport in the boundary-driven dissipative Klein-Gordon chain
• Scaling at the OTOC Wavefront: free versus chaotic models
• Control of spatiotemporal chaos by stochastic resetting
• Spatio-temporal fluctuations in the passive and active Riesz gas on the circle
• Out-of-time-ordered correlator in the one-dimensional Kuramoto-Sivashinsky and Kardar-Parisi-Zhang equations
• Dynamical regimes of finite temperature discrete nonlinear Schrödinger chain
• Manifestation of strange nonchaotic attractors in extended systems: A study through out-of-time-ordered correlators
• Extreme dynamics and relaxation of quantum gases: A hydrodynamic approach