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Temporal correlations and chaos from spacetime kernel

Rathindra Nath Das, Arnab Kundu, Matheus H. Martins Costa, Nemai Chandra Sarkar

TL;DR

The paper develops a finite-dimensional spacetime density kernel, extended to a generalized spacetime density kernel (GSDK), to encode multi-point, two-time correlations and timelike entanglement. By constructing two-leg and four-leg kernels and their higher-point generalizations, it shows that Haar-averaged $2N$-point correlators yield the $(2N)$-th moment of the spectral form factor (SFF) at an effective temperature $\beta_{\rm eff}=\beta/(2N)$, thereby unifying scrambling diagnostics with spectral statistics. It also derives bounds on chaos via Hölder inequalities and ETH-based arguments, and introduces a local-norm framework that recovers a chaos bound for connected OTOCs. The framework provides a unified, operator-based perspective linking short-time information scrambling with long-time spectral universality, with potential extensions to quantum field theories and holography. Overall, GSDK offers a versatile tool to probe dynamical information across temporal orders and scales, connecting temporal correlations to detailed spectral data.

Abstract

We develop a finite-dimensional formulation of the recently introduced notion of ``timelike entanglement'', defined in terms of two-point functions between operators supported on different Cauchy slices. Using a local orthonormal operator basis, we recast this construction in terms of a generalized response tensor. Building on this, we introduce a generalized spacetime density kernel (GSDK) corresponding to higher-point correlation functions, including time-ordered as well as out-of-time-ordered correlators. We show that the Haar-averaged $(2N)$-point function yields the $(2N)$-th moment of the spectral form factor (SFF), evaluated at an $N$-enhanced effective temperature. The correlation functions of the GSDK operators also yield the SFF, with an effective $(1/N)$-reduction of the physical time-scales. The GSDK places both scrambling diagnostics and spectral statistics on a similar footing and clarifies how higher-point correlators and non-trivial time ordering capture fine-grained dynamical information of a quantum system.

Temporal correlations and chaos from spacetime kernel

TL;DR

The paper develops a finite-dimensional spacetime density kernel, extended to a generalized spacetime density kernel (GSDK), to encode multi-point, two-time correlations and timelike entanglement. By constructing two-leg and four-leg kernels and their higher-point generalizations, it shows that Haar-averaged -point correlators yield the -th moment of the spectral form factor (SFF) at an effective temperature , thereby unifying scrambling diagnostics with spectral statistics. It also derives bounds on chaos via Hölder inequalities and ETH-based arguments, and introduces a local-norm framework that recovers a chaos bound for connected OTOCs. The framework provides a unified, operator-based perspective linking short-time information scrambling with long-time spectral universality, with potential extensions to quantum field theories and holography. Overall, GSDK offers a versatile tool to probe dynamical information across temporal orders and scales, connecting temporal correlations to detailed spectral data.

Abstract

We develop a finite-dimensional formulation of the recently introduced notion of ``timelike entanglement'', defined in terms of two-point functions between operators supported on different Cauchy slices. Using a local orthonormal operator basis, we recast this construction in terms of a generalized response tensor. Building on this, we introduce a generalized spacetime density kernel (GSDK) corresponding to higher-point correlation functions, including time-ordered as well as out-of-time-ordered correlators. We show that the Haar-averaged -point function yields the -th moment of the spectral form factor (SFF), evaluated at an -enhanced effective temperature. The correlation functions of the GSDK operators also yield the SFF, with an effective -reduction of the physical time-scales. The GSDK places both scrambling diagnostics and spectral statistics on a similar footing and clarifies how higher-point correlators and non-trivial time ordering capture fine-grained dynamical information of a quantum system.

Paper Structure

This paper contains 19 sections, 147 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Spacetime density kernel
  3. Spacetime kernel from two-point correlator
  4. Generalised spacetime density kernel from higher point correlators
  5. Disjoint two-region OTOC AA–BB
  6. Spectral form factor from generalised spacetime density kernel
  7. 2-point GSDK and SFF
  8. 4-point GSDK, OTOC, TOC and SFF
  9. Trace Relations for Temporally Separated GSDKs
  10. Bounds from the spacetime kernel
  11. Bounds using Hölder's inequality
  12. Kernel norms and the Eigenstate Thermalization Hypothesis
  13. Discussions
  14. Acknowledgements
  15. Proof of the Haar averaging of the operators
  16. ...and 4 more sections

Figures (2)

  • Figure 1: Normalised $F_2(t,\beta)$ as a function of time for Hamiltonian chosen from GOE, GUE and GSE distributions for $\beta=0,0.5,1$. The blue plot is for a single instance, and the red plot is for the average $\overline{F_2(t,\beta)}$ over 200 instances of random matrices.
  • Figure 2: Normalised $F_4(t,\beta)$ as a function of time for Hamiltonian chosen from GOE, GUE and GSE distributions for $\beta=0,0.5,1$. The blue plot is for a single instance, and the red plot is for the average $\overline{F_4(t,\beta)}$ over 200 instances of random matrices.