From operator statistics to wormholes

TL;DR

The work develops an operator-resolvent effective field theory based on causal symmetry breaking to describe ergodic behavior in chaotic quantum many-body systems, with the SYK model serving as a concrete testbed. It derives universal ramp-plateau behavior in operator correlations from a sigma-model topological expansion and interprets the ramp in terms of bulk wormhole configurations, while non-universal massive modes govern the approach to ergodicity with observable-dependent Thouless times. By linking quantum chaos, random-matrix statistics, Haar averaging, and holography, the paper provides quantitative predictions for ergodic times and suggests a unified framework for wormhole contributions across observables. These insights help bridge microscopic dynamics, spectral statistics, and gravitational duals, with potential implications for understanding late-time unitarity in holographic theories like SYK and beyond.

Abstract

For a generic quantum many-body system, the quantum ergodic regime is defined as the limit in which the spectrum of the system resembles that of a random matrix theory (RMT) in the corresponding symmetry class. In this paper we analyse the time dependence of correlation functions of operators. We study them in the ergodic limit as well as their approach to the ergodic limit which is controlled by non-universal massive modes. An effective field theory (EFT) corresponding to the causal symmetry and its breaking describes the ergodic phase. We demonstrate that the resulting Goldstone-mode theory has a topological expansion, analogous to the one described in arXiv:2008.02271 with added operator sources, whose leading non-trivial topologies give rise to the universal ramp seen in correlation functions. The ergodic behaviour of operators in our EFT is seen to result from a combination of RMT-like spectral statistics and Haar averaging over wave-functions. Furthermore we analytically capture the plateau behaviour by taking into account the contribution of a second saddle point. Our main interest are quantum many-body systems with holographic duals and we explicitly establish the validity of the EFT description in the SYK-class of models, starting from their microscopic description. By studying the tower of massive modes above the Goldstone sector we get a detailed understanding of how the ergodic EFT phase is approached and derive the relevant Thouless time scales. We point out that the topological expansion can be reinterpreted in terms of contributions of bulk wormholes and baby-universes.

Figures (8)

• Figure 1: Late time behaviour of a generic observable in a quantum system (in the universality class of Gaussian unitary ensembles for specificity). From the holographic perspective, the conventional gravitational contribution computes the decay in the non-ergodic regime. Region I corresponds to the fast decay determined by the smooth energy dependence of coarse-grained DoS. This corresponds to the contribution of the disk geometry to the correlation function. In region II, correlation function decay exponentially because of presence of non-ergodic and non-universal massive modes. The part of the curve in purple is the universal ramp (region III) and plateau (region IV) behaviour and is governed by the mean level statistics of the eigenstates. The timescales at which the linear ramp becomes conspicuous is known as $t_{\rm er}$ or the ergodic time. Lastly, $t_H$ is the time at which the plateau begins and is of the order of Heisenberg time. The universal ramp-plateau corresponds to the contribution of the sum over disk geometries with handles around the standard and the Andreev-Altshuler saddle points. Also see \ref{['fig.massive-massless']} for more accurate representation of operator correlation functions in the SYK model.
• Figure 2: The leading contribution to the operator 2-point correlation function arises from the diagram demonstrated in (a). Here, the differently coloured lines demonstrate different causalities. The dots represent the projectors on the bosonic sector as well as the insertion of the sources for the operators. It arises from computing the resolvent with $\mathrel{\vcenter{\hbox{\tiny+}\hbox{\tiny+}}}$-causality in \ref{['eq:resolvent']} and therefore does not receive contributions from the ergodic modes, $B,\tilde{B}$. This diagram can be understood to arise from the integration over the eigenfunctions of the Hamiltonian. In this diagram we have refrained from explicitly demonstrating the disorder average. The figure (b) is a depiction of the diagram as a bulk geometry, which is a disk in this case. The bulk is understood to emerge out of disorder averaging.
• Figure 3: The subleading contribution to the operator correlation function arises from the non-planar diagram of figure (a). The coloured fat-line propagators demonstrate matrix fields $B,\tilde{B}$, respectively, in the double line notation (refer to \ref{['eq.pertT-param']} below). These $B,\tilde{B}$ fields have one index each in the advanced and the retarded sector, which is demonstrated by different colours of the double lines. The dots represent the projectors on the bosonic sector as well as the insertion of the sources for the operators. The blue dotted lines are the Hamiltonian exchange, or the disorder average. We invite the reader to refer to \ref{['app.details']} for a microscopic understanding of the above diagrams. This diagram is dual to a bulk geometry demonstrated in (b): a torus with a boundary. The dotted blue lines span the two dimensional surface, constructing the bulk geometry, albeit this time they wind around the different cycles of the genus-1 Riemann surface.
• Figure 4: Comparing the numerically computed value of the resolvent as defined in \ref{['eq.resolvent']} with the prediction of the $\sigma$-model for the SYK model with $N$ Majorana fermions. In (a), for $N = 14$ we have computed the resolvent for hopping operator between sites 3 and 5. In (b), for $N=18$, the resolvent of the hopping operator between sites 4 and 6 is computed. In our conventions, a system of $N$ Majorana fermions has $N/2$sites.
• Figure 5: The correlation function $C(t)$ for the two-body hopping operator, $\mathcal{O} = \psi_i \psi_j$ ($i \neq j$), shown versus (a) physical time $t$ and (b) scaled time $t/t_H$ for three choices of $N$: black curve ($N=14$), red ($N=18$) and grey ($N=22$).