From operator statistics to wormholes

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 analyze 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 nonuniversal 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 by Altland and Sonner [SciPost Phys. 11, 034 (2021)] with added operator sources, whose leading nontrivial topologies give rise to the universal ramp seen in correlation functions. The ergodic behavior 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 capture analytically the plateau behavior by taking into account the contribution of a second saddle point. Our main interest is quantum many-body systems with holographic duals, and we explicitly establish the validity of the EFT description in the Sachdev-Ye-Kitaev 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 we derive the relevant Thouless timescales. We point out that the topological expansion can be reinterpreted in terms of contributions of bulk wormholes and baby universes.

Figure 1

Late-time behavior 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 nonergodic regime. Region I corresponds to the fast decay determined by the smooth energy dependence of the coarse-grained DOS. This corresponds to the contribution of the disk geometry to the correlation function. In region II, correlation function decays exponentially because of the presence of nonergodic and nonuniversal massive modes. The part of the curve in purple is the universal ramp (region III) and plateau (region IV) behavior and is governed by the mean level statistics of the eigenstates. The timescales at which the linear ramp becomes conspicuous is known as $t_{\mathrm{er}}$ 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 Fig. 5 for a more accurate representation of operator correlation functions in the SYK model.

Figure 2

The leading contribution to the operator two-point correlation function arises from the diagram demonstrated in (a). Here, the differently colored 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 $‡$-causality in Eq. (2.4) 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. Part (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 nonplanar diagram of (a). The colored fat-line propagators demonstrate matrix fields $B,\tilde{B}$, respectively, in the double line notation [refer to Eq. (2.30) below]. These $B,\tilde{B}$ fields have one index each in the advanced and the retarded sector, which is demonstrated by different colors 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 Appendix pp2 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 Eq. (2.1) 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 $C\left(t\right)$ for the two-body hopping operator, $\mathcal{O}={\psi }_i{\psi }_j$ ($i\ne j$), shown vs (a) physical time $t$ and (b) scaled time $t/t_H$ for three choices of $N$: black curve ($N=14$), red ($N=18$), and gray ($N=22$).

Figure 6

In the boson-boson block, only one of the saddle points $y_0^{\pm }$ can be reached by a deformation of the original integration contour $\mathrm{Im}\left({\lambda }_b^{r/a}\right)=0$, denoted by the gray line, without crossing the pole. In the above figures, the poles are denoted by black crosses, and the saddle points by black points. Part (a) demonstrates the deformation of the contour in the retarded space, while (b) demonstrates the contour deformation for the advanced space.

Figure 7

For the eigenvalue integrals in the fermionic sector, the contour of the stationary phase passes through both the saddle points $y_0^{\pm }$ and the zero ${\lambda }_0$. However, the contribution of one of the saddle points is subleading with respect to the other. The original integration contour, $\mathrm{Re}\left({\lambda }_f^{r/a}\right)=0$, is denoted by the gray line.

Figure 8

The contribution to the operator correlation function arises from the nonplanar diagram.