Late time physics of holographic quantum chaos

Abstract

Quantum chaotic systems are often defined via the assertion that their spectral statistics coincides with, or is well approximated by, random matrix theory. In this paper we explain how the universal content of random matrix theory emerges as the consequence of a simple symmetry-breaking principle and its associated Goldstone modes. This allows us to write down an effective-field theory (EFT) description of quantum chaotic systems, which is able to control the level statistics up to an accuracy ${\cal O} \left(e^{-S} \right)$ with $S$ the entropy. We explain how the EFT description emerges from explicit ensembles, using the example of a matrix model with arbitrary invariant potential, but also when and how it applies to individual quantum systems, without reference to an ensemble. Within AdS/CFT this gives a general framework to express correlations between "different universes" and we explicitly demonstrate the bulk realization of the EFT in minimal string theory where the Goldstone modes are bound states of strings stretching between bulk spectral branes. We discuss the construction of the EFT of quantum chaos also in higher dimensional field theories, as applicable for example for higher-dimensional AdS/CFT dual pairs.

Figures (7)

• Figure 1: The generic late-time behavior described by the effective theory of quantum chaos. The top line shows the ramp and plateau behavior of a simple observable -- the spectral form factor $K(\tau)$ -- and the bottom line shows how the different qualitative regimes are obtained in the sigma model. The ramp behavior is initially well described by small perturbative corrections around the standard saddle (the north pole of the sphere in the sigma model geometry $H_2\times S^2$). In this phase causal symmetry is spontaneously broken. Around the Heisenberg time $\tau = 1$, one must also include the non-perturbative contributions of the second saddle, as described in \ref{['eq.TheFullStructure']}. At very late times the system explores the full Goldstone manifold $H_2\times S^2$ and the causal symmetry is restored.
• Figure 2: Bulk sigma model: we start from the configuration comprising 'sea branes', $\mu=1,\ldots L$, and spectral branes, $a=1,\ldots 4$. As a microscopic realization, for example in the context of minimal string theory, we can take the $L$ 'sea branes' to be a stack of $L$ coincident ZZ branes and the spectral branes to be FZZT branes. In the corresponding matrix theory, pairs of strings connecting sea branes $\mu$ and $\nu$, respectively, to the same spectral brane, $a$, are represented by the bilinears $\bar{\psi}_\mu^a H_{\mu\nu} \psi_\nu^a$, with Chan-Paton factors $\psi_\mu^a$. Integration over the stack of sea branes leads to a dual picture, with 'string bound states' $\bar{\psi}_\mu^a A^{a,b}\psi_\mu^b$ correlating spectral branes, $a,b$, via an induced bulk geometry with each other. The previously exact symmetry between different $a$'s (for spectral branes at coinciding energy-coordinates, $E$) is spontaneously broken and leads to the emergence of the $Q^{ab}$ pseudo Goldstone degrees of freedom. The reader may note that this is a microscopic description of an open/closed duality in the presence of 'flavor' branes Karch:2002sh giving rise to an additional open string sector (the FZZT-ZZ strings). In section \ref{['sec.ContinuumGravityFormulation']} we show how to extract the Goldstone contributions from a world-sheet calculation of the FZZT-ZZ strings.
• Figure 3: The annulus contribution to the spectral two-point function. Its universal part gives the correct EFT singular contribution of $-1/2s^2$. The diagram is identical to \ref{['eq.GoldstoneCylinder']}, although here it represents a minimal-string worldsheet interpreted as a 2D spacetime with two branes, also known as the Laplace transform of a Euclidean wormhole. The 2D gravity amplitude gives the same leading singularity including the numerical coefficient as \ref{['eq.GoldstoneCylinder']}, coming from the Goldstone sector of causal symmetry.
• Figure 4: Alternative ways of representing building blocks of matrix theories. Left: double line notation customary in matrix field theory. Right: 'impurity diagram representation' customary in the physics of random systems.
• Figure 5: Alternative ways of representing building blocks of matrix theories. Left: double line notation customary in matrix field theory. Right: 'impurity diagram representation' customary in the physics of random systems.