Extended Eigenstate Thermalization and the role of FZZT branes in the Schwarzian theory

TL;DR

This work provides a universal description of Schwarzian pure-state correlators, proving an extended ETH in the Schwarzian sector by classifying operator behavior via Virasoro coadjoint orbits and monodromy. By embedding the Schwarzian in 2D boundary Liouville theory and incorporating ZZ/FZZT boundary conditions, the authors derive both semiclassical and exact results for bilocal operators and OTOCs, and show how FZZT branes encode coherent Schwarzian states. A detailed phase structure emerges: eigenstates and certain coherent states thermalize with well-defined ETH temperatures, while other coherent states (r>√2) exhibit non-ergodic, oscillatory dynamics; heavy-light configurations further modify the effective temperature. The results illuminate the role of boundary states in low-dimensional holography and offer precise handles on thermalization, chaos, and microstate physics in Eden-like quantum gravity models.

Abstract

In this paper we provide a universal description of the behavior of the basic operators of the Schwarzian theory in pure states. When the pure states are energy eigenstates, expectation values of non-extensive operators are thermal. On the other hand, in coherent pure states, these same operators can exhibit ergodic or non-ergodic behavior, which is characterized by elliptic, parabolic or hyperbolic monodromy of an auxiliary equation; or equivalently, which coadjoint Virasoro orbit the state lies on. These results allow us to establish an extended version of the eigenstate thermalization hypothesis (ETH) in theories with a Schwarzian sector. We also elucidate the role of FZZT-type boundary conditions in the Schwarzian theory, shedding light on the physics of microstates associated with ZZ branes and FZZT branes in low dimensional holography.

Figures (8)

• Figure 1: We obtain results on Schwarzian correlation functions by taking an appropriate limit of 2D Liouville theory with boundary conditions, or equivalently (as shown on the left) with the corresponding boundary states (details in section \ref{['sec.stateInterpretation']}). The full cylinder is obtained when we implement the ZZ boundary condition via a doubling trick, allowing us to extend the field periodically. The resulting theory on the torus is then reduced to one dimension and gives rise to correlation functions in the Schwarzian theory with respect to the pure-state density operator $\rho_\Psi = | \Psi \rangle \langle \Psi|$.
• Figure 2: Phase diagram of the chaotic properties of the Schwarzian theory. The theory behaves thermally for the parameter range $-\sqrt{2}<r<r_c$ with $r_c = \sqrt{2}$. The range $r<r_c$ corresponds to elliptic coherent state insertions, while the critical case $r=r_c$ can also be interpreted as the parabolic orbit corresponding to the eigenstates $|E(k)\rangle$ of the theory. In the full range $-\sqrt{2}<r\leq r_c$ we find that the model scrambles with the maximal Lyapunov exponent predicted by ETH temperature, $\lambda = 2\pi T_{\rm ETH}$. In the hyperbolic range $r>r_c$ the Schwarzian theory behaves non-ergodically, and in particular its OTOC is oscillatory. See also table \ref{['fig.SummaryTable']} for more details.
• Figure 3: An open string Partition function with generic boundary conditions is the same as a closed string amplitude between corresponding boundary states.
• Figure 4: Illustration of our computation of the semiclassical OTO by saddle point. In order to get the operator in OTO order as shown, we must twist two of the insertions around each other. Quantum mechanically this corresponds to the 'second-sheet' analytic continuation from Euclidean to Lorentzian times and is equivalent to the insertion of a commutator $\left[f(\sigma_1),f(\sigma_2) \right]$ when writing the operators in terms of integrals over the Goldstone mode $f$. This commutator can be computed semiclassically using the AS symplectic form to generate the Poisson bracket $\left\{f(\sigma_1),f(\sigma_2) \right\}$ living on the appropriate Virasoro coadjoint orbit (Details in section \ref{['sec.SemiclassicalChaos']}).
• Figure 5: The configuration of various insertions on the two-dimensional complex plane that evaluates the bilocal two-point functions with ZZ and FZZT boundary conditions. Operators $B$ are those that impose the ZZ and FZZT boundary conditions on positive and negative real axis respectively. While the tilde-operators are the mirror operators after the doubling trick.