Generalized Eigenstate Thermalization Hypothesis in 2D Conformal Field Theories

Infinite-dimensional conformal symmetry in two dimensions leads to integrability of 2D conformal field theories (CFTs) by giving rise to an infinite tower of local conserved quantum Korteweg–de Vries (qKdV) charges in involution. We discuss how the presence of conserved charges constrains equilibration in 2D CFTs. We propose that in the thermodynamic limit large central charge 2D CFTs satisfy generalized eigenstate thermalization, with the values of qKdV charges forming a complete set of thermodynamically relevant quantities, which unambiguously determine expectation values of all local observables from the vacuum family. Equivalence of ensembles further provides that local properties of an eigenstate can be described by the generalized Gibbs ensemble that includes only qKdV charges. In the case of a general initial state, upon equilibration, the emerging generalized Gibbs ensemble will necessarily include negative chemical potentials and holographically will be described by a quasiclassical black hole with quantum soft hair.

Figure 1

Plot of $q_{2k-1}/q_1^k-1$ in the units of $1/c$ as a function of $\tau =\beta \mathbf{(}{\pi }^2/(6c{\mu }_3){\mathbf{)}}^{1/3}$ for $k=2$, 3. It approaches zero as $|\tau {|}^{-3}$ for all $k$ when $\tau \to -\infty$. The opposite limit $\tau \to \infty$ corresponds to the Gibbs ensemble, $q_1\sim {\beta }^{-1}$, ${\mu }_3\to 0$, and $c(q_{2k-1}/q_1^k-1)$, for $k=2$, 3 approaching $22/5$ and $302/21$, correspondingly.