Quantum thermalization and Virasoro symmetry

TL;DR

The paper addresses whether eigenstate thermalization can hold in 2d CFTs by computing high-energy matrix elements between Virasoro descendants using a novel oscillator representation of the Virasoro algebra. In the heavy-light regime, diagonal elements form a smooth energy-dependent curve while off-diagonal elements are power-law suppressed relative to the diagonal, signaling a refined but nontrivial ETH; restricting to states with near-thermal KdV charge $T_2$ yields ETH$_{T_2}$ behavior, suggesting generalized thermalization within Virasoro modules. Numerical results corroborate the analytic heavy-light perturbation theory and reveal structured, non-random off-diagonal patterns that become negligible at leading order. The findings are consistent with holographic expectations in AdS$_3$/CFT$_2$, where bulk thermalization via black-hole formation aligns with boundary ETH-like behavior, while more KdV charges could strengthen ETH statements further.

Abstract

We initiate a systematic study of high energy matrix elements of local operators in 2d CFT. Knowledge of these is required in order to determine whether the eigenstate thermalization hypothesis (ETH) can hold in such theories. Most high energy states are high level Virasoro descendants, and by employing an oscillator representation of the Virasoro algebra we develop an efficient method for computing matrix elements of primary operators in such states. In parameter regimes where we expect (e.g. from AdS/CFT intuition) thermalization to occur, we observe striking patterns in the matrix elements: diagonal matrix elements are smoothly varying and off-diagonal elements, while nonzero, are power-law suppressed compared to the diagonal elements. We discuss the implications of these universal properties of 2d CFTs in regard to their compatibility with ETH.

Figures (10)

• Figure 1: [Left] Absolute values of matrix elements, $\langle h_U, \lbrace m_i \rbrace |O_h | h_V, \lbrace n_i \rbrace \rangle$, up to descendant level 12 for $h_U=h_V=300$, $h=5$ and central charge $c=30$. [Right] The diagonal matrix elements after filtering out outliers (see text) using $\sum_k\langle L_{-k}L_k \rangle$ (each descendant level from 1 to 12 is labelled by the colors red to violet).
• Figure 2: Perturbation theory matrices -- The left hand side is the numerical solution of the matrix elements \ref{['matrix-elem']} contained within $F(U,\bar{V})$ up to descendant level 9 for the parameters $c=5000$, $h_U=h_V=5000$ and $h=1$. The right hand side shows the matrix elements which get activated order-by-order in $1/\lambda_U$ perturbation theory -- equations \ref{['1stord']} and \ref{['2ndord']}. Clearly these non-zero matrix elements from the first few orders appear more prominently than the rest in the numerical solution.
• Figure 3: Absolute values of matrix elements till descendant level 12 at intermediate values of central charge.
• Figure 4: Absolute values of matrix elements till descendant level 12 for light probes in heavy states.
• Figure 5: Absolute values of matrix elements till descendant level 12 for light probes having conformal dimensions of the same order as the primaries in the external states.