Probing beyond ETH at large $c$

TL;DR

This work analyzes probe corrections to the Eigenstate Thermalization Hypothesis in large-$c$ 2D CFTs with a sparse spectrum, focusing on HL four-point functions dominated by the vacuum Virasoro block. By using the monodromy method, it shows that finite probe strength $ ext{O}(h_L/c)$ replaces thermal poles with branch cuts and reveals an infinite family of saddles connected on an infinite-sheeted Riemann surface, implying a nonperturbative, finite-$c$ resolution. The authors extend the analysis to micro-canonical ensembles and Renyi entropies, deriving a WKB framework that yields a linear Renyi entropy in the subsystem arc length and an entanglement spectrum peaked at a characteristic energy, while demonstrating a deep connection to Stokes phenomena and Whittaker functions. Finite-$c$ studies via Zamolodchikov recursions corroborate the branch-cut picture, showing zeros condense along predicted paths and suggesting a universal finite-$c$ resolution mechanism across blocks, with implications for ETH and black hole information in holographic CFTs.

Abstract

We study probe corrections to the Eigenstate Thermalization Hypothesis (ETH) in the context of 2D CFTs with large central charge and a sparse spectrum of low dimension operators. In particular, we focus on observables in the form of non-local composite operators $\mathcal{O}_{obs}(x)=\mathcal{O}_L(x)\mathcal{O}_L(0)$ with $h_L\ll c$. As a light probe, $\mathcal{O}_{obs}(x)$ is constrained by ETH and satisfies $\langle \mathcal{O}_{obs}(x)\rangle_{h_H}\approx \langle \mathcal{O}_{obs}(x)\rangle_{\text{micro}}$ for a high energy energy eigenstate $| h_H\rangle$. In the CFTs of interests, $\langle \mathcal{O}_{obs}(x)\rangle_{h_H}$ is related to a Heavy-Heavy-Light-Light (HL) correlator, and can be approximated by the vacuum Virasoro block, which we focus on computing. A sharp consequence of ETH for $\mathcal{O}_{obs}(x)$ is the so called "forbidden singularities", arising from the emergent thermal periodicity in imaginary time. Using the monodromy method, we show that finite probe corrections of the form $\mathcal{O}(h_L/c)$ drastically alter both sides of the ETH equality, replacing each thermal singularity with a pair of branch-cuts. Via the branch-cuts, the vacuum blocks are connected to infinitely many additional "saddles". We discuss and verify how such violent modification in analytic structure leads to a natural guess for the blocks at finite $c$: a series of zeros that condense into branch cuts as $c\to\infty$. We also discuss some interesting evidences connecting these to the Stoke's phenomena, which are non-perturbative $e^{-c}$ effects. As a related aspect of these probe modifications, we also compute the Renyi-entropy $S_n$ in high energy eigenstates on a circle. For subsystems much larger than the thermal length, we obtain a WKB solution to the monodromy problem, and deduce from this the entanglement spectrum.

Figures (24)

• Figure 1: Forbidden singularites in the leading order results (left), via a conformal mapping, are related to the thermal images of the OPE singularity (right), a sign of emergent thermality.
• Figure 2: Solutions ($\epsilon_H=36, \epsilon_L=5*10^{-3}$) for the accessory parameter $p_n(x)$ for $n=0,1,2$ (solid), compared against the leading order in $\mathcal{O}(\epsilon_L)$ result $p_0(x)=p_{\text{therm}}(x)$ that exhibits thermal singularities.
• Figure 3: Examples of the monodromy of $p_{n-1}(x)$ around the branch point above the forbidden singularities: $x^+_n = x_n + i\frac{4\epsilon_L}{|p_{n-1}(x_n)|}$ for $n=1,2,3$
• Figure 4: After the re-summation of probe corrections, the leading order forbidden thermal poles are "resolved" into a series of branch-cuts. Through them a chain of additional saddles $p_n(x)$ are sewn together, and form an infinite Riemann surface $\mathcal{M}_p$.
• Figure 5: plots of the saddle-point equation $\;\epsilon_H-\frac{\pi^2}{\beta^2}-\frac{2\epsilon_L}{\beta}+2\epsilon_L\cot{\left(\frac{\pi\tau}{\beta}\right)}\frac{\pi\tau}{\beta^2}\;$ as a function of $\beta$, for $\;\epsilon_L=10^{-1}, \epsilon_H=6\mathcal{E}=36$. Left: for $\tau<\beta_{\text{thermal}}(\mathcal{E})$, the dominant saddle agrees well with $\beta_{\text{thermal}}(\mathcal{E})$ (grey line); Right: for $\tau>\beta_{\text{thermal}}$, the dominant saddle is replaced by a $\tau$-dependent new one.