Probing beyond ETH at large $c$

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 $⟨{\mathcal{O}}_{obs}(x){⟩}_{h_H}\approx ⟨{\mathcal{O}}_{obs}(x){⟩}_{\text{micro}}$ for a high energy energy eigenstate $|h_H⟩$. In the CFTs of interests, $⟨{\mathcal{O}}_{obs}(x){⟩}_{h_H}$ 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.

35 pages, 40 figures, additional results and comments added

References in corpus (6)

• Thermalization and its mechanism for generic isolated quantum systems
• Holography from Conformal Field Theory
• Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches
• Universality of Quantum Information in Chaotic CFTs
• Dissimilarities of reduced density matrices and eigenstate thermalization hypothesis
• Thermality and excited state Rényi entropy in two-dimensional CFT

Cited by in corpus (31)

• The Quantum Gravity Dynamics of Near Extremal Black Holes
• Quantum Regge Trajectories and the Virasoro Analytic Bootstrap
• Universal Dynamics of Heavy Operators in CFT${}_2$
• Probing thermality beyond the diagonal
• Heavy Operators and Hydrodynamic Tails
• Typicality and thermality in 2d CFT
• Generalized Eigenstate Thermalization in 2d CFTs
• Phases of scrambling in eigenstates
• Thermal Correlation Functions of KdV Charges in 2D CFT
• Leading Multi-Stress Tensors and Conformal Bootstrap
• Semi-classical BMS${}_3$ blocks and flat holography
• Generalized Gibbs Ensemble of 2d CFTs at large central charge in the thermodynamic limit
• Thermalization in Large-N CFTs
• Universality in asymptotic bounds and its saturation in $2$D CFT
• Holographic Renyi Entropy at High Energy Density
• Lightcone Commutator and Stress-Tensor Exchange in $d>2$ CFTs
• Quantum thermalization and Virasoro symmetry
• Emergent classical spacetime from microstates of an incipient black hole
• Universal Dynamics of Heavy Operators in Boundary CFT${}_2$
• Thermalization, Viscosity and the Averaged Null Energy Condition
• Eigenstate Thermalisation in the conformal Sachdev-Ye-Kitaev model: an analytic approach
• Multi-stress tensors and next-to-leading singularities in the Regge limit
• CFT correlators, $\mathcal{W}$-algebras and Generalized Catalan Numbers
• Corrections in the relative entropy of black hole microstates
• Zero modes of local operators in 2d CFT on a cylinder
• Subsystem entropy in 2d CFT and KdV ETH
• Rényi entropy at large energy density in 2D CFT
• Subsystem eigenstate thermalization hypothesis for translation invariant systems
• Classical conformal blocks, Coulomb gas integrals and Richardson-Gaudin models
• Finite temperature corrections to black hole quasinormal modes from 2D CFT
• Generalized monodromy method in gauge/gravity duality