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Typicality and thermality in 2d CFT

Shouvik Datta, Per Kraus, Ben Michel

TL;DR

The paper investigates whether typical high-energy eigenstates in 2d CFTs at finite central charge $c$ exhibit thermal behavior for stress-tensor and current correlators. It shows that typical states are level $h/c$ descendants of primaries with $h_p=(c-1)/c\,h$, and that their correlators reproduce the thermal values in the thermodynamic limit, while primary states generally do not unless $c$ is large. The authors demonstrate that discrepancies in primary-state thermality can be resolved by invoking a generalized Gibbs ensemble for the KdV charges, but typical Descendant states render the ordinary canonical ensemble sufficient. The results support a holographic interpretation where black holes in AdS$_3$/CFT$_2$ arise from coarse-graining over typical microstates and clarify when generalized ensembles are necessary for describing high-energy eigenstates.

Abstract

We identify typical high energy eigenstates in two-dimensional conformal field theories at finite $c$ and establish that correlation functions of the stress tensor in such states are accurately thermal as defined by the standard canonical ensemble. Typical states of dimension $h$ are shown to be typical level $h/c$ descendants. In the AdS$_3$/CFT$_2$ correspondence, it is such states that should be compared to black holes in the bulk. We also discuss the discrepancy between thermal correlators and those computed in high energy primary states: the latter are reproduced instead by a generalized Gibbs ensemble with extreme values chosen for the chemical potentials conjugate to the KdV charges.

Typicality and thermality in 2d CFT

TL;DR

The paper investigates whether typical high-energy eigenstates in 2d CFTs at finite central charge exhibit thermal behavior for stress-tensor and current correlators. It shows that typical states are level descendants of primaries with , and that their correlators reproduce the thermal values in the thermodynamic limit, while primary states generally do not unless is large. The authors demonstrate that discrepancies in primary-state thermality can be resolved by invoking a generalized Gibbs ensemble for the KdV charges, but typical Descendant states render the ordinary canonical ensemble sufficient. The results support a holographic interpretation where black holes in AdS/CFT arise from coarse-graining over typical microstates and clarify when generalized ensembles are necessary for describing high-energy eigenstates.

Abstract

We identify typical high energy eigenstates in two-dimensional conformal field theories at finite and establish that correlation functions of the stress tensor in such states are accurately thermal as defined by the standard canonical ensemble. Typical states of dimension are shown to be typical level descendants. In the AdS/CFT correspondence, it is such states that should be compared to black holes in the bulk. We also discuss the discrepancy between thermal correlators and those computed in high energy primary states: the latter are reproduced instead by a generalized Gibbs ensemble with extreme values chosen for the chemical potentials conjugate to the KdV charges.

Paper Structure

This paper contains 16 sections, 106 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Current correlators
  3. Thermal correlator
  4. Microstate correlator
  5. Stress tensor correlators
  6. Two-point functions
  7. Typical states
  8. Typical state two-point function
  9. Statistics of Virasoro generator expectation values
  10. Discussion
  11. Current two-point function
  12. Equivalence of two forms of thermal correlator
  13. Minimal size of thermal correlator
  14. Technical results
  15. Higher point functions of the stress tensor
  16. ...and 1 more sections

Figures (1)

  • Figure 1: [Left] A random partition, i.e. a set of $N_n$ drawn from the distribution $P(N_n)$ in \ref{['Pn']} with $\beta~=~1, \, L=3\times 10^6$. [Right] Comparison of the term that differs between the thermal and microstate correlators (the third terms in \ref{['currb']} and \ref{['currc']} respectively). The microstate on the plot is defined by the $\{N_n\}$ from the left panel.