Fermionic pole-skipping in holography

TL;DR

This work extends the holographic pole-skipping paradigm to minimally coupled fermions, showing that at negative imaginary fermionic Matsubara frequencies there exist special momenta where the bulk Dirac equation admits multiple ingoing solutions and the boundary retarded Green's function becomes multivalued. The authors develop systematic horizon-based methods—via second-order and first-order formulations—to locate all pole-skipping points in AdS$_3$ and higher dimensions, including detailed BTZ and Schwarzschild-AdS examples. They demonstrate explicit pole-zero intersections that match exact fermionic Green's functions, including half-integer conformal dimensions, and discuss anomalous points where the pole-skipping form acquires subtle behavior. The results reveal universal near-horizon structures for fermionic correlators, highlight differences from bosonic cases, and suggest broader generalizations to other bulk fields and potential links to chaotic dynamics.

Abstract

We examine thermal Green's functions of fermionic operators in quantum field theories with gravity duals. The calculations are performed on the gravity side using ingoing Eddington-Finkelstein coordinates. We find that at negative imaginary Matsubara frequencies and special values of the wavenumber, there are multiple solutions to the bulk equations of motion that are ingoing at the horizon and thus the boundary Green's function is not uniquely defined. At these points in Fourier space a line of poles and a line of zeros of the correlator intersect. We analyze these `pole-skipping' points in three-dimensional asymptotically anti-de Sitter spacetimes where exact Green's functions are known. We then generalize the procedure to higher-dimensional spacetimes. We also discuss the special case of a fermion with half-integer mass in the BTZ background. We discuss the implications and possible generalizations of the results.

Figures (3)

• Figure 1: Plots of the locations of pole-skipping points for the fermionic Green's function in the BTZ black hole background. The top row is for $m=0$ and the bottom row shows the locations for $m=1$. The left column shows only the locations of the pole-skipping points as predicted from the near horizon analysis. The gray points correspond to the momentum written with a positive sign in and the hollow points correspond to the momenta with a negative sign as written in \ref{['eq:pspointsBTZ']}. Comparing the top left and bottom left panel we notice that, by increasing the mass, the gray points get rigidly translated to the right by the value of $m$ and the hollow points get translated by an equal amount to the left. The right column has superimposed the lines of zeros (red, dashed) from \ref{['eq:btzgzeros']} and lines of poles (blue) from \ref{['eq:btzgpoles']}. For both values of the mass the near-horizon analysis predicts the location of the intersections of lines of zeros and lines of poles.
• Figure 2: Pole-skipping points as predicted from the near-horizon analysis for half-integer mass values. The left plot are the locations for $m = \tfrac12$ and the right plot contains the locations for $m = \tfrac32$. The gray points correspond to the momentum written with a positive sign in and the hollow points correspond to the momenta with a negative sign as written in \ref{['eq:pspointsBTZ']}. We see that at half-integer values of the mass, some of the locations overlap (black circles with gray filling). These cases correspond to so-called anomalous points (see appendix \ref{['app:form']} for details) and signal that a more thorough analysis of the boundary Green's function is needed.
• Figure 3: Comparison of the locations of the pole-skipping points predicted by the near-horizon analysis (gray and hollow points) and the locations of the intersections of the lines of poles (blue) and lines of zeros (red, dashed) of the exact boundary retarded Green's function for half-integer values of the conformal dimension. We see that the non-anomalous pole-skipping points (either hollow or gray, but not gray with black circle) perfectly match the locations of the intersections. The anomalous pole-skipping points (gray with black boundary) correspond to the locations where two lines of poles intersect. The physical interpretation of these anomalous points is still unclear.