Influence Through Mixing: Hotspots as Benchmarks for Basic Black-Hole Behaviour

Abstract

Effective theories are being developed for fields outside black holes, often with an unusual open-system feel due to the influence of large number of degrees of freedom that lie out of reach beyond the horizon. What is often difficult when interpreting such theories is the absence of comparisons to simpler systems that share these features. We propose here such a simple model, involving a single external scalar field that mixes in a limited region of space with a 'hotspot' containing a large number of hot internal degrees of freedom. Since the model is at heart gaussian it can be solved explicitly, and we do so for the mode functions and correlation functions for the external field once the hotspot fields are traced out. We compare with calculations that work perturbatively in the mixing parameter, and by doing so can precisely identify its domain of validity. We also show how renormalization-group EFT methods can allow some perturbative contributions to be resummed beyond leading order, verifying the result using the exact expression.

Qubit heating near a hotspot

“…Such benchmarks can be useful both as laboratories for exploring new EFTs features in these new kinds of environments, and as comparison points for calculations done with more realistic but harder-to-solve practical systems. It was with the view to providing one of these benchmarks that reference [23] proposed a solvable Caldeira-Leggett style [24,25] model consisting of an external massless quantum field φ interacting with many unseen gapless thermal fields in a very small spatial volume (called a 'hotspot'). Some implications of this model, such as for the coherence of the state of the field φ, are explored in a companion paper [26].…”

“…Among other things, equilibrium turns out to require both proximity to the hotspot -with |x Q | g/4π -and sufficiently high hotspot temperatures -T ω. For qubits much further from the hotspot than the hotspot's size these conditions require the response of the field φ to the hotspot to be itself understood in a regime beyond the JHEP08(2021)132 domain of perturbation theory in g; a regime for which this response is nonetheless known (and given in [23]) because of the solvability of the model.…”

“…This section briefly reviews the main features of the benchmark hotspot model proposed in [23], whose interactions with the Unruh-DeWitt detector are to be studied.…”

“…The two types of fields only couple to one another in the localized throat region, which can be taken to be a small sphere of radius ξ, or effectively a point in the limit that ξ is much smaller than all other scales of interest. (figure taken from [23]. )…”

“…A simple solvable Caldeira-Leggett type model of a quantum field interacting within a small area with many unmeasured thermal degrees of freedom was recently proposed in ref. [23] to provide a toy model of this kind of dynamics against which more complete black hole calculations might be compared. We here compute the response of a simple Unruh-DeWitt detector (or qubit) interacting with a massless quantum field φ coupled to such a hotspot.…”