[Paper Review] Exactly Stable Collective Oscillations in Conformal Field Theory

Benjamin W. Freivogel, John McGreevy+1 more|DSpace@MIT (Massachusetts Institute of Technology)|Sep 27, 2011

Cold Atom Physics and Bose-Einstein Condensates3 citations

TL;DR

This paper demonstrates that any conformal field theory (CFT) on a sphere supports exactly stable, undamped collective oscillations with frequencies that are integer multiples of the inverse sphere radius, due to $sl(2,\mathbb{R})$ subalgebras of the conformal algebra. These oscillations persist indefinitely from generic initial conditions and are robust in the large-$N$ limit, challenging conventional thermalization expectations in holographic CFTs.

ABSTRACT

Any conformal field theory (CFT) on a sphere supports completely undamped collective oscillations. We discuss the implications of this fact for studies of thermalization using AdS/CFT. Analogous oscillations occur in Galilean CFT, and they could be observed in experiments on ultracold fermions.

Motivation & Objective

• To identify and characterize a class of non-stationary, exactly stable collective oscillations in conformal field theories on a sphere.
• To explain how these oscillations arise from $sl(2,\mathbb{R})$ subalgebras of the conformal algebra and are tied to conserved charges from conformal Killing vectors.
• To demonstrate that these oscillations survive in the large-$N$ limit, contradicting conventional expectations of thermalization in interacting quantum field theories.
• To connect these oscillations to holographic descriptions, showing that boosted black holes in AdS correspond to oscillating CFT states.
• To explore experimental realizations in ultracold fermionic atoms with spherical harmonic traps, where such modes could be observed if symmetry is preserved.

Proposed method

• Constructing oscillating states explicitly via ladder operators derived from the conformal algebra, particularly $L_{\pm} = H - \omega_0^2 C \pm i\omega_0 D$, which act on primary states.
• Using the $sl(2,\mathbb{R})$ algebra to show that energy eigenstates evolve periodically with fixed frequency $2\omega_0$, analogous to coherent states in a harmonic oscillator.
• Analyzing the stress-energy tensor moments in CFT on a sphere, showing that $\ell=1$ components behave as harmonic oscillators with undamped oscillations.
• Applying the holographic principle to show that a boosted Schwarzschild-AdS black hole corresponds to a CFT state with periodic correlators and conserved oscillation amplitude.
• Extending the analysis to Galilean CFTs with a harmonic potential, where the Hamiltonian $H_{\text{osc}} = H + \omega_0^2 C$ supports stable nonlinear modes.
• Using linearized hydrodynamics and symmetry-based arguments to show that damping vanishes for the $2\omega_0$ mode in spherical traps, consistent with the prediction of infinite lifetime.

Experimental results

Research questions

• RQ1 Do conformal field theories on a sphere support non-stationary, undamped collective oscillations due to conformal symmetry?
• RQ2 What is the role of $sl(2,\mathbb{R})$ subalgebras in generating stable, periodic time evolution in CFTs?
• RQ3 How do these oscillations affect the thermalization process in holographic CFTs, particularly in the large-$N$ limit?
• RQ4 Can these oscillations be realized experimentally in ultracold fermionic atoms under spherical harmonic trapping?
• RQ5 Why do certain collective modes in trapped Fermi gases remain undamped when standard hydrodynamic models predict dissipation?

Key findings

• Any CFT on a sphere supports a class of non-stationary states whose time evolution is periodic with frequencies that are integer multiples of the inverse sphere radius.
• The oscillations are exactly stable and undamped due to conserved charges associated with conformal Killing vectors, with the amplitude of oscillation itself being a conserved quantity.
• In holographic CFTs, these oscillations correspond to boosted black holes in AdS, which 'slosh' back and forth forever due to exact conformal symmetry.
• The $2\omega_0$ breathing mode in a spherical harmonic trap is predicted to have infinite lifetime in the absence of anisotropy, consistent with the absence of shear viscosity effects in the symmetric case.
• These modes are superuniversal and generalize to any CFT, including Galilean CFTs, where they are constructed using the Hamiltonian $H_{\text{osc}} = H + \omega_0^2 C$ and ladder operators.
• The existence of these modes contradicts the conventional expectation that generic initial states in interacting theories thermalize, as these states never approach a stationary configuration.