Holographic Quantum Scars: Boson Stars as Islands of Integrability in a Chaotic Sea ✦
Mini-boson stars in asymptotically AdS are considered as a holographic candidate for many-body quantum scars. Their spectrum exhibits features of chaos from random matrices but contains embedded integrable branches. The entire gravitational system, including black holes, is chaotic, and most states obey the Eigenstate Thermalization Hypothesis (ETH). However, the boson star macrostate belongs to a near-integrable sector, manifested by anomalously low entanglement compared to black holes and persistent revivals in Krylov complexity. The spectral, entropic, and dynamical evidence points to the realization of holographic quantum scars in a self-gravitating system and opens a new connection between scar physics, quantum chaos, and horizonless dynamics.
Context
Understanding how isolated quantum systems reach thermal equilibrium, and how this process can be violated, remains a central problem at the intersection of quantum chaos, statistical mechanics, and gravity. A classic exception to the ergodic hypothesis is provided by quantum many-body scars — atypical non-thermal states immersed in a chaotic spectrum that exhibit hidden integrability, suppressed entanglement, and anomalous revivals. First discovered in Rydberg atoms, scars are now identified in many models. In parallel, holographic systems offer an arena to study thermalization in strongly coupled quantum matter: black holes correspond to maximally chaotic thermal states obeying the eigenstate thermalization hypothesis (ETH), while horizonless geometries can encode non-thermal sectors. This distinction raises the question: can quantum scars have a gravitational or holographic realization?
Methods
The authors investigated the simplest horizonless solution — a spherically symmetric static mini-boson star in 3+1-dimensional anti-de Sitter (AdS) space within Einstein gravity minimally coupled to a complex scalar field. Families of regular background solutions were first constructed via a shooting method. Then a linear perturbation analysis was performed in the parity-even sectors with multipole numbers ℓ = 0 and ℓ ≥ 2. The normal mode spectrum was obtained numerically and analyzed in the WKB approximation for high frequencies. To diagnose quantum chaos, random matrix theory (RMT) was applied: the average ratio of consecutive spacings ⟨r⟩ was computed for low-lying modes. Entanglement entropy was estimated holographically via the Ryu–Takayanagi (RT) formula, and dynamics was probed using Krylov complexity, where operator growth was modeled by the infall of a massive test particle into the star along a timelike geodesic.
Results
The normal mode spectrum of the boson star shows a striking duality. At low mode numbers, the level spacing distribution conforms to random matrix statistics: the mean gap ratio ⟨r⟩ falls in the regime between the Gaussian Orthogonal Ensemble (GOE, ≈0.536) and the Symplectic Ensemble (GSE, ≈0.676), indicating global chaos. However, at high frequencies, the modes split into distinct branches with nearly equidistant levels — a hallmark of hidden integrability. This coexistence is the spectral signature of scars. A comparison of holographic entanglement entropy for a hemisphere showed that boson stars have significantly lower entropy than Schwarzschild–AdS black holes of the same mass: for mass M ≈ 5, the star's entropy is about one-third that of the black hole. Finally, Krylov complexity for boson stars experiences clear periodic revivals with period Trev ≥ Tg, where Tg is the oscillation period of the ground state. For comparison, in a black hole, complexity grows monotonically. The revival period increases with the star's mass and is larger for excited states, reflecting the 'dressed' collective dynamics of the scarred subsector.
Implications
This work presents, for the first time, compelling evidence that mini-boson stars in AdS serve as holographic realizations of quantum many-body scars. It unifies three seemingly disparate fields: the physics of scars in many-body systems, quantum chaos in holography, and horizonless gravitational dynamics. The discovered coexistence of chaotic low-frequency statistics and integrable high-frequency tower structures echoes the axiom that scars are not isolated integrable systems but rather weak violations of ETH in overwhelmingly chaotic environments. The suppressed entanglement entropy and revivals of Krylov complexity further confirm the non-thermal, scar-like character of these configurations, prompting a reassessment of the role of coherent gravitational structures in thermalizing quantum matter.
Future development
The results pave the way to search for other horizonless geometries capable of realizing scars: rotating boson stars, fermion stars, wormholes, and even dynamical collapse scenarios. One can expect a whole zoo of scarred states in holographic systems, each with its own spectral, entanglement, and dynamical portrait. Studying transitions between these states and black holes could shed light on mechanisms of thermal equilibrium violation in quantum gravity and resolve paradoxes related to information loss.
Impact
The results will impact quantum chaos theory, condensed matter physics (where scars are studied experimentally), and quantum gravity, especially in the context of the AdS/CFT correspondence.
Next steps
Immediate tasks include studying the stability of scars under rotation, exploring dynamical revivals in nonlinear regimes, and seeking analogies in flat spacetime, potentially linked to astrophysical objects.
Key open problems
The discovered connection directly addresses unresolved problems: the mechanism of thermalization in closed quantum systems, the nature of quantum chaos in gravity, and the black hole information paradox, as scars offer natural non-thermal channels of evolution.
🎯 Boson stars — hypothetical objects made of a self-gravitating condensate of ultra-light bosons — may constitute part of dark matter, and their observational signatures (microlensing, gravitational waves) are actively searched for.
🎬 In the movie Interstellar, the heroes enter a black hole; a boson star, however, lacks a horizon, so a traveler could pass through and return — almost like a wormhole, but without singularities.
Key numbers
- Mean gap ratio for GOE: 0.536
- Mean gap ratio for GSE: 0.676
- Maximum mass of studied boson stars: about 5 in dimensionless units
- Revival period of Krylov complexity for the ground state: T_g = \pi in the vacuum limit
- Entanglement entropy of boson star relative to black hole of the same mass: approximately 70% smaller