Niels Bohr Institute Defines Black Hole States Using Thermal Correlators

Researchers at the Niels Bohr Institute and Nordita have defined a black hole state using a quantum spin chain model, calculating an effective central charge of approximately 5.2 for the entanglement entropy, which grows logarithmically with this central charge. By applying the Heisenberg model, the team offers an integrable system for investigating these phenomena, revealing a connection between condensed matter physics and the study of black holes. They also find evidence for thermalization at infinite temperature. This work builds upon the idea that at the heart of the gauge, gravity duality lies a remarkable identification of a thermal ensemble in field theory with a black hole in the dual gravitational description.

Holographic Definition of Black Hole States

A calculated effective central charge of approximately 5.2 was determined. The entanglement entropy grows logarithmically with this effective central charge, as derived from studying the spin chain model. The researchers state that this work builds upon the idea that at the heart of the gauge, gravity duality lies a remarkable identification of a thermal ensemble in field theory with a black hole in the dual gravitational description. The team, based at the Niels Bohr Institute, Copenhagen University and Nordita, KTH Royal Institute of Technology and Stockholm University, utilized the Heisenberg model, a well-established system in condensed matter physics, as the specific quantum spin chain to investigate these states, creating a link between seemingly disparate fields. They constructed this black hole state by summing all non-crossing pairings of spins, a method rooted in the quantum mechanical theory of valence originally proposed by Rumer.

This approach reflects the chaotic nature inherent to black holes. The resulting state is not merely a mathematical construct; it is designed to break the integrability of the Heisenberg model, mirroring the chaotic behavior expected within a black hole. “The black-hole state should be a spin singlet, as black holes have no hair, and it should be sufficiently generic to reflect a black hole’s chaotic nature,” the team explains. This condition, observed through the properties of the spin chain, provides a testable prediction about the behavior of matter under the intense gravitational forces near a black hole’s event horizon. They are now probing the black-hole state with standard diagnostics of quantum chaos, including entanglement entropy and eigenvalue thermalization, to further refine their understanding of these complex systems and their holographic counterparts.

Integrability Breaking in the Heisenberg Model

The pursuit of understanding black holes through quantum mechanics has led researchers to explore connections between condensed matter physics and gravity. Current investigations focus on modeling black hole states using quantum spin chains, specifically the Heisenberg model, an approach that leverages the model’s inherent integrability as a starting point for studying more complex, chaotic systems. This allows for quantifiable analysis of properties like entanglement entropy and complexity, offering insights into the quantum structure of black holes. Recent work at the Niels Bohr Institute and Nordita, detailed in a pre-print publication, centers on defining black hole states within this Heisenberg model framework. Crucially, the Heisenberg model, while initially integrable, undergoes a transition to chaos when non-zero λ breaks integrability.

The researchers deliberately introduced a non-zero λ, breaking integrability and creating a fully chaotic system for λ greater than or equal to 0.5. This manipulation is designed to assess how much the thermalization of the black hole state depends on the chaotic nature of the Hamiltonian. The team found that the black-hole state is expected to break integrability, and they will confirm this expectation by demonstrating that Q₃|BH⟩ is not equal to 0 at length six and higher. The number of pairings for L spins is the Catalan number C_L/2, which defines the dimension of the singlet subspace for the spin chain of length L.

Their work, detailed in a recent preprint, constructs black hole states within the framework of quantum spin chains, offering a novel approach to understanding these enigmatic objects. A key element of their methodology involves the Rumer relation, a result originally developed for quantum mechanical valence, to systematically enumerate spin singlets. This approach allows for the creation of a state that is both mathematically manageable and representative of the complex quantum environment surrounding a black hole. Further analysis focused on quantifying the entanglement within these constructed states. The Heisenberg model, while integrable in its basic form, exhibits chaotic behavior when non-zero λ is introduced, mirroring the expected dynamics of a real black hole.

The researchers discovered that the number of possible non-crossing pairings of spins within the chain, which dictates the dimension of the singlet subspace, is precisely described by the Catalan number C_L/2. Further analysis revealed that the black-hole state is manifestly SU(2)-invariant, changes sign under cyclic permutations, and evenly covers the singlet subspace of the spin chain’s Hilbert space.

Conventional depictions of black holes often portray them as simple gravitational sinks, yet recent work suggests a far more complex quantum structure is at play. This isn’t merely an abstract mathematical exercise; the team’s formulation leverages the established framework of integrability to probe the chaotic nature inherent to these cosmic objects. The team’s approach utilizes a matrix integral formulation, where the wavefunction of the black hole state is expressed as a limit of an integral over Hermitian matrices with Grassmann entries. This integration produces all possible spin pairings, with only planar contractions surviving in the large-N limit.

The research reveals a quantifiable property of the quantum structure within simulated black holes, a result achieved by researchers at the Niels Bohr Institute and Nordita, KTH Royal Institute of Technology and Stockholm University through a novel application of spin chain modeling.