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arXiv:2512.23432 · Physical Review Letters

Black Hole States in Quantum Spin Chains

Charlotte Kristjansen, Konstantin Zarembo

Published Jun 4, 2026• 1 citation
arXivHTMLPDF

Abstract

We define a black hole state in a spin chain by studying thermal correlators in holography. Focusing on the Heisenberg model we investigate the thermal and complexity properties of the black hole state by evaluating its entanglement entropy, emptiness formation probability, and Krylov complexity. The entanglement entropy grows logarithmically with effective central charge c≃5.2. We also find evidence for thermalization at infinite temperature.

DOI
10.1103/5myp-jksn

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We define a black hole state in a spin chain by studying thermal correlators in holography. Focusing on the Heisenberg model we investigate the thermal and complexity properties of the black hole state by evaluating its entanglement entropy, emptiness formation probability, and Krylov complexity. The entanglement entropy grows logarithmically with effective central charge c≃5.2. We also find evidence for thermalization at infinite temperature.
To model the interior of a black hole, a study is made of a spin system with long-range random four-spin couplings that exhibits quantum chaos. The black hole limit corresponds to a system where the microstates are approximately degenerate and equally likely, corresponding to the high temperature limit of the spin system. At the leading level of approximation, reconstruction of bulk physics implies that local probes of the black hole should exhibit free propagation and unitary local evolution. We test the conjecture that a particular mean field Hamiltonian provides such a local bulk Hamiltonian by numerically solving the exact Schrodinger equation and comparing the time evolution to the approximate mean field time values. We find excellent agreement between the two time evolutions for timescales smaller than the scrambling time. In earlier work, it was shown bulk evolution along comparable time slices is spoiled by the presence of the curvature singularity, thus the matching found in the present work provides evidence of the success of this approach to interior holography. The numerical solutions also provide a useful testing ground for various measures of quantum chaos and global scrambling. A number of different observables, such as entanglement entropy, out-of-time-order correlators, and trace distance are used to study these effects. This leads to a suitable definition of scrambling time, and evidence is presented showing a logarithmic variation with the system size.
We study the quantum correlations and complexity of simulation, characterized by quantum mutual information and entanglement entropy in operator space, respectively, for thermal states in critical, noncritical, and quantum chaotic spin chains. A simple general relation between the two quantities is proposed. We show that in all cases mutual information and entanglement entropy saturate with system size, whereas as a function of the inverse temperature, we find logarithmic divergences for critical cases and uniform bounds in noncritical cases. A simple efficient quasiexact method for computation of arbitrary entropy-related quantities in thermalized XYXYXY spin chains is proposed.
We study the entanglement entropy of blocks of contiguous spins in non-periodic (quasi-periodic or more generally aperiodic) critical Heisenberg, XX and quantum Ising spin chains, e.g. in Fibonacci chains. For marginal and relevant aperiodic modulations, the entanglement entropy is found to be a logarithmic function of the block size with log-periodic oscillations. The effective central charge, c eff , defined through the constant in front of the logarithm may depend on the ratio of couplings and can even exceed the corresponding value in the homogeneous system. In the strong modulation limit, the ground state is constructed by a renormalization group method and the limiting value of c eff is exactly calculated. Keeping the ratio of the block size and the system size constant, the entanglement entropy exhibits a scaling property, however, the corresponding scaling function may be nonanalytic.
The entanglement entropy between quantum fields inside and outside a black hole horizon is a promising candidate for the microscopic origin of black hole entropy. We show that the entanglement entropy may be defined in loop quantum gravity, and compute its value for spin network states. The entanglement entropy for an arbitrary region of space is expressed as a sum over punctures where the spin network intersects the region's boundary. Our result agrees asymptotically with results previously obtained from the isolated horizon framework, and we give a justification for this agreement. We conclude by proposing a new method for studying corrections to the area law and its implications for quantum corrections to the gravitational action.
This thesis studies the complexity, thermalization, and entanglement properties of quantum mechanical systems and the implications of their behavior in high energy and condensed matter models. In Chapter 2, we use the geometry of Lie groups to study the circuit complexity of quantum mechanical systems and develop the technology to analytically predict the appearance of the obstruction to the complexity growth of integrable systems. We simulate the spin and fermionic systems, in which the amount of chaos can be controlled, and conjecture the close relation between quantum complexity and thermalization. In Chapter 3, we apply quantum information theory to an evaporating black hole in Jackiw-Teitelboim gravity in the context of AdS/CFT as a quantum error correction protocol. By calculating the entropy measures using gravitational path integrals, we establish the criterion for an error in the Hawking radiation to be correctable and identify the transition in the reversibility of the error channel as the Page transition. In Chapter 4, we study a 1+1 dimensional random circuit with measurement and dissipation. We map this type of quantum system to a classical Ising model analytically and show that the entanglement dynamics can be understood as the thermodynamics of the domain wall formed in the spin model, as one of the spatial dimensions stretches. We then confirm the calculated time scales in the classical theory qualitatively with simulations of realistic qubit circuits. Overall, we present a holistic picture of quantum systems, from those evolving with pure unitary, to those evolving also with influence from the environment. The complexity of the theory and entanglement of the states exhibit universal behavior at early times (∼ O(N)) and distinct long-time behavior, determined by their thermalization properties in spin chains, toy models of black holes in AdS/CFT, and monitored random circuits with dissipation.