Quantum Simulation of Fermions in $AdS_2$ Black Hole: Chirality, Entanglement, and Spectral Crossovers

We consider free Dirac fermions on a discretized $AdS_2$ black hole background, and analyze how curved space redshift, horizons, and the spin connection induced chiral gravitational effect shape spectral, transport, and scrambling phenomena. The system is discretized via staggered fermions followed by the Jordan-Wigner transform to encode the model in qubit degrees of freedom, whose Hamiltonian carries site dependent warp factors and bond chirality terms encoding the redshift and spin connection effects. We calculate the ground state and first excited states energies, their local charge profiles, and their half-chain entanglement entropies, showing how redshift and chirality affect the transition from criticality to a gapped regime. Probing operator growth via out-of-time-order correlators, we find that horizons and the chiral coupling accelerate scrambling, yet remain within a non-chaotic regime. Finally, we map out an integrable to ergodic crossover via level-spacing statistics and Brody fits, and introduce on-site disorder to drive a many body localization transition.

💡 Research Summary

This paper presents a comprehensive study of free Dirac fermions propagating on a discretized two‑dimensional anti‑de Sitter (AdS₂) black‑hole background and demonstrates how such a system can be mapped onto qubits for quantum simulation. The authors begin by reviewing the continuum theory of a massive Dirac spinor in a curved spacetime with metric (ds^{2}= -f(r)dt^{2}+f^{-1}(r)dr^{2}) where (f(r)=r^{2}-r_{h}^{2}) (in units with AdS radius (L=1)). The spin connection (\omega^{t}_{01}=-r/L^{2}) generates a chiral gravitational term that will later appear as a two‑site antisymmetric hopping in the lattice model.

To obtain a lattice description, the authors employ the staggered‑fermion formalism. A two‑component Dirac field is encoded in a single Grassmann variable (\chi_{n}) per lattice site, thereby halving the degrees of freedom while preserving the correct continuum limit. The staggered fields are then transformed into Pauli operators ((X_{n},Y_{n},Z_{n})) via the Jordan‑Wigner map, which yields a spin‑½ Hamiltonian that can be directly implemented on a quantum processor.

The resulting qubit Hamiltonian contains three distinct contributions: (i) a site‑dependent hopping term (\propto \alpha_{n}^{2}(X_{n}X_{n+1}+Y_{n}Y_{n+1})) where (\alpha_{n}= \sqrt{r_{n}^{2}-r_{h}^{2}}/L) is the red‑shift factor; (ii) on‑site mass and chemical‑potential terms each weighted by a single power of (\alpha_{n}); and (iii) a chiral term (\propto \frac{1}{8L^{2}}\sum_{n} n (X_{n}Y_{n+1}-Y_{n}X_{n+1})) that is independent of the red‑shift. The red‑shift therefore modulates the kinetic energy and the effective local chemical potential, while the chiral term encodes the spin‑connection‑induced gravitational Chern–Simons‑like effect.

Exact diagonalization of systems with up to thirty sites is performed. The authors compute the ground‑state energy, the first excited‑state energy, and the corresponding charge density profiles (\langle Z_{n}\rangle). As the horizon radius (r_{h}) grows (i.e., as the red‑shift becomes stronger near the horizon), a gap opens in the spectrum, signalling a crossover from a critical, gapless Dirac liquid to a gapped regime. The charge profile becomes increasingly asymmetric, and the chiral term drives a net spin current localized near the boundary, reminiscent of the chiral vortical effect.

Entanglement is probed via the half‑chain von‑Neumann entropy (S_{\mathrm{EE}}). In the presence of both strong red‑shift and the chiral coupling, the usual logarithmic scaling of a 1D critical system is suppressed and the entropy saturates at a lower value, indicating a reduction of the effective central charge.

Dynamical scrambling is investigated through out‑of‑time‑order correlators (OTOCs) of local Pauli operators. Near the horizon and for larger chiral coupling, the early‑time growth of the OTOC is faster, but the growth remains polynomial rather than exponential; the Lyapunov exponent is effectively zero, confirming that the free‑fermion model stays integrable despite the gravitational background.

Spectral statistics are analyzed using the adjacent‑gap ratio (r) and Brody parameter (\beta). For weak red‑shift the level spacings follow Poisson statistics ((\beta\approx0)), while increasing red‑shift and chiral strength drives the distribution toward Wigner‑Dyson ((\beta\approx1)), although full level repulsion is never achieved. This demonstrates an integrable‑to‑ergodic crossover that is driven purely by geometric effects rather than interactions.

Finally, the authors introduce on‑site random potentials (\epsilon_{n}) to study many‑body localization (MBL). Because the red‑shift creates an intrinsic inhomogeneity in the effective hopping amplitudes, the disorder threshold for localization is reduced compared with a homogeneous chain. Numerical diagnostics (e.g., decay of a Néel‑state imbalance) confirm a transition from an ergodic to an MBL phase as the disorder strength is increased, with the critical disorder decreasing as the horizon radius grows.

In summary, the work provides a concrete, experimentally relevant qubit Hamiltonian that captures key gravitational phenomena—red‑shift, spin‑connection‑induced chirality, horizon‑driven scrambling, and disorder‑enhanced localization. It bridges high‑energy concepts such as holographic black‑hole physics with quantum‑information tools (entanglement entropy, OTOCs, level statistics) and offers a realistic pathway for tabletop quantum simulations of curved‑spacetime fermion dynamics.