Breaking the Entanglement-Structure Trade-off: Many-Body Localization Protects Emergent Holographic Geometry in Random Tensor Networks

Abstract

We present a systematic numerical investigation of the "entanglement geometry gravity" chain in random tensor networks (RTN) established by the ER EPR conjecture and Jacobson's thermodynamic derivation. First, we verify the kinematic foundation: the entanglement first law $δ\langle K\rangle=δS$ (slope=1.000), the encoding of geometry by mutual information (correlation=0.92), and the locality of holographic perturbations (3.3x). We also confirm that gravitational dynamics (JT gravity) does not emerge, identifying a sharp kinematics-dynamics boundary. Second, and more importantly, we discover that many-body localization (MBL) is the mechanism that protects emergent holographic geometry from thermalization. Replacing Haar-random evolution (geometry lifetime $t\sim6$) with an XXZ Hamiltonian plus on-site disorder, we observe a finite-size crossover at disorder strength $W_c\approx10-12$ above which mutual-information-lattice correlations persist indefinitely ($r>0.5$ for $t>50$). We map the full parameter space: the optimal regime is a near-Ising anisotropy $Δ\approx50$ with $W=30$ yielding $r=0.779\pm0.002$ (confirmed by a fine scan over $Δ\in[30,70]$); only holographic (RTN) initial states sustain geometry, while product, Néel, and Bell-pair states do not. MBL preserves the spatial structure of entanglement (adjacent/non-adjacent MI ratio ~2.6-4.2x vs. 1.0x in the thermal phase), rather than its total amount. A comparison with classical cellular automata reveals that MBL uniquely breaks the entanglement-structure trade-off imposed by quantum monogamy: classical systems achieve spatial structure only at the cost of negligible mutual information, while MBL sustains both.

Figures (6)

• Figure 1: Geometric encoding of entanglement. (a) MI distance $1/I(i,j)$ vs. lattice Manhattan distance on the $3\times3$ lattice at $\chi=4$, showing strong correlation ($r=0.92$). (b) RT ratio and MI/lattice correlation as functions of $\chi$. The RT ratio increases monotonically, while MI/lattice correlation saturates at $r=0.92$ for $\chi\geq3$.
• Figure 2: Entanglement first law and locality. (a) $\delta\langle K_A\rangle$ vs. $\delta S(A)$ across all subsystem sizes and boundary positions (slope${}=1.000$, $r=1.000$). (b) $|\delta S(i)|$ at each boundary site when the bulk tensor at $(1,1)$ is perturbed. Red bars indicate sites adjacent to the perturbation, demonstrating $2.8\times$ locality.
• Figure 3: JT gravity false-positive diagnosis. (a) Regge deficit angle changes $\delta\epsilon$ vs. dilaton changes $\delta\Phi$: most data points are locked at $\delta\epsilon=0$ (blue), with only 3 non-zero points (red crosses) producing the spurious $r=0.968$. (b) Ollivier--Ricci curvature changes $\delta\kappa$ vs. $\delta\Phi$: all vertices contribute, revealing no significant correlation ($r\approx 0$).
• Figure 4: MBL protects holographic geometry. (a) Late-time MI/lattice correlation vs. disorder $W$ for $\Delta=1$, showing the crossover at $W_c \approx 10$--$12$. Error bars are standard errors of the mean over $N_{\rm dis}=100$ disorder realizations. (b) Late-time correlation vs. anisotropy $\Delta$ at $W=30$, peaking at $\Delta \approx 50$ ($r = 0.78$). The pure Ising limit ($\Delta\to\infty$) drops due to insufficient entanglement. (c) Entanglement spectrum effective dimension $N_{\rm eff}$ vs. time for $W=0$ (thermal) and $W=30$ (MBL).
• Figure 5: The entanglement--structure trade-off. Only MBL occupies the "golden quadrant" (high $S/S_{\rm max}$ and high $\mathcal{L}$). Color encodes geometry correlation $r$. Classical systems (not shown) achieve $\mathcal{L}>1$ only with negligible MI. Thermal evolution melts geometry ($\mathcal{L}\to 1$); the pure Ising limit lacks entanglement. Points colored by late-time geometry correlation.