Chaos, Diffusivity, and Spreading of Entanglement in Magnetic Branes, and the Strengthening of the Internal Interaction

TL;DR

This work uses holography to study chaos, diffusion, and entanglement spread in a finite-temperature gauge theory with a constant magnetic field, implementing a gravity dual that flows from a UV AdS$_5$ fixed point to an IR BTZ×R$^2$ fixed point as controlled by $\mathcal{B}/T^2$. By analyzing shock waves, mutual information, and entanglement entropy, the authors show that the entanglement and butterfly velocities become anisotropic and tend to increase in the IR, while still respecting causality; the Lyapunov exponent saturates $\lambda_L=2\pi/\beta$ and the diffusion bound $D_c \geq (3/2) v_B^2 \tau_L$ holds along all flows. The magnetic field strengthens internal interactions, as evidenced by enhanced two-sided mutual information and altered entanglement spreading, with IR limits reflecting dimensional reduction. The study also clarifies the relation between chaos and diffusion, finding that diffusion eigenvalues do not generally bound chaos velocities, though Blake-type bounds on $D_c$ remain valid. Overall, the paper provides a coherent holographic picture of how magnetic fields shape chaos, diffusion, and entanglement in strongly coupled plasmas and highlights their utility as probes of RG flow.

Abstract

We use holographic methods to study several chaotic properties of a super Yang-Mills theory at temperature $T$ in the presence of a background magnetic field of constant strength $\mathcal{B}$. The field theory we work on has a renormalization flow between a fixed point in the ultraviolet and another in the infrared, occurring in such a way that the energy at which the crossover takes place is a monotonically increasing function of the dimensionless ratio $\mathcal{B}/T^2$. By considering shock waves in the bulk of the dual gravitational theory, and varying $\mathcal{B}/T^2$, we study how several chaos-related properties of the system behave while the theory they live in follows the renormalization flow. In particular, we show that the entanglement and butterfly velocities generically increase in the infrared theory, violating the previously suggested upper bounds but never surpassing the speed of light. We also investigate the recent proposal relating the butterfly velocity with diffusion coefficients. We find that electric diffusion constants respect the lower bound proposed by Blake. All our results seam to consistently indicate that the global effect of the magnetic field is to strengthen the internal interaction of the system.

Figures (17)

• Figure 1: Penrose diagram for the two-sided black branes we consider. This geometry is dual to a thermofield double state $|TFD \rangle$ made by entangling two copies of the boundary theory.
• Figure 2: Penrose diagram for the shock wave geometry. This geometry is dual to a thermofield double state perturbed at a time $t_0$ in the far past.
• Figure 3: Butterfly velocity squared $v_\textrm{\tiny B}^2$ versus the dimensionless parameter $\mathcal{B}/T^2$. The blue curve represents the butterfly velocity along the direction of the magnetic field, while the red curve stands for the butterfly velocity along any direction perpendicular to the magnetic field. The horizontal lines represent either the conformal result $v_\textrm{\tiny B}^2=2/3$ or the the speed of light.
• Figure 4: (a) Schematic representation of the $t=0$ slice of the two-sided black brane geometry. (b) Schematic representation of the shock wave geometry, in which the wormhole becomes longer. In both cases the blue curves represent the U-shaped extremal surfaces $\gamma_A$ (in the left side of the geometry) and $\gamma_B$ (in the right side of the geometry). The red curves represent the extremal surfaces $\gamma_1$ and $\gamma_2$ connecting the two sides of the geometry . The extremal surface $\gamma_\text{wormhole}$ defined in the text is given by $\gamma_\text{wormhole}=\gamma_1 \cup \gamma_2$.
• Figure 5: (a) Mutual Information (in units of $V_2/G_\textrm{\tiny N}$) as a function of the strip's width $\ell$ for several values of $\mathcal{B}/T^2$. The curves correspond to $\mathcal{B}/T^2 = 0$ (black curve), $\mathcal{B}/T^2 = 14.8$ (blue curves), $\mathcal{B}/T^2 = 21.2$ (purple curves), $\mathcal{B}/T^2 = 27.5$ (red curves). (b) Mutual information (in units of $V_2/G_\textrm{\tiny N}$) versus $\mathcal{B}/T^2$ for strips of fixed width $\ell/\ell_{AdS} =0.75$. The continuous (dashed) curves correspond to the results for orthogonal (parallel) strips.