Holographic Butterfly Effect at Quantum Critical Points

TL;DR

This work proposes that the butterfly velocity $v_B$, a key quantity in quantum chaos, can diagnose quantum phase transitions in holographic theories by probing IR fixed points through horizon data. Focusing on a holographic Q-lattice model that exhibits metal–insulator transitions, the authors show that the zero-temperature QCPs correspond to local extrema of $∂_k v_B$, and that $v_B$ scales differently with temperature in metallic ($v_B \sim T^{1/2}$) versus insulating phases, reflecting distinct IR geometries. The analysis is extended to anisotropic settings, where $\mathfrak{v}_B(θ)$ depends on direction but still signals QCPs via $∂_k \mathfrak{v}_B(θ)$ maxima, highlighting an information-theoretic route to identifying QPTs. The results suggest experimental tests using OTOC measurements and motivate applying this diagnostic to a broader class of holographic and non-holographic chaotic systems, with potential practical implications for probing quantum criticality in strongly correlated materials.

Abstract

When the Lyapunov exponent $λ_L$ in a quantum chaotic system saturates the bound $λ_L\leqslant 2πk_BT$, it is proposed that this system has a holographic dual described by a gravity theory. In particular, the butterfly effect as a prominent phenomenon of chaos can ubiquitously exist in a black hole system characterized by a shockwave solution near the horizon. In this paper we propose that the butterfly velocity can be used to diagnose quantum phase transition (QPT) in holographic theories. We provide evidences for this proposal with an anisotropic holographic model exhibiting metal-insulator transitions (MIT), in which the derivatives of the butterfly velocity with respect to system parameters characterizes quantum critical points (QCP) with local extremes in zero temperature limit. We also point out that this proposal can be tested by experiments in the light of recent progress on the measurement of out-of-time-order correlation function (OTOC).

Figures (7)

• Figure 1: MIT phase diagram at T = 0.001 Ling:2015dma.
• Figure 2: $v_B$ v.s. $k$ at different low temperatures $T=10^{-4},10^{-5},10^{-9},10^{-11}$ respectively. In each plot the dotted line in red represents the location of QCP, separating the insulating phase (left side) and metallic phase (right side).
• Figure 3: The left plot is $\partial_k v_B \,v.s.\, k$ at $T=10^{-11}$, in which the red vertical line represents the position of the critical point while the blue line denotes the position of the local maximum of $\partial_k v_B$. The right plot is for the temperature dependence of $\Delta k$.
• Figure 4: $Tv'_B/v_B \,v.s.\, T$ for different phases ($k=0.500,\,0.805$ corresponds to metallic phases and $k=1.31,\,1.50$ corresponds to insulating phases). The purple dashed line points to $Tv'_B/v_B=0.5$.
• Figure 5: $\mathfrak{v}_B (\theta)\,v.s.\,\theta$ at $\lambda=2, T = 10^{-11}$ with $k$ specified by the plot legend.