Butterfly Velocity in Quantum Systems

• Butterfly velocity is a measure of how quickly local quantum perturbations spread, defining an effective state-dependent light cone.
• It is derived from out-of-time-order correlators and shockwave analyses, linking operator growth to quantum chaos and scaling behavior near criticality.
• Experimental protocols and holographic models use butterfly velocity to diagnose transport bounds and phase transitions in many-body quantum systems.

The butterfly velocity, typically denoted $v_B$, quantifies the speed at which a local quantum perturbation scrambles information in many-body quantum systems. It sets the effective “light cone” for the space-time propagation of operator commutators or, equivalently, the spread of chaos as measured by out-of-time-order correlators (OTOCs). The butterfly velocity is both a diagnostic of quantum information dynamics and a fundamental velocity in quantum and holographic theories, connecting bounds on chaos, transport, and the geometry of emergent spacetime.

1. Definition, Core Formulation, and the Lieb-Robinson Bound

The butterfly velocity %%%%1%%%% emerges from analyzing the space-time growth of the commutator between local operators. For two local operators $V(0)$ and $W(x,t)$, the squared commutator, often thermal or state averaged,

$C(x,t) = -\langle [W(x,t), V(0)]^2 \rangle,$

grows as

$$C(x,t) = \frac{K}{N^2} e^{\lambda_L (t - x/v_B)} + \text{higher order},$$

where $\lambda_L$ is the Lyapunov exponent and $N$ is a system-size or large-$N$ parameter (Roberts et al., 2016).

This is strongly analogous to the Lieb-Robinson (LR) bound for local Hamiltonians, which asserts

$$\| [W(x,t), V(0)] \| \leq K_0 \|W\| \|V\| \exp\left[-\frac{|x| - v_{LR} t}{\xi_0}\right],$$

with $v_{LR}$ the state-independent Lieb-Robinson velocity. In contrast, $v_B$ is state-dependent and emerges as an effective lower-energy LR velocity, generically smaller than $v_{LR}$. $v_B$ therefore determines when operator non-commutativity, and hence influence or information transfer, becomes $\mathcal{O}(1)$ at distant spacetime points, signaling the advance of the butterfly effect front.

2. Butterfly Velocity in Holographic and Quantum Chaotic Systems

In quantum chaotic and holographic systems, $v_B$ is defined by the ballistic (linear-in-space-time) propagation front in the OTOC: $$F(t, \vec{x}) = 1 - \alpha \exp\left[\lambda_L (t - t^* - |\vec{x}|/v_B)\right] + \cdots,$$ where $t^*$ is the scrambling time (Ling et al., 2016). For strongly coupled or large-$N$ systems with holographic duals, $\lambda_L$ typically saturates the chaos bound ($\lambda_L = 2\pi T$), and butterfly velocity can be computed geometrically via shockwave solutions in black hole backgrounds.

In generic holographic geometries with hyperscaling-violating metrics, the butterfly velocity is determined by the IR scaling exponents: $$v_B = (\beta_0/\beta)^{1-1/z} \sqrt{\frac{d+z-\theta}{2(d-\theta)}},$$ where $z$ is the dynamical exponent, $\theta$ is the hyperscaling violation exponent, and $d$ the spatial dimension. For $z>1$, $v_B \sim T^{1-1/z}$ and thus encodes temperature-dependent slowing of information spreading at low temperatures; in relativistic systems ($z=1, \theta=0$), $v_B$ becomes a universal constant $v_B = \sqrt{(d+1)/(2d)}$ (Roberts et al., 2016).

Shockwave analysis in gravitational duals expresses $v_B$ as

$v_B = \frac{2\pi}{\beta m},$

where $m$ is extracted from the linearized Einstein equations governing the shockwave perturbation at the horizon (Ling et al., 2016).

3. Role in Quantum Criticality and Thermodynamic Bounds

The butterfly velocity is highly sensitive to quantum phase transitions. Near generic quantum critical points (QCPs), in holographic models, $v_B$ is maximized at criticality and decreases away from the QCP. This results in a pronounced extremum in $\partial_P v_B$, where $P$ is a tuning parameter such as lattice strength, wavevector, or magnetic field (Ling et al., 2016, Ling et al., 2017, Fu et al., 2022, Zhao et al., 13 Mar 2025). Table 1 summarizes qualitative diagnostic behavior:

Setting	Butterfly velocity behavior	Diagnostic
Isotropic QCP	$v_B$ maximum at QCP	Maximum in $v_B$ at QCP
Anisotropic QCP	Max/min in different directions; screening length ℒ maximized	extremum in $\partial_P v_B$, ℒ

In quantum critical regions with scaling exponent $z$,

$$v_B^2 \sim T^{2-2/z} (1 - \gamma \kappa^2 / T^{2/\nu} + \ldots),$$

where $\kappa$ is the relevant operator deformation and $\nu$ the correlation length exponent (Ling et al., 2017).

At holographic BKT transitions, this universal scaling can break down, with $v_B$ becoming nonanalytic or even discontinuous (Ling et al., 2017). In some anisotropic models, the dimensionless information screening length $ℒ$ is a more universal indicator of criticality than $v_B$ itself (Baggioli et al., 2018).

4. Relations to Transport, Diffusion, and Bounds

The butterfly velocity is closely associated with lower bounds on diffusion constants for energy and, sometimes, charge. In homogeneous systems,

$D = \mathcal{C} \frac{v_B^2}{T},$

with $\mathcal{C}$ an order-one constant (Lucas et al., 2016, Kim et al., 2017). However, in spatially inhomogeneous or disordered systems (e.g., with stripes), the relation becomes an inequality: $2 T D \leq \mathcal{C} v_B^2,$ so $v_B$ does not set a sharp lower bound for charge diffusion in general (Lucas et al., 2016). Energy diffusion bound saturation is more robust in strongly coupled models, while the relevant velocity for charge diffusion may differ from $v_B$ in such contexts (Kim et al., 2017).

5. Operator Dependence, Bulk Causal Structure, and Generalizations

Butterfly velocity is generally operator- and state-dependent. Its precise value may vary for different observables or code subspaces. In holographic systems, $v_B$ is tied directly to the causal structure of the emergent bulk via entanglement wedge reconstruction or the behavior of extremal surfaces:

• For a local bulk operator mapped to a boundary region, the butterfly velocity is set by the minimal expansion speed of the entanglement wedge necessary to enclose the operator after a time $\Delta t$ (Qi et al., 2017, Nomura et al., 2017).
• In AdS geometries, all butterfly velocities are upper-bounded by the speed of light, saturating this bound when the entanglement wedge coincides with the causal wedge (Qi et al., 2017).
• For non-AdS (e.g., flat space) holography, the butterfly velocity can diverge for small boundary regions, suggesting that any local holographic dual must be nonlocal (Qi et al., 2017, Nomura et al., 2017).

A monotonic decrease of $v_B$ away from the UV into the IR is generically enforced by the null energy condition in Einstein gravity (Qi et al., 2017).

6. Experimental Measurement and Physical Implications

Modern OTOC-based protocols enable direct experimental measurement of $v_B$ in quantum simulators, cold atom arrays, and condensed matter systems. Protocols such as Hamiltonian sign reversal and measurement of squared commutators are realizable in, e.g., optical lattices (Roberts et al., 2016, Ling et al., 2016). The shockwave and pole-skipping analyses provide tools for extracting $v_B$ in theoretical models, with the latter linking zero determinant points of the retarded Green’s function to chaos features (Lilani et al., 21 May 2025).

The physical implications of the butterfly velocity are broad:

• It sets a speed limit for quantum information or signal propagation in nonrelativistic and relativistic settings (Roberts et al., 2016).
• It provides an independent diagnostic of quantum phase transitions, including those lacking local order parameters (Ling et al., 2016, Zhao et al., 13 Mar 2025).
• In systems with topologically protected edge states, $v_B$ governs ballistic bulk scrambling, but certain edge states can trap information, decoupling their dynamics from the bulk (Sedlmayr et al., 2023).
• In curved backgrounds, such as de Sitter space, the butterfly velocity can become complex, signaling suppression of chaos by spacetime curvature effects (Ageev, 2021).

7. Extensions, Limitations, and Open Directions

While the butterfly velocity offers a sharp probe of scrambling, its interpretation, universality, and quantitative relation to transport and diffusivity depend on underlying symmetry, disorder, anisotropy, and coupling regime. Key findings include:

• In generic anisotropic or disordered systems, $v_B$ may not correspond to bounds on all transport coefficients, and the notion of a universal information speed can break down (Lucas et al., 2016, Baggioli et al., 2018).
• Higher-derivative gravitational corrections, such as Einstein–Gauss–Bonnet or quadratic curvature terms, modify the shockwave equations and can give rise to two distinct butterfly velocities under certain parameter regimes (Huang, 2017, Huang, 2018).
• Non-equilibrium, non-thermal, or microcanonical generalizations of $v_B$ reveal sensitivity to the Hilbert space subregion, further connecting butterfly dynamics to the structure of operator growth and the geometry of quantum codes (Qi et al., 2017).

Ongoing research addresses how the butterfly velocity interfaces with quantum error correction, the structure of operator entanglement, higher-form symmetries, and the microscopic details of many-body localization and ergodicity breaking.

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In summary, the butterfly velocity $v_B$ codifies the ballistic frontier of chaos and information spreading in quantum systems, acting as a dynamical, often state-dependent effective light cone. It serves as a unifying diagnostic at the intersection of quantum chaos, criticality, transport, and spacetime geometry, and is closely linked to various experimental and theoretical advances in strongly coupled (including holographic) quantum matter.