Butterfly Velocity (vB) in Many-Body Chaos

• Butterfly velocity (vB) is the critical speed at which a local operator becomes non-local, defining the effective light-cone for chaos propagation.
• It is determined by the sign change of the velocity-dependent Lyapunov exponent, offering insights into the transition from exponential to stretched-exponential operator growth.
• Extraction of vB is achieved through fitting out-of-time-order correlators and tracking operator fronts, making it a key diagnostic in analyzing many-body dynamics.

The butterfly velocity, $v_B$, is a central diagnostic of operator spreading and many-body quantum chaos in spatially extended systems. It quantifies the emergent ballistic light-cone within which an initially local operator becomes non-local under time evolution and can be defined for both quantum and classical many-body systems. $v_B$ is rigorously codified as the critical velocity that separates exponential growth from exponential decay of out-of-time-order correlators (OTOCs) or analogous classical decorrelators along spacetime rays, and is deeply connected to the propagation of information, chaos, hydrodynamics, and underlying transport in strongly interacting systems.

1. Foundational Definition: Velocity-Dependent Lyapunov Exponent and the Butterfly Cone

The protocol for defining the butterfly velocity starts from the OTOC: $$C(\mathbf{x}, t) = \frac{1}{2}\langle [O_0(t),\, W_{\mathbf{x}}]^\dagger [O_0(t),\, W_{\mathbf{x}}] \rangle$$ where $O_0$ is a local operator at the origin and $W_{\mathbf{x}}$ a probe at position $\mathbf{x}$. For spatially local systems, one studies the asymptotic behavior along rays $\mathbf{x} = \mathbf{v} t$: $$C(\mathbf{x} = \mathbf{v} t, t) \sim \exp[\lambda(\mathbf{v})\, t]$$ Here, $\lambda(\mathbf{v})$ is the velocity-dependent Lyapunov exponent (VDLE). The butterfly velocity in direction $\hat{\mathbf{n}}$ is the critical velocity for which the growth rate changes sign: $$\lambda(\mathbf{v} = v_B(\hat{\mathbf{n}})\hat{\mathbf{n}}) = 0$$ The operator-spreading "light cone" is thus $|\mathbf{x}| = v_B(\hat{\mathbf{n}}) t$ (Khemani et al., 2018). This definition extends naturally to classical systems using appropriate decorrelators, and to anisotropic models with direction-dependent $v_B$.

2. Scaling Form of $\lambda(v)$ Near the Front and Geometric Interpretation

In all local systems, the Lieb–Robinson bound ensures exponential suppression of operator growth outside the front ($v > v_B$), so $\lambda(v) < 0$ there. Near the front,

$$\lambda(v) \sim - (v - v_B)^\alpha \quad (v \to v_B^+)$$

The scaling exponent $\alpha$ encodes the broadening of the operator front and is related to the broadening exponent $\beta$ by $\beta = (\alpha - 1)/\alpha$ (Khemani et al., 2018). For classical many-body chaos and large-$N$ systems, typically $\alpha=1$; fully quantum spin chains and random circuits display $\alpha > 1$ (e.g., $\alpha=2$ in 1d random circuits), with the operator front broadening sub-ballistically. Integrable models show stretched-exponential tails with $\alpha = 3/2$.

3. Direction Dependence and Anisotropy

In anisotropic or inhomogeneous systems, $\lambda(\mathbf{v})$ depends not only on speed but also on direction $\hat{\mathbf{n}}$. The butterfly velocity becomes a function on the unit sphere, $v_B(\hat{\mathbf{n}})$. The shape of the operator-spreading front is generically determined by a Wulff construction, relating $v_B(\hat{\mathbf{n}})$ to the normal velocity of the growing operator front (Khemani et al., 2018).

Models can realize asymmetric butterfly velocities, $v_B^\text{left} \neq v_B^\text{right}$, via explicit breaking of inversion symmetry, either in local Hamiltonians with chiral interactions or in deeply chiral random circuits. This introduces chirality into operator spreading, highlighting that $v_B$ is not universally isotropic and must be labeled by direction (Stahl et al., 2018).

4. Regime Classification: Quantum, Semi-Classical, Classical, and Integrable Systems

The behavior of $\lambda(v)$ and the dynamical regime inside the light cone ($v < v_B$) is highly system dependent:

• Classical many-body systems: $\lambda(v)$ crosses from positive (ballistic exponential growth, $\alpha=1$) to negative; stretched exponentials are absent (Khemani et al., 2018, Bilitewski et al., 2018).
• Large-$N$/semiclassical quantum systems and holographic duals: $C \sim \epsilon \exp[\lambda_0 (t - |x|/v_B)]$ with $\lambda(v) = \lambda_0(1 - v/v_B)$; exponential regime exists for $v < v_B$ over significant timescales (Khemani et al., 2018).
• "Fully quantum" random circuit models ($d<4$): Broadening of the front ($\beta>0$), $\lambda(v)$ defined only for $v > v_B$ ($\lambda<0$), $C(x, t)$ shows stretched-exponential decay outside the front ($\alpha>1$).
• Integrable systems: No $\lambda(v) > 0$ regime; $\lambda(v) \sim - (v - v_B)^{3/2}$, front broadening is diffusive ($\beta=1/2, \alpha=2$), similar to random circuit universality.

A summary of $\alpha$ and broadening exponents for key classes: | System/Class | $\alpha$ | $\beta$ | Notes | |-------------------------------|-----------|---------------------|--------------------------------| | 1D random circuit | 2 | 1/2 | KPZ front, diffusive | | 2D random circuit | 3/2 | 1/3 | | | Free fermion chain | 3/2 | --- | Airy tail | | Holography (large-$N$ chains) | 1 | 0 | Ballistic, exponential regime | | Classical systems | 1 | 0 | Simple exponential | | Integrable (interacting) | 2 | 1/2 | Diffusive |

5. Concrete Formulas for $v_B$ in Model Systems

Random Unitary 1D Quantum Circuit:

$$C_{1d}(x, t) \simeq \frac14 \left[1 + \operatorname{erf}\left(\frac{v_B t + x}{\sqrt{2Dt}}\right)\right] \left[1 + \operatorname{erf}\left(\frac{v_B t - x}{\sqrt{2Dt}}\right)\right]$$

For $|x| - v_B t \gg \sqrt{Dt}$: $$\lambda(v) = - \frac{(v - v_B)^2}{2D}, \qquad \alpha = 2$$

SYK/large-$N$ chain at low $T$:

$$\lambda(v) \simeq 2\pi T (1 - v/v_B),\qquad v_B=\text{model dependent}, \qquad \alpha=1$$

Exponential regime, positive $\lambda(v)$ for $v < v_B$ (Khemani et al., 2018).

Free Fermion Chains:

$\lambda(v) \sim - (v - v_B)^{3/2}$

Relation to Transport:

In classical and semiclassical regimes, the spin/energy/charge diffusion constant $D$ is related to chaos parameters via

$D \sim \frac{v_B^2}{\lambda}$

or, in SYK/large-$N$ chains, $v_B^2 = 2\pi D T$. This chaos-transport connection holds numerically in classical models as well (Bilitewski et al., 2018).

6. Measurement Protocols and Extraction of $v_B$

$v_B$ can be extracted by several methods:

• Fitting OTOCs/Decorrelators: Determine the locus $v$ where $\lambda(v)$ crosses zero. In classical/spin chain/automaton simulations, the front position versus time gives a direct $v_B$ (Khemani et al., 2018, Liu et al., 2021).
• Operator Fronts: Track the spatial profile of operator weight (e.g., right-weight, left-weight in Pauli string decomposition); the speed of the propagating peak is $v_B$ (Stahl et al., 2018).
• Holographic and Black Hole Settings: Compute the shockwave profile decay or pole-skipping condition on the horizon; $v_B$ is determined kinematically from near-horizon data (Khemani et al., 2018).

7. Theoretical Significance, Universality, and Physical Interpretation

The butterfly velocity defines the (state-dependent) effective Lieb–Robinson velocity for chaos, providing a sharp, dynamical light-cone for information spreading at finite temperature. Its precise determination is essential for delineating the regime of local quantum chaos and diagnosing the phase structure in holographic metals, spin glasses, and classical spin liquids. The dependence of $v_B$ on temperature, interaction strength, spatial anisotropy, and quantum versus classical statistics is model dependent but follows sharply categorized behaviors as summarized above.

The interplay between $\lambda(v)$ and $v_B$ encodes subtle distinctions between quantum/semiclassical/classical dynamics, reflected in the nature and scaling of the chaos front. These distinctions are crucial for understanding the universality class of operator spreading and the onset of hydrodynamics from chaos.

In summary, the butterfly velocity $v_B$ systematically quantifies the speed of spatial growth of chaos in many-body systems, providing a unifying technical framework for ballistic light-cone dynamics, chaos-transport relations, and the demarcation between exponential and stretched-exponential operator propagation regimes across quantum, semiclassical, and classical domains (Khemani et al., 2018).