Operator Spreading and Scrambling

Operator spreading asks where a local Heisenberg operator has acquired noncommuting support. Relativistic microcausality fixes an exact outer light cone, while an interacting state can have a slower butterfly front, a broadening law, and a separate saturation scale inside it. “Scrambling” is a useful physical interpretation only after the operator pair, state, regulator, front definition, and information-theoretic handoff are stated.

Required background. OTOCs and contour regularization define the squared commutator used to locate a front. Microcausality fixes the continuum causal boundary. Helpful background. Long-time tails explain why conserved modes can modify late-time relaxation behind the front.

Causal support, butterfly front, and saturation

For local operators $W(t,\mathbf x)$ and $V(0)$, define

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

Three boundaries must remain distinct.

1. Causal support: in a local relativistic QFT, the commutator vanishes at spacelike separation, $t^2-\lvert\mathbf x\rvert^2<0$, as an operator-valued distribution away from contact subtleties.
2. Butterfly front: inside the causal cone, a chosen level set $C(t,x)=C_*$ may propagate as $x\simeq v_Bt$.
3. Saturation: behind the front, $C$ approaches a state- and normalization-dependent value after local operator information has spread over the accessible degrees of freedom.

$v_B$ need not equal the speed of light, a sound speed, an entanglement velocity, or a diffusion constant. It is defined from a particular operator-growth diagnostic.

Regulated lattice bounds and continuum causality

For a short-range lattice Hamiltonian, a Lieb–Robinson estimate has the schematic form

$$\lVert[W_X(t),V_Y]\rVert \le c\,\lVert W_X\rVert\lVert V_Y\rVert \exp\!\left[-\mu\bigl(d(X,Y)-v_{\mathrm{LR}}\lvert t\rvert\bigr)\right].$$

$v_{\mathrm{LR}}$ is a nonunique bound set by the regulator, interaction norm, and proof; it is not a measured butterfly velocity Lieb and Robinson 1972. A lattice discretization of a relativistic theory can therefore have

$$v_B\le v_{\mathrm{front}}\le v_{\mathrm{LR}},$$

while the continuum limit must recover the physical causal speed for operator support. Comparing velocities across regulators requires common physical units and a demonstrated scaling limit.

For continuum QFT, smearing or point splitting controls ultraviolet products. A nonzero result outside the light cone can be a discretization, smearing, truncation, or numerical error rather than superluminal physics.

Front scaling and broadening

A sharp-velocity ansatz is often too simple. Write a scaling form

$$C(t,x) \simeq \mathcal F\!\left( \frac{x-v_Bt}{t^\chi} \right),$$

where $t^\chi$ is the front width. Different dynamics produce different $\chi$. In one-dimensional Haar-random unitary circuits, operator endpoints execute a biased random walk, giving diffusive broadening $\chi=1/2$ and an exactly controlled coarse-grained front Nahum, Vijay, and Haah 2018. This is a solvable universality example, not a theorem that every Hamiltonian front diffuses.

A reproducible extraction uses several thresholds $C_*$, fits $x_*(t;C_*)$, and checks that the inferred $v_B$ converges while threshold differences grow with the proposed width. A single threshold cannot distinguish drift in amplitude from motion of the front.

Operator-basis growth

On a regulated spin or finite local Hilbert space, expand

$$W(t)=\sum_{\mathcal S}a_{\mathcal S}(t)\,\mathcal S$$

in orthonormal operator strings $\mathcal S$. Unitarity preserves the operator norm,

$$\sum_{\mathcal S}\lvert a_{\mathcal S}(t)\rvert^2 =\text{constant},$$

while weight moves from short strings to longer, spatially extended strings. The squared commutator with a local $V_x$ measures the fraction of strings that act nontrivially near $x$, weighted by their local algebra.

This representation clarifies two limitations. First, operator size depends on the chosen local tensor-product regulator. Second, growth of operator support is not literal loss of information: the coefficients evolve unitarily and can in principle be reversed. Recoverability, decoding cost, and channel capacity are developed in Volume XIII.

A front-extraction protocol

Record:

• the state and thermal regularization;
• the local operators and their normalization;
• spatial smearing or lattice spacing;
• $C$‘s disconnected and saturation values;
• all thresholds or a full-profile likelihood;
• the time interval excluding the microscopic transient and finite-size wraparound;
• covariance across $x$ and $t$; and
• competing sharp, diffusive, KPZ-like, and exponential-tail forms when relevant.

Then vary system size, boundaries, operator pair, smearing, and regulator. A butterfly velocity is licensed only where these choices converge. The thermalization and chaos evidence matrix supplies the common claim ceiling.

Common failure modes

Calling $v_{\mathrm{LR}}$ a measured velocity. It is a proof-dependent upper bound.

Using one threshold. Amplitude drift and broadening bias a single level-set velocity.

Treating operator spreading as thermalization. A front can propagate in integrable, localized, or otherwise nonthermal systems.

Ignoring conserved tails. Diffusive or hydrodynamic modes can leave long-lived structure behind an otherwise ballistic front.

The figure locates operator-front measurements beside, but not upstream of, the other diagnostics in the lower row. Inspect how front velocity and broadening answer a different question from an OTOC fit or late spectral statistics.

Operator spreading is characterized by support, front velocity, broadening, conserved tails, and regulator dependence. The lower row is deliberately arrow-free: OTOCs can sample commutator growth, but no implication runs from a front to a Lyapunov fit, spectral statistics, or thermalization.

The text equivalent is to reconstruct the operator-weight profile or commutator front, vary the local basis and cutoff, and separate ballistic motion from broadening and hydrodynamic tails. Recoverability and thermal ensemble agreement require additional observables.

Exercises

Assume $C(t,x)=\frac12\operatorname{erfc}[(x-v_Bt)/(4Dt)^{1/2}]$. How does the separation between two fixed thresholds scale?

Solution

Solving for a threshold gives $x_*(t)=v_Bt+(4Dt)^{1/2}\operatorname{erfc}^{-1}(2C_*)$. The difference between two thresholds is proportional to $t^{1/2}$, while their common leading slope is $v_B$.

Why may a lattice measurement yield $v_B<v_{\mathrm{LR}}$ without approaching a bound-saturating regime?

Solution

$v_{\mathrm{LR}}$ is an upper bound assembled from interaction norms and is generally not tight. $v_B$ measures the state- and operator-dependent front. No saturation is expected unless separately demonstrated.

Continue to late-time and information questions

Spectral statistics tests late-time energy correlations that a front does not determine. Chaos bounds constrain temporal growth under thermal analyticity hypotheses. Information velocities and recoverability continue in Volume XIII.

References

• Lieb, Elliott H., and Derek W. Robinson. “The Finite Group Velocity of Quantum Spin Systems.” Communications in Mathematical Physics 28 (1972): 251–257. doi:10.1007/BF01645779.
• Nahum, Adam, Sagar Vijay, and Jeongwan Haah. “Operator Spreading in Random Unitary Circuits.” Physical Review X 8 (2018): 021014. doi:10.1103/PhysRevX.8.021014. Open preprint.
• Roberts, Daniel A., Douglas Stanford, and Leonard Susskind. “Localized Shocks.” Journal of High Energy Physics 2015, no. 3 (2015): 051. doi:10.1007/JHEP03(2015)051. Open preprint.