Inhomogeneous Lieb-Robinson Cone

• Inhomogeneous Lieb-Robinson cone is a framework that generalizes the standard light-cone by incorporating spatially varying velocities and interaction strengths in quantum systems.
• It features algebraic spatial decay and non-linear, sublinear, or super-logarithmic temporal scaling, crucial in cMBL, power-law interacting models, and inhomogeneous Luttinger liquids.
• The approach provides practical error bounds and insights for operator spreading, assisting stability analyses and dynamic phase diagram studies in disordered quantum media.

The inhomogeneous Lieb-Robinson (LR) cone describes the maximal spacetime region within which quantum correlations or information can propagate in spatially or interaction-parameter-varying quantum systems, generalizing the standard LR “light-cone” paradigm. In contrast to conventional homogeneous systems—where a linear, causality-like bound delimits the spread of operator support—the inhomogeneous LR cone accommodates contexts with spatially varying velocities, power-law long-range interactions, or many-body constraints. This leads to highly nontrivial, non-linear, and even sublinear propagation fronts, algebraic spatial decay, and variable temporal scaling of correlation growth. Inhomogeneous LR cones have become essential for understanding locality and information flow in constrained many-body localization (cMBL), power-law interacting spin models, and inhomogeneous Luttinger liquids.

1. General Formalism and Definitions

The LR bound places a rigorous upper limit on the norm of the commutator between two initially disjoint local observables $A(x, t)$ and $B(y, 0)$. Conventionally, in homogeneous, short-range interacting quantum systems, this bound assumes the form

$$\| [A(x, t), B(y, 0)] \| \leq C\, \exp\left[ -\mu(|x-y| - v_{\mathrm{LR}} t) \right],$$

with $C, \mu, v_{\mathrm{LR}}$ constants. This defines a linear light-cone $|x-y| \leq v_{\mathrm{LR}} t$ outside of which correlations are exponentially suppressed.

In inhomogeneous settings, this picture generalizes fundamentally. Three broad structural features arise:

• Spatially varying velocities: The maximal group velocity at which information can propagate depends on position, leading to curved spacetime cones.
• Power-law interacting systems: Absence of a unique light-cone slope, with algebraic decay outside the front and effective propagation speed depending on both time and distance.
• Constraint-induced nonlocality: Constraints such as in cMBL generate algebraic spatial suppression with logarithmic (rather than exponential or linear) time dependence.

The general inhomogeneous LR bound may involve:

• A position-dependent velocity profile $v(x)$.
• An effective metric distance $d(x, y) = \int_x^y \! du / v(u)$.
• Spatial and temporal exponents, sometimes defined via the structure of the effective integrals of motion or the decay of interaction strength.

Operator norm bounds, OTOC light cones, and commutator decays display highly non-uniform, sometimes superlogarithmic or sublinear, dependence on $t$ and $x$.

2. Inhomogeneous LR Cones in Constrained Many-Body Localization

In the cMBL scenario, exemplified by constrained Rydberg-blockade spin chains with strong quasirandom disorder, the LR bound takes a profoundly inhomogeneous structure (Chen et al., 2020):

$$\| [A(x, t), B(0, 0)] \| \lesssim c\, |x|^{-\eta}\, [\ln |t|]^{\xi}$$

where:

• $\| \cdot \|$ denotes the operator norm,
• $|x|$ is the graph distance,
• $c > 0$, $\eta > 0$, $\xi > \eta > 0$ are constants dependent on model specifics.

The resulting propagation front for a fixed small commutator threshold $\epsilon$ is determined by

$$x \sim [\ln t]^{\xi/\eta} \qquad \text{with}\quad \xi/\eta > 1$$

This is a super-logarithmic but strictly sub-linear light-cone in $t$: spatial growth with $\ln t$ raised to a power.

The derivation combines:

• OTOC front analysis, numerically mapping “light-cones” in the Frobenius norm,
• A splitting of the Hamiltonian into short-range and long-range components,
• Iterative application of Hastings–Koma-style Lieb–Robinson bounds to the short-range part,
• Dynamically chosen cutoffs and resummation to capture the algebraically decaying tails arising from the constrained Hilbert space.

The cMBL cone differs both qualitatively and quantitatively from:

• Standard LR cone: Linear $x \propto t$.
• Unconstrained MBL (uMBL): Logarithmic $x \sim \ln t$.
• Diagonal MBL (dMBL): Sub-logarithmic $x \sim (\ln t)^{1/k}$, $k>1$.

Extracted numerical OTOC front exponents $\mu_c = \eta/\xi$ in cMBL are consistently less than unity ($\mu_c \simeq 0.54$–$0.68$), confirming super-logarithmic growth (Chen et al., 2020).

3. Inhomogeneous Cones in Power-Law Interacting Systems

For spin Hamiltonians on a $d$-dimensional lattice with arbitrary $k$-body terms decaying as $R^{-\alpha}$ for $\alpha > d$, the inhomogeneous LR cone is governed by bounds of the type (Else et al., 2018):

$$\| [\tau_t^H(A), B] \| \leq \|A\|\,\|B\|\,\Big\{ 2\,|X|\,e^{v\,t - r^{1-\sigma}} + C_1\,|X|\,e^{v\,t} \, \frac{\mathcal{G}(v\,t)}{r^{\sigma\alpha}} \Big\}$$

where:

• $r = d(X, Y)$,
• $C_1$, $v$—constants,
• $\sigma \in ((d+1)/(\alpha+1), 1)$,
• $\mathcal{G}(v\,t)$—polynomial in $t$ and $|X|$ (Else et al., 2018).

The LC1 and LC2 cone notions distinguish regions where, respectively, the commutator vanishes at late times or where Hamiltonian perturbations do not affect operator evolution up to time $t$.

Algebraic (rather than exponential) spatial decay persists outside any linear cone. The propagation front is determined by:

$$r \gtrsim t^{\beta}\,,\quad \beta = \frac{\alpha+1}{\alpha-d} > 1$$

implying a super-ballistic cone for $\alpha$ slightly larger than $d$. In the short-range limit ($\alpha \to \infty$), $\beta \to 1$ and the usual linear light-cone is recovered. The bound remains valid for arbitrary-order $k$-body interactions.

The inhomogeneity here refers to the non-uniform dependence of the effective velocity $v_{\text{eff}}(r)$ on $r$ due to the $r^{1-\sigma}$ scaling in the exponent.

4. Curved Light-Cones in Inhomogeneous Luttinger Liquids

For quantum critical systems such as Luttinger liquids with spatially varying Hamiltonian density $f(x)$, the LR cone is geometrically curved reflecting the nonuniform local velocity $v(x)=v_0 f(x)$ (Dubail et al., 2017). The effective metric is:

$ds^2 = dx^2 + v^2(x)\, d\tau^2$

Light-like geodesics then solve:

$$\frac{dx}{dt} = v(x)\,,\qquad t = \int_{x_0}^{x(t)} \frac{du}{v(u)}$$

The front $x_c(t)$ is defined implicitly by this integral.

The corresponding commutator/LR bound for two local observables $A_x$, $B_y$ becomes

$$\| [A_x(t), B_y(0)] \| \leq c\, \|A\|\,\|B\|\,\exp\left[ -\mu \left( \int_x^y \frac{du}{v(u)} - t \right) \right]$$

Away from the “curved cone” $t < \int_x^y du/v(u)$, spatial commutators are exponentially suppressed. At the CFT level, commutators vanish identically in the spacelike region, while in the lattice realization, exponentially decaying tails persist (Dubail et al., 2017).

A universal “curved light-cone” structure is seen in inhomogeneous critical systems with constant Luttinger parameter $K$, both in equilibrium and nonequilibrium quenched settings.

5. Comparison Table: Growth Fronts in Different Settings

Regime	Spatial Front $x(t)$	Spatial Decay	Temporal Scaling in Bound
Standard (Linear) LR	$x \sim v_{\mathrm{LR}} t$	Exponential	Linear in $t$ (e.g., $e^{v t}$)
cMBL (constrained MBL)	$x \sim [\ln t]^{\xi/\eta}$, $\xi/\eta > 1$	Algebraic ($|x|^{-\eta}$)	$[\ln |t|]^\xi$ (Chen et al., 2020)
Unconstrained MBL (uMBL)	$x \sim \ln t$	Exponential	$[\ln |t|]^\nu$
Diagonal MBL (dMBL)	$x \sim (\ln t)^{1/k}, k>1$	Sub-exponential	$e^{-\theta|x|^k}$
Power-law Interactions	$x \sim t^{\beta}$, $\beta>1$	Algebraic ($1/r^{\sigma\alpha}$)	Time-polynomial prefactors (Else et al., 2018)
Inhomogeneous Luttinger Liquids	$x_c(t)$ via $t = \!\int du/v(u)$	Exponential	Exponential in “metric distance” (Dubail et al., 2017)

6. Physical Origins and Consequences

The inhomogeneity of the LR cone can stem from:

• Constraints (cMBL): Blockade or hard-core constraints induce emergent quasi-local integrals of motion with algebraic decay—generating algebraic tails and slow, logarithmic cone expansion. This signals delocalization beyond local integrals of motion phenomenology.
• Power-law interactions: Absence of a unique locality scale leads to non-uniform front propagation, with algebraic leakage of information even arbitrarily far from the nominal front.
• Spatially varying Hamiltonians: Inhomogeneous velocity fields produce geodesics in an effective metric, leading to curved propagation fronts that govern the spread of entanglement and correlation.

Consequences include distinctive regimes for operator spreading detected via OTOCs and deficits in the validity of Magnus (Floquet) expansions, thermalization rates in isolated quantum simulators, and dynamic phase diagram stability—particularly in experimental architectures such as trapped ions or Rydberg arrays.

7. Applications and Broader Significance

The inhomogeneous LR cone is crucial whenever rigorous spatiotemporal control over quantum information spreading is needed in:

• Stability analyses of phase diagrams under long-range or inhomogeneous perturbations.
• Rigorous error bounds in effective Floquet-Magnus expansions in driven systems with generated many-body terms decaying with distance.
• Quantitative bounds on prethermalization and thermalization in cold atom, trapped-ion, and polar molecule platforms.
• Certified structure of OTOC fronts and entanglement light-cones in disordered and constrained quantum matter.
• Understanding the structure of curved correlations in Tonks–Girardeau gases and other spatially inhomogeneous critical models.

The mathematical formalism and concrete bounds unify and refine previous two-body and multi-body LR results and establish the foundation for future works on information propagation in highly nonuniform quantum media (Chen et al., 2020, Else et al., 2018, Dubail et al., 2017).