Logarithmic Light Cone Dynamics

• Logarithmic Light Cone is a regime in quantum many-body systems where information and operator spread grow logarithmically with time, as dictated by modified Lieb–Robinson bounds.
• It is observed in contexts like many-body localization and long-range interactions, leading to ultra-slow entanglement growth and suppressed scrambling.
• The universal logarithmic behavior, supported by analytical, numerical, and holographic studies, has significant implications for quantum memory stability and causal dynamics.

The logarithmic light cone describes regimes in quantum many-body systems where the propagation of information, operator spread, or entanglement is confined within a region whose radius grows logarithmically with time. This sharply contrasts with the conventional (linear) light cone of relativistic or short-range-interacting non-relativistic systems, where such propagation is bounded by a maximal velocity and expands linearly in time. Logarithmic light cones (LLCs) arise in various contexts, including many-body localization (MBL) phases, long-range interacting systems with specific decay exponents, and in the study of entanglement entropy in conformal field theories (CFTs) on null and light-cone geometries. LLCs fundamentally alter the causal structure and dynamical properties of quantum systems, leading to ultra-slow entanglement growth, suppressed scrambling, and fundamentally different behaviors in quantum memory and information propagation.

1. Emergence of Logarithmic Light Cones: Lieb–Robinson Type Bounds

Logarithmic light cones originate from modified Lieb–Robinson bounds, which, in contrast to their short-range counterparts, constrain the commutator of local observables $O_X(t)$ and $O_Y$ by

$$\|[O_X(t), O_Y]\| \leq C \|O_X\| \|O_Y\| |t|^{\alpha} \exp(-d/\xi),$$

where $d$ is the separation between the supports of $O_X$ and $O_Y$, $\xi$ is a localization length, $C$ a system-dependent constant, and $\alpha$ a polynomial exponent determined by the microscopic or phenomenological model. Inverting this bound for a fixed threshold gives $d \lesssim \alpha\xi \ln t$, delineating a region of appreciable operator non-commutativity expanding logarithmically in time—the logarithmic light cone (Zeng et al., 2023).

In systems with power-law interactions ($1/r^{\alpha}$) on a $D$-dimensional lattice, Hastings–Koma established the bound

$\|[A(t),B]\| \leq c\, e^{vt} / r^{\alpha}$

with $v = O(J)$ and $c=O(1)$. For exponents $D<\alpha\le 2D$, the light cone implied by this bound also grows logarithmically with $r$ for fixed time $t$, resulting in $t \sim (\alpha/v)\ln r$ (Foss-Feig et al., 2014). Thus, the causal region within which information and correlations can propagate is only logarithmically slow as a function of distance or time in these regimes.

2. Physical Realizations and Regimes

LLCs are realized in at least three distinct settings:

• Many-Body Localization (MBL): Disordered quantum spin chains such as the random-field XXZ model display LLCs in both the growth of entanglement and operator spread, with evidence from exact analytical bounds and supported by numerics (Zeng et al., 2023).
• Long-Range Interacting Systems: For power-law interactions in the intermediate regime $D<\alpha\le 2D$, the logarithmic cone dominates the operator spread, beyond which algebraic and eventually linear cones emerge as $\alpha$ increases (Foss-Feig et al., 2014).
• Conformal Field Theory on Light Cones: In higher-dimensional CFTs, the study of entanglement entropy (EE) in regions bounded on a null plane or light cone reveals that all Rényi entropies are determined by local invariants, with the Markov property implying a product state across rays. For light-cone regions, the universal logarithmic term in the EE is generated by the Wess–Zumino anomaly action of a “dilaton” field defined on $S^{d-2}$, with robust consequences for holography and conformal anomaly matching (Casini et al., 2018).

3. Information Propagation and Entanglement Growth

A hallmark of the LLC regime is logarithmically slow entanglement growth. For a bipartition with boundary $|\partial|$, the change in entanglement for any pure state $\ket\Psi$ obeys, after minimization,

$$\Delta S(t)\leq 2|\partial|\,\xi \left[\,(\alpha+1)\ln t + \ln\ln t + \text{const} + O(1)\right],$$

revealing both a leading $\ln t$ and a subleading $\ln\ln t$ scaling, where the latter is attributable to number entropy in models with conserved charges (Zeng et al., 2023). This logarithmic growth matches numerical results in disordered spin chains and is fundamentally distinct from the ballistic ($\sim t$) or diffusive ($\sim t^{1/2}$) growth seen in ergodic or integrable systems.

In systems governed by power-law interactions, the nature of the light cone strongly affects the speed of entanglement generation. In the log-cone regime ($D<\alpha\leq 2D$), the instantaneous effective velocity for information transfer grows exponentially in time. As $\alpha$ increases, the system transitions to algebraic ($r\sim t^{\zeta}$) and ultimately linear ($r\sim vt$) cones, restoring a notion of finite maximum speed (Foss-Feig et al., 2014).

4. Operator Spreading, Scrambling, and Quantum Memory

In the LLC regime, the propagation of operators and corresponding growth of out-of-time-ordered correlators (OTOCs) are severely slowed:

$C_0(r, t) \leq C' t^{2\alpha} e^{-2r/\xi},$

with the contour $C_0(r_\theta, t)=\theta$ yielding $r_\theta(t)\gtrsim \alpha\xi \ln t$. Thus, operator growth is only logarithmic in $t$, in stark contrast to polynomial or linear growth in less constrained systems (Zeng et al., 2023).

A key implication is the stability of topological quantum memories: logical qubits encoded in topologically ordered systems with code distance $L^*$ remain robust under unitary evolution for times up to $t^*\approx \exp(L^*/(4\alpha\xi))$. Local errors remain localized within neighborhoods small enough to preserve correctability, yielding exponentially long code lifetimes under purely unitary evolution. This property is inherent to many-body-localized systems exhibiting the LLC and is not achievable in ballistic quantum systems (Zeng et al., 2023).

5. Entanglement Entropy and the Universal Logarithmic Term on the Light Cone

For arbitrary regions with boundary on a light cone in even $d$-dimensional CFTs, the universal logarithmic contribution to the entanglement entropy is generated by a two-dimensional Wess–Zumino (WZ) anomaly action of a “dilaton” field $\phi=\log(\gamma(\Omega)/\epsilon)$, living on $S^{d-2}$:

$$S_{\rm WZ}[\phi] =\int_{S^{d-2}} d^{d-2}\Omega \sqrt{g}\; \left\{(\nabla\phi)^2 + R\,\phi + \cdots\right\}$$

The coefficient of this log term is fixed by the type-$A$ conformal anomaly, and its existence reflects the deep link between geometry, anomaly matching, and quantum information (Casini et al., 2018). The full entanglement entropy takes the structured local-plus-universal form:

$$S(\gamma) = \int_{S^{d-2}} d\Omega\,\sqrt{g}\left[\beta_0(\gamma/\epsilon)^{d-2} + \beta_2(\gamma/\epsilon)^{d-4}(\cdots) + \cdots\right] + S_{\rm WZ}[\phi]+ F,$$

with $F$ a cutoff-independent constant. These results connect directly to the entropic $a$-theorem in $d=4$, whereby the concavity of the entanglement RG flow and the dominance of the $-4\Delta A\log(r/\epsilon)$ term yield an independent proof that $\Delta A\leq 0$ (Casini et al., 2018).

6. Transitions from Logarithmic to Algebraic and Linear Light Cones

In long-range systems, the logarithmic cone applies in the intermediate regime of exponents $D<\alpha\leq 2D$. For $\alpha>2D$, operator propagation is governed by an algebraic (polynomial) light cone, $t\sim r^{\zeta}$, with $1/\zeta = 1 + (1+D)/(\alpha-2D)$. As $\alpha$ increases further ($\alpha\to\infty$), $\zeta\to 1$, recovering the linear Lieb–Robinson light cone characteristic of strictly finite-range models (Foss-Feig et al., 2014). The boundaries between these regimes are sharp, dictated by the summability of couplings and key convolution bounds in the expansion of the time-evolution operator. In short, the progression log-cone $\to$ algebraic $\to$ linear encapsulates the emergence of locality as interactions become shorter-range.

Interaction Decay $\alpha$	Causal Structure	Light Cone Behavior
$\alpha \leq D$	None	No decay bound
$D<\alpha\leq2D$	Logarithmic	$t\sim(\alpha/v)\ln r$
$\alpha>2D$	Algebraic	$t\sim r^{\zeta}$
$\alpha\to\infty$	Linear	$t\sim r/v$

7. Holography and Robustness of Logarithmic Terms

In the AdS/CFT context, the universal logarithmic term in the EE for boundaries on the light cone has a holographic origin as the regularized area of extremal surfaces in pure AdS. For the null-plane, these surfaces are independent of the boundary shape, reproducing the Markov property and the constancy of the entropy. On the light cone, the same AdS isometry ensures the emergence of the WZ log term in even dimensions. Stringy and quantum corrections, including higher-derivative and bulk entanglement contributions, are shown to preserve the robust shape independence and the universal structure of the logarithmic term, revealing a high degree of universality (Casini et al., 2018).

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Logarithmic light cones provide the unifying description for ultra-slow information propagation and entanglement growth in disordered, long-range, and holographic systems, with profound implications for the structure of correlation spreading, entropy production, quantum memory stability, and fundamental bounds in quantum many-body dynamics (Foss-Feig et al., 2014, Casini et al., 2018, Zeng et al., 2023).