Conservation Laws and Information Transport

A conserved charge introduces a slow mode that transports information about its density even when other local degrees of freedom equilibrate rapidly. Conservation can change correlation tails, front broadening, symmetry-resolved entanglement, and accessible recovery. It does not force the total entanglement entropy to grow diffusively.

Required background. Mutual-information and correlation spreading supplies the direct diagnostics.

Helpful background. Charge-resolved entanglement separates fluctuations among sectors from entanglement within sectors.

Continuity and hydrodynamic scale

For a locally conserved density $q(x,t)$ and current $j(x,t)$,

$$\partial_t q+\nabla\!\cdot j=0.$$

If the late-time constitutive relation is diffusive, $j=-D\nabla q+\cdots$, then

$$\partial_t q=D\nabla^2q$$

and a localized perturbation spreads over $x\sim\sqrt{Dt}$. Ballistic transport instead yields peaks near $x=\pm vt$. Determine the transport law from density and current observables before using it to interpret entanglement.

For a global pure state with total charge fixed, the reduced density matrix commutes with the regional charge $Q_A$ and decomposes into sectors:

$$\rho_A=\bigoplus_q p_q\,\rho_{A,q}, \qquad S(\rho_A)=H(\{p_q\})+\sum_qp_qS(\rho_{A,q}).$$

The first term records charge fluctuations; the second records within-sector entanglement. A slowly broadening $p_q$ can give hydrodynamic corrections while the leading total entropy grows ballistically.

Matched symmetry test

Compare two local Hamiltonians with similar energy density and interaction scale, one exactly $U(1)$ symmetric and one with a weak symmetry-breaking term. Prepare the same localized charge profile where meaningful. Measure:

• $q(x,t)$ and current correlations;
• $I(A{:}B;t)$ for separated regions;
• total and charge-resolved entropies;
• front position and width over nested time windows.

First confirm diffusion or ballistic transport directly. Then determine which entanglement correction disappears when the symmetry is broken. A small breaking rate creates a crossover time; data at $t$ below that time can look conserved even though no asymptotic charge exists.

Charge conservation produces diffusive structures in operator and correlation spreading in random circuits and chaotic chains; Rakovszky, Pollmann, and von Keyserlingk 2018, §§ II–IV provide a controlled example. The precise consequence for a continuum QFT depends on its hydrodynamic regime.

Conservation changes the dynamical-class input and can supply a hydrodynamic effective mode. The resulting charge front must still be compared with direct entropy and mutual-information diagnostics. The map is schematic.

Sector mixing and inference

If operations must respect the charge, coherence between sectors may be inaccessible, changing operational entanglement and recovery. An environment that exchanges charge breaks the closed-system continuity equation even if the Hamiltonian part is symmetric. Report whether the bath conserves total system-plus-environment charge and whether the measurement resolves $Q_A$.

Weak symmetry breaking, unnoticed sector mixing, or an exchanging bath can turn an apparent asymptotic conservation law into a finite-time crossover. The map is schematic.

Exercises

For a one-dimensional diffusion kernel $q(x,t)=Q(4\pi Dt)^{-1/2}e^{-x^2/(4Dt)}$, find the position where $q$ falls to a fixed fraction $e^{-c}$ of its peak.

Solution

$x_c(t)=2\sqrt{cDt}$. The contour grows as $\sqrt t$, so assigning a constant ballistic velocity would be inappropriate.

References

• Rakovszky, Tibor, Frank Pollmann, and C. W. von Keyserlingk. “Diffusive Hydrodynamics of Out-of-Time-Ordered Correlators with Charge Conservation.” Physical Review X 8 (2018): 031058. DOI.