Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws

TL;DR

This work shows that chaotic quantum systems with locally conserved densities can exhibit dissipative hydrodynamics despite unitary evolution. Using a constrained random circuit with S_z^{tot} conservation, the authors derive a two-component operator picture: a diffusing conserved part and a rapidly spreading nonconserved flux that yields ballistic fronts and power-law tails. The dynamics are captured by coupled diffusion equations with a source term set by the conserved current, explaining slow dissipation of observable entropy and diffusive tails in OTOCs. The framework is validated through analysis of physical Floquet and Hamiltonian spin chains, highlighting the universal role of conservation laws in shaping scrambling and information propagation.

Abstract

We study the scrambling of local quantum information in chaotic many-body systems in the presence of a locally conserved quantity like charge or energy that moves diffusively. The interplay between conservation laws and scrambling sheds light on the mechanism by which unitary quantum dynamics, which is reversible, gives rise to diffusive hydrodynamics, which is a dissipative process. We obtain our results in a random quantum circuit model that is constrained to have a conservation law. We find that a generic spreading operator consists of two parts: (i) a conserved part which comprises the weight of the spreading operator on the local conserved densities, whose dynamics is described by diffusive charge spreading. This conserved part also acts as a source that steadily emits a flux of (ii) non-conserved operators. This emission leads to dissipation in the operator hydrodynamics, with the dissipative process being the conversion of operator weight from local conserved operators to nonconserved, at a rate set by the local diffusion current. The emitted nonconserved parts then spread ballistically at a butterfly speed, thus becoming highly nonlocal and hence essentially non-observable, thereby acting as the "reservoir" that facilitates the dissipation. In addition, we find that the nonconserved component develops a power law tail behind its leading ballistic front due to the slow dynamics of the conserved components. This implies that the out-of-time-order commutator (OTOC) between two initially separated operators grows sharply upon the arrival of the ballistic front but, in contrast to systems with no conservation laws, it develops a diffusive tail and approaches its asymptotic late-time value only as a power of time instead of exponentially. We also derive these results within an effective hydrodynamic description which contains multiple coupled diffusion equations.

Figures (7)

• Figure 1: Left: a diagram of the random unitary circuit. Each site (black dot) is the direct product of a two-state qubit and a $q$-state qudit. Each gate (blue box) locally conserves $S_z^{\rm tot}$, the total $z$ component of the two qubits it acts upon, and is thus a block-diagonal unitary of the form shown on the right, with each block of each gate independently Haar-random. The smaller blocks do not flip the qubits and thus operate only on the two qudits, while the larger block also produces $S_z^{\rm tot}$-conserving qubit "flip-flops".
• Figure 2: A Pauli string $S$ (green) with its rightmost non-identity operator on site $i$ at time $t$ has its right-front on the gate $(i,i+1)$ (colored red). Under the action of the circuit, the front moves forwards if the endpoint of the string moves to $(i+1)$ at time $t+1$ (left), while the front moves backwards if the action of the circuit leaves the endpoint at site $i$ (right).
• Figure 3: (a,b): Right/left-weight profiles $\rho_{R/L}(x,t)$ showing the spreading of an initially local conserved charge $(z\mathbb{I})_0(t)$ in a random circuit model with $S_z^{\rm tot}$ conservation in a system of size $L=1000$ at different times $t$. These profiles depict three regimes: (i) a "lump" in the region $|x|\lesssim \sqrt{D_c t}$ reflecting the weight of the operator on diffusively spreading conserved charges (shaded purple). This lump emits ballistically spreading nonconserved operators at a slow power-law rate. This emission creates (ii) the leading ballistic "fronts" near $|x| \sim v_B t$ within which the majority of the operator right- and left-weight is contained (shaded red for the latest time). These leading fronts are from nonconserved operators emitted at early times and they are perfectly sharp at $q=\infty$ where $v_B=1$ (b), and have a width $\sqrt{D_\rho t}$ for finite $q$ (a); Finally, the slow emission also leads to (iii) diffusive tails $\sim(v_B t -|x|)^{-3/2}$ behind the leading fronts which reflect the operator weight in "lagging" fronts of nonconserved operator strings that were emitted at later times (shaded blue for the latest time). The curves in (a) are obtained via a simulation at $q=3$ which takes into account the different processes (diffusion of charges, emission of nonconserved operators and the biased diffusion of the nonconserved right- and left-weights) to order $1/q^2$. The red dashed curve is the exact infinite $q$ answer for the "tail" \ref{['eq:tail']}. (c,d): For comparision, $\rho_{R/L}(x,t)$ in an unconstrained random circuit modelopspreadAdamopspreadCurt where $z_0(t)$ isn't "special". Regimes (i) and (iii) do not exist in an unconstrained circuit, and the ballistically spreading operator fronts describe the entire right- and left-weight profiles. The fronts are again infinitely sharp at $q=\infty$ (d) and have a finite width $\sim \sqrt{D_\rho t}$ for $q< \infty$ (c).
• Figure 4: Densities of local operators charged under "spin" and "raising action" behind the ballistic front of a spreading operator at $q=\infty$ and late times \ref{['eq:deltars_exact']}. $\rho_{u/d}(x,t)$ measures the local operator weight on site $x$ on $u/d$ operators that are charged under spin, while $\rho_{r/l}(x,t)$ measures the local weight on $r/l$ operators that have raising charge. Outside the ballistic operator front (at $|x|>v_B t$), the spreading operator locally is purely identities which contribute equally to $\rho_{u/d}=1/2$, but do not contribute to $\rho_{r/l}=0$. The arrival of the front at a given site turns on a noisy coupled diffusion process between the spin and raising charges which relaxes the initially imbalanced densities of these charges to the final "equilibrium" value where all local operators are equally likely. Note that $\rho_u = \rho_d$ and $\rho_r = \rho_l$ in this regime since any initial imbalances in raising/spin charges spread only diffusively and are not present near the ballistic front at late times. Dashed lines plot the "coarse-grained" densities in the scaling limit \ref{['eq:deltars']}.
• Figure 5: One minus the out-of-time-order commutator (OTOC) between $z_0(t)$ and $r_x$ at zero chemical potential, ${ \mathcal{{C}}}^0_{zr}$, plotted against $x$ for a system of length $L=1000$ at different times $t$ showing the different regimes discussed in the text. For $|x| >t$ (outside the dashed vertical lines), the OTOC is strictly zero due to the locality of the circuit. In the region $v_B t <|x|<t$, which is inside the causal light cone but before the leading front arrives, the OTOC is exponentially small (green shaded area for the latest time). The arrival of the ballistic operator front ($|x| \sim v_B t$) leads to a strong increase in the OTOC from a value exponentially small to an $O(1)$ value (shaded red area for the latest time). However, diffusive tails in the operator shape or internal structure lead to diffusive power-law tails in space and time $\sim (x-v_B t)^{-1/2}$ in the late-time approach of the OTOC to its final value of 1 (shaded blue area for the latest time). By contrast, for an unconstrained random circuit (not shown), the OTOC at a given site approaches one exponentially quickly after the leading front passesopspreadAdamopspreadCurt. The diffusive region near the origin $|x| \lesssim \sqrt{D_c t}$ (shaded purple) receives a subleading $1/t$ contribution from the conserved charges which shows up as a "dimple" in the curves at early times which becomes weaker at late times. All curves are obtained via a simulation using $q=3$ and taking into account all processes to order $1/q^2$. The dashed red curve is the $q=\infty$ prediction for the functional form of the tail \ref{['eq:otoc_tail']}.