Operator Spreading and the Emergence of Dissipative Hydrodynamics under Unitary Evolution with Conservation Laws

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 slow 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 slow 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.

Resilience of scrambling measurements

“…Perhaps the physics of scrambling resilience can be related to known types of robustness, e.g., the robustness of renormalization-group fixed points. It would be interesting to probe resilience in many other recently studied models, including noninteracting, weakly coupled, and semiclassical systems [15,[48][49][50][51][52], many-body-localized states [53][54][55][56][57], the SYK model [6,[43][44][45], open systems [37], local random-circuit models [58][59][60][61], other special solvable models [28], and much else.…”

A random unitary circuit model for black hole evaporation

“…Second, we consider random gates U i,j with a block structure determined by the presence of a U (1) charge, as done for local RUCs in Refs. [39,42]. The two cases are treated separately in the next subsections.…”

“…We implement a dynamics with a U (1) conserved charge by imposing that the two-qudit unitary gates U i,j have some special structure, as done in Refs. [39,42] for the case of spatially local circuits. For the rest of this section, we will focus on the case of qubits, namely d = 2.…”

“…For the rest of this section, we will focus on the case of qubits, namely d = 2. Then, following [39,42] we consider gates of the form…”

“…It turns out that the formalism introduced in the previous section is adequate to compute numerically the mutual information in Eqs. (42) and (43), for both RUCs without and with a conserved U (1) charge. To see this, we can exploit the fact that the Rényi entropy of a subsystem K is equal to that of its complementary one (with respect to the whole space), and rewrite…”