From Dual Unitarity to Generic Quantum Operator Spreading

Dual-unitary circuits are paradigmatic examples of exactly solvable yet chaotic quantum many-body systems, but solvability naturally goes along with a degree of nongeneric behavior. By investigating the effect of weakly broken dual unitarity on the spreading of local operators, we study whether, and how, small deviations from dual unitarity recover fully generic many-body dynamics. We present a discrete path-integral formula for the out-of-time-order correlator and recover a butterfly velocity smaller than the light-cone velocity, $v_B<v_{\mathrm{LC}}$, and a diffusively broadening operator front, two generic features of ergodic quantum spin chains absent in dual-unitary circuit dynamics. The butterfly velocity and diffusion constant are determined by a small set of microscopic quantities, and the operator entanglement of the gates has a crucial role.

Figure 1

(a) The unitary time evolution operator is constructed as a brickwork circuit composed out of identical two-site gates. One row of gates corresponds to a time step $\mathrm{\Delta }t=1$. (b) Graphical depiction of processes contributing to the OTOC. When dual unitarity is broken, the edge of the operator string can scatter into the light cone. The OTOC is then given by a weighted sum of all paths. “Folded” dual-unitary gates are depicted in red and dual-unitary-breaking gates in purple.

Figure 2

(a) Broadening of the OTOC front as a function of time. (b) Comparison of the asymptotic front for different approximations (the shift in the center of the front is corrected for better comparability).

Figure 3

Early time relaxation of the dual-unitary form to the generic shape on the geometric light cone for a range of perturbation strengths. The timescale is indicated by vertical dotted lines.

Figure 4

Plots of (a) butterfly velocity and (b) diffusion constant as function of perturbation strength and comparison to the analytical predictions of the one- and two-step approximations.