Fig. 4: Weak ergodicity breaking from quantum scarring.
a Time-averaged projection \(Q(\left\vert \psi (t)\right\rangle )\) over the manifold of IS states, for various models and initial conditions. In the top row, the system is initialized in \(\vert {{{{\rm{IS}}}}}_{y}\rangle\), namely, on the IS UPO aligned along y (ϕ = θ = π/2, marked by a star), and is more likely to be found on the same UPO at long times. In the bottom row, the system is initialized in \(\vert {{{{\rm{IS}}}}}_{\mu }\rangle\), namely, on the IS UPO aligned along μ (also marked by a star), and is more likely to be found there at long times. That is, the system retains some information on its initial condition and weakly breaks ergodicity. This is a quantum effect: the classical projection (see Supplemental Information), shown for the XXZ model, is insensitive to the initial condition and almost uniform. b, c Scaling of \(\bar{Q}\) with system size N, with focus on the time-averaged return probabilities (\(\bar{Q}({{{\mathbf{\ y}}}})\) for \(\left\vert {\psi }_{0}\right\rangle=\left\vert {{{{\rm{IS}}}}}_{y}\right\rangle\), denoted y → y, and \(\bar{Q}({{{\boldsymbol{\mu }}}})\) for \(\left\vert {\psi }_{0}\right\rangle=\left\vert {{{{\rm{IS}}}}}_{\mu }\right\rangle\), denoted μ → μ) and cross probability (\(\bar{Q}({{{\mathbf{\ y}}}})\) for \(\left\vert {\psi }_{0}\right\rangle=\left\vert {{{{\rm{IS}}}}}_{\mu }\right\rangle\), and viceversa, denoted y ↔ μ). While \(\bar{Q}\) decays exponentially in N, scarring makes the return probabilities consistently larger than the cross probabilities, indeed by a factor >4 for all considered system sizes. In (a) we considered \(s=\frac{1}{2}\), N = 20, and μ = (2.4, 0, 0.4), and chose J ensuring that the system is fully thermal: Jzz = −1.8 for the Ising model, Jxx = Jyy = −0.4 and Jzz = −1.8 for the XXZ model, and Jxx = Jyy = −1.4 for the XX model. In (b, c) we considered the Ising model.
