Quantum many-body attractors


 Real-world complex systems often show robust, persistent oscillatory dynamics, e.g.~non-trivial attractors. On the quantum level this behaviour has only been found in semi-classical or weakly correlated systems under restrictive assumptions. However, strongly interacting systems without classical limits, e.g.~electrons on a lattice or spins, typically relax quickly to a stationary state (trivial attractors). This raises the puzzling question of how non-trivial attractors can arise from the quantum laws. Here, we introduce strictly local dynamical symmetries that lead to extremely robust and persistent oscillations in quantum many-body systems without a classical limit. Observables that do not have overlap with the symmetry operators can relax, losing memory of their initial conditions. The remaining observables enter complex dynamical cycles, signalling the emergence of a quantum many-body attractor. We provide a recipe for constructing Hamiltonians featuring local dynamical symmetries. As an example, we introduce the spin lace – a model of a quasi-1D quantum magnet.

Constructing a many-body system with strictly local dynamical symmetries. (a) Schematic close-up of an arbitrary lattice showing the supports of a strictly local dynamical symmetryÂ and its associated local HamiltonianĤA. The remainder of the system can be governed by any HamiltonianĤ A without affecting the local dynamical symmetry. (b) Connecting local Hamiltonians together as shown builds a many-body system with an extensive number of strictly local dynamical symmetries.
generally such quantities are highly non-local operators 89 that are inaccessible to local measurements in a large 90 system.

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The situation becomes non-trivial when there exists an eigenoperatorÂ that is strictly local, i.e. supported on only a few neighbouring lattice sites. We can then divide the system into two parts, such thatĤ =Ĥ A +Ĥ A with such that its support is strictly larger than that ofÂ (see at frequency ω q and never relax to a stationary value.

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Crucially, moreover, the connectivity of the lattice im-116 plies that [Ĥ p ,Ĥ p+1 ] = 0. As shown below, this allows 117 local observables that overlap withÂ p to undergo par-118 tial relaxation and tend to a limit cycle at long times.
value, up to small fluctuations that we attribute to nu-208 merical finite-size effects. The disparity between these 209 two behaviours is clearly seen in the Fourier domain 210 (Fig. 3(b)), where a dominant peak at frequency 2B dis-   In our numerical simulation for Fig.3(a), we considered The frequency response shown in Fig. 3(b) is numerically obtained from the finite-time evolution data in Fig. 3(a) by calculating the function where T is the maximum simulation time and C O (t) = 279 Ô (t)ŝ x r is the correlation function describing the re-280 sponse ofÔ to a perturbationĤ pert ∝ŝ x r .

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Generation of initial states 282 In Fig. 3(c) we demonstrate the convergence of local observables to a non-stationary quantum many-body attractor starting from different initial conditions. Specifically, we consider global magnetisation configurations that correspond to initial preparations of the product formρ = rρ r , withρ r the local state describing one unit cell. For concreteness, we takê ρ r = 1 8 Î 2 +σ x 2r−1 ⊗ I 2 + aσ x 2r,1 + bσ y 2r,1 + cσ z 2r,1 ⊗ Î 2 , sults over all cells in the lattice, we obtain Fig. 3(c)