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1. Hilbert-Space Ergodicity in Driven Quantum Systems: Obstructions and Designs

   https://doi.org/10.1103/PhysRevX.14.041059  

   Pilatowsky-Cameo, Saúl; Marvian, Iman; Choi, Soonwon; Ho, Wen Wei (December 2024, Physical Review X)

   Despite its long history, a canonical formulation of quantum ergodicity that applies to general classes of quantum dynamics, including driven systems, has not been fully established. Here we introduce and study a notion of quantum ergodicity for closed systems with time-dependent Hamiltonians, defined as statistical randomness exhibited in their longtime dynamics. Concretely, we consider the temporal ensemble of quantum states (time-evolution operators) generated by the evolution, and investigate the conditions necessary for them to be statistically indistinguishable from uniformly random states (operators) in the Hilbert space (space of unitaries). We find that the number of driving frequencies underlying the Hamiltonian needs to be sufficiently large for this to occur. Conversely, we show that statistical —indistinguishability up to some large but finite moment—can already be achieved by a quantum system driven with a single frequency, i.e., a Floquet system, as long as the driving period is sufficiently long. Our work relates the complexity of a time-dependent Hamiltonian and that of the resulting quantum dynamics, and offers a fresh perspective to the established topics of quantum ergodicity and chaos from the lens of quantum information. Published by the American Physical Society2024 

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2. Robust Effective Ground State in a Nonintegrable Floquet Quantum Circuit

   https://doi.org/10.1103/PhysRevLett.133.030401  

   Ikeda, Tatsuhiko N; Sugiura, Sho; Polkovnikov, Anatoli (July 2024, Physical Review Letters)

   An external periodic (Floquet) drive is believed to bring any initial state to the featureless infinite temperature state in generic nonintegrable isolated quantum many-body systems in the thermodynamic limit, irrespective of the driving frequency $\mathrm{\Omega }$. However, numerical or analytical evidence either proving or disproving this hypothesis is very limited and the issue has remained unsettled. Here, we study the initial state dependence of Floquet heating in a nonintegrable kicked Ising chain of length up to $L=30$with an efficient quantum circuit simulator, showing a possible counterexample: the ground state of the effective Floquet Hamiltonian is exceptionally robust against heating, and could stay at finite energy density even after infinitely many Floquet cycles, if the driving period is shorter than a threshold value. This sharp energy localization transition or crossover does not happen for generic excited states. The exceptional robustness of the ground state is interpreted by (i) its isolation in the energy spectrum and (ii) the fact that those states with $L$-independent $\hslash \mathrm{\Omega }$energy above the ground state energy of any generic local Hamiltonian, like the approximate Floquet Hamiltonian, are atypical and viewed as a collection of noninteracting quasiparticles. Our finding paves the way for engineering Floquet protocols with finite driving periods realizing long-lived, or possibly even perpetual, Floquet phases by initial state design. Published by the American Physical Society2024 

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3. Topological defects in Floquet circuits

   https://doi.org/10.21468/SciPostPhys.16.3.075  

   Tan, Mao Tian; Wang, Yifan; Mitra, Aditi (March 2024, SciPost Physics)

   We introduce a Floquet circuit describing the driven Ising chain with topological defects. The corresponding gates include a defect that flips spins as well as the duality defect that explicitly implements the Kramers-Wannier duality transformation. The Floquet unitary evolution operator commutes with such defects, but the duality defect is not unitary, as it projects out half the states. We give two applications of these defects. One is to analyze the return amplitudes in the presence of “space-like” defects stretching around the system. We verify explicitly that the return amplitudes are in agreement with the fusion rules of the defects. The second application is to study unitary evolution in the presence of “time-like” defects that implement anti-periodic and duality-twisted boundary conditions. We show that a single unpaired localized Majorana zero mode appears in the latter case. We explicitly construct this operator, which acts as a symmetry of this Floquet circuit. We also present analytic expressions for the entanglement entropy after a single time step for a system of a few sites, for all of the above defect configurations. 

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4. Nearly Periodic Maps and Geometric Integration of Noncanonical Hamiltonian Systems

   https://doi.org/10.1007/s00332-023-09891-4  

   Burby, J_W; Hirvijoki, E.; Leok, M. (February 2023, Journal of Nonlinear Science)

   Abstract M. Kruskal showed that each continuous-time nearly periodic dynamical system admits a formalU(1)-symmetry, generated by the so-called roto-rate. When the nearly periodic system is also Hamiltonian, Noether’s theorem implies the existence of a corresponding adiabatic invariant. We develop a discrete-time analog of Kruskal’s theory. Nearly periodic maps are defined as parameter-dependent diffeomorphisms that limit to rotations along aU(1)-action. When the limiting rotation is non-resonant, these maps admit formalU(1)-symmetries to all orders in perturbation theory. For Hamiltonian nearly periodic maps on exact presymplectic manifolds, we prove that the formalU(1)-symmetry gives rise to a discrete-time adiabatic invariant using a discrete-time extension of Noether’s theorem. When the unperturbedU(1)-orbits are contractible, we also find a discrete-time adiabatic invariant for mappings that are merely presymplectic, rather than Hamiltonian. As an application of the theory, we use it to develop a novel technique for geometric integration of non-canonical Hamiltonian systems on exact symplectic manifolds. 

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5. Matrix product operator approach to nonequilibrium Floquet steady states

   https://doi.org/10.1103/PhysRevB.106.L220307  

   Cheng, Zihan; Potter, Andrew C. (December 2022, Physical Review B)

   We present a numerical method to simulate nonequilibrium Floquet steady states of one-dimensional periodically driven many-body systems coupled to a dissipative bath, based on a matrix product operator ansatz for the Floquet density matrix in frequency space. This method enables access to large systems beyond the reach of exact simulations, while retaining the periodic micromotion information. An excited-state extension of this technique allows computation of the dynamical approach to the steady state. We benchmark our method with a driven-dissipative Ising model and apply it to study the possibility of stabilizing prethermal discrete time-crystalline order by coupling to a cold bath. 

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