Abstract The action of any local operator on a quantum system propagates through the system carrying the information of the operator. This is usually studied via the out-of-time-order correlator (OTOC). We numerically study the information propagation from one end of a periodically driven spin-1/2 XY chain with open boundary conditions using the Floquet infinite-temperature OTOC. We calculate the OTOC for two different spin operators, σ x and σ z . For sinusoidal driving, the model can be shown to host different types of edge states, namely, topological (Majorana) edge states and non-topological edge states. We observe a localization of information at the edge for both σ z and σ x OTOCs whenever edge states are present. In addition, in the case of non-topological edge states, we see oscillations of the OTOC in time near the edge, the oscillation period being inversely proportional to the gap between the Floquet eigenvalues of the edge states. We provide an analytical understanding of these effects due to the edge states. It was known earlier that the OTOC for the spin operator which is local in terms of Jordan–Wigner fermions (σ z ) shows no signature of information scrambling inside the light cone of propagation, while the OTOC for the spin operator which is non-local in terms of Jordan–Wigner fermions (σ x ) shows signatures of scrambling. We report a remarkable ‘unscrambling effect’ in the σ x OTOC after reflections from the ends of the system. Finally, we demonstrate that the information propagates into the system mainly via the bulk states with the maximum value of the group velocity, and we show how this velocity is controlled by the driving frequency and amplitude.
Abstract The action of any local operator on a quantum system propagates through the system carrying the information of the operator. This is usually studied via the out-of-time-order correlator (OTOC). We numerically study the information propagation from one end of a periodically driven spin-1/2 XY chain with open boundary conditions using the Floquet infinite-temperature OTOC. We calculate the OTOC for two different spin operators, σ x and σ z . For sinusoidal driving, the model can be shown to host different types of edge states, namely, topological (Majorana) edge states and non-topological edge states. We observe a localization of information at the edge for both σ z and σ x OTOCs whenever edge states are present. In addition, in the case of non-topological edge states, we see oscillations of the OTOC in time near the edge, the oscillation period being inversely proportional to the gap between the Floquet eigenvalues of the edge states. We provide an analytical understanding of these effects due to the edge states. It was known earlier that the OTOC for the spin operator which is local in terms of Jordan–Wigner fermions (σ z ) shows no signature of information scrambling inside the light cone of propagation, while the OTOC for the spin operator which is non-local in terms of Jordan–Wigner fermions (σ x ) shows signatures of scrambling. We report a remarkable ‘unscrambling effect’ in the σ x OTOC after reflections from the ends of the system. Finally, we demonstrate that the information propagates into the system mainly via the bulk states with the maximum value of the group velocity, and we show how this velocity is controlled by the driving frequency and amplitude.
The out-of-time-order correlator (OTOC) measures the propagation and scrambling of local quantum information. For the transverse field Ising model with open boundaries, the local operator ${\ensuremath{\sigma}}^{x}$ shows an interesting picture of the ground state OTOC where the local information alternately gets scrambled and then ``unscrambled'' upon reflection at the ends. Earlier discussions of OTOCs did not explain the physical processes responsible for such scrambling and unscrambling. We explicitly show that in the paramagnetic phase, the scrambling and unscrambling is due to the scattering of a pair of low-energy spin-flip excitations, and extend it to incorporate integrability-breaking interactions. In the ferromagnetic phase these are explained by the motion of a domain-wall excitation, and we find that unscrambling survives up to moderate values of interactions. Thus, in different limits of the system parameters, we provide a simple understanding of the OTOCs, including the unscrambling, in terms of the low-energy excitations like spin flips or domain walls. We mention a system where our results can be experimentally tested.
Out-of-time-ordered correlators (OTOC) have been proposed to characterize quantum chaos in generic systems. However, they can also show interesting behavior in integrable models, resembling the OTOC in chaotic systems in some aspects. Here we study the OTOC for different operators in the exactly-solvable one-dimensional quantum Ising spin chain. The OTOC for spin operators that are local in terms of the Jordan-Wigner fermions has a ``shell-like'' structure: After the wavefront passes, the OTOC approaches its original value in the long-time limit, showing no signature of scrambling; the approach is described by a ${t}^{\ensuremath{-}1}$ power law at long time t. On the other hand, the OTOC for spin operators that are nonlocal in the Jordan-Wigner fermions has a ``ball-like'' structure, with its value reaching zero in the long-time limit, looking like a signature of scrambling; the approach to zero, however, is described by a slow power law ${t}^{\ensuremath{-}1/4}$ for the Ising model at the critical coupling. These long-time power-law behaviors in the lattice model are not captured by conformal field theory calculations. The mixed OTOC with both local and nonlocal operators in the Jordan-Wigner fermions also has a ``ball-like'' structure, but the limiting values and the decay behavior appear to be nonuniversal. In all cases, we are not able to define a parametrically large window around the wavefront to extract the Lyapunov exponent.
A closed quantum system never forgets its initial state, but the encoded information can get scrambled and become inaccessible without measuring a large fraction of all the system degrees of freedom. This scrambling can be diagnosed by studying the spatial spreading of initially local operators under the Heisenberg time evolution, and the decay of the out-of-time-ordered correlators (OTOC). What insights can OTOCs provide to understand the dynamics of quantum many-body systems? What are the characteristic behaviors of OTOCs during the time evolution? How is information scrambling affected by the dissipation in open quantum many-body systems? We first study slow scrambling in many-body localized systems via calculating various correlators, two-point retarded correlators and OTOCs. Comparing with retarded correlators, OTOCs provide more information about the dynamics. We find that disorder slows and partially halts the onset of information scrambling. Instead of ballistic spreading, propagation of information forms a logarithmic light cone. Next, we study the finite-size scaling of OTOCs at late times in generic thermalizing quantum many-body systems. When energy is conserved, the late-time saturation value of the OTOC of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation. We also study information scrambling in open quantum many-body systems. We define a dissipative version of OTOC and study its behaviors in a prototypical chaotic quantum chain with dissipation. We find that dissipation leads to not only the overall decay of the scrambled information due to leaking, but also structural changes so that the information light cone can only reach a finite distance even when the effect of overall decay is removed. Finally, we construct a family of local Hamiltonians for understanding the asymmetric information scrambling. Our models live on a one-dimensional lattice and exhibit asymmetric butterfly light cone between the left and right spatial directions.
We investigate the integrability-to-chaos transition and information scrambling in Ising spin networks via a graph-theoretic formulation. Modeling spins as vertices and interactions as edges encoded by adjacency matrices across path, Erdos-Renyi, and Watts-Strogatz topologies, we demonstrate that long-range couplings and heterogeneous degree distributions markedly accelerate quantum information propagation. The Hamiltonian comprises local and normalized non-local interactions; tuning the non-local coupling and field heterogeneity drives integrability breaking. To quantify scrambling, we employ bipartite mutual and tripartite information. Increasing non-local interactions drives tripartite information to large negative values, signaling deep information scrambling. The squared commutators constructed from out-of-time-order correlators (OTOCs) exhibit early-time exponential growth; equivalently, the OTOC four-point functions themselves decay exponentially, yielding quantum Lyapunov exponents that scale systematically with parameters governing the chaotic regime. Complementing this, Krylov complexity reveals rapid operator growth in the chaotic phase, synchronizing with OTOC and mutual information dynamics. Spectrally, the transition manifests as a shift from Poissonian to Wigner-Dyson level spacing statistics. The spectral form factor (SFF) exhibits the characteristic slope-dip-ramp-plateau structure, enabling the extraction of Thouless and Heisenberg times. Crucially, a reduced Thouless time strongly correlates with accelerated information and operator scrambling. Ultimately, this work establishes a unified framework bridging network topology with information-theoretic, operator, and spectral diagnostics, offering insights into thermalization and non-equilibrium dynamics in quantum many-body systems.
We report a numerical observation where the infinite-temperature out-of-time-order correlators (OTOCs) directly probe quantum phase transitions at zero temperature, in contrast to common intuition where low-energy quantum effects are washed away by strong thermal fluctuations at high temperature. By comparing numerical simulations with exact analytic results, we determine that this phenomenon has a topological origin and is highly generic, as long as the underlying system can be mapped to a 1D Majorana chain. Using the Majorana basis, we show that the infinite-temperature OTOCs probe zero-temperature quantum phases via detecting the presence of Majorana zero modes at the ends of the chain that is associated with 1D Z2 topological order. Hence, we show that strong zero modes also affect OTOCs and scrambling dynamics. Our results demonstrate an intriguing interplay between information scrambling and topological order, which leads to a new phenomenon in the scrambling of generic nonintegrable models: topological order induced prescrambling, paralleling the notion of prethermalization of two-time correlators that defines a timescale for the restricted scrambling of topologically protected quantum information.
Inhomogeneous systems have become increasingly relevant in experimentally engineered quantum many-body systems, where spatially varying interaction strengths can significantly influence nonequilibrium dynamics. Motivated by this, we numerically study the out-of-time-ordered correlators (OTOCs) in inhomogeneous XXZ spin chains. Using two types of spatially varying interaction profiles, namely, linear-gradient and stepwise profiles, we show that spatial inhomogeneity in the interactions can induce pronounced asymmetry in information scrambling, with OTOC operators located at different sites exhibiting distinct dynamical behavior. In particular, OTOCs associated with the strongly interacting side exhibit suppressed scrambling compared to those associated with the weakly interacting side, with the difference becoming more pronounced when the interaction strengths differ significantly. To elucidate the origin of the long-time saturation of OTOCs, we analyze the diagonal matrix elements of the OTOC observables in the energy eigenbasis and their overlap with the Hamiltonian, which explicitly incorporates the spatial interaction profile. The resulting analytical expression is consistent with the numerical results. Finally, we extend our analysis to periodically driven inhomogeneous spin chains and show that the interaction-gradient-induced asymmetry persists under periodic driving.