Abstract We study the out-of-time-order correlation (OTOC) of the Floquet transverse Ising model and use it to verify the phase diagram of the system. First, we present the exact analytical solution of the transverse magnetization OTOC using the Jorden-Wigner transformation. We calculate the speed of correlation propagation and analyze the behavior of the revival time with the separation between the observables. In order to get the phase structure of the Floquet transverse Ising system, we use the longitudinal magnetization OTOC as it is known to serve as an order parameter of the system. We show the phase structure numerically in the transverse Ising Floquet system by using the long time average of the longitudinal magnetization OTOC. In both the open and the closed chain systems, we find distinct phases out of which two are paramagnetic (0-paramagnetic and π -paramagnetic), and two are ferromagnetic (0-ferromagnetic and π -ferromagnetic) as defined in the literature.
Abstract We study the out-of-time-order correlation (OTOC) of the Floquet transverse Ising model and use it to verify the phase diagram of the system. First, we present the exact analytical solution of the transverse magnetization OTOC using the Jorden-Wigner transformation. We calculate the speed of correlation propagation and analyze the behavior of the revival time with the separation between the observables. In order to get the phase structure of the Floquet transverse Ising system, we use the longitudinal magnetization OTOC as it is known to serve as an order parameter of the system. We show the phase structure numerically in the transverse Ising Floquet system by using the long time average of the longitudinal magnetization OTOC. In both the open and the closed chain systems, we find distinct phases out of which two are paramagnetic (0-paramagnetic and π -paramagnetic), and two are ferromagnetic (0-ferromagnetic and π -ferromagnetic) as defined in the literature.
We study characteristic, dynamic, and saturation regimes of the out-of-time-order correlation (OTOC) in the constant-field Floquet system with and without longitudinal field. In the calculation of OTOC, we take local spins in longitudinal and transverse directions as observables which are local and nonlocal in terms of Jordan-Wigner fermions, respectively. We use the exact analytical solution of OTOC for the integrable model (without the longitudinal field term) with transverse direction spins as observables and numerical solutions for other integrable and nonintegrable cases. OTOCs in both cases depart from unity at a kick equal to the separation between the observables when the local spins are in the transverse direction, and one additional kick is required when the local spins are in the longitudinal direction. The number of kicks required to depart from unity depends on the separation between the observables and is independent of the Floquet period and system size. In the dynamic region, OTOCs show power-law growth in both models, the integrable one (without longitudinal field) and the nonintegrable one (with longitudinal field). The exponent of the power-law increases with increasing separation between the observables. Near the saturation region, OTOCs grow linearly at a very low rate.
In recent times out-of-time-order correlators (OTOC) have been established as a tool to understand butterfly effects, quantum information scrambling, and many-body localization. They can also be useful in determining different phases of quantum critical systems. OTOCs can identify the quantum chaos within a system undergoing time evolution; and therefore, they can distinguish between chaotic and regular dynamics. This motivates us to study OTOCs in integrable and nonintegrable periodically kicked quantum spin models. A periodically kicked quantum Ising spin system, known as the quantum Ising Floquet system, is a variant of the transverse Ising model. In place of constant transverse magnetic fields in the transverse Ising system, time-periodic fields are applied in the form of delta pulses in the quantum Ising Floquet spin system. It provides very interesting and peculiar dynamics separate from that of the transverse Ising system.
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.
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-order correlators (OTOCs) progressively play an important role in different fields of physics, particularly in the nonequilibrium quantum many-body systems. In this paper, we show that OTOCs can be used to probe the Floquet dynamical quantum phase transitions (FDQPTs). We investigate the OTOCs of two exactly solvable Floquet spin models, namely, Floquet XY chain and synchronized Floquet XY model. We show that the border of driving frequency range, over which the Floquet XY model shows FDQPT, is signaled by the global minimum of the infinite-temperature time averaged OTOC. Moreover, our results manifest that OTOCs decay algebraically in the long time, for which the decay exponent in the FDQPT region is different from that in the region where the system does not show FDQPTs. In addition, for the synchronized Floquet XY model, which reveals FDQPT at any driving frequency depending on the initial infinite or finite temperature, the imaginary part of the OTOCs becomes zero whenever the system shows FDQPT.
There has been much recent interest in the study of out of equilibrium quantum systems due to experimental advances in trapped cold atomic gases that allow systematic studies of these effects.In this work we study a quantum Ising model in the presence of transverse and longitudinal magnetic fields.The model lacks an analytical solution in the most general case and has to be treated numerically.We first investigate the static properties of the model, including the ground state energy and the zero temperature phase diagram, using the Density Matrix Renormalization Group (DMRG) method.We identify ordered and disordered phases separated by a quantum phase transition.We then consider the time evolution of the system following an abrupt quench of the fields.We study the response of the order parameter to the quenches within the ordered and disordered phases as well as quenches between the phases using the time-dependent DMRG method.The results are in qualitative agreement with experiments and the implemented numerical method in this work can be extended to allow for more general interactions.iii I would like to thank my supervisor, Dr. Malcolm Kennett, whose constant support and hope in the most disappointing situations, his helpful guidance, and endless energy were all new and very invaluable to me.I like to thank my