Determining the border between ergodic and localized behavior is of central interest for interacting many-body systems. We consider here the recently quite popular kicked Ising chain. A convenient description of such a many-body system is achieved by the dual operator that evolves the system in contrast to the time-evolution operator not in time but particle direction. We focus in this paper on the largest eigenvalue of a function of the dual operator. It is identified to be a convenient tool to determine if the system shows ergodic or many-body localized features. We analytically explain the eigenvalue structure by perturbation theory in the vicinity of the noninteracting system. This result agrees with the numerics for small times in a previous work [Braun et al., Phys. Rev. E 101, 052201 (2020)]. Given that a direct numerical computation of this largest eigenvalue is only possible for small times, another achievement of this paper is to provide a way to determine this eigenvalue by the ratio of spectral form factors for adjacent particle numbers. By extensive large-time numerical computations of the spectral form factor, we can then distinguish between localized and ergodic system features and compute the Thouless time, i.e., the transition time between these regimes in the thermodynamic limit.
Determining the border between ergodic and localized behavior is of central interest for interacting many-body systems. We consider here the recently quite popular kicked Ising chain. A convenient description of such a many-body system is achieved by the dual operator that evolves the system in contrast to the time-evolution operator not in time but particle direction. We focus in this paper on the largest eigenvalue of a function of the dual operator. It is identified to be a convenient tool to determine if the system shows ergodic or many-body localized features. We analytically explain the eigenvalue structure by perturbation theory in the vicinity of the noninteracting system. This result agrees with the numerics for small times in a previous work [Braun et al., Phys. Rev. E 101, 052201 (2020)]. Given that a direct numerical computation of this largest eigenvalue is only possible for small times, another achievement of this paper is to provide a way to determine this eigenvalue by the ratio of spectral form factors for adjacent particle numbers. By extensive large-time numerical computations of the spectral form factor, we can then distinguish between localized and ergodic system features and compute the Thouless time, i.e., the transition time between these regimes in the thermodynamic limit.
The main goal of this thesis is to study the fluctuations of the kicked Ising spin chain and then compare the results to the predictions of random matrix theory (RMT), which is believed to be an indicator of quantum chaos. This is achieved by applying a recently discovered method based on the duality between the propagation in time and in space. Before the results are presented, some relevant concepts from RMT will be introduced. In addition to the general introduction of this theory, the connection to quantum chaos is also discussed. The thesis continues with the definition of the kicked Ising model and the space-time duality.
By investigating the model at the self-dual point, where transfer matrix is unitary in both space and time, we calculate the first power of the trace of the Floquet propagator and it further confirms the previously obtained result for the averaged spectral form factor (SFF). Then the same steps are used to estimate the variance of the SFF. These results are further tested by two numerical methods, the Monte-Carlo simulations and the power method, which is applied to the dual quantum propagator. The predictions for all higher-order moments are also presented at the end.
Our results show that, contrary to the expectations, the fluctuations of the self-dual kicked Ising model do not agree with the results obtained in the scope of RMT. These findings are interesting due to the fact that it is believed that non-integrable chaotic quantum many-body systems generally agree with random matrix theory predictions.
We analytically study the time evolution of the expectation values of observables in periodically kicked many-body quantum systems. Starting from an initial state, we compute both the transient and the long-time properties of the observables. Our derivation explains the criteria and the mechanism that lead to the infinite-temperature statistical average of observables at long times, irrespective of the initial state. When the criteria are violated, the observables oscillate with time. These oscillations are subharmonic and robust to small perturbations, suggesting the emergence of a discrete time crystal phase. We demonstrate these features explicitly in periodically kicked nonintegrable spin chains. For a spin chain with two kicks per cycle, we show that the kicked chain can exhibit an ergodic or a discrete-time crystal phase for the same kicking strengths, depending on the initial state preparation. We complement our time-evolution study of observables with the spectral form factor of these kicked models.
Abstract We demonstrate that the dynamics of kicked spin chains possess a remarkable duality property. The trace of the unitary evolution operator for N spins at time T is related to one of a non-unitary evolution operator for T spins at time N . We characterize this dual operator by its spectrum. Using the duality relation we obtain the oscillating part of the density of states for a large number of spins. Furthermore, the duality relation explains the anomalous short-time behavior of the spectral form factor previously observed in the literature.
We compute the full probability distribution of the spectral form factor in the self-dual kicked Ising model by providing an exact lower bound for each moment and verifying numerically that the latter is saturated.We show that at long enough times the probability distribution agrees exactly with the prediction of random-matrix theory if one identifies the appropriate ensemble of random matrices.We find that this ensemble is not the circular orthogonal one-composed of symmetric random unitary matrices and associated with time-reversal-invariant evolution operators-but is an ensemble of random matrices on a more restricted symmetric space [depending on the parity of the number of sites this space is either S p(N )/U (N ) or O(2N )/O(N )×O(N )].Even if the latter ensembles yield the same averaged spectral form factor as the circular orthogonal ensemble, they show substantially enhanced fluctuations.This behavior is due to a recently identified additional antiunitary symmetry of the self-dual kicked Ising model.
A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT).Most prominent features of such RMT behavior with respect to a random spectrum, both encompassed in spectral pair correlation function, are statistical suppression of small level spacings (correlation hole) and enhanced stiffness of the spectrum at large spectral ranges.For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry [Proc.R. Soc.London, A400, 229 (1985)] within the so-called diagonal approximation of semiclassical periodic-orbit sums, while the derivation of the full RMT spectral form factor K(t) (Fourier transform of spectral pair correlation function), from semiclassics has been completed by Müller et al. [Phys.Rev. Lett.93, 014103 (2004)].In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains.Such systems display two universal types of behaviour which are termed as the 'many-body localized phase' and 'ergodic phase'.In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder.Here we provide a clear theoretical explanation for these observations.We compute K(t) in the leading two orders in t and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.In particular, we relate K(t) to partition functions of a class of twisted classical Ising models on a ring of size t, hence the leading order RMT behavior K(t)2t is a consequence of translation and reflection symmetry of the Ising partition function.