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 Sp(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.Received 15 September 2020Accepted 30 November 2020DOI:https://doi.org/10.1103/PhysRevResearch.2.043403Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum chaosPhysical SystemsFloquet systemsQuantum spin modelsTechniquesRandom matrix theoryStatistical Physics
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 Sp(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.Received 15 September 2020Accepted 30 November 2020DOI:https://doi.org/10.1103/PhysRevResearch.2.043403Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum chaosPhysical SystemsFloquet systemsQuantum spin modelsTechniquesRandom matrix theoryStatistical Physics
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.
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.
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 the 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. A 400, 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 the 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 to 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 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 nonintegrable, 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.Received 5 February 2018Revised 12 April 2018DOI:https://doi.org/10.1103/PhysRevX.8.021062Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum chaosQuantum statistical mechanicsStatistical PhysicsNonlinear DynamicsCondensed Matter, Materials & Applied Physics
We show the emergence of random matrix theory (RMT) spectral correlations in the chaotic phase of generic, periodically kicked, interacting, quantum many-body systems by analytically calculating the spectral form factor (SFF), K ( t ) , up to two leading orders in time, t . We explicitly consider the presence or absence of time-reversal ( T ) symmetry to investigate all three Dyson symmetry classes, namely, the circular orthogonal ensemble (COE) for T -invariant systems with T 2 = 1 , the circular unitary ensemble (CUE) in the absence of T symmetry, and the circular symplectic ensemble (CSE) for T -invariant systems with T 2 = − 1 . Our derivation applies the random phase approximation to perform ensemble averages, and we give its justification for generic many-body interactions. We express the SFF as a sum of contributions of certain diagrams. While the number of diagrams that contribute to the SFF in the first and second orders in t for the COE and CUE classes is finite, the number of such diagrams is exponentially large in t for the CSE class. Thus, we have developed a new diagrammatic technique that uses reduced diagrams to incorporate contributions from many diagrams. For the COE class, we show that beyond Thouless time t * , the SFF takes the form K ( t ) ≃ 2 t − 2 t 2 / N up to second order in time, where N is the Hilbert space dimension. For the CUE class, we show that K ( t ) ≃ t beyond t * , and there is no universal term in the second order, unlike the COE case. For the CSE class, we show that K ( t ) ≃ 2 t + 2 t 2 / N up to two orders in time beyond t * . The above findings of the SFF up to second order in time are in agreement with the RMT predictions for these classes. In all three cases, the scaling of t * with system size L is determined by the eigenvalues of a doubly stochastic matrix M . For strongly interacting fermionic chains, M is S U ( 2 ) invariant in all three cases, leading to t * ∝ L 2 in the presence of U ( 1 ) symmetry. In the absence of U ( 1 ) symmetry, we find t *
Quantum many-body systems are commonly considered as quantum chaotic if their spectral statistics, such as the level spacing distribution, agree with those of random matrix theory (RMT). Using the example of the kicked Ising chain we demonstrate that even if both level spacing distribution and eigenvector statistics agree well with random matrix predictions, the entanglement entropy deviates from the expected RMT behavior, i.e., the Page curve. To explain this observation we propose a quantity that is based on the effective Hamiltonian of the kicked system. Specifically, we analyze the distribution of the strengths of the effective spin interactions and compare them with analytical results that we obtain for circular ensembles. Thereby we group the effective spin interactions corresponding to the number k of spins which contribute to the interaction. By this the deviations of the entanglement entropy can be attributed to significantly different behavior of the k-spin interactions compared with RMT.
Spectral form factor (SFF), one of the key quantity from random matrix theory, serves as an important tool to probe universality in disordered quantum systems and quantum chaos. In this work, we present exact closed-form expressions for the second- and third-order SFFs in the circular unitary ensemble (CUE), valid for all real values of the time parameter, and analyze their asymptotic behavior in different regimes. In particular, for the second-order SFF, we derive an exact closed-form expression in terms of polygamma functions. In the limit of infinite matrix size, and when the time parameter is restricted to integer values, the second-order SFF reproduces the standard result established in earlier studies. When the time parameter is of order one relative to the matrix size, we demonstrate that the second-order SFF grows logarithmically with the ensemble dimension. For the third-order SFFs, a closed-form result in a special case is obtained by exploiting the translational invariance of CUE.