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Fluctuations of the spectral form factor in the kicked Ising chain

Ana Flack

Published Jan 1, 2020• 0 citations
OpenAlex

Abstract

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.

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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 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.
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.
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.
Many-body quantum chaos at ℏ∼1\hbar\sim 1ℏ∼1: Analytic connection to random matrix theory
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). For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry (1985) within the so-called diagonal approximation of semiclassical periodic-orbit sums. Derivation of the full RMT spectral form factor K(t)K(t)K(t) from semiclassics has been completed only much later in a tour de force by Mueller et al (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 `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 the first theoretical explanation for these observations. We compute K(t)K(t)K(t) explicitly in the leading two orders in ttt 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.
The most general and versatile defining feature of quantum chaotic systems is that they possess an energy spectrum with correlations universally described by random matrix theory (RMT). This feature can be exhibited by systems with a well-defined classical limit as well as by systems with no classical correspondence, such as locally interacting spins or fermions. Despite great phenomenological success, a general mechanism explaining the emergence of RMT without reference to semiclassical concepts is still missing. Here we provide the example of a quantum many-body system with no semiclassical limit (no large parameter) where the emergence of RMT spectral correlations is proven exactly. Specifically, we consider a periodically driven Ising model and write the Fourier transform of spectral density's two-point function, the spectral form factor, in terms of a partition function of a two-dimensional classical Ising model featuring a space-time duality. We show that the self-dual cases provide a minimal model of many-body quantum chaos, where the spectral form factor is demonstrated to match RMT for all values of the integer time variable t in the thermodynamic limit. In particular, we rigorously prove RMT form factor for an odd t, while we formulate a precise conjecture for an even t. The results imply ergodicity for any finite amount of disorder in the longitudinal field, rigorously excluding the possibility of many-body localization. Our method provides a novel route for obtaining exact nonperturbative results in nonintegrable systems.