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arXiv:1602.07130 · Journal of Physics A Mathematical and Theoretical

Particle-time duality in the kicked Ising spin chain

M Akila, D Waltner, B Gutkin, T Guhr

Published Aug 19, 2016• 124 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8113/49/37/375101

Related papers

Fluctuations of the spectral form factor in the kicked Ising chain

Ana Flack

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.

•Jan 1, 2020•0 citations

Localization in the kicked Ising chain

Daniel Waltner, Petr A. Braun

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.

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ERGODIC PROPERTIES OF QUANTUM SPIN CHAINS: KICKED TRANSVERSE ISING MODEL

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The problem of calculation of time-correlation functions and relaxation phenomena in interacting quantum many-body systems or quantized fields is put into the framework of ergodic theory and dynamical systems. In this spirit we discuss spectral analysis of the adjoint propagator in a suitable Hilbert space of quantum observables in Heisenberg picture as an alternative approach to characterize many-body dynamics. We illustrate our approach by working out the time-autocorrelation functions at infinite temperature of a large class of observabIes for a spin s = 1/2 Ising chain in a periodically kicking transverse magnetic field, and show that they decay to their asymptotic values as ex: t3/ 2 General understanding of time-correlation functions and relaxation phenomena of interacting quantum many-body systems is a challenging problem with only very partial existing solutions. Situation is much better if one considers the motion of a one-body or few-body systems where the classical ergodic theory (see e.g. [1]) describes a variety of different types of motion, ranging from quasi-periodic integrable motion characterized by a discrete spectrum of the (Liouville) evolution operator, through KAM quasi-integrability, to ergodicity, mixing and chaos characterized by a continuous evolution spectrum. Mixing, which is equivalent of saying that correlation functions of an arbitrary pair of observables (A,B) decay in time, limt-HXJ ((A(t)B) (A)(B)) = 0, is the necessary dynamical property needed to justify the relaxation to equilibrium (micro/grand/canonical) state and the laws of statistical mechanics, such as the fluctuation-dissipation theorem and transport laws. Mixing in quantum many-body system has been investigated by numerical experiment only quite recently [2]. 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Our aim is to include description of dynamics of periodically time-dependent (e.g. kicked) quantum systems so we will adopt Floquet formalism and integrate equations of motion over one period of time T (for autonomous systems, T may be arbitrary, e.g. infinitesimally small), defining the so called quantum Floquet map or one-step propagatorU := T exp(-i J; dtH(t) )which induces the adjoint Floquet map U ad acting on the (sub)space 6 of observables A(t +T) = U ad A(t), U A:= utAU. We should require the subspace 6 to be invariat under dynamics, namely A E 6 :::} U ad A E 6. Linear space 6 is turned into the Hilbert space by introducing an innner product of two extensive observables in thermodynamic limit (size L -+ 00) as (AlB) = (AtB), (A):= lim .!.. tr A £--+00 L tr 1 (1) (2) which has a meaning of a canonical average at infinite temperature. 1 We note that the adjoint Floquet map U ad is a unitary operator in 6 w.r.t. inner product (1), or equivalently, that ad H is selfadjoint. Thus a pair (U ad , 6) defines a unitary evolution to which we apply standard terminology and results of spectral theory of dynamical systems (see e.g. part III of ref. [1]). To any observable A E 6 we assign spectral measure dI-'A(fJ), where the spectral parameter lies on a unit circle () E 8 1 == [0, 21f), which is directly related to the time-autocorrelation function of A (t) (AtA(mT)) = (AI[U 1 A) = / dI-'A(fJ)e • So, ergodic properties of the unitary evolution are dual to its spectral properties: IThis is the simplest choice which makes sense also for time-dependent systems. For autonomous systems one may work with more general finite-temperature canonical inner product (see e.g. [6]). Ergodic Properties ofQuantum Spin Chains: Kicked Transverse Ising Model 319 • Quantum dynamics is ergodic in 6 iff 1 is (at most) simple (nondegenerate) eigenvalue of Uad (with eigenvector 1 if 1 E 6). 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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.

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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.

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Statistics of the spectral form factor in the self-dual kicked Ising model

Ana Flack, Bruno Bertini, Tomaž Prosen

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

•Jan 29, 2021•37 citations
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.
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.
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 problem of calculation of time-correlation functions and relaxation phenomena in interacting quantum many-body systems or quantized fields is put into the framework of ergodic theory and dynamical systems. In this spirit we discuss spectral analysis of the adjoint propagator in a suitable Hilbert space of quantum observables in Heisenberg picture as an alternative approach to characterize many-body dynamics. We illustrate our approach by working out the time-autocorrelation functions at infinite temperature of a large class of observabIes for a spin s = 1/2 Ising chain in a periodically kicking transverse magnetic field, and show that they decay to their asymptotic values as ex: t3/ 2 General understanding of time-correlation functions and relaxation phenomena of interacting quantum many-body systems is a challenging problem with only very partial existing solutions. Situation is much better if one considers the motion of a one-body or few-body systems where the classical ergodic theory (see e.g. [1]) describes a variety of different types of motion, ranging from quasi-periodic integrable motion characterized by a discrete spectrum of the (Liouville) evolution operator, through KAM quasi-integrability, to ergodicity, mixing and chaos characterized by a continuous evolution spectrum. Mixing, which is equivalent of saying that correlation functions of an arbitrary pair of observables (A,B) decay in time, limt-HXJ ((A(t)B) (A)(B)) = 0, is the necessary dynamical property needed to justify the relaxation to equilibrium (micro/grand/canonical) state and the laws of statistical mechanics, such as the fluctuation-dissipation theorem and transport laws. Mixing in quantum many-body system has been investigated by numerical experiment only quite recently [2]. In another numerical experiment [3], a suggestive possibility of dynamical phase transition from non-ergodic/non-mixing to ergodic/mixing dynamics in a non-integrable periodically kicked chain of interacting spinless fermions (kicked t-V model) has been proposed. General absence of mixing in completely integrable quantum many-body systems (such as those solvable by Bethe ansatz), being the consequence of an infinite sequence of conservation 317 J. Bonca et al. (eds.), Open Problems in Strongly Correlated Electron Systems. 317-322. © 2001 Kluwer Academic Publishers. 318 OPEN PROBLEMS IN STRONGLY CORRELATED ELECTRON SYSTEMS laws, is directly responsible for finite temperature anomalous transport properties of such systems [4, 5]. Quantum mixing property is defined (here) relative with respect to a certain (Hilbert) (sub)space 6 of observables and is most conveniently discussed in Heisenberg picture where the equations of motion for quantum observable A(t) are addressed. Our approach may be interpreted also as a kind ofgeneralization ofLee's recurrence-relations methodof constructing continued-fraction representation of time-correlation functions [6]. After a short outline of the main ideas we will illustrate the method by describing a recent calculation [8] of the time-correlation functions for Ising s = 1/2 spin chain periodically kicked with transverse magnetic field. Quantum dynamics in Heisenberg picture is governed by the Heisenberg equation (d/dt)A(t) = i( adH(t))A(t), where (adH)A:= [H, A1 =HA AH. Our aim is to include description of dynamics of periodically time-dependent (e.g. kicked) quantum systems so we will adopt Floquet formalism and integrate equations of motion over one period of time T (for autonomous systems, T may be arbitrary, e.g. infinitesimally small), defining the so called quantum Floquet map or one-step propagatorU := T exp(-i J; dtH(t) )which induces the adjoint Floquet map U ad acting on the (sub)space 6 of observables A(t +T) = U ad A(t), U A:= utAU. We should require the subspace 6 to be invariat under dynamics, namely A E 6 :::} U ad A E 6. Linear space 6 is turned into the Hilbert space by introducing an innner product of two extensive observables in thermodynamic limit (size L -+ 00) as (AlB) = (AtB), (A):= lim .!.. tr A £--+00 L tr 1 (1) (2) which has a meaning of a canonical average at infinite temperature. 1 We note that the adjoint Floquet map U ad is a unitary operator in 6 w.r.t. inner product (1), or equivalently, that ad H is selfadjoint. Thus a pair (U ad , 6) defines a unitary evolution to which we apply standard terminology and results of spectral theory of dynamical systems (see e.g. part III of ref. [1]). To any observable A E 6 we assign spectral measure dI-'A(fJ), where the spectral parameter lies on a unit circle () E 8 1 == [0, 21f), which is directly related to the time-autocorrelation function of A (t) (AtA(mT)) = (AI[U 1 A) = / dI-'A(fJ)e • So, ergodic properties of the unitary evolution are dual to its spectral properties: IThis is the simplest choice which makes sense also for time-dependent systems. For autonomous systems one may work with more general finite-temperature canonical inner product (see e.g. [6]). Ergodic Properties ofQuantum Spin Chains: Kicked Transverse Ising Model 319 • Quantum dynamics is ergodic in 6 iff 1 is (at most) simple (nondegenerate) eigenvalue of Uad (with eigenvector 1 if 1 E 6). Then limT-too ~ JOT dt {(A(t)IB) (All)(IIB)} = O. • Quantum dynamics is mixing in 6 iflimt-too(A(t)IB) = (All)(IIB) for arbitrary A, B E 6. Then U ad has an absolutely continuous spectrum, except possibly for a simple eigenvalue 1. • If 1 is a multiply degenerate eigenvalue of U ad then the corresponding eigenvectors Qn are the conservation laws of 6, U ad Qn = Qn Let Qn be a complete set of orthonormalized (QnIQm) = dnm eigenvectors with eigenvalue 1. The averaged time-autocorrelation function is the spectral weight at point 1 1 jT DA:= lim -T dt(A(t)IA) = L. I(QnIA )1 • T-too 2 -T n (3) For example, if A = J is the particle/spin current D J is proportional to the charge/spin stiffness at infinite temperature and D J > 0 indicates ballistic transport. Even when the set of known conservation laws is incomplete, the above relation gives rigorous lower bound on DA [7, 5]. Next we propose a simple but nontrivial application of the above picture. We consider aHilbert space 6 ofobservables over infinite spin 1/2 chains described by Pauli operators a}, s E {x, y, z}, j E Z, which is spanned by two infinite sequences of selfadjoint observables Un and Vn,nEZ, namely n ~ 1, n =0, n <-1 ,
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.
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.
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