Topological classification of quasiperiodically driven quantum systems

Crowley, Philip J. D.; Martin, Ivar; Chandran, Anushya

doi:10.1103/physrevb.99.064306

Cited by 76 publications

(73 citation statements)

References 70 publications

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“…The topological invariants discussed in Ref. [61] can also be understood in a similar fashion. One might think that the nontrivial invariant of this kind could also be realized in the frame-twisted high-frequency limit through a bootstrap construction as in Sec.…”

Section: Relation To Previous Workmentioning

confidence: 89%

“…Before we proceed, let us note that the study of the dynamics of quantum systems under quasiperiodic driving has a venerable history, encompassing diverse applications from experiments in chemistry and physics, to the basic structure of first-order differential equations [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58]. Interesting dynamical behavior related to topology in fewbody or noninteracting scenarios have also been reported [59][60][61][62][63]. Many-body quasiperiodically driven quantum systems with interactions have received increasing attention recently [64][65][66].…”

Section: Introductionmentioning

confidence: 99%

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Long-Lived Interacting Phases of Matter Protected by Multiple Time-Translation Symmetries in Quasiperiodically Driven Systems

Else

Dumitrescu

2020

Phys. Rev. X

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We show how a large family of interacting nonequilibrium phases of matter can arise from the presence of multiple time-translation symmetries, which occur by quasiperiodically driving an isolated, quantum many-body system with two or more incommensurate frequencies. These phases are fundamentally different from those realizable in time-independent or periodically driven (Floquet) settings. Focusing on high-frequency drives with smooth time dependence, we rigorously establish general conditions for which these phases are stable in a parametrically long-lived "preheating" regime. We develop a formalism to analyze the effect of the multiple time-translation symmetries on the dynamics of the system, which we use to classify and construct explicit examples of the emergent phases. In particular, we discuss time quasicrystals which spontaneously break the time-translation symmetries, as well as time-translation symmetry-protected topological phases.

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Section: Relation To Previous Workmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Long-Lived Interacting Phases of Matter Protected by Multiple Time-Translation Symmetries in Quasiperiodically Driven Systems

Else

Dumitrescu

2020

Phys. Rev. X

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“…We are interested in QPD systems which can be defined via a Fourier expansion of the Hamiltonian [34][35][36]…”

mentioning

confidence: 99%

“…their mutual ratios are irrational. There are some approaches to analyse QPD systems by mappings to higher dimensions [35,37], constructing effective Hamiltonians to describe the system perturbatively [36], for analysing their asymptotic behaviour [38][39][40] or using them for enhanced performance of quantum simulations [41,42] and creation of Majorana edge modes [43]. While in general for QPD systems we are lacking efficient methods for calculating their full time evolution, the FTSs introduced here are a special class corresponding to periodic systems in a rotating frame.…”

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confidence: 99%

Floquet time spirals and stable discrete-time quasicrystals in quasiperiodically driven quantum many-body systems

2019

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We analyse quasi-periodically driven quantum systems that can be mapped exactly to periodically driven ones and find Floquet Time Spirals in analogy with spatially incommensurate spiral magnetic states. Generalising the mechanism to many-body systems we discover that a form of discrete time-translation symmetry breaking can also occur in quasi-periodically driven systems. We construct a discrete time quasi-crystal stabilised by many-body localisation. Crucially, it persists also under perturbations that break the equivalence with periodic systems. As such it provides evidence of a stable quasi-periodically driven many-body quantum system which does not heat up to the featureless infinite temperature state.The dynamics of a quantum system governed by a generic time-dependent Hamiltonian is generally hard to analyse because to reach a certain point in time the entire history of evolution is required. Hence, simulations quickly become computationally unfeasible, especially for many-body systems. There are some exceptions such as periodically driven (PD) systems, i.e. whose Hamiltonian is periodic in time H(t + T ) = H(t), for which Floquet theory significantly reduces the effort to simulate the dynamics by decomposing the wave function to Floquet modes periodic over one time period T up to phases. PD systems are of immense recent interest as they have been exploited extensively for engineering sought after Hamiltonians [1-5], controlling quantum systems [6][7][8][9][10][11][12][13][14]and realising non-equilibrium quantum phases [15][16][17]. For instance, the proper combination of periodic driving and many-body localisaton [18] gave birth to the discovery of discrete time crystals (DTCs) [19][20][21][22][23][24][25][26], a new quantum phase with broken discrete time translation symmetry.Here, we address the intriguing question whether quasi-periodically driven (QPD) quantum systems can show features similar to time-crystalline behaviour of periodically driven ones? Normally it is expected -apart for special integrable systems with momentum conservation [27] -that QPD systems generically heat up to featureless infinite-temperature states [28,29]. Hence, a related important question we address is whether there are interacting QPD quantum systems with a stable nontrivial steady state? Of course, generally such questions are hard to answer because Floquet theory, which provides an elegant framework for the analysis of PD quantum systems normally does not generalise to aperiodic time-dependencies. Here, we address the above questions by studying a class of QPD systems which can be efficiently treated within Floquet theory. This allows us to find QPD steady states with time-crystalline behaviour at special fine-tuned points whose stability to generic perturbations is studied with exact diagonalisation.The main idea is that special cases of originally QPD Hamiltonian H can be mapped exactly to a periodic(a) (b) Δθ 2Δθ FIG. 1. (a) Mapping a quasi-periodic magnetic spiral to a periodic ferromagnet. (b)Illustration of the tw...

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“…The case of incommensurate frequencies (see, for example, Ref. 's [13] and [14]) is beyond the scope of this work.…”

Section: Introductionmentioning

confidence: 99%

Validity of many-mode Floquet theory with commensurate frequencies

Poertner

Martin

2020

Phys. Rev. A

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Many-mode Floquet theory [T.-S. Ho, S.-I. Chu, and J. V. Tietz, Chem. Phys. Lett. 96, 464 (1983)] is a technique for solving the time-dependent Schrödinger equation in the special case of multiple periodic fields, but its limitations are not well understood. We show that for a Hamiltonian consisting of two time-periodic couplings of commensurate frequencies (integer multiples of a common frequency), many-mode Floquet theory provides a correct expression for unitary time evolution. However, caution must be taken in the interpretation of the eigenvalues and eigenvectors of the corresponding many-mode Floquet Hamiltonian, as only part of its spectrum is directly relevant to time evolution. We give a physical interpretation for the remainder of the spectrum of the Hamiltonian. These results are relevant to the engineering of quantum systems using multiple controllable periodic fields.

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Topological classification of quasiperiodically driven quantum systems

Cited by 76 publications

References 70 publications

Long-Lived Interacting Phases of Matter Protected by Multiple Time-Translation Symmetries in Quasiperiodically Driven Systems

Long-Lived Interacting Phases of Matter Protected by Multiple Time-Translation Symmetries in Quasiperiodically Driven Systems

Floquet time spirals and stable discrete-time quasicrystals in quasiperiodically driven quantum many-body systems

Validity of many-mode Floquet theory with commensurate frequencies

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