Dynamical quantum phase transitions in systems with broken continuous time and space translation symmetries

Kosior, Arkadiusz; Syrwid, Andrzej; Sacha, Krzysztof

doi:10.1103/physreva.98.023612

Cited by 39 publications

(21 citation statements)

References 96 publications

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“…A crystalline structure in time is related to the periodic dynamics of a system. It has been shown that periodically driven quantum many-body systems can spontaneously self-reorganize their motion and start moving with a period different from the period of the driving [12][13][14] (see also [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]). This kind of spontaneous formation of a new crystalline structure in time is dubbed discrete or Floquet time crystals and has been already realized experimentally [30][31][32][33][34].…”

Section: Introductionmentioning

confidence: 99%

Topological time crystals

et al. 2019

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By analogy with the formation of space crystals, crystalline structures can also appear in the time domain. While in the case of space crystals we often ask about periodic arrangements of atoms in space at a moment of a detection, in time crystals the role of space and time is exchanged. That is, we fix a space point and ask if the probability density for detection of a system at this point behaves periodically in time. Here, we show that in periodically driven systems it is possible to realize topological insulators, which can be observed in time. The bulk-edge correspondence is related to the edge in time, where edge states localize. We focus on two examples: Su-Schrieffer-Heeger model in time and Bose Haldane insulator which emerges in the dynamics of a periodically driven many-body system. 5 Note that due to the negative effective mass m eff , the first energy band is the highest in energy. 2 New J. Phys. 21 (2019) 052003 where = ¢ J J i 2and J 2i−1 =J, which is identical to the SSH model [86]. The latter describes spinless fermions hopping on a 1D-lattice with staggered hopping amplitudes. Changing the ratio λ 1 /λ in (1), allows one to control the ratio ¢ J J . This effective Hamiltonian belongs to the BDI class of the periodic table of the topological insulators and superconductors [87] and is characterized by a  topological invariant, the winding number ν. For an infinite system with ¢ > J J ( ¢ < J J ), the system is in a topological (trivial) phase with winding number ν=1 (ν=0). For a finite system, the topological phase exhibits zero energy edge states protected by the topology of the bulk. The SSH model has been experimentally realized in quantum simulators and both the presence of edge states and the winding number have been measured [63][64][65]. Let us emphasize that the Wannier states w i (x, t) of equation (2) are localized wavepackets of the effective SSH Hamiltonian of equation (3). We then discuss how such states allow one to detect the topology of the SSH Hamiltonian. corresponding to quasi-energies closest to zero, see top panel. In the topological phase, these eigenstates have zero quasi-energy and are linear combinations of the edge states localized at the edge of time. That is, in the laboratory frame, a detector is placed close to the oscillating mirror (x≈0) and the probability density of clicking of the detector is shown versus time for different values of the ratio ¢ J J of the tunneling amplitudes in (3). This behavior is repeated with the period 2π/ω. At sωt/(2π)=1 and sωt/(2π)=40 there are edges where the eigenstate localizes if ¢ > J J 1. The results correspond to ω=0.067, λ=0.06 in (1) and are obtained within the quantum secular approach [84], see the appendix.

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Section: Introductionmentioning

confidence: 99%

Topological time crystals

et al. 2019

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“…In particular, because the dynamics of systems under time-periodic modulations has led to fascinating discoveries like Floquet topological states [20][21][22][23][24] and discrete time crystals [25][26][27], it is urgent to investigate how DQPTs may occur in such Floquet systems. Along this avenue, there have been scattered studies, but still with the notion that DQPTs are best aroused by a quench to some system parameters [28,29]. Here we introduce a novel class of DQPTs, termed Floquet DQPTs, which can be regarded as intrinsic features of systems with time-periodic modulations.…”

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confidence: 99%

Floquet dynamical quantum phase transitions

et al. 2019

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Dynamical quantum phase transitions (DQPTs) are manifested by time-domain nonanalytic behaviors of many-body systems. Introducing a quench is so far understood as a typical scenario to induce DQPTs. In this work, we discover a novel type of DQPTs, termed "Floquet DQPTs", as intrinsic features of systems with periodic time modulation. Floquet DQPTs occur within each period of continuous driving, without the need for any quenches. In particular, in a harmonically driven spin chain model, we find analytically the existence of Floquet DQPTs in and only in a parameter regime hosting a certain nontrivial Floquet topological phase. The Floquet DQPTs are further characterized by a dynamical topological invariant defined as the winding number of the Pancharatnam geometric phase versus quasimomentum. These findings are experimentally demonstrated with a single spin in diamond. This work thus opens a door for future studies of DQPTs in connection with topological matter.DQPTs are often associated with quantum quenches, a protocol in which parameters of a Hamiltonian are suddenly changed [1,2]. A quantum quench across an equilibrium quantum critical point may induce a DQPT. If pre-quench and post-quench systems are in topologically distinct phases, DQPTs may also be characterized by dynamical topological invariants [3][4][5]. As a promising approach to classify quantum states of matter in nonequilibrium situations, DQPTs have been theoretically explored in both closed and open quantum systems at different physical dimensions [2,7,8]. Experimentally, DQPTs have been observed in trapped ions [3,10], cold atoms [11,12], superconducting qubits [13], nanomechanical oscillators [14], and photonic quantum walks [15,16].To date, in most studies of DQPTs, a quantum quench acts as a trigger for initiating nonequilibrium dynamics and then exposing the underlying topological features. However, DQPTs under more general nonequilibrium manipulations are still largely unexplored [17][18][19]. In particular, because the dynamics of systems under time-periodic modulations has led to fascinating discoveries like Floquet topological states [20-24] and discrete time crystals [25][26][27], it is urgent to investigate how DQPTs may occur in such Floquet systems. Along this avenue, there have been scattered studies, but still with the notion that DQPTs are best aroused by a quench to some system parameters [28,29]. Here we introduce a novel class of DQPTs, termed Floquet DQPTs, which can be regarded as intrinsic features of systems with time-periodic modulations. As schematically shown in Floquet DQPT Time Time Rate function Rate function H i H f (a) H(t) (b) Conventional DQPT T T Quench T FIG. 1. Comparison between (a) DQPTs following a quantum quench, and (b) Floquet DQPTs without any quenches.Here Hi and H f denote the Hamiltonians before and after the quench, H(t) denotes a periodically and continuously modulated Hamiltonian. Fig. 1, the Floquet DQPTs we discovered occur within each period of a continuous driving field, without the need for any...

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“…3, where the doubly critical phase disappears at large α. On the other hand, the purely long-range hopping case β → ∞ always contains a doubly critical phase for α 3.6223, as follows from (38) and (39), but it does not display the triply critical phase for any α value.…”

Section: The α > 3 Regimementioning

confidence: 97%

“…Additionally, this dynamical critical point was shown to coincide with that of the type of dynamical phase transition based on a local order parameter [18,[20][21][22][23][24][25]. Also recently, the theory of DQPT has been extended to Floquet systems [38][39][40] and models with many-body localized phases [41,42]. DQPTs in nonintegrable models have been studied in, for instance, Ref.…”

Section: Introductionmentioning

confidence: 99%

Out-of-equilibrium phase diagram of long-range superconductors

et al. 2020

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Within the ultimate goal of classifying universality in quantum many-body dynamics, understanding the relation between out-of-equilibrium and equilibrium criticality is a crucial objective. Models with power-law interactions exhibit rich well-understood critical behavior in equilibrium, but the out-of-equilibrium picture has remained incomplete, despite recent experimental progress. We construct the rich dynamical phase diagram of free-fermionic chains with power-law hopping and pairing and provide analytic and numerical evidence showing a direct connection between nonanalyticities of the return rate and zero crossings of the string order parameter. Our results may explain the experimental observation of so-called accidental dynamical vortices, which appear for quenches within the same topological phase of the Haldane model, as reported by Fläschner et al. [Nat. Phys. 14, 265 (2018)]. Our work is readily applicable to modern ultracold-atom experiments, not least because state-of-the-art quantum gas microscopes can now reliably measure the string order parameter, which, as we show, can serve as an indicator of dynamical criticality.

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Dynamical quantum phase transitions in systems with broken continuous time and space translation symmetries

Cited by 39 publications

References 96 publications

Topological time crystals

Topological time crystals

Floquet dynamical quantum phase transitions

Out-of-equilibrium phase diagram of long-range superconductors

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