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arXiv:2209.12510 · Quantum

Dynamical many-body delocalization transition of a Tonks gas in a quasi-periodic driving potential

Vincent Vuatelet, Adam Rançon

Published Feb 9, 2023• 6 citations
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Abstract

The quantum kicked rotor is well-known for displaying dynamical (Anderson) localization. It has recently been shown that a periodically kicked Tonks gas will always localize and converge to a finite energy steady-state. This steady-state has been described as being effectively thermal with an effective temperature that depends on the parameters of the kick. Here we study a generalization to a quasi-periodic driving with three frequencies which, without interactions, has a metal-insulator Anderson transition. We show that a quasi-periodically kicked Tonks gas goes through a dynamical many-body delocalization transition when the kick strength is increased. The localized phase is still described by a low effective temperature, while the delocalized phase corresponds to an infinite-temperature phase, with the temperature increasing linearly in time. At the critical point, the momentum distribution of the Tonks gas displays different scaling at small and large momenta (contrary to the non-interacting case), signaling a breakdown of the one-parameter scaling theory of localization.

DOI
10.22331/q-2023-02-09-917

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•Sep 6, 2022•0 citations
The quantum kicked rotor is well-known for displaying dynamical (Anderson) localization. It has recently been shown that a periodically kicked Tonks gas will always localize and converge to a finite energy steady-state. This steady-state has been described as being effectively thermal with an effective temperature that depends on the parameters of the kick. Here we study a generalization to a quasi-periodic driving with three frequencies which, without interactions, has a metal-insulator Anderson transition. We show that a quasi-periodically kicked Tonks gas goes through a dynamical many-body delocalization transition when the kick strength is increased. The localized phase is still described by a low effective temperature, while the delocalized phase corresponds to an infinite-temperature phase, with the temperature increasing linearly in time. At the critical point, the momentum distribution of the Tonks gas displays different scaling at small and large momenta (contrary to the non-interacting case), signaling a breakdown of the one-parameter scaling theory of localization.
It is widely recognized that finite temperatures degrade quantum coherence and can induce thermalization. Here, we study the effect of finite temperature on a kicked Tonks--Girardeau gas, which is known to exhibit many--body dynamical localization and delocalization under periodic and quasiperiodic kicks, respectively. We find that many--body dynamical localization persists at finite--and even high--temperatures, although the coherence of the localized state is further degraded. In particular, we demonstrate a modified effective thermalization of the localized state by considering the initial temperature. Moreover, we show many--body dynamical localization transition at intermediate temperature. Our work extends the study of many--body dynamical localization and delocalization to the finite--temperature regime, providing guidance for cold-atom experiments, particularly in the strongly-interacting regime.
Dynamical localization is the analog of Anderson localization in momentum space, where the system's energy saturates and the single-particle wave functions are exponentially localized in momentum space. In the presence of interactions, in the context of a periodically kicked Bose gas, it has been argued that dynamical localization persists. Focusing on the Tonks (strongly interacting) regime, we show that the many-body dynamically localized phase is effectively thermal, a clear deviation from the breaking of ergodicity observed in standard many-body localized systems. We relate the effective temperature to the driving parameters, and thus quantitatively describe the loss of coherence at large distances in this phase. Contrary to the noninteracting case, the momentum distribution decays as a power-law at large momenta, characterized by an effectively thermal Tan's contact. This is a rare example where driving and many-body (dynamical) localization lead to an effectively ergodic state.
Abstract We consider a finite-size periodically driven quantum system of coupled kicked rotors which exhibits two distinct regimes in parameter space: a dynamically-localized one with kinetic-energy saturation in time and a chaotic one with unbounded energy absorption (dynamical delocalization). We provide numerical evidence that the kinetic energy grows subdiffusively in time in a parameter region close to the boundary of the chaotic dynamically-delocalized regime. We map the different regimes of the model via a spectral analysis of the Floquet operator and investigate the properties of the Floquet states in the subdiffusive regime. We observe an anomalous scaling of the average inverse participation ratio (IPR) analogous to the one observed at the critical point of the Anderson transition in a disordered system. We interpret the behavior of the IPR and the behavior of the asymptotic-time energy as a mark of the breaking of the eigenstate thermalization in the subdiffusive regime. Then we study the distribution of the kinetic-energy-operator off-diagonal matrix elements. We find that in presence of energy subdiffusion they are not Gaussian and we propose an anomalous random matrix model to describe them.
We investigate the Lieb-Liniger model of one-dimensional bosons subjected to periodic kicks. In both the noninteracting and strongly interacting limits, the system undergoes dynamical localization, leading to energy saturation at long times. However, for finite interactions, we reveal an interaction-driven transition from an insulating to a metallic phase at a critical kicking strength, provided the number of particles is three or more. Using the Bethe ansatz solution of the Lieb-Liniger gas, we establish a formal correspondence between its dynamical evolution and an Anderson model in N spatial dimensions, where N is the number of particles. This theoretical prediction is supported by extensive numerical simulations for three particles, complemented by finite-time scaling analysis, demonstrating that this transition belongs to the orthogonal Anderson universality class.
The kicked rotor system is a textbook example of how classical and quantum dynamics can drastically differ. The energy of a classical particle confined to a ring and kicked periodically will increase linearly in time whereas in the quantum version the energy saturates after a finite number of kicks. The quantum system undergoes Anderson localization in angular-momentum space. Conventional wisdom says that in a many-particle system with short-range interactions the localization will be destroyed due to the coupling of widely separated momentum states. Here we provide evidence that for an interacting one-dimensional Bose gas, the Lieb-Liniger model, the dynamical localization can persist at least for an unexpectedly long time.
Le Kicked Rotor, paradigme du chaos quantique, est l'analogue dans l'espace des impulsions d'un système d'Anderson désordonné, de part le fait qu'on y retrouve le phénomène de localisation dites dynamique. Nous étudions l'effet des interactions sur la localisation dynamique dans un gaz de Bose en interaction forte, appelé gaz de Tonks-Girardeau. En exploitant cette limite et la correspondance de Bose-Fermi, nous étudions la physique d'un gaz de Tonks-Girardeau soumis à des kicks au travers de la matrice réduite à un corps du système. Nous trouvons une similitude du système avec un gaz de Bose à température finie, nous caractérisons les observables du système, notamment son énergie, sa distribution en impulsion, ainsi que la cohérence spatiale. Nous relions la température effective émergente aux caractéristiques de localisation du système, alors dans une phase localisée dynamiquement à N-corps. Nous présentons également une caractérisations de la version quasi-périodique du Kicked Rotor, analogue d'un système d'Anderson tridimensionnel, marqué par une transition de phase entre un régime localisé et diffusif. Nous analysons enfin la distribution en impulsion au seuil critique, et caractérisons le pic central, et le comportement à petites et grandes impulsions, dotés de trois comportements différents.