Fate of current, residual energy, and entanglement entropy in aperiodic driving of one-dimensional Jordan-Wigner integrable models

Somnath Maity, Utso Bhattacharya, and Amit Dutta

Phys. Rev. B 98, 064305 – Published 20 August 2018

Abstract

We investigate the dynamics of two Jordan Wigner solvable models; namely, the one-dimensional chain of hard-core bosons and the one-dimensional transverse field Ising model under coin-toss-like aperiodically driven staggered on-site potential and the transverse field, respectively. It is demonstrated that both the models heat up to the infinite temperature ensemble for a minimal aperiodicity in driving. Consequently, in the case of the hard-core bosons chain, we show that the initial current generated by the application of a twist vanishes in the asymptotic limit for any driving frequency. For the transverse Ising chain, we establish that the system not only reaches the diagonal ensemble but the entanglement entropy also attains the thermal value in the asymptotic limit following initial ballistic growth. All these findings, contrasted with that of the perfectly periodic situation, are analytically established in the asymptotic limit within an exact disorder matrix formalism developed using the uncorrelated binary nature of the coin-toss aperiodicity.

Received 16 May 2018
Revised 6 August 2018

DOI:https://doi.org/10.1103/PhysRevB.98.064305

Physics Subject Headings (PhySH)

Quantum phase transitions Quantum thermodynamics

Floquet systems

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Somnath Maity, Utso Bhattacharya, and Amit Dutta

Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India

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Issue

Vol. 98, Iss. 6 — 1 August 2018

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Images

Figure 1
Brillouin zone of the HCB chain with zero staggered potential, showing the ground state (blue dashed line) and the excited state (red solid line) for the twist $ν = 0$ (top) and $ν \neq 0$ (bottom).
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Figure 2
(a) The residual energy (RE) and (b) current ( $J (N)$ ) plotted as a function of stroboscopic intervals ( $N$ ) for HCB chain with periodically kicked ( $p = 1$ ) staggered potential with several values of frequency ( $ω$ ). Both the RE as well as the current reach frequency-dependent steady-state values. The mean value of the current decreases monotonically with increasing frequency and vanishes in the asymptotic limit of $N$ for very large value of $ω$ resulting in the dynamical localization. In (c) and (d), we plotted the disorder averaged RE and current, respectively, with aperiodically kicked (with bias $p = 0.5$ ) staggered potential. We observed that for all frequencies, RE goes to a universal value (corresponding to the infinite temperature) while the current vanishes in the asymptotic limit of $N$ . Here, the amplitude of kicking $α = π / 16$ , twist parameter $ν = 0.2$ , and the system size $L = 1000$ and we observe similar behavior for all values of $p \neq 0, 1$ . In the case of aperiodic drive, the current vanishes very fast as a function of $N$ compared to the dynamical localization situation in the periodic high-frequency $δ$ -kicking.
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Figure 3
(a) The residual energy (RE) and (b) the current ( $J (N)$ ) plotted as a function of stroboscopic intervals ( $N$ ) for HCB chain with perfectly periodic ( $p = 1$ ) sinusoidal driving of the staggered potential. Unlike the delta kicking, both the RE and the current stick to their initial value for very high frequency of driving resulting implying the absence of dynamical localization. In (c) and (d), we plotted the disorder averaged RE and current, respectively, for aperiodically sinusoidal driving (with bias $p = 0.5$ ) of staggered potential. We also observed the saturation of RE to infinite temperature value and vanishing of the current in this case for all values of $ω$ similar to the aperiodic delta kicked situation. Here, we have chosen amplitude of driving $α = 5$ , twist parameter $ν = 0.2$ , and the system size $L = 1000$ .
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Figure 4
(a) The entanglement entropy for the TFIM as function of stroboscopic intervals with different values of subsystem size $l$ for perfectly periodic drive ( $p = 1$ ). (b) Configuration averaged EE plotted as function of stroboscopic intervals for the aperiodic drive ( $p = 0.5$ ). We have chosen sinusoidal driving with frequency $ω = 10$ , amplitude $α = 5$ , total system size $L = 100$ in both cases and, for the aperiodic case, the EE is averaged over 1000 configurations. Note that we use logarithmic scale in both the axes to show the initial linear growth as well as the saturation in large time limit. In both cases, a linear growth of EE is present up to a crossover time $t^{*} = l / 2$ (as $v_{max} = 1$ in this case) and the quasirevivals occur at $t_{r} = L$ , which is independent of $l$ . See the gridlines along $y$ axis to note the saturation value.
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