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arXiv:2006.10054 · Physical Review Research

Fine structure of heating in a quasiperiodically driven critical quantum system

Bastien Lapierre, Kenny Choo, Apoorv Tiwari, Clément Tauber, Titus Neupert, R. Chitra

Published Sep 22, 2020• 41 citations
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Abstract

The authors study a one-dimensional critical quantum system subject to a Fibonacci quasiperiodic drive that alternates between two spatially modulated coupling profiles and find regions of fast heating as well as ultra-slow heating regions where the system remains non-heated within an experimentally accessible time.
DOI
10.1103/physrevresearch.2.033461

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We study the heating dynamics of a generic one-dimensional critical system when driven quasiperiodically. Specifically, we consider a Fibonacci drive sequence comprising the Hamiltonian of uniform conformal field theory (CFT) describing such critical systems and its sine-square deformed counterpart. The asymptotic dynamics is dictated by the Lyapunov exponent which has a fractal structure embedding Cantor lines where the exponent is exactly zero. Away from these Cantor lines, the system typically heats up fast to infinite energy in a nonergodic manner where the quasiparticle excitations congregate at a small number of select spatial locations resulting in a buildup of energy at these points. Periodic dynamics with no heating for physically relevant timescales is seen in the high-frequency regime. As we traverse the fractal region and approach the Cantor lines, the heating slows enormously and the quasiparticles completely delocalize at stroboscopic times. Our setup allows us to tune between fast and ultraslow heating regimes in integrable systems.
We analyze the anisotropic Dicke model in the presence of a periodic drive and under a quasiperiodic drive. The study of drive-induced phenomena in this experimentally accessible model is important since, although it is simpler than full-fledged many-body quantum systems, it is still rich enough to exhibit many interesting features. We show that under a quasiperiodic Fibonacci (Thue-Morse) drive, the system features a prethermal plateau that increases as an exponential (stretched exponential) with the driving frequency before heating to an infinite-temperature state. In contrast, when the model is periodically driven, the dynamics reaches a plateau that is not followed by heating. In either case, the plateau value depends on the energy of the initial state and on the parameters of the undriven Hamiltonian. Surprisingly, this value does not always approach the infinite-temperature state monotonically as the frequency of the periodic drive decreases. We also show how the drive modifies the quantum critical point and discuss open questions associated with the analysis of level statistics at intermediate frequencies.
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In this paper, we analytically study phase transitions in quasiperiodically driven one-dimensional quantum critical systems that are described by conformal field theories (CFTs). The phase diagrams and phase transitions can be analytically obtained by using Avila's global theory in one-frequency quasiperiodic cocycles. Compared to previous works where the quasiperiodicity was introduced in the driving time and no phase transitions were observed [Wen et al., Phys. Rev. Res. 3, 023044 (2021)], here we propose a setup where the quasiperiodicity is introduced in the driving Hamiltonians. In our setup, one can observe the heating phases, nonheating phases, and phase transitions. The phase diagram as well as the Lyapunov exponents that determine the entanglement entropy evolution can be analytically obtained. In addition, based on Avila's theory, we prove there is no phase transition in the previously proposed setup of quasiperiodically driven CFTs [Wen et al., Phys. Rev. Res. 3, 023044 (2021)]. We verify our field theory results by studying the time evolution of entanglement entropy on lattice models.
In this paper and its upcoming sequel, we study nonequilibrium dynamics in driven (1 + 1)-dimensional conformal field theories (CFTs) with periodic, quasiperiodic, and random driving.We study a soluble family of drives in which the Hamiltonian only involves the energy-momentum density spatially modulated at a single wavelength.The resulting time evolution is then captured by a Möbius coordinate transformation.In this paper, we establish the general framework and focus on the first two classes.In periodically driven CFTs, we generalize earlier work and study the generic features of entanglement and energy evolution in different phases, i.e., the heating and nonheating phases and the phase transition between them.In quasiperiodically driven CFTs, we mainly focus on the case of driving with a Fibonacci sequence.We find that (i) the nonheating phases form a Cantor set of measure zero; (ii) in the heating phase, the Lyapunov exponents (which characterize the growth rate of the entanglement entropy and energy) exhibit self-similarity, and can be arbitrarily small; (iii) the heating phase exhibits periodicity in the location of spatial structures at the Fibonacci times; (iv) one can find exactly the nonheating fixed point, where the entanglement entropy and energy oscillate at the Fibonacci numbers, but grow logarithmically and polynomially at the non-Fibonacci numbers; (v) for certain choices of driving Hamiltonians, the nonheating phases of the Fibonacci driving CFT can be mapped to the energy spectrum of electrons propagating in a Fibonacci quasicrystal.In addition, another quasiperiodically driven CFT with an Aubry-André-type sequence is also studied.We compare the CFT results to lattice calculations and find remarkable agreement.