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arXiv:1807.09767 · Physical review. B./Physical review. B

Strong-disorder renormalization group for periodically driven systems

William Berdanier, Michael Kolodrubetz, S. A. Parameswaran, Romain Vasseur

Published Nov 9, 2018• 17 citations
arXivHTMLPDF

Abstract

Quenched randomness can lead to robust nonequilibrium phases of matter in periodically driven (Floquet) systems. Analyzing transitions between such dynamical phases requires a method capable of treating the twin complexities of disorder and discrete time-translation symmetry. We introduce a real-space renormalization group approach, asymptotically exact in the strong-disorder limit, and exemplify its use on the periodically driven interacting quantum Ising model. We analyze the universal physics near the critical lines and multicritical point of this model, and demonstrate the robustness of our results to the inclusion of weak interactions.

DOI
10.1103/physrevb.98.174203

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Quenched randomness can lead to robust nonequilibrium phases of matter in periodically driven (Floquet) systems. Analyzing transitions between such dynamical phases requires a method capable of treating the twin complexities of disorder and discrete time-translation symmetry. We introduce a real-space renormalization group approach, asymptotically exact in the strong-disorder limit, and exemplify its use on the periodically driven interacting quantum Ising model. We analyze the universal physics near the critical lines and multicritical point of this model, and demonstrate the robustness of our results to the inclusion of weak interactions.
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Abstract When random quantum spin chains are submitted to some periodic Floquet driving, the eigenstates of the time-evolution operator over one period can be localized in real space. For the case of periodic quenches between two Hamiltonians (or periodic kicks), where the time-evolution operator over one period reduces to the product of two simple transfer matrices, we propose a block-self-dual renormalization procedure to construct the localized eigenstates of the Floquet dynamics. We also discuss the corresponding strong disorder renormalization procedure, that generalizes the RSRG-X procedure to construct the localized eigenstates of time-independent Hamiltonians.
We develop a comprehensive renormalization group (RG) approach to criticality in open Floquet systems, where dissipation enables the system to reach a well-defined Floquet steady state of finite entropy, and all observables are synchronized with the drive. We provide a detailed description of how to combine Keldysh and Floquet formalisms to account for the critical fluctuations in the weakly and rapidly driven regime. A key insight is that a reduction to the time-averaged static sector is not possible close to the critical point. This guides the design of a perturbative dynamic RG approach, which treats the time-dependent, dynamic sector associated to higher harmonics of the drive, on an equal footing with the time-averaged sector. Within this framework, we develop a weak drive expansion scheme, which enables to systematically truncate the RG flow equations in powers of the inverse drive frequency ${\mathrm{\ensuremath{\Omega}}}^{\ensuremath{-}1}$. This allows us to show how a periodic drive inhibits scale invariance and critical fluctuations of second-order phase transitions in rapidly driven open systems. Although criticality emerges in the limit ${\mathrm{\ensuremath{\Omega}}}^{\ensuremath{-}1}=0$, any finite drive frequency produces a scale that remains finite all through the phase transition. This is a universal mechanism that relies on the competition of the critical fluctuations within the static and dynamic sectors of the problem.