Operator growth and eigenstate entanglement in an interacting integrable Floquet system

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Operator growth and eigenstate entanglement in an interacting integrable Floquet system

Sarang Gopalakrishnan

Phys. Rev. B 98, 060302(R) – Published 14 August 2018

Abstract

We analyze a simple model of quantum dynamics, which is a discrete-time deterministic version of the Fredrickson-Andersen model. This model is integrable, with a quasiparticle description related to the classical hard-rod gas. Despite the integrability of the model, commutators of physical operators grow with a diffusively broadening front, in this respect resembling generic chaotic models. In addition, local operators behave consistently with the eigenstate thermalization hypothesis (ETH). However, large subsystems violate ETH; as a function of subsystem size, eigenstate entanglement first increases linearly and then saturates at a scale that is parametrically smaller than half the system size.

Received 21 June 2018
Revised 4 August 2018

DOI:https://doi.org/10.1103/PhysRevB.98.060302

Physics Subject Headings (PhySH)

Eigenstate thermalization Entanglement entropy Quantum chaotic transport

Floquet systems

Integrable systems

Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Sarang Gopalakrishnan

Department of Physics and Astronomy, CUNY College of Staten Island, Staten Island, New York 10314, USA and Physics Program and Initiative for the Theoretical Sciences, The Graduate Center, CUNY, New York, New York 10016, USA

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Issue

Vol. 98, Iss. 6 — 1 August 2018

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Images

Figure 1
Left: Temporal growth of state-averaged out-of-time-order commutator for system size $L = 600$ , averaged over 10 000 initial states. Dashed red lines show the causal light-cone velocity, outside which the commutator strictly vanishes. Right: Second Renyi entropy $S_{2}$ vs subsystem size, averaged over 30 random eigenstates, for various system sizes $L$ . The dashed black line has a slope of $ln 2$ , the thermal prediction. There is a clear crossover scale beyond which subsystem entanglement saturates.
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Figure 2
Plots of spin dynamics; black cells are spin up and white cells spin down. (a) Collision of a right-moving and left-moving quasiparticle. (b) An initial state with four adjacent up spins is a three-quasiparticle state. (c) Time evolution of a product state in which the left and right third are generic, whereas the middle third has only one quasiparticle. (d) OTOC for this product state, obtained by moving one of the quasiparticles; note the absence of chaos.
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Figure 3
Upper panel: Value of the OTOC at a fixed time vs position (left), and at a fixed position vs time (right), for $L = 600$ systems. The operator front is Gaussian. Middle right: Squared width (variance) of operator front vs time, indicating diffusive broadening. Lower left: Density plot of the OTOC vs time, for a smaller system ( $L = 200$ ), showing the unexpected overshooting effect at times longer than system size; cross sections at $t = 50$ , 100, and 150 are shown in the lower right panel.
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Figure 4
Upper left: An example of the dynamics of an initial product state, illustrating how configurations recur after a period of order $L$ up to an overall translation (see arrows; in this case the period is 45 and the shift is four sites to the left). Upper right: Typical $\sqrt{T_{r}}$ vs system size, as predicted by the quasiparticle picture. Left: State-to-state fluctuations of Renyi entropy $S_{2}$ vs subsystem size for $L = 80$ ; each line represents a randomly chosen eigenstate. The dashed black line has a slope of $ln 2$ . Right: Histogram of the expectation value of the spin on the first site $〈 Z_{1} 〉$ across eigenstates, for various system sizes. The histograms narrow with increasing system size.
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Figure 5
Left: The fraction of nonzero off-diagonal matrix elements falls off exponentially with system size; the slope of the fit line is $- ln 2 / 2$ . Right: The distributions of the remaining nonzero matrix elements broaden (histograms are rescaled to have the same weight).
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