Prethermal Fragmentation in a Periodically Driven Fermionic Chain

Somsubhra Ghosh, Indranil Paul, and K. Sengupta

Phys. Rev. Lett. 130, 120401 – Published 23 March 2023

Abstract

We study a fermionic chain with nearest-neighbor hopping and density-density interactions, where the nearest-neighbor interaction term is driven periodically. We show that such a driven chain exhibits prethermal strong Hilbert space fragmentation (HSF) in the high drive amplitude regime at specific drive frequencies $ω_{m}^{*}$ . This constitutes the first realization of HSF for out-of-equilibrium systems. We obtain analytic expressions of $ω_{m}^{*}$ using a Floquet perturbation theory and provide exact numerical computation of entanglement entropy, equal-time correlation functions, and the density autocorrelation of fermions for finite chains. All of these quantities indicate clear signatures of strong HSF. We study the fate of the HSF as one tunes away from $ω_{m}^{*}$ and discuss the extent of the prethermal regime as a function of the drive amplitude.

Received 14 December 2022
Revised 9 February 2023
Accepted 3 March 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.120401

Physics Subject Headings (PhySH)

Eigenstate thermalization Spinless fermions

Floquet systems

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Somsubhra Ghosh ¹, Indranil Paul ², and K. Sengupta ¹

¹School of Physical Sciences, Indian Association for the Cultivation of Science, Kolkata 700032, India
²Université Paris Cité, CNRS, Laboratoire Matériaux et Phénomènes Quantiques, 75205 Paris, France

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Vol. 130, Iss. 12 — 24 March 2023

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Images

Figure 1
(a) Plot of $S (n T) / S_{p}$ as a function of $n$ for $ω_{1}^{*}$ and several $V_{1}$ starting from a random Fock state. The red dotted line corresponds to $S_{p}^{f} / S_{p}$ of the largest fragment of $H_{F}^{(1)}$ with HSD 1008 to which the initial state belongs. $S (n T)$ saturates to $S_{p}^{f} = 0.8 S_{p}$ for large $V_{1}$ and $n \leq 200$ . (b) Same as (a) but at $2 ω_{1}^{*}$ ; here $S$ saturates to $S_{p}$ for all $V_{1}$ . (c) Plot of $Δ S = (S (n T) - S_{p}^{f}) / S_{p}$ as a function of $1 / V_{1}$ for $n = 200$ . $Δ S \to 0$ at $ω_{1}^{*}$ and for large $V_{1}$ . (d) Plot of $S (n T) / S_{p}$ for $V_{1} = 25$ and at $ω_{1}^{*}$ corresponding to several initial Fock states belonging to fragments of $H_{F}^{(1)}$ with HSD 1008 (green), 288 (black), 144 (brown), and 56 (pink). The dotted lines correspond to $S_{p}^{f} / S_{p}$ to which $S (n T) / S_{p}$ saturates for $n \leq 200$ . All plots indicate clear signature of prethermal strong HSF at $ω_{1}^{*}$ . For all plots $L = 16$ , $V_{0} = V_{2} = 2$ and all energies are scaled in units of $J$ .
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Figure 2
(a) $C_{L} (n T)$ as a function of $n$ at $ω_{1}^{*}$ computed using exact $H_{F}$ for several $V_{1}$ starting from a random thermal state. For large $V_{1}$ , $C_{L} (n T)$ saturates to the black dotted line which corresponds to its value computed using $H_{F}^{(1)}$ indicating prethermal HSF. (b) Plot of $C_{L} (n T)$ as a function of $n$ for several $ω_{D}$ for $V_{1} = 19$ showing ETH predicted thermalization away from $ω_{1}^{*}$ . (c) Plot of $C_{L} (n T)$ for $n = 5000$ as a function of $V_{1}$ at $ω_{1}^{*}$ for several $L$ showing a clear crossover to a prethermal HSF regime at large $V_{1}$ . The inset shows the number of cycles $n_{th}$ required for $C_{L} (n T)$ to reach its ETH predicted value at $ω_{1}^{*}$ for $L = 16$ ; $n_{th}$ scales exponentially with $V_{1}$ showing a long prethermal regime at large $V_{1}$ . (d) Phase diagram obtained from plot of $C_{L} (n T)$ for $n = 5000$ as a function of $V_{1}$ and $ℏ ω_{D} / V_{1}$ showing clear signature of finite value of $C_{L}$ at large $V_{1}$ at $ℏ ω_{1 (2)}^{*} = V_{1} / 2 (4)$ . For all plots $V_{0} = V_{2} = 2$ and all energies are scaled in units of $J$ . $L = 16$ for (a) and (b) and $L = 14$ for (d).
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Figure 3
(a) Plot of $χ_{1} (n T)$ as a function of $n$ for $V_{1} = 19$ at $ω_{1}^{*}$ (blue curve) and $6 ω_{1}^{*}$ (red curve) starting from a random frozen state showing lack of ETH predicted thermalization at $ω_{1}^{*}$ . (b) Plot of $⟨ χ_{1} ⟩$ as a function of $V_{1}$ at $ω_{1}^{*}$ ; $⟨ χ_{1} ⟩$ stays close to its initial value for large $V_{1}$ which is consistent with prethermal HSF. (c) Same as in (a) but for initial $| Z_{2} ⟩$ state showing slow oscillations at $ω_{1}^{*}$ . (d) Schematic diagram for the Floquet quasienergies showing doubly degenerate $| Z_{2} ⟩$ and $| {\bar{Z}}_{2} ⟩$ with $N_{d} = 0$ and other states with $N_{d} \neq 0$ . The arrows indicate transition to $| {\bar{Z}}_{2} ⟩$ from $| Z_{2} ⟩$ using intermediate states with $N_{d} \neq 0$ leading to slow oscillations. For all plots $V_{0} = 10 V_{2} = 2$ , $L = 14$ , and all energies are scaled in units of $J$ .
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