Fractionalized Prethermalization in the One-Dimensional Hubbard Model

Abstract

Prethermalization phenomena in driven systems are generally understood via a local Floquet Hamiltonian obtained from a high-frequency expansion. Remarkably, recently it has been shown that a driven Kitaev spin liquid with fractionalized excitations can realize a quasi-stationary state that is not captured by this paradigm. Instead distinct types of fractionalized excitations are characterized by vastly different temperatures-a phenomenon dubbed "fractionalized prethermalization". In our work, we analyze fractionalized prethermalization in a driven one-dimensional Hubbard model at strong coupling which hosts spin-charge fractionalization. At intermediate frequencies quasi-steady states emerge which are characterized by a low spin and high charge temperature with lifetimes set by two competing processes: the lifetime of the quasiparticles determined by Fermi's Golden rule and the exponentially long lifetime of a Floquet prethermal plateau. We classify drives into three categories, each giving rise to distinct (fractional) prethermalization dynamics. Resorting to a time-dependent variant of the Schrieffer-Wolff transformation, we systematically analyze how these drive categories are linked to the underlying driven Hubbard model, thereby providing a general understanding of the emergent thermalization dynamics. We discuss routes towards an experimental realization of this phenomenon in quantum simulation platforms.

Figures (11)

• Figure 1: Fractionalized prethermalization arising from spin-charge separation in the one-dimensional Hubbard model: a) Fractionalization of fermions causes a periodic drive with frequency $\omega$ to typically couple asymmetrically to the fast charge quasi-particles of scale $t$ and slow spin quasi-particles of scale $J = 4t^2/U$, where $U$ is the Hubbard interaction. The latter experiences a higher relative frequency due to the smaller intrinsic energy scale at strong coupling, $J\ll t$. This results in a fractional prethermal plateau characterized by fast heating of the charge sector and a low effective temperature prethermal state in the spin sector. b) The lifetime of the resulting plateau is determined by the coupling strength of the drive to the spin quasi-particles. Three classes should be distinguished: (I) For drives that do not directly couple to the spinons, a fractionalized prethermal regime exists and its lifetime is limited only by the quasi-particle lifetime $\mathcal{O} (U^2/t^2)$. (II) For a weak coupling of the drive to the spin degrees of freedom on the order of $J$, the lifetime of the fractionalized prethermal plateau is determined by the competition between the quasi-particle lifetime $\mathcal{O} (U^2/t^2)$ and the exponential lifetime of a prethermal plateau in the spin sector $\mathcal{O} (e^{\omega U/t^2})$. (III) When both charge and spin is strongly driven the system heats rapidly. A fractional plateau is not observable.
• Figure 2: a) Squeezed space formalism: Spin-charge separation in the 1D tJ-model is understood by a parton construction, where holes reside on the bonds of a squeezed spin chain. Multiple adjacent holes are mapped to the same bond. Within the parton description, the kinetic energy $\mathcal{T}$ then corresponds to bosonic chargons hopping between links of the chain and Heisenberg interactions $\tilde{S}_i \tilde{S}_{i+1}$ between fermionic spinons are turned off when chargons reside on the bond in between. The density interactions $\kappa(h^\dagger h)$ are related to the number of holes on each bond, see main text for details.
• Figure 3: Fractionalized prethermalization with Fermi Golden Rule lifetime. The heating dynamics of a driven tJ-model shows a two-step structure. (a) Due to the strong coupling of the chargon with the drive, the charge sector heats up rapidly as indicated by the kinetic energy, which fluctuates around zero after an initial relaxation time. (b) In contrast, energy cannot be absorbed into the spin sector effectively, resulting in a long lived quasi-steady state. This state is captured by the spin-spin coupling $\tilde{Z}$ which retains most of its initial value for times up to $t\tau \sim 10^4$. (c,d) The lifetime of the quasi-steady spin state is determined by the breakdown of quasiparticles, resulting in a Fermi Golden Rule scaling of the lifetime $\tau_\mathrm{th} \sim U^2$ at large $U$ and an exponential decay $\sim e^{-\tau/\tau_\mathrm{th}}$ of the spin energy.
• Figure 4: Absence of conventional prethermalization. (a) Kinetic energy $\tilde{K}$ and spin correlations $\tilde{Z}$ in thermal equilibrium, representing hole and spin energies, respectively. (b) During the heating process, the effective temperature of the spin and charge sector differ by multiple orders of magnitude over a large temporal regime of $10^2 \lesssim t\tau \lesssim 10^4$ leading to an unconventional fractional prethermal quasi-steady state. Simulation parameters are the same as in Fig.$\,$(\ref{['fig:staggered-pot-drive']}).
• Figure 5: Fractionalized prethermalization with exponential lifetime: (a) While heating occurs rather slowly in the charge sector at small frequencies, we see an opposite trend with increasing frequency: Heating rates in the charge sector increase up to frequencies $\omega \approx 4t$ as captured by the kinetic energy. For better visibility, we show data for a few frequencies and apply a moving average conserving filter (unfiltered data in light colors below). (b) The spin correlations show that heating is strongly suppressed in the spin sector leading to a fractionalized prethermal plateau with exponential lifetime at smaller frequencies, before approaching a constant value determined by the quasiparticle lifetime. (c) The prethermal nature of the spin state is also captured by the half-chain entanglement entropy of the squeezed spin chain. (d) At intermediate frequencies, the lifetime of the fractional prethermal plateau shows exponential dependence on frequency $\omega$. Numerical data is obtained for $U=26t$.