Ferromagnetism in tilted fermionic Mott insulators

TL;DR

This work investigates magnetism in tilted fermionic Mott insulators, revealing that a large linear tilt induces Wannier-Stark localization of charges and converts the effective magnetic exchange to ferromagnetic. The authors derive the tilt-dependent spin Hamiltonian both perturbatively and via Floquet theory, showing $J_{\mathrm{eff}}(E)=\frac{J_0}{1-(E/U)^2}$ with $J_0=\frac{4t_h^2}{U}$, and confirm ferromagnetism through real-time simulations of the 1D Hubbard chain. They demonstrate that tilt-controlled dynamics accelerate or reverse spin evolution through a scale factor $f(E)$, enabling time-reversal protocols useful for measuring out-of-time-ordered correlators, and discuss experimental platforms such as ultracold atoms in optical lattices. The findings offer a pathway to tunable magnetic interactions in strongly correlated systems and motivate experimental exploration in AMO setups and quantum-dot architectures. Overall, the paper connects nonequilibrium control with emergent ferromagnetic order in a tunable, strongly interacting lattice model.

Abstract

We investigate the magnetism in tilted fermionic Mott insulators. With a small tilt, the fermions are still localized and form a Mott-insulating state, where the localized spins interact via antiferromagnetic exchange coupling. While the localized state is naively expected to be broken with a large tilt, in fact, the fermions are still localized under a large tilt due to the Wannier-Stark localization and it can be regarded as a localized spin system. We find that the sign of the exchange coupling is changed and the ferromagnetic interaction is realized under the large tilt. To show this, we employ perturbation theory and real-time numerical simulation of the fermionic Hubbard chain. Our simulation exhibits that it is possible to effectively control the speed and time direction of the dynamics by changing the size of the tilt, which may be useful for experimentally measuring out-of-time-ordered correlators. Finally, we discuss experimental platforms, such as ultracold atoms in an optical lattice, to observe these phenomena.

Figures (9)

• Figure 1: (a, b) Schematic picture of (a) the Mott insulating regime ($E \ll U$) and (b) the Wannier-Stark localized regime ($U \ll E$). (c) Time average of the double occupancy per site $\overline{\mathcal{N}}_\mathrm{double}$ [Eq. \ref{['eq:ave_double_occ']}], starting from the singly occupied state $\ket{\uparrow\downarrow\uparrow\downarrow\cdots}$ for $L=10$. The time step $\Delta t$ is set to $1/800$. $\overline{\mathcal{N}}_\mathrm{double}$ takes a large value around the resonant condition $nE=U$ ($n=1,2,\cdots$) where the insulating state easily breaks down. (d) Effective interaction $J_\mathrm{eff}(E)/J_\mathrm{eff}(0)$ defined in Eq. \ref{['eq:J_eff']}. $t_h$ ($t_h^{-1}$) is used as the unit of energy (time).
• Figure 2: Time evolution of the doublon number per site $\mathcal{N}_\mathrm{double}(t)$ [Eq. \ref{['eq:double_occ']}], where electric field $E$ is switched on at $t=50$. The data for $E\leq (\geq)~50$ are shown in the left (right) panel. We set $U=50$, $L=10$, and $\Delta t=1/3200$, and choose the singly occupied state $\vert\uparrow\downarrow\uparrow\downarrow\cdots\uparrow\downarrow\rangle$ as the initial state. $t_h$ ($t_h^{-1}$) is used as the unit of energy (time).
• Figure 3: (a) Perturbation processes relevant to the kinetic exchange. The two processes (i) and (ii) are inequivalent only when an electric field is applied. (b) $E$-dependence of $J_\mathrm{\mathrm{eff}}(E)$ [Eq. \ref{['eq:J_eff']}], $J_+(E)$, and $J_-(E)$ [Eq. \ref{['eq:Jpm']}]. The plotted value is normalized by $J_0/2=2t_h^2/U$.
• Figure 4: Time evolution of the local spin imbalance $\mathcal{I}_{L/2}$ [Eq. \ref{['eq:imbalance']}] for $L=10$ before and after the field $E$ is switched on at $t_\mathrm{on} = 50$. We set $U=50$ and $\Delta t = 1/3200$, and choose the singly occupied state $\vert\uparrow\downarrow\uparrow\downarrow\cdots\uparrow\downarrow\rangle$ as the initial state. We vary $E$ for $0\leq E\leq 2U$ and the data for $E \leq U$ and $E \geq U$ are shown in panels (a) and (b) respectively. Note that the data for $E=70$ in panel (b) exhibits almost perfect time-reversal dynamics because $J_\mathrm{eff}(E)/J_0$ ($\simeq -1.04$) is nearly minus one. In panel (c), we show all the data with a rescaled time defined in Eq. \ref{['eq:rescaled_time']}. $t_h$ ($t_h^{-1}$) is used as the unit of energy (time).
• Figure 5: Time evolution of the local spin imbalance $\mathcal{I}_{L/2}$ [Eq. \ref{['eq:imbalance']}] for $L=10$ when the field $E = 70.71 \sim \sqrt{2}U$ is turned on at $t_\mathrm{on}$ and switched off at $t_\mathrm{off} = t_\mathrm{on} + \delta t$ with $\delta t = 20$. We set $U=50$ and $\Delta t = 1/3200$, and choose the singly occupied state $\vert\uparrow\downarrow\uparrow\downarrow\cdots\uparrow\downarrow\rangle$ as the initial state. We vary $t_\mathrm{on}$ for $t_\mathrm{on}=50, 75, 100$ and they are shown with different line types. The time duration when $E$ is finite are shown with the shaded regions respectively. The dynamics without $E$ is shown in a gray dashed curve. $t_h$ ($t_h^{-1}$) is used as the unit of energy (time).