Asymptotic transport exponent in a strongly tilted two-dimensional Hubbard lattice
In plain words
When a lattice of atoms is strongly tilted, the centre of mass of the atoms can barely move, so density spreads more slowly than ordinary diffusion. Experiments see this slowing, but whether it is the final law at the longest times and distances is not established.
Precise statement
In the Fermi-Hubbard model on a square lattice with hopping $J$, interaction $U$ and uniform tilt $\Delta$ per site $(\Delta >> J)$, dipole-conserving hydrodynamics predicts that a density modulation of wavelength $\lambda$ relaxes in time $\tau \sim \lambda^{4}$ (dynamical exponent $z = 4$) instead of $\tau \sim \lambda^{2}$. Guardado-Sanchez et al. (PRX 10, 011042, 2020) observed the crossover from $\tau \sim \lambda^{2}$ to $\tau \sim \lambda^{4}$ with increasing tilt over a limited range of $\lambda$. Determine whether $z = 4$ is the asymptotic exponent as $\lambda \to \infty$ at finite energy density, and the wavelength beyond which higher-order dipole-breaking processes change it. Answer: $z$ and the crossover scale as functions of $\Delta/J$ and $U/J$.
What would settle it
Quantum-gas measurements or large-scale numerics of $\tau(\lambda)$ over at least a decade in $\lambda$ at several $\Delta/J$, matched by an effective hydrodynamic theory including dipole-breaking terms.
Status in the literature
Subdiffusion with $\tau \sim \lambda^4$ was observed in 2020 over a limited range of wavelengths; the asymptotic law is untested.