Diffusion in interacting two-dimensional systems under a uniform magnetic field

Abstract

The dynamics of interacting particles in orbital magnetic fields are notoriously difficult to study, as this physics is inherently connected to electronic correlations in two-dimensional systems, for which no straightforward theoretical methods are available. Here, we report on the diffusive relaxation dynamics of two-dimensional interacting fermionic systems under a uniform magnetic field in the infinite temperature regime. We first show that the fermionic truncated Wigner approximation captures the equilibration dynamics unexpectedly well for intermediate interaction strengths when going beyond one dimension. This high accuracy holds at least for relatively small ladder systems, which are accessible to the Lanczos method that we use to benchmark the reliability of the Wigner approximation. We find that strong interactions, which exceed the hopping energy, suppress magnetic-field effects on diffusive transport. However, when the interactions are comparable to the kinetic energy, the diffusion is significantly reduced by the magnetic flux. This is observed for sufficiently large systems (above approximately 400 lattice sites), where finite-size effects weakly affect particle transport. We suggest that our results should be directly accessible on current optical lattice platforms.

Figures (7)

• Figure 1: Schematic visualization of (a) the system described in Eq. \ref{['hamiltonian']}, where blue (red) dotes show an occupied (unoccupied) sites in the 2D lattice. The magnetic flux per plaquette is given by $\Phi=\gamma/2\pi$. (b) Construction of the sine-profile density by averaging over an ensemble of initial states.
• Figure 2: Time evolution of the amplitude $A$ for a $10\times 2$ ladder system with $1000$ initial states, comparing two methods: Lanczos (red) and fTWA (blue). Panels (a), (c) show results without a magnetic field, while (b), (d) correspond to $\gamma=\pi$. Panels (a), (b) are plotted for interaction strength $V/J=0.5$, and panels (c), (d) for $V/J=2.5$. All panels represent data for the longest wavelength mode, i.e., $\lambda = L_x= 10$ in units of the lattice spacing in the $x$-direction. For fTWA, $5000$ trajectories were sampled.
• Figure 3: The corresponding deviations of fTWA from the Lanczos results are shown for different system sizes $L_x$ (a)-(b) and interaction strengths (c)-(d). Here, $\sigma$ and $\delta$ denote the absolute and relative errors, respectively (see Eqs. \ref{['absolute']}-\ref{['relative']}). The system and simulation parameters correspond to those used in Fig. \ref{['fig2']}, and the errors are calculated for the time range $(0,T)=(0,50)$ in units of $J$.
• Figure 4: Time evolution of the amplitude $A$ for (a) $10\times 2$ ladder system and (b) $20\times 1$ one-dimensional system, comparing Lanczos (red) and fTWA (blue). Panel (a) shows the shortest wavelength mode $q=\pi$ with interaction strength $V/J=0.5$ and $600$ initial states, while panel (b) displays the longest wavelength mode $q=2\pi/L$ with interaction strength $V/J=1.5$ and $5000$ initial states. For fTWA, $5000$ trajectories were sampled.
• Figure 5: Panels (a) and (b) show the time evolution of the amplitude $A$ for system size $L=22\times 2$ and $L=34\times 2$, respectively, with phase $\gamma=0$ and $\gamma=\pi$. Panels (c) and (d) present the diffusion constant $D/J$ as a function of system size $L_x$: panel (c) without a magnetic field, $\gamma=0$ and panel (d) with $\gamma=\pi$. In panels (a) and (b), the red dashed line indicates the fitted function $\exp(-\alpha t)$, with the fitting region marked by a gray rectangle spanning $t \in [0;45]$. The insets in panels (c) and (d) display the coefficient $\alpha$ as a function of system size $L_x$, obtained from fitting $\exp(-\alpha t)$ in the same time window $t \in [0;45]$. The number of initial states is chosen as $100 \times L_{x}$ (e.g., $34\times 2$ corresponds to $3400$ states). For fTWA, the number of trajectories varies from $5000$ for the smallest system to $100$ for the largest system.