Quantum Hall effect in momentum space

We theoretically discuss a momentum-space analog of the quantum Hall effect, which could be observed in topologically nontrivial lattice models subject to an external harmonic trapping potential. In our proposal, the Niu-Thouless-Wu formulation of the quantum Hall effect on a torus is realized in the toroidally shaped Brillouin zone. In this analogy, the position of the trap center in real space controls the magnetic fluxes that are inserted through the holes of the torus in momentum space. We illustrate the momentum-space quantum Hall effect with the noninteracting trapped Harper-Hofstadter model, for which we numerically demonstrate how this effect manifests itself in experimental observables. Extension to the interacting trapped Harper-Hofstadter model is also briefly considered. We finally discuss possible experimental platforms where our proposal for the momentum-space quantum Hall effect could be realized.

Figure 1

The analog magnetic fluxes that are inserted through the holes of the toroidally shaped Brillouin zone, whose length in the $p_x$ direction is ${\mathcal{L}}_x$ and in the $p_y$ direction is ${\mathcal{L}}_y$. In this analogy, the Berry curvature $\mathrm{\Omega }\left(\mathbf{p}\right)$ acts as an effective magnetic field in momentum space. The trap center is located at $(x_0,y_0)$. (a) The toroidal structure of the Brillouin zone. (b) The flux corresponding to $x_0\ne 0$. (c) The flux corresponding to $y_0\ne 0$.

Figure 2

Persistent current contribution to the Hall current $-\langle {\tilde{j}}_y\rangle /\kappa$ as a function of $y_0$ for $x_0=0,\alpha =\frac{1}{4}$, and $\kappa =0.01J$. The solid line is the geometrical right-hand side of (31), and dots are the observable middle expression of (31) calculated from the noninteracting ground state.

Figure 3

Simulation of the momentum-space Hall conductivity for $\alpha =\frac{1}{4}$ and $\kappa =0.01J$. (a) The position of the center of the trap $x_0$ (solid line and the left axis) and the velocity of the trap motion ${\partial }_t{x}_0$ (dashed line and the right axis) are plotted as a function of time in units of the total sweeping time $T_{\mathrm{total}}$. (b), (c) The mean position of the wave function $\langle y\rangle$ as a function of the center of the trap $x_0$. The solid line is the numerically obtained value of the observable $\langle y\rangle$ from the real-time simulation starting from the ground state, whereas the dashed line is the geo-metrical prediction $-v\tilde{\mathrm{\Omega }}\left({\mathbf{r}}_0\right)/\kappa$. The total sweep time is $T_{\mathrm{total}}={10}^5/J$ for (b) and $T_{\mathrm{total}}=5\times {10}^4/J$ for (c). As predicted by Eq. (33), the two lines agree well for (b) in the region where the velocity is almost constant. The deviation of the two lines seen in (c) is due to the faster sweep causing the transition to excited states.

Figure 4

Estimated charge transport $P_y/{\mathcal{L}}_y$. The lines are the geometrical contribution to the charge transport [the middle expression of (35)], and the dots are estimated from the numerical calculation using the observable expression of the charge transport [the right expression of (35)]. The lower line and dots are for $y_0=0$ and the upper line and dots are for $y_0=0.5$. (a) $\alpha =\frac{1}{4}$ is fixed and $\kappa$ is varied. (b) $\kappa =0.01J$ is fixed and $\alpha$ is varied.

Figure 5

Spatial density profile of one of the four degenerate ground states of the trapped Harper-Hofstadter model with $\kappa =0.02J$ and $UN=0.5J$ when the trap center is located at (a) ${\mathbf{r}}_0=(0,0)$ and (b) ${\mathbf{r}}_0=(\frac{1}{2},\frac{1}{2})$. (c) Schematic drawing of the real-space lattice displaying the two trajectories ${\gamma }_1$ (along solid black arrows) and ${\gamma }_2$ (along a dotted green arrow) discussed in the main text for moving the center of the trap.