Momentum-space Harper-Hofstadter model

We show how the weakly trapped Harper-Hofstadter model can be mapped onto a Harper-Hofstadter model in momentum space. In this momentum-space model, the band dispersion plays the role of the periodic potential, the Berry curvature plays the role of an effective magnetic field, the real-space harmonic trap provides the momentum-space kinetic energy responsible for the hopping, and the trap position sets the boundary conditions around the magnetic Brillouin zone. Spatially local interactions translate into nonlocal interactions in momentum space: within a mean-field approximation, we show that increasing interparticle interactions leads to a structural change of the ground state, from a single rotationally symmetric ground state to degenerate ground states that spontaneously break rotational symmetry.

Figure 1

Momentum-space HH model for the lowest band of $\alpha =1/4$ which has $\mathcal{C}=-1$. The MBZ enclosed by a solid line is the natural MBZ corresponding to the magnetic gauge of our choice (3). Differently shaped MBZs, described by nonsolid lines, can be obtained by using different magnetic gauges. The hoppings are given up to a constant factor.

Figure 2

Numerical estimate of $J^{\prime }$ as a function of $\kappa$ (both in units of $J)$ for an $\alpha =1/4$ real-space HH model. Numerical calculation was performed on a $80\times 80$ real-space lattice.

Figure 3

The wave function of the $\alpha =1/4$ ground state in the MBZ for $n_0=0$ and $m_0=0$, 0.5, and 1, from left to right, numerically calculated on an $80\times 80$ real-space lattice with $\kappa =0.01J$.

Figure 4

The weight $|{\alpha }_0{|}^2$ as a function of $m_0$ for $n_0=0$ (lower line and dots) and as a function of $n_0$ for $m_0=0$ (upper line and dots) for four different values of $\kappa$: (a) $\kappa =0.005J$, (b) $\kappa =0.01J$, (c) $\kappa =0.02J$, and (d) $\kappa =0.03J$. The solid lines are the prediction from the momentum-space HH model. The dots are numerical results, calculated on a $80\times 80$ real-space lattice.

Figure 5

Density profile $|a_{m,n}{|}^2$ of the ground-state wave function numerically obtained by the imaginary-time propagation method on a $40\times 40$ real-space lattice with $\kappa =0.01J$ for different values of $UN$: (a) $UN=0.04J,Z_4$ state, (b) $UN=0.11J,Z_2$ state, (c) $UN=0.14J,Z_{\times }$ state, and (d) $UN=0.40J,Z_2$ state. Only the central region is plotted.

Figure 6

The weights of rotational eigenstates $|{\beta }_0{|}^2+{\left|{\beta }_1\right|}^2$ (blue solid line), $|{\beta }_2{|}^2$ (green dashed line), and $|{\beta }_3{|}^2$ (red dotted line) as a function of $U^{\prime }/J^{\prime }$, obtained by minimizing the momentum-space Harper-Hofstadter Hamiltonian.

Figure 7

Diagram showing how the symmetry of the ground state of the trapped interacting HH model changes as a function of the trap and the interaction strengths, with both axes in units of $J$. The solid lines are the predictions of the momentum-space HH model for the transition lines, while the dots show the corresponding predictions from numerical simulations performed via the imaginary-time propagation method on a $40\times 40$ lattice.