Artificial magnetic fields in momentum space in spin-orbit-coupled systems

The Berry curvature is a geometrical property of an energy band which can act as a momentum-space magnetic field in the effective Hamiltonian of a wide range of systems. We apply the effective Hamiltonian to a spin-$\frac{1}{2}$ particle in two dimensions with spin-orbit coupling, a Zeeman field, and an additional harmonic trap. Depending on the parameter regime, we show how this system can be described in momentum space as either a Fock-Darwin Hamiltonian or a one-dimensional ring pierced by a magnetic flux. With this perspective, we interpret important single-particle properties, and identify analog magnetic phenomena in momentum space. Finally, we discuss the extension of this work to higher-spin systems, as well as experimental applications in ultracold atomic gases and photonic systems.

Figure 1

A schematic indicating the key characteristics of the energy dispersion and the Berry curvature of the lower band in (a) the single minimum regime for $\zeta ={\lambda }^{2}M/\Delta <1$ and (b) the ring minima regime for $\zeta >1$.

Figure 2

The low-energy numerical spectrum adjusted for the minimum energy of the lower band $\text{min}\left(E_{-}\right)$ as a function of $\zeta ={\lambda }^{2}M/\Delta$, for $\chi =0.2$. We have included states with a radial quantum number $n=0,1$ and an azimuthal quantum number $-3\le m\le +3$. The energy of each state is obtained by numerically diagonalizing (6) over a basis of 150 states. To emphasize key features, we then subtract the minimum energy of the lower band (7) from the total energy of each state. The vertical dotted line marks the crossover from the single minimum $(\zeta <1)$ to the ring minima regime $(\zeta >1)$.

Figure 3

The energy splitting between states with $+m$ and $-m$ for (a) $\chi =0.01$ and (b) $\chi =0.1$. Solid lines are the analytical Fock-Darwin results [Eq. (29)]. Dashed lines are numerical calculations from the full Hamiltonian (6). Multiple dashed lines of the same color represent states with different values of $n$. The agreement between analytics and numerics is best for small $\chi ,\zeta ,n$, and $m$.

Figure 4

Quantitative comparison between the analytical (An.) Fock-Darwin eigenstates (26) and numerical (Nu.) eigenstates of the full Hamiltonian (6) for (a) $n=0,m=0$ and (b) $n=0,m=1$ for $\chi =0.1$ and over a range of $\zeta$.

Figure 5

Left-hand side: the orientation of the local spin vector for (above) intermediate and (below) large $\zeta$ around the ring of minima for $\Delta >0$. Right-hand side: the mapping of the local spin vector onto the spin-$\frac{1}{2}$ Bloch sphere. As $\zeta \to \infty$, the spins at $p_0$ lie in the $xy$ plane, and the mapped path traverses the equator on the Bloch sphere. This corresponds to the maximum possible Berry phase $\left|\Phi \right|=\pi$ for this model.

Figure 6

A segment of the energy spectrum as a function of the momentum-space magnetic flux, tuned via the Zeeman field. In the limit $\Delta \to 0,\Phi /{\Phi }_0\to -\frac{1}{2}$, for $\zeta /\chi =100$. The dashed lines indicate the analytical predictions from Eq. (40). The solid lines are extracted from numerics by calculating computationally the total energy and then subtracting the theoretical radial energy $E_n/\omega$ [Eq. (47)].

Figure 7

The azimuthal “velocity” in momentum space as a function of the flux for the five lowest-energy states $-2\le m\le 2$ and $n=0$. The flux is varied by tuning the Zeeman field $\Delta$, while holding the ratio $\zeta /\chi =100$ fixed. Dashed lines are analytical, while solid lines are numerical results.

Figure 8

Comparison between Eq. (67) (An.) and numerical (Nu.) real-space wave function for $\Phi /{\Phi }_0=-0.45$ and $\Phi /{\Phi }_0=-0.25$, with $\zeta /\chi =40$ and $m=0$. (a) The density of spin-up atoms and (b) the density of spin-down atoms for a cut along $y=0$. As the density profiles are rotationally symmetric, the full distribution can be found by rotation around the $z$ axis.

Figure 9

(a) The analytical angular energy (75) for $\zeta /\chi =20$ and $F=3$, showing the successive ground states with different values of $m$. The degeneracies between states occur at half-integer values of the flux. (b) The analytical “velocity” for the ground state (77) for the same parameters.