Topological phase transitions driven by non-Abelian gauge potentials in optical square lattices

We analyze a tight-binding model of ultracold fermions loaded in an optical square lattice and subjected to a synthetic non-Abelian gauge potential featuring both a magnetic field and a translationally invariant SU(2) term. We consider in particular the effect of broken time-reversal symmetry and its role in driving nontrivial topological phase transitions. By varying the spin-orbit coupling parameters, we find both a semimetal-insulator phase transition and a topological phase transition between insulating phases with different numbers of edge states. The spin is not a conserved quantity of the system, and the topological phase transitions can be detected by analyzing its polarization in time-of-flight images, providing a clear diagnostic for the characterization of the topological phases through the partial entanglement between spin and lattice degrees of freedom.

Figure 1

Square-lattice model with non-Abelian gauge potential. The tunneling operators $U_x$ and $U_y$, defined in Eqs. (3) and (4), are represented for $\Phi ={\Phi }_0/3$. $U_x=e^{iq{\sigma }_x}$ is independent of the position, whereas $U_y$ is characterized by a phase dependent on $x$, which determines the Abelian flux $\Phi$ per plaquette.

Figure 2

The spectrum of the Hamiltonian (7) is shown for $q=\pi /4$. The Dirac cones connecting energy bands 3 and 4 are clearly visible in the upper part of the plot at $E=2$, whereas the gap between bands 2 and 3 closes at $(k_x,k_y)=(\pm \pi /2,\pm \pi /2)$ and $E=0$. The spectrum is symmetric about $E=0$, and no energy gap appears for any value of $q$. The energy is expressed in units of $t$.

Figure 3

The energies of the two lowest states of the Hamiltonian (16) are plotted for $q=\pi /5$ (blue dashed line), $q=\pi /3$ (red solid line), and $q=2\pi /5$ (green dotted line), for $k_y=\pi /3$, and as a function of $k_x$. For $q=\pi /5$ and $q=2\pi /5$ the lowest bands are separated by a gap, and for filling $\nu =1/3$, the system is in two different topological insulating phases. At $q=\pi /3$ the gap closes, causing a topological phase transition. The energy is in units of the tunneling amplitude $t$, and the chemical potential is fixed at $\mu =0$.

Figure 4

Phase diagram at filling $1/3$ as a function of $q_x$ and $q_y$ for the gauge potential (18). The insulating regions characterized by Chern numbers ${\mathcal{C}}_{-2}=-2,1$ are shown in green (dark gray) and orange (light gray), respectively. Uncolored regions correspond to indirect overlap between bands (topological semimetal phase), whereas the black lines show the position of the Dirac cones corresponding to a discontinuity in the Chern number ${\mathcal{C}}_{-2}$. The red dots represent topological phase transitions at $q=\pi /3,2\pi /3$ along the diagonal $q_x=q_y=q$.

Figure 5

Comparison between the spin polarization at $q=\pi /5$ (top row), which represents the topological insulating phase with $\mathcal{W}=0$, and $q=2\pi /5$ (bottom row), which describes the second insulating phase with $\mathcal{W}=-1$. The component $S_z$ of the polarization (third column) behaves very differently in the two cases. For $q=\pi /5$ (as generally for $q<\pi /3$), $S_z$ is negative in the whole Brillouin zone (top right panel), whereas for $q=2\pi /5$ (as, in general, for $\pi /3<q<\pi /2$) the polarization exhibits a skyrmion structure. When $q=2\pi /5$, the skyrmion interpolates from $S_z\simeq 0.067$ at $\vec{k}=(0,0)$ to $S_z\simeq -0.067$ at $\vec{k}=(\pi /3,\pi /3)$ (bottom right panel). The $S_x$ and $S_y$ components display opposite signs in the two cases.

Figure 6

Effects of discretization on the relative error in the computation of the winding number. (a) Estimated spin winding number $\mathcal{W}$ for various values of the discretization $L$: $L=100$ (solid red line), $L=10$ (dashed blue line), and $L=8$ (dashed-dotted black line). The finer grid ($L=100$) reproduces almost exactly the theoretical value, whereas coarser grids feature deviations at the transition points (mainly at $q=\pi /3$ because of the relative positions of the grid and the skyrmion) plus a significant error at the maximally entangled $q=\pi /2$ point. (b) Winding number computed at $q=\pi /2+\epsilon$ for different discretizations. (c) The minimum of the norm of the polarization vector $|\vec{S}|$ as a function of $q$. This minimum is reached in the skyrmionic region, showing that the detection of the nontrivial winding number requires more statistical power. In particular, at $q=\pi /2$ the norm vanishes at the skyrmion core, rendering the measurement protocol useless at that particular point.

Figure 7

In-plane spin polarization of the edge modes of the lowest-energy band in the two different topological phases at $q=\pi /5$ and $q=2\pi /5$. The average spin densities along $x$ and $y$ are shown for a single edge state in the first energy gap for a hard-wall potential with a 40-site radius. The top and bottom rows show the results for $q=\pi /5$ and $q=2\pi /5$. The spin distribution is qualitatively the same in the two cases.

Figure 8

Total mass density $|{\Psi }_{\uparrow }{|}^2+{\left|{\Psi }_{\downarrow }\right|}^2$ for two edge states of the lowest-energy band at $q=\pi /5$ and $q=2\pi /5$ and under a harmonic trapping potential. A typical interference pattern breaking the axial symmetry appears. The system size is $L=102$, and the trapping radius is $R=2L/5$.

Figure 9

Total density $|{\Psi }_{\uparrow }{|}^2+{\left|{\Psi }_{\downarrow }\right|}^2$ of an eigenstate with energy lying in the first band gap in a lattice with harmonic potential along the $y$ direction for (left) $q=\pi /5$ and (right) $q=2\pi /5$. In the upper and lower edges of the right panel, two edge modes are clearly distinguishable. On the vertical edges the two modes can no longer be distinguished due to their stronger localization dictated by the hard-wall condition on the boundary.

Figure 10

Density distribution of the edge states in the first gap, shown for $q=\pi /5$ in a square system of size $102\times 102$ as a function of energy and position. The dark central zone in each panel is the band gap separating the two lower bands [central yellow (gray) regions at the top and the bottom]. The edge states are the yellow (gray) lateral lines. The left panel represents a system with a harmonic trapping potential, the middle one refers to a quartic potential, and the right one refers to a hard-wall potential. The trappings are only along the $y$ direction. The densities are calculated by averaging the total wave-function density along the $x$ direction in the central region of the system (for $-30\le x\le 30$), far from the vertical edges.