Spontaneous Formation of a Nonuniform Chiral Spin Liquid in a Moat-Band Lattice

A number of lattices exhibit moatlike band structures, i.e., a band with infinitely degenerate energy minima attained along a closed line in the Brillouin zone. If such a lattice is populated with hard-core bosons, the degeneracy prevents their condensation. At half-filling, the system is equivalent to the $s=1/2XY$ model at a zero magnetic field, while the absence of condensation translates into the absence of magnetic order in the $XY$ plane. Here, we show that the ground state breaks time reversal as well as inversion symmetries. This state, which may be identified with the chiral spin liquid, has a bulk gap and chiral gapless edge excitations. The applications of the developed analytical theory include an explanation of recent numerical findings and a suggestion for the chiral spin liquid realizations in experiments with cold atoms in optical lattices.

Figure 1

Unit cell of honeycomb lattice with NN $J_1$ and next NN $J_2$ couplings. The CS fluxes associated with spontaneously broken time-reversal symmetry are shown.

Figure 2

Phase diagram in the space of dimensionless parameters $(A^0/J_2;\varphi )$. The shadowed region, bounded by the bold line $A^0/J_2=3\sqrt{3}\sin \varphi$, represents topologically nontrivial states with Chern number $C=1$ [22]. The solutions of the self-consistency conditions (3), shown by the dashed line, all fall into the topological sector with $C=1$. Such solutions exist in the interval $0.2<J_2/J_1<0.36$, and are confined between the two full thin lines representing Eq. (5) for the boundaries of this interval $J_2/J_1=0.2;0.36$. Upper (lower) inset: staggered flux, $\varphi$, (chirality, $\chi$,) vs $J_2/J_1$, as obtained from Eqs. (3).

Figure 3

Bare band spectrum $E_{\mathbf{p}}(0,0)$ plotted from Eq. (4) for $J_2/J_1=0.1;0.2;0.3;1/3;0.37$ from bottom up. Horizontal line represents chemical potentials at half filling for $J_2/J_1=0.37$. For $J_2/J_1\le 1/3$, the chemical potential is at $\mu =0$. For $J_2/J_1=1/3$ the maximum of the lowest branch at $\mathrm{\Gamma }$ point reaches the height of Dirac points $K$ and $K^{\prime }$. A phase transition is, thus, expected for $J_2/J_1\gtrsim 1/3$, towards the spin density wave (SDW) state [27].

Figure 4

Schematic phase diagram of the $XY$ model across the range of the ratio $J_2/J_1$. Phases are marked by broken symmetries and corresponding Chern numbers, $C$.