Symmetric $Z_2$ spin liquids and their neighboring phases on triangular lattice

Motivated by recent numerical discovery of a gapped spin liquid phase in spin-$1/2$ triangular-lattice $J_1-J_2$ Heisenberg model, we classify symmetric $Z_2$ spin liquids on triangular lattice in the Abrikosov-fermion representation. We find 20 phases with distinct spinon symmetry quantum numbers, eight of which have their counterparts in the Schwinger-boson representation. Among them we identify two promising candidates (#1 and #20), which can realize a gapped $Z_2$ spin liquid with up to next nearest neighbor mean-field amplitudes. We analyze their neighboring magnetic orders and valence bond solid patterns, and find one state (#20) that is connected to 120-degree Neel order by a continuous quantum phase transition. We also identify gapped nematic $Z_2$ spin liquids in the neighborhood of the symmetric states and find three promising candidates (#1, #6, and #20).

Figure 1

Crystal symmetry of triangular lattice, and nearest-neighbor (NN) hopping amplitudes of fermionic spinons in the $\pi$-flux state on the triangle lattice. All NN hoppings are real with the same magnitude $\left|\alpha \right|>0$, where thick (thin) bonds have negative (positive) hopping signs. Each magnetic unit cell (denoted by the green oval) is chosen to contain four lattice sites labeled by red/blue diamonds/dots.

Figure 2

Mean-field hopping amplitudes (up to NNN) of nematic $Z_2$ spin liquid #20 in Table 1. There are two types of NN hoppings denoted by black and red colors: for black ones, thick and thin lines have the same amplitude $|{\alpha }_1|$ but opposite signs; for red ones, solid and dashed lines have the same amplitude $|{\alpha }_2|$ but opposite signs. NNN amplitudes are labeled by blue color: solid and dashed lines have the same amplitude $|{\beta }_1|$ but opposite sings. Onsite real paring is also allowed by symmetry.

Figure 3

Mean-field hopping amplitudes (up to NNN) of nematic $Z_2$ spin liquid #6 in Table 1. Nonzero NN hoppings are denoted by black thick lines: for black ones, solid and dashed lines have the same amplitude $|{\alpha }_1|$ but opposite signs. The thin lines connecting NNs have zero hopping amplitude. There are two types of NNN amplitudes: for blue (or red) colors, solid and dashed lines have the same amplitude $|{\beta }_1|$ (or $|{\beta }_2|$) but opposite sings. Onsite real paring is also allowed by symmetry.