Spin-liquid states on the triangular and Kagomé lattices: A projective-symmetry-group analysis of Schwinger boson states

A symmetry-based analysis (projective symmetry group) is used to study spin-liquid phases on the triangular and Kagomé lattices in the Schwinger boson framework. A maximum of eight distinct $Z_2$ spin-liquid states are found for each lattice, which preserve all symmetries. Out of these only a few have nonvanishing nearest-neighbor amplitudes, which are studied in greater detail. On the triangular lattice, only two such states are present—the first (zero-flux state) is the well-known state introduced by Sachdev, which on condensation of spinons leads to the 120° ordered state. The other solution, which we call the $\pi$-flux state has not previously been discussed. Spinon condensation leads to an ordering wave vector at the Brillouin zone edge centers, in contrast to the 120° state. While the zero-flux state is more stable with just nearest-neighbor exchange, we find that the introduction of either next-neighbor antiferromagnetic exchange or four-spin ring exchange (of the sign obtained from a Hubbard model) tends to favor the $\pi$-flux state. On the Kagomé lattice four solutions are obtained—two have been previously discussed by Sachdev, which on spinon condensation give rise to the $q=0$ and $\sqrt{3}\times \sqrt{3}$ spin-ordered states. In addition we find two states with significantly larger values of the quantum parameter at which magnetic ordering occurs. For one of them this even exceeds unity ${\kappa }_c\approx 2.0$ in a nearest-neighbor model, indicating that if stabilized, could remain spin disordered for physical values of the spin. This state is also stabilized by ring-exchange interactions with signs as derived from the Hubbard model.

Figure 1

Oblique coordinates system of triangular lattice.

Figure 2

An example of a relation between space-group elements $T_1^{-1}{T}_2{T}_1{T}_2^{-1}=\mathcal{I}$, that leads to an algebraic constraint on the PSG.

Figure 3

Ansatz of the zero-flux state. An arrow from site $i$ to site $j$ means $A_{ij}>0$. All $A_{ij}$ have the same magnitude. All $B_{ij}$ are real and of uniform magnitude (not shown).

Figure 4

Ansatz of the $\pi$-flux state. All nearest-neighbor $A_{ij}$ are real and of the same magnitude and are positive in the directions shown. All nearest-neighbor $B_{ij}$ are zero. The unit cell for the Schwinger bosons is doubled due to the flux, and $u$ and $v$ are the two sites in this unit cell.

Figure 5

Triangular lattice Brillouin zone (BZ) and $k$-space coordinate system. The large hexagon is the BZ of the original lattice and the zero-flux ansatz. The large rectangle is the BZ of the $\pi$-flux ansatz. Black triangles at $K$-points: $\pm (2\pi ∕3,2\pi ∕3)$ are the minima of the spinon dispersion of the zero-flux state, and are also the wave vectors of magnetic ordering of the zero-flux state (120° state). Black diamonds at $\pm (\pi ∕2,-\pi ∕2)$ are the minima of the spinon dispersion of the $\pi$-flux state. Black hexagons at $M$-points: $\pm (\pi ,-\pi ),\pm (-\pi ,0),\pm (\pi ,0)$ are the magnetic-ordering wave vectors of the $\pi$-flux state.

Figure 6

The lower edge of the two-spinon spectrum for the zero-flux state at $\kappa =0.3$. Axes are in dimensionless units $k_{x,y}a$ where $a$ is the lattice constant. The darker regions have lower energy. The 120° magnetic order arises from magnon condensation at the zone corners (see also Fig. 5).

Figure 7

The lower edge of the two-spinon spectrum of the $\pi$-flux state with $\kappa =0.3$. Axes are in dimensionless units $k_{x,y}a$ where $a$ is the lattice constant. The darker regions have lower energy. Magnetic order arises from magnon condensation at the centers of the zone edges (see also Fig. 5).

Figure 8

(Color online) The mean-field phase diagram of the triangular lattice antiferromagnet with nearest-neighbor $J_1$ and next-neighbor $J_2$ interactions, as a function of the ratio $\alpha =J_2∕J_1$ and quantum parameter $\kappa$. The diagram is a quantitatively reliable solution of mean-field equations in the region near to and below the red line (dashed and solid) where spinon condensation of the $\pi$-flux state occurs. Just above the solid red line, magnetic ordering at the Brillouin zone $M$ point occurs, while below it the $\pi$-flux spin-liquid phase is obtained. The black line is the first-order phase boundary between the zero-flux and $\pi$-flux spin-liquid phases, while the blue line represents the onset of 120° magnetic order. The quick rise of this line to large values of $\kappa$ is due to the frustration of the 120° Néel order by next-neighbor interactions. If uninterrupted by magnetic order (of the $M$ type) this may open a window of spin liquid for physical spin values in this model.

Figure 9

Coordinates system of Kagomé lattice. Rhombus enclosed by dashed lines is the unit cell at origin. Three sites with black dots form the basis of the lattice. Black rhombus is an example of length-8 loops.

Figure 10

(Color online). $\mathbf{k}$ space of the Kagomé lattice. The large hexagon is the Brillouin zone of the original Kagomé lattice and the two $p_1=0$ Ansätze. The large rectangle is the Brillouin zone of the two $p_1=1$ Ansätze, which enclose a flux of $\pi$ in their unit cells. The red dots are the minima of the spinon dispersion for both $p_1=1$ states within the reduced Brillouin zone. The blue triangles are the locations of the minima of the lower edge of the two-spinon spectrum for both $p_1=1$ states.

Figure 11

Ansatz of the $p_1=1$, $p_2=1$, $p_3=0$ state. The presence of $\pi$ flux in the Kagomé lattice unit-cell leads to unit cell doubling, which is now defined by the dashed rhombus. Six sites with black dots form the basis. All nearest-neighbor $A_{ij}$ are real and of the same magnitude, and are positive in the directions shown. This spin-liquid Ansatz has zero flux in the length-6 hexagonal loops but $\pi$ flux in the length-8 loops such as the rhombus, hence $[0\mathrm{Hex},\pi \mathrm{Rhom}]$. It has a large critical quantum parameter ${\kappa }_c=2.0$.

Figure 12

Two-spinons spectrum lower edge of the $p_1=1$, $p_2=1$, $p_3=0$ state at $\kappa ={\kappa }_c=2.0$. Axes are in dimensionless unit $k_{x,y}a$ where $a$ is lattice constant (two times nearest-neighbor distance). Darker regions have lower energy. Although the two-spinon minima occur at the same wave-vector locations as in the previous $p_1=1$, $p_2=0$, $p_3=1$ state, the differences in the dispersion are apparent.

Figure 13

Two-spinons spectrum lower edge of the $p_1=1$, $p_2=0$, $p_3=1$ state at $\kappa ={\kappa }_c=0.9$. Axes are in dimensionless unit $k_{x,y}a$ where $a$ is the lattice constant (two times nearest-neighbor distance). The darker regions have lower energy.

Figure 14

Coordinates system of anisotropic triangular lattice.

Figure 15

Ansatz of the zero-flux state on the anisotropic triangular lattice. The different arrows represent different amplitudes of $A$. Single-arrow $A$ are real. $B$ are not represented in this picture.

Figure 16

Ansatz of the $\pi$-flux state on anisotropic triangular lattice. The different arrows represent different amplitudes of $A$. Single-arrow $A$ are real. $B$ are not represented in the picture.

Figure 17

Ansatz of the $p_1=0,p_2=0,p_3=1$ state, which is the $Q_1=-Q_2$ state of Sachdev, which on spinon condensation leads to the $\sqrt{3}\times \sqrt{3}$ state.

Figure 18

Ansatz of the $p_1=0,p_2=1,p_3=0$ state, which is the $Q_1=Q_2$ state in Sachdev’s notation. On spinon condensation this leads to the $q=0$ magnetically ordered state.

Figure 19

Ansatz of the $p_1=1,p_2=0,p_3=1$ state. The Rhombus enclosed by dashed lines is the unit cell. Six sites with black dots form the basis.