Schwinger boson mean field theories of spin liquid states on a honeycomb lattice: Projective symmetry group analysis and critical field theory

Motivated by the recent numerical evidence [Z. Meng, T. Lang, S. Wessel, F. Assaad, and A. Muramatsu, Nature (London) 464, 847 (2010)] of a short-range resonating valence bond state in the honeycomb lattice Hubbard model, we consider Schwinger boson mean field theories of possible spin liquid states on honeycomb lattice. From general stability considerations the possible spin liquids will have gapped spinons coupled to $Z_2$ gauge field. We apply the projective symmetry group method to classify possible $Z_2$ spin liquid states within this formalism on honeycomb lattice. It is found that there are only two relevant $Z_2$ states, differed by the value of gauge flux, zero or $\pi$, in the elementary hexagon. The zero-flux state is a promising candidate for the observed spin liquid and continuous phase transition into commensurate Néel order. We also derive the critical field theory for this transition, which is the well-studied O(4) invariant theory [A. V. Chubukov, T. Senthil, and S. Sachdev, Phys. Rev. Lett. 72, 2089 (1994); A. V. Chubukov, S. Sachdev, and T. Senthil, Nucl. Phys. B 426, 601 (1994); S. V. Isakov, T. Senthil, and Y. B. Kim, Phys. Rev. B 72, 174417 (2005)], and has an irrelevant coupling between Higgs and boson fields with cubic power of spatial derivatives as required by lattice symmetry. This is in sharp contrast to the conventional theory [S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991)], where such transition generically leads to incommensurate magnetic order. In this scenario the $Z_2$ spin liquid could be close to a tricritical point. Soft boson modes will exist at seven different wave vectors. This will show up as low-frequency dynamical spin susceptibility peaks not only at the $\Gamma$ point (the Néel order wave vector) but also at Brillouin-zone-edge center $M$ points and twelve other points. Some simple properties of the $\pi$-flux state are studied as well. Symmetry allowed further neighbor mean field ansatz are derived in appendices which can be used in future theoretical works along this direction.

Figure 1

(Color online) Mean field phase diagram of the zero-flux state. Horizontal axis is the variational parameter, ratio between next-nearest-neighbor and nearest-neighbor mean field couplings, $A_2/A_1$. Vertical axis is the average boson density ${⟨\widehat{n}⟩}_{\text{MF}}$. The dash line ${⟨\widehat{n}⟩}_{\text{MF}}=1$ indicates the boson density of spin-1/2 system. Solid lines are phase boundaries. The red solid line between the zero-flux $Z_2$ spin liquid and the Néel order is a continuous transition described by the field theory Eq. (22). The vertical solid black line between the two ordered states is a first-order transition. The blue line between the $Z_2$ spin liquid and the incommensurate magnetic order has yet to be studied but is likely a continuous transition. There is a very small parameter range of $0.493<A_2/A_1<0.516$ (see also the inset) such that a spin-1/2 system will be a gapped $Z_2$ spin liquid, which is a promising explanation of the observed spin liquid (Ref. 19). The variational parameter $A_2/A_1$ can, in principle, be tuned by physical parameters. For example, as argued in Sec. 4, increase in $U/t$ will decrease $A_2/A_1$, which can drive a continuous magnetic-ordering transition at the crossing point (black dot) of the dash line and the red solid line.

Figure 2

The honeycomb lattice is shown on the left. Open (filled) circles indicate the two sublattices. ${\mathbf{a}}_1,{\mathbf{a}}_2$ are primitive vectors. For simplicity we assume the lattice constant $a=\left|{\mathbf{a}}_1\right|=\left|{\mathbf{a}}_2\right|=1$. $u,v$ denote the two sites within one unit cell. The hexagon on the right is the enlarged unit cell with schematic illustration of the space-group generators, translations $T_1$ and $T_2$, sixfold rotation $C_6$, and reflection $\sigma$.

Figure 3

(Color online) The zero-flux ansatz. Left part shows the nearest-neighbor ansatz. Single arrow from $i$ to $j$ means $A_{⟨ij⟩}=-A_{⟨ji⟩}=A_1>0$. All nearest-neighbor $B_{⟨ij⟩}$ must be zero according to Appendix . Blue dash rhombus is the unit cell of the mean field theory, containing two sites $u,v$. The large hexagon on the right is the enlarged mean field unit cell showing the next-nearest-neighbor bonds. Double arrow from $i$ to $j$ means $A_{⟪ij⟫}=-A_{⟪ji⟫}=A_2$. All next-nearest-neighbor $B_{⟪ij⟫}=+B_2$ are real according to Eqs. (B5a, B5b, B5c, B5d, B5e, B5f).

Figure 4

(Color online) The $\pi$-flux ansatz. Left part shows the nearest-neighbor ansatz. Single arrow from $i$ to $j$ means $A_{⟨ij⟩}=-A_{⟨ji⟩}=A_1>0$. All nearest-neighbor $B_{⟨ij⟩}$ must be zero according to Appendix . Blue dash rhombus is the doubled unit cell of the mean field theory, containing four sites $u,v,p,q$. The large double hexagon on the right is the enlarged mean field unit cell showing the next-nearest-neighbor bonds. Double arrow from $i$ to $j$ means $A_{⟪ij⟫}=-A_{⟪ji⟫}=A_2$. All next-nearest-neighbor $B_{⟪ij⟫}=\pm B_2$ are real, with the $\pm$ signs given in Eqs. (B5a, B5b, B5c, B5d, B5e, B5f). Red thick bonds are those different from the zero-flux ansatz Fig. 3.

Figure 5

The zero-flux mean field boson dispersion $E_{\pm }$ Eq. (19), with $A_2/A_1=1/2$ and average boson density ${⟨\widehat{n}⟩}_{\text{MF}}=1$ (for spin-1/2 model), along high-symmetry directions $\Gamma \text{−}K\text{−}M\text{−}\Gamma$ [see Fig. 6a]. Note the very low-energy boson modes at $T$ point.

Figure 6

(Color online) (a) Hexagon is the Brillouin zone of the zero-flux ansatz. Central black dot is the $\Gamma$ point $\left(k_1,k_2\right)=\left(0,0\right)$, where boson condensation happens when $A_2/A_1<1/2$. When $A_2/A_1$ increases from 1/2 to $+\infty$, the boson condensation momenta move along the red short lines, $\pi <\left|\mathbf{k}\right|<4\pi /3$, from $T$ to $K\left(K^{\prime }\right)$. The $\Gamma$ point, three BZ edge center $M$ points, six $T$ points (filled red circle, $\left|\mathbf{k}\right|=\pi$) on $\Gamma \text{−}K\left(K^{\prime }\right)$ lines, and six $D$ points (open blue diamond, $\left|\mathbf{k}\right|=\sqrt{3}\pi /2$) on $\Gamma \text{−}M$ lines are the would-be magnetic Bragg peaks for zero-flux spin liquid with $A_2/A_1\sim 1/2$, namely, peaks in dynamical spin susceptibility at low frequency around spin gap. (b) Hexagon is the Brillouin zone of the original lattice. Dash rectangle is the reduced Brillouin zone for the $\pi$-flux ansatz. In the $\pi$-flux state, bosons can condense at the momenta indicated by the filled red small triangles and produce magnetic Bragg peaks with possible wave vectors indicated by the open blue small hexagons.