Algebraic vortex liquid theory of a quantum antiferromagnet on the kagome lattice

There is growing evidence from both experiment and numerical studies that low half-odd integer quantum spins on a kagome lattice with predominant antiferromagnetic near-neighbor interactions do not order magnetically or break lattice symmetries even at temperatures much lower than the exchange interaction strength. Moreover, there appears to be a plethora of low-energy excitations, predominantly singlets but also spin carrying, which suggests that the putative underlying quantum spin liquid is a gapless “critical spin liquid” rather than a gapped spin liquid with topological order. Here, we develop an effective field theory approach for the spin-$\frac{1}{2}$ Heisenberg model with easy-plane anisotropy on the kagome lattice. By employing a vortex duality transformation, followed by a fermionization and flux smearing, we obtain access to a gapless yet stable critical spin liquid phase, which is described by $(2+1)$-dimensional quantum electrodynamics $\left({\mathrm{QED}}_3\right)$ with an emergent SU(8) flavor symmetry. The specific heat, thermal conductivity, and dynamical structure factor are extracted from the effective field theory, and contrasted with other theoretical approaches to the kagome antiferromagnet.

Figure 1

(Color online) (Top) The quantum spin model with easy-plain anisotropy, Eq. (2), on the kagome lattice (thick line) supplemented by an extra site at the center of each hexagon. There is a half-integer spin for each kagome site whereas integer spins are placed at each center of hexagons. Four spins in a unit cell are labeled by $j=1,2,3,4$. For this triangular extension of the kagome lattice, the dual lattice is the honeycomb lattice indicated in the upper left-hand part. (Bottom) The $q=0$ state (left-hand side) and the $\sqrt{3}\times \sqrt{3}$ state (right-hand side). Positive-negative vorticities (chiralities) for each triangle are denoted by +/−.

Figure 2

(Color online) The flux-smeared mean-field background for fermionized vortices. Each hexagon is threaded by either zero or $\pi$-flux. (Hexagons with zero-flux are specified by “0” whereas all the other hexagons are pierced by $\pi$-flux.) A convenient choice of a gauge is also shown: for links labeled by ↔, we assign the Peierls phase factor $e^{-i{a}_{\mathbf{x}{\mathbf{x}}^{\prime }}^0}=e^{i\pi }$. The unit cell in this gauge consists of 16 sites as labeled in the figure. For links represented by a solid line we assign hopping amplitude $t(=1)$ whereas for links denoted by a broken line we assign hopping $t^{\prime }$. We take × in the figure as an origin.

Figure 3

(Color online) The first Brillouin zone for the flux-smeared mean-field ansatz and the location of the nodes ${\mathbf{Q}}_{1,\dots ,4}$. The hexagonal physical Brillouin zone is also presented. Small hexagons are a guide for the eyes.

Figure 4

(Color online) Summary of the main characterizations of the kagome AVL phase. The low-energy excitations are located at wave vectors $\mathbf{0}$, $\pm \mathbf{Q}$, ${\mathbf{M}}_{1,2,3}$, and $\pm {\mathbf{P}}_{1,2,3}$ in the physical Brillouin zone of the kagome lattice, where $\mathbf{Q}=(\pi ∕3,\pi ∕\sqrt{3})$; ${\mathbf{M}}_1=\mathbf{(}\pi ∕2,-\pi ∕\left(2\sqrt{3}\right)\mathbf{)}$, ${\mathbf{M}}_2=\mathbf{(}\pi ∕2,\pi ∕\left(2\sqrt{3}\right)\mathbf{)}$, ${\mathbf{M}}_3=(0,\pi ∕\sqrt{3})$; ${\mathbf{P}}_1=(\pi ∕3,0)$, ${\mathbf{P}}_2=\mathbf{(}-\pi ∕6,\pi ∕\left(2\sqrt{3}\right)\mathbf{)}$, and ${\mathbf{P}}_3=\mathbf{(}-\pi ∕6,-\pi ∕\left(2\sqrt{3}\right)\mathbf{)}$. Both $S^z$ and $S^{+}$ exhibit power law correlations at all these momenta, Eqs. (19, 22). The $S^z$ correlations are “enhanced” $({\eta }_z\approx 2.46)$ for the subset $\pm \mathbf{Q}$, $\pm {\mathbf{P}}_{1,2,3}$ and not enhanced $({\eta }_z=3)$ for the remaining wave vectors. On the other hand, all $S^{+}$ correlations are characterized by the same exponent ${\eta }_{\pm }\approx 3.24$.

Figure 5

(Color online) (Top) Lower edge of the vortex-antivortex continuum along several cuts in the Brillouin zone as shown in Fig. 4 for parameter values $t^{\prime }∕t=0.5$ and $t^{\prime }∕t=0.2$ from the higher to lower curve. (Bottom) Spin-wave excitations, Eq. (17), of the kagome $XY$ antiferromagnet with the nearest-neighbor exchange $J$ only along the same cuts in the Brillouin zone.