Edge magnetization and spin transport in an SU(2)-symmetric Kitaev spin liquid

We investigate the edge magnetism and the spin transport properties of an SU(2)-symmetric Kitaev spin liquid (KSL) model put forward by Yao and Lee [Phys. Rev. Lett. 107, 087205 (2011)] on the honeycomb lattice. In this model, the spin degrees of freedom fractionalize into a ${\mathbb{Z}}_2$ static gauge field and three species of either gapless (Dirac) or gapped (chiral) Majorana fermionic excitations. We find that, when a magnetic field is applied on a zigzag edge, the Dirac KSL exhibits a nonlocal magnetization associated with the existence of zero-energy edge modes. The application of a spin bias $V={\mu }_{\uparrow }-{\mu }_{\downarrow }$ at the interface of the spin system with a normal metal produces a spin current into the KSL, which depends as a power law on $V$, in the zero-temperature limit, for both Dirac and chiral KSLs, but with different exponents. Lastly, we study the longitudinal spin Seebeck effect, in which a spin current is driven by the combined action of a magnetic field perpendicular to the plane of the honeycomb lattice and a thermal gradient at the interface of the KSL with a metal. Our results suggest that edge magnetization and spin transport can be used to probe the existence of charge-neutral edge states in quantum spin liquids.

Figure 1

Schematic representation of the finite honeycomb lattice containing the degrees of freedom of the Yao-Lee model. Throughout this work, we consider a slab geometry with zigzag edges, such that the lattice obeys periodic boundary conditions in the $\widehat{\mathit{x}}$ direction, and $L_{\perp }$ and $L_{\parallel }$ are defined as the unit-cell dimensions along the $\widehat{\mathit{x}}$ and $\widehat{\mathit{y}}$ axes, respectively. As depicted in the inset, the upper edge contains only sites of the A sublattice (blue circles, “$•$”), whereas the lower edge has sites of the B sublattice (red circles, “$•$”). The two vectors ${\mathbf{n}}_1=\frac{\sqrt{3}a}{2}(1,\sqrt{3}), {\mathbf{n}}_2=\frac{\sqrt{3}a}{2}(-1,\sqrt{3})$ refer to the lattice vectors. The allowed nearest-neighbor spin-exchange interactions $J_{x,y,z}$ are indicated by the letters $x, y$, and $z$.

Figure 2

Energy dispersion curves $E_{\ell }\left(k_{\perp }\right)$ (in units of $J_K$) as a function of the $k_{\perp }$ momentum for the KSL on the honeycomb lattice with zigzag surfaces along the $\widehat{\mathit{y}}$ direction: (a) in the absence of the chiral interaction (i.e., $J_{\chi }=0$) and (b) for a finite chiral interaction (i.e., $J_{\chi }=0.3J_K$). Here, the lattice constant $a$ has been set to unity.

Figure 3

Magnetization per unit cell $\langle S_{\ell }^z\left(B_z\right)\rangle /L_{\perp }$ in the limit of $T\to 0$ for the sublattices $\ell =(b,L_{\parallel }-y+1)$ parallel to the $\widehat{\mathit{x}}$ direction. This magnetization is induced by a local magnetic field acting on the spin-$1/2$ magnetic moments at the upper zigzag edge of the honeycomb lattice shown in Fig. 1. (a) and (b) refer to the behavior exhibited by the Dirac KSL, whereas (c) and (d) refer to the behavior of the chiral KSL. In (a) and (c), we set $B_z=0.5J_K$. Note that a nonlocal (long-distance) edge magnetization appears just for the spin system described by the Dirac KSL. These results were obtained by substituting the momentum integral in Eq. (11) by a sum over a mesh with 1500 points.

Figure 4

Schematic representation of the experimental setup used for the injection of spin currents into the KSL described by the Yao-Lee model. In the presence of strong spin-orbital coupling or disordered skew scattering, an electronic current density ${\mathbf{J}}_{\text{C}}$ flowing in a metallic plate produces a transverse spin current density ${\mathbf{J}}_{\text{S}}$ at the KSL-metal interface. As a result, the spin current is injected into the KSL and carried here by the Majorana fermionic excitations, originated from the fractionalization of the magnetic moments. This unusual spin current can be detected by measuring a finite charge current ${\mathbf{J}}_{\text{C}}$ produced in a second metallic plate by the inverse spin Hall effect.

Figure 5

Zero-temperature limit of the spin current $I_{\text{spin}}\left[V\right]$ [in units of $\pi J^2{L}_{\perp }\nu \left({\epsilon }_F\right)J_K/4$] for chiral interactions $J_{\chi }=0,0.05J_K,0.10J_K,0.15J_K$. The spin current for the Dirac KSL behaves as a linear function of the spin bias $V$. On the other hand, a finite $J_{\chi }$ (i.e., chiral KSL) leads to the cubic dependence of $I_{\text{spin}}\left[V\right]$ on the spin bias $V$ for $V\ll J_K$.

Figure 6

Illustration of the experimental setup for the observation of the LSS effect on an SU(2)-symmetric KSL system described by the Yao-Lee model. In order for this effect to occur, the spin system and a metal are placed in a region with finite magnetic field $\mathit{B}=B_z\widehat{\mathit{z}}$ and kept at different temperatures $T_s$ and $T_m$, respectively. As a result, a finite spin current $I_{\text{spin}}$ can be driven at their interface and transformed in the metal into an electric current with an associated voltage $\mathcal{E}$ by means of the inverse spin Hall effect.

Figure 7

Spin current $I_{\text{spin}}\left[B_z\right]$ generated by the LSS effect in a system consisting of a normal metal in contact with a Dirac or chiral KSL described by the Yao-Lee model. For magnetic fields $|B_z|\ll J_K, I_{\text{spin}}\left[B_z\right]$ exhibits a continuous change in the flow direction as the chiral interaction $J_{\chi }$ varies from zero to positive finite values. This feature can be used to identify the topological nature of KSLs, since the most relevant contribution to the spin current comes from the edge states. Here, $I_{\text{spin}}\left[B_z\right]$ is given in units of $2\sqrt{2}{J}^2{L}_{\perp }{\chi }_0{J}_K$, the metal spin relaxation is ${\tau }_s=1/\left(2J_K\right)$, and metal and KSL temperatures are set to $T_m=0.1J_K$ and $T_s=0$, respectively.