Topological spinon bands and vison excitations in spin-orbit coupled quantum spin liquids

Spin liquids are exotic quantum states characterized by the existence of fractional and deconfined quasiparticle excitations, referred to as spinons and visons. Their fractional nature establishes topological properties such as a protected ground-state degeneracy. This work investigates spin-orbit coupled spin liquids where, additionally, topology enters via nontrivial band structures of the spinons. We revisit the ${\mathbb{Z}}_2$ spin-liquid phases that have recently been identified in a projective symmetry-group analysis on the square lattice when spin-rotation symmetry is maximally lifted [J. Reuther et al., Phys. Rev. B 90, 174417 (2014)]. We find that in the case of nearest-neighbor couplings only, ${\mathbb{Z}}_2$ spin liquids on the square lattice always exhibit trivial spinon bands. Adding second-neighbor terms, the simplest projective symmetry-group solution closely resembles the Bernevig-Hughes-Zhang model for topological insulators. Assuming that the emergent gauge fields are static, we investigate vison excitations, which we confirm to be deconfined in all investigated spin phases. Particularly, if the spinon bands are topological, the spinons and visons form bound states consisting of several spinon-Majorana zero modes coupling to one vison. The existence of such zero modes follows from an exact mapping between these spin phases and topological $p+ip$ superconductors with vortices. We propose experimental probes to detect such states in real materials.

Figure 1

Spinon-band structure of the nearest-neighbor Hamiltonian in Eq. (20) using the parameters $\alpha =\beta =\gamma =1$ and $\delta =2$. The $h_{\mathit{k}}^1$ bands ($h_{\mathit{k}}^2$ bands) are plotted in red (blue) color. (a) Spinon bands for periodic boundary conditions in $x$ and $y$ directions. (b) Band structure of the $h_{\mathit{k}}^1$ block for a cylinder edge along the $x$ direction. (c) Band structure of the $h_{\mathit{k}}^2$ block for a cylinder edge along the $x$ direction. Note that for an edge along one of the lattice directions, only one block shows topologically trivial edge states.

Figure 2

Top: phase diagram of the topological spinon bands for the mean-field Hamiltonian in Eq. (23) using $\alpha =1$. Chern numbers $n_{\uparrow }$ of the block $h_{\mathit{k}}$ are indicated in each phase. Note that the phase diagram is independent of $\gamma$. (a)–(d) Spinon band structures for $\beta =0,1,2,3$ and $\alpha =\gamma =1$ where the upper plots correspond to a cylinder geometry with an edge along the $x$ axis and the lower plots are for a torus geometry. All spinon bands are doubly degenerate with respect to the spin degree of freedom.

Figure 3

Effect of a local gauge transformation ${\mathcal{S}}_{\mathit{r}}$ [Eq. (32)] acting on a site $\mathit{r}$ (black dot): all bond variables ${\sigma }_{\mathit{r}{\mathit{r}}^{\prime }}^z$ are flipped (${\sigma }_{\mathit{r}{\mathit{r}}^{\prime }}^z\to -{\sigma }_{\mathit{r}{\mathit{r}}^{\prime }}^z$) on links forming a star around $\mathit{r}$ (red lines). (a) For a nearest-neighbor mean-field Hamiltonian, the star consists of four bonds emanating from $\mathit{r}$. When second-neighbor mean-field amplitudes are added (dashed lines), the gauge fields ${\sigma }_{\mathit{r}{\mathit{r}}^{\prime }}^z$ are also defined on diagonal links, such that the gauge transformation acts on all bonds highlighted in (b).

Figure 4

(a) Possible gauge-field configuration for a pair of separated visons (marked by stars) in the case of nearest-neighbor mean-field models. The full red lines indicate bonds with ${\sigma }_{\mathit{r}{\mathit{r}}^{\prime }}^z=-1$ while ${\sigma }_{\mathit{r}{\mathit{r}}^{\prime }}^z=1$ otherwise. Gauge fields ${\sigma }_{\mathit{r}{\mathit{r}}^{\prime }}^z=-1$ occur on all bonds crossing the “gauge string” (dashed red line). The plaquettes at which the visons reside are threatened by a finite local flux. (b) Vison pair-excitation energy $E\left(d\right)$ as a function of the distance $d$ for the nearest-neighbor PSG solution in Eq. (20) using the parameters $\alpha =\beta =\gamma =1$, $\delta =2$. Here, $d$ is defined in units of the nearest-neighbor lattice spacing and the separation is along a lattice direction. For comparison, the spinon bulk gap $\mathrm{\Delta }$ is indicated in the figure. (c) Same as (a) but with additional second-neighbor mean-field amplitudes. In this case also diagonal bond variables ${\sigma }_{\mathit{r}{\mathit{r}}^{\prime }}^z$ need to be flipped along the gauge string. (d) Vison potentials $E\left(d\right)$ for the BHZ-type model [see Eq. (23)] in the phases with Chern numbers $n_{\uparrow }=1$ (blue line with parameters $\alpha =\beta =\gamma =1$) and $n_{\uparrow }=2$ (yellow line with parameters $\alpha =\gamma =1$, $\beta =3$).

Figure 5

(a)–(c) Excitation energies $\epsilon \left(d\right)$ of in-gap states as a function of the vison-pair separation $d$ for (a) the nearest-neighbor spin liquid in Eq. (20), (b) the BHZ-type spin liquid in the phase with $n_{\uparrow }=1$ [see Eq. (23)], and (c) the BHZ-type spin liquid in the phase with $n_{\uparrow }=2$. The corresponding mean-field parameters are the same as in Fig. 4. The plots in (b) and (c) use a logarithmic energy axis and only show the positive part of the spectrum (the negative part is an exact mirror image). Note that all depicted data points are degenerate with respect to the spin degree of freedom. In (b) we find indications for an oscillating modulation as indicated by a fit of $\epsilon \left(d\right)$ to a function $\sim e^{-ad}cos(bd+\varphi )$ with fit parameters $a$, $b$, $\varphi$ (blue line). The quality of the fit becomes worse at larger $d$ (see dashed blue line). (d) Real-space probability distribution of the spinon wave function corresponding to the zero-energy mode of the $n_{\uparrow }=1$ BHZ-type system in (b). The locations of the vison cores are given by the red spots where the wave function is sharply peaked.

Figure 6

Possible gauge-field configuration ${\sigma }_{\mathit{r}{\mathit{r}}^{\prime }}^z$ for a single vison, located at the origin (marked by a red star). Blue (red) bonds correspond to ${\sigma }_{\mathit{r}{\mathit{r}}^{\prime }}^z=1$ (${\sigma }_{\mathit{r}{\mathit{r}}^{\prime }}^z=-1$). Bonds with ${\sigma }_{\mathit{r}{\mathit{r}}^{\prime }}^z=-1$ form a string along the $\phi =0$ line. For large $\mathit{r},{\mathit{r}}^{\prime }\gg 1$, the angle ${\phi }_{\mathit{r}}-{\phi }_{{\mathit{r}}^{\prime }}$ between two coupled lattice sites $\mathit{r}$, ${\mathit{r}}^{\prime }$ vanishes (see text for details).

Figure 7

Fermionic spinon density of states $N\left(\epsilon \right)$ in the bulk gap for a dilute gas of randomly arranged visons. The vison density is indicated for each curve. Using a system size of $100\times 100$ lattice sites, $N\left(\epsilon \right)$ counts the number of states in each energy interval of length 0.01. All results are averaged over 100 different realizations. The thick gray line indicates the onset of the bulk gap. The data in (a)–(c) correspond to the nearest-neighbor spin liquid, the BHZ-type model with $n_{\uparrow }=1$, and the BHZ-type model with $n_{\uparrow }=2$, respectively, with the same parameters as in Fig. 4. The insets in (b) and (c) show the density of states in the low-energy region.