Quantum spin liquid signatures in Kitaev-like frustrated magnets

Motivated by recent experiments on $\alpha -{\mathrm{RuCl}}_3$, we investigate a possible quantum spin liquid ground state of the honeycomb-lattice spin model with bond-dependent interactions. We consider the $K-\mathrm{\Gamma }$ model, where $K$ and $\mathrm{\Gamma }$ represent the Kitaev and symmetric-anisotropic interactions between spin-1/2 moments on the honeycomb lattice. Using the infinite density matrix renormalization group, we provide compelling evidence for the existence of quantum spin liquid phases in an extended region of the phase diagram. In particular, we use transfer-matrix spectra to show the evolution of two-particle excitations with well-defined two-dimensional dispersion, which is a strong signature of a quantum spin liquid. These results are compared with predictions from Majorana mean-field theory and used to infer the quasiparticle excitation spectra. Further, we compute the dynamical structure factor using finite-size cluster computations and show that the results resemble the scattering continuum seen in neutron-scattering experiments on $\alpha -{\mathrm{RuCl}}_3$. We discuss these results in light of recent and future experiments.

Figure 1

Ground-state energy density, $E_{GS}$, of the $K-\mathrm{\Gamma }$ model, and the corresponding entanglement entropy, $S_E$, for a bipartition of the cylinder into two infinite cylinders, as determined using iDMRG for cylinders with circumference $L=6$. Kinks in $E_{GS}$ are indications of first-order phase transitions. Inset shows the cylinder's three-unit-cell circumference.

Figure 2

Staggered magnetization (top) and static spin structure factor (bottom), calculated using iDMRG. Inset: Brillouin zone with labeled positions of symmetry points.

Figure 3

Transfer-matrix spectrum ${\lambda }_i$ in the Kitaev limit ($\varphi \to 0$) for $L=6$. Plotted is the spectrum $E\left(k\right)=-log\left|\lambda \right|$ with each point corresponding to a single eigenvalue $\lambda$, where $\lambda =\left|\lambda \right|e^{i\eta }$ and $\eta$ is identified with the momentum $k_x$ along the cylinder. $k_y$ denotes the transverse momentum obtained as a quantum number with respect to translation along the cylinder. See Appendix pp1 for more details.

Figure 4

(a) Single Majorana fermion spectrum ${\varepsilon }_c\left(\mathbf{k}\right)$ for the isotropic Kitaev model, $K_x=K_y=K_z=-1$, on a cylinder with a three-unit-cell circumference. (b) Corresponding minimum energy for two-particle excitations, ${\mathrm{\Omega }}_{\min }\left(\mathbf{q}\right)$.

Figure 5

Same as Fig. 3 for different $\varphi$.

Figure 6

Energy density per bond, as obtained using iDMRG, for systems with a three- ($L=6$) and six-unit-cell ($L=12$) circumference.

Figure 7

Allowed momentum cuts for the cylindrical geometry. Symbols indicate the positions of the soft two-particle excitations, as deduced from the TM spectrum. $•$ indicate the leading soft modes at $K, K^{\prime }$, and $M$ points. Additional soft modes are plotted for $\varphi =0.03\pi$ ($▴$) and $\varphi =0.2\pi$ ($\blacksquare$). As $\varphi$ is tuned between these values, we expect that these minima smoothly move from the squares ($\blacksquare$) to the triangles ($▴$).

Figure 8

Band structure of $c$ (red) and $b$ (blue) Majorana fermions, plotted along high-symmetry lines of the Brillouin zone, for several values of $\varphi$ in the two-dimensional thermodynamic limit. The flat bands for $\varphi =0$ are threefold degenerate; for $\varphi /\pi =0.5$, the lower-energy flat bands are twofold degenerate. When $\varphi /\pi \sim 0.2$, there is a finite density of zero-energy $b$ fermion states, around the $K$ (and $\mathrm{\Gamma }$) points in the Brillouin zone. At $\varphi /\pi =0.4$ the $b$ fermion bands are still dispersive, but gapped.

Figure 9

$b+c$ (solid lines) and $c+c$ (dashed lines) two-Majorana-fermion spectrum plotted along the three cuts in momentum space allowed on a cylinder with a circumference of three unit cells (see Fig. 7).

Figure 10

Intensity plot of the dynamic structure factor, $S(\mathbf{q},\omega )$, presented along high-symmetry lines of the Brillouin zone for several values of $\varphi$, as obtained using the ED method. Color mapping indicates relative intensity. Inset: Brillouin zone with labels of symmetry points used in this figure.

Figure 11

Dynamical structure factor, $S(\mathbf{q},\omega )$, integrated over low (top) and high (bottom) frequencies, as obtained from exact diagonalization. The discrete set of peaks in momentum space, resulting from the finite size of the studied cluster, was broadened in order to improve the visualization of the obtained pattern. The black lines depict the boundaries of the first and second Brillouin zones.

Figure 12

Equal-time structure factor $S\left(\mathbf{q}\right)$, obtained using the Majorana mean-field theory.

Figure 13

Dynamic structure factor for $\varphi /\pi =0.2, J_3=0.05$, and several values of the magnetic field $h_{\perp c^{*}}$. Symmetry points labeled as in Fig. 10.

Figure 14

Magnetization and second derivative of the energy as a function of magnetic field, parallel, $h_{\left|\right|c^{*}}$, and perpendicular, $h_{\perp c^{*}}$, to $c^{*}=(1,1,1)$.

Figure 15

Quasienergies $E\left(k\right)=-log{\lambda }_i$ of the regular transfer-matrix spectrum ${\lambda }_i$ compared to analytical excitation energies for single- (thin dotted), two- (solid), four- (dashed), and six-particle excitations (dash-dotted) of the anisotropic $XY$ Heisenberg chain with transverse field. The parameters are $\gamma =0.1, h=0.75$. The analytical results are scaled by a factor $a=1.73$ in order to match the $min\left(E_i\right)$ with $min\left(E_{\text{2p}}\left(k\right)\right)$, where $E_{\text{2p}}\left(k\right)$ is the lower edge of the two-particle excitation band.

Figure 16

Quasienergies $E\left(k\right)$ of the anisotropic Kitaev model with $K_x=-1, K_y=-1.2$, and $K_z=-0.9$. Lines display the analytic results of the single-particle (solid) and two-particle (dashed) lowest-energy excitation [6] on the corresponding momentum cuts: $k_y=0$ (blue), $k_y=2\pi /3$ (red), and $k_y=-2\pi /3$ (green). The analytic results are scaled by a factor $a=0.38$.

Figure 17

Schematic representation of (a) the regular and (b) the mixed transfer matrix $\mathcal{T}$ with translation $T$ applied along the circumference. (c) Dominant eigenvector ${\tilde{\mathrm{\Lambda }}}_0$ of ${\mathcal{T}}^T$ determines the $q$ quantum numbers associated with each bond leg.

Figure 18

Quasienergies $E\left(k\right)$ of the anisotropic $K-\mathrm{\Gamma }$ model for $\varphi =0.1$ (top) and $\varphi =0.35$ (bottom) with an anisotropy $a=1.1$ according to Eq. (A2).

Figure 19

Spin-spin correlations in real space. The open black circles indicate site $i=0$ and the solid circles indicate $i$. The circle size and color represent the magnitude and value of the corresponding correlation $\langle S_i^x{S}_0^x\rangle$. (a) Mean-field results for $\varphi /\pi =0.4$. (b) ED results for $\varphi /\pi =0.5$. In both cases one finds the same spin component correlations only among a subset of all sites.

Figure 20

Left: ${\varepsilon }_c\left(\mathbf{k}\right)$ (dashed) and ${\varepsilon }_b\left(\mathbf{k}\right)$ (solid) for anisotropic mean-field amplitudes $A_x=1, A_y=1.1, A_z=1.2$ and $B_x=0.45, B_y=0.5, B_z=0.55$. Right: Minimum energy for $b+c$ excitations. The spectra are plotted along the allowed momentum cuts, as in Fig. 7.

Figure 21

${\mathrm{\Omega }}_{\min }\left(\mathbf{q}\right)$ for $c+c$ (dashed) and $b+c$ (solid) excitations, plotted along the momentum cuts in Fig. 7 for different values of $\varphi$.