Field-angle anisotropy of proximate Kitaev systems under an in-plane magnetic field

We have investigated the field-angle behaviors of magnetic excitations under an in-plane magnetic field for proximate Kitaev systems. By employing the exact diagonalization method in conjunction with the linear spin wave theory, we have demonstrated that the magnetic excitation gap in the polarized phase is determined by the magnon excitation at $M$ points and has a strong anisotropy with respect to the field direction in the vicinity of the critical field limit. The specific heat from this magnon excitation bears qualitatively the same anisotropic behaviors as the expected one for the non-Abelian spin liquid phase in the Kitaev model and the experimentally observed one of the intermediate phases in $\alpha \text{−}{\mathrm{RuCl}}_3$.

Figure 1

(a) The connection of three types of bond-dependent Ising interactions, and coordinate axes of local spins ($x$, $y$, $z$) and global honeycomb lattice ($a$, $b$, $c$) in the Kitaev model. The Ising directions in the $x$-, $y$-, and $z$-type bonds drawn with green, blue, and red lines are parallel to the $x$, $y$, and $z$ coordinate axes, respectively. The anticipated field-angle variations of (b) Majorana gap of the Kitaev model and (c) magnon gap in the polarized phase of proximate Kitaev systems under an in-plane field. $\phi$ is the angle between the in-plane field and the $a$ axis. The radius of curves refers to the size of gap. The dark violet (sea green) curve denotes the Chern number $C=+1$ ($-1$). When the gap is determined at the $M_1$, $M_2$, and $M_3$ points (see the Brillouin zone in inset), the curve is drawn with green, red, and blue colors in (c), respectively.

Figure 2

(a) Schematic diagram of a periodic 24-site cluster. A and B refer to two base sites of a honeycomb lattice, respectively. The bonds between first, second, and third nearest neighboring (NN) spins are drawn with solid, fine dotted, and thick dotted lines, respectively. $x$, $y$, and $z$ types of bonds are highlighted by green, blue, and red colors, respectively. $x_1$, $x_2$, and $x_3$ mean the $x$-type first, second, and third NN bonds, respectively. Static spin structure factors of the ground state $\langle {\mathbf{S}}_{-\mathbf{q}}·{\mathbf{S}}_{\mathbf{q}}\rangle$ at $\mathbf{q}=\mathrm{\Gamma }$, $M_1$, and $M_2$ points under the (b) $a$- and (c) $b$-axis fields ($\phi =0^{\circ }$ and $\phi ={90}^{\circ }$). Solid black curves in (b) and (c) represent the second derivatives of the ground state energy ($-d^2{E}_g/d{h}^2$) with respect to the field strength for given field direction. The critical field strength of the phase transition from the zigzag order to the polarized phase are indicated with vertical dotted line. Dynamical spin structure factors (DSSFs) ${\chi }_{\mathbf{q}}\left(\omega \right)$ of the $K\text{−}\mathrm{\Gamma }\text{−}{J}_3$ model with $J_3/\left|K\right|=\mathrm{\Gamma }/\left|K\right|=0.1$ and $K<0$ at (d) $\mathrm{\Gamma }$, (e) $M_1$, and (f) $M_2$ points as a function of the field strength $h$ when a magnetic field is along the $a$ axis. The $\mathrm{\Gamma }$, $K_1$, $M_1$, and $M_2$ points are defined in Fig. 1. DSSF at (g) $M_1$ and (h) $M_2$ points as a function of the field angle $\phi$, the angle between the in-plane field and the $a$ axis, when $g{\mu }_{B}h/\left|K\right|=0.12$. All results are calculated by the ED method with a periodic 24-site cluster. The circle data represent the lowest seven excitation energies calculated by the thick-restarted Lanczos method [49].

Figure 3

(a) Theoretical specific heat $C_m$ of the $K\text{−}\mathrm{\Gamma }\text{−}{J}_3$ model with $J_3/\left|K\right|=\mathrm{\Gamma }/\left|K\right|=0.1$ and $K<0$ when $g{\mu }_{B}h/\left|K\right|=0.12$ under the $a$- and $b$-axis fields ($\phi =0^{\circ }$ and $\phi ={90}^{\circ }$). (b) $\phi$ variation of the specific heat for $g{\mu }_{B}h/\left|K\right|=0.12$ and 0.2 at $k_{B}T/\left|K\right|=0.008$ indicated by red arrow in (a). Results are calculated by the finite-temperature Lanczos method (FTLM) [50, 51] with a periodic 24-site cluster. Dotted lines in (a) and error bars in (b) represent standard deviations of FTLM calculation (see Appendix pp1 for details).

Figure 4

Expectation values of the order parameters of the zigzag order and polarized phase of the classical $K\text{−}\mathrm{\Gamma }\text{−}{J}_3$ model with $\mathrm{\Gamma }/\left|K\right|=J_3/\left|K\right|=0.1$ and $K<0$ under the (a) $a$- and (b) $b$-axis fields ($\phi =0^{\circ }$ and $\phi ={90}^{\circ }$). They are calculated with the classical Monte Carlo method at $k_{B}T/\left|K\right|=0.02$. The lowest magnon energies at the $M_1$ and $M_2$ points in the polarized phase are presented with orange and red lines, respectively. Vertical dotted lines represent the phase boundary.

Figure 5

Magnon bands and magnon specific heats of the $K\text{−}\mathrm{\Gamma }\text{−}{J}_3$ model with $J_3/\left|K\right|=0.1$, $\mathrm{\Gamma }/\left|K\right|=0.1$, and $K<0$ when $g{\mu }_{B}h/\left|K\right|=0.3048$. (a) Magnon bands under the $a$- and $b$-axis fields ($\phi =0^{\circ }$ and $\phi ={90}^{\circ }$). (b) Magnon bands as a function of $\phi$ at the $M_1$, $M_2$, and $M_3$ points. (c) Specific heats as a function of $T$ under the $a$- and $b$-axis fields. Zoom-in of the low-temperature specific heat is presented in the inset of (c). (d) Specific heats as a function of $\phi$ when $k_{B}T/\left|K\right|=0.02$ indicated by red arrow in the inset of (c). All data are calculated by the linear spin-wave theory.

Figure 6

Second derivative of the ground state energy with respect to the field strength ($-d^2{E}_g/d{h}^2$) for models 1 to 7 in Table 1 under (a) $a$- and (b) $b$-axis fields ($\phi =0^{\circ }$ and $\phi ={90}^{\circ }$). The ground state energy is obtained with the ED calculation of the periodic 24-site cluster shown in Fig. 2.

Figure 7

Dynamical spin structure factors ${\chi }_{M_1}\left(\omega \right)$ under the $a$-axis field ($\phi =0^{\circ }$) and ${\chi }_{M_2}\left(\omega \right)$ under the $b$-axis field ($\phi ={90}^{\circ }$), and the magnon dispersions and magnon specific heats under the $a$- and $b$-axis fields at the critical field strength for (a)–(d) model 1, (e)–(h) model 2, (i)–(l) model 3, and (m)–(p) model 4 in Table 1. The critical field strength, at which the magnon bands condense under the $b$-axis field, is $g{\mu }_{B}h/\left|K\right|\approx 0.343$ for model 1, 0.242 for model 2, 0.204 for model 3, and 0.293 for model 4. Circle data in (a), (b), (e), (f), (i), (j), (m), and (p) represent the lowest seven excitation energies calculated by the thick-restarted Lanczos method [49].

Figure 8

Dynamical spin structure factors ${\chi }_{M_1}\left(\omega \right)$ under the $a$-axis field ($\phi =0^{\circ }$) and ${\chi }_{M_2}\left(\omega \right)$ under the $b$-axis field ($\phi ={90}^{\circ }$), and the magnon dispersions and magnon specific heats under the $a$- and $b$-axis fields at the critical field strength for (a)–(d) model 5, (e)–(h) model 6, and (i)–(l) model 7 in Table 1. The critical field strength, at which the magnon bands condense under the $b$-axis field, is $g{\mu }_{B}h/\left|K\right|\approx 0.202$ for model 5, 0.178 for model 6, and 0.106 for model 7. Circle data in (a), (b), (e), (f), (i), and (j) represent the lowest seven excitation energies calculated by the thick-restarted Lanczos method [49].

Figure 9

Dynamical spin structure factors (a) ${\chi }_{M_1}\left(\omega \right)$ and (b) ${\chi }_{M_2}\left(\omega \right)$ as a function of a field angle $\phi$ in the $K\text{−}{J}_3$ model with $J_3/\left|K\right|=0.1$, $K<0$, and $g{\mu }_{B}h/\left|K\right|=0.16$. (c) Magnon dispersions and (d) magnon specific heats under both $a$- and $b$-axis fields ($\phi =0^{\circ }$ and $\phi ={90}^{\circ }$) at the critical field strength ($g{\mu }_{B}h/\left|K\right|=0.15$) in the $K\text{−}{J}_3$ model. Circle data in (a) and (b) represent the lowest seven excitation energies calculated by the thick-restarted Lanczos method [49].