Kitaev model in a magnetic field: stable emergent structure, degenerate classical ground states, and reentrant topology

Abstract

We have studied the anti-ferromagnetic Kitaev model on a honeycomb lattice under the Zeeman field, using an extensive Majorana mean-field analysis. When the magnetic field is along a specific Cartesian axis, we find that the emergent fields exhibit direction-dependent stabilization up to a certain critical strength of the external field. For a conical magnetic field, the characteristics of the emergent intermediate state are elusive. Our mean-field analysis reveals the existence of two distinct phases in the intermediate region. First, the system enters a disordered phase, where emergent-field densities converge to random values, and the Chern number is ill-defined. The magnitude of magnetization also fluctuates and remains less than unity, indicating a strong quantum effect. In the second phase, emergent-field densities attain vanishingly small values. In this phase, the magnetization components fluctuate heavily, but the magnitude of the magnetization vectors becomes unity, indicating highly degenerate classical ground states. We perform exact diagonalization calculations that qualitatively support some of the mean-field results. We extend our study to the anisotropic limit of the Kitaev coupling parameters. When the couplings are beyond the triangular inequality, the pure Kitaev model is known to host a topologically trivial gapped quantum spin liquid. We find that, for intermediate strengths of a conical magnetic field, topology shows a reentrant behavior.

Figures (9)

• Figure 1: (a) Representation of the momentum space vectors (magenta arrows) $\hat{q}_1$ and $\hat{q}_2$ in the rhombic Brillouin zone (in black line). The two-site unit cell is shaded in magenta. The dashed gray lines are shown to easily identify the correspondence between the hexagon and rhombus. (b) 24-site Kitaev honeycomb model with periodic boundary conditions used for the exact diagonalization. The direction-dependent Ising-like $x$, $y$, $z$-type interactions are indicated in red, blue, and green colours. One of the plaquette operators $\hat{B}_p$, is shown in magenta shade.
• Figure 2: Mean-field results for AFM Kitaev model with magnetic field in (a-c) [0,0,1] and (d-f) [1,1,1] direction. (a,d) Ground state energy, and its first and second order derivative with respect to magnitude of the magnetic field. (b,e) Components of magnetization on sub-lattices $A$ and $B$. (c,f) Gauge and matter fields' density. The emergent fields $f_\gamma$ and $g_\gamma$ under this MF scheme behave as conjugate to each other. So, for example in the pure AFM Kitaev limit, if they take simultaneously positive or negative values, both provide the same valid solution. Because in this limit we have, $\sigma^z_j\sigma^z_k\less0$$\implies$$\sigma^z_j\sigma^z_k$ = ${\langle} ib_j^zc_j~ib_k^zc_k{\rangle}$ = $-{\langle} ib_j^zb_k^z~ic_jc_k{\rangle}=-gf\less0$$\implies$ sign($g$) sign($f$)$>0$. That's why we have plotted absolute value of $f_\gamma$ and $g_\gamma$. Chern number $C_n$ is mentioned inside the colored top bar in (a,d). Legends for (a,d), (b,e) and (c,f) are indicated inside the legend box of (d), (e), and (f), respectively.
• Figure 3: Exact diagonalization results for magnetic field in (a-d) [0,0,1], and (e-h) [1,1,1] direction. (a,e) Ground state energy and its derivatives with respect to magnetic field strength. MF energy is shown in black line for comparison. (b,f) Magnetization components (not averaged, and explicitly plotted for all sites). (c,g) Expectation value of the flux operator ${\langle} \hat{B}_p{\rangle}$, and quantum distance $d_Q$. The set of twelve flux operator values are not averaged, and they have been plotted one top of another, to explicitly show the fluctuation in between $h_c^{(2)}$ and $h_c^{(3)}$. (d,h) Spin-spin correlations on the bonds: 21-22 ($z$-type), 22-23 ($x$-type), and 23-24 ($y$-type). Legends for (a,e), (b,f) and (c,g) are indicated inside the legend box of (a), (b), and (c), respectively. Correlations with different colors are split into multiple legend boxes, and are kept in (d) and (h).
• Figure 4: (a) Gapless ($B$) and gapped ($A_{x,y,z}$) phases (in absence of magnetic field) in the Kitaev coupling parameter space on the triangle having $\kappa_x+\kappa_y+\kappa_z=1$. The chosen coupling constants when projected on this plane, are indicated by colored dots. (b) Chern number $C_n$ as we vary strength of the magnetic field. The color coding indicates the coupling strengths as indicated on the Kitaev's parameter triangle in the left panel. We have given a slight shift along the vertical axis for visual clarity. The black dots on the field axis indicate the $h_{111}$ values for which the band spectra is shown in FIG. \ref{['figure_anisotropic_kitaev_energy_band']}, for the parameter values $\kappa_x=\kappa_y=1$, $\kappa_z=2.5$ (red line here).
• Figure 5: The eight band energy spectra in momentum space for $\kappa_x=\kappa_y=1$, $\kappa_z=2.5$ (corresponds to red line in FIG. \ref{['figure_anisotropic_kitaev_topology']}). Various strengths of the magnetic field are $h_{111}/\sqrt{3}$ = (a) 0.58, (b) 1.29, (c) 1.73, (d) 2.19, (e) 2.60, indicated by different black dots on the horizontal axis of FIG. \ref{['figure_anisotropic_kitaev_topology']}b. The system makes topological transition at (b) and (d).