Heisenberg-Kitaev model in a magnetic field: $1/S$ expansion

The exact solution of Kitaev's spin-$1/2$ honeycomb spin-liquid model has sparked an intense search for Mott insulators hosting bond-dependent Kitaev interactions, of which ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ and $\alpha \text{−}\mathrm{Ru}{\mathrm{Cl}}_3$ are prime examples. Subsequently, it has been proposed that also spin-1 and spin-$3/2$ analogs of Kitaev interactions may occur in materials with strong spin-orbit coupling. As a minimal model to describe these Kitaev materials, we study the Heisenberg-Kitaev Hamiltonian in a consistent $1/S$ expansion, with $S$ being the spin size. We present a comprehensive study of this model in the presence of an external magnetic field applied along two different directions, [001] and [111], for which an intricate classical phase diagram has been reported. In both settings, we employ spin-wave theory in a number of ordered phases to compute phase boundaries at the next-to-leading order in $1/S$ and show that quantum corrections substantially modify the classical phase diagram. More broadly, our work presents a consistent route to investigate the leading quantum corrections in spin models that break spin-rotational symmetry.

Figure 1

Spin configurations of ordered phases of the HK model in a $\left[111\right]$ field, projected onto a plane perpendicular to [111]. The respective magnetic unit cells are shown in dashed lines. Unequal lengths of the projected spins in the canted zigzag, canted stripy, and AF star configurations reflect the occurrence of nonuniform canting.

Figure 2

Magnetization $m_h$ as a function of field $h$ in the HK model with $J=Acos\phi$ and $K=2Asin\phi$ in a magnetic field $\mathbf{h}\parallel \left[001\right]$, at leading (black) and next-to-leading (blue) order in $1/S$ for $S=1/2$ and different values of $\phi$. To aid the comparison, the horizontal axes have been rescaled by the respective classical critical fields $h_{\mathrm{c}0}$. Red arrows highlight an unphysical saturation of the magnetization curve, suggesting that, except in the Heisenberg limit (a), phase transitions occur below the classical critical field in (b)–(d). Green arrows indicate the positions of the corrected critical fields according to Secs. 2e and 2f.

Figure 3

(a) Leading-order and (b) NLO contributions to the magnon gap in the polarized phase for $\phi =0.3\pi$ (blue) and $\phi =0.7\pi$ (red) with $\mathbf{h}\parallel \left[001\right]$. The two values of $\phi$ are representative for the cases of vanishing (blue) and finite (red) instability wave vectors $\mathbf{Q}$, respectively. These different behaviors justify the need for different conditions to determine the expansion of $1/h_{\mathrm{c}}$.

Figure 4

$\mathcal{O}(1/S)$ coefficient $c_1$ in the expansion of the inverse of the critical field, see Eq. (26), as a function of $\phi$ in the HK model with couplings $J=Acos\phi$ and $K=2Asin\phi$ in a magnetic field $\mathbf{h}\parallel \left[001\right]$. Results using Eq. (16) in the ordered phases (blue open circles) are fully equivalent within our numerical precision with those that follow from applying Eq. (22) for instability wave vector $\mathbf{Q}=0$ and Eq. (23) for $\mathbf{Q}\ne 0$ in the disordered phase (black diamonds). The inset shows the locations in the first Brillouin zone of the various instability wave vectors corresponding to different intervals of $\phi$ (${\text{M}}_1$, ${\text{M}}_3$, and $\mathrm{\Gamma }$) and different field directions ($\mathrm{K}$, ${\mathrm{K}}^{\prime }$, and ${\mathrm{M}}_2$). The blue line is a guide to the eye. Green dots at $\phi =0$ and $\phi \approx 0.83\pi$ denote points where the leading-order correction to the critical field vanishes.

Figure 5

Phase diagram of the HK model at $T=0$ with $J=Acos\phi$ and $K=2Asin\phi$ in a magnetic field $\mathbf{h}\parallel \left[001\right]$, at next-to-leading order in $1/S$ for (a) $S=2$, (b) $S=3/2$, (c) $S=1$, and (d) $S=1/2$. Dot-dashed and solid lines mark continuous and first-order phase transitions, respectively, whereas the light dotted lines represent the classical phase boundaries, which formally correspond to the limit $S\to \infty$ [55]. The yellow dots added to the $S=1$ and $S=1/2$ diagrams show the $h=0$ phase boundaries according to (c) an infinite density matrix renormalization group study [45] and (d) 24-site exact diagonalization results [26]. In both cases, the red stripes below the horizontal axis indicate the domains of spin liquid phases. Note that the AF Kitaev spin liquid near $\phi =\pi /2$ is expected to cover a sizable field range [16, 55], which is not contemplated by our semiclassical expansion [73].

Figure 6

Magnetization per site $m_h$ in units of $S$ as a function of the field $h$ in units of $h_{\mathrm{c}0}$ in the HK model with $J=Acos\phi$, $K=2Asin\phi$, and a magnetic field $\mathbf{h}\parallel \left[001\right]$, at NLO in $1/S$. Left panels: $\phi =0.4\pi$ above Néel phase for (a) $S=3/2$, (b) $S=1$, (c) $S=1/2$. Right panels: $\phi =0.7\pi$ above zigzag phase for (d) $S=3/2$, (e) $S=1$, (f) $S=1/2$. The vertical dashed lines mark the positions of the $1/S$-corrected and classical critical fields, $h_{\mathrm{c}}$ and $h_{\mathrm{c}0}$, respectively. Red curves correspond to the partially polarized phase, whereas blue curves were obtained for the ordered phases below. The dashed portions of the blue curves should therefore be discarded, for they lie in the interval $\left[h_{\mathrm{c}},h_{\mathrm{c}0}\right]$, which is now occupied by the partially polarized phase. Still, one cannot extend the red curve below $h_{\mathrm{c}0}$ because the classical polarized state is unstable in this region.

Figure 7

$\mathcal{O}(1/S)$ coefficient $c_1$ in the expansion of the inverse of the critical field, see Eq. (26), as a function of $\phi$ in the HK model with couplings $J=Acos\phi$ and $K=2Asin\phi$ in a magnetic field $\mathbf{h}\parallel \left[111\right]$, obtained from the spin-wave calculation in the ordered phase [Eq. (16)]. The blue line is a guide to the eye. Green dots at $\phi =0$ and $\phi \approx -0.47\pi$ denote points where the leading-order correction to the critical field vanishes. Gaps in the data correspond to intervals of $\phi$ in which the transition to the polarized phase is discontinuous.

Figure 8

Phase diagrams of the HK model as in Fig. 5 but now for a magnetic field $\mathbf{h}\parallel \left[111\right]$. Note that the dot-dashed lines representing the critical fields fall below the lower classical boundaries of the vortex and AF vortex phases for small $S$ and an increasing range of $\phi$ values, leading to a complete disappearance of the AF vortex phase and a strong suppression of the vortex order for $S\le 1$. Note also that the AF Kitaev spin liquid near $\phi =\pi /2$ is expected to cover a sizable field range [16, 55], which is not contemplated by our semiclassical expansion [73].

Figure 9

Spin-wave-theory results for $\phi =0.3\pi$ and $\mathbf{h}\parallel \left[111\right]$. (a) Corrections to the classical canting angle for spins in the two sublattices of the canted Néel phase. Although the individual angles diverge in the opposing limits of $h\to 0$ and $h\to h_{\mathrm{c}0}$, the ratio of $\delta {\theta }_1$ to $\delta {\theta }_2$ shows that the spins tend, respectively, to an antiparallel and a parallel state. (b) Magnetization curves in leading (black) and NLO (blue with markers) order for $S=1/2$. Note that the divergence of $\delta {\theta }_1$ and $\delta {\theta }_2$ as $h\to 0$ does not manifest itself in the magnetization.

Figure 10

Left panels: NLO contribution to the ground-state energies of the vortex and AF vortex phases of the HK model in a $\left[111\right]$ field from LSW theory, illustrating the order-by-disorder mechanism. The blue lines are fits of cosine functions to guide the eyes. Right panels: Projections of the ${120}^{\circ }$ spin configurations selected by quantum fluctuations onto the plane perpendicular to the $\left[111\right]$ direction.

Figure 11

Linear spin-wave spectra in the ordered phases in a $\mathbf{h}\parallel \left[001\right]$ magnetic field for different values of $h$ and $\phi$. The right column of the panel represents data immediately below the classical critical field $h_{\mathrm{c}0}$. The corresponding path along high-symmetry lines of the Brillouin zone is shown in Fig. 12. The plots related to the canted zigzag and canted stripy superimpose the spectra of two degenerate magnetic domains.

Figure 12

Linear spin-wave spectra in several of the ordered phases in a $\left[111\right]$ magnetic field. The corresponding path along high-symmetry lines of the Brillouin zone is shown on the top left. The plots related to the canted zigzag and canted stripy superimpose the spectra of three degenerate magnetic domains. The only spectra that remain gapless under the application of a $\left[111\right]$ magnetic field are those of the canted Néel and vortex phases.

Figure 13

Representation of the domains of the canted stripy and canted zigzag phases used to obtain the results in Eqs. (C24) and (C23). Each of the four magnetic sublattices is labeled by a number from 1 to $N_{\mathrm{s}}=4$.

Figure 14

Nonlinear spin-wave spectra (dots) in the $\left[001\right]$ high-field polarized phase including NLO contributions in $1/S$ for $S=1/2$. The dashed lines correspond to the LSW results. Each row illustrates the effect of lowering the magnetic field from 130% to 100.1% of the classical critical field $h_{\mathrm{c}0}$ at a constant value of $\phi$. Plots (a)–(c) show data for $\phi =0.3\pi$, whereas (d)–(f) and (g)–(i) correspond to $\phi =0.62\pi$ and $\phi =1.687\pi$, respectively. The spectrum acquires a finite and nonzero gap as $h\to h_{\mathrm{c}0}^{+}$ above the canted Néel phase, whereas the gap diverges as one approaches the transition to the canted zigzag or canted stripy phases. This is consistent with the discussion regarding the reduction of the critical field in Sec. 2f.