Bilayer Kitaev models: Phase diagrams and novel phases

Kitaev's honeycomb-lattice spin-$1/2$ model has become a paradigmatic example for ${\mathbb{Z}}_2$ quantum spin liquids, both gapped and gapless. Here we study the fate of these spin-liquid phases in differently stacked bilayer versions of the Kitaev model. Increasing the ratio between the interlayer Heisenberg coupling $J_{\perp }$ and the intralayer Kitaev couplings $K^{x,y,z}$ destroys the topological spin liquid in favor of a paramagnetic dimer phase. We study phase diagrams as a function of $J_{\perp }/K$ and Kitaev coupling anisotropies using Majorana-fermion mean-field theory, and we employ different expansion techniques in the limits of small and large $J_{\perp }/K$. For strongly anisotropic Kitaev couplings, we derive effective models for the different layer stackings that we use to discuss the quantum phase transition out of the Kitaev phase. We find that the phase diagrams depend sensitively on the nature of the stacking and anisotropy strength. While in some stackings and at strong anisotropies we find a single transition between the Kitaev and dimer phases, other stackings are more involved. Most importantly, we prove the existence of two novel macrospin phases, which can be understood in terms of Ising chains that can be either coupled ferromagnetically or remain degenerate, thus realizing a classical spin liquid. In addition, our results suggest the existence of a flux phase with spontaneous interlayer coherence. We discuss prospects for experimental realizations.

Figure 1

Illustration of different stackings AA, AB, and AC (with two inequivalent choices of anisotropy) and schematic phase diagrams. The critical $J_{\perp }/K$ for the transition from the MAC-AF and MAC-L phases to the DIM phase is obtained exactly at $\lambda =0$ to be $J_{\perp }=K/2$ with $K=\max (K^x,K^y,K^z)$. As explained in Sec. 5d, the DIM phase in the AA stacking is expected to be of greater stability compared to the AB- and AC-stacked models because triplons in the AA stacking are fully localized. For various labels and further details we refer the reader to the text.

Figure 2

Mean-field phase diagram for the AA stacking. First- (second-) order transitions are marked with thick (thin) lines. The KSL phase becomes gapped when $\lambda <0.5$ (marked by a dashed line), while all other phases occurring are gapped for all parameter regimes. As discussed in Sec. 3c, we consider the DIM'-phase to be an artifact of mean-field theory, and the transition at $J_{\perp }/K^z=0.58$ can be expected to become a crossover when going beyond mean-field theory.

Figure 3

Mean-field parameters [34] obtained from the Majorana mean-field theory as a function of $J_{\perp }/K$ with Kitaev couplings $K^x=K^y=\lambda K^z$. In the interest of clarity, we only show MFT parameters $u_0^{0,\alpha }$ in the lower layer, since, employing a ${\mathbb{Z}}_2$ redundancy, $u_1^{0,\alpha }=-u_0^{0,\alpha }$. We denote the weak $x\text{and}y$ bonds and corresponding Majorana flavors with $\lambda$. Also note that $w^{11}=w^{22}$ for the chosen anisotropy. [(a)–(c)] AA stacking with varying anisotropies $\lambda =0,0.75,1$. Close to the isotropic point, a phase (FLUX) with interlayer-coherence is observed. At strong anisotropies, the DIM' dominates, which is separated from DIM by a second-order transition, which is expected to become a crossover beyond mean-field theory.

Figure 4

Mean-field phase diagram for the AB stacking with an anisotropy $K^x=K^y=\lambda K^z$. First- (second-) order transitions are marked with thick (thin) lines.

Figure 5

Mean-field parameters [34] for the AB stacking obtained from the Majorana mean-field theory as a function of $J_{\perp }/K$ with anisotropic Kitaev couplings $K^x=K^y=\lambda K^z$ with $\lambda =0.25$. The mean-field parameters in the MAC-phase describe decoupled chains. Since the chains are decoupled on mean-field level, the MFT parameters for MAC in the AC-stacking are identical.

Figure 6

Illustration of an effective zigzag chain in the MAC phase for the AB-stacked model with strong anisotropy on the $x$ bonds (for visual clarity, a different anisotropy was chosen than in the main text).

Figure 7

Band structure in the MAC phase for the AB-stacked model at $J_{\perp }=0.3$ and $\lambda =0.28$, showing a one-dimensional character due to hopping on chains. (a) Cut along high-symmetry lines. (b) Energy of the lowest dispersing quasiparticle band (obtained by removing flat bands resulting from localized excitations).

Figure 8

Mean-field phase diagram for the $\sigma \mathrm{AC}$ stacking with a symmetry-compatible anisotropy $K^y=K^z=\lambda K^x$. First- (second-) order transitions are marked with thick (thin) lines.

Figure 9

Phase diagram for the $\overline{\sigma }\mathrm{AC}$ stacking with anisotropy $K^x=K^y=\lambda K^z$, which spoils the reflection symmetry $\sigma$ of the AC stacking. First- (second-) order transitions are marked with thick (thin) lines.

Figure 10

Dispersion $E{\left(\mathbf{k}\right)}_{\alpha ,\mu }$ for AB stacking, $\alpha =x$, and $\mu =1,2$ (Ocher, black) along high-symmetry directions in BZ. (Insets) Dispersion contours for $\mu =1,2$ (blue, orange).

Figure 11

[(a)–(c)] $O\left(9\right)$ gap for all stackings and all distinct triplet types vs $K$ (in units of $J_{\perp }\equiv 1$) for the isotropic case: Bare gap-series (bold solid black), Padé (solid red), and dLog-Padé (dashed blue). (b) Bare upper $\mu =2$-triplet band-gap (thin solid black) and noninteracting two-particle continuum (gray hatched).

Figure 12

Density of dLog-Padé approximant poles vs $\lambda ,J_{\perp }/K^z$ for AB- (left) and $\overline{\sigma }\mathrm{AC}$ stacking (right). Extracted from $K^z$ series for fixed $\lambda =K^{x,y}/K^z$. Bin size $\delta \lambda ,\delta J_{\perp }/K^Z\sim 0.03$ with color coding from single poles (dark blue) to $O\left(100\right)$ poles (green to red) per bin.

Figure 13

The left picture shows the original bilayer brick wall lattice with plaquettes $p$ in each layer $m=1,2$. The spin-1/2 degrees of freedom of the bilayer Kitaev model reside on the black and white circles. The mapping replaces the two spins 1/2 degrees of each $K^z$ dimer into a hard-core boson and a pseudospin 1/2. These degrees of freedom reside on the blue circles of the bilayer square lattice depicted on the right side.

Figure 14

(a) Effective bilayer square lattice. (b) Notation of the four sites associated to a plaquette $p$. (c) Effective two-dimensional square lattice. The dual pseudospin 1/2 operators act in the centers of the plaquettes. These centers also form a square lattice shown with the dashed lines. Little orange squares indicate the sites where the sublattice rotation is performed.

Figure 15

The mean-field phase diagram of (39) as a function of ${\tilde{J}}_1^{xx}$, ${\tilde{J}}_2^{xx}$, and ${\tilde{J}}_4$. The blue (orange) regions are phase transitions of second- (first-) order. The red (black) points indicate the exact (mean-field) critical values for the 1d-TFIM on the ${\tilde{J}}_1^{xx}$ axis, the 2D-TFIM on the dashed cyan line, and the toric code in a transverse field on the ${\tilde{J}}_4$ axis. The purple line is the rescaled physical path generated for $K^x=K^y\approx 0.25$, which pierce the phase transition surface at ${\tilde{J}}_2^{xx}\approx {\tilde{J}}_4\approx 0$ and ${\tilde{J}}_1^{xx}\approx 1$. As a consequence, it is the second-order region which is relevant for the quantum phase transition in the anisotropic AA-stacked bilayer Kitaev model.

Figure 16

Illustration of the four different anisotropic limits: (a) AA stacking, (b) AB stacking, (c) $\sigma \mathrm{AC}$ stacking when replacing $K^z$ bonds by supersites (shown as filled blue circles), and (d) $\overline{\sigma }\mathrm{AC}$ stacking when replacing $K^x$ bonds by supersites (shown as filled red circles). The grey lines represent interlayer $J_{\perp }$ interactions. Note that the thick grey lines in (a) and (d) refer to the effective interaction from two $J_{\perp }$ couplings between the supersites. The green, red, and blue lines refer to $K^x$, $K^y$, and $K^z$ interactions in the two Kitaev layers, respectively.

Figure 17

Illustration of the plaquette operators ${\widehat{W}}_{m,p}$ for Kitaev layer $m=1$ (left) and $m=2$ (right) for the different anisotropic limits: (a) AA stacking, (b) AB stacking, (c) $\overline{\sigma }\mathrm{AC}$ stacking when replacing $K^z$ bonds by supersites (shown as filled blue circles), and (d) $\sigma \mathrm{AC}$-stacking when replacing $K^x$ bonds by supersites (shown as filled red circles). The numbering 1–4 of the plaquette sites refers to the definition of the Wen-plaquette operator ${\widehat{W}}_p={\tau }_1^y{\tau }_2^z{\tau }_3^y{\tau }_4^z$ in all cases.

Figure 18

Effective chains consisting of dimers coupled by $K^z$. Effective brick-wall models are obtained by replacing each dimer by a pseudospin. (1) For the $\overline{\sigma }\mathrm{AC}$ stacking, there is an intrachain coupling $K^y$, and the chains are coupled via $V_{d,d^{\prime }}$, which consists of $K^x$ couplings. (2) In the case of AB stacking, both inter and intrachain couplings are due to $K^x$ and $K^y$ interactions.