Dynamical and topological properties of the Kitaev model in a [111] magnetic field

The Kitaev model exhibits a quantum spin liquid hosting emergent fractionalized excitations. We study the Kitaev model on the honeycomb lattice coupled to a magnetic field along the [111] axis. Utilizing large-scale matrix product-based numerical models, we confirm three phases with transitions at different field strengths depending on the sign of the Kitaev exchange: a non-Abelian topological phase at low fields, an enigmatic intermediate regime only present for antiferromagnetic Kitaev exchange, and a field-polarized phase. For the topological phase, we numerically observe the expected cubic scaling of the gap and extract the quantum dimension of the non-Abelian anyons. Furthermore, we investigate dynamical signatures of the topological and the field-polarized phase using a matrix product operator-based time evolution method.

Figure 1

(a) Bonds labeled with x, y, and $z$ and an exemplaric $S_i^y{S}_j^y$ Kitaev-exchange (orange); (b) a single three-spin term $S_i^x{S}_j^z{S}_k^y$ of the three-spin interaction in $H_{K_3}$.

Figure 2

Geometries used for iDMRG and their corresponding accessible momenta (orange lines) in reciprocal space with respect to the first Brillouin zone (inner hexagon). The second Brillouin zone is shown partially. The roman numbers label links across the boundary. (a) rhombic geometry with three unit cells, $L_{\text{circ}}=6$ sites, along the circumference and (b) its corresponding reciprocal space. (c), (d) rhombic-2 geometry with five unit cells circumference, $L_{\text{circ}}=10$ sites.

Figure 3

Several observables of the Kitaev model with antiferromagnetic coupling, $K>0$, in a magnetic field along [111]. From top to bottom: Second derivative $-d^{2}E/d{h}^2$ of the energy with respect to the field $h$, entanglement entropy $S_E$ of a bipartition of the cylinder divided by the number $L_y$ of bonds cut, correlation length $\xi$, average of plaquette fluxes $W_p$, and flux $W_l$ of a noncontractible loop around the cylinder. At least three phases are observed: topological phase for $h<h_{c1,\text{AF}}\approx 0.22$, intermediate possibly gapless $h_{c1,\text{AF}}<h<h_{c2,\text{AF}}\approx 0.36$, and a subsequent field-polarized phase. Solid blue lines are for the $W_l=-1$ sector, dashed blue lines for $W_l=+1$, and its intensity encodes the bond dimension $\chi$ used, where dark blue refers to a large $\chi$. Thin dashed black lines depict the phase transitions obtained from the peaks in $-d^{2}E/d{h}^2$.

Figure 4

Comparison of correlation length $\xi$ between (a) the rescaled magnetic field $h\to 32h^3$ and (b) the three-spin exchange $K_3$. The solid red line is a guide-to-the-eye corresponding to a $1/K_3$ or $1/\left(32h^3\right)$ scaling. Within $0.05<K_3,32h^3<0.2$, that is, where numerical convergence is achieved, the behavior of $\xi$ is consistent with a predicted opening of the gap as $\mathrm{\Delta }E\propto h^3$ or as $\mathrm{\Delta }E\propto K_3$, respectively.

Figure 5

Remainder $\mathrm{\Delta }{S}_E$ of the entanglement entropy of a bipartition of the cylinder after subtracting a fermionic and a gauge field contribution following Eq. (7). The magnetic field has been rescaled, $h\to 32h^3$, based on the behavior of the correlation length in Fig. 4. The vertical dashed lines signal the transitions in a magnetic field. The horizontal lines correspond to $log(\mathcal{D}/d_a)$ as discussed in the main text.

Figure 6

Dynamical spin-structure factor ${\mathcal{S}}^{xx}(k,\omega )$ along $\mathrm{\Gamma }$–$M$–${\mathrm{\Gamma }}^{\prime }$ at (a) $h=0.0$, (b) $h=0.1$, (c) $h=0.2$, (d) $h=0.0,K_3=-0.25$, (e) $h=0.1,K_3=-0.25$, (f) $h=0.175,K_3=-0.25$ within the topological phase. (a)–(f) have a logarithmic color scale ranging over two decades. (g), (h) ${\mathcal{S}}^{xx}(k,\omega )$ at high-symmetry points $\mathrm{\Gamma },K,M$, and ${\mathrm{\Gamma }}^{\prime }$ for different $h$ and $K_3$. An vertical offset is used for better visibility. In all plots, ${\mathcal{S}}^{xx}(k,\omega )$ is normalized as given in the main text.

Figure 7

Dynamical spin-structure factor $\mathcal{S}(k,\omega )$ in the field-polarized phase along $\mathrm{\Gamma }$–$M$–${\mathrm{\Gamma }}^{\prime }$ at (a) $h=0.58$, (b) $h=0.5$, (c) $h=0.425$, and (d) $h=0.375$. (e) $\mathcal{S}(k,\omega )$ at high-symmetry points $K,M$, and ${\mathrm{\Gamma }}^{\prime }$ for different $h$. An vertical offset is used for better visibility. In all plots, $\mathcal{S}(k,\omega )$ is normalized as given in the main text.

Figure 8

Comparison of the ground state energy $E_{\text{GS}}$ vs bond-dimension $\chi$ for different geometries and sizes of the iDMRG cluster at a single field strength of $h=0.275$. For small $\chi$, larger iDMRG cluster have a smaller $E_{\text{GS}}$. However, in the limit $1/\chi \to 0,E_{\text{GS}}$ is of very similar value for all geometries used. In fact, large iDMRG cluster show a phase transition from an ordered ground state at small $\chi$ to a translational invariant ground state at large $\chi$ captured by the smallest iDMRG cluster.

Figure 9

Entanglement entropy $S$ of a bipartition of the cylinder over correlation length $\xi$ for various bond dimension $\chi$ to check for possible finite-$\chi$ scaling. Data is shown for different magnetic field strength $h=0.25,0.275,0.3,0.325,0.35$ across the intermediate regime. Lines are fits to the five points with largest $\xi$ at each field.