Robustness of Vacancy-Bound Non-Abelian Anyons in the Kitaev Model in a Magnetic Field

Abstract

Non-Abelian anyons in quantum spin liquids (QSLs) provide a promising route to fault-tolerant topological quantum computation. In the exactly solvable Kitaev honeycomb model, such anyons of the QSL state can be bound to nonmagnetic spin vacancies and endowed with non-Abelian statistics by an infinitesimal magnetic field. Here, we investigate how this approach for stabilizing non-Abelian anyons extends to a finite magnetic field represented by a proper Zeeman term. Through large-scale density-matrix renormalization group (DMRG) simulations, we compute the vacancy-anyon binding energy as a function of magnetic field for both the ferromagnetic (FM) and antiferromagnetic (AFM) Kitaev models. We find that anyon binding remains robust within the entire QSL phase for the FM Kitaev model but breaks down already inside this phase for the AFM Kitaev model. To compute a binding energy several orders of magnitude below the magnetic energy scale, we introduce both a refined definition and an extrapolation scheme based on carefully tailored perturbations.

Figures (6)

• Figure 1: Honeycomb lattice in a narrow cylindrical geometry with (a) zigzag, (b) twisted, and (c) armchair boundary conditions, as well as a single spin vacancy in the center. The length $L_X$, the width $L_Y$, and the partial lengths $L_{\rm left}$ and $L_{\rm right}$ of the cylinder are indicated, with (a) $L_Y=3$ and (b,c) $L_Y=4$. This figure shows $L_{\rm left}=L_{\rm right}=3$ and $L_X=L_{\rm left}+L_{\rm right}+1=7$ for compactness, but the calculations correspond to $L_{\rm left}=L_{\rm right}=7$ and $L_X=15$ in order to reduce edge effects. Each loop operator $W_{\mathcal{P}}$, $W_{\mathcal{O}}$, and $W_{X}$ introduced in the text is a simple product of appropriate spin operators ($x,y,z$) specified along the given green, purple, or red path. Anisotropic Kitaev spin interactions $S^{\alpha} S^{\alpha}$ along $\alpha = x,y,z$ bonds are also shown.
• Figure 2: (a-d) Expectation value of the topological loop operator, $\langle W_X \rangle$, against the coordinate $1 \leq X \leq 15$ along the cylinder length for the perturbed Hamiltonians (a) $H^{\lambda}(+1, +1)$, (b) $H^{\lambda}(+1, -1)$, (c) $H^{\lambda}(-1, +1)$, and (d) $H^{\lambda}(-1, -1)$ in Eq. (\ref{["eq:H'"]}) with perturbation strength $\lambda=0.06$, FM Kitaev exchange $J=1$, as well as magnetic fields $h=0.002$ (colorful solid symbols) and $h=0.012$ (gray empty symbols) on a narrow cylinder with $L_X=15$, $L_Y=3$, $L_{\rm left}=L_{\rm right}=7$, and twisted boundary conditions. The light blue shading marks the non-existent topological loop operator going through the vacancy at coordinate $X=X_{\mathcal{O}}=8$. (e) Extraction of the unperturbed energy coefficient $E_{\mathcal{O}}$ in Eq. (\ref{['eq:binding-2']}) through a linear extrapolation of the perturbed energy coefficient $E_{\mathcal{O}}^{\lambda}$ in Eq. (\ref{['eq:binding-4']}) from $0.06 \leq \lambda \leq 0.1$ to $\lambda = 0$ for a variety of magnetic fields $0 \leq h \leq 0.014$. The gray cross marks an obvious outlier value that is excluded from the extrapolation.
• Figure 3: Vacancy-flux binding energy, $E_{\rm binding} = -2 E_{\mathcal{O}}$, as a function of the external magnetic field $h$ for the (a) FM and (b) AFM Kitaev model on a width $L_Y=3$ cylinder with zigzag (teal triangles) and twisted (blue circles) boundary conditions as well as a width $L_Y=4$ cylinder with armchair (purple diamonds) and twisted (red squares) boundary conditions. The orange triangle, circle, diamond, and square represent the corresponding exact values at $h=0$, while the orange star shows the exact value at $h=0$ in the TDL. For the FM Kitaev model, the dash-dotted line marks the approximate critical field, $h_C^{\rm FM} \approx 0.014$, for the transition out of the non-Abelian QSL phase Gohlke2018. For the AFM Kitaev model, the analogous critical field, $h_C^{\rm AFM} \approx 0.22$, lies outside the simulated field range. The cylinder dimensions are $L_X=15$ and $L_{\rm left}=L_{\rm right}=7$ throughout this figure.
• Figure 4: Energy coefficients $E_{\rm const}$, $E_{\rm left}$, $E_{\rm right}$, and $E_{\mathcal{O}}$ [see Eq. (\ref{['eq:E']})] against (a) the partial cylinder length on the left side of the vacancy, $L_{\rm left}$, for fixed $L_{\rm right} = L_Y = 4$, (b) the partial cylinder length on the right side of the vacancy, $L_{\rm right}$, for fixed $L_{\rm left} = L_Y = 4$, and (c) the cylinder width $L_Y$ for fixed $L_{\rm left} = L_{\rm right} = 20$. These coefficients are calculated through the exact solution of the pure Kitaev model at zero magnetic field ($h=0$). Note that certain coefficients are plotted with an appropriate scaling factor (see the legend of each panel) in order to be comparable with other coefficients.
• Figure 5: Extraction of the vacancy-flux binding energy in the TDL of the pure Kitaev model via a standard finite-size scaling in the cylinder dimension $L = L_{\rm left} = L_{\rm right} = L_Y$ with $11 \leq L \leq 40$, using (a) the refined definition in Eqs. (\ref{['eq:binding-1']}) and (\ref{['eq:binding-2']}) as well as (b,c) the naive definitions in Eq. (\ref{['eq:binding-3']}). The solid, dashed, and dotted lines represent quadratic extrapolations in $1/L$ from finite-size binding energies with specific moduli of $L$ with respect to $3$, as indicated by different colors.