Perturbative approach to an exactly solved problem: Kitaev honeycomb model

We analyze the gapped phase of the Kitaev honeycomb model perturbatively in the isolated-dimer limit. Our analysis is based on the continuous unitary transformations method, which allows one to compute the spectrum as well as matrix elements of operators between eigenstates at high order. The starting point of our study consists of an exact mapping of the original honeycomb spin system onto a square-lattice model involving an effective spin and a hard-core boson. We then derive the low-energy effective Hamiltonian up to order 10 which is found to describe an interacting-anyon system, contrary to the order 4 result which predicts a free theory. These results give the ground-state energy in any vortex sector and thus also the vortex gap, which is relevant for experiments. Furthermore, we show that the elementary excitations are emerging free fermions composed of a hard-core boson with an attached spin- and phase-operator string. We also focus on observables and compute, in particular, the spin-spin correlation functions. We show that they admit a multiplaquette expansion that we derive up to order 6. Finally, we study the creation and manipulation of anyons with local operators, show that they also create fermions, and discuss the relevance of our findings for experiments in optical lattices.

Figure 1

(Color online) Mapping of the honeycomb (brick-wall) lattice onto a square lattice with unit basis vectors ${\mathit{n}}_1$ and ${\mathit{n}}_2$. Each $z$ dimer with four-spin configurations is replaced by a single site with 4 degrees of freedom: the occupation number of hard-core boson (0 or 1) and the effective spin ($\Uparrow$, or $\Downarrow$), which is chosen as the spin of the black site of the considered $z$ dimer. The numbering of the sites of a plaquette $p$ is shown in both cases.

Figure 2

(Color online) The periodic boundary conditions used in this work (a) on the original brick-wall lattice and (b) on the effective square lattice of $z$ dimers (see Fig. 1). In the figure $p=1$. In both cases, the finite-size system is put on a torus obtained by identifying the opposite sides of the dashed (magenta) square. For clarity, the site at the point chosen as the origin has been depicted bigger. Half the square plaquettes in (b) have been colored in cyan (gray) to show that the periodic boundary conditions allow us to bicolor the lattice.

Figure 3

(Color online) Illustration of the conserved plaquette quantity $W_p$. The thick yellow (lightest gray) line delimitates the plaquette $p$. The thick red (gray), green (light gray), or blue (dark gray) segments represent the Pauli matrices ${\sigma }_i^{\text{out}\left(i\right)}$.

Figure 4

(Color online) Illustration of the relation ${\mathcal{L}}^{\prime }=W_p\mathcal{L}$, with $\mathcal{L}={\prod }_{i∊\mathcal{C}}{\sigma }_i^{\text{out}\left(i\right)}$, $W_p$ already shown in Fig. 3 and ${\mathcal{L}}^{\prime }={\prod }_{i∊{\mathcal{C}}^{\prime }}{\sigma }_i^{\text{out}\left(i\right)}$. The thick yellow (lightest gray) line in (a) represents the contour $\mathcal{C}$ and the one in (b) represents ${\mathcal{C}}^{\prime }$. As in Fig. 3, the thick red (gray), green (light gray), or blue (dark gray) segments represent the Pauli matrices ${\sigma }_i^{\text{out}\left(i\right)}$.

Figure 5

(Color online) Examples of conserved loop operators $\mathcal{L}={\prod }_{i∊\mathcal{C}}{\sigma }_i^{\text{out}\left(i\right)}$ in a finite-size system with PBC.

Figure 6

(Color online) On the four figures, the thick yellow (lightest gray) line represents the contours $\mathcal{C}$. The operators ${\omega }_{\mathit{i}}$ which are such that the loop operators read $\mathcal{L}={\prod }_{\mathit{i}∊\mathcal{C}}{\omega }_{\mathit{i}}$ are indicated in the figures. In the present case, they can take the following values $x={\tau }^x$, $\overline{x}={\left(-1\right)}^{b^{†}b}{\tau }^x$ (and the same for $y$ and $z$), and $-\overline{x}=-{\left(-1\right)}^{b^{†}b}{\tau }^x$. These figures are the same as the ones of Fig. 5 but on the effective lattice.

Figure 7

(Color online) Two-anyon configurations (gray central plaquette and one of the numbered plaquettes) on a vortex-free background. $\Delta E_{1v}$ $\left(\Delta E_{2v}\right)$ is the energy cost (at order 10) for adding one vortex (two vortices) to the vortex-free state. $\Delta E_{1v}$ reads $\Delta E_{1v}=J^4+8J^6+75J^8+784J^{10}$. For simplicity, we have set here $J_x=J_y=J$ but the results, in the general case, are easily obtained from the coefficients given in Appendix .

Figure 8

(Color online) Illustration of the two different points of view one can have of a bicolored lattice: sites are at vertices in (a) and on the bonds in (b). The plaquette $m$ in (a) remains a plaquette in (b), while the plaquette operator $e$ transforms into a star operator.

Figure 9

(Color online) Two of the states entering the equal-weight superposition in Eq. (24). Crosses on the sites indicate a spin flip with respect to the reference state $|\Uparrow ⟩$, where all spins point upward (the $A_e$ operators are still pictured by thick crosses at the vertices). The loops, in dotted lines, are obtained by joining the flipped spins.

Figure 10

(Color online) The two loop operators ${\mathcal{L}}^x$ and ${\mathcal{L}}^y$ and their contours in yellow (lightest gray) thick lines, corresponding to ${\mathcal{C}}_b$ and ${\mathcal{C}}_c$ in Fig. 6. As in the latter figure, but for $s$ spins instead of $\tau$ spins, $y$ means $s^y$, etc. In the 0-QP sector, $\overline{y}={\left(-1\right)}^{b^{†}b}{s}^y$ is the same as $y$.

Figure 11

(Color online) Two of the states entering the equal-weight (in absolute value) superposition in Eq. (25), involving the ${\mathcal{L}}^x$ loop operator. Graphical conventions are the same as in Figs. 9, 10.

Figure 12

(Color online) Two of the states entering the equal-weight (in absolute value) superposition for the state having two magnetic vortices. The latter are represented with little gray squares marked with the letter $m$ and are linked with a string of spin flips [blue (dark gray) thick line]. The other graphical conventions are the same as in Fig. 9.

Figure 13

(Color online) Illustration of operators involved in the braiding of a magnetic vortex around an electric vortex. $X$ is the product ${\prod }_{\mathit{i}}{s}_{\mathit{i}}^x$ for $\mathit{i}$ belonging to the blue (dark gray) thick path linking the two $m$ vortices (orthogonal to bonds). $Z$ is the product ${\prod }_{\mathit{i}}{s}_{\mathit{i}}^z$ for $\mathit{i}$ belonging to the green (light gray) thick path linking the two $e$ vortices (drawn on the bonds). $X^{\prime }$ is the product ${\prod }_{\mathit{i}}{s}_{\mathit{i}}^x$ for $\mathit{i}$ belonging to the dashed loop around the leftmost $e$ vortex and is also equal to a product of the $A_e$ operators encircled by the loop (denoted as thick crosses).

Figure 14

(Color online) Illustration of states of the 1-QP basis, built from the ground state. (a) The action of $b_O^{†}$ is to create a particle at site $O$ (large black filled circle) as well as to create a pair of $e$ and $m$ vortices. The action of the string operator $S$ represented with an oriented thick (cyan) line (a) on the state, then yields the state with the same vortices (b) but a particle at site $\mathit{i}$. We do not bicolor the lattice in cyan (gray) and white anymore so that the figures are easier to read.

Figure 15

(Color online) [(a)–(c)] States and (d) contour needed to compute the matrix element [Eq. (32)] (see text).

Figure 16

(Color online) [(a)–(c)] States and (d) contour needed to compute the matrix element of a hopping in the ${\mathit{n}}_2$ direction when site $\mathit{i}$ is “one site away from the edge” of the dashed (magenta) square and (d) thus involving a loop operator (see text for details).

Figure 17

(Color online) Illustration of the phase factors appearing at site ${\mathit{j}}_2$ in the string operators $S_{\dots ,{\mathit{j}}_1,{\mathit{j}}_2,{\mathit{j}}_3,\dots }$. The phase factors ${\left(-1\right)}^{b_{{\mathit{j}}_2}^{†}{b}_{{\mathit{j}}_2}}$ are denoted as large (cyan) dots and are only involved in the three top processes, which are, from left to right, $S_{\dots ,\mathit{j},\mathit{j}+{\mathit{n}}_1,\mathit{j}+2{\mathit{n}}_1,\dots }$, $S_{\dots ,\mathit{j},\mathit{j}+{\mathit{n}}_2,\mathit{j}+{\mathit{n}}_1+{\mathit{n}}_2,\dots }$, and $S_{\dots ,\mathit{j},\mathit{j}-{\mathit{n}}_2,\mathit{j}+{\mathit{n}}_1-{\mathit{n}}_2,\dots }$.

Figure 18

(Color online) Illustration of the exchange of two particles discussed in the text for $\mathit{j}=\mathit{i}+{\mathit{n}}_2$, $\mathit{k}=\mathit{i}+{\mathit{n}}_1$, and $\mathit{l}=\mathit{i}-{\mathit{n}}_2$.

Figure 19

(Color online) A possible gauge choice realizing the vortex-full lattice $\nu =1$. The thin (bold) links are associated to $u_{jk}=+1$ $\left(u_{jk}=-1\right)$, where $j$ belongs to the black sublattice and $k$ to the white one (see Appendix ). The eigenvalue of the plaquette operator is then simply given by $w_p={\prod }_{\left(j,k\right)∊p}{u}_{jk}$.

Figure 20

(Color online) A possible gauge choice realizing the vortex-half lattice $\nu =1/2$. Notations are the same as in Fig. 19. In this configuration, vortices are localized in alternance on horizontal rows.

Figure 21

(Color online) Another possible gauge choice realizing the vortex-half lattice $\nu =1/2$. Notations are the same as in Fig. 19. In this configuration, vortices are localized, in alternance, on diagonal bands.

Figure 22

(Color online) Behavior of the spectral weights $I_n^z$ for fermion numbers $n=0,2,4$ as a function of the coupling $J=J_x=J_y$ for $J_z=1/2$. Gray plaquettes in the insets show the positions $p_1^z$ and $p_2^z$ at which the anyons are created under the action of ${\tau }_{\mathit{i}}^z$.

Figure 23

(Color online) Action, in the vortex-free sector, of the string operator $S={\sigma }_{{\mathit{i}}_3,●}^z{\sigma }_{{\mathit{i}}_2,●}^z{\sigma }_{{\mathit{i}}_1,●}^z$. Each operator flips the two plaquettes adjacent to the $z$ dimer; it is attached to but also creates fermionic excitations (not shown).

Figure 24

(Color online) $\mathcal{P}$ as a function of $J_x=J_y=J$ computed for $m=25$. Top: nonresummed (bare) expression (65) at order 2 [red (bottom)], 4 [green (top)], and 6 [blue (middle)]. Bottom: exponentiated form $\text{exp}\left(A-mB\right)$ at order 2 [red (top)], 4 [green (middle)], and 6 [blue (bottom)].

Figure 25

(Color online) Labeling of the plaquettes by a site index $\left(\mathit{i}\right)$ and a position $u,d,l,r$ with respect to that site.