Ising anyons in frustration-free Majorana-dimer models

Dimer models have long been a fruitful playground for understanding topological physics. Here, we introduce a class, termed Majorana-dimer models, wherein bosonic dimers are decorated with pairs of Majorana modes. We find that the simplest examples of such systems realize an intriguing, intrinsically fermionic phase of matter that can be viewed as the product of a chiral Ising theory, which hosts deconfined non-Abelian quasiparticles, and a topological $p_x-i{p}_y$ superconductor. While the bulk anyons are described by a single copy of the Ising theory, the edge remains fully gapped. Consequently, this phase can arise in exactly solvable, frustration-free models. We describe two parent Hamiltonians: one generalizes the well-known dimer model on the triangular lattice, while the other is most naturally understood as a model of decorated fluctuating loops on a honeycomb lattice. Using modular transformations, we show that the ground-state manifold of the latter model unambiguously exhibits all properties of the $\text{Ising}\times (p_x-i{p}_y)$ theory. We also discuss generalizations with more than one Majorana mode per site, which realize phases related to Kitaev's 16-fold way in a similar fashion.

Figure 1

Left panel: bilayer of an Ising phase and a $p_x-i{p}_y$ topological superconductor with opposite chirality, which together give rise to the topological phase discussed in this paper. This phase is characterized by three distinct topological sectors, but has a fully gapped edge. Right panel: the Hilbert space of Majorana-dimer models consists of bosonic dimers on the edges of the lattice and Majorana modes on the lattice sites. In the low-energy subspace, the Majoranas are paired according to the placement of the dimers: e.g., the fermion wave function corresponding to the dimer configuration shown is the ground state of $H_F=-i{\gamma }_1{\gamma }_2-i{\gamma }_3{\gamma }_4$.

Figure 2

Kasteleyn orientation (arrows) and reference dimer configuration (blue bonds) on the triangular lattice (left panel) and the Fisher lattice (right panel).

Figure 3

Illustration of Majorana pairings on the Fisher lattice. The green highlighted strip illustrates part of a transition-graph loop. Away from the loop, Majorana modes pair into the reference configuration; the corresponding fermion state has each of the $f_q$ fermion unoccupied. Along the transition-graph loop, dimers are not in the reference configuration, and the Majoranas instead pair between neighboring complex fermions $f_q,f_{q^{\prime }}$. The precise state of the fermions along the transition graph loop is the ground state of the Kitaev chain formed using Majoranas from neighboring sites. In the above example, this chain has the form $h=\cdots +i{\gamma }_0^A{\gamma }_1^A-i{\gamma }_1^B{\gamma }_2^B-i{\gamma }_2^A{\gamma }_3^A+i{\gamma }_3^B{\gamma }_4^B+...$. The arrow orientation on the reference edges determines the identification of the two Majoranas at each site as ${\gamma }_q^A$ or ${\gamma }_q^B$. The other arrow orientations determine the sign of the coupling between Majoranas.

Figure 4

Upper left: the entanglement cut (red dashed line) used for the numerical calculations of the modular $S$ and $T$ matrices. Upper right: a dimer configuration belonging to the $(1,0)$ topological sector. Bottom: the Dehn twist $T$ permutes the sectors $(1,0)$ and $(1,1)$ while preserving the sector $(0,1)$.

Figure 5

An example of tetragonalization for $t=4$. A flip operator for the original 10-sided polygon may be decomposed into a series of elementary flips for each tetragon. In this representation, the tetragonal plaquettes, from top to bottom right, respectively, correspond to $U_{1,3}, U_{1,5}, U_{1,7}$, and $U_{1,9}$ in Eq. (C1).

Figure 6

Illustration of ${\mathcal{B}}_{p^{\prime }}^{⎔}{\mathcal{B}}_p^{⎔}$ acting on an initial configuration without any loops in the overlap area between neighboring plaquettes $p$ and $p^{\prime }$. The dashed lines in the middle figure are auxiliary lines for tetragonalization.

Figure 7

Variation of Fig. 6 in which the plaquette $p$ is occupied by a loop.