Majorana-Hubbard model on the honeycomb lattice

Phase diagram of a Hubbard model for Majorana fermions on the honeycomb lattice is explored using a combination of field theory, renormalization group, and mean-field arguments, as well as exact numerical diagonalization. Unlike the previously studied versions of the model we find that even weak interactions break symmetries and lead to interesting topological phases. We establish two topologically nontrivial phases at weak coupling, one gapped with chiral edge modes and the other gapless with antichiral edge modes. At strong coupling a mapping onto a novel frustrated spin-$\frac{1}{2}$ model suggests a highly entangled spin liquid ground state.

Figure 1

(a) Lattice structure of the model. The arrow indicates the sign of the hopping terms. (b),(c) The ordering of the four operators in $g_1$ and $g_2$ terms, respectively.

Figure 2

(a) Phase diagram of the interacting Majorana model at weak coupling. MF Hamiltonian Eq. (7) on a strip with the zig-zag boundary (with ${\tau }_1=\pm 0.1,{\tau }_2=-0.15$) is used to illustrate the energy spectra in the gapped Majorana Chern insulator phase (b) and the gapless Majorana metal phase (c).

Figure 3

(a) Reflection symmetry ${\mathcal{R}}_{1,2}$ and inversion symmetry $\mathcal{P}$. (b) The sign convention of the MF Hamiltonian. (c) The definition of the order parameters ${\mathrm{\Delta }}_a$.

Figure 4

Self-consistent solution of the MF equations (12).

Figure 5

Order parameters ${\mathrm{\Delta }}_a$ (a)–(c) from MF calculations and (d)–(f) from ED using a cluster of 32 lattice sites with periodic boundary conditions. (g)–(i) Quantitative comparison between MF (blue) and ED (red) results, with solid line $g_1=g_2=g$ and dash line $g_1=-g_2=g$.

Figure 6

Lowest many-body energy levels of $\mathcal{H}$ computed using ED for different values of $g$.

Figure 7

Lowest many-body energy levels in the large $g$ limit, where $g_1=(-)g_2=g$ for the blue (red) curve as in Fig. 6.

Figure 8

Jordan-Wigner transformation. (a) The path used to define the Jordan-Wigner transformation Eq. (13). (b) Each four-Majorana interaction term maps onto a three-spin interaction. (c) The resulting triangular lattice spin model.