Majorana-Hubbard model on the square lattice

We study a tight-binding model of interacting Majorana (Hermitian) modes on a square lattice. The model may have an experimental realization in a superconducting-film–topological-insulator heterostructure in a magnetic field. We find a rich phase diagram, as a function of interaction strength, including an emergent superfluid phase with spontaneous breaking of an emergent $\mathrm{U}\left(1\right)$ symmetry, separated by a supersymmetric transition from a gapless normal phase.

Figure 1

The mean-field field phase diagram for $t=1$ and $t_2=0$ as a function of $g$. The solid (dashed) lines represent second-order (first-order) transitions. The broken symmetry states are sketched in Figs. 2, 8, and 12.

Figure 2

Sketch of the two mean-field ground states occurring for $g>0$. The blue dots represent the lattice sites. The open circles appear on bonds on which the two Majorana modes combine to form Dirac fermions, which are unoccupied.

Figure 3

The values of $A$ and $B$ that minimize the energy density (5.11) $\left(t_2=0\right)$ as a function of $g$. In addition to a first-order transition at $g=0^{+}$, we find a critical point near $g\approx 0.9$. For a finite $t_2=0.5$, we also find a critical point and a phase transition to an emergent superfluid phase $\left|B\right|>0$. However, this transition is in a different universality class than the $t_2=0$ case as the normal phase is gapped (gapless) for $t_2\ne 0$ $\left(t_2=0\right)$.

Figure 4

$\mathcal{F}\left(g\right)$, Eq. (5.14), as a function of $g$. The critical point corresponds to $\mathcal{F}\left(g\right)=0$.

Figure 5

$\mathcal{F}(A_c\left(g_c\right),B)$ for small $B$.

Figure 6

The energy density (5.23) for $A=B=C=0$ as a function of $D$ for various values of $g$. The minimum energy always corresponds to $D=0$.

Figure 7

Sketch of the mean-field ground state arising from diagonal factorization of the interaction term for $g>0$, for only $D\ne 0$. An arrow pointing along a diagonal from site ${\vec{r}}_1$ to site ${\vec{r}}_2$ indicates that $i\langle {\gamma }_{{\vec{r}}_1}{\gamma }_{{\vec{r}}_2}\rangle >0$.

Figure 8

Sketch of the two mean-field ground states occurring for $g<0$. The blue dots represent the lattice sites. The open circles appear on bonds on which the two Majorana modes combine to form Dirac fermions, which are unoccupied.

Figure 9

The mean-field parameters minimizing the energy density for the vertical decoupling case with $g<0$. There appears to be two mean-filed transitions; first to a solution with only nonvanishing $D$ and then to a state with both $D$ and $C$ nonzero.

Figure 10

The energy density for various values of $g$ and $C=0$ as a function of $D$, supporting a first-order mean-filed transition.

Figure 11

The value of $B$ that minimizes the energy density for the case of diagonal decoupling with $g<0$, indicating a second-order mean-field transition. The minimization yields $A=C=D=0$ for all $g$.

Figure 12

Diagonal ground state for $g<0$. An arrow pointing along a diagonal from site ${\vec{r}}_1$ to site ${\vec{r}}_2$ indicates that $i\langle {\gamma }_{{\vec{r}}_1}{\gamma }_{{\vec{r}}_2}\rangle >0$.