Electronic structure of topological superconductors in the presence of a vortex lattice

Certain types of topological superconductors and superfluids are known to host protected Majorana zero modes in cores of Abrikosov vortices. When such vortices are arranged in a dense periodic lattice one expects zero modes from neighboring vortices to hybridize and form dispersing bands. Understanding the structure of these bands is essential for the schemes that aim to employ the zero modes in quantum computation applications and in studies of their strongly interacting phases. We investigate here the band formation phenomenon in two concrete models, describing a two-dimensional $p_x+i{p}_y$ superconductor and a superconducting surface of a three-dimensional strong topological insulator (Fu-Kane model), using a combination of analytical and numerical techniques. We find that the physics of the Majorana bands is well described by tight-binding models of Majorana fermions coupled to a static $Z_2$ gauge field with a nontrivial gauge flux through each plaquette, in accord with expectations based on very general arguments. In the case of the Fu-Kane model we also find that, irrespective of the lattice geometry, the Majorana band becomes completely flat at the so-called neutrality point (chemical potential coincident with the Dirac point) where the model exhibits an extra chiral symmetry. In this limit the low-energy physics will be dominated by four-fermion interaction terms which are permitted by symmetries and may arise from the Coulomb interaction between the constituent electron degrees of freedom.

Figure 1

Phase factors and branch cuts in a triangular vortex lattice. Oriented solid lines indicate integration paths between the reference points located just to the right of each vortex center. These are used to evaluate the gauge invariant phase factors ${\omega }_{ij}$. Dashed lines represent a specific choice of the branch cuts discussed in the text.

Figure 2

Vortex lattice geometries: (a) square and (b) triangular. Two-vortex unit cell is shaded. The arrows specify the $Z_2$ gauge factors for the MZM tight-binding models and satisfy the Grosfeld-Stern rule Eq. (3). Hopping in the direction of the arrow incurs a phase factor of $+i$ while hopping in the opposite direction $-i$.

Figure 3

Band structure for the continuum model of a $p_x+i{p}_y$ superconductor. In the top panel solid (dotted) lines indicate the band structure calculated in the presence (absence) of the square vortex lattice. The parameters are chosen as follows: ${\mu }_0={\mathrm{\Delta }}_0=1, {\mu }^{\prime }=40$, and $\xi /a=0.2$. The two bands closest to zero energy are the MZM bands. They are enlarged in the bottom panel where the dashed line represents the best fit to the Majorana tight-binding model (34) with the hopping parameters $t=0.0305$ and $t^{\prime }=0.0076$. The inset shows the path taken in the first Brillouin zone.

Figure 4

Band structure for the continuum Fu-Kane model, square vortex lattice. In the top panel solid (dashed) lines indicate the band structure calculated in the presence (absence) of the vortex lattice. The parameters are chosen as follows: $v=0.14, m_0=7.0, \mu =0$, and $\xi /a=0.14$. The high-energy cutoff $\mathrm{\Lambda }=8$ and all the quantities are in units of ${\mathrm{\Delta }}_0=1$. As in Fig. 4 the two bands closest to zero energy are the MZM bands. They are enlarged in the bottom panel where the dashed line represents the best fit to the Majorana tight-binding model (34) with the hopping parameters $t=0.0153$ and $t^{\prime }=0.0032$.

Figure 5

Band structure for the continuum Fu-Kane model, triangular vortex lattice. Solid (dotted) lines indicate the band structure calculated in the presence (absence) of the vortex lattice. The parameters are chosen as follows: $v=0.33, m=1.0, \mu =0$ and $\xi /a=0.17$. The high energy cutoff $\mathrm{\Lambda }=20$ and all the quantities are in units of ${\mathrm{\Delta }}_0=1$. As in Fig. 4 the two bands closest to zero energy are the MZM bands. The dashed line represents the best fit to the Majorana tight binding model (35) with the hopping parameter $t=0.0425$.

Figure 6

Band structure for the lattice $p_x+i{p}_y$ superconductor with a square vortex lattice. (a) Full band structure in a $10\times 10$ magnetic unit cell with ${\mathrm{\Delta }}_0=0.50$ and ${\varepsilon }_F=-2.2$. (b) Detail of the Majorana band (solid squares) and the best fit to the tight-binding Majorana dispersion Eq. (34) with $t=4.3027\times {10}^{-3}$ and $t^{\prime }=1.9375\times {10}^{-4}$ (solid line).

Figure 7

The Majorana overlap amplitudes $t$ and $t^{\prime }$ as a function of ${\varepsilon }_F$ extracted from the lattice model $p_x+i{p}_y$ superconductor (solid symbols). The magnetic unit cell is $50\times 50$ and ${\mathrm{\Delta }}_0=0.50$. The thin (blue) line represents the analytical result of Ref. [48] while the thick (red) line corresponds to the simple phenomenological expression (53) discussed in the text. The fit parameters are $A=2.43\times {10}^{-4}, b=0.0163$ and $A^{\prime }=5.91\times {10}^{-5}, b^{\prime }=0.0066$.

Figure 8

Band structure for the lattice version of the Fu-Kane model with a square vortex lattice. (a) Full band structure in a $30\times 30$ magnetic unit cell with $\tau =0.5, {\mathrm{\Delta }}_0=0.4, m=0.5$, and ${\varepsilon }_F=0.25$. (b) Detail of the Majorana band (solid squares) and the best fit to the tight-binding Majorana dispersion Eq. (34) with $t=2.36\times {10}^{-3}$ and $t^{\prime }=3.45\times {10}^{-4}$ (solid line).

Figure 9

The Majorana overlap amplitudes $t$ and $t^{\prime }$ as a function of ${\varepsilon }_F$ extracted from the Fu-Kane model formulated on the lattice (solid symbols). The magnetic unit cell is $30\times 30, \tau =0.5, {\mathrm{\Delta }}_0=0.4$, and $m=0.5$. The solid (red) line corresponds to the simple phenomenological expression (57) discussed in the text. The fit parameters are $A=7.85\times {10}^{-3}, b=0.224$ and $A^{\prime }=5.87\times {10}^{-4}, b^{\prime }=0.170$.

Figure 10

Band structure for the lattice version of the Fu-Kane model with a deformed square vortex lattice, as explained in the text. A $30\times 30$ magnetic unit cell is used with vortices located at $(5,5)$ and $(-5,-5)$ basis vectors. Also $\tau =0.5, {\mathrm{\Delta }}_0=0.4, m=0.5$, and ${\varepsilon }_F=0.0$.