Surface transport coefficients for three-dimensional topological superconductors

We argue that surface spin and thermal conductivities of three-dimensional topological superconductors are universal and topologically quantized at low temperature. For a bulk winding number $\nu$, there are $\left|\nu \right|$ “colors” of surface Majorana fermions. Localization corrections to surface transport coefficients vanish due to time-reversal symmetry (TRS). We argue that Altshuler-Aronov interaction corrections vanish because TRS forbids color or spin Friedel oscillations. We confirm this within a perturbative expansion in the interactions, and to lowest order in a large-$\left|\nu \right|$ expansion. In both cases, we employ an asymptotically exact treatment of quenched disorder effects that exploits the chiral character unique to two-dimensional, time-reversal-invariant Majorana surface states.

Figure 1

In a disordered metal (a), electrons scatter off of both impurities and density Friedel oscillations; the latter produce Altshuler-Aronov (AA) quantum conductance corrections [15, 16, 17]. The Majorana fluid (b) at the surface of a TSC remains featureless in any realization of nonmagnetic disorder, as no relevant (spin, mass, or color) density can become nonzero without breaking time-reversal symmetry. We therefore expect that surface transport coefficients for bulk TSCs are free of AA corrections. By contrast, AA corrections are ubiquitous in other 2D Dirac systems, including the surface states of 3D topological insulators [4], and nodal quasiparticles in time-reversal-invariant, nontopological superconductors [24, 25, 26, 27].

Figure 2

First-order Hartree (a) and Fock (b) interaction corrections to the spin conductivity in classes CI and AIII, and examples of second-order corrections [(c), (d)] for class AIII. The dashed lines indicate the spin current operators. The solid lines represent the exact noninteracting Matsubara Green's functions in an arbitrary, fixed realization of quenched surface disorder. The wavy lines correspond to the interaction potentials. Both current operators and interaction potentials are local in space. Panels (c) and (d) depict categories of second-order corrections that are subleading in the inverse winding number. Each correction in (a) and (b) vanishes individually, while those in each category (c) and (d) sum to zero. In Sec. 4b, we show that a similar cancellation occurs for all second-order corrections. We sketch a proof that the same mechanism works to all orders in Sec. 4c, implying that Altshuler-Aronov corrections do not exist for Majorana surface transport coefficients.

Figure 3

Class AIII model on the diamond lattice. Sublattice A (B) sites are indicated by red (blue) spheres, and are subject to the potential ${\mu }_s \left(-{\mu }_s\right)$. (a) Nearest-neighbor hopping with amplitude $t^{\prime }$. (b) BCS pairing within each of the sublattices. The red lines indicate $d$-wave spin-singlet pairing in the $xy$ plane. The blue lines indicate $p$-wave $z$-axial spin-triplet pairing in the $yz$ plane. (c) and (d) Next-nearest-neighbor hopping in the $xz$ plane. The solid and dashed lines mean that the hopping amplitudes are $+t$ and $-t$, respectively.

Figure 4

Diagram showing the emergence of the Dirac points of the Hamiltonian (3.15) in the first Brillouin zone of the diamond lattice, obtained from Eq. (3.20). The red dashed lines indicate the Fermi surface at half filling when $t={\mu }_s=\Delta =0$. Nonzero $\Delta$ gaps most of the the Fermi surface except for the massless Dirac nodes indicated by the red spots [Eq. (3.21)]. Mass gaps open at those Dirac nodes for nonzero $t$ or ${\mu }_s$, so that topologically trivial or nontrivial superconductors can be obtained depending on the values of these. See Fig. 5.

Figure 5

Topological superconducting phases of Eq. (3.11) as a function of the next-nearest-neighbor hopping strength $t$ and the staggered chemical potential ${\mu }_s$. (a) Phase diagram. The blue lines indicate the phase boundaries where the bulk gap closes. The gray area indicates the topologically nontrivial phases with $\left|\nu \right|=2$, where the superconductor possesses two gapless surface bands. (b) Numerical results for the winding number [Eq. (3.25)], with $\Delta =2$ and $t^{\prime }=4$.

Figure 6

Topological phases of the BCS Hamiltonian (3.11) with pairing potential in Eq. (3.28) and next-nearest-neighbor hopping in Eq. (3.29). (a) Phase diagram. The blue and red lines indicate the phase boundaries where the bulk gap closes. The gray and yellow areas indicate the topological phase with $\left|\nu \right|=1$ (one surface color) and $\left|\nu \right|=2$ (two surface colors), respectively. (b) Numerical results of the winding number [Eq. (3.25)] for $\Delta =2$ and $t^{\prime }=4$.

Figure 7

The second-order interaction corrections to the spin current-current correlation function in class AIII. The Feynman rules are interpreted in the caption for Fig. 2 of the main text.

Figure 8

High-order interaction corrections to the spin current-current correlation function in class AIII. The equations shown are valid up to order of $\Omega$. (a) The current-current correlation function $\delta {\Pi }_{\alpha \alpha ,mn}^{\left(\mathrm{a}\right)}\left(i\Omega \right)$, where the two current density operators $j_{\alpha }\left({\mathbf{r}}_a\right)$ and $j_{\alpha }\left({\mathbf{r}}_b\right)$ locate on different fermion loops. (b) The current-current correlation function $\delta {\Pi }_{\alpha \alpha ,mn}^{\left(\mathrm{b}\right)}\left(i\Omega \right)$, where the current density operators are on one fermion loop.

Figure 9

Feynman rules for class AIII. (a)–(c) Fast field propagators taking the corresponding expressions in Eq. (5.30). (d) Stiffness vertex arising from the coupling between fast and slow fields represented by Eq. (5.34). The numeric labels appearing at the terminals of fermion lines encode replica and frequency indices of fast field $\widehat{Y}$. For the vertex in panel (d) the index “$3^{\prime }\text{""}$ can carry a fast frequency, while “$1\text{""}$ and “$2\text{""}$ carry only the slow ones.

Figure 10

Category ${\mathfrak{D}}_{\text{AIII}}$: Diagrams renormalizing $\lambda$ and ${\lambda }_{\mathrm{A}}$ for class AIII in the replica limit $n\to 0$. ${\mathfrak{D}}_{\text{AIII}}$(a) renormalizes ${\lambda }_{\mathrm{A}}$, and ${\mathfrak{D}}_{\text{AIII}}$(b) and ${\mathfrak{D}}_{\text{AIII}}$(c) renormalize $\lambda$.

Figure 11

Feynman rules for class DIII. (a) The propagator ${\mathsf{P}}_{\lambda }$ in Eq. (5.46b). The first (second) diagram is associated with the contraction rule ${\delta }_{1,4}{\delta }_{2,3}{\delta }_{ad}{\delta }_{bc} \left({\delta }_{1,-3}{\delta }_{2,-4}{\delta }_{ac}{\delta }_{bd}\right)$. We use the “blob” to indicate sign flips of frequencies. (b) The propagator ${\mathsf{P}}_{\mathrm{c}}$ in Eq. (5.46c). The diagram represents the term with the frequency conservation law ${\delta }_{1+3,2+4}$. (c) The stiffness vertex represented by Eq. (5.34) supplemented with a minus sign.

Figure 12

Category ${\mathfrak{D}}_{\text{DIII}}$: Diagrams renormalizing $\lambda$ in class DIII. (a) and (b) Weak antilocalization contribution [Eq. (5.47)]. (c)–(f) Altshuler-Aronov correction [Eq. (5.48)].

Figure 13

Feynman rules for class CI. (a) Disorder-only propagator ${\mathsf{P}}_{\mathrm{O}}$ in Eq. (5.59). The three diagrams represent the three terms (5.59a)–(5.59c) in respective order. We use blob and square nodes to distinguish spin exchange channels. (b) Interaction-dressed propagator ${\mathsf{P}}_{\mathrm{I}}$ in Eq. (5.59). The two diagrams represent the two terms (5.59d) and (5.59e) in respective order. (c) The stiffness vertex that arises from the coupling between fast and slow fields represented by Eq. (5.34).

Figure 14

Category ${\mathfrak{D}}_{\text{CI}}1$: Disorder-only (weak localization) diagrams renormalizing $\lambda$ in class CI. The sum of the diagrams (b)–(e) gives the same contribution as (a). Summation is implied over repeated spin indices. The amplitude is evaluated in Eq. (5.60).

Figure 15

Category ${\mathfrak{D}}_{\text{CI}}2$: Interaction-dressed (Altshuler-Aronov) diagrams renormalizing $\lambda$ in class CI. The diagrams in the left and right columns give identical contributions. Moreover, the sum of the diagrams (c), (e), (g), and (i) gives the same contribution as (a). Summation is implied over repeated spin indices. The amplitude is evaluated in Eq. (5.61).