Majorana surface states of superfluid ${}^3\mathrm{He}$ $A$ and $B$ phases in a slab

Motivated by experiments on the superfluid ${}^3\mathrm{He}$ confined in a thin slab, we design a concrete experimental setup for observing the Majorana surface states. We solve the quasiclassical Eilenberger equation, which is quantitatively reliable, to evaluate several quantities, such as local density of states (LDOS), mass current for the $A$ phase, and spin current for the $B$ phase. In connection with realistic slab samples, we consider the upper and lower surfaces and the side edges including the corners with several thicknesses. Consequently, the influence on the Majorana zero modes from the spatial variation of the $l$ vector for the $A$ phase in thick slabs and the energy splitting of the zero-energy quasiparticles for the $B$ phase confined in thin slabs are demonstrated. The corner of slabs in the $B$ phase is accompanied by the unique zero-energy LDOS of corner modes. On the basis of the quantitative calculation, we propose several feasible and verifiable experiments to check the existence of the Majorana surface states, such as the measurement of specific heat, edge current, and anisotropic spin susceptibility.

Figure 1

Stereographic views of typical dispersion relation. (a) Dirac valley for the $A$ phase in a thin slab $D=8{\xi }_0$ at the left side edge “L” and the right side edge “R.” (b) Majorana cone for the $B$ phase in a thick slab $D=30{\xi }_0$ at the upper or lower surface. (c) Split Majorana cone for the $B$ phase in a thin slab $D=14{\xi }_0$ at the upper or lower surface. The units of energy and wavelength are $\pi k_B{T}_c$ and $k_F$, respectively.

Figure 2

(a) A schematic configuration of a slab with thickness $D$ along the $z$ direction and macroscopic length $L$ along the $x$ direction. We consider a cross section $A$ $(0⩽x⩽L,0⩽z⩽D)$. (b) Indexes of positions in the cross section $A$. We label $(x,z)=(0,0)$, $(0,D/2)$, $(0,D)$, $(L/2,0)$, and $(L/2,D/2)$ as “1,” “2,” “3,” “4,” and “5.” Paths $z=0$, $z=D/2$, $z=D$, and $z=0.5{\xi }_0$ along the $x$ axis are labeled as “I,” “II,” “III,” and “IV.”

Figure 3

Calculated results for the $A$ phase with $D=8$. Profiles of OP (a) and mass current $j_y$ (b) along the $x$ axis. (c) LDOS $N(E,x)$ at “2” and “5” defined in Fig. 2b. (d) Zero-energy LDOS $N(E=0,x)$ from the edge at $x=0$ to the bulk at $x=L/2$. Angle-resolved LDOS $N(E,x=0,\theta )$ with $\varphi =0^{°}$ (e) and $N(E,x=0,\varphi )$ with $\theta ={90}^{°}$ (f). The peak energy of angle-resolved LDOS at the side edges as a function of $k_z$ for $k_y=0$ (g) and as a function of $k_y$ for $k_z=0$ (h), where “L” (“R”) is for the left (right) edge. In this and the following figures, the units of length, energy, and $A_{\mu i}$, LDOS, and wavelength are ${\xi }_0$, $\pi k_B{T}_c$, $N_0$, and $k_F$, respectively.

Figure 4

The OP for the $A$ phase with $D=14$. (a) Amplitude of each component. Profiles of each component along “I” (b) and along “II” (c). (d) Orientations of $l$-vector texture in the $x$-$z$ plane of a slab.

Figure 5

Calculated results for the $A$ phase with $D=14$. (a) Mass current $j_y(\mathit{r})$ in the $x$-$z$ plane. (b) Profiles of the mass current $j_y(\mathit{r})$ along paths “I,” “II,” and “III.” (c) LDOS $N(E,\mathit{r})$ at “1,” “2,” “3,” and “5.” (d) Zero-energy LDOS $N(E=0,\mathit{r})$ along “I,” “II,” and “III.” Angle-resolved LDOS $N(E,\mathit{r},\theta )$ with $\varphi =0^{°}$ at “1” (e), “2” (f), and “3” (g). (h) Peak energy of angle-resolved LDOS as a function of $k_z$ for $k_y=0$ at “1.”

Figure 6

The OP for the $B$ phase with $D=30$. (a) Amplitude of each component. Profiles of each component along “I” (b) and along “II” (c).

Figure 7

(a) The spin current for the $B$ phase with $D=30$. The unit of the spin current is ${10}^{-9}$ $\mathrm{J}/{\mathrm{m}}^2$. (b) A schematic flow of the spin current for $S_x$, $S_y$, and $S_z$ components.

Figure 8

The LDOS for the $B$ phase with $D=30$. (a) LDOS $N(E,\mathit{r})$ at “1,” “2,” “4,” and “5.” (b) Zero-energy LDOS $N(E=0,\mathit{r})$ along “I,” “II,” and “IV” ($z=0.5$). The peak energy of angle-resolved LDOS as a function of $k_y$ for $k_z=0$ and as a function of $k_z$ for $k_y=0$ at “2” (c), and as a function of $k_x$ for $k_y=0$ and as a function of $k_y$ for $k_x=0$ at “4” (d). (e) and (f) are the same plots as (c) and (d), but at “1.”

Figure 9

The same as in Fig. 8, but for the $B$ phase with $D=14$.

Figure 10

Calculated results for the $B$ phase at “4.” (a) The peak energy of angle-resolved LDOS as a function of $k_x$ for $k_y=0$ and as a function of $k_y$ for $k_x=0$ with several thicknesses $D=30$, $12$, and $8$. (b) The dependence of the energy splitting on the thickness. The dashed line is proportional to $\exp (-D/3{\xi }_0)$.