Superfluid ${}^3$He in a restricted geometry with a perpendicular magnetic field

We theoretically investigate the role of surface Andreev bound states (SABSs) on the phase diagram and spin susceptibilities of superfluid ${}^3$He confined to a restricted geometry. We first explicitly derive the dispersion of the SABS in ${}^3$He-B in the presence of a magnetic field, where the Majorana Ising spin and the spin susceptibility contributed from the SABS are associated with the SO$\left(3\right)$ order-parameter manifold. Subsequently, based on the quasiclassical Eilenberger theory with Fermi-liquid corrections, we discuss the nonlinear effect of a magnetic field on the SABS, where the magnetic field is perpendicular to the specular surface. It is directly demonstrated that a gapped SABS strongly enhances the magnetization density and spin susceptibility at the surface, compared with that in the normal ${}^3$He. To capture the characteristics of the SABS, we show the field and temperature dependencies of the spatially averaged susceptibility which is detectable through NMR experiments. It turns out that the contribution of the SABS leads to nonmonotonic temperature dependence of the spin susceptibility. Furthermore, we present the superfluid phase diagram, where the B phase undergoes a first-order (second-order) phase transition to A phase or planar phase at low (high) temperatures.

Figure 1

Schematic picture of the system which is considered in this work. The superfluid ${}^3$He is sandwiched by two specular walls, the distance of which is set to be $D$. In and after Sec. 3, a magnetic field is applied to the $z$ axis which is normal to the wall.

Figure 2

Schematic picture on the relation between ${\widehat{\ell }}_z$ and $\mathit{S}$, where $\mathit{S}$ and ${\mathit{S}}^{\mathrm{M}}$ denote the orientation of the Majorana Ising spins for an arbitrary $\widehat{\mathit{n}}$ and $\mathit{S}$ for $\widehat{\mathit{n}}\parallel \widehat{\mathit{z}}$. The $\widehat{\mathit{z}}$ axis is normal to the specular surfaces as displayed in Fig. 1.

Figure 3

One-to-one correspondence between two points on the three-dimensional Fermi sphere. The quasiclassical Green's function at an arbitrary $O\widehat{\mathit{k}}$ belonging to the path (ii) is obtained by the $\text{SO}{\left(2\right)}_{L_z+S_z}$ rotation of $\underline{g}(\widehat{\mathit{k}},z;i{\omega }_n)$ calculated on the path (i).

Figure 4

(a) Spatial profiles of ${\Delta }_{\parallel }\left(z\right)$ (solid lines) and ${\Delta }_{\perp }\left(z\right)$ (dashed lines) for $D/{\xi }_0=10$, 20, and 40 at $H=0$ and $T=0.2T_{\mathrm{c}0}$. The horizontal axis is scaled with $D$. (b) Field dependence of ${\Delta }_{\parallel }\left(z\right)$ (solid lines) and ${\Delta }_{\perp }\left(z\right)$ (dashed lines) for $D=20{\xi }_0$ and $T=0.2T_{\mathrm{c}0}$, where ${\mu }_{\mathrm{n}}H/\pi T_{\mathrm{c}0}=0$, $0.061$, and $0.096$.

Figure 5

(a) Spatial profiles of the local magnetization density $M_{\mu }\left(z\right)/M_{\mathrm{N}}$, corresponding to the ratio of the local spin susceptibility ${\chi }_{zz}\left(z\right)/{\chi }_{\mathrm{N}}$, for various $D$'s at $T=0.2T_{\mathrm{c}0}$ and ${\mu }_{\mathrm{n}}H/\pi T_{\mathrm{c}0}=0.0122$. (b) Field dependence of ${\chi }_{zz}\left(z\right)$ on the surface $z=0$ (solid line) and $z=10{\xi }_0$ (dashed line) at $T=0.2T_{\mathrm{c}0}$ and $0.6T_{\mathrm{c}0}$ where $D=20{\xi }_0$ is fixed.

Figure 6

$\widehat{\mathit{k}}$-resolved surface density of states $\mathcal{N}({\widehat{k}}_{\parallel },z=0,E)$ for $D=20{\xi }_0$ (a), $12{\xi }_0$ (b), and $10{\xi }_0$ (c), where $H=0$. $\mathcal{N}({\widehat{k}}_{\parallel },z=0,E)$ for $D=20{\xi }_0$ at ${\mu }_{\mathrm{n}}H=0.0488\pi T_{\mathrm{c}0}$ (d) and $0.0854\pi T_{\mathrm{c}0}$ (e). (f) and (g) are for $D=12{\xi }_0$ and $10{\xi }_0$ at ${\mu }_{\mathrm{n}}H=0.0488\pi T_{\mathrm{c}0}$. In all the data, the temperature is set to be $T=0.2T_{\mathrm{c}0}$.

Figure 7

Local density of states $\mathcal{N}(z,E)$ at $z=0$ (the solid line) and $z=10{\xi }_0$ (the dashed line) for ${\mu }_{\mathrm{n}}H=0$ (a), $0.0488\pi T_{\mathrm{c}0}$ (b), and $0.0854\pi T_{\mathrm{c}0}$ (d), where $\mathit{H}$ is perpendicular to the surface. (c), (e) $-\mathrm{Im}{\langle g_{\mu }^{\mathrm{R}}(\widehat{\mathit{k}},z;E)\rangle }_{\widehat{\mathit{k}}}$ in the same condition as (b), (d). In all the data, the temperature and thickness are set to be $T=0.2T_{\mathrm{c}0}$ and $D=20{\xi }_0$.

Figure 8

Local density of states $\mathcal{N}(z,E)$ at $z=0$ (the solid line) and $z=10{\xi }_0$ (the dashed line) for ${\mu }_{\mathrm{n}}H=0.0488\pi T_{\mathrm{c}0}$ (b), where $\mathit{H}\parallel \widehat{\mathit{x}}$ and $\widehat{\mathit{n}}\parallel \widehat{\mathit{z}}$. (b) $-\mathrm{Im}{\langle g_{\mu }^{\mathrm{R}}(\widehat{\mathit{k}},z;E)\rangle }_{\widehat{\mathit{k}}}$ for the same situation as (a). The other parameters are the same as those in Fig. 7.

Figure 9

Field dependence of ${\Delta }_{\mu }\left(z\right)/\pi T_{\mathrm{c}0}$ at $z=20{\xi }_0$ (a) and of thermodynamic potential of the B phase relative to the A phase $\delta {\Omega }_{\mathrm{AB}}$ (b), where $D=40{\xi }_0$ and $T=0.2T_{\mathrm{c}0}$. The solid (dashed) line denotes $\delta {\Omega }_{\mathrm{AB}}$ with (without) the Fermi-liquid corrections and the arrows in (b) point out the first- and second-order transition fields.

Figure 10

Field dependence of ${\Delta }_{\perp }(z=D/2)$ for various values of the thickness $D$ at $T=0.2\pi T_{\mathrm{c}0}$ (solid lines). The open (filled) circles denote the first-order (second-order) phase transition points. The bottom describes the phase boundary between the distorted B phase (the shaded area) and the planar (or A) phase, where the thick (thin) line corresponds to the first-order (second-order) line.

Figure 11

Superfluid phase diagram in the space spanned by temperature $T$, perpendicular magnetic field $H$, and thickness $D$. The definition of the open (filled) circles and thick (thin) lines are same as those in Fig. 10. The shaded area is occupied by the distorted B phase and the other is covered by the planar (or A) phase.

Figure 12

(a) $T$ dependence of the spatially averaged spin susceptibility $\langle {\chi }_{zz}\rangle$ at ${\mu }_{\mathrm{n}}H/\pi T_{\mathrm{c}0}=0.003$ (solid lines), $0.03$ (dashed lines), and $0.06$ (dotted-dashed lines) for $D=12{\xi }_0$, $20{\xi }_0$, and $40{\xi }_0$. The arrows denote $T_{\min }$ in the lowest field (see the text). (b), (c) $T$ dependence of local spin susceptibilities ${\chi }_{zz}\left(z\right)$ at $z=0$ and $D/2$, (b) $D=20{\xi }_0$, and (c) $12{\xi }_0$ at ${\mu }_{\mathrm{n}}H=0.003\pi T_{\mathrm{c}0}$. The thin solid line in (a) and dashed line in (b) and (c) depict the spin susceptibility ${\chi }_{zz}^{\left(\mathrm{bulk}\right)}$ in the bulk B phase given in Eq. (42).