Effective field theory of boson-fermion mixtures and bound fermion states on a vortex of boson superfluid

We construct a Galilean invariant low-energy effective field theory of boson-fermion mixtures and study bound fermion states on a vortex of boson superfluid. We derive a simple criterion to determine for which values of the fermion angular momentum $l$ there exist an infinite number of bound energy levels. We apply our formalism to two boson-fermion mixed systems: the dilute solution of ${}^3\mathrm{He}$ in ${}^4\mathrm{He}$ superfluid and the cold polarized Fermi gas on the BEC side of the “splitting point.” For the ${}^3\mathrm{He}\text{−}{}^4\mathrm{He}$ mixture, we determine parameters of the effective theory from experimental data as functions of pressure. We predict that infinitely many bound ${}^3\mathrm{He}$ states on a superfluid vortex with $l=-2,-1,0$ are realized in a whole range of pressure $0–20\mathrm{atm}$, where experimental data are available. As for the cold polarized Fermi gas, while only $S$-wave $(l=0)$ and $P$-wave $(l=\pm 1)$ bound fermion states are possible in the BEC limit, those with higher negative angular momentum become available as one moves away from the BEC limit.

Figure 1

(Color online) Solutions of Eq. (38) as functions of $r∕\xi$ for $w=1,2$, and $3$.

Figure 2

(Color online) Solutions of Eq. (42) in the $S$-wave channel $(l=0)$ for $\gamma =1.47$. Left panel: Bound energy levels ${ϵ}_n$ for $n=0,1,\dots ,6$ (dotted lines) and the potential energy $\gamma \left\{h\right(x{)}^2-1\}$ (solid curve) are shown as functions of $x=r∕\xi$. The normalized wave function of the ground state $(n=0)$ is also shown by the dashed curve. Right panel: $\mid {ϵ}_n\mid$ (dotted lines) are shown in the log scale.

Figure 3

(Color online) Solutions of Eq. (42) in the $P$-wave channel $(l=\pm 1)$ for $\gamma =1.47$. Left panel: Bound energy levels ${ϵ}_n$ for $n=0,1,2,3$ (dotted lines) and the potential energy $1∕x^2+\gamma \left\{h\right(x{)}^2-1\}$ (solid curve) are shown as functions of $x=r∕\xi$. The normalized wave function of the ground state $(n=0)$ is also shown by the dashed curve. Right panel: $\mid {ϵ}_n\mid$ (dotted lines) are shown in the log scale.

Figure 4

(Color online) $l_{+}$ and $l_{-}$ in Eq. (60) as functions of $m^{*}∕m$ for $\partial \Delta ∕\partial \mu =4\gamma$. $m^{*}∕m=1$ corresponds to the BEC limit, while $m^{*}∕m\to \infty$ corresponds to the splitting point in Ref. 34. The curves are shown for $m^{*}∕m\ge 0$.