Impurity in a Fermi sea on a narrow Feshbach resonance: A variational study of the polaronic and dimeronic branches

We study the problem of a single impurity of mass $M$ immersed in a Fermi sea of particles of mass $m$. The impurity and the fermions interact through an $s$-wave narrow Feshbach resonance, so that the Feshbach length $R_{*}$ naturally appears in the system. We use simple variational ansätze, limited to at most one pair of particle-hole excitations of the Fermi sea, and we determine for the polaronic and dimeronic branches the phase diagram between absolute ground state, local minimum, thermodynamically unstable regions (with negative effective mass), and regions of complex energies (with negative imaginary part). We also determine the closed-channel population which is experimentally accessible. Finally, we identify a nontrivial weakly attractive limit where analytical results can be obtained, in particular for the crossing point between the polaronic and dimeronic energy branches.

Figure 1

The polaronic and dimeronic problems correspond to a discrete state coupled to a continuum. (a) When the discrete state is on the border of the continuum it is expelled by the coupling and remains a discrete state. When the unperturbed energy of the discrete state is inside the continuum the discrete state is either (b) expelled or (c) washed out in the continuum where it gives rise to a resonance with a complex energy. The symbol $+$ corresponds to a discrete state while the symbol $*$ corresponds to a resonance.

Figure 2

Polaronic energy as a function of total momentum $\mathbf{P}=\hslash \mathbf{K}$ and dimeronic energy at $\mathbf{P}=\mathbf{0}$. The thick black line is the solution of the implicit Eq. (15) over the interval of $K$ where it admits a real solution, which corresponds to the discrete state discussed in the text. The dashed black line is the lower border of the continuum for the polaron obtained by solving Eq. (23) with $\mathbf{q}$ on the Fermi surface [39]. The red horizontal solid line is the solution of the implicit Eq. (22) for the dimeronic energy at $\mathbf{P}=\mathbf{0}$, and the red horizontal dashed line corresponds to the upper bound (26) for the lower border of the $\mathbf{P}=\mathbf{0}$ dimeron continuum. The physical parameters are $k_{\mathrm{F}}{R}_{*}=1$, $r=0.1505$, and $1/\left(k_{\mathrm{F}}a\right)=4$. Note that, for this set of parameters, the polaron has a negative effective mass at zero momentum, and the upper bound for the lower border of the $\mathbf{P}=\mathbf{0}$ dimeron continuum is negative.

Figure 3

Phase diagram of system at $\mathbf{P}=\mathbf{0}$ and zero total angular momentum in $\left(k_{\mathrm{F}}{R}_{*}\right)$-$(1/k_{\mathrm{F}}a)$ plane. The left panels describe the polaronic sector while the right panels belong to the dimeronic sector. GS indicates that the corresponding branch is the ground state, and the black solid line marks the polaron-dimeron transition. $m_{\mathrm{pol}\left(\dim \right)}^{*}>0$ indicates that the corresponding branch is metastable with a positive effective mass, the gray-shaded area is the region where the corresponding branch is thermodynamically unstable due to a negative effective mass $m_{\mathrm{pol}\left(\dim \right)}^{*}<0$, the red solid line indicates where the inverse effective mass is zero. For the dimeron, $\mathrm{Im}\left(\Delta E_{\dim }\right)<0$ indicates that its energy is complex with negative imaginary part (which implies a positive real part, so that the dashed red lines correspond to $\Delta E_{\dim }=0$). Note that, in the ground state of our variational ansatz, the effective mass is always positive, as expected, except in a narrow region in the upper-right panel (see Sec. 4f).

Figure 4

Crossing between the polaronic energy Eq. (15) and dimeronic energy Eq. (22), for different values of the mass ratio: $r=0.1505$ (column a), $r=1$ (column b), $r=6.6439$ (column c), and of the Feshbach length, respectively $k_{\mathrm{F}}{R}_{*}=0$ (row 1), $k_{\mathrm{F}}{R}_{*}=1$ (row 2), and $k_{\mathrm{F}}{R}_{*}=10$ (row 3). Thick lines correspond to the ground-state branch while the thin lines indicate that the corresponding branch is metastable. Instead the dashed lines correspond to a negative value of the effective mass. The red circles (dimeron) and black triangles (polaron) with error bars are the Monte Carlo results of Ref. [10].

Figure 5

Dimeron-polaron crossing point $1/{\left(k_{\mathrm{F}}a\right)}_c$ as a function of various parameters. (a) As a function of the Feshbach length $R_{*}$, for three values of the mass ratio ($r=0.1505$, $r=1$, $r=6.6439$ from top to bottom). (b) As a function of the mass ratio $r=M/m$ for three values of the Feshbach length: (b1) $k_{\mathrm{F}}{R}_{*}=0$, (b2) $k_{\mathrm{F}}{R}_{*}=1$, (b3) $k_{\mathrm{F}}{R}_{*}=10$. For each of the figures, $1/{\left(k_{\mathrm{F}}a\right)}_c$ is the line in the plane $\left(k_{\mathrm{F}}{R}_{*}\right)$-$(1/k_{\mathrm{F}}a)$ or $(M/m)$-$(1/k_{\mathrm{F}}a)$ that separates the region where the dimeron is the ground state (above that line) from the region where the polaron is the ground state (below that line). Solid lines show the numerical solution from the full polaronic and dimeronic ansätze. Dashed lines show the asymptotic analytical prediction for $k_{\mathrm{F}}{R}_{*}\to +\infty$ (59). The dashed line is barely distinguishable from the solid line for $r=1$ in (a) and for (b3). Although it does not a priori make sense, we have plotted this asymptotic prediction even in the regimes $k_{\mathrm{F}}{R}_{*}\lesssim 1$ or $-1\lesssim 1/{\left(k_{\mathrm{F}}a\right)}_c$ to show that it remains quite close to the numerical results even in these nonasymptotic regimes.

Figure 6

Inverse effective mass of dimeron (red lines) and of polaron (black lines) for different values of Feshbach length, respectively $k_{\mathrm{F}}{R}_{*}=0$ (row 1), $k_{\mathrm{F}}{R}_{*}=1$ (row 2), $k_{\mathrm{F}}{R}_{*}=10$ (row 3), and different values of mass ratio: $r=0.1505$ (column a), $r=1$ (column b), and $r=6.6439$ (column c). The vertical dotted lines are in correspondence of the crossing value $1/{\left(k_{\mathrm{F}}a\right)}_c$: On the left of this line, the ground state is polaronic while on the right the system has a dimeronic ground state. Thick lines correspond to the ground state, thin lines indicate where the system is metastable, and dashed lines indicate where the effective mass is negative. The red circles (dimeron) and black triangles (polaron) with error bars correspond to the Monte Carlo results of Ref. [10]. The inverse effective mass of the polaron tends to $m/m_{\mathrm{pol}}^{*}\to 1/r$ in the limit where $a\to 0^{-}$, while the inverse effective mass of the dimeron tends to $m/m_{\dim }^{*}\to 1/(1+r)$ in the limit where $a\to 0^{+}$. The gray-shaded area in (a1) indicates an instability of the system (within our $\mathbf{P}=\mathbf{0}$, $\ell =0$ variational calculation): The ground branch is dimeronic and its effective mass is negative.

Figure 7

Closed-channel population (with $\mathbf{P}=\mathbf{0}$) as a function of $1/\left(k_{\mathrm{F}}a\right)$ for different values of the mass ratio: $r=0.1505$ (a), $r=1$ (b), and $r=6.6439$ (c). These correspond to the three bottom panels of Fig. 4, where $k_{\mathrm{F}}{R}_{*}=10$. Thick lines correspond to the ground state while thin lines indicate that the corresponding branch is metastable. The vertical dotted lines are in correspondence of the polaron-to-dimeron crossing point: On the left, the ground state is polaronic and on the right it is dimeronic.

Figure 8

Energy of polaron calculated at $1/\left(k_{\mathrm{F}}a\right)=0$, as a function of $k_{\mathrm{F}}{R}_{*}$ for different values of the mass ratio: $r=1$, $r=0.1505$, and $r=6.6439$. Thick lines indicate a polaronic ground state, while thin lines indicate that the polaronic branch is metastable and the ground state is dimeronic. Dashed lines indicate that the effective mass is negative. Vertical dotted lines indicate the crossing point between the dimeronic and polaronic branches (from left to right: $r$ in decreasing order). The inset is a magnification.

Figure 9

Energy of dimeron calculated at $1/\left(k_{\mathrm{F}}a\right)=0$, as a function of $k_{\mathrm{F}}{R}_{*}$ for different values of mass ratio: $r=1$, $r=0.1505$, and $r=6.6439$. Thick lines indicate a dimeronic ground state, while thin lines indicate that the dimeronic branch is metastable and the ground state is polaronic. An abrupt stop of the energy lines indicate that the dimeronic branch becomes complex. Vertical dotted lines indicate the crossing point between the dimeronic and polaronic branches (from left to right: $r$ in decreasing order). The inset shows the corresponding closed-channel occupations for the three different mass ratios; the curves overlap and cannot be distinguished at the scale of the figure.

Figure 10

Energy of the dimeronic branch $\Delta E_{\dim }\left(\mathbf{0}\right)=E_{\dim }\left(\mathbf{0}\right)-E_{\mathrm{FS}}\left(N\right)$ at $\mathbf{P}=\mathbf{0}$, calculated for equal masses $r=1$ and at $k_{\mathrm{F}}{R}_{*}=1$ (in red): (a) real part, (b) imaginary part (with error bars indicating the numerical uncertainty due to discretization of the momentum variables). The real negative branch for $\Delta E_{\dim }\left(\mathbf{0}\right)$ is either below (red thick line) or above (red thin line) the polaron energy $\Delta E_{\mathrm{pol}}\left(\mathbf{0}\right)$ (black line). As soon as $\Delta E_{\dim }\left(\mathbf{0}\right)$ crosses the zero threshold the dimeronic energy becomes complex with a small imaginary part that tends to $0^{-}$ when $a\to 0^{-}$; in (a) the real part of the dimeronic energy is then indicated with a red dashed line.

Figure 11

Polaron and dimeron inverse effective mass calculated for $r=0.125$ and for two different values of the Feshbach length: (a) $k_{\mathrm{F}}{R}_{*}=0$ and (b) $k_{\mathrm{F}}{R}_{*}=1$. The convention for the lines is as in Fig. 6. The gray-shaded area indicates what we call the instability region, where the ground state of the system within our variational ansatz is dimeronic with a corresponding negative effective mass.

Figure 12

Dimeron-to-cooperon energy difference (61) divided by ${\left(k_{\mathrm{F}}a\right)}^2$ as a function of $k_{\mathrm{F}}a$, calculated for a mass ratio $r=6.6439$ at $s=s_c={[r/(1+r)]}^{1/2}$ where $s$ is defined by Eq. (41). To leading order in the nontrivial weakly attractive limit, $s=s_c$ corresponds to the polaron-to-dimeron crossing point. The horizontal line is the analytic Eq. (66), which is reached by the full numerical calculation only in the limit $a\to 0^{-}$ (with $a{R}_{*}$ fixed). The dashed line is the slope of the linear terms in the numerical data.