Normal-superfluid interface for polarized fermion gases

Bert Van Schaeybroeck and Achilleas Lazarides

Phys. Rev. A 79, 053612 – Published 8 May 2009

Abstract

Recent experiments on imbalanced fermion gases have proved the existence of a sharp interface between a superfluid and a normal phase. We show that, at the lowest experimental temperatures, a temperature difference between normal (N) and superfluid (SF) phases can appear as a consequence of the blocking of energy transfer across the interface. Such blocking is a consequence of the existence of a SF gap, which causes low-energy normal particles to be reflected from the N-SF interface. Our quantitative analysis is based on the Hartree-Fock–Bogoliubov–de Gennes formalism, which allows us to give analytical expressions for the thermodynamic properties and characterize the possible interface scattering regimes, including the case of unequal masses. Our central result is that the thermal conductivity is exponentially small at the lowest experimental temperatures.

Received 23 July 2008

DOI:https://doi.org/10.1103/PhysRevA.79.053612

Authors & Affiliations

Bert Van Schaeybroeck and Achilleas Lazarides

Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan 200 D, B-3001 Leuven, Belgium

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 79, Iss. 5 — May 2009

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
The N-SF interface (thick vertical line) with the spectra of the species $↑$ and $↓$ on the N side and the gapped $α, β$ spectra in the SF. The long-dashed line cuts the spectra at particlelike (filled dots) and holelike (empty dots) quasiparticle states, all having the same energy. The minimal values of the $α$ and the $β$ spectrum are $2 (\sqrt{\tilde{m}} Δ - ε_{α, β}^{0}) / (\tilde{m} + 1)$ where $ε_{α, β}^{0} \equiv [\pm h_{S} (1 + \tilde{m}) + μ_{S} (1 - \tilde{m})] / 2$ . An incident particle of species $↑$ (curly arrow) with momentum $k_{p}$ and energy $E$ can have up to four scattering channels: the Andreev reflected $k_{h}$ hole, the specularly reflected $- k_{p}$ particle and the transmitted holelike $- k^{h}$ and particlelike $k^{p}$ states.Reuse & Permissions
Figure 2
The various scattering regimes for a particle of species $↑$ which is incident on the interface from the N side, as a function of its “energy” $ε_{α}$ [see Eq. (16a)] and $χ_{p} = ℏ^{2} k_{p}^{2} / 2 m_{↑}$ (where $k_{p}$ is its momentum along the $z$ axis). We assumed here no species mass asymmetry $m_{↑} = m_{↓}$ and no HF potentials $U_{↑, ↓, S} = 0$ ; we depict a generic diagram for the more general case ( $m_{↑} \neq m_{↓}$ and $U_{↑, ↓, S} \neq 0$ ) in Fig. 3. The heavily shaded regimes are energetically forbidden while complete reflection occurs in the lightly shaded regimes. Particles in regime VI may scatter to all states indicated in Fig. 1 by curly arrows. Above line 4 (regimes IV and V), Andreev reflection does not occur, and above curve 3 (regime IV), holelike excitations are also impossible. The functions of the curves 1–4 are given in Eq. (17).Reuse & Permissions
Figure 3
The same applies as in Fig. 2 except here $m_{↑} \neq m_{↓}$ and $U_{↑, ↓, S} \neq 0$ . Regime I is bound by threshold line $ε_{α} = \sqrt{\tilde{m}} Δ$ and the lower limit $ε_{α} = ε_{α}^{0} = [h_{S} (1 + \tilde{m}) + μ_{S} (1 - \tilde{m})] / 2$ . Compared to Fig. 2, it can be seen that curves 1, 2, 3, and 4 are shifted to the right or to the left.Reuse & Permissions
Figure 4
The transmission coefficient $S_{α}^{p}$ for $↑$ particles scattering to the $α$ channel as a function of the energy $ε_{α}$ [see Eq. (16a)] and the variable $χ_{p} \equiv ℏ^{2} k_{p}^{2} / (2 m_{↑})$ , both in units of $μ_{↓}$ , in the case of $m_{↑} = m_{↓}$ and with the HF potentials set to zero. One can distinguish the regimes IV, V, and VI (see diagram of Fig. 2) of nonzero $S_{α}^{p}$ . We choose the value $Δ = 0.5 μ_{↓}$ and used the coexistence condition $h_{S} = Δ / \sqrt{2}$ . Regime I where $ε_{α} < Δ$ is not depicted in this figure since there $S_{α}^{p} = 0$ . For values slightly above the threshold $ε_{α} = Δ$ , one observes the square-root behavior of relation (20). We give the analytical expressions for $S_{α}^{p}$ in Eq. (A5) in the Appendix.Reuse & Permissions
Figure 5
The thermal conductivity $κ$ across the N-SF interface divided by the normal-phase conductivity $κ_{N}$ against $k_{B} T / Δ$ for the unitary, BCS and deep BCS regimes when $m_{↑} = m_{↓}$ and $U_{↑, ↓, S} = 0$ . For $k_{B} T ≲ 0.1 Δ$ , $κ / κ_{N}$ drops dramatically (notice the logarithmic scale). The curve represents the analytical result obtained using of the Andreev approximation, Eq. (23). At unitarity, $Δ = 0.69 k_{B} T_{F}$ , in the BCS case $Δ = 0.25 k_{B} T_{F}$ and in the deep BCS case $Δ = 0.002 k_{B} T_{F}$ . For large values of the temperature, one finds that $κ / κ_{N} = 1$ .Reuse & Permissions
Figure 6
The thermal conductivity $κ$ across the N-SF interface divided by the normal-phase conductivity $κ_{N}$ against $T / T_{F}$ at the interface in the unitary regime. The full line depicts the Andreev approximation wherein the coexistence condition is taken from the one-channel model. The dotted line is the Andreev approximation with the coexistence condition is taken from Monte Carlo simulations [22] and a superfluid local-density approximation [33]. At the lowest experimental values, that is when $T < 0.03 T_{F}$ , the conductivity is already very low.Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information