Theory of superfluids with population imbalance: Finite-temperature and BCS-BEC crossover effects

In this paper we present a very general theoretical framework for addressing fermionic superfluids over the entire range of BCS to Bose Einstein condensation (BEC) crossover in the presence of population imbalance or spin polarization. Our emphasis is on providing a theory which reduces to the standard zero temperature mean-field theories in the literature, but necessarily includes pairing fluctuation effects at nonzero temperature within a consistent framework. Physically, these effects are associated with the presence of preformed pairs (or a fermionic pseudogap) in the normal phase, and pair excitations of the condensate, in the superfluid phase. We show how this finite $T$ theory of fermionic pair condensates bears many similarities to the condensation of point bosons. In the process we examine three different types of condensate: the usual breached pair or Sarma phase and both the one- and two-plane-wave Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) states. The last of these has been discussed in the literature albeit only within a Landau-Ginzburg formalism, generally valid near $T_c$. Here we show how to arrive at the two-plane-wave LOFF state in the ground state as well as at general temperature $T$.

Figure 1

Diagrams for Sarma states or alternatively one-plane-wave LOFF states including (a) bare Green’s function $G_0$. (b) vertices from pairing, (c) full Green's function $G$, and (d) anomalous Green's function $⟨c_{\mathbf{k}\uparrow }{c}_{-\mathbf{k}\downarrow }⟩$ for Sarma states and $⟨c_{\mathbf{k}\uparrow }{c}_{-\mathbf{k}+\mathbf{q}\downarrow }⟩$ for single-plane-wave LOFF states. Here the thin and thick lines represent bare and full Green's functions, respectively. The momenta for Sarma states are $(\mathbf{k}\downarrow ,-\mathbf{k}\uparrow )$ and $(\mathbf{k}\uparrow ,-\mathbf{k}\downarrow )$, while for single-plane-wave LOFF states, they are $(\mathbf{k}\downarrow ,-\mathbf{k}+\mathbf{q}\uparrow )$ and $(\mathbf{k}\uparrow ,-\mathbf{k}+\mathbf{q}\downarrow )$.

Figure 2

Diagrams representing the original Ovchinnikov-Larkin theory for (a) noninteracting Green’s function $G_0$, (b) vertices from pairing with momentum $\mathbf{q}$, (c) vertices from pairing with momentum $-\mathbf{q}$, (d) Green’s function $G$, and (e) anomalous Green's function $⟨c_{\mathbf{k}\uparrow }{c}_{-\mathbf{k}+\mathbf{q}\downarrow }⟩$.

Figure 3

Diagrams associated with the $3\times 3$ Bogoliubov diagonalization theory for (a) noninteracting Green’s function $G_0$, (b) vertices from pairing with momentum$\mathbf{q}$, (c) vertices from pairing with momentum $-\mathbf{q}$, (d) Green’s function $G$, (e) anomalous Green's function $⟨c_{\mathbf{k}\uparrow }{c}_{-\mathbf{k}+\mathbf{q}\downarrow }⟩$. Here the “+” and “−” signs on the vertices indicate momenta $+\mathbf{q}$ and $-\mathbf{q}$, respectively.

Figure 4

$T$-matrix and self-energy diagrams for the present $T$-matrix scheme. The self-energy comes from contributions of both condensed $\left({\Sigma }_{\mathit{sc}}\right)$ and noncondensed $\left({\Sigma }_{\mathit{pg}}\right)$ pairs. Note that there is one dressed Green’s function in the $T$-matrix. Here $t_{\mathit{pg}}$ represents the propagator for the noncondensed pairs.