Multiparticle instability in a spin-imbalanced Fermi gas

Weak attractive interactions in a spin-imbalanced Fermi gas induce a multiparticle instability, binding multiple fermions together. The maximum binding energy per particle is achieved when the ratio of the number of up- and down-spin particles in the instability is equal to the ratio of the up- and down-spin densities of states in momentum at the Fermi surfaces, to utilize the variational freedom of all available momentum states. We derive this result using an analytical approach, and verify it using exact diagonalization. The multiparticle instability extends the Cooper pairing instability of balanced Fermi gases to the imbalanced case, and could form the basis of a many-body state, analogously to the construction of the Bardeen-Cooper-Schrieffer theory of superconductivity out of Cooper pairs.

Figure 1

Idealized representation of the spin-imbalanced system showing Fermi surfaces for the down- (light blue circle) and up-spin (light red fragment of circle) species, with shaded areas denoting the allowed momentum states extending outwards by the Debye momentum $k_{\mathrm{D}}$. Also shown are Fermi surface arcs, bounded by thick blue lines for the down-spin species and thick red lines for the up-spin species, for (a) the simplest instability of one up-spin and one down-spin particle, and (b) a proposed multiparticle instability with $(N_{\uparrow },N_{\downarrow })=(2,1)$, indicating the momentum states used in the trial wave functions.

Figure 2

The binding energy per critical species particle as a function of the normalized ratio of number of particles per species at intermediate interaction strength $g=E_{\uparrow \mathrm{F}}$. Results are shown for a free dispersion in $D=2$ dimensions and for different imbalance ratios, with the infinite-interaction-strength limit indicated by a dashed line.

Figure 3

Example discretized momentum states (gray points) for use in exact diagonalization calculations, showing up- and down-spin Fermi surfaces (red and blue curves) with a ratio of ${\nu }_{\uparrow }/{\nu }_{\downarrow }=2$. The origin is marked by the large black point. The subsets of momentum states used in calculations are colored and circled, in this case showing a $(L_{\uparrow },L_{\downarrow })=(16,8)$ system. These states are shown larger, for clarity, on the right-hand side, with sample particle occupations.

Figure 4

The normalized binding energy per critical species particle obtained using exact diagonalization (lines), including the Cooper pair values (orange points).

Figure 5

System size dependence of the binding energy per critical species particle, for a ratio of the number of momentum states $L_{\uparrow }/L_{\downarrow }=2$. The different lines correspond to different ratios of numbers of particles in the instabilities.

Figure 6

The weighting of basis states for the $(N_{\uparrow },N_{\downarrow })=(2,1)$ instability of the $(L_{\uparrow },L_{\downarrow })=(14,7)$ system. Colored lines between the momentum states [represented by large red (up-spin) and blue (down-spin) points] indicate the basis states: each down-spin momentum state is part of a basis state with the two up-spin momentum states joined to it by lines of the same color and thickness. Thicker lines indicate higher weighting [larger $\alpha ({\mathsf{\text{K}}}_{\uparrow },{\mathsf{\text{K}}}_{\downarrow })]$ of the basis states; thinner lines indicate lower weighting. Color indicates the separation of the up-spin momentum states in each basis state, with yellow indicating small separation and purple large separation, with the color key indicating the separation in number of momentum states. Only the 35 highest weighted basis states are shown for clarity. Above the up-spin momentum states and below the down-spin momentum states are the integrated weights of the basis states at each momentum state, indicating the separation of the momentum states into Fermi surface arcs.