Signatures of the BCS-BEC crossover in the yrast spectra of Fermi quantum rings

We study properties of the lowest energy states at nonzero total momentum (yrast states) of the Hubbard model for spin-$\frac{1}{2}$ fermions in the quantum ring configuration with attractive on-site interaction at low density. In the one-dimensional (1D) case we solve the Hubbard model using the Bethe ansatz, while for the crossover into the 2D regime we use the Full-Configuration-Interaction Quantum Monte Carlo method to obtain the yrast states for the spin-balanced Fermi system. We show how the yrast excitation spectrum changes from the 1D to the 2D regime and how pairing affects the yrast spectra. We also find signatures of fragmented condensation for certain yrast states usually associated with dark solitons.

Figure 1

Comparison of yrast state excitation energies for a 1D Hubbard chain of length $L=21$ with $N=10$ fermions, obtained for interaction strengths $U/t=0,-1,-2,-3,-4,-5$ using the Bethe ansatz (empty symbols, dashed lines) and FCIQMC (filled symbols, solid lines). For $|U/t|\le 5$, the agreement is sufficient for reproducing the main features of the yrast dispersion. The lines are merely a guide to the eye.

Figure 2

Yrast spectrum for a lattice of $21\times 11$ sites and interaction values $U/t=0,-1,-2,-3,-4,-5$. The spectrum changes from having linear segments to a parabolic shape as in the 1D case depicted in Fig. 1.

Figure 3

Opposite-spin pair-correlation function $g_{\uparrow \downarrow }(k_1,k_2)=\langle c_{k_1,\uparrow }^{†}{c}_{k_2,\downarrow }^{†}{c}_{k_2,\downarrow }{c}_{k_1,\uparrow }\rangle$ for the ground state (left) and the half umklapp point (right) at $U/t=-1$ (top) and $U/t=-5$ (bottom). The crossover from a weakly interacting Fermi gas to a BEC of bound pairs is clearly visible, as the half umklapp point changes its characteristics from one spin component shifted in momentum space with respect to the other to a translation of the whole system. The half umklapp point of the Fermi system becomes the first full umklapp point of a fully paired superfluid with 5 bosonic pairs.

Figure 4

Excitation energy for the half umklapp point $P_h=5P_0$ (purple data set at lower energies) and for the full umklapp point $P_u=10P_0$ (green data set at higher energies) for different values of interaction strength $U$ and width $W$. Solid lines are calculated for $W=1$ using the Bethe ansatz and symbols denote FCIQMC results for the 1D system in (a) and for 2D systems in (b). The horizontal dashed lines show the behavior expected in the continuum 1D Fermi gas (YG model) as explained in the main text, while the dotted lines show the expected asymptotic behavior for $-U/t\gg 1$. While the excitation energy at the full umklapp point appears to be largely independent of the transverse lattice size indicating a full translation of both Fermi seas in momentum space, the half umklapp point shows additional features of a mesoscopic system for different lattice sizes. Interestingly, the values for $W=11$ approach the other points from below. For this particular value of $W$, a rearrangement rather than a simple shift of the fermions in momentum space takes place.

Figure 5

Inertial mass of the first yrast maximum vs interaction strength. Unlike in free space, the inertial mass in the lattice asymptotically approaches $-\infty$ linearly with the interaction strength $U/t$. For quantum rings with larger width ($W=9,11$) and at smaller interaction strength, shell effects distort the shape of the yrast dispersion to be nonparabolic such that extraction of the inertial mass is not possible (and hence no data are shown in this regime). In the 2D regime with $W=11$ and $|U/t|⪆4$, a parabolic yrast dispersion shape reappears (see Fig. 2) and the inertial mass is larger by a factor of two. Lines are a guide to the eye.

Figure 6

Pair densities for 2D lattices with $21\times 11$ sites. The weakly interacting case ($U/t=-1$, top row) shows a small peak in the pair density at zero momentum both for the ground state (top left) and the yrast state at $P/P_0=2$ (top right). For strong interactions ($U/t=-5$, bottom row), the peak at zero momentum dominates at $P=0$ (bottom left), while at $P/P_0=2$ (bottom right), a second peak appears, with the system showing condensation into both zero momentum and momentum $\hslash k_x=2P_0$.

Figure 7

Pair densities for the 1D ground state (top left), the yrast states with $P/P_0=2$ (top right), $P/P_0=3$ (bottom left), and the half umklapp point (bottom right). The ground state shows a rapid growth of the pair density at zero momentum, as expected for Bose condensation of pairs, while the half umklapp point is identical to the ground state but shifted by one lattice momentum unit. For the yrast states, however, we observe near equal growth of both pair momenta $P=0$ and $P=P_0$, meaning that fragmented condensation into zero and nonzero momentum states is taking place.