Exact diagonalization study of trionic crossover and trion liquid in the attractive three-component Hubbard model

We investigate the trion formation and the effective trionic properties in the attractive Hubbard model with three fermionic colors using exact diagonalization. The crossover to the trionic regime with colorless compound fermions upon increasing strength of the onsite attraction parameter $U$ features smoothly evolving ground-state properties and exhibits clear similarities to the BCS/BEC crossover for two colors. In the excitation spectrum, there is a clear gap opening between a band of well-defined trions and excitations of broken-up trions at $U_c\sim 1.8t$. This picture remains the same away from the SU(3)-symmetric point. The spatial pairing correlations for colored Cooper pairs are compatible with a power law at small attractions and change to an exponential decay above the trionic crossover. Furthermore, we show that the effective trionic liquid for $U>U_c$ can be well modeled with spinless “heavy” fermions interacting with a strong nearest-neighbor repulsion.

Figure 1

(Color online) Data for a 14-site chain, one fermion/color. Right vertical scale: empty circles: energies of the lowest states from ED, $U=t$. Full circles: same for $U=6t$. Left vertical scale: trionic weight $w_t$ for the same states, empty squares $U=t$, and filled squares $U=6t$.

Figure 2

(Color online) Data for a 19-site chain, one fermion/color. Right vertical scale: empty circles: interaction energies of the lowest states from ED, $U=t$. Full circles: same for $U=6t$. Left vertical scale: trionic anticommutator for these states, empty squares $U=t$, and filled squares $U=6t$.

Figure 3

(Color online) Data for a 14-site chain, one fermion/color. Location of the trion gap opening for asymmetric interactions $U_{12}=U_{13}=U>0$ and $U_{23}\ne U$. Inset: trionic gap vs $U/t$ for the symmetric case $U_{12}=U_{13}=U_{23}=U$.

Figure 4

(Color online) Trionic weight and pairing weight in the ground state of 111 (short for $n_1=1,n_2=1,n_3=1$) and 110 systems for a 10-site chain and a 20-site chain.

Figure 5

(Color online) Data for a 12-site chain, 333 filling. Left: logarithmic plot of the pairing correlations $\Pi$ vs distance $d_x$ in the ground state for different $U$. For very weak interaction ($U=0.01$, open squares) the data show a dip, a $k_F{d}_x=\pi$, as expected from the noninteracting case. One can also observe the nearly exponential decay for strong interactions ($U=6.00$, filled squares). For intermediate attraction ($U=1.60$, open circles) near the crossover to trions the pairing correlations are largest. Right: logarithmic mapping of the pairing correlation for distances up to half the system size and attractions $0\le U\le 5$ and contour lines. The solid line shows the $U$ dependence of the pairing correlations maximum. Top: fitting parameter $\lambda$ (with error bars) discriminating the weakly interacting regime from the regime with exponential decay of pairing. Also shown are the fit results for the exponential decay factor $f$, which have huge error bars for small interaction where $\lambda$ is near zero and the exponential decay plays only a negligible role.

Figure 6

(Color online) Data for a six-site chain, one fermion/color for respective upper graph of (a) or (b), and two fermions/color in the lower plots. The dashed line marks the trionic Fermi level, separating the trion emission from the trion addition spectrum. The trionic spectral function (lowest feature in the plots) shows a cosine band of trion states. In the lower plots in (a) and (b) this band shows a gap opening above the Fermi level at $\left|k\right|=\pi /2$ and some backfolding for the case of two trions. This gap [in the lower plot of (a) between $\omega \approx -12.5t$ and $\omega \approx -12t$] is due to the nearest-neighbor interaction between the trions. It scales with $t^2/\left|U\right|$ and is larger than typical finite-size effects of the system measured by energy differences in the band between neighbored $k$ values. The width of the trion band with the gap subtracted scales as $t^3/U^2$. In the upper plots of (a) and (b) we observe a second, flatter band in the trion addition spectrum which is split up from the lower ones again by an energy $\sim t^2/\left|U\right|$. The split occurs due to the interaction of the added trion with the initial state trion. A similar third band is found in the lower plots of (a) and (b), arising from adding a third trion. There is also an additional split up of the $k=\pm \pi /3$ excitations that seems to decrease in the same way as the bandwidth gets smaller with larger $\left|U\right|$.

Figure 7

(Color online) Data for a 16-site chain vs $4\times 4$ square cluster, 111 filling for effective hopping (left vertical scale), 222 filling to find effective repulsion (right vertical scale). Left vertical scale, open symbols: the 2D hopping (circles) shows an extra term compared to the 1D hopping (squares). Right vertical scale, filled symbols: the effective repulsion in 2D (circles) indicates the larger $U_c$ for the trion regime than in 1D (squares).

Figure 8

(Color online) Left top: data for a six-site chain, two fermions/color, $U=8t$. The spatial trionic correlation function shows a strong next-neighbor repulsion. The value of $⟨n^t\left(0\right)n^t\left(0\right)⟩<1$ reflects the circumstance that the trions are composite particles that are broken up to some degree by quantum fluctuations. Left bottom: data for a $4\times 4$ square cluster, two fermions/color, $U=8t$. Trionic correlation function shows strong next-neighbor repulsion similar to expectations for spinless fermions. Right: enhanced expectation value for trions on (2,2) relative to expectation for spinless fermions and $U$ dependence of enhancement. Best fit uses a $U^{-1}$ term plus an additional $U^{-2}$ term with negative prefactor and has excellent agreement with the data.