Emergent mode and bound states in single-component one-dimensional lattice fermionic systems

We study the formation of bound states in a one-dimensional, single-component Fermi chain with attractive interactions. The phase diagram, computed from DMRG (density matrix renormalization group), shows not only a superfluid of paired fermions (pair phase) and a liquid of three-fermion bound states (trion phase), but also a phase with two gapless modes. We show that the latter phase is described by a two-component Tomonaga-Luttinger liquid (TLL) theory, consisting of one charged and one neutral mode. We argue based on our numerical data, that the single, pair, and trion phases are descendants of the two-component TLL theory. We speculate on the nature of the phase transitions amongst these phases.

Figure 1

Central charge as a function of interactions in the lattice model (1) computed at filling fraction $1/5$. Four phases labeled are identified with further analysis of correlation functions. We have checked that all the reported phases exist for $V_1=2V_2$ and we expect the qualitative features of the phase diagram to hold for generic values of $V_1\ne V_2$ in the vicinity of $V_1=V_2$.

Figure 2

Verification of predicted decay exponents. Left: leading decay exponent (${\eta }_2$) of pair correlator in pair phase at $V_1=V_2=-0.8, V_3=1.4$. Right: leading decay exponent (${\eta }_3$) of trion correlator in trion phase at $V_1=V_2=-1, V_3=1.4$. The two lines are predictions from TLL theory (cf. Table 1), ${\eta }_2=\frac{1}{2K}$ and ${\eta }_3=\frac{1}{2}(K+\frac{1}{K})$. The values of Luttinger parameter $K$ are extracted from the neutral sector [29]. In order to cover a larger range of $K$, we use DMRG data from fillings (left to right) $\frac{1}{5},\frac{1}{6},..., \frac{1}{10}$.

Figure 3

Spectra $G_1, G_2$, and $G_3$ (from top to bottom) as a function of wave vector and interaction strength ($V_1=V_2$), showing agreement of peak locations between DMRG and theory. The data is taken at cuts shown in Fig. 1. Plots (a)–(c) taken at $V_3=1.56$ show the trion, 2M, and pair phases; plots (d)–(f) taken at $V_3=1.3$ show the trion and single phases (with a possible 2M phase in between). Darker (blue) colors represent larger values of amplitudes. The peak in the data of $G_1$, which continuously varies between 0 and $k_{\text{F}}/3$ in the 2M/single phase, is identified as $k^{\prime }$. The lines added to the color plot are theoretic predictions with the determined parameter $k^{\prime }$. The solid lines denote several long-distance $k_{\text{osc}}$ associated with algebraic decay; the dotted lines denote several exponential-decay “peaks,” which are possibly visible if the decay length scale is large. The parameters for the plots are explained in the Supplemental Material [29].