Emergence of a molecular quantum liquid in one dimension

Abstract

We investigate the fate of a one-dimensional lattice superfluid formed by hard-core bosons, aka `atoms' (alternatively, a free spinless Fermi sea) subjected to nearest-neighbor attractive Hubbard-like interactions only in subgroups of two sites. The system, as expected, stabilizes a fluid of dimerized molecules at large attractive interactions. However, the composite molecules have an effective meek hopping scale and dominant repulsive interactions solely due to virtual quantum fluctuations. Interestingly, at an intermediate attractive potential, the system realizes a phase-separated region where the system is in an absorbing state. We show that this phase-separated region is due to an emergent attractive interaction between the dimers which leads to a local charge-density wave puddle where particles effectively cluster with local half-filling. Moreover the molecular superfluid gets spontaneously charge-ordered in the addition of an unpaired atom, reflecting the extreme sensitivity of the system to the existence of lone atoms. Using density-matrix renormalization group studies and effective low-energy Hamiltonians, we isolate the quantum processes to uncover the physics behind molecule formation in a strongly interacting one-dimensional system.

Figures (14)

• Figure 1: Formation of molecular superfluid: (a) Fidelity susceptibility $\chi_F$ as a function of $U$ for $L=32$ under OBC at different particle fillings ($N$ particles). (b) Dimer density $\langle N_d \rangle/L_p$ (red circles), SF momentum distribution $\langle N_b(K=0)\rangle$ (blue squares), and DSF momentum distribution $\langle N_d(K=0)\rangle$ (green diamonds) as a function of $U$ for a system of size $L=80$ at $\rho=1/4$ filling. $\langle N_d \rangle/L_p$ and $\langle N_d(K=0)\rangle$ are multiplied by $10$ and $3$ respectively for better visibility. (c) Particle number as a function of chemical potential $\mu$ for $U=2.6$ and $U=5.8$, respectively, for $L=32$ under PBC. The region indicated by the red rectangle is enlarged in the inset. (d) Single-particle correlation function $\Gamma(r)$ and pair correlation function $\Gamma^{d}(r)$ on a log-log scale at $\rho=1/4$ for $U=2.0$ and $U=6.0$, for $L=80$ under OBC.
• Figure 2: Phase diagram: (a) Phase diagram in the $\rho-U$ plane for different system sizes under PBC. The horizontal solid line at $\rho=0.5$ indicates the gapped phases. (b) Phase diagram of the model in the $U$–$\mu$ plane with PBC. Shaded regions are the gapped phases, and the white regions are gapless phases. The zoomed-in portion of the magenta rectangular region is shown in the inset. The red star in (a) and (b) marks the transition point from the BO to the d-CDW phase at half-filling. (c) shows the particle number as a function of $\mu$ for $U=3.0$ and $U=5.3$, respectively, for $L=56$ under PBC. The zoomed-in version of the red rectangular region is shown in the inset.
• Figure 3: Intermediate-$U$ range: Fidelity susceptibility, $\chi_F/L$, as a function of $U$ for (a) odd particle filling shown for a system of size $L=32$ calculated under OBC. (b) The ground state energy for odd particle sectors ($E_{2N+1}$) is compared against the ground state energy of $N$ dimers and one lone particle for a system of size $L=80$. (c) The ground state energy for even particle sectors ($E_{2N}$) is compared against the ground state energy of $N$ dimers for a system of size $L=80$. (d) shows the on-site particle occupation for two different values of $U$, when both even and odd numbers of particles are present for a system of size $L=80$.
• Figure 4: Local charge order:$\chi_F/L$ (red squares) and the CDW structure factor $S^d(\pi)$ (blue circles) as functions of $U$, (a) at quarter filling, and (b) for one particle more than quarter filling with $L=48$. All results are obtained with OBC.
• Figure 5: Ground state energies obtained from three different approaches for a system of size $L=32$ at $1/4$ filling under PBC are shown on the left axis, namely, the low-$U$ Fermi-sea expression given in Eq. \ref{['eq:lowUeq']} (red diamonds), the energies from the effective dimer Hamiltonian in Eq. \ref{['eq:highUeq']} (green squares), and the results from the actual Hamiltonian in Eq. \ref{['eq:Ham']} (red circles). $\chi_F/L$ is plotted as a function of $U$ and displayed on the right axis using blue stars.