Trimers in the Extended Hubbard Model

Technical Summary: Trimer Instability in the Extended Hubbard Model

Problem Statement
This work investigates the stability of the Lieb insulating ferrimagnetic phase within the extended Hubbard model on a bipartite coupled trimer chain at half-filling (one electron per site). While the standard Hubbard model on bipartite lattices with nearest-neighbor hopping and on-site repulsion $U$ is known to exhibit a robust ferrimagnetic ground state predicted by Lieb's theorem, the introduction of a nearest-neighbor Coulomb interaction $V$ is expected to induce charge instabilities. The central problem addressed is the evolution of this ferrimagnetic state as $V$ increases, specifically identifying the conditions under which the homogeneous insulating phase becomes unstable to phase separation (PS) and characterizing the nature of the resulting coexisting phases.

Methodology
The authors employ the Density Matrix Renormalization Group (DMRG) method to solve the extended Hubbard Hamiltonian for coupled trimer chains with open boundary conditions. The system consists of $L$ trimers, where each trimer contains three sites labeled $A$, $B_1$, and $B_2$. The Hamiltonian includes:

• Intra-trimer hopping amplitude $t$ (set to $t=1$).
• Inter-trimer hopping amplitude $t'$ (set to $t'=0.8$).
• On-site Coulomb repulsion $U$.
• Nearest-neighbor Coulomb repulsion $V$.

Calculations were performed on chains with sizes up to $L=60$ (total sites $N=180$), utilizing maximum bond dimensions between 512 and 2048, with a truncation error of $\sim 1 \times 10^{-6}$. To probe the ground state properties, the authors analyzed:

1. Spatial averages and local distributions of particle occupancy ($\langle n_i \rangle$) and doublon occupancy ($\langle d_i \rangle = \langle n_{i\uparrow} n_{i\downarrow} \rangle$).
2. Total spin per trimer ($S/L$), calculated via spin correlation functions in the $S_z=0$ sector.
3. Compressibility ($\kappa$), derived from finite-size energy differences to identify incompressible (gapped) versus compressible (metallic/phase-separated) regimes.
4. Homogeneity metrics, specifically $\Delta_{max}$, defined as the maximum deviation of the particle number in any trimer from the homogeneous density of 3 electrons per trimer.

Key Contributions and Results

• Atomic Limit Analysis ($t=t'=0$): The authors first analyzed the static configurations in the atomic limit. They identified that increasing $V$ from zero favors a transition from the standard half-filled state (one electron per site) to a state where $A$ sites are empty and $B$ sites are doubly occupied. This transition occurs at a critical value $V = U/4$, where the energy of the half-filled state ($E/L = 4V$) equals that of the charge-transferred state ($E/L = U$).

• The $U=0$ Case: For vanishing on-site repulsion, any finite $V$ induces phase separation. The system separates into two distinct phases: one phase is rich in doublons on the $B$ sublattice (insulating), and the other is rich in doublons on the $A$ sublattice (metallic). The total spin per trimer is zero ($S/L = 0$) in this regime.

• Stability of the Lieb Ferrimagnetic Phase ($U \neq 0$):

  • For $V=0$, the system exhibits the expected Lieb ferrimagnetic insulating state with $S/L = 1/2$.
  • As $V$ increases, the system remains in a homogeneous insulating phase up to a critical point. However, there is a significant transfer of charge from $A$ sites to $B$ sites, increasing the doublon density on the $B$ sublattice.
  • The authors identify a crossover line near $V \approx U/4$. Below this line, the system is homogeneous. Between the crossover line and the phase-separation line, the system remains homogeneous but exhibits a "doublon-rich" character with increased charge transfer, while maintaining the ferrimagnetic order ($S/L = 1/2$).

• Phase Separation and Coexistence:

  • Above a critical phase-separation (PS) line (observed at $V \gtrsim U/4$), the homogeneous state becomes unstable. The system enters a coexistence region where two phases with distinct trimer occupancies exist spatially.
  • For $U \gtrsim 1$: The coexistence region consists of:
    1. An insulating phase where $B$ sites are predominantly doubly occupied (approx. 4 electrons per trimer), corresponding to a doublon-rich state.
    2. A metallic, unsaturated ferromagnetic phase where the charge distribution is more uniform but lacks the strict half-filling condition. This phase exhibits a net magnetic moment ($S/L > 0$), arising from the interplay of charge itinerancy and the ferrimagnetic alignment of moments on $A$ and $B$ sites.
  • For $U \lesssim 1$: The coexistence region resembles the $U=0$ case, consisting of two paramagnetic phases ($S/L \approx 0$), one doublon-rich on $B$ sites and the other on $A$ sites.

• High-$U$ Behavior: The authors estimate that the crossover line and the PS line merge at approximately $U \sim 16$. For $U > 16$, the system transitions directly from the homogeneous ferrimagnetic phase to the phase-separated region without an intermediate crossover regime, indicated by a jump in spatial averages rather than an inflection point.

Significance and Claims
The paper claims to demonstrate that the Lieb ferrimagnetic insulating phase is remarkably robust against nearest-neighbor repulsion $V$, persisting well beyond the point where doublon density significantly increases. The primary contribution is the mapping of the phase diagram for coupled trimer chains, distinguishing between a homogeneous "doublon-rich" ferrimagnetic regime and the subsequent phase-separated regime.

The authors assert that their results clarify the nature of the ground state instability in this specific lattice geometry. They highlight that above the PS line, the system supports a unique coexistence of an insulating paramagnetic phase and a metallic unsaturated ferromagnetic phase for intermediate to large $U$. The work provides a quantitative estimate of the PS and crossover lines using DMRG, confirming that the transition is driven by the competition between the energy cost of double occupancy ($U$) and the energy gain from avoiding nearest-neighbor repulsion ($V$). The authors conclude that these findings provide a foundation for future investigations into doped models and the topological aspects of such phase transitions, without proposing specific new experimental realizations beyond the theoretical framework.