Interacting spinless fermions in a diamond chain

We study spinless fermions in a flux threaded AB${}_2$ chain taking into account nearest-neighbor Coulomb interactions. The exact diagonalization of the spinless AB${}_2$ chain is presented in the limiting cases of infinite or zero nearest-neighbor Coulomb repulsion for any filling. Without interactions, the AB${}_2$ chain has a flat band even in the presence of magnetic flux. We show that the respective localized states can be written in the most compact form as standing waves in one or two consecutive plaquettes. We show that this result is easily generalized to other frustrated lattices such as the Lieb lattice. A restricted Hartree-Fock study of the $V/t$ versus filling phase diagram of the AB${}_2$ chain has also been carried out. The validity of the mean-field approach is discussed taking into account the exact results in the case of infinite repulsion. The ground-state energy as a function of filling and interaction $V$ is determined using the mean-field approach and exactly for infinite or zero $V$. In the strong coupling limit, two kinds of localized states occur: one-particle localized states due to geometry and two-particle localized states due to interaction and geometry. These localized fermions create open-boundary regions for itinerant carriers. At filling $\rho =2/9$ and in order to avoid the existence of itinerant fermions with positive kinetic energy, phase separation occurs between a high-density phase ($\rho =2/3$) and a low-density phase ($\rho =2/9$) leading to a metal-insulator transition. The ground-state energy reflects such phase separation by becoming linear on filling above $2/9$. We argue that for filling near or larger than 2/9, the spectrum of the t-V AB${}_2$ chain can be viewed as a mix of the spectra of Luttinger liquids (LL) with different fillings, boundary conditions, and LL velocities.

Figure 1

(a) The AB${}_2$ chain consists of a one-dimensional array of diamond rings. Under magnetic field, each diamond ring is threaded by a magnetic flux $\phi$ and the inner star is threaded by a flux ${\phi }_i$. This system can be pictured as an outer ring with flux ${\phi }_o$ and an inner ring with a flux ${\phi }_i$. (b) The diamond chain when ${\phi }^{\prime }=0$.

Figure 2

The Hamiltonian of the diamond chain threaded by an arbitrary flux can be mapped into a noninteracting periodic Anderson-like model with a basis of three sites per unit cell and with hopping amplitudes controlled by magnetic flux.

Figure 3

Eigenstates of the tight-binding AB${}_2$ chain for (a) $\phi =0$ and (b) $\phi =\pi$. For zero flux, itinerant (red dashed line) as well as localized eigenstates (blue dotted line) are present. For $\phi =\pi$, all states are localized.[34]

Figure 4

Dispersion relation of the diamond chain for different flux values (we have omitted the flat band ${\epsilon }_k=0$ since it is not affected by the flux). For $\phi =\pi$, all bands are flat reflecting the local $Z$${}_2$ symmetry of the Hamiltonian.

Figure 5

Ground-state energy in the thermodynamic limit as a function of filling for several values of flux. $E_{\mathrm{GS}}$ remains constant when filling the flat band, $\rho \in [1/3,2/3]$, achieving its minimum value in that interval for $\phi =\pi$. For $\phi =\pi$, $E_{\mathrm{GS}}$ has linear behavior in $\rho$ since all bands are flat.

Figure 6

Dispersion relation of a tight-binding ring with eight sites: (a) without flux and (b) threaded by a magnetic flux $\phi =\pi /5$. For zero flux, states $k$ and $-k$ are doubly degenerate (with the obvious exception of $k=0$ and $k=\pi$). When $\phi \ne 0$, the eigenstates of the tight-binding ring with opposite momenta $k$ and $-k$ do not have the same energy, reflecting the loss of time-reversal symmetry.

Figure 7

In the presence of flux, considering two consecutive rings and following first the continuous path (in red) and then the dotted path (in black), the total Peierls phase in that path is zero (the path is clockwise for the left ring and counterclockwise for the right ring). If one constructs standing waves for a ring of $2N_r$ sites with the additional condition of extra nodes at sites A and B, this state can be folded to give an eigenstate of the two rings threaded by flux with nodes at the extremities and at site A.

Figure 8

Localized states in the case of the Lieb lattice with or without magnetic flux. The introduction of magnetic flux extends the localized state to two plaquettes. These plaquettes are equivalent to the quantum rings discussed in the text and the localized state is a standing wave with nodes in the plaquettes vertices.

Figure 9

$V/t$ versus filling mean-field phase diagram for the spinless diamond chain considering nearest-neighbor Coulomb interactions and $N_c=128$. For $\rho <1/3$, we can have a uniform density phase ($\Delta \rho =0$) or a phase with excess of density at A or BC pseudosites ($\Delta \rho >0$ or $\Delta \rho <0$, respectively). For $\rho >1/3$, Pauli's exclusion principle breaks the symmetry between A and BC pseudosites. A uniform density phase is not possible anymore and the density of particles can no longer be situated only at A sites. It can however, for $\rho <2/3$, be situated only at BC pseudosites. For $\rho >1/3$, due to Pauli's exclusion principle, the density of particles is required to be spread among A and BC sites and for $\rho =1$ the order parameter is $\Delta \rho =-1/3$, which implies a uniform density of particles between the real A, B, and C sites.

Figure 10

Order parameter as a function of filling for $V/t=0.5$, 3, and 20 and $N_c=200$. The changes of slope of the order parameter indicate the phase transitions. Note that the absence of a positive solution of $\Delta \rho$ for $\rho >1/3$ reflects the fact that the A and BC pseudosites may be occupied by one and two fermions, respectively. As a consequence, for $\rho <1/3$, the order parameter can be double valued since we are able to localize all fermions at A sites ($\Delta \rho >0$) or at BC pseudosites ($\Delta \rho <0$). Since for $\rho >1/3$, we can no longer localize all fermions at A sites, the order parameter can no longer be positive.

Figure 11

Ground-state energy per site as a function of $\rho$ for $V/t=0$, 1, 3, and 20 and $N_c=200$, obtained in the mean-field approach. For finite $V$, no flat region appears and the minimum energy is shifted to lower filling. The ground-state energy remains negative for $\rho <2/3$. For $\rho >2/3$, the ground-state energy is almost linear in the filling with a large slope since nearest-neighbor pairs are being created.

Figure 12

(a) Two-particle localized state and one-particle (b) itinerant, and (c) localized state involving B and C sites. The filled and empty circles indicate occupied and unoccupied sites, respectively. Gray circles indicate bonding (+) and antibonding ($-$) occupations. The particles in (a) are localized due to the interaction ($V/t=\infty$) and in (c) due to geometric frustration. In (b), the particle is free to hop to the neighboring A sites.

Figure 13

(a) State with localized as well as itinerant particles. (b) Open boundary region where the itinerant fermions in the state given in (a) may move. Note that the open-boundary regions terminate always at B and C sites. (c) Corresponding state of the linear lattice.

Figure 14

(a) Ground-state energy as a function of filling of the AB${}_2$ chain in the strong-coupling limit $V=\infty$ and with $N_c=6$. The curves are the analytical results given by Eqs. (28), (29), and (31) and the dots are the energy levels obtained by numerical diagonalization of the $V=\infty$ AB${}_2$ chain. (b) Ground-state energy in the thermodynamic limit as a function of filling in the strong-coupling limit $V=\infty$. The transition between a metallic and an insulating ground state occurs exactly at $\rho =2/9$ and the minimum energy is obtained when $\rho \approx 0.2$.