Spontaneous particle-hole symmetry breaking of correlated fermions on the Lieb lattice

We study spinless fermions with nearest-neighbor repulsive interactions ($t\text{−}V$ model) on the two-dimensional three-band Lieb lattice. At half-filling, the free electronic band structure consists of a flat band at zero energy and a single cone with linear dispersion. The flat band is expected to be unstable upon inclusion of electronic correlations, and a natural channel is charge order. However, due to the three-orbital unit cell, commensurate charge order implies an imbalance of electron and hole densities and therefore doping away from half-filling. Our numerical results show that below a finite-temperature Ising transition a charge density wave with one electron and two holes per unit cell and its partner under particle-hole transformation are spontaneously generated. Our calculations are based on recent advances in auxiliary-field and continuous-time quantum Monte Carlo simulations that allow sign-free simulations of spinless fermions at half-filling. It is argued that particle-hole symmetry breaking provides a route to access levels of finite doping, without introducing a sign problem.

Figure 1

The bipartite lattice structure of the Lieb lattice (a) with the three orbitals $A,B$, and $C$ per unit cell indicated in gray and the dispersion of the noninteracting system (b) which features a single linearly dispersing cone at the corner of the Brillouin zone.

Figure 2

Distribution of the (squared) order parameter for unbinned data from AF-QMC and CT-INT simulations. The data from the CT-INT simulation show slowly decaying tails which renders the variance ill-defined.

Figure 3

Ground-state energy $E$ for fixed electron density $n$ and various interaction strength $V$ on a $L=2$ lattice from exact diagonalization. The distribution of the charge density for the ground state with $1/3$- and $2/3$-filling is illustrated in the insets. Lines are guide to the eye only.

Figure 4

Density correlation function for $L=12$ and at $V/t=2$ as a function of spatial separation along the lattice axes (a). The inset displays the growing correlation length on a semi-logarithmic scale. (b) shows the finite-size behavior of the (squared) order parameter on a double-logarithmic scale. The data are compatible with a phase transition between $\beta t=2.4$ and 2.5.

Figure 5

Finite-size data collapse (a)–(e) of the (squared) order parameter across the thermal phase transition using the critical exponents of the two-dimensional Ising model for different interaction strength $V$. Panel (f) shows the behavior of the extracted critical temperatures as a function of $V$ compared to mean-field theory (MFT) results and the classical Ising-limit for strong Coulomb repulsion [51].

Figure 6

The one-particle spectral function at temperature (a) $T=0.5$ above $T_c$ and (b) $T=0.2$ below the Ising transition for a $L=10$ lattice and $V/t=2$. The spectrum in (b) can be interpreted as the superposition of the dispersions for (c) 1/3-filling and (d) 2/3 filling (here obtained from mean-field calculations).