Phase diagram of hard-core bosons on a zigzag ladder

We study hard-core bosons with unfrustrated nearest-neighbor hopping $t$ and repulsive interaction $V$ on a zigzag ladder. As a function of the boson density $\rho$ and $V/t$, the ground state displays different quantum phases. A standard one-component Tomonaga-Luttinger liquid is stable for $\rho <1/3$ (and $\rho >2/3$) at any value of $V/t$. At commensurate densities $\rho =1/3$, $1/2$, and $2/3$ insulating (crystalline) phases are stabilized for a sufficiently large interaction $V$. For intermediate densities $1/3<\rho <2/3$ and large $V/t$, the ground state shows a clear evidence of a bound state of two bosons, implying gapped single-particle excitations but gapless excitations of boson pairs. These properties can be understood by the fact that the antisymmetric sector acquires a gap and a single gapless mode survives. Finally, for the same range of boson densities and weak interactions, the system is again a one-component Tomonaga-Luttinger liquid with no evidence of any breaking of discrete symmetries, in contrast to the frustrated case, where a ${\mathcal{Z}}_2$ symmetry breaking has been predicted.

Figure 1

The two-leg ladder (a) is topologically equivalent to a one-dimensional chain with nearest and next-nearest connections (b).

Figure 2

Examples of classical ground states for different boson densities. Full (empty) circles indicate particles (empty sites). (a) $\rho =1/3$, where the ground state is threefold degenerated; (b) $1/3<\rho <1/2$, where the ground state has a finite entropy; (c) $\rho =1/2$, where the ground state is fourfold degenerated and corresponds to pairs of nearest-neighbor particles.

Figure 3

Boson density $\rho$ as a function of the chemical potential $\mu$ for three different values of the interaction $V/t$. Calculations have been done on a lattice with $L=108$ sites and open boundary conditions.

Figure 4

Entanglement entropy $S$ as a function of the (reduced) block length $x$, for $\rho =5/12$ and for different values of the interaction strength $V/t$. The dashed line indicates the slope $1/6$ (equivalent to $c=1$). The number of sites is $L=216$.

Figure 5

Difference between the energy needed to add a particle ${\mu }^{+}=E(M+1)-E(M)$ and the one needed to remove it ${\mu }^{-}=E(M)-E(M-1)$ [where $E(M)$ is the energy of $M$ particles] as a function of the density $\rho$, for different sizes of a system with $V/t=5$ (upper panel) and $V/t=10$ (lower panel).

Figure 6

Boson density $\rho$ as a function of the chemical potential $\mu$ for positive and negative values of the hopping parameters. Calculations have been done on a system with $L=108$ sites with open boundary conditions. The existence of a plateau at $\rho =1/2$ is clear for $t=-1$, whereas no plateau is present for $t=1$.

Figure 7

Density structure factor $N(q)$ for different boson densities. Calculations have been done on lattices with $L=108$ (solid curves) and $L=60$ (dashed curves) for $V/t=20$.

Figure 8

Position of the peak $\mathrm{q̄}$ of the density-density correlations as a function of the density. The value of $\mathrm{q̄}$ does not depend on the interaction strength $V/t$.

Figure 9

Tomonaga-Luttinger parameter $K$ as a function of the boson density $\rho$ for $V/t=10$.

Figure 10

DMRG results for the density-density correlation functions on systems with open boundary conditions. Left panel: Small-$q$ behavior of $N(q)$ for a $216$-site system. Data for $36$ sites with periodic boundary conditions obtained by Lanczos diagonalizations are also shown. The boson density is $\rho =5/12$, while $V/t=10$. A linear fit of numerical DMRG data gives $K=0.27\pm 0.01$, while for Lanczos data $K=0.29\pm 0.02$. Apart from the system sizes, the small discrepancies could be ascribed to different boundary conditions imposed in the two cases. Right panel: Size scaling of the peak of the structure factor $N(\mathrm{q̄})\propto L^{\alpha }$. The exponent $\alpha =0.71\pm 0.01$ is in good agreement with the value $1-K$ obtained from the previous fits. Dashed lines are fits of the numerical results.