Quantum phases of hardcore bosons in two coupled chains: A density matrix renormalization group study

We consider hardcore bosons in two coupled chains of one dimensional lattices at half filling with repulsive intrachain interaction and interchain attraction. This can be mapped onto a coupled chain of spin-1/2 $XXZ$ model with interchain ferromagnetic coupling. We investigate various phases of hardcore bosons (and related spin model) at zero temperature by a density matrix renormalization group method. Apart from the usual superfluid and density wave phases, pairing of interchain bosons leads to the formation of phases like pair superfluid and density wave of strongly bound pairs. We discuss the possible experimental realization of such correlated phases in the context of cold dipolar gas.

Figure 1

Schematic of the two chains with dipolar bosons. There is nearest-neighbor repulsive interaction $V$, and nearest-neighbor hopping parameter $t$, in each of the chains. Both chains are coupled with on-site attractive interaction $U$, while there is no hopping between the two chains.

Figure 2

Two-dimensional phase diagram in the phase space of two parameters, $U$ and $V$. The phase diagram is quite rich with phases, namely, superfluid (SF), pair-superfluid (PSF), and density wave (DW) phases.

Figure 3

Plot of correlation function $C_1\left(r\right)$, as a function of $r$, at $U=0.5$ and different values of $V$ [$V=0.4$ (square), $V=1.0$ (triangle), and $V=1.4$ (circle)]. Inset shows scaling of L/${\xi }_L^1$ as a function of $V$ for $U=0.5$. Coalescence of the data points of different L shows SF-DW transition at $V=1.1\pm 0.05$.

Figure 4

Plot of correlation function ${(-1)}^r{G}_1\left(r\right)$, as a function of $r$, at $U=0.5$ and different values of $V$ [$V=0.4$ (circle), $V=1.0$ (square), $V=1.2$ (triangle), and $V=1.4$ (diamond)]. Inset shows finite-size scaling of $O_G\left(L\right)$, at $U=0.5$, and for different values of $V$.

Figure 5

(a) Finite-size scaling of $K_C\left(L\right)$, at $U=0.5$ and different values of $V$. (b) Plot of the extrapolated values of $K_C\left(L\right)$ vs $V$ for $U=0.5$, showing SF to DW transition at $V=1.12\pm 0.04$. (c) Power law fitting of $C_1\left(r\right)$ for $V=1.0$, on a log-log scale.

Figure 6

Plot of correlation function $C_1\left(r\right)$, as a function of $r$, at $V=0.1$ and different values of $U$. In the inset, scaling of L/${\xi }_L^1$ as a function of $U$ for $V=0.1$. Coalescence of the data points of different system sizes shows SF-PSF transition at $U=1.6\pm 0.1$.

Figure 7

Plot of pair-correlation functions $P\left(r\right)$, as a function of $r$, on a log-log scale, at $U=2.0$ and different values of $V$. The plot shows PSF to DW transition at $V=0.4\pm 0.05$.

Figure 8

Plot of correlation function ${(-1)}^r{G}_1\left(r\right)$, as a function of $r$, at $U=2.0$ and different values of $V$: $V=0.1$ (circle), $V=0.3$ (diamond), $V=0.4$ (triangle left), $V=0.5$ (triangle down), and $V=0.6$ (square). Inset shows finite-size scaling of $O_G$, at $U=2.0$ and different values of $V$.

Figure 9

Plot of dimer-dimer correlation function $D\left(r\right)$, as function of $r$, for $V=2.2$ and different values of $U$. Inset shows plot of $N_{av}$ as a function of $U$, and different values of $V$.

Figure 10

(a) Plot of spin density $〈s_l^z〉$, as function of $l$, for $U=2.0$ and $V=1.5$. Spin density of the first chain is denoted with a square, while the second chain is denoted with a circle. Inset shows the schematic of dimerization of spins in two chains. (b) Plot of density correlation $g\left(l\right)$, as a function of $l$, for $U=2.0$ and $V=1.5$.