Quantum phase transitions and the hidden order in a two-chain extended boson Hubbard model at half-odd-integer fillings

We study the phase diagram of two weakly coupled one-dimensional dipolar boson chains at half-odd-integer fillings. We find that the system contains a rich phase diagram. Four different phases are found. They are the Mott insulators, the single-particle resonant superfluid, the paired superfluid, and the bond-density or interchain density waves. Moreover, the Mott insulating phase can be further classified according to a hidden string order parameter, which is analogous to the one investigated recently in the one-dimensional boson Mott insulator at integer fillings.

Figure 1

(Color online) The two-chain lattice model described by the Hamiltonian [Eq. (1)]. The arrows denote the possible orientations of the dipole moments. (a) corresponds to the IDW phase and (b) represents the BDW phase considered in the paper (see Sec. 3).

Figure 2

(Color online) The phase diagram of the two weakly coupled half-filled one-dimensional EBHM obtained in the hard-core boson limit. Notice that for the hard-core bosons, only the BDW phase can occur, and this is because the two conditions of $(K_s<1,K_a<1∕2)$ and $V_{\perp }>0$ cannot be satisfied simultaneously. However, we expect that this is an artifact of the hard-core condition, and for more general cases, depending on the sign of $V_{\perp }$, both the IDW and the BDW phases can be stabilized (see the discussions in Sec. 5). Notice also that the phase transition across the red line is of the Ising universality class, while the phase transition across the blue line belongs to the KT universality class. Mott insulator II below the green line contains a nontrivial string order, and it is separated from the Mott insulating I state by a $U\left(1\right)$ Gaussian critical theory.