Rise and fall of hidden string order of lattice bosons

We investigate the ground-state properties of a newly discovered phase of one-dimensional lattice bosons with extended interactions [E. G. Dalla Torre et al., Phys. Rev. Lett. 97, 260401 (2006)]. The new phase, termed the Haldane insulator in analogy with the gapped phase of spin-1 chains, is characterized by a nonlocal order parameter, which can only be written as an infinite string in terms of the bosonic densities. We show that the string order can nevertheless be probed with physical fields that couple locally, via the effect those fields have on the quantum phase transitions separating the exotic phase from the conventional Mott and density wave phases. Using a field theoretical analysis, we show that a perturbation that breaks lattice inversion symmetry gaps the critical point separating the Mott and Haldane phases and eliminates the sharp distinction between them. This is remarkable given that neither of these phases involves broken inversion symmetry. We also investigate the evolution of the phase diagram with the tunable coupling between parallel chains in an optical lattice setup. We find that interchain tunneling destroys the direct phase transition between the Mott and Haldane insulators by establishing an intermediate superfluid phase. On the other hand, coupling the chains only by weak repulsive interactions does not modify the structure of the phase diagram. The theoretical predictions are confirmed with numerical calculations using the density matrix renormalization group.

Figure 1

(Color online) Array of one-dimensional chains formed by a two-dimensional optical lattice. The polarizing field E is directed perpendicular to the chains and at an angle $\theta$ which can be used to tune the interchain interaction $V_{\perp }$ in (2).

Figure 2

Typical configurations in the (a) HI and (b) MI ground states. The numbers represent $\delta n$ (the deviation of the local occupation from the average density). The HI can be described as a charge ordered $+,-,+,\dots$ state with an undetermined number of 0 sites between each $+$ and $-$. The MI is a dilute gas of particle-hole pairs (indicated by the dotted line).

Figure 3

Evolution of the phase diagram with interchain coupling predicted from bosonization. (a) Plane of the nearest-neighbor interaction $V$ versus $t_{\perp }$ (interchain tunneling) at fixed on-site interaction $U$ and no interchain repulsion. (b) In the plane of $V$ versus $V_{\perp }$ (interchain repulsion) at $t_{\perp }=0$ and fixed $U$.

Figure 4

(Color online) (a) Phase diagram of a single unperturbed chain in the space of $\left(U/t,V/t\right)$, obtained from the numerical calculations. (b) The charge $(◇)$ and neutral $(\square )$ gaps along the same cut through the three insulating phases. The charge gap vanishes at the MI-HI transition, whereas only the neutral gap vanishes at the transition to the DW phase. (c) The parity $(◇)$, string $(\square )$ and Density wave $(○)$ order parameters as a function of $V/t$ along a line of constant $U/t=6$. The string and parity orders are defined as the square roots of Eqs. (12, 13), respectively.

Figure 5

(Color online) Decay of the parity and string correlations and of the single-particle density matrix at the HI-MI transition. The power-law fits are consistent with the field theoretical predictions for the relations between the different decay exponents (see text).

Figure 6

(Color online) Charge gap as a function of $V/t$ at constant $U/t=6$ for different values of the inversion-symmetry-breaking perturbation. $\lambda$ is defined in Eq. (26).

Figure 7

(Color online) Effect of interchain interaction coupling on the phase diagram. (a) Calculated DW order parameter as a function of $V/t$ at constant $U/t=6$ for different values of the interchain interaction $V_{\perp }$. (b) Calculated charge gap on the same cut through the phase diagram and the same values of $V_{\perp }$. (c) Evolution of the phase boundaries with $V_{\perp }$ inferred from the calculations.

Figure 8

(Color online) Effect of interchain tunnel coupling on the phase diagram. (a) Calculated charge gap as a function of $V/t$ at constant $U/t=6$ for different values of the interchain tunneling $t_{\perp }$. The gaps were obtained by extrapolation to an infinite chain through finite-size scaling analysis. The fact that the estimate for the gap are sometimes slightly negative is due to errors in the extrapolation. Note the gapless phase that appears between the two insulating phases. (b) Power-law fit for the spatial decay of the single-particle density matrix for the same cuts through the phase diagram. The fact that $\nu <1/4$ in the gapless phase confirms that it is indeed a superfluid. The transition to the gapped phase seems to occur at the universal exponent $\nu =1/4$ consistent with a Kosterlitz-Thouless transition. (c) Evolution of the phase boundaries with $t_{\perp }$ inferred from the calculations.