Erratum: Mott-insulator phases of nonlocally coupled one-dimensional dipolar Bose gases [Phys. Rev. A75, 053613 (2007)]

Abstract

In our paper we analyzed nonlocally coupled bosons in ladder lattices, or equivalently two-component bosons in onedimensional ͑1D͒ lattices. We showed that due to the direct Mott-insulator ͑MI͒ to pair-superfluid ͑PSF͒ transition, the MI lobes become severely distorted, leading to a counterintuitive behavior of the MI spatial regions in harmonic traps.In our paper, we characterized the possible phases, i.e., MI, PSF, and a phase of two superfluids ͑2SF͒, by employing as order parameters ͗b i,␣ ͘ and ͗b i,1 b i,2 ͘, where b i,␣ denotes the annihilation operator for the rung i and the wire ␣ =1,2. These order parameters characterize indeed the phases in the grand-canonical ensemble, since for PSF ͗b i,1 b i,2 ͘ 0 and ͗b i,␣ ͘ = 0, for 2SF both are nonzero, and for MI all are zero. Although our matrix-product-state ͑MPS͒ calculations conserve in principle the number of particles ͑and hence work in the canonical ensemble͒, a slight numerical inaccuracy in our simulations allowed for the violation of the number conservation. This effectively allowed for the identification of the MI, PSF, and 2SF phases. However, more accurate simulations do conserve the number of particles, and hence ͗b i,1 b i,2 ͘ = ͗b i,␣ ͘ = 0 in any case. Consequently, we have to rely in number conserving order parameters for the phases. In particular we employ the correlationsWe have calculated these correlation functions as a function of the site distance ⌬. For the MI phase both correlations decay exponentially with ⌬, for the PSF G 1 decays exponentially and G 2 polynomically, whereas for 2SF both decay polynomically. By fitting the correlation functions using exponential functions or power laws ͑see Fig. 1͒, we have identified for different values of the chemical potential and the tunneling rate J the different regions ͑see Fig. 2͒, which coincide almost exactly with those in our paper.

Ultracold dipolar gases in optical lattices

“…We focus on the physical situation in which the two layers are very close to one another (d ⊥ ≪ d) such that (U + W ) ≪ U , and small in-plane dipolar interactions U NN ≪ U , because in these limits particles at the same lattice site i of different layers pair into composites. The composites localize in a MI state for small values of the tunneling coefficient, while for larger values of J the pairs hop around in the optical lattice forming a pair-superfluid (PSF) phase [47]. Furthermore, the presence of the in-plane long-range interactions leads to the formation of a novel pair-supersolid phase (PSS), namely, a supersolid of composites [55].…”

“…The relative strength between U NN and W can be tuned by changing the spacing d ⊥ between the two layers, relative to the 2D optical lattice spacing d. Because of the dependence of the dipole-dipole interaction like the inverse cubic power of the distance, the ratio |W |/U N N can be tuned over a wide range. While it can be negligible for d ⊥ ≫ d making the system asymptotically similar to a single 2D lattice layer, it can also become relevant and give rise to interesting physics, not existing in the single layer model as pointed out in [47,48,49,50,51,52,53,54].…”

“…which after expansion of the field operators in the basis of Wannier functions, becomeŝ 47) and the matrix elements V ijkl are given by the integral…”

Formation of classical crystals of dipolar particles in a helical geometry

“…Some recent examples of the numerical methods applied to dipolar particles can be found in Refs. [58][59][60][61].…”

Dipoles on a two-leg ladder

“…These problems can be overcome in low-dimensional geometries where the dipolar particles are confined to either twodimensional (2D) planes or one-dimensional (1D) tubes with and without the presence of lattice potentials. A number of interesting predictions have been made for the phases of system with dipolar interactions, including exotic superfluids [15][16][17][18][19] , Luttinger liquids [20][21][22][23][24][25] , Mott insulators 26,27 , interlayer pairing [28][29][30][31] , non-trivial quantum critical points 32,33 , modified confinement-induced resonances [34][35][36][37] , roton modes and stripe instabilities [38][39][40][41][42][43][44][45] , and crystallization [46][47][48][49][50][51][52][53][54] , as well as formation of chain complexes [55][56][57][58][59][60]…”