Phases of the one-dimensional dipolar $t$-$J_\perp$ chain of ultracold polar molecules: the spin-gapped intermediate phase, the $K_\rho=1$ line, and the unexplored $t$-$J$-$V$-$W$ parameter space

Ultracold polar molecules in optical lattices, with two microwave-dressed rotational states playing the role of a pseudospin, realize a highly tunable long-range generalization of the $t$-$J$ model (the $t$-$J$-$V$-$W$ model) in which XXZ-type spin exchange, density-density, and density-spin interactions all decay as dipolar $1/r^3$ and are independently controllable. The 2011 discovery paper computed the one-dimensional phase diagram of the simplest dipolar $t$-$J_\perp$ chain ($V=W=J_z=0$) with the density matrix renormalization group but, within its finite-size precision, could not locate the Luttinger-parameter line $K_\rho=1$ (conservatively $0.85\le K_\rho\le1.15$), could not identify a gapless superfluid region in the dipolar chain, and could not rule out an exotic intermediate phase with a spin gap, $K_\rho<1$, and dominant superfluid correlations. Later work (2015 and 2017) confirmed that an intermediate spin-gapped phase with $K_\rho<1$ exists and exhibits algebraic correlation tails, identifying its dominant correlations as charge-density-wave-like, but the high-filling region, the precise $K_\rho=1$ line, the definitive nature of this phase, the phase diagram for nonzero $V$, $W$, $J_z$ in 1D, and the 2D $t$-$J$-$V$-$W$ phase diagram remain undetermined, and no experiment has yet detected the equilibrium phases.

Background

Polar molecules (e.g. bialkali dimers) carry large electric dipole moments, so two molecules separated by $r$ interact through a dipole-dipole potential decaying as $1/r^3$. In an optical lattice — a standing-wave light field trapping molecules at regular sites — the rotational states of a molecule can be mixed by static electric and microwave fields; choosing two such 'dressed' rotational states as a pseudospin $\uparrow/\downarrow$ and letting molecules hop between sites yields an effective Hubbard-like model whose strong-coupling limit is a generalization of the $t$-$J$ model of doped Mott insulators (hopping $t$ plus spin exchange $J$). Gorshkov, Manmana, Chen, Ye, Demler, Lukin, and Rey showed that with dipolar molecules this becomes the $t$-$J$-$V$-$W$ model: XXZ-type spin exchange ($J_z$ longitudinal and $J_\perp$ transverse, i.e. $S^z_iS^z_j$ and $S^+_iS^-_j+S^-_iS^+_j$ terms), density-density interactions $V$, and density-spin interactions $W$, all long-ranged and all controllable in magnitude and sign independently of each other and of $t$ (Gorshkov et al., Phys. Rev. Lett. 107, 115301 (2011); the microscopic derivation is in the companion paper Gorshkov et al., Phys. Rev. A 84, 033619 (2011)).

In one dimension, such chains are described by Luttinger-liquid theory: the charge sector is a gapless Luttinger liquid characterized by the Luttinger parameter $K_\rho$, with $K_\rho<1$ ('repulsive') and $K_\rho>1$ ('attractive') fixed points; when the spin sector is gapped ('spin gap', the energy cost of a spin-triplet excitation) the system is a one-channel (Luther-Emery-like) liquid, while an ungapped spin sector gives a two-channel liquid. The dominant long-distance correlations — singlet or triplet superfluid (pairing) correlations, spin-density-wave (SDW) correlations, or charge-density-wave (CDW) correlations — are determined by the Luttinger parameters and exponents. For the nearest-neighbor $t$-$J$ and $t$-$J_\perp$ chains the full phase diagram was established by Moreno, Muramatsu, and Manmana (Phys. Rev. B 83, 205113 (2011)) using the density matrix renormalization group (DMRG, a variational method for 1D quantum many-body ground states): repulsive Luttinger liquid with dominant SDW correlations, an attractive gapless superfluid, a singlet superfluid with spin gap, and phase separation (collapse of the inverse compressibility to zero).

Gorshkov et al. (2011) computed the same phase diagram for the dipolar $t$-$J_\perp$ chain (the simplest experimentally realizable slice $V=W=J_z=0$, with $J_\perp\propto1/r^3$) on up to 100 sites and found an enhanced superfluid region and a spin gap up to four times that of the standard $t$-$J$ chain. However, the long-range interactions severely degrade the finite-size extraction of $K_\rho$: the authors conservatively placed the true $K_\rho=1$ line inside a band $0.85\le K_\rho\le1.15$, could not identify the gapless (attractive-liquid) superfluid region in the dipolar chain within their numerical precision, and explicitly could not rule out an exotic intermediate phase combining a spin gap, $K_\rho<1$, and dominant superfluid correlations — a combination absent from the nearest-neighbor model. The Letter also proposed probing the phase diagram by Bloch oscillations (oscillations of the center of mass of particles in a tilted lattice) in small systems and listed the 2D $t$-$J$-$V$-$W$ phase diagram as an open direction.

Problem Statement

Determine the ground-state phase structure of the one-dimensional dipolar $t$-$J_\perp$ chain realized with microwave-dressed polar molecules in an optical lattice (the $t$-$J$-$V$-$W$ model in its simplest slice $V=W=J_z=0$, with transverse spin exchange $J_\perp$ decaying as $1/r^3$), and of its generalizations. Concretely, the originating work left open: (i) whether a gapless superfluid (attractive Luttinger-liquid, $K_\rho>1$) region exists in the dipolar chain, and where the $K_\rho=1$ line actually lies beyond the conservative $0.85\le K_\rho\le1.15$ band — in particular at fillings $n>0.7$ approaching phase separation, where finite-size Luttinger-parameter extraction becomes unreliable; (ii) the definitive nature of the intermediate phase with a spin gap and $K_\rho<1$: whether its dominant long-distance correlations are charge-density-wave-like (as the 2017 follow-up suggests), superfluid/pairing-like (the exotic possibility the 2011 Letter could not rule out), or otherwise, and the mechanism of the algebraic correlation tails in this gapped phase; (iii) the phase diagram of the full $t$-$J$-$V$-$W$ model with nonzero density-density $V$, density-spin $W$, and longitudinal $J_z$ interactions in 1D, where all couplings are independently sign- and magnitude-tunable; and (iv) the 2D phase diagram of the $t$-$J$-$V$-$W$ model, which the Letter predicted would similarly show enhanced superfluidity but which has not been computed. These four items are independently standing open sub-questions rather than a fixed checklist: (i) and (ii) determine the phase structure of the dipolar $t$-$J_\perp$ chain and are the core deliverable of the task, while (iii), (iv), and the experimental-observation question below are the broader open generalizations a fuller answer also addresses; because the $V=W=J_z=0$ slice of the 1D $t$-$J$-$V$-$W$ model in (iii) is exactly the dipolar $t$-$J_\perp$ chain of (i) and (ii), a determination of (iii) subsumes them, and settling any single item is a self-contained contribution that does not depend on settling the others. The overarching objective is to establish which phases the long-range dipolar interactions produce, how they differ from the nearest-neighbor $t$-$J$ chain, and how the enhanced spin gap and superfluidity can be observed experimentally.

References

1. A. V. Gorshkov, S. R. Manmana, G. Chen, J. Ye, E. Demler, M. D. Lukin, and A. M. Rey, 'Tunable Superfluidity and Quantum Magnetism with Ultracold Polar Molecules', Phys. Rev. Lett. 107, 115301 (2011), DOI 10.1103/PhysRevLett.107.115301, https://arxiv.org/abs/1106.1644
2. A. V. Gorshkov, S. R. Manmana, G. Chen, E. Demler, M. D. Lukin, and A. M. Rey, 'Quantum magnetism with polar alkali-metal dimers', Phys. Rev. A 84, 033619 (2011), DOI 10.1103/PhysRevA.84.033619, https://arxiv.org/abs/1106.1655 (companion paper containing the microscopic derivation deferred from the PRL)
3. A. Moreno, A. Muramatsu, and S. R. Manmana, 'Ground-state phase diagram of the one-dimensional $t$-$J$ model', Phys. Rev. B 83, 205113 (2011), DOI 10.1103/PhysRevB.83.205113, https://arxiv.org/abs/1012.4028
4. Chen Cheng, Bin-Bin Mao, Fu-Zhou Chen, and Hong-Gang Luo, 'Phase diagram of the one-dimensional $t$-$J$ model with long-range dipolar interactions', EPL 110, 37002 (2015), DOI 10.1209/0295-5075/110/37002, https://doi.org/10.1209/0295-5075/110/37002
5. S. R. Manmana, M. Möller, R. Gezzi, and K. R. A. Hazzard, 'Correlations and enlarged superconducting phase of $t$-$J_\perp$ chains of ultracold molecules on optical lattices', Phys. Rev. A 96, 043618 (2017), DOI 10.1103/PhysRevA.96.043618, https://arxiv.org/abs/1707.00343
6. T. N. De Silva, 'Asymptotic behavior of correlation functions of one-dimensional polar-molecules on optical lattices', arXiv:1904.08700 (2019), https://arxiv.org/abs/1904.08700
7. S. Fazzini, L. Barbiero, and A. Montorsi, 'Interaction-Induced Fractionalization and Topological Superconductivity in the Polar Molecules Anisotropic $t$-$J$ Model', Phys. Rev. Lett. 122, 106402 (2019), DOI 10.1103/PhysRevLett.122.106402, https://arxiv.org/abs/1809.05007
8. A. N. Carroll, H. Hirzler, C. Miller, D. Wellnitz, S. R. Muleady, J. Lin, K. P. Zamarski, R. R. W. Wang, J. L. Bohn, A. M. Rey, and J. Ye, 'Observation of generalized $t$-$J$ spin dynamics with tunable dipolar interactions', Science 388, 381 (2025), DOI 10.1126/science.adq0911, https://arxiv.org/abs/2404.18916
9. L. Bellinato Giacomelli, T. Bland, L. Lafforgue, F. Ferlaino, M. J. Mark, and L. Barbiero, 'Topology meets superconductivity in a one-dimensional $t$-$J$ model of magnetic atoms', Nat. Commun. 17, 5328 (2026), DOI 10.1038/s41467-026-71248-8, https://arxiv.org/abs/2509.03387