- Letter
- Open Access
Realizing multiband states with ultracold dipolar quantum simulators
Phys. Rev. Research 6, L042024 – Published 24 October, 2024
DOI: https://doi.org/10.1103/PhysRevResearch.6.L042024
Abstract
The manipulation of dipolar interactions within ultracold molecular ensembles represents a pivotal advancement in experimental physics, aiming at the emulation of quantum phenomena unattainable through mere contact interactions. Our study uncovers regimes of multiband occupation which allow to probe more realistic, complex long-range interacting lattice models with ultracold dipolar simulators. By mapping out experimentally relevant ranges of potential depths, interaction strengths, particle fillings, and geometric configurations, we calculate the agreement between the state prepared in the quantum simulator and a target lattice state. We do so by separately calculating numerically exact many-body wave functions in the continuous space and single- or multiband lattice representations, and building their many-body state overlaps. Our findings reveal that for shallow lattices and stronger interactions above half filling, multiband population increases, resulting in fundamentally different ground states than the ones observed in simple lowest-band descriptions, e.g., striped vs checkerboard states. A wide range of probed parameter regimes in its turn provides a systematic and quantitative blueprint for realizing multiband states with two-dimensional quantum simulators employing ultracold dipolar molecules.
Physics Subject Headings (PhySH)
Article Text
Supplemental Material
References (113)
- W. Zwerger, Mott–Hubbard transition of cold atoms in optical lattices, J. Opt. B: Quantum Semiclass. Opt. 5, S9 (2003).
- I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).
- T. Esslinger, Fermi-Hubbard physics with atoms in an optical lattice, Annu. Rev. Condens. Matter Phys. 1, 129 (2010).
- I. Bloch, J. Dalibard, and S. Nascimbène, Quantum simulations with ultracold quantum gases, Nat. Phys. 8, 267 (2012).
- H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, Cold atoms in cavity-generated dynamical optical potentials, Rev. Mod. Phys. 85, 553 (2013).
- T. Langen, R. Geiger, and J. Schmiedmayer, Ultracold atoms out of equilibrium, Annu. Rev. Condens. Matter Phys. 6, 201 (2015).
- C. Gross and I. Bloch, Quantum simulations with ultracold atoms in optical lattices, Science 357, 995 (2017).
- J. A. Blackmore, L. Caldwell, P. D. Gregory, E. M. Bridge, R. Sawant, J. Aldegunde, J. Mur-Petit, D. Jaksch, J. M. Hutson, B. E. Sauer, M. R. Tarbutt, and S. L. Cornish, Ultracold molecules for quantum simulation: rotational coherences in CaF and RbCs, Quantum Sci. Technol. 4, 014010 (2018).
- N. R. Cooper, J. Dalibard, and I. B. Spielman, Topological bands for ultracold atoms, Rev. Mod. Phys. 91, 015005 (2019).
- F. Schäfer, T. Fukuhara, S. Sugawa, Y. Takasu, and Y. Takahashi, Tools for quantum simulation with ultracold atoms in optical lattices, Nat. Rev. Phys. 2, 411 (2020).
- F. Mivehvar, F. Piazza, T. Donner, and H. Ritsch, Cavity QED with quantum gases: new paradigms in many-body physics, Adv. Phys. 70, 1 (2021).
- I. Ferrier-Barbut, M. Schmitt, M. Wenzel, H. Kadau, and T. Pfau, Liquid quantum droplets of ultracold magnetic atoms, J. Phys. B: At. Mol. Opt. Phys. 49, 214004 (2016).
- S. Baier, M. J. Mark, D. Petter, K. Aikawa, L. Chomaz, Z. Cai, M. Baranov, P. Zoller, and F. Ferlaino, Extended Bose-Hubbard models with ultracold magnetic atoms, Science 352, 201 (2016).
- M. A. Norcia and F. Ferlaino, Developments in atomic control using ultracold magnetic lanthanides, Nat. Phys. 17, 1349 (2021).
- L. Chomaz, I. Ferrier-Barbut, F. Ferlaino, B. Laburthe-Tolra, B. L. Lev, and T. Pfau, Dipolar physics: a review of experiments with magnetic quantum gases, Rep. Prog. Phys. 86, 026401 (2023).
- T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, The physics of dipolar bosonic quantum gases, Rep. Prog. Phys. 72, 126401 (2009).
- P. K. Molony, P. D. Gregory, Z. Ji, B. Lu, M. P. Köppinger, C. R. Le Sueur, C. L. Blackley, J. M. Hutson, and S. L. Cornish, Creation of ultracold molecules in the rovibrational ground state, Phys. Rev. Lett. 113, 255301 (2014).
- L. Reichsöllner, A. Schindewolf, T. Takekoshi, R. Grimm, and H.-C. Nägerl, Quantum engineering of a low-entropy gas of heteronuclear bosonic molecules in an optical lattice, Phys. Rev. Lett. 118, 073201 (2017).
- B. Gadway and B. Yan, Strongly interacting ultracold polar molecules, J. Phys. B: At. Mol. Opt. Phys. 49, 152002 (2016).
- S. A. Moses, J. P. Covey, M. T. Miecnikowski, D. S. Jin, and J. Ye, New frontiers for quantum gases of polar molecules, Nat. Phys. 13, 13 (2017).
- I. Stevenson, A. Z. Lam, N. Bigagli, C. Warner, W. Yuan, S. Zhang, and S. Will, Ultracold gas of dipolar NaCs ground state molecules, Phys. Rev. Lett. 130, 113002 (2023).
- P. D. Gregory, L. M. Fernley, A. L. Tao, S. L. Bromley, J. Stepp, Z. Zhang, S. Kotochigova, K. R. A. Hazzard, and S. L. Cornish, Second-scale rotational coherence and dipolar interactions in a gas of ultracold polar molecules, Nat. Phys. (2024).
- N. Bigagli, W. Yuan, S. Zhang, B. Bulatovic, T. Karman, I. Stevenson, and S. Will, Observation of Bose-Einstein condensation of dipolar molecules, Nature 631, 289 (2024).
- F. Böttcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn, M. Guo, T. Langen, and T. Pfau, Transient supersolid properties in an array of dipolar quantum droplets, Phys. Rev. X 9, 011051 (2019).
- L. Tanzi, E. Lucioni, F. Famà, J. Catani, A. Fioretti, C. Gabbanini, R. N. Bisset, L. Santos, and G. Modugno, Observation of a dipolar quantum gas with metastable supersolid properties, Phys. Rev. Lett. 122, 130405 (2019).
- L. Tanzi, S. M. Roccuzzo, E. Lucioni, F. Famà, A. Fioretti, C. Gabbanini, G. Modugno, A. Recati, and S. Stringari, Supersolid symmetry breaking from compressional oscillations in a dipolar quantum gas, Nature (London) 574, 382 (2019).
- L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Trautmann, C. Politi, G. Durastante, R. M. W. van Bijnen, A. Patscheider, M. Sohmen, M. J. Mark, and F. Ferlaino, Long-lived and transient supersolid behaviors in dipolar quantum gases, Phys. Rev. X 9, 021012 (2019).
- M. Guo, F. Böttcher, J. Hertkorn, J.-N. Schmidt, M. Wenzel, H. P. Büchler, T. Langen, and T. Pfau, The low-energy Goldstone mode in a trapped dipolar supersolid, Nature (London) 574, 386 (2019).
- G. Natale, R. M. W. van Bijnen, A. Patscheider, D. Petter, M. J. Mark, L. Chomaz, and F. Ferlaino, Excitation spectrum of a trapped dipolar supersolid and its experimental evidence, Phys. Rev. Lett. 123, 050402 (2019).
- L. Tanzi, J. G. Maloberti, G. Biagioni, A. Fioretti, C. Gabbanini, and G. Modugno, Evidence of superfluidity in a dipolar supersolid from nonclassical rotational inertia, Science 371, 1162 (2021).
- M. A. Norcia, C. Politi, L. Klaus, E. Poli, M. Sohmen, M. J. Mark, R. N. Bisset, L. Santos, and F. Ferlaino, Two-dimensional supersolidity in a dipolar quantum gas, Nature (London) 596, 357 (2021).
- M. Sohmen, C. Politi, L. Klaus, L. Chomaz, M. J. Mark, M. A. Norcia, and F. Ferlaino, Birth, life, and death of a dipolar supersolid, Phys. Rev. Lett. 126, 233401 (2021).
- M. Schmidt, L. Lassablière, G. Quéméner, and T. Langen, Self-bound dipolar droplets and supersolids in molecular Bose-Einstein condensates, Phys. Rev. Res. 4, 013235 (2022).
- J. Sánchez-Baena, C. Politi, F. Maucher, F. Ferlaino, and T. Pohl, Heating a dipolar quantum fluid into a solid, Nat. Commun. 14, 1868 (2023).
- A. Recati and S. Stringari, Supersolidity in ultracold dipolar gases, Nat. Rev. Phys. 5, 735 (2023).
- L. Su, A. Douglas, M. Szurek, R. Groth, S. F. Ozturk, A. Krahn, A. H. Hébert, G. A. Phelps, S. Ebadi, S. Dickerson, F. Ferlaino, O. Marković, and M. Greiner, Dipolar quantum solids emerging in a Hubbard quantum simulator, Nature (London) 622, 724 (2023).
- J.-R. Li, K. Matsuda, C. Miller, A. N. Carroll, W. G. Tobias, J. S. Higgins, and J. Ye, Tunable itinerant spin dynamics with polar molecules, Nature (London) 614, 70 (2023).
- Y. Bao, S. S. Yu, L. Anderegg, E. Chae, W. Ketterle, K.-K. Ni, and J. M. Doyle, Dipolar spin-exchange and entanglement between molecules in an optical tweezer array, Science 382, 1138 (2023).
- L. Christakis, J. S. Rosenberg, R. Raj, S. Chi, A. Morningstar, D. A. Huse, Z. Z. Yan, and W. S. Bakr, Probing site-resolved correlations in a spin system of ultracold molecules, Nature (London) 614, 64 (2023).
- A. L. Gaunt, T. F. Schmidutz, I. Gotlibovych, R. P. Smith, and Z. Hadzibabic, Bose-Einstein condensation of atoms in a uniform potential, Phys. Rev. Lett. 110, 200406 (2013).
- A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M. Greiner, A cold-atom Fermi–Hubbard antiferromagnet, Nature (London) 545, 462 (2017).
- M. Gall, N. Wurz, J. Samland, C. F. Chan, and M. Kóhl, Competing magnetic orders in a bilayer Hubbard model with ultracold atoms, Nature (London) 589, 40 (2021).
- N. Navon, R. P. Smith, and Z. Hadzibabic, Quantum gases in optical boxes, Nat. Phys. 17, 1334 (2021).
- C.-F. Chen and A. Lucas, Finite speed of quantum scrambling with long range interactions, Phys. Rev. Lett. 123, 250605 (2019).
- F. Machado, D. V. Else, G. D. Kahanamoku-Meyer, C. Nayak, and N. Y. Yao, Long-range prethermal phases of nonequilibrium matter, Phys. Rev. X 10, 011043 (2020).
- N. Defenu, T. Donner, T. Macrì, G. Pagano, S. Ruffo, and A. Trombettoni, Long-range interacting quantum systems, Rev. Mod. Phys. 95, 035002 (2023).
- I. Morera, R. Ołdziejewski, G. E. Astrakharchik, and B. Juliá-Díaz, Superexchange liquefaction of strongly correlated lattice dipolar bosons, Phys. Rev. Lett. 130, 023602 (2023).
- M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Boson localization and the superfluid-insulator transition, Phys. Rev. B 40, 546 (1989).
- D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Cold bosonic atoms in optical lattices, Phys. Rev. Lett. 81, 3108 (1998).
- D. Jaksch and P. Zoller, The cold atom Hubbard toolbox, Ann. Phys. 315, 52 (2005).
- H. A. Gersch and G. C. Knollman, Quantum cell model for bosons, Phys. Rev. 129, 959 (1963).
- D. Das and S. Doniach, Existence of a Bose metal at =0, Phys. Rev. B 60, 1261 (1999).
- I. V. Yurkevich and I. V. Lerner, Granular superconductors: From the nonlinear model to the Bose-Hubbard description, Phys. Rev. B 64, 054515 (2001).
- R. Fazio and H. van der Zant, Quantum phase transitions and vortex dynamics in superconducting networks, Phys. Rep. 355, 235 (2001).
- M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature (London) 415, 39 (2002).
- C. Bruder, R. Fazio, and G. Schön, The Bose–Hubbard model from Josephson junction arrays to optical lattices, Ann. Phys. 517, 566 (2005).
- W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet, and M. Greiner, Probing the superfluid–to–Mott insulator transition at the single-atom level, Science 329, 547 (2010).
- T. Tomita, S. Nakajima, I. Danshita, Y. Takasu, and Y. Takahashi, Observation of the Mott insulator to superfluid crossover of a driven-dissipative Bose-Hubbard system, Sci. Adv. 3, e1701513 (2017).
- R. Lin, L. Papariello, P. Molignini, R. Chitra, and A. U. J. Lode, Superfluid–Mott-insulator transition of ultracold superradiant bosons in a cavity, Phys. Rev. A 100, 013611 (2019).
- R. Lin, C. Georges, J. Klinder, P. Molignini, M. Büttner, A. U. J. Lode, R. Chitra, A. Hemmerich, and H. Kessler, Mott transition in a cavity-boson system: A quantitative comparison between theory and experiment, SciPost Phys. 11, 030 (2021).
- D. van Oosten, P. van der Straten, and H. T. C. Stoof, Mott insulators in an optical lattice with high filling factors, Phys. Rev. A 67, 033606 (2003).
- J. Larson, A. Collin, and J.-P. Martikainen, Multiband bosons in optical lattices, Phys. Rev. A 79, 033603 (2009).
- O. Dutta, A. Eckardt, P. Hauke, B. Malomed, and M. Lewenstein, Bose–Hubbard model with occupation-dependent parameters, New J. Phys. 13, 023019 (2011).
- A. Mering and M. Fleischhauer, Multiband and nonlinear hopping corrections to the three-dimensional Bose-Fermi-Hubbard model, Phys. Rev. A 83, 063630 (2011).
- D.-S. Lühmann, O. Jürgensen, and K. Sengstock, Multi-orbital and density-induced tunneling of bosons in optical lattices, New J. Phys. 14, 033021 (2012).
- J.-P. Martikainen and J. Larson, Multiorbital bosons in bipartite optical lattices, Phys. Rev. A 86, 023611 (2012).
- M. Łącki, D. Delande, and J. Zakrzewski, Dynamics of cold bosons in optical lattices: effects of higher Bloch bands, New J. Phys. 15, 013062 (2013).
- W. Xu, M. Olshanii, and M. Rigol, Multiband effects and the Bose-Hubbard model in one-dimensional lattices, Phys. Rev. A 94, 031601(R) (2016).
- K.-Y. Li, Y. Zhang, K. Yang, K.-Y. Lin, S. Gopalakrishnan, M. Rigol, and B. L. Lev, Rapidity and momentum distributions of 1d dipolar quantum gases, Phys. Rev. A 107, L061302 (2023).
- M. Hughes and D. Jaksch, Dipolar Bose-Hubbard model in finite-size real-space cylindrical lattices, Phys. Rev. A 105, 053301 (2022).
- M. Hughes, A. U. J. Lode, D. Jaksch, and P. Molignini, Accuracy of quantum simulators with ultracold dipolar molecules: A quantitative comparison between continuum and lattice descriptions, Phys. Rev. A 107, 033323 (2023).
- Z.-W. Xie and W. M. Liu, Superfluid–Mott-insulator transition of dipolar bosons in an optical lattice, Phys. Rev. A 70, 045602 (2004).
- O. Dutta, M. Gajda, P. Hauke, M. Lewenstein, D.-S. Lühmann, B. A. Malomed, T. Sowiński, and J. Zakrzewski, Non-standard Hubbard models in optical lattices: A review, Rep. Prog. Phys. 78, 066001 (2015).
- K. Biedroń, M. Łącki, and J. Zakrzewski, Extended Bose-Hubbard model with dipolar and contact interactions, Phys. Rev. B 97, 245102 (2018).
- C. Lagoin, U. Bhattacharya, T. Grass, R. W. Chhajlany, T. Salamon, K. Baldwin, L. Pfeiffer, M. Lewenstein, M. Holzmann, and F. Dubin, Extended Bose–Hubbard model with dipolar excitons, Nature (London) 609, 485 (2022).
- K. Tamura, S. Watabe, and T. Nikuni, Analysis of shape change of droplet in dipolar Bose–Hubbard model, J Low Temp Phys 208, 365 (2022).
- E. G. Dalla Torre, E. Berg, and E. Altman, Hidden order in 1D Bose insulators, Phys. Rev. Lett. 97, 260401 (2006).
- T. Giamarchi, C. Rüegg, and O. Tchernyshyov, Bose–Einstein condensation in magnetic insulators, Nat. Phys. 4, 198 (2008).
- L. Pollet, J. D. Picon, H. P. Büchler, and M. Troyer, Supersolid phase with cold polar molecules on a triangular lattice, Phys. Rev. Lett. 104, 125302 (2010).
- B. Capogrosso-Sansone, C. Trefzger, M. Lewenstein, P. Zoller, and G. Pupillo, Quantum phases of cold polar molecules in 2D optical lattices, Phys. Rev. Lett. 104, 125301 (2010).
- V. Zapf, M. Jaime, and C. D. Batista, Bose–Einstein condensation in magnetic insulators, Rev. Mod. Phys. 86, 563 (2014).
- M. Marciniak, M. Lebek, J. Kopycnski, W. Gorecki, R. Ołdziejewski, and K. Pawlowski, Super-Tonks-Girardeau quench in the extended Bose-Hubbard model, Phys. Rev. A 108, 043304 (2023).
- For an odd number of sites centered around 0 in the direction, we set . For an even number of sites, we set .
- S. Sinha and L. Santos, Cold dipolar gases in quasi-one-dimensional geometries, Phys. Rev. Lett. 99, 140406 (2007).
- U. R. Fischer, A. U. J. Lode, and B. Chatterjee, Condensate fragmentation as a sensitive measure of the quantum many-body behavior of bosons with long-range interactions, Phys. Rev. A 91, 063621 (2015).
- B. Chatterjee and A. U. J. Lode, Order parameter and detection for a finite ensemble of crystallized one-dimensional dipolar bosons in optical lattices, Phys. Rev. A 98, 053624 (2018).
- B. Chatterjee, M. C. Tsatsos, and A. U. J. Lode, Correlations of strongly interacting one-dimensional ultracold dipolar few-boson systems in optical lattices, New J. Phys. 21, 033030 (2019).
- B. Chatterjee, C. Lévêque, J. Schmiedmayer, and A. U. J. Lode, Detecting one-dimensional dipolar bosonic crystal orders via full distribution functions, Phys. Rev. Lett. 125, 093602 (2020).
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.6.L042024 for more information about the methods and the units of our simulations, as well as a more systematic analysis of different geometries and particle numbers, including energy and fidelity calculations.
- A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, General variational many-body theory with complete self-consistency for trapped bosonic systems, Phys. Rev. A 73, 063626 (2006).
- A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Role of excited states in the splitting of a trapped interacting Bose-Einstein condensate by a time-dependent barrier, Phys. Rev. Lett. 99, 030402 (2007).
- O. E. Alon, A. I. Streltsov, and L. S. Cederbaum, Unified view on multiconfigurational time propagation for systems consisting of identical particles, J. Chem. Phys. 127, 154103 (2007).
- O. E. Alon, A. I. Streltsov, and L. S. Cederbaum, Multiconfigurational time-dependent Hartree method for bosons: Many-body dynamics of bosonic systems, Phys. Rev. A 77, 033613 (2008).
- A. U. J. Lode, Multiconfigurational time-dependent Hartree method for bosons with internal degrees of freedom: Theory and composite fragmentation of multicomponent Bose-Einstein condensates, Phys. Rev. A 93, 063601 (2016).
- E. Fasshauer and A. U. J. Lode, Multiconfigurational time-dependent Hartree method for fermions: Implementation, exactness, and few-fermion tunneling to open space, Phys. Rev. A 93, 033635 (2016).
- R. Lin, P. Molignini, L. Papariello, M. C. Tsatsos, C. Lévêque, S. E. Weiner, E. Fasshauer, and R. Chitra, MCTDH-X: The multiconfigurational time-dependent Hartree method for indistinguishable particles software, Quantum Sci. Technol. 5, 024004 (2020).
- A. U. J. Lode, C. Lévêque, L. B. Madsen, A. I. Streltsov, and O. E. Alon, Colloquium: Multiconfigurational time-dependent Hartree approaches for indistinguishable particles, Rev. Mod. Phys. 92, 011001 (2020).
- A. U. J. Lode, M. C. Tsatsos, E. Fasshauer, S. E. Weiner, R. Lin, L. Papariello, P. Molignini, C. Lévêque, M. Büttner, J. Xiang, S. Dutta, and Y. Bilinskaya, MCTDH-X: The multiconfigurational time-dependent Hartree method for indistinguishable particles software (2024), http://ultracold.org/.
- G. H. Wannier, The structure of electronic excitation levels in insulating crystals, Phys. Rev. 52, 191 (1937).
- N. Marzari and D. Vanderbilt, Maximally localized generalized Wannier functions for composite energy bands, Phys. Rev. B 56, 12847 (1997).
- I. Souza, N. Marzari, and D. Vanderbilt, Maximally localized Wannier functions for entangled energy bands, Phys. Rev. B 65, 035109 (2001).
- M. Modugno and G. Pettini, Maximally localized Wannier functions for ultracold atoms in one-dimensional double-well periodic potentials, New J. Phys. 14, 055004 (2012).
- J. R. Y. I. S. Nicola Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Maximally localized Wannier functions: Theory and applications, Rev. Mod. Phys. 84, 1419 (2012).
- R. Walters, G. Cotugno, T. H. Johnson, S. R. Clark, and D. Jaksch, Ab initio derivation of Hubbard models for cold atoms in optical lattices, Phys. Rev. A 87, 043613 (2013).
- S. Dutta, A. Buyskikh, A. J. Daley, and E. J. Mueller, Density matrix renormalization group for continuous quantum systems, Phys. Rev. Lett. 128, 230401 (2022).
- J. M. Zhang and R. X. Dong, Exact diagonalization: the Bose–Hubbard model as an example, Eur. J. Phys. 31, 591 (2010).
- P. Weinberg and M. Bukov, QuSpin: a Python package for dynamics and exact diagonalisation of quantum many body systems part I: spin chains, SciPost Phys. 2, 003 (2017).
- P. Weinberg and M. Bukov, QuSpin: a Python Package for Dynamics and Exact Diagonalisation of quantum many body systems. Part II: bosons, fermions and higher spins, SciPost Phys. 7, 020 (2019).
- C. Trefzger, C. Menotti, B. Capogrosso-Sansone, and M. Lewenstein, Ultracold dipolar gases in optical lattices, J. Phys. B: At. Mol. Opt. Phys. 44, 193001 (2011).
- M. Maik, P. Hauke, O. Dutta, M. Lewenstein, and J. Zakrzewski, Density-dependent tunneling in the extended Bose–Hubbard model, New J. Phys. 15, 113041 (2013).
- K. Suthar, R. Kraus, H. Sable, D. Angom, G. Morigi, and J. Zakrzewski, Staggered superfluid phases of dipolar bosons in two-dimensional square lattices, Phys. Rev. B 102, 214503 (2020).
- C. Staudinger, D. Hufnagl, F. Mazzanti, and R. E. Zillich, Striped dilute liquid of dipolar bosons in two dimensions, Phys. Rev. A 108, 033303 (2023).
- Y. Kora and M. Boninsegni, Dipolar bosons in one dimension: The case of longitudinal dipole alignment, Phys. Rev. A 101, 023602 (2020).