In extended Bose-Hubbard settings where a correlated pair-hopping amplitude is the dominant transport mechanism — the regime engineered in the state-dependent optical-lattice proposals of Eckholt and Garcia-Ripoll (2008, 2009) — the nature of the pair-condensed phase and of its phase transitions is only partially characterized. The founding studies, working on one-dimensional chains, established a pair-condensed phase but left its correlator asymptotics uncontrolled: the 2008 study states explicitly that its simulation sizes did not yet allow determining the thermodynamic-limit behavior of the two-particle correlator (fits suggesting an algebraic decay with a nonuniversal exponent α<0.7\alpha < 0.7), and the 2009 follow-up reports α≈0.5\alpha \approx 0.5–0.60.6 from finite-bond-dimension iTEBD while affirmatively claiming that all phases are connected by second-order transitions, again without controlled error estimates. This problem asks for a controlled characterization of this one-dimensional model family: the asymptotic spatial decay of the single-particle and pair correlators, a controlled test of the order and universality class of the transitions from Mott-insulating and density-wave phases into the pair condensate, and unbiased phase boundaries beyond site-factorized Gutzwiller mean field. Later literature settles substantial parts of the one-dimensional and constrained-limit physics (paired Luttinger liquids, BKT transitions, an exact spin-chain mapping in the hard-core limit) and covers hard-core and attractive three-body-constrained variants, including in two dimensions, but the soft-core source model's correlator exponent, the verification of its second-order claim, and its quantitative boundaries still lack a controlled determination.

Background

The Bose-Hubbard model — bosons hopping on a lattice, repelled on site by energy U0U_0 and controlled by chemical potential μ\mu — is the paradigmatic lattice model of the Mott-insulator-to-superfluid quantum phase transition. A Mott insulator is a gapped phase at commensurate integer filling with exponentially decaying correlations; a conventional superfluid has a coherent single-particle condensate. Extensions relevant here add a nearest-neighbor density interaction VV of either sign, with V>0V>0 favoring a charge-density-wave (CDW) insulator at commensurate fillings, and — the key ingredient — correlated hopping terms generated in state-dependent optical lattices where two internal atomic states feel different lattice potentials and the interspecies scattering length differs from the intraspecies ones. For two-state atoms with interaction asymmetries parameterized by g↑↑,g↓↓,g↑↓g_{\uparrow\uparrow}, g_{\downarrow\downarrow}, g_{\uparrow\downarrow}, the deep state-dependent superlattice limit produces an effective pair hopping t∝2g↑↓−g↑↑−g↓↓t \propto 2g_{\uparrow\downarrow}-g_{\uparrow\uparrow}-g_{\downarrow\downarrow} (a boson pair coherently moving between neighboring sites) and a density-induced correlated hopping j∝g↑↑−g↓↓j \propto g_{\uparrow\uparrow}-g_{\downarrow\downarrow} (single bosons hopping only conditioned on the occupation of the sites involved), yielding the effective one-dimensional Hamiltonian of the 2009 study (its Eq. (1), treated in the grand-canonical picture F=H−μNF = H - \mu N)

H=∑i[U4:(ni+ni+1)2:+V nini+1−t(ci†2ci+12+H.c.)−j((ni−1)ci†(ci−1+ci+1)+H.c.)]−μ∑ini,H = \sum_i \left[ \frac{U}{4}:(n_i+n_{i+1})^2: + V\, n_i n_{i+1} - t\left(c_i^{\dagger 2} c_{i+1}^{2} + \mathrm{H.c.}\right) - j\left((n_i-1)c_i^{\dagger}(c_{i-1}+c_{i+1}) + \mathrm{H.c.}\right) \right] - \mu\sum_i n_i,

with cic_i the boson annihilation operator, ni=ci†cin_i=c_i^{\dagger}c_i, and :⋯::\cdots: denoting normal ordering. In the superlattice realization U=(2U0+U1)/4U=(2U_0+U_1)/4, V=−U1/8V=-U_1/8, t=U1/16t=U_1/16 and j=U2/8j=U_2/8, with U>0U>0 and V>−UV>-U required for stability against collapse; the 2008 study is the j=0j=0 case of the same family. Crucially, the model contains no independent single-particle hopping term: for j=0j=0 atoms move only in pairs, so that the occupation parity (−1)ni(-1)^{n_i} of each site is conserved. The natural condensed phase is therefore a pair superfluid (PSF): a phase-coherent condensate of composite bosons with strong pair correlations but suppressed single-particle coherence. Its diagnostic is the pair correlator C2(r)=⟨ci†2ci+r2⟩C_2(r)=\langle c_i^{\dagger 2} c_{i+r}^{2}\rangle versus the single-particle correlator C1(r)=⟨ci†ci+r⟩C_1(r)=\langle c_i^{\dagger} c_{i+r}\rangle. In one dimension a continuous symmetry cannot be broken even at zero temperature; instead a superfluid shows quasi-long-range order, meaning these correlators decay algebraically ∼r−α\sim r^{-\alpha}, with a nonuniversal exponent fixed by the interactions. Both source studies concern such one-dimensional chains at zero temperature (an array of alternating ∣+⟩|+\rangle/∣−⟩|-\rangle superlattice sites); the 2009 conclusions remark that the engineering could in principle be extended to two-dimensional lattice potentials, but no two-dimensional analysis is performed there, and the open questions identified below are one-dimensional.

Eckholt and Garcia-Ripoll (Phys. Rev. A 77, 063603 (2008)) found the pair-condensed phase with a Gutzwiller ansatz — a site-factorized mean field in which each lattice site carries an independent variational boson state, a method known to become quantitatively unreliable near criticality — and with a variational matrix-product-state (MPS) ground-state search on chains of up to 40 sites with bond dimension D=40D=40 (MPS is the variational ansatz underlying DMRG, density-matrix renormalization group, a quasi-exact method for one-dimensional quantum states; the infinite-lattice variant used in the 2009 follow-up is iTEBD, infinite time-evolving block decimation, an MPS algorithm acting directly on the infinite chain). The 2008 paper states explicitly that the simulation sizes did not yet allow determining the behavior of C2C_2, with fits suggesting algebraic decay with a nonuniversal exponent α<0.7\alpha < 0.7 depending on U0U_0 and μ\mu, and that the Gutzwiller ansatz 'does not reproduce accurately neither the location of phase transitions nor the behavior of correlators'. The 2009 study (New J. Phys. 11, 093028), adding the density-induced hopping jj, obtained α≈0.5\alpha \approx 0.5–0.60.6 from iTEBD with bond dimension D=16D=16–6464 and on-site cutoff nmax⁡=8n_{\max}=8, and affirmatively claimed that all phases are connected by second-order transitions — a claim resting on mean field plus moderate-DD MPS without controlled error estimates. Related pair superfluids were known from other routes (counterflow dynamics of two-species mixtures, Feshbach-resonance pairing — associating atom pairs via a tunable scattering resonance — and attractive bosons under a three-body constraint forbidding triply occupied sites), and mean-field treatments of pair-tunneling models were developed contemporaneously by Zhou et al. and later by Iskin. What was and remains missing for this model family is a controlled determination of the pair correlator exponent and its parameter dependence, a controlled test of the claimed second-order character and of the universality classes of the insulator-to-PSF transitions, and quantitative phase boundaries. In one dimension a Berezinskii-Kosterlitz-Thouless (BKT) transition — a topological transition driven by vortex-antivortex unbinding, with exponentially diverging correlation length — is the generic superfluid-insulator mechanism, and a Luttinger liquid (the universal low-energy theory of 1D gapless systems, characterized by a Luttinger parameter setting all algebraic exponents) describes the pair superfluid as a paired (molecular) Luttinger liquid; bosonization — the exact low-energy rewriting of a 1D model in terms of these density-fluctuation fields — is the standard tool from which its exponents would be computed.

Problem Statement

For the one-dimensional correlated-hopping model defined in the Background — soft-core bosons on a chain governed by the Hamiltonian HH above, with pair hopping tt, density-induced correlated hopping jj (the j=0j=0 case of the 2008 study included), on-site scale U>0U>0, nearest-neighbor interaction VV with V>−UV>-U, and chemical potential μ\mu, at zero temperature — provide a controlled characterization of the pair-condensed phase and its boundaries, jointly addressing three linked questions arising from the source studies: (1) Correlator asymptotics: determine the thermodynamic-limit spatial decay of the single-particle correlator C1(r)C_1(r) and the pair correlator C2(r)C_2(r) inside the pair superfluid, and the parameter dependence of the algebraic decay exponent of C2C_2 that the 2008 study left undetermined and the 2009 study estimated as α≈0.5\alpha\approx0.5–0.60.6 at finite bond dimension, as a controlled determination with estimable errors. (2) Transition order and universality: the sources claim, on the basis of Gutzwiller mean field and finite-bond-dimension iTEBD without controlled error estimates, that all phases are connected by second-order transitions; establish whether the quantum phase transitions from the Mott-insulating and charge-density-wave phases into the pair superfluid are indeed continuous and, if so, their universality classes (for example BKT, Luttinger-liquid, or discrete-symmetry-breaking/Ising criticality in one dimension), or identify where the second-order claim fails. (3) Phase boundaries: locate the quantitative phase boundaries in the (U,V,t,j,μ)(U, V, t, j, \mu) parameter space beyond site-factorized Gutzwiller mean field, which the 2008 study states does not accurately reproduce the transition locations, with controlled or estimable errors. These three deliverables jointly support one objective — a controlled phase diagram and critical description of pair-hopping bosons in one dimension.

References

  1. María Eckholt and Juan José García-Ripoll, "Pair condensation of bosonic atoms induced by optical lattices", Phys. Rev. A 77, 063603 (2008). DOI: 10.1103/PhysRevA.77.063603; arXiv:0707.0383. URL: https://doi.org/10.1103/PhysRevA.77.063603
  2. María Eckholt and Juan José García-Ripoll, "Correlated hopping of bosonic atoms induced by optical lattices", New J. Phys. 11, 093028 (2009). DOI: 10.1088/1367-2630/11/9/093028; arXiv:0908.2383. URL: https://doi.org/10.1088/1367-2630/11/9/093028
  3. M. J. Bhaseen, S. Ejima, F. H. L. Essler, H. Fehske, M. Hohenadler, and B. D. Simons, "Discrete symmetry breaking transitions between paired superfluids", Phys. Rev. A 85, 033636 (2012). DOI: 10.1103/PhysRevA.85.033636. URL: https://doi.org/10.1103/PhysRevA.85.033636
  4. Dirk-Sören Lühmann, "Twisted superfluid phase in the extended one-dimensional Bose-Hubbard model", Phys. Rev. A 94, 011603(R) (2016). DOI: 10.1103/PhysRevA.94.011603. URL: https://doi.org/10.1103/PhysRevA.94.011603
  5. Manpreet Singh, Sebastian Greschner, and Tapan Mishra, "Anomalous pairing of bosons: Effect of multibody interactions in an optical lattice", Phys. Rev. A 98, 023615 (2018). DOI: 10.1103/PhysRevA.98.023615. URL: https://doi.org/10.1103/PhysRevA.98.023615
  6. Suman Mondal, Augustine Kshetrimayum, and Tapan Mishra, "Two-body repulsive bound pairs in a multibody interacting Bose-Hubbard model", Phys. Rev. A 102, 023312 (2020). DOI: 10.1103/PhysRevA.102.023312. URL: https://doi.org/10.1103/PhysRevA.102.023312
  7. Lars Bonnes and Stefan Wessel, "Half-vortex unbinding and Ising transition in constrained superfluids", Phys. Rev. B 85, 094513 (2012). DOI: 10.1103/PhysRevB.85.094513. URL: https://doi.org/10.1103/PhysRevB.85.094513
  8. Alvin J. R. Heng, Wenan Guo, Anders W. Sandvik, and Pinaki Sengupta, "Pair hopping in systems of strongly interacting hard-core bosons", Phys. Rev. B 100, 104433 (2019). DOI: 10.1103/PhysRevB.100.104433. URL: https://doi.org/10.1103/PhysRevB.100.104433
  9. Manali Malakar, Sudip Sinha, and S. Sinha, "Formation of paired phases of bosons and their excitations in a square lattice", Phys. Rev. B 108, 054518 (2023). DOI: 10.1103/PhysRevB.108.054518. URL: https://doi.org/10.1103/PhysRevB.108.054518
  10. Ákos Rapp, Xiaolong Deng, and Luis Santos, "Ultracold lattice gases with periodically modulated interactions", Phys. Rev. Lett. 109, 203005 (2012). DOI: 10.1103/PhysRevLett.109.203005. URL: https://doi.org/10.1103/PhysRevLett.109.203005
  11. Qizhong Zhu, Qi Zhang, and Biao Wu, "Extended two-site Bose–Hubbard model with pair tunneling: spontaneous symmetry breaking, effective ground state and fragmentation", J. Phys. B 48, 045301 (2015). DOI: 10.1088/0953-4075/48/4/045301. URL: https://doi.org/10.1088/0953-4075/48/4/045301
  12. M. Iskin, "Mean-field theory for the Mott-insulator–paired-superfluid phase transition in the two-species Bose-Hubbard model", Phys. Rev. A 82, 055601 (2010). DOI: 10.1103/PhysRevA.82.055601. URL: https://doi.org/10.1103/PhysRevA.82.055601
  13. Xiang-Fa Zhou, Yong-Sheng Zhang, and Guang-Can Guo, "Pair tunneling of bosonic atoms in an optical lattice", Phys. Rev. A 80, 013605 (2009). DOI: 10.1103/PhysRevA.80.013605. URL: https://doi.org/10.1103/PhysRevA.80.013605
  14. Julia Stasińska, Omjyoti Dutta, Luca Barbiero, Maciej Lewenstein, and Ravindra W. Chhajlany, "Clustered superfluids in the one-dimensional Bose-Hubbard model with extended correlated hopping", Phys. Rev. B 103, 134513 (2021). DOI: 10.1103/PhysRevB.103.134513. URL: https://doi.org/10.1103/PhysRevB.103.134513
  15. Nitya Cuzzuol, Luca Barbiero, and Arianna Montorsi, "Nonlocal order parameter of pair superfluids", SciPost Phys. Core 8, 020 (2025). DOI: 10.21468/SciPostPhysCore.8.1.020. URL: https://doi.org/10.21468/SciPostPhysCore.8.1.020
  16. Domantas Burba, Giedrius Žlabys, Dzmitry Viarbitski, Thomas Busch, and Gediminas Juzeliūnas, "Chiral and pair superfluidity in triangular ladder produced by state-dependent Kronig-Penney lattice", arXiv:2603.04498 (2026). URL: https://arxiv.org/abs/2603.04498