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arXiv:2601.16630

$d$-wave FFLO state and charge-2e supersolidity in the $t$-$t'$-$J$ model under Zeeman fields

Xing-Zhou Qu, Dai-Wei Qu, Qiaoyi Li, Wei Li, Gang Su

Published Jan 23, 2026• 0 citations
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Abstract

Unconventional superconductivity under strong Zeeman fields--particularly beyond the Pauli paramagnetic limit--remains a central challenge in condensed matter physics. The exotic Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, in particular, remains in need of definitive study within fundamental electronic models. Here we employ state-of-the-art finite-temperature and ground-state tensor network approaches to systematically explore the superconducting (SC) phase diagram of the $t$-$t'$-$J$ model subjected to Zeeman fields. We find that zero-momentum $d$-wave superconductivity persists until the spin gap closes, coexisting with charge density waves. A novel $d$-wave FFLO phase emerges under a higher Zeeman field even above the Pauli limit, concomitant with a field-enhanced spin density waves. We identify these phases, characterized by the simultaneous presence of pairing condensate and density wave orders, as charge-2e supersolids. Analysis of Matsubara Green's function reveals that the FFLO pairing momentum is locked to the underlying Fermi surface. Our results provide microscopic insights into field-induced unconventional pairing mechanisms and reveal the long-sought FFLO state in a fundamental correlated electron model, offering a promising route for its realization in ultracold atom optical lattice.

DOI
10.48550/arxiv.2601.16630

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ddd-wave FFLO state and charge-2e supersolidity in the ttt-t′t't′-JJJ model under Zeeman fields
Unconventional superconductivity under strong Zeeman fields--particularly beyond the Pauli paramagnetic limit--remains a central challenge in condensed matter physics. The exotic Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, in particular, remains in need of definitive study within fundamental electronic models. Here we employ state-of-the-art finite-temperature and ground-state tensor network approaches to systematically explore the superconducting (SC) phase diagram of the ttt-t′t't′-JJJ model subjected to Zeeman fields. We find that zero-momentum ddd-wave superconductivity persists until the spin gap closes, coexisting with charge density waves. A novel ddd-wave FFLO phase emerges under a higher Zeeman field even above the Pauli limit, concomitant with a field-enhanced spin density waves. We identify these phases, characterized by the simultaneous presence of pairing condensate and density wave orders, as charge-2e supersolids. Analysis of Matsubara Green's function reveals that the FFLO pairing momentum is locked to the underlying Fermi surface. Our results provide microscopic insights into field-induced unconventional pairing mechanisms and reveal the long-sought FFLO state in a fundamental correlated electron model, offering a promising route for its realization in ultracold atom optical lattice.
The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase is an unconventional superconducting state found under the influence of a strong Zeeman field. This phase is identified by finite center-of-mass momenta in the Cooper pairs, causing the pairing amplitude to oscillate in real space. Repulsive correlations, on the other hand, smear out spatial inhomogeneities in ddd-wave superconductors. We investigate the FFLO state in a strongly correlated ddd-wave superconductor within a consolidated framework of Hartree-Fock-Bogoliubov theory and the Gutzwiller approximation. We find that the crucial effects of strong correlations lie in shifting the BCS-FFLO phase boundary towards a lower Zeeman field and thereby enlarging the window of the FFLO phase. In the FFLO state, our calculation features a sharp midgap peak in the density of states, indicating the formation of strongly localized Andreev bound states. We also find that the signatures of the FFLO phase survive even in the presence of an additional translational-symmetry-breaking competing order in the ground state. This is demonstrated by considering a broken-symmetry ground state with a simultaneous presence of the ddd-wave superconducting order and a spin-density wave order, often found in unconventional superconductors.
We numerically study the ground-state properties of a one-dimensional lattice model in the presence of a quasiperiodic Zeeman field, by means of the density matrix renormalization group algorithm. We map out a global phase diagram, which encloses a Bardeen-Cooper-Schrieffer (BCS) phase, a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase where the Cooper pairs obtain a finite center-of-mass momentum, and a localized phase. The FFLO phase is promoted by the quasiperiodic Zeeman field, which demonstrates a power-law pairing correlation with a critical exponent ${\ensuremath{\eta}}_{\mathrm{FFLO}}<1$ decaying much slower than the charge density and spin density correlations. By tuning the interaction and the filling factor, we find the FFLO phase is suppressed in the strong interaction and low filling regions. A large Zeeman field destroys the FFLO state and drives a superconductor-insulator transition. Furthermore, we propose this FFLO phase and the superconductor-insulator transition could be observed in the optical lattice experiment.
We systematically investigate the emergence of finite-momentum Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconductivity in a square lattice Hubbard model with finite filling, driven by either dxyd_{xy}dxy​ -wave or dx2−y2d_{x^{2}-y^{2}}dx2−y2​ -wave altermagnetic order in the presence of on-site sss -wave attractive interactions. Our study combines mean-field calculation in the superconducting phase with pairing instability analysis of the normal state, incorporating the next-nearest-neighbor hopping in the single-particle dispersion relation. We demonstrate that the two types of altermagnetism have markedly different impacts on the stabilization of FFLO states. Specifically, dxyd_{xy}dxy​ -wave altermagnetism supports FFLO superconductivity over a broad parameter regime at low fillings, whereas dx2−y2d_{x^{2}-y^{2}}dx2−y2​ -wave altermagnetism only induces FFLO pairing in a narrow range at high fillings. Furthermore, we find that the presence of a Van Hove singularity in the density of states tends to suppress FFLO superconductivity. These findings may provide guidance for experimental exploration of altermagnetism-induced FFLO states in real materials with more complex electronic structures.
Fulde-Ferrell-Larkin-Ovchinnikov Superfluids with PPP-wave Symmetry in Spinless Fermionic Optical Lattices
The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, whose Cooper pairing hosts a non-zero center-of-mass momentum, is predicted and well studied in spin-singlet superfluids in past decades. Different from previous works that engineer FFLO superfluids in spinful cold atoms, we show that the FFLO superfluids, with ppp-wave symmetry, can emerge in a spinless environment associated with nearest-neighbor-site interactions. The novel mechanism of the FFLO superfluids lies on the split Fermi surfaces at low filling factor, which naturally gives rise to Cooper pairings with finite momentum. The phase transition is accompanied by changed Chern numbers, in which, different from the conventional picture, the band gap does not close. The exotic topological transition can be explained by the spontaneously-broken translational invariance of order parameters. Finally, we investigate the Berezinskii-Kosterlitz-Thouless (BKT) transition temperature of the FFLO state, and show its possibility to be observed at finite temperature.
We present an analytical study of the ground-state phase diagram for dilute two-dimensional spin-1/2 Fermi gases exhibiting ddd-wave altermagnetic spin splitting under sss-wave pairing. Within the Bogoliubov-de Gennes mean-field framework, four distinct phases are identified: a Bardeen-Schrieffer-Cooper-type superfluid, a normal metallic phase, a nodal superfluid with topological Bogoliubov Fermi surfaces (TBFSs), and Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states with finite center-of-mass momentum. Among these, the FFLO states and TBFSs exemplify two unconventional forms of superconductivity. Considering the simplicity of this model, with only one band, zero net magnetization, and sss-wave paring, the emergence of both unconventional phases underscores the pivotal role of altermagnetic spin splitting in enabling exotic pairing phenomena. This analytical study not only offers a valuable benchmark for future numerical simulations, but also provides a concrete experimental roadmap for realizing FFLO states and TBFSs in altermagnets.