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arXiv:2509.02688 · Proceedings of the National Academy of Sciences

In search of exotic pairing in the Hubbard model: Many-body computation and quantum gas microscopy

Chunhan Feng, Thomas R. Hartke, Yuan-Yao He, Botond Oreg, Carter Turnbaugh, Ningyuan Jia, Martin W. Zwierlein, Shiwei Zhang

Published Jun 17, 2026• 2 citations
arXivHTMLPDF

Abstract

Finite-momentum pairing, exemplified by Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states, represents a paradigmatic form of unconventional superfluidity driven by Fermi-surface mismatch, but its detection in two-dimensional systems has remained elusive. Here we study a doped, spin-imbalanced attractive Hubbard model using a combined experimental and computational approach, based on quantum gas microscopy and constrained-path auxiliary-field quantum Monte Carlo, with direct comparisons showing quantitative agreement for short-range correlations at experimentally accessible temperatures. We identify broad regimes in density and magnetization where FFLO correlations emerge, and establish their finite-temperature evolution, with clear signatures of finite-momentum pairing already appearing at experimentally accessible temperatures. Spin-XY correlations are identified as a robust, directly measurable proxy for FFLO physics. Quantitative characterizations are obtained on the temperature dependence of a variety of observables and correlations, which elucidate the interplay of pairing with competing orders.

DOI
10.1073/pnas.2600303123

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Finite-momentum pairing, exemplified by Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states, represents a paradigmatic form of unconventional superfluidity driven by Fermi-surface mismatch, but its detection in two-dimensional systems has remained elusive. Here we study a doped, spin-imbalanced attractive Hubbard model using a combined experimental and computational approach, based on quantum gas microscopy and constrained-path auxiliary-field quantum Monte Carlo, with direct comparisons showing quantitative agreement for short-range correlations at experimentally accessible temperatures. We identify broad regimes in density and magnetization where FFLO correlations emerge, and establish their finite-temperature evolution, with clear signatures of finite-momentum pairing already appearing at experimentally accessible temperatures. Spin-XY correlations are identified as a robust, directly measurable proxy for FFLO physics. Quantitative characterizations are obtained on the temperature dependence of a variety of observables and correlations, which elucidate the interplay of pairing with competing orders.
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconductivity, Cooper pairing at finite-momentum driven by spin imbalance, remains difficult to study numerically in two dimensions. The fermion sign problem in the spin-imbalanced attractive Hubbard model severely limits quantum Monte Carlo access to the low temperatures where pairing correlations are most pronounced. We show that the Holstein model, in which the effective attraction is retarded by phonon dynamics, strongly suppresses sign fluctuations under spin imbalance, enabling unbiased determinant quantum Monte Carlo simulations deep in this regime. With increasing magnetization, zero-momentum pairing evolves into finite-momentum FFLO pairing, evidenced by oscillating real-space pair correlations and a shift of the pairing susceptibility peak to finite Q⃗\vec{Q}Q​. The finite-momentum susceptibility substantially exceeds its zero-momentum value. At high density n∼0.875n \sim 0.875n∼0.875, near magnetization m≈1−nm \approx 1-nm≈1−n where the majority-spin density approaches half filling, FFLO pairing coexists with checkerboard charge-density-wave order at short range, while staggered longitudinal spin correlations emerge as a magnetic halo at larger distances. These spatial textures provide experimentally accessible signatures of finite-momentum pairing and motivate cold-atom realizations of FFLO physics.
We study sss-wave superconductivity in the two-dimensional attractive Hubbard model in an applied magnetic field, assume the extreme Pauli limit, and examine the role of spatial fluctuations in the coupling regime corresponding to BCS-BEC crossover. We use a decomposition of the interaction in terms of an auxiliary pairing field, retain the static mode, and sample the pairing field via a Monte Carlo approach. The method requires iterative solution of the Bogoliubov--de-Gennes equations for amplitude- and phase-fluctuating configurations of the pairing field. We establish the full thermal phase diagram of this strong-coupling problem. At low field we observe the magnetized but homogeneous ``breached pair'' superfluid phase. It reveals that Tc{T}_{c}Tc​ scales an order of magnitude below the mean-field estimate, spontaneous inhomogeneity in the field-induced magnetization, and a strong nonmonotonicity in the temperature dependence of the low-energy density of states. We compare our results to the experimental phase diagram of the imbalanced Fermi gas at unitarity. At higher field we obtain the modulated Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phases. The thermal transition from the FFLO phases to the normal state is strongly first order. We track the fermionic momentum distribution, the density of states, and the pairing structure factor deep into the normal state. The pairing structure factor retains weak signature of finite momentum pairing to a high temperature despite the low Tc{T}_{c}Tc​ itself, while the spin-resolved density of states changes from the ``pseudogapped'' FFLO character to gapless and pseudogapped again with increasing temperature.
The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase was first predicted in two-dimensional superconductors about 50 years ago, but so far unambiguous experimental evidence is still lacking. The recently experimentally realized spin-imbalanced Fermi gases may potentially unveil this elusive state, but they require very stringent experimental conditions. In this Rapid Communication, we show that FFLO phases may be observed even in a three-dimensional (3D) degenerate Fermi gas with spin-orbit coupling and an in-plane Zeeman field. The FFLO phase is driven by the interplay between the asymmetry of the Fermi surface and the superfluid order, instead of the interplay between magnetic and superconducting order in solid materials. The predicted FFLO phase exists in a giant parameter region, possesses a stable long-range superfluid order due to the 3D geometry, and can be observed with an experimentally already achieved temperature ($T\ensuremath{\sim}0.05{E}_{F}$), thus opening a fascinating avenue for exploring FFLO physics.
The recent development of experimental techniques in ultracold atomic Fermi gases is extremely helpful in the progress of the realization of the unconventional Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid phase in quasi-one dimensional systems (Liao et al 2010 Nature 467 567). Due to a Fermi surface nesting, which is enhanced in 1D, the low-dimensional systems are particularly good candidates to find the FFLO phase stable. We investigate the influence of a dimensional crossover (from one dimension (1D) to two dimensions (2D) or three dimensions (3D)) on the stability of the FFLO state in the spin-imbalanced attractive Hubbard model.
We investigate properties of exotic pairing states in a three-dimensional Fermi gas with three-dimensional spin-orbit coupling and an effective Zeeman field. The interplay of spin-orbit coupling, effective Zeeman field, and pairing can lead to first-order phase transitions between different phases, and to interesting nodal superfluid states with gapless surfaces in the momentum space. We then demonstrate that pairing states with zero center-of-mass momentum are unstable against Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states, with a finite center-of-mass momentum opposite to the direction of the effective Zeeman field. Unlike conventional FFLO states, these FFLO states are induced by the coexistence of spin-orbit coupling and Fermi surface deformation, and have intriguing features like first-order transitions between different FFLO states, nodal FFLO states with gapless surfaces in momentum space, and exotic fully gapped FFLO states. With the recent theoretical proposals for realizing three-dimensional spin-orbit coupling in ultracold atomic gases, our work is helpful for future experimental studies and provides valuable information for the general understanding of pairing physics in spin-orbit-coupled fermionic systems.