High-field transport measurements up to 33 T on the organic-ion-intercalated, quasi-two-dimensional superconductor (TBA⁺)ₓFeSe (zero-resistance transition temperature above 40 K, FeSe layers separated by about 1.6 nm) revealed cuprate-like exotic vortex phenomenology: fragile superconductivity whose irreversibility field is strongly suppressed by small measurement currents, a phase-fluctuating vortex state with finite longitudinal but vanishing Hall resistance, power-law 2D vortex-glass scaling above about 31 T, and a quantum-critical-like magnetoresistance crossing near 31 T. The authors attribute this phenomenology to a conjectured emergent electronic order nucleating in vortex halos that fragments global superconductivity into Josephson-coupled puddles, and explicitly pose the unresolved questions: the microscopic identity of the vortex-halo order (charge-density wave, pair-density wave, or another competing phase); the zero-temperature ground state at fields beyond 33 T up to the extrapolated upper critical field of about 75 T (persistent 2D vortex glass versus anomalous metal, superconductor-insulator or superconductor-metal transition, or quantum Griffiths regime); whether the coincident ~31 T field scales share a microscopic origin; and the mechanism of strong effective pinning in a structurally clean quasi-2D pancake-vortex system. Adjacent 2025–2026 results on other iron-based superconductors make vortex-halo charge order plausible without confirming it here.

Background

Iron selenide (FeSe) is the structurally simplest iron-based superconductor: superconducting FeSe layers are electrically neutral and held together by weak van der Waals forces, so foreign species can be inserted (intercalated) between them. Electrochemical intercalation of the organic cation tetrabutylammonium (TBA⁺) into FeSe single crystals yields (TBA⁺)ₓFeSe, a bulk superconductor with zero resistance above 40 K (transition temperature up to 50 K) in which the superconducting FeSe layers are spaced about 1.6 nm apart (Shi, M. Z. et al., New J. Phys. 20, 123007 (2018)). The resulting electronic structure is effectively two-dimensional (quasi-2D): the superconducting anisotropy γ (ratio of in-plane to out-of-plane upper critical fields) exceeds 60, and the Ginzburg number Gi — a dimensionless measure of the strength of superconducting thermal fluctuations — exceeds 320, comparable to the most anisotropic cuprates. In such an extreme type-II superconductor (one that admits magnetic flux as quantized vortices), a field along the layer normal creates pancake vortices: 2D circulating currents confined to individual FeSe layers and only weakly coupled between layers.

The organization of vortex matter controls high-field transport. The standard phases are the Bragg glass (a vortex lattice weakly distorted by disorder, with quasi-long-range positional order and little bulk pinning), the vortex liquid (a fluctuating, resistive state), and the vortex glass (a disorder-frozen state whose linear resistance vanishes as T→0). When quantum fluctuations dominate at low temperature, 2D and quasi-2D superconductors may instead fail to establish global phase coherence, producing an anomalous metal or 'failed superconductor' — a metallic state with finite, saturating resistance as T→0 — or a field-tuned quantum phase transition such as the superconductor-insulator transition (SIT) or superconductor-metal transition (SMT); in disordered systems, slowly relaxing rare regions can further generate a quantum Griffiths regime with activated rather than power-law scaling (reviewed in Wang, Z. et al., Rep. Prog. Phys. 87, 014502 (2024); Kapitulnik, Kivelson and Spivak, Rev. Mod. Phys. 91, 011002 (2019)).

In charge-ordered cuprates, high-field transport has additionally revealed fragile superconductivity: a superconducting regime whose critical temperature and critical current density are strongly degraded by magnetic field and which collapses under very small currents. Yu and Kivelson (Phys. Rev. B 99, 144513 (2019)) attributed this to superconductivity intertwined with weakly disordered charge-density waves (CDWs — spatially periodic modulations of the electron density): superconductivity survives locally near CDW dislocations, and global phase coherence is limited by the weakest Josephson link between such superconducting islands. Related cuprate experiments found anomalous vortex liquids coupled to charge order (Hsu, Y.-T. et al., PNAS 118, e2016275118 (2021); Shi, Z. et al., Sci. Adv. 6, eaay8946 (2020)), field-tuned two-stage SITs in La₂₋ₓSrₓCuO₄ (Shi, X. et al., Nat. Phys. 10, 437 (2014)), and charge order nucleating in and around vortex cores — the surrounding region is called the vortex halo. Heavily electron-doped FeSe-derived superconductors host analogous competing orders: charge order in hydrothermal (Li,Fe)OHFeSe (Chen, Z. et al., Nat. Commun. 14, 2023 (2023)), an incommensurate pair-density wave (PDW — a superconducting state whose order-parameter amplitude oscillates in space) at domain walls of monolayer Fe(Se,Te)/SrTiO₃ (Liu, Y. et al., Nature 618, 934 (2023)), and a pair-density modulation in FeTe₀.₅₅Se₀.₄₅ flakes (Kong, L. et al., Nature 640, 55 (2025)).

Against this backdrop, Li, X. et al. (arXiv:2601.18531, 2026) mapped the vortex phase diagram of (TBA⁺)ₓFeSe to 33 T via simultaneous longitudinal (RxxR_{xx}) and Hall (RxyR_{xy}) magnetoresistance (MR) with excitation currents spanning 2–1000 μA. They report: a Berezinskii–Kosterlitz–Thouless (BKT) transition — the topological vortex-unbinding transition of a 2D superfluid — at 46 K; an extrapolated out-of-plane upper critical field μ0Hc2c(0)≈75 T\mu_0H_{c2}^c(0)\approx75\ \mathrm{T} with in-plane coherence length ξab(0)≈2.1 nm\xi^{ab}(0)\approx2.1\ \mathrm{nm} (the coherence length is the spatial size of the superconducting order-parameter nucleus); fragile superconductivity (the irreversibility field at 1.75 K drops by about 5 T when the current rises from 20 μA to 1000 μA, with Joule heating excluded by control experiments); a phase-fluctuating vortex state (PFVS) — a broad region of the H–T plane with Rxx>0R_{xx}>0 but Rxy≈0R_{xy}\approx0 (below the ∼10−4 Ω\sim10^{-4}\ \Omega instrument resolution), whose onset fields are nearly current-independent between 20 and 1000 μA, implying unusually strong global vortex pinning; power-law resistance R(H,T)=R0(H)Tα(H)R(H,T)=R_0(H)T^{\alpha(H)} over 1.55–2.2 K with R0(H)R_0(H) turning on above μ0HVG≈31 T\mu_0H_{VG}\approx31\ \mathrm{T}, interpreted as a 2D vortex-glass ground state; and a crossing of MR isotherms (10 K and 15 K) near 31 T with finite-size scaling T∗∝∣δ∣zνT^*\propto|\delta|^{z\nu}, zν≈0.78z\nu\approx0.78, suggesting a quantum critical region near 31 T. To explain how fragile superconductivity, current-sensitive irreversibility, and the zero-Hall PFVS coexist in a structurally clean system, the authors propose that an emergent electronic order (CDW, PDW, or another competing phase) nucleates in the vortex halos, so that with increasing vortex density the condensate fragments into phase-randomized superconducting puddles joined by weak Josephson couplings. This scenario is a transport-level conjecture, and the paper itself identifies what remains unresolved.

Problem Statement

Establish, microscopically, the origin and the ultimate fate of the exotic high-field vortex states in (TBA⁺)ₓFeSe. The source work frames this as four linked questions:

  1. Vortex-halo order. Does an emergent electronic order nucleate in the vortex halos of (TBA⁺)ₓFeSe, and if so what is it — a charge-density wave (a spatially periodic modulation of the electron density), a pair-density wave (a superconducting state whose order-parameter amplitude oscillates in space), or another competing order? An answer must establish its existence, symmetry, coherence length, and interplay with local superconducting pairing using spatially resolved or symmetry-sensitive probes at high magnetic fields (e.g., scanning tunneling spectroscopy, nuclear magnetic resonance, resonant inelastic X-ray scattering, or high-field quantum oscillation measurements), or else rule out the halo-order scenario.

  2. High-field ground state. What is the zero-temperature ground state in the field range beyond the experimentally covered 33 T and below the extrapolated μ0Hc2c(0)≈75 T\mu_0H_{c2}^c(0)\approx75\ \mathrm{T}: does the 2D vortex-glass (fragile superconducting) state persist up to the destruction of superconductivity, or does the system enter a stable anomalous metal, a superconducting-insulator or superconducting-metal transition, or a quantum-Griffiths critical regime? This requires measurements at higher fields and lower temperatures, with current-resolved current–voltage characterization sufficient to distinguish a dissipationless glass from a saturating metallic resistance, an insulating state, or activated scaling.

  3. Common origin of the ~31 T scales. Are the magnetoresistance crossing near μ0H∗≈31 T\mu_0H^{*}\approx31\ \mathrm{T} (the apparent quantum critical point), the onset of 2D vortex-glass scaling at μ0HVG≈31 T\mu_0H_{VG}\approx31\ \mathrm{T} microscopically connected, or is their coincidence accidental? An answer must discriminate the two possibilities, for example by probes that image vortices and electronic order at these fields, or by systematic disorder or field-angle tuning that tests whether the characteristic fields shift together or independently.

  4. Pinning mechanism. What produces the strong effective vortex pinning and the current-independence of the phase-fluctuating-vortex-state onset fields over 20–1000 μA in a structurally clean quasi-2D pancake-vortex system where weak pinning is expected — intrinsic emergent granularity from competing order, subtle extrinsic inhomogeneity, or novel vortex-dislocation or hexatic (a vortex phase with bond-orientational but no long-range positional order) dynamics?

These questions share one objective: deciding whether the cuprate-like fragile superconductivity and phase-fluctuating vortex state of this iron-based superconductor arise from vortex-halo-induced emergent electronic order and the resulting granular Josephson network, and what quantum ground state that mechanism produces in the high-field limit. Question (1) is the linchpin — the proposed mechanism stands or falls on whether the halo order exists — while (3) and (4) test the same mechanism from independent directions and (2) fixes the ultimate high-field fate. A complete resolution establishes the microscopic mechanism and the ground state; partial answers (for example, a spectroscopic identification of the halo order, or a beyond-33 T ground-state determination) resolve the corresponding sub-question.

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