Fermi systems in the cross-over regime between weakly coupled Bardeen–Cooper–Schrieffer (BCS) and strongly coupled Bose–Einstein-condensate (BEC) limits are among the most fascinating objects to study the behavior of an assembly of strongly interacting particles. The physics of this cross-over has been of considerable interest both in the fields of condensed matter and ultracold atoms. One of the most challenging issues in this regime is the effect of large spin imbalance on a Fermi system under magnetic fields. Although several exotic physical properties have been predicted theoretically, the experimental realization of such an unusual superconducting state has not been achieved so far. Here we show that pure single crystals of superconducting FeSe offer the possibility to enter the previously unexplored realm where the three energies, Fermi energy ${\epsilon }_{\mathrm{F}}$, superconducting gap Δ, and Zeeman energy, become comparable. Through the superfluid response, transport, thermoelectric response, and spectroscopic-imaging scanning tunneling microscopy, we demonstrate that ${\epsilon }_{\mathrm{F}}$ of FeSe is extremely small, with the ratio $\mathrm{\Delta }/{\epsilon }_{\mathrm{F}}\sim 1\left(\sim 0.3\right)$ in the electron (hole) band. Moreover, thermal-conductivity measurements give evidence of a distinct phase line below the upper critical field, where the Zeeman energy becomes comparable to ${\epsilon }_{\mathrm{F}}$ and Δ. The observation of this field-induced phase provides insights into previously poorly understood aspects of the highly spin-polarized Fermi liquid in the BCS-BEC cross-over regime. Significance The BCS-BEC (Bardeen–Cooper–Schrieffer––Bose–Einstein-condensate) cross-over bridges the two important theories of bound particles in a unified picture with the ratio of the attractive interaction to the Fermi energy as a tuning parameter. A key issue is to understand the intermediate regime, where new states of matter may emerge. Here, we show that the Fermi energy of FeSe is extremely small, resulting in that this system can be regarded as an extraordinary “high-temperature” superconductor located at the verge of a BCS-BEC cross-over. Most importantly, we discover the emergence of an unexpected superconducting phase in strong magnetic fields, demonstrating that the Zeeman splitting comparable to the Fermi energy leads to a strong modification of the properties of fermionic systems in such a regime.

We have observed Shubnikov–de Haas oscillations in FeSe. The Fermi surface deviates significantly from predictions of band-structure calculations and most likely consists of one electron and one hole thin cylinder. The carrier density is in the order of 0.01 $\mathrm{carriers}/\mathrm{Fe}$, an order of magnitude smaller than predicted. Effective Fermi energies as small as 3.6 meV are estimated. These findings call for elaborate theoretical investigations incorporating both electronic correlations and orbital ordering.

We report the measurements of the de Haas–van Alphen (dHvA) effect for field ranges below and above the metamagnetic transition field (${\mathit{H}}_{\mathit{m}}$) in ${\mathrm{CeRu}}_2$ ${\mathrm{Si}}_2$. The dHvA frequency branches and the effective masses ranging from 1.5${\mathit{m}}_0$ to 120${\mathit{m}}_0$ observed below ${\mathit{H}}_{\mathit{m}}$ agree with the predictions of the itinerant f electron model, whereas those above ${\mathit{H}}_{\mathit{m}}$ can be explained well with the localized f electron model. All the dHvA frequencies change abruptly around ${\mathit{H}}_{\mathit{m}}$. The effective masses decrease considerably around ${\mathit{H}}_{\mathit{m}}$ and then continue to decrease with increasing field.

We investigate by electrical transport the field-induced superconducting state (FISC) in the organic conductor $\lambda -(\mathrm{BETS}{)}_2{\mathrm{FeCl}}_4$. Below 4 K, antiferromagnetic-insulator, metallic, and eventually superconducting (FISC) ground states are observed with increasing in-plane magnetic field. The FISC state survives between 18 and 41 T and can be interpreted in terms of the Jaccarino-Peter effect, where the external magnetic field compensates the exchange field of aligned ${\mathrm{Fe}}^{3+}$ ions. We further argue that the ${\mathrm{Fe}}^{3+}$ moments are essential to stabilize the resulting singlet, two-dimensional superconducting state.

We report the results of SQUID and torque magnetometry of an organic spin-$1/2$ triangular-lattice $\kappa -{\mathrm{H}}_3(\text{Cat-EDT-TTF}{)}_2$. Despite antiferromagnetic exchange coupling at 80–100 K, we observed no sign of antiferromagnetic order down to 50 mK owing to spin frustration on the triangular lattice. In addition, we found nearly temperature-independent susceptibility below 3 K associated with Pauli paramagnetism. These observations suggest the development of gapless quantum spin liquid as the ground state. On the basis of a comparative discussion, we point out that the gapless quantum spin liquid states in organic systems share a possible mechanism, namely the formation of a band with a Fermi surface possibly attributed to spinons.