Introduction

Many unconventional superconductors are highly anisotropic (low-dimensional), strongly type II, and often possess a non-s-wave order parameter. Such a unique combination of properties can give rise to novel vortex dynamics and thermodynamics, the study of which has created a rich field of both fundamental and technological interest1,2,3,4. Type-II superconductors, by definition, possess both a lower critical field, Hc1, beyond which magnetic flux first enters the material, and an upper critical field, Hc2, beyond which the vortex cores overlap and the normal state is restored. The region between Hc1 and Hc2 is composed of vortices which, in a perfectly homogeneous system, are arranged in a periodic (Abrikosov) lattice. On applying an external current, the flux lines experience a Lorentz force FL that causes the whole vortex lattice to move, and that can be counteracted only by the friction force. In such an unpinned vortex solid (VS), dissipationless transport is impossible, i.e., there is a finite resistivity at all temperatures below the SC transition temperature Tc1 (see Fig. 1a).

Fig. 1: Schematic H–T phase diagrams of type-II superconductors.
figure 1

a In a system with no pinning or fluctuations, the region between the Meissner phase (shaded blue below the Hc1(T) line) and the normal phase beyond Hc2(T) is occupied by the so-called unpinned vortex solid (VS) which has a finite resistivity for all T below Tc. b In case of a finite pinning and negligible fluctuations, the Hirr(T) curve lies close to Hc2(T), and the region between Hc1(T) and Hc2(T) is a pinned VS with zero-resistivity for all T < Tc. (In panels b–d, the Meissner phase has been omitted for clarity). c In the presence of both pinning and thermal fluctuations, the VS is destroyed, creating a wide vortex liquid (VL) regime and a marked separation between Hirr(T) and Hc2(T). At low-T, however, the two lines coincide. d The VS can also be destroyed by quantum fluctuations, giving rise to a quantum vortex liquid (QVL) phase. In this case, Hirr(T) lies below Hc2(T) even at T ≪ Tc. Here it is worth mentioning that all phase diagrams refer to the limit j → 0 where Hirr line coincides with the melting line (see text).

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Defects introduce pinning forces Fp into the system that can counteract FL without invoking the friction force and thus preserve dissipationless transport over an extended field H and temperature T range. Since the system is no longer perfectly homogeneous, the vortex lattice usually transforms into a vortex glass. For simplicity, we refer here to both the vortex lattice and the vortex glass as VS phases. In order to move the vortices and thus destroy the dissipationless transport, FL has to exceed Fp, which can occur either above a critical depinning current density jdp or above an irreversibility (or depinning) field Hirr1. (Here, we focus on the limit j → 0 so that only Hirr is the important field scale, and any complications with self-fields can be ignored.) In the case of strong pinning, often realized by artificial treatment of a superconductor, FL < Fp and, as a result, Hirr(T) ~ Hc2(T). In the absence of fluctuations, the region between Hc1 and Hc2 is occupied by the pinned VS, which exhibits a zero-resistive state for all T < Tc (see Fig. 1b). The situation becomes more complicated if the pinning is weak, since then in general FL can become larger than Fp at some Hirr < Hc2 in which case the VS can become unpinned. Such complications, however, can be discarded in the limit j → 0 we are interested in, where we can safely assume that FL is always smaller than Fp.

H–T phase diagrams become much richer when we also take thermal fluctuations of the SC order parameter into account. These fluctuations can be understood as fluctuations in the position of the vortex lines, i.e., in the phase field of the SC order parameter. (Besides phase, there are also amplitude fluctuations of the SC order parameter not related to a vortex motion. Those fluctuations, however, are usually confined to a narrow region around Tc and Hc2.) At the melting field Hm(T), these thermal fluctuations become of order the vortex spacing, and the vortex lattice melts into a VL. In the VL phase, vortices move freely so that, on average, Fp = 0, FL > Fp, and dissipationless transport is lost. In addition to Hc1 and Hc2, Hm(T) is a fundamental quantity that shapes the H–T phase diagram of type-II superconductors. It is worth mentioning here that Hm(T) refers to a real solid-liquid transition, i.e., it is a thermodynamic quantity, while Hirr(T) refers to the onset of a finite resistivity arising from either the solid-liquid transition or from a simple unpinning of the VS. In other words, Hirr(T) does not necessarily point towards the VS-VL transition and is, in general, always ≤Hm(T). In the limit j → 0, Hm(T) = Hirr(T), thus enabling the determination of Hm(T) from dc transport measurements. A schematic representation of the H–T phase diagram in the presence of thermal fluctuations is shown in Fig. 1c. As we can see, the thermal fluctuations destroy the VS across a large portion of the phase diagram at elevated temperatures.

If in addition to these thermal fluctuations, the system is also subject to quantum fluctuations of the SC order parameter, the VS can melt even at T ≪ Tc, resulting in the formation of a quantum VL (QVL) (see Fig. 1d). While initial theoretical work explored the role of quantum fluctuations at finite temperatures5, later work considered quantum melting of the vortex lattice even in the zero-temperature limit6,7,8. Experimental evidence for QVL formation, however, has only been reported for a small number of systems, including certain amorphous thin films9,10,11, the organic conductor κ-(BEDT-TTF)2Cu(NCS)212,13,14,15, and low-doped high-Tc cuprates16, suggesting that conditions for its realization are rather stringent.

What is not evident from previous studies is whether the QVL and associated SC quantum fluctuations can be influenced by a nearby phase. Studies on amorphous films have shown that an increase in the normal state resistivity ρn leads to an expansion of the QVL regime17, but it is not clear whether a similar correlation exists in the more strongly correlated organic and cuprate superconductors. Many correlated metals also harbor a quantum critical point (QCP) somewhere in their T vs. g phase diagram, where g is some non-thermal tuning parameter, such as pressure, chemical substitution, or magnetic field. Moreover, while there have been studies of the evolution of the critical fields Hc118 and Hc219,20 across putative QCPs, the role of associated fluctuations in destabilizing the vortex lattice at low T has not been explored, despite the fact that they can have a profound influence on both the normal and SC state properties. In order to explore these questions, it is necessary to identify a material class that harbors both a QVL and a QCP somewhere within its tunable range of superconductivity. In this report, we consider one such candidate material and study the evolution of the low-T vortex dynamics in the iron chalcogenide family FeSe1−xXx where X = S, Te.

FeSe is unique among iron-based superconductors in that it develops nematic order, characterized by a spontaneous rotational symmetry breaking of the electronic state, without accompanying magnetic order21,22,23,24. With increasing x, the nematic phase transition is suppressed, terminating at a critical S (Te) concentration xc ~ 0.17 (0.50), respectively25,26. Superconductivity persists across the entire series with Tc peaking in the S-substituted family around 10 K for x ≈ 0.10, where the spin fluctuations are also enhanced27, and around 14 K in the Te-substituted family near the nematic end point20,28. In FeSe1−xSx, the magnitude of the SC gap Δ roughly halves outside of the nematic phase29,30, suggesting that nematicity has a profoundly distinct influence on the SC properties of the two systems. Finally, the low carrier density coupled with its relatively high Tc has led to speculation that superconductivity in FeSe and its cousins sits proximate to a Bardeen–Cooper–Schrieffer Bose-Einstein condensate (BCS–BEC) crossover24.

The normal state transport properties of FeSe1−xSx exhibit many features synonymous with quantum criticality and non-Fermi-liquid behavior43, while \({j}_{0}=4\tilde{{H}_{c}}/(3\sqrt{6}\lambda )\), where \(\tilde{{H}_{c}}\) is the thermodynamic critical field and λ is the London penetration depth. Taking \({\mu }_{0}\tilde{{H}_{c}}(0)=0.21\,{{{{{{{\rm{T}}}}}}}}\)44 and λ(0) ≈ 400 nm45, we obtain jc ≈ 2 × 107 A/cm2 and jc/j0 ≈ 2 × 10−3, a value similar to that found in cuprates and ~10–100 times smaller than in conventional superconductors1. Such a value suggests that pinning in FeSe is weak so that the melting transition between the VS and the VL phase will be only weakly perturbed by the presence of disorder1. According to ref. 43, however, jc in FeSe1−xSx follows a similar power-law decay H−0.5 to that observed in iron-pnictides and attributed to strong pinning by sparse nm-sized defects46. Nevertheless, it seems reasonable to assume that at the current densities applied in this work FL < Fp and that Hirr(T) ≈ Hm(T), i.e., a finite resistivity is unambiguously related to the presence of a VL phase (see Supplementary Note 1). This is especially true for the longitudinal orientation H∥ab where j∥H and, therefore FL = 0.

The strength of thermal fluctuations is usually quantified by the Ginzburg number Gi = (1/8πμ0) × ((kBTcΓ)/\({({H}_{c}^{2}(0){\xi }_{\parallel }^{3}(0))}^{2}\)—itself, a measure of the relative size of the thermal and condensation energies within a coherence volume1,47. Here, kB is the Boltzmann constant, Γ = ξ∥(0)/ξ⊥(0) is the anisotropy of Hc2, and ξ∥(0) and ξ⊥(0) are the in- and out-of-plane coherence lengths at 0 K, respectively. From existing thermodynamic data, Gi in FeSe is estimated to be 5 × 10−4 44, which is lower than in the cuprates where thermal fluctuations are very strong (10−3 < Gi < 10−1)48, but around four orders of magnitude larger than in classical superconductors1. Indeed, evidence of strong thermal SC fluctuations has been found in several normal state properties for pure FeSe41, as well as for FeSe1−xSx beyond xc\(-{H}_{\max }\) to \(+{H}_{\max }\). Complementary Hall effect measurements were also performed at HFML in magnetic fields up to 33 T and at fixed temperatures down to 0.3 K. At each temperature, the transverse MR signal Vx (Hall signal Vy) was symmetrized (antisymmetrized) in order to eliminate any finite Hall (MR) component. Analysis of the Hall data is shown in Supplementary Note 3, while additional sample details are given in Supplementary Note 7.