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 behavior31,32,33,34,35,36,37. In particular, the limiting low-T resistivity ρ(T), exposed by the application of a large magnetic field, traces out a fan-like region of T-linearity centered on the nematic QCP31,32,36 below which a T2 resistivity is recovered, with a coefficient that is strongly enhanced upon approaching the QCP (from the high-x side)38. In FeSe1−xTex near the nematic QCP, there is evidence for a poorly metallic or incoherent normal state that may also indicate a non-Fermi-liquid ground state though high-field, low-T transport measurements have not yet been reported.

In this report, we investigate the evolution of the vortex state in a series of FeSe1−xSx and FeSe1−xTex single crystals via detailed magnetotransport measurements down to T/Tc ≈ 0.03 and in fields up to 60 T. These measurements cover a wide range of concentrations x spanning the nematic QCPs in both systems and for FeSe1−xSx in both longitudinal and transverse field orientations. Our study reveals an expanded QVL regime over a broad region of the cojoined FeSe1−xXx phase diagram. Subsequent analysis indicates that the extent of the QVL regime is directly correlated with the strength of the SC order parameter, being anomalously broadened in the concentration ranges where Tc, Hc2, and/or the SC energy gap Δ are suppressed. By extending the study across two families, we can rule out a unique or dominant role for nematic or spin fluctuations, critical or otherwise, as well as for the BCS-BEC crossover in creating the QVL. Nevertheless, it is likely that their combined presence, coupled with the low dimensionality, creates the necessary conditions for quantum SC fluctuations to have a profound destabilizing effect on the VS, one that is amplified wherever the intrinsic superconductivity is weakened.

Results

In this section, we will focus predominantly on the FeSe1−xSx study, for which the bulk of the measurements were performed, and will summarize the corresponding findings for FeSe1−xTex in the “Discussion” section that follows. The key parameters for quantifying the extent of the QVL regime are Hirr(0) and Hc2(0), which are determined from magnetoresistance (MR) measurements using a low current excitation in order to stay within the j → 0 limit where Hirr(T) ≈ Hm(T) (see Supplementary Note 1 where we also discuss the current dependence of Hirr(0)). Note that we use the same criteria for determining Hirr(T) and Hc2(T) for both families.

For the longitudinal field orientation (H∥ab), the MR in FeSe1−xSx is negligible for x ≥ 0.16 and small and linear for x < 0.1632, making the identification of Hc2(T) relatively straightforward. Here, we define Hc2(T) as the field at which dρ/d(μ0H) returns to a constant value, i.e., when SC fluctuations are fully suppressed, and the MR assumes its normal state form, while Hirr(T) is defined as the field above which the derivative dρ/d(μ0H) rises above a threshold set by the noise level.

Panels a–e of Fig. 2 show dρ/d(μ0H) for H∥ab at a fixed temperature T = 1.3 K for FeSe1−xSx crystals with nominal x values of 0, 0.1, 0.16, 0.18, and 0.25 (labeled hereafter FSS00, FSS10, FSS16, FSS18, and FSS25), respectively. The noise floor for each panel is given in the Figure caption and indicated by the colored horizontal dashed lines. The vertical black arrows in panels a-e indicate the corresponding values for Hirr and Hc2. A marked expansion of the QVL regime is clearly evident beyond xc ~ 0.17 that is due predominantly to a marked reduction in Hirr rather than an enhancement in Hc2. This observation contrasts with a recent pressure study on FeSe1−xSx (x = 0.11)39, where the expansion of the VL regime across the nematic QCP occurs due to an enhancement in Hc2.

Fig. 2: Determination of Hirr and Hc2 in FeSe1−xSx with H∥ab at T = 1.3 K.
figure 2

Field derivatives of the MR curves for a FSS00, b FSS10, c FSS16, d FSS18 and e FSS25. For all samples, Hirr is determined from the field value above which dρ/d(μ0H) rises above a threshold—indicated by the colored horizontal dashed lines—set by the noise level, while Hc2 is the field value beyond which dρ/d(μ0H) returns to a constant value within the noise floor. The noise threshold is 0.05 μΩcm/T for FSS00 and FSS25 and 0.1 μΩcm/T for FSS10, FSS16, and FSS18, respectively. The marked expansion of the low-T vortex liquid regime for x > xc is clear.

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In the transverse field orientation, the MR curves exhibit a complex H, T dependence33 that makes a determination of Hc2(T) more challenging. Nevertheless, as shown in Supplementary Notes 2 and 3, by normalizing the field derivatives at each temperature and comparing them with the corresponding Hall response, a robust estimate of Hc2(T) could be obtained for each MR curve measured (with the exception of FSS00—see Supplementary Note 2 for details).

Resultant Hirr(T) and Hc2(T) curves for H∥ab are plotted in panels a–c of Fig. 3 for FSS00, FSS16, and FSS25, respectively. In all cases, Hirr(T) lies well below Hc2(T) at intermediate T, indicating the presence of a broad VL regime that narrows with decreasing temperature. In FSS00, there is a small but sharp change of slope in Hirr(T) below T ~ 1 K that has been attributed previously to the emergence of a Fulde–Ferrell–Ovchinnikov–Larkin phase40. For FSS16, Hirr(T) ≈ 0.8-0.9 Hc2(T) as T → 0, while for FSS25, Hirr(0) < 0.5Hc2(0), as demonstrated in Fig. 2e.

Fig. 3: Paradigmatic H–T phase diagrams of FeSe1−xSx in longitudinal and transverse magnetic fields.
figure 3

Hirr(T) (orange circles) and Hc2(T) (green squares) curves extracted from the MR derivatives for a FSS00, b FSS16, and c FSS25 with H∥ab and for e FSS13, f FSS16, and g FSS25 with H∥c. The VL state is shaded green. The Meissner phase at low fields has been omitted for clarity. Error bars for Hirr and Hc2 are estimated from the threshold noise level shown, for example, in Fig. 2. The larger error bars for Hc2 at low T in b are due to the limited field range of our experiment. (For this reason, Hc2 could not be determined below 1.3 K in all samples). Larger error bars for Hc2 at low T in panels e and f are due to the emergence of quantum oscillations as indicated in Supplementary Fig. 2. d, h The ratio Hirr(0)/Hc2(0) deduced at the temperatures indicated for H∥ab and H∥c, respectively. In both panels, the black dashed line is a guide to the eye, while the dark (light) gray backgrounds refer to the nematic (non-nematic) phases. The vertical dotted line spanning panels d and h locates the position of the nematic QCP at x = xc ≈ 0.17. Finally, the error bars in panels d and h are composite errors obtained from the errors in Hirr and Hc2.

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Resultant Hirr(T) and Hc2(T) curves for H∥c are plotted in panels e-g of Fig. 3 for FSS13, FSS16, and FSS25, respectively. In contrast to what is found in parallel fields, Hirr(T) is suppressed relative to Hc2(T) down to the lowest temperatures at all concentrations. It is also evident that Hc2(T) follows a distinct T dependence in the two field orientations, which for H∥c does not follow the standard Ginzburg–Landau expression \({H}_{c2}(T)={H}_{c2}(0)(1-{(T/{T}_{c})}^{2})\). This non-standard T-dependence is likely to reflect the multi-band character of FeSe1−xSx, though it may also result from proximity to the BCS–BEC crossover regime41,42 (to be discussed in more detail below).

In order to quantify the extent of the VL regime, we define Hirr(0)/Hc2(0) as the ratio of the two field scales as T → 0 (determined at the lowest measured temperatures—typically 0.3 K or 1.3 K). The evolution of Hirr(0)/Hc2(0) with x is shown in Fig. 3d for H∥ab at three representative temperatures (0.3, 1.3, and 2.0 K) and in Fig. 3h for H∥c at T = 0.3 K and 1.3 K. The overall trend is clear and appears to be independent of temperature. (Two other procedures for the determination of Hc2(T), presented in Supplementary Fig. 5, demonstrate a qualitatively similar evolution of the ratio with x.) For H∥c, Hirr(0)/Hc2(0) ≤ 0.6 for all finite x. The most striking feature, however, is the marked, twofold drop in Hirr(0)/Hc2(0) at high x, revealing an expansion of the QVL phase beyond xc in both field orientations. This is the main finding of the FeSe1−xSx study. The corresponding result for FeSe1−xTex is shown together with FeSe1−xSx in Fig. 4 and will be discussed in the following section.

Fig. 4: Comparison of the QVL regimes in FeSe1−xSx and FeSe1−xTex for H∥c.
figure 4

Symbols represent the ratio Hirr(0)/Hc2(0) deduced at the lowest temperatures T = 0.3 K for FeSe1−xSx and T = 0.6 − 1.5 K for FeSe1−xTex. The data for FeSe1−xSx are taken from Fig. 3h and for FeSe1−xTex from a similar analysis of data from our previous study20 (see Supplementary Note 5). The green dashed line is the x-evolution of the corresponding SC transition temperature Tc which for FeSe1−xTex is reproduced from Fig. 3 in ref. 20. Vertical dotted lines indicate the positions of the nematic QCPs. The error bars are composite errors obtained from the errors in Hirr and Hc2.

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Discussion

According to Ginzburg–Landau theory, the H–T phase diagram of a type-II superconductor depends on three fundamental parameters: (i) pinning strength, (ii) thermal fluctuations, and (iii) quantum fluctuations (see Fig. 1). The pinning strength represents the effective energy barrier a vortex has to overcome in order to become mobile. It can be evaluated from the ratio jc/j0 of the depinning current density (above which the vortices start to move) and depairing current density (above which Cooper pairs are destroyed)1. Magnetization measurements in pure FeSe give jc ≈ 4 × 104A/cm2 43, 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 xc49, in line with the broad VL regime found here (see Fig. 3).

As T → 0, the effect of thermal fluctuations inevitably diminishes, leaving only quantum fluctuations of the SC order parameter to influence the vortex dynamics. The persistence of a broad QVL regime in Fig. 3 thus supports the notion that the vortex lattice in FeSe1−xSx is destabilized at very low temperatures by strong quantum fluctuations inherent in the system. According to the Lindemann criterion, quantum melting occurs when the zero-point amplitude of the vortices becomes an appreciable fraction (typically 0.1–0.2) of the vortex spacing. At that point, there are strong fluctuations in the positions of the vortex lines, i.e., fluctuations in the phase field of the SC order parameter, which leads to the formation of a QVL. (The corresponding amplitude fluctuations can be ignored since they are usually confined to a narrow region around Tc and Hc2.)

The strength of such quantum fluctuations can be estimated by the dimensionless quantum resistance Qu = e2/(ℏηs), where e is the elementary charge, ℏ the reduced Planck constant, η the friction coefficient, and s = 5.5 Å the interlayer spacing5. Using the conventional BCS approach in which the friction coefficient η is proportional to the normal state conductivity 1/ρn1,5,6, we obtain a small value of Qu for pure FeSe. Theories that go beyond the mean-field approach50,51, however, suggest that the friction coefficient can be drastically reduced due to details of the electronic structure and/or proximity to other non-SC orders, leading to a significant enhancement of Qu in line with the broad QVL regime observed here.

With this in mind, let us now turn to consider the QVL regime in more detail. The two-fold drop in Hirr(0)/Hc2(0) beyond xc indicates a marked broadening of the QVL beyond the nematic QCP. Thermal conductivity and specific heat measurements in FeSe1−xSx30 indicate a similarly abrupt change in the SC gap structure at xc while scanning tunneling spectroscopy29 detected a twofold drop in Δ beyond xc29, in remarkable agreement with the x-evolution of Hirr(0)/Hc2(0). Such a correspondence between Hirr(0)/Hc2(0) and Δ clearly demonstrates an intimate relationship between the extent of the QVL regime and the strength of superconductivity in FeSe1−xSx.

This correspondence becomes even more evident when we compare these ratios for the two chalcogenide families, as is done in Fig. 4. The most striking feature of this plot is that the marked downturns in Hirr(0)/Hc2(0) (purple circles)—corresponding to an expansion of the QVL regime—coincide with a dip in their respective Tc(x) curves (green dashed line). This correlation provides strong evidence that the QVL regime is the most extended wherever the intrinsic superconductivity is the weakest. It should be stressed, however, that this is not a trivial finding since such an expanded QVL is never observed in conventional but weakly coupled superconductors. This then begs the question: what is the cause of strong SC quantum fluctuations and associated QVL phase that permeate the chalcogenide phase diagram?

As already mentioned, SC quantum fluctuations can be significantly enhanced due to proximity to a non-SC order50,51, fluctuations of which can cause an additional scattering of quasiparticles within the vortex cores52. In underdoped La2−xSrxCuO4, for example, a QVL phase is formed close to the antiferromagnetic (AFM) endpoint, where spin fluctuations are known to be prevalent16. In FeSe1−xSx, although no magnetic order exists at ambient pressure, strong AFM fluctuations are present53,54,55, associated with the static order that develops under applied pressures56. Such fluctuations could affect the vortex dynamics and possibly establish the QVL state. With increasing x, however, AFM fluctuations in FeSe1−xSx are progressively suppressed and become negligible beyond the nematic QCP27,57,58, exactly where the QVL state seems to be the most pronounced. Although a detailed Te composition dependence of spin fluctuations for 0 ≤ x ≤ 0.50 is currently lacking, a related pressure study showed that magnetic interactions do become weaker in the higher-Tc dome region59. Thus it seems unlikely that spin fluctuations can account for the development of the QVL phase in either system.

Strong signatures of nematic quantum criticality have been seen in the elasto-resistivity, Hall and MR responses in FeSe1−xSx25,32,33,35,37. Nematic fluctuations diverge as x → xc25, while the fan of T-linear resistivity and enhanced T2 coefficient (on approach to xc from the high x side38) have both been attributed to enhanced quasiparticle scattering off critical nematic fluctuations that may also enhance vortex dissipation at low-T beyond xc. The absence of any sharp dip in Hirr(0)/Hc2(0) near the QCP in FeSe1−xTex, however, coupled with the fact that the QVL regime is most pronounced inside the nematic phase (Fig. 4), appears to rule that possibility out. This latter point also rules out the loss of orthorhombicity and the associated disappearance of twin boundaries as pinning centers as the origin for the expanded QVL regime in FeSe1−xSx60,61.

In pure FeSe, the ratio Δ/ϵF ≈ 0.3 − 1.0, where ϵF is the Fermi energy24,42,45, placing FeSe close to the BCS-BEC crossover (see Supplementary note 6). Proximity to this crossover will inevitably lead to strong SC fluctuations and potentially to an enhancement of both the VL and QVL regimes. ARPES measurements on FeSe1−xSx62 claim to show strong evidence for BEC-like superconductivity beyond xc, while recent thermodynamic studies have revealed that the shape of the heat capacity jump at Tc changes across xc, exhibiting non-mean-field behavior that is reminiscent of a BEC transition49. Certainly, the marked drop in Hirr(0)/Hc2(0) across xc shown in Fig. 3d, h seems to correlate with enhanced SC fluctuations associated with BCS–BEC crossover49,62. Again, the comparison with FeSe1−xTex turns out to be highly instructive. In the latter, the BCS–BEC crossover is most pronounced at x ≈ 0.52, i.e., where Hirr(0)/Hc2(0) reaches almost the highest value, and the QVL regime is least pronounced. This suggests that proximity to the BCS–BEC crossover while creating an ideal environment for strong SC fluctuations to proliferate, cannot be the dominant reason for an expanded QVL regime in the chalcogenides. Nevertheless, given the profound influence that spin fluctuations, nematic fluctuations, and the putative BCS–BEC crossover appear to have on the normal state, it is highly likely that they play some role in the mixed state (vortex) dynamics of FeSe1−xXx.

The finding that the drop in Hirr(0)/Hc2(0) in FeSe1−xSx coincides with a similar reduction in Δ arguably provides the key to unraveling the origin of the expanded QVL regime beyond xc. Generally63, pinning strength is determined by the product of the condensation energy Uc (∝Δ2) and the minimum volume change associated with vortex motion (∝ξ 3). Hence, the twofold reduction in Δ beyond xc29 must have a sizeable effect on the overall pinning strength and, thus, on the stability of the vortex liquid, i.e., the melting line. While a comparable study of Δ(x) in the Te-substituted family has not yet been reported, the correlation between the expanded QVL regime and the reduced Tc and Hc2(0) suggests that a similar relation between Hirr(0)/Hc2(0) and Δ may exist in FeSe1−xTex too. Finally, we remark that the possible influence of the so-called Bogoliubov Fermi surface64, invoked to account for the high residual density of states found beyond xc in FeSe1−xSx30, should also be considered.

Before closing, we turn to consider why the QVL phase is much reduced, if present at all, for x < xc in parallel fields. In this field orientation, vortices will be configured to lie between the SC planes, and while the SC state in FeSe1−xSx is not strongly anisotropic, they may still become susceptible to an additional pinning potential created by the lamellar nature of the crystal. This in turn, will raise the overall barrier height, thereby reducing the propensity for quantum fluctuations to destabilize the lattice. Beyond xc, however, even this additional barrier becomes ineffective.

In summary, by carrying out a comparative transport study of the mixed state in both FeSe1−xSx and FeSe1−xTex, we have uncovered two markedly enhanced QVL regimes that are concomitant with a reduced SC pairing strength. The persistence of this broad QVL regime over such an extended region of the phase diagram, however, suggests that key elements of the electronic state of iron chalcogenides, such as the presence of strong nematic and magnetic fluctuations and proximity to the BCS–BEC crossover, conspire to destabilize the VS to such an extent that any further weakening of the SC order leads to the QVL regime being amplified. Whatever the origin of this reduced SC strength beyond xc = 0.17 in FeSe1−xSx and at x ≈ 0.30 in FeSe1−xTex, the existence of such an extended QVL regime provides an unprecedented opportunity to study the dynamics of this elusive state in great detail. At the same time, it enables us to identify new guiding principles for its emergence that go beyond the simple requirement of an SC state of reduced dimensionality, specifically the propensity to fluctuating order and proximity to a BCS-BEC crossover. While in amorphous films, the disorder is likely to be playing a key role, in the underdoped cuprates and organic superconductors, fluctuating order is also present. Thus, from a theoretical perspective, the results reported here should help to motivate future investigations into how strong order parameter fluctuations act to destabilize the vortex lattice in unconventional superconductors.

Methods

FeSe1−xSx (0 ≤ x ≤ 0.25) and FeSe1−xTex (0 ≤ x ≤ 0.48) crystals were grown by the chemical-vapor-transport (CVT) technique, while FeSe1−xTex crystals with 0.52 ≤ x ≤ 0.90 were obtained by the Bridgman method. For the latter, the Te annealing procedure was applied to minimize excess Fe65. The actual Te composition x of crystals synthesized by the CVT method is determined for each sample from the c-axis length measured by X-ray diffraction. The x values of the FeSe1−xSx crystals, as well as the FeSe1−xTex crystals grown by the Bridgman method, are taken from the nominal values. The MR measurements on FeSe1−xSx and FeSe1−xTex were performed at the High Field Magnet Laboratory (HFML) and the International MegaGauss Science Laboratory at the University of Tokyo, respectively. The orientation of the samples was determined by using a Hall probe mounted on the rotating sample platform. All measurements were performed at fixed temperatures while sweeping the field from \(-{H}_{\max }\) 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.