Evidence for nematic fluctuations in FeSe superconductor: a57Fe Mössbauer spectroscopy study

There has been controversy about the driving force of the nematic order in the FeSe superconductor. Here, we present a detailed study of the57Fe Mössbauer spectra of FeSe single-crystal powders, focusing on the temperature dependences of the hyperfine parameters in the vicinity of the nematic transition temperature,Ts∼ 90 K. The nematicity-induced splitting ofdxzanddyzbands, obtained from the anomalous increase in quadrupole splitting nearTs, starts at 143 K. The temperature evolution of the lattice dynamics, deduced from the recoilless fractions and second-order Doppler shifts, is found to undergo successively two segments of phonon-softening (160 K-105 K) and phonon-hardening (105 K-90 K), related to the appearance of local orthorhombic distortions aboveTsand the establishing way of the associated nematic correlations. Analysis of the linewidths shows that spin fluctuations occur not only below 70 K but also acrossTs(105 K-70 K), accompanied by the non-Fermi liquid behavior of the electrons. The results demonstrate the strong interactions between lattice, spin, and electron degrees of freedom in the vicinity ofTsand that the lattice degrees of freedom may play an essential role in driving the nematic order for FeSe.

Tomonaga-Luttinger liquid and quantum criticality in spin- 1 2 antiferromagnetic Heisenberg chain C 14 H 18 CuN 4 O 10 via Wilson ratio

The ground state of a one-dimensional spin-1 2 uniform antiferromagnetic Heisenberg chain (AfHc) is a Tomonaga-Luttinger liquid which is quantum-critical with respect to applied magnetic fields up to a saturation field μ 0 H s beyond which it transforms to a fully polarized state. Wilson ratio has been predicted to be a good indicator for demarcating these phases [Phys. Rev. B 96, 220401 (2017)]. From detailed temperature and magnetic field-dependent magnetization, magnetic susceptibility and specific heat measurements in a metalorganic complex and comparisons with field theory and quantum transfer matrix method calculations, the complex was found to be a very good realization of a spin-1 2 AfHc. Wilson ratio obtained from experimentally obtained magnetic susceptibility and magnetic contribution of specific heat values was used to map the magnetic phase diagram of the uniform spin-1 2 AfHc over large regions of phase space demarcating Tomonaga-Luttinger liquid, saturation field quantum critical, and fully polarized states. Luttinger parameter and spinon velocity were found to match very well with the values predicted from conformal field theory.

Spin Echo, Fidelity, and the Quantum Critical Fan in TmVO_{4}

Using spin-echo nuclear magnetic resonance in the model transverse field Ising system TmVO_{4}, we show that low frequency quantum fluctuations at the quantum critical point have a very different effect on ^{51}V nuclear spins than classical low-frequency noise or fluctuations that arise at a finite temperature critical point. Spin echoes filter out the low-frequency classical noise but not the quantum fluctuations. This allows us to directly visualize the quantum critical fan and demonstrate the persistence of quantum fluctuations at the critical coupling strength in TmVO_{4} to high temperatures in an experiment that remains transparent to finite temperature classical phase transitions. These results show that while dynamical decoupling schemes can be quite effective in eliminating classical noise in a qubit, a quantum critical environment may lead to rapid entanglement and decoherence.

Emergent tetragonality in a fundamentally orthorhombic material

Symmetry plays a key role in determining the physical properties of materials. By Neumann's principle, the properties of a material remain invariant under the symmetry operations of the space group to which the material belongs. Continuous phase transitions are associated with a spontaneous reduction in symmetry. Less common are examples where proximity to a continuous phase transition leads to an increase in symmetry. We find signatures of an emergent tetragonal symmetry close to a charge density wave (CDW) bicritical point in a fundamentally orthorhombic material, ErTe3, for which the two distinct CDW phase transitions are tuned via anisotropic strain. We first establish that tension along the a axis favors an abrupt rotation of the CDW wave vector from the c to a axis and infer the presence of a bicritical point where the two continuous phase transitions meet. We then observe a divergence of the nematic elastoresistivity approaching this putative bicritical point, indicating an emergent tetragonality in the critical behavior.

Perspective: probing elasto-quantum materials with x-ray techniques andin situanisotropic strain

Anisotropic lattice deformation plays an important role in the quantum mechanics of solid state physics. The possibility of mediating the competition and cooperation among different order parameters by applyingin situstrain/stress on quantum materials has led to discoveries of a variety of elasto-quantum effects on emergent phenomena. It has become increasingly critical to have the capability of combining thein situstrain tuning with x-ray techniques, especially those based on synchrotrons, to probe the microscopic elasto-responses of the lattice, spin, charge, and orbital degrees of freedom. Herein, we briefly review the recent studies that embarked on utilizing elasto-x-ray characterizations on representative material systems and demonstrated the emerging opportunities enabled by this method. With that, we further discuss the promising prospect in this rising area of quantum materials research and the bright future of elasto-x-ray techniques.

Nematic Spin Correlations Pervading the Phase Diagram of FeSe_{1-x}S_{x}

We use resonant inelastic x-ray scattering (RIXS) at the Fe-L_{3} edge to study the spin excitations of uniaxial-strained and unstrained FeSe_{1-x}S_{x} (0≤x≤0.21) samples. The measurements on unstrained samples reveal dispersive spin excitations in all doping levels, which show only minor doping dependence in energy dispersion, lifetime, and intensity, indicating that high-energy spin excitations are only marginally affected by sulfur doping. RIXS measurements on uniaxial-strained samples reveal that the high-energy spin-excitation anisotropy observed previously in FeSe is also present in the doping range 0200  K in x=0.18 and reaches a maximum around the nematic quantum critical doping (x_{c}≈0.17). Since the spin-excitation anisotropy directly reflects the existence of nematic spin correlations, our results indicate that high-energy nematic spin correlations pervade the regime of nematicity in the phase diagram and are enhanced by the nematic quantum criticality. These results emphasize the essential role of spin fluctuations in driving electronic nematicity and highlight the capability of uniaxial strain in tuning spin excitations in quantum materials hosting strong magnetoelastic coupling and electronic nematicity.

Quantum phase transitions in two-dimensional superconductors: a review on recent experimental progress

Superconductor-insulator/metal transition (SMT) as a paradigm of quantum phase transition has been a research highlight over the last three decades. Benefit from recent developments in the fabrication and measurements of two-dimensional (2D) superconducting films and nanodevices, unprecedented quantum phenomena have been revealed in the quantum phase transitions of 2D superconductors. In this review, we introduce the recent progress on quantum phase transitions in 2D superconductors, focusing on the quantum Griffiths singularity (QGS) and anomalous metal state. Characterized by a divergent critical exponent when approaching zero temperature, QGS of SMT is discovered in ultrathin crystalline Ga films and subsequently detected in various 2D superconductors. The universality of QGS indicates the profound influence of quenched disorder on quantum phase transitions. Besides, in a 2D superconducting system, whether a metallic ground state can exist is a long-sought mystery. Early experimental studies indicate an intermediate metallic state in the quantum phase transition of 2D superconductors. Recently, in high-temperature superconducting films with patterned nanopores, a robust anomalous metal state (i.e. quantum metal or Bose metal) has been detected, featured as the saturated resistance in the low temperature regime. Moreover, the charge-2equantum oscillations are observed in nanopatterned films, indicating the bosonic nature of the anomalous metal state and ending the debate on whether bosons can exist as a metal. The evidences of the anomalous metal states have also been reported in crystalline epitaxial thin films and exfoliated nanoflakes, as well as granular composite films. High quality filters are used in these works to exclude the influence of external high frequency noises in ultralow temperature measurements. The observations of QGS and metallic ground states in 2D superconductors not only reveal the prominent role of quantum fluctuations and dissipations but also provide new perspective to explore quantum phase transitions in superconducting systems.

Signatures of Cooper pair dynamics and quantum-critical superconductivity in tunable carrier bands

Different superconducting pairing mechanisms are markedly distinct in the underlying Cooper pair kinematics. Quantum-critical soft modes drive pairing interactions in which the pair scattering processes are highly collinear and can be classified into two categories: forward scattering and backscattering. Conversely, in conventional phonon mechanisms, Cooper pair scattering is of a generic noncollinear character. In this study, we present a method to discern the kinematic type by observing the evolution of superconductivity while adjusting the Fermi surface geometry. To demonstrate our approach, we utilize the recently reported phase diagrams of untwisted graphene multilayers. Our analysis connects the emergence of superconductivity at "ghost crossings" of Fermi surfaces in distinct valleys to the pair kinematics of a backscattering type. Together with the observed nonmonotonic behavior of superconductivity near its onset (sharp rise followed by a drop), it lends strong support to a particular quantum-critical superconductivity scenario in which pairing is driven by intervalley coherence fluctuations. These findings offer direct insights into the genesis of pairing in these systems, providing compelling evidence for the electron-electron interactions driving superconductivity. More broadly, our work highlights the potential of tuning bands via ghost crossings as a promising means of boosting superconductivity.

Colossal transverse magnetoresistance due to nematic superconducting phase fluctuations in a copper oxide

Electronic anisotropy ("nematicity") has been detected in cuprate superconductors by various experimental techniques. Using angle-resolved transverse resistance (ARTR) measurements, a very sensitive and background-free technique that can detect 0.5% anisotropy in transport, we have observed it also in La2-xSrxCuO4 (LSCO) for 0.02 ≤ x ≤ 0.25. A central enigma in LSCO is the rotation of the nematic director (orientation of the largest longitudinal resistance) with temperature; this has not been seen before in any material. Here, we address this puzzle by measuring the angle-resolved transverse magnetoresistance (ARTMR) in LSCO. We report the discovery of colossal transverse magnetoresistance (CTMR)-an order-of-magnitude drop in the transverse resistivity in the magnetic field of 6 T. We show that the apparent rotation of the nematic director is caused by anisotropic superconducting fluctuations, which are not aligned with the normal electron fluid, consistent with coexisting bond-aligned and diagonal nematic orders. We quantify this by modeling the (magneto-)conductivity as a sum of normal (Drude) and paraconducting (Aslamazov-Larkin) channels but extended to contain anisotropic Drude and Cooper-pair effective mass tensors. Strikingly, the anisotropy of Cooper-pair stiffness is much larger than that of the normal electrons. It grows dramatically on the underdoped side, where the fluctuations become quasi-one-dimensional. Our analysis is general rather than model dependent. Still, we discuss some candidate microscopic models, including coupled strongly-correlated ladders where the transverse (interladder) phase stiffness is low compared with the longitudinal intraladder stiffness, as well as the anisotropic superconducting fluctuations expected close to the transition to a pair-density wave state.

Optimized superconductivity in the vicinity of a nematic quantum critical point in the kagome superconductor Cs(V1-xTix)3Sb5

CsV3Sb5 exhibits superconductivity at Tc = 3.2 K after undergoing intriguing two high-temperature transitions: charge density wave order at ~98 K and electronic nematic order at Tnem ~ 35 K. Here, we investigate nematic susceptibility in single crystals of Cs(V1-xTix)3Sb5 (x = 0.00-0.06) where double-dome-shaped superconducting phase diagram is realized. The nematic susceptibility typically exhibits the Curie‒Weiss behaviour above Tnem, which is monotonically decreased with x. Moreover, the Curie‒Weiss temperature is systematically suppressed from ~30 K for x = 0 to ~4 K for x = 0.0075, resulting in a sign change at x = ~0.009. Furthermore, the Curie constant reaches a maximum at x = 0.01, suggesting drastically enhanced nematic susceptibility near a putative nematic quantum critical point (NQCP) at x = ~0.009. Strikingly, Tc is enhanced up to ~4.1 K with full Meissner shielding realized at x = ~0.0075-0.01, forming the first superconducting dome near the NQCP. Our findings directly point to a vital role of nematic fluctuations in enhancing the superconducting properties of Cs(V1-xTix)3Sb5.

Modulations in Superconductors: Probes of Underlying Physics

The importance of modulations is elevated to an unprecedented level, due to the delicate conditions required to bring out exotic phenomena in quantum materials, such as topological materials, magnetic materials, and superconductors. Recently, state-of-the-art modulation techniques in material science, such as electric-double-layer transistor, piezoelectric-based strain apparatus, angle twisting, and nanofabrication, have been utilized in superconductors. They not only efficiently increase the tuning capability to the broader ranges but also extend the tuning dimensionality to unprecedented degrees of freedom, including quantum fluctuations of competing phases, electronic correlation, and phase coherence essential to global superconductivity. Here, for a comprehensive review, these techniques together with the established modulation methods, such as elemental substitution, annealing, and polarization-induced gating, are contextualized. Depending on the mechanism of each method, the modulations are categorized into stoichiometric manipulation, electrostatic gating, mechanical modulation, and geometrical design. Their recent advances are highlighted by applications in newly discovered superconductors, e.g., nickelates, Kagome metals, and magic-angle graphene. Overall, the review is to provide systematic modulations in emergent superconductors and serve as the coordinate for future investigations, which can stimulate researchers in superconductivity and other fields to perform various modulations toward a thorough understanding of quantum materials.

Real-Space Observation of Unidirectional Charge Density Wave and Complex Structural Modulation in the Pnictide Superconductor Ba1-xSrxNi2As2

Here we use low-temperature and variable-temperature scanning tunneling microscopy to study the pnictide superconductor, Ba_1-xSr_xNi₂As₂. In the low-temperature phase (triclinic phase) of BaNi₂As₂, we observe the unidirectional charge density wave (CDW) with Q = 1/3 on both the Ba and NiAs surfaces. On the NiAs surface of the triclinic BaNi₂As₂, there are structural-modulation-induced chain-like superstructures with distinct periodicities. In the high-temperature phase (tetragonal phase) of BaNi₂As₂, the NiAs surface appears as the periodic 1 × 2 superstructure. Interestingly, in the triclinic phase of Ba_0.5Sr_0.5Ni₂As₂, the unidirectional CDW is suppressed on both the Ba/Sr and NiAs surfaces, and the Sr substitution stabilizes the periodic 1 × 2 superstructure on the NiAs surface, which enhance the superconductivity in Ba_0.5Sr_0.5Ni₂As₂. Our results provide important microscopic insights for the interplay among the unidirectional CDW, structural modulation, and superconductivity in this class of pnictide superconductors.

Nematic-Fluctuation-Mediated Superconductivity Revealed by Anisotropic Strain in Ba(Fe_{1-x}Co_{x})_{2}As_{2}

Anisotropic strain is an external field capable of selectively addressing the role of nematic fluctuations in promoting superconductivity. We demonstrate this using polarization-resolved elasto-Raman scattering by probing the evolution of nematic fluctuations under strain in the normal and superconducting state of the paradigmatic iron-based superconductor Ba(Fe_{1-x}Co_{x})_{2}As_{2}. In the parent compound BaFe_{2}As_{2} we observe a strain-induced suppression of the nematic susceptibility which follows the expected behavior of an Ising order parameter under a symmetry breaking field. For the superconducting compound, the suppression of the nematic susceptibility correlates with the decrease of the critical temperature T_{c}, indicating a significant contribution of nematic fluctuations to electron pairing. Our results validate theoretical scenarios of enhanced T_{c} near a nematic quantum critical point.

Tracking the nematicity in cuprate superconductors: a resistivity study under uniaxial pressure

Overshadowing the superconducting dome in hole-doped cuprates, the pseudogap state is still one of the mysteries that no consensus can be achieved. It has been suggested that the rotational symmetry is broken in this state and may result in a nematic phase transition, whose temperature seems to coincide with the onset temperature of the pseudogap stateT∗around optimal doping level, raising the question whether the pseudogap results from the establishment of the nematic order. Here we report results of resistivity measurements under uniaxial pressure on several hole-doped cuprates, where the normalized slope of the elastoresistivityζcan be obtained as illustrated in iron-based superconductors. The temperature dependence ofζalong particular lattice axis exhibits kink feature atT_kand shows Curie-Weiss-like behavior above it, which may suggest a spontaneous nematic transition. WhileT_kseems to be the same asT∗around the optimal doping and in the overdoped region, they become very different in underdoped La2-xSr_xCuO₄. Our results suggest that the nematic order, if indeed existing, is an electronic phase within the pseudogap state.

Pure nematic quantum critical point accompanied by a superconducting dome

SignificanceThe notion of the quantum critical point (QCP) is at the core of modern condensed matter physics. Near a QCP of the symmetry-breaking order, associated quantum-mechanical fluctuations are intensified, which can lead to unconventional superconductivity. Indeed, dome-shaped superconducting phases are often observed near the magnetic QCPs, which supports the spin fluctuation-driven superconductivity. However, the fundamental question remains as to whether a nonmagnetic QCP of electronic nematic order characterized by spontaneous rotational symmetry breaking can promote superconductivity in real materials. Here, we provide an experimental demonstration that a pure nematic QCP exists near the center of a superconducting dome in nonmagnetic FeSe[Formula: see text] Te_x. This result evidences that nematic fluctuations enhanced around the nematic QCP can boost superconductivity.

Comparison of temperature and doping dependence of elastoresistivity near a putative nematic quantum critical point

Strong electronic nematic fluctuations have been discovered near optimal doping for several families of Fe-based superconductors, motivating the search for a possible link between these fluctuations, nematic quantum criticality, and high temperature superconductivity. Here we probe a key prediction of quantum criticality, namely power-law dependence of the associated nematic susceptibility as a function of composition and temperature approaching the compositionally tuned putative quantum critical point. To probe the ‘bare’ quantum critical point requires suppression of the superconducting state, which we achieve by using large magnetic fields, up to 45 T, while performing elastoresistivity measurements to follow the nematic susceptibility. We performed these measurements for the prototypical electron-doped pnictide, Ba(Fe_1−xCo_x)₂As₂, over a dense comb of dopings. We find that close to the putative quantum critical point, the elastoresistivity appears to obey power-law behavior as a function of composition over almost a decade of variation in composition. Paradoxically, however, we also find that the temperature dependence for compositions close to the critical value cannot be described by a single power law.

Evidence for quantum criticality in Fe-based superconductors is still being accumulated. Here, the authors observe power-law behavior of the elastoresistivity as a function of composition in Ba(Fe_1−xCo_x)₂As₂ near a putative nematic quantum critical point, consistent with expectations for quantum criticality, while the temperature dependence near the critical doping deviates from a power law.

Charge-density-wave-driven electronic nematicity in a kagome superconductor

Electronic nematicity, in which rotational symmetry is spontaneously broken by electronic degree of freedom, has been demonstrated as a ubiquitous phenomenon in correlated quantum fluids including high-temperature superconductors (HTS) and quantum Hall systems^1,2. More strikingly, the electronic nematicity in HTS exhibits an intriguing entanglement with superconductivity, generating complicated superconducting pairing and intertwined electronic orders. Recently, an unusual competition between superconductivity and a charge-density-wave (CDW) order has been found in AV₃Sb₅ (A = K, Rb, Cs) family with two-dimensional vanadium kagome nets^3-8. Whether these phenomena involve electronic nematicity is still elusive. Here, we report compelling evidence for the existence of electronic nematicity in CsV₃Sb₅, using a combination of elastoresistance measurements, nuclear magnetic resonance (NMR) and scanning tunnelling microscopy/spectroscopy (STM/S). The temperature-dependent elastoresistance coefficient (m₁₁-m₁₂) and NMR spectrum clearly demonstrate that, besides a C₂ structural distortion of 2a₀×2a₀ supercell due to out-of-plane modulation, significant nematic fluctuations emerge immediately below the CDW transition (T_CDW ~ 94 K) and finally a nematic transition occurs below T_nem ~ 35 K. STM experiment directly visualizes the C₂-structure-pinned long-range nematic order below T_nem, suggesting a novel nematicity described by a three-state Potts model. Our findings unambiguously prove an intrinsic electronic nematicity in the normal state of CsV₃Sb₅, which sets a new paradigm for revealing the role of electronic nematicity on pairing mechanism in unconventional superconductors.

Iron pnictides and chalcogenides: a new paradigm for superconductivity

Superconductivity is a remarkably widespread phenomenon that is observed in most metals cooled to very low temperatures. The ubiquity of such conventional superconductors, and the wide range of associated critical temperatures, is readily understood in terms of the well-known Bardeen-Cooper-Schrieffer theory. Occasionally, however, unconventional superconductors are found, such as the iron-based materials, which extend and defy this understanding in unexpected ways. In the case of the iron-based superconductors, this includes the different ways in which the presence of multiple atomic orbitals can manifest in unconventional superconductivity, giving rise to a rich landscape of gap structures that share the same dominant pairing mechanism. In addition, these materials have also led to insights into the unusual metallic state governed by the Hund's interaction, the control and mechanisms of electronic nematicity, the impact of magnetic fluctuations and quantum criticality, and the importance of topology in correlated states. Over the fourteen years since their discovery, iron-based superconductors have proven to be a testing ground for the development of novel experimental tools and theoretical approaches, both of which have extensively influenced the wider field of quantum materials.

Signatures of a Quantum Griffiths Phase Close to an Electronic Nematic Quantum Phase Transition

In the vicinity of a quantum critical point, quenched disorder can lead to a quantum Griffiths phase, accompanied by an exotic power-law scaling with a continuously varying dynamical exponent that diverges in the zero-temperature limit. Here, we investigate a nematic quantum critical point in the iron-based superconductor FeSe_{0.89}S_{0.11} using applied hydrostatic pressure. We report an unusual crossing of the magnetoresistivity isotherms in the nonsuperconducting normal state that features a continuously varying dynamical exponent over a large temperature range. We interpret our results in terms of a quantum Griffiths phase caused by nematic islands that result from the local distribution of Se and S atoms. At low temperatures, the Griffiths phase is masked by the emergence of a Fermi liquid phase due to a strong nematoelastic coupling and a Lifshitz transition that changes the topology of the Fermi surface.

The transport-structural correspondence across the nematic phase transition probed by elasto X-ray diffraction

Electronic nematicity in iron pnictide materials is coupled to both the lattice and the conducting electrons, which allows both structural and transport observables to probe nematic fluctuations and the order parameter. Here we combine simultaneous transport and X-ray diffraction measurements with in-situ tunable strain (elasto X-ray diffraction) to measure the temperature dependence of the shear modulus and elastoresistivity above the nematic transition and the spontaneous orthorhombicity and resistivity anisotropy below the nematic transition, all within a single sample of Ba(Fe_0.96Co_0.04)₂As₂. The ratio of transport to structural quantities is nearly temperature independent over a 74 K range and agrees between the ordered and disordered phases. These results show that elasto X-ray diffraction is a powerful technique to probe the nemato-elastic and nemato-transport couplings, which have important implications to the nearby superconductivity. It also enables the measurement in the large strain limit, where the breakdown of the mean-field description reveals the intertwined nature of nematicity.

Elastocaloric signature of nematic fluctuations

The elastocaloric effect (ECE) relates changes in entropy to changes in strain experienced by a material. As such, ECE measurements can provide valuable information about the entropy landscape proximate to strain-tuned phase transitions. For ordered states that break only point symmetries, bilinear coupling of the order parameter with strain implies that the ECE can also provide a window on fluctuations above the critical temperature and hence, in principle, can also provide a thermodynamic measure of the associated susceptibility. To demonstrate this, we use the ECE to sensitively reveal the presence of nematic fluctuations in the archetypal Fe-based superconductor Ba([Formula: see text])₂[Formula: see text] By performing these measurements simultaneously with elastoresistivity in a multimodal fashion, we are able to make a direct and unambiguous comparison of these closely related thermodynamic and transport properties, both of which are sensitive to nematic fluctuations. As a result, we have uncovered an unanticipated doping dependence of the nemato-elastic coupling and of the magnitude of the scattering of low-energy quasi-particles by nematic fluctuations-while the former weakens, the latter increases dramatically with increasing doping.

Discovery of mesoscopic nematicity wave in iron-based superconductors

[Figure: see text].

Emerging symmetric strain response and weakening nematic fluctuations in strongly hole-doped iron-based superconductors

Electronic nematicity is often found in unconventional superconductors, suggesting its relevance for electronic pairing. In the strongly hole-doped iron-based superconductors, the symmetry channel and strength of the nematic fluctuations, as well as the possible presence of long-range nematic order, remain controversial. Here, we address these questions using transport measurements under elastic strain. By decomposing the strain response into the appropriate symmetry channels, we demonstrate the emergence of a giant in-plane symmetric contribution, associated with the growth of both strong electronic correlations and the sensitivity of these correlations to strain. We find weakened remnants of the nematic fluctuations that are present at optimal doping, but no change in the symmetry channel of nematic fluctuations with hole doping. Furthermore, we find no indication of a nematic-ordered state in the AFe2As2 (A = K, Rb, Cs) superconductors. These results revise the current understanding of nematicity in hole-doped iron-based superconductors.

Multiple Charge Density Waves and Superconductivity Nucleation at Antiphase Domain Walls in the Nematic Pnictide Ba_{1-x}Sr_{x}Ni_{2}As_{2}

How superconductivity interacts with charge or nematic order is one of the great unresolved issues at the center of research in quantum materials. Ba_{1-x}Sr_{x}Ni_{2}As_{2} (BSNA) is a charge ordered pnictide superconductor recently shown to exhibit a sixfold enhancement of superconductivity due to nematic fluctuations near a quantum phase transition (at x_{c}=0.7) [1]. The superconductivity is, however, anomalous, with the resistive transition for 0.4<x<x_{c} occurring at a higher temperature than the specific heat anomaly. Using x-ray scattering, we discovered a new charge density wave (CDW) in BSNA in this composition range. The CDW is commensurate with a period of two lattice parameters, and is distinct from the two CDWs previously reported in this material [1,2]. We argue that the anomalous transport behavior arises from heterogeneous superconductivity nucleating at antiphase domain walls in this CDW. We also present new data on the incommensurate CDW, previously identified as being unidirectional [2], showing that it is a rotationally symmetric "4Q" state with C_{4} symmetry. Our study establishes BSNA as a rare material containing three distinct CDWs, and an exciting test bed for studying coupling between CDW, nematic, and SC orders.

Nematic quantum criticality in an Fe-based superconductor revealed by strain-tuning

Quantum criticality may be essential to understanding a wide range of exotic electronic behavior; however, conclusive evidence of quantum critical fluctuations has been elusive in many materials of current interest. An expected characteristic feature of quantum criticality is power-law behavior of thermodynamic quantities as a function of a nonthermal tuning parameter close to the quantum critical point (QCP). Here, we observed power-law behavior of the critical temperature of the coupled nematic/structural phase transition as a function of uniaxial stress in a representative family of iron-based superconductors, providing direct evidence of quantum critical nematic fluctuations in this material. These quantum critical fluctuations are not confined within a narrow regime around the QCP but rather extend over a wide range of temperatures and compositions.

Mössbauer spectroscopy study of nematicity in Ba(Fe0.962Cu0.038)2As2single crystal: enhanced orbital effect

The origin of the nematic order remains unclear due to the strong coupling between orbital, spin and lattice degrees of freedom in iron-based superconductors. Although the driving force of hole-doped BeFe₂As₂is still controversial, the nematic fluctuation of electron-doped compounds is generally believed to be spin fluctuation driven. Here, we present a comprehensive study of the nematic phase transition in Ba(Fe_0.962Cu_0.038)₂As₂single crystal by using Mössbauer spectroscopy. The electric field gradient and its in-plane asymmetry on Fe nucleus, which are directly determined by the occupation of individualt_2gorbital, are sensitive to the local nematicity of Fe ions. The nematic phase transition happens atT_nem≈ 73.8 K in the compound while the band splitting betweend_xz/d_yzorbitals begins far aboveT_nemand reaches 18.8 meV at 30 K. The temperature evolution of the hyperfine parameters proves the existence of electron-phonon interaction and non-Fermi-liquid behaviour nearT_nem. However, the spin-lattice relaxation signal is only evident belowT_nem. These observations show that the role of orbital degrees of freedom is more active in driving nematicity than in Co- or Ni-doped BaFe₂As₂compounds, and can be attributed to enhanced electronic localization caused by Cu doping.

Short-Range Nematic Fluctuations in Sr_{1-x}Na_{x}Fe_{2}As_{2} Superconductors

Interactions between nematic fluctuations, magnetic order and superconductivity are central to the physics of iron-based superconductors. Here we report on in-plane transverse acoustic phonons in hole-doped Sr_{1-x}Na_{x}Fe_{2}As_{2} measured via inelastic x-ray scattering, and extract both the nematic susceptibility and the nematic correlation length. By a self-contained method of analysis, for the underdoped (x=0.36) sample, which harbors a magnetically ordered tetragonal phase, we find it hosts a short nematic correlation length ξ∼10  Å and a large nematic susceptibility χ_{nem}. The optimal-doped (x=0.55) sample exhibits weaker phonon softening effects, indicative of both reduced ξ and χ_{nem}. Our results suggest short-range nematic fluctuations may favor superconductivity, placing emphasis on the nematic correlation length for understanding the iron-based superconductors.

Nanoscale decoupling of electronic nematicity and structural anisotropy in FeSe thin films

In a material prone to a nematic instability, anisotropic strain in principle provides a preferred symmetry-breaking direction for the electronic nematic state to follow. This is consistent with experimental observations, where electronic nematicity and structural anisotropy typically appear hand-in-hand. In this work, we discover that electronic nematicity can be locally decoupled from the underlying structural anisotropy in strain-engineered iron-selenide (FeSe) thin films. We use heteroepitaxial molecular beam epitaxy to grow FeSe with a nanoscale network of modulations that give rise to spatially varying strain. We map local anisotropic strain by analyzing scanning tunneling microscopy topographs, and visualize electronic nematic domains from concomitant spectroscopic maps. While the domains form so that the energy of nemato-elastic coupling is minimized, we observe distinct regions where electronic nematic ordering fails to flip direction, even though the underlying structural anisotropy is locally reversed. The findings point towards a nanometer-scale stiffness of the nematic order parameter.

Momentum dependent [Formula: see text] band splitting in LaFeAsO

The nematic phase in iron based superconductors (IBSs) has attracted attention with a notion that it may provide important clue to the superconductivity. A series of angle-resolved photoemission spectroscopy (ARPES) studies were performed to understand the origin of the nematic phase. However, there is lack of ARPES study on LaFeAsO nematic phase. Here, we report the results of ARPES studies of the nematic phase in LaFeAsO. Degeneracy breaking between the [Formula: see text] and [Formula: see text] hole bands near the [Formula: see text] and M point is observed in the nematic phase. Different temperature dependent band splitting behaviors are observed at the [Formula: see text] and M points. The energy of the band splitting near the M point decreases as the temperature decreases while it has little temperature dependence near the [Formula: see text] point. The nematic nature of the band shift near the M point is confirmed through a detwin experiment using a piezo device. Since a momentum dependent splitting behavior has been observed in other iron based superconductors, our observation confirms that the behavior is a universal one among iron based superconductors.

Dominant In-Plane Symmetric Elastoresistance in CsFe_{2}As_{2}

We study the elastoresistance of the highly correlated material CsFe_{2}As_{2} in all symmetry channels. Neutralizing its thermal expansion by means of a piezoelectric-based strain cell is demonstrated to be essential. The elastoresistance response in the in-plane symmetric channel is found to be large, while the response in the symmetry-breaking channels is weaker and provides no evidence for a divergent nematic susceptibility. Rather, our results can be interpreted naturally within the framework of a coherence-incoherence crossover, where the low-temperature coherent state is sensitively tuned by the in-plane atomic distances.

Unusual Pressure-Induced Quantum Phase Transition from Superconducting to Charge-Density Wave State in Rare-Earth-Based Heusler LuPd_{2}In Compound

We investigate the pressure effects on the electronic structures and phonon properties of rare-earth-based cubic-Heusler compound LuPd_{2}In, on the basis of ab initio density functional theory. We find the occurrence of intriguing phase transition from the superconducting (SC) to charge-density wave (CDW) state under pressure (P), which is quite unusual in that the pressure is detrimental to the CDW state in usual systems. The SC transition temperature T_{C} of LuPd_{2}In increases first with increasing pressure, up to P_{C}≈28  GPa, above which a quantum phase transition into the CDW state takes place. This extraordinary transition originates from the occurrence of phonon softening instability at a special q=M in the Brillouin zone. We thus propose that LuPd_{2}In is a quite unique material, in which the CDW quantum critical point is realized under the SC dome by applying the pressure.

Simple analytical method for determining electrical resistivity and sheet resistance using the van der Pauw procedure

This work reports an analytical method for determining electrical resistivity (ρ) and sheet resistance (RS) of isotropic conductors. The method is compared with previous numerical solutions and available experimental data showing a universal behavior for isotropic conductors. An approximated solution is also reported allowing one to easily determine ρ and RS for samples either with regular or arbitrary shapes.

Evolution of the Nematic Susceptibility in LaFe_{1-x}Co_{x}AsO

We report a systematic elastoresistivity study on LaFe_{1-x}Co_{x}AsO single crystals, which have well separated structural and magnetic transition lines. All crystals show a Curie-Weiss-like nematic susceptibility in the tetragonal phase. The extracted nematic temperature is monotonically suppressed upon cobalt doping, and changes sign around the optimal doping level, indicating a possible nematic quantum critical point beneath the superconducting dome. The amplitude of the nematic susceptibility shows a peculiar double-peak feature. This could be explained by a combined effect of different contributions to the nematic susceptibility, which are amplified at separated doping levels of LaFe_{1-x}Co_{x}AsO.

Rigid platform for applying large tunable strains to mechanically delicate samples

Response to uniaxial stress has become a major probe of electronic materials. Tunable uniaxial stress may be applied using piezoelectric actuators, and so far two methods have been developed to couple samples to actuators. In one, actuators apply force along the length of a free, beam-like sample, allowing very large strains to be achieved. In the other, samples are affixed directly to piezoelectric actuators, allowing the study of mechanically delicate materials. Here, we describe an approach that merges the two: thin samples are affixed to a substrate, which is then pressurized uniaxially using piezoelectric actuators. Using this approach, we demonstrate the application of large elastic strains to mechanically delicate samples: the van der Waals-bonded material FeSe and a sample of CeAuSb₂ that was shaped with a focused ion beam.

Magnetism and superconductivity in Fe1+y Te1-xSex

Neutron scattering has played a significant role in characterizing magnetic and structural correlations in Fe1+y Te1-xSexand their connections with superconductivity. Here we review several key aspects of the physics of iron chalcogenide superconductors where neutron studies played a key role. These topics include the phase diagram of Fe1+y Te1-xSex, where the doping-dependence of structural transitions can be understood from a mapping to the anisotropic random field Ising model. We then discuss orbital-selective Mott physics in the Fe chalcogenide series, where temperature-dependent magnetism in the parent material provided one of the earliest cases for orbital-selective correlation effects in a Hund's metal. Finally, we elaborate on the character of local magnetic correlations revealed by neutron scattering, its dependence on temperature and composition, and the connections to nematicity and superconductivity.

Electronic nematicity in Sr2RuO4

We have measured the angle-resolved transverse resistivity (ARTR), a sensitive indicator of electronic anisotropy, in high-quality thin films of the unconventional superconductor Sr₂RuO₄ grown on various substrates. The ARTR signal, heralding the electronic nematicity or a large nematic susceptibility, is present and substantial already at room temperature and grows by an order of magnitude upon cooling down to 4 K. In Sr₂RuO₄ films deposited on tetragonal substrates the highest-conductivity direction does not coincide with any crystallographic axis. In films deposited on orthorhombic substrates it tends to align with the shorter axis; however, the magnitude of the anisotropy stays the same despite the large lattice distortion. These are strong indications of actual or incipient electronic nematicity in Sr₂RuO₄.

Nematic Correlation Length in Iron-Based Superconductors Probed by Inelastic X-Ray Scattering

Nematicity is ubiquitous in electronic phases of high-T_{c} superconductors, particularly in the Fe-based systems. We used inelastic x-ray scattering to extract the temperature-dependent nematic correlation length ξ from the anomalous softening of acoustic phonon modes in FeSe, underdoped Ba(Fe_{0.97}Co_{0.03})_{2}As_{2}, and optimally doped Ba(Fe_{0.94}Co_{0.06})_{2}As_{2}. In all cases, we find that ξ is well described by a power law (T-T_{0})^{-1/2} extending over a wide temperature range. Combined with the previously reported Curie-Weiss behavior of the nematic susceptibility, these results point to the mean-field character of the nematic transition, which we attribute to a sizable nematoelastic coupling that is likely detrimental to superconductivity.

Novel electronic nematicity in heavily hole-doped iron pnictide superconductors

Electronic nematicity, a correlated state that spontaneously breaks rotational symmetry, is observed in several layered quantum materials. In contrast to their liquid-crystal counterparts, the nematic director cannot usually point in an arbitrary direction (XY nematics), but is locked by the crystal to discrete directions (Ising nematics), resulting in strongly anisotropic fluctuations above the transition. Here, we report on the observation of nearly isotropic XY-nematic fluctuations, via elastoresistance measurements, in hole-doped Ba_1-x Rb _x Fe₂As₂ iron-based superconductors. While for [Formula: see text], the nematic director points along the in-plane diagonals of the tetragonal lattice, for [Formula: see text], it points along the horizontal and vertical axes. Remarkably, for intermediate doping, the susceptibilities of these two symmetry-irreducible nematic channels display comparable Curie-Weiss behavior, thus revealing a nearly XY-nematic state. This opens a route to assess this elusive electronic quantum liquid-crystalline state.

Measurements of elastoresistance under pressure by combining in-situ tunable quasi-uniaxial stress with hydrostatic pressure

Uniaxial stress, as well as hydrostatic pressure are often used to tune material properties in condensed matter physics. Here, we present a setup that allows for the study of the combined effects of quasi-uniaxial stress and hydrostatic pressure. Following earlier designs for measurements under finite stress at ambient pressures [e.g., Chu et al., Science 337, 710 (2012)], the present setup utilizes a piezoelectric actuator to change stress in situ inside the piston-cylinder pressure cell. We show that the actuator can be operated over the full temperature (from 30 K up to 260 K) and pressure range (up to ≈2 GPa), resulting in a clear and measurable quasi-uniaxial strain. To demonstrate functionality, measurements of the elastoresistance (i.e., the change of resistance of a sample as a response to quasi-uniaxial strain) under finite hydrostatic pressure on the iron-based compound BaFe₂As₂ are presented as a proof-of-principle example and discussed in the framework of electronic nematicity. Overall, this work introduces the combination of in situ tunable quasi-uniaxial stress and large (up to ≈2 GPa) hydrostatic pressure as a powerful combination in the study of novel electronic phases. In addition, it also points toward further technical advancements which can be made in the future.

Sixfold enhancement of superconductivity in a tunable electronic nematic system

The electronic nematic phase-in which electronic degrees of freedom lower the crystal rotational symmetry-is commonly observed in high-temperature superconductors. However, understanding the role of nematicity and nematic fluctuations in Cooper pairing is often made more complicated by the coexistence of other orders, particularly long-range magnetic order. Here we report the enhancement of superconductivity in a model electronic nematic system that is not magnetic, and show that the enhancement is directly born out of strong nematic fluctuations associated with a quantum phase transition. We present measurements of the resistance as a function of strain in Ba_1-x Sr _x Ni₂As₂ to show that strontium substitution promotes an electronically driven nematic order in this system. In addition, the complete suppression of that order to absolute zero temperature leads to an enhancement of the pairing strength, as evidenced by a sixfold increase in the superconducting transition temperature. The direct relation between enhanced pairing and nematic fluctuations in this model system, as well as the interplay with a unidirectional charge-density-wave order comparable to that found in the cuprates, offers a means to investigate the role of nematicity in strengthening superconductivity.

Strain-Induced Spin-Nematic State and Nematic Susceptibility Arising from 2×2 Fe Clusters in KFe_{0.8}Ag_{1.2}Te_{2}

Spin nematics break spin-rotational symmetry while maintaining time-reversal symmetry, analogous to liquid crystal nematics that break spatial rotational symmetry while maintaining translational symmetry. Although several candidate spin nematics have been proposed, the identification and characterization of such a state remain challenging because the spin-nematic order parameter does not couple directly to experimental probes. KFe_{0.8}Ag_{1.2}Te_{2} (K_{5}Fe_{4}Ag_{6}Te_{10}, KFAT) is a local-moment magnet consisting of well-separated 2×2 Fe clusters, and in its ground state the clusters order magnetically, breaking both spin-rotational and time-reversal symmetries. Using uniform magnetic susceptibility and neutron scattering measurements, we find a small strain induces sizable spin anisotropy in the paramagnetic state of KFAT, manifestly breaking spin-rotational symmetry while retaining time-reversal symmetry, resulting in a strain-induced spin-nematic state in which the 2×2 clusters act as the spin analog of molecules in a liquid crystal nematic. The strain-induced spin anisotropy in KFAT allows us to probe its nematic susceptibility, revealing a divergentlike increase upon cooling, indicating the ordered ground state is driven by a spin-orbital entangled nematic order parameter.

Evidence of precursor orthorhombic domains well above the electronic nematic transition temperature in Sr(Fe1-x Co x )2As2

Raman scattering, synchrotron x-ray diffraction, specific heat, resistivity and magnetic susceptibility measurements were performed in Sr(Fe_1-x Co _x )₂As₂ [[Formula: see text]] single crystals with superconducting critical temperature [Formula: see text] K and two additional transitions at 132 and 152 K observed in both specific heat and resistivity data. A quasielastic Raman signal with B _2g symmetry (tetragonal cell) associated with electronic nematic fluctuations is observed. Crucially, this signal shows maximum intensity at [Formula: see text] K, marking the nematic transition temperature. X-ray diffraction shows evidence of coexisting orthorhombic and tetragonal domains between [Formula: see text] and [Formula: see text]  ∼  152 K, implying that precursor orthorhombic domains emerge over an extended temperature range above [Formula: see text]. While the height of the quasielastic Raman peak is insensitive to [Formula: see text], the temperature-dependence of the average nematic fluctuation rate indicates a slowing down of the nematic fluctuations inside the precursor orthorhombic domains. These results are analogous to those previously reported for the LaFeAsO parent oxypnictide (Kaneko et al 2017 Phys. Rev. B 96 014506). We propose a scenario where the precursor orthorhombic phase may be generated within the electronically disordered regime ([Formula: see text]) as long as the nematic fluctuation rate is sufficiently small in comparison to the optical phonon frequency range. In this regime, the local atomic structure responds adiabatically to the electronic nematic fluctuations, creating a net of orthorhombic clusters that, albeit dynamical for [Formula: see text], may be sufficiently dense to sustain long-range phase coherence in a diffraction process up to [Formula: see text].

Two-Dimensional Non-Fermi-Liquid Metals: A Solvable Large-N Limit

Significant effort has been devoted to the study of "non-Fermi-liquid" (NFL) metals: gapless conducting systems that lack a quasiparticle description. One class of NFL metals involves a finite density of fermions interacting with soft order parameter fluctuations near a quantum critical point. The problem has been extensively studied in a large-N limit (N corresponding to the number of fermion flavors) where universal behavior can be obtained by solving a set of coupled saddle-point equations. However, a remarkable study by Lee revealed the breakdown of such approximations in two spatial dimensions. We show that an alternate approach, in which the fermions belong to the fundamental representation of a global SU(N) flavor symmetry, while the order parameter fields transform under the adjoint representation (a "matrix large-N" theory), yields a tractable large N limit. At low energies, the system consists of an overdamped boson with dynamical exponent z=3 coupled to a non-Fermi-liquid with self-energy Σ(ω)∼ω^{2/3}, consistent with previous studies.

Momentum Dependence of the Nematic Order Parameter in Iron-Based Superconductors

The momentum dependence of the nematic order parameter is an important ingredient in the microscopic description of iron-based high-temperature superconductors. While recent reports on FeSe indicate that the nematic order parameter changes sign between electron and hole bands, detailed knowledge is still missing for other compounds. Combining angle-resolved photoemission spectroscopy with uniaxial strain tuning, we measure the nematic band splitting in both FeSe and BaFe_{2}As_{2} without interference from either twinning or magnetic order. We find that the nematic order parameter exhibits the same momentum dependence in both compounds with a sign change between the Brillouin center and the corner. This suggests that the same microscopic mechanism drives the nematic order in spite of the very different phase diagrams.

Anisotropic spin fluctuations in detwinned FeSe

Superconductivity in FeSe emerges from a nematic phase that breaks four-fold rotational symmetry in the iron plane. This phase may arise from orbital ordering, spin fluctuations or hidden magnetic quadrupolar order. Here we use inelastic neutron scattering on a mosaic of single crystals of FeSe, detwinned by mounting on a BaFe2As2 substrate to demonstrate that spin excitations are most intense at the antiferromagnetic wave vectors QAF = (±1, 0) at low energies E = 6-11 meV in the normal state. This two-fold (C2) anisotropy is reduced at lower energies, 3-5 meV, indicating a gapped four-fold (C4) mode. In the superconducting state, however, the strong nematic anisotropy is again reflected in the spin resonance (E = 3.6 meV) at QAF with incommensurate scattering around 5-6 meV. Our results highlight the extreme electronic anisotropy of the nematic phase of FeSe and are consistent with a highly anisotropic superconducting gap driven by spin fluctuations.

In-plane and out-of-plane properties of a BaFe2As2 single crystal

Anisotropy of transport and magnetic properties of parent compounds of iron based superconductors is a key ingredient of superconductivity. In this work, we investigate in-plane and out-of-plane properties, namely thermal, electric, thermoelectric transport and magnetic susceptibility in a high quality BaFe₂As₂ single crystal of the 122 parent compound, using a combined experimental and theoretical approach. Combining the ab initio calculation of the band structure and the measured in-plane and out-of-plane resistivity, we evaluate the scattering rates which turn out to be strongly anisotropic and determined by spin excitations in the antiferromagnetic state. The observed anisotropy of thermal conductivity is discussed in terms of anisotropy of sound velocities which we estimate to be [Formula: see text]. Remarkably, we find that thermal conductivity is characterized by a sizeable electronic contribution at low temperature, which is ascribed to the high purity of our crystal.

Ultrafast nematic-orbital excitation in FeSe

The electronic nematic phase is an unconventional state of matter that spontaneously breaks the rotational symmetry of electrons. In iron-pnictides/chalcogenides and cuprates, the nematic ordering and fluctuations have been suggested to have as-yet-unconfirmed roles in superconductivity. However, most studies have been conducted in thermal equilibrium, where the dynamical property and excitation can be masked by the coupling with the lattice. Here we use femtosecond optical pulse to perturb the electronic nematic order in FeSe. Through time-, energy-, momentum- and orbital-resolved photo-emission spectroscopy, we detect the ultrafast dynamics of electronic nematicity. In the strong-excitation regime, through the observation of Fermi surface anisotropy, we find a quick disappearance of the nematicity followed by a heavily-damped oscillation. This short-life nematicity oscillation is seemingly related to the imbalance of Fe 3d_xz and d_yz orbitals. These phenomena show critical behavior as a function of pump fluence. Our real-time observations reveal the nature of the electronic nematic excitation instantly decoupled from the underlying lattice.

Several experiments have shown evidence for unusual nematic electronic behaviour in unconventional superconductors. Here the authors use pump-probe spectroscopy to observe out-of-equilibrium behaviour of coupled nematic-orbital excitations in iron selenide.

Ferroelasticity, anelasticity and magnetoelastic relaxation in Co-doped iron pnictide: Ba(Fe0.957Co0.043)2As2

The hypothesis that strain has a permeating influence on ferroelastic, magnetic and superconducting transitions in 122 iron pnictides has been tested by investigating variations of the elastic and anelastic properties of a single crystal of Ba(Fe_0.957Co_0.043)₂As₂ by resonant ultrasound spectroscopy as a function of temperature and externally applied magnetic field. Non-linear softening and stiffening of C ₆₆ in the stability fields of both the tetragonal and orthorhombic structures has been found to conform quantitatively to the Landau expansion for a pseudoproper ferroelastic transition which is second order in character. The only exception is that the transition occurs at a temperature (T _S  ≈  69 K) ~10 K above the temperature at which C ₆₆ would extrapolate to zero ([Formula: see text]  ≈  59 K). An absence of anomalies associated with antiferromagnetic ordering below T _N  ≈  60 K implies that coupling of the magnetic order parameter with shear strain is weak. It is concluded that linear-quadratic coupling between the structural/electronic and antiferromagnetic order parameters is suppressed due to the effects of local heterogeneous strain fields arising from the substitution of Fe by Co. An acoustic loss peak at ~50-55 K is attributed to the influence of mobile ferroelastic twin walls that become pinned by a thermally activated process involving polaronic defects. Softening of C ₆₆ by up to ~6% below the normal-superconducting transition at T _c  ≈  13 K demonstrates an effective coupling of the shear strain with the order parameter for the superconducting transition which arises indirectly as a consequence of unfavourable coupling of the superconducting order parameter with the ferroelastic order parameter. Ba(Fe_0.957Co_0.043)₂As₂ is representative of 122 pnictides as forming a class of multiferroic superconductors in which elastic strain relaxations underpin almost all aspects of coupling between the structural, magnetic and superconducting order parameters and of dynamic properties of the transformation microstructures they contain.

Divergence of the quadrupole-strain susceptibility of the electronic nematic system YbRu2Ge2

A wide range of strongly correlated quantum materials, including some high-temperature superconductors, exhibit “electronic nematic” phases, in which the electronic properties spontaneously break the rotational symmetry of the crystal. However, the role that the corresponding nematic fluctuations play in these complicated systems is unclear, motivating the search for simpler model systems. Here, we identify a particular 4f intermetallic material which exhibits ferroquadrupole order, YbRu2Ge2, as just such a model system. We also provide a robust and accurate method to probe the divergence of an important associated quantity, the quadrupole-strain susceptibility. The temperature dependence of this quantity provides insight into the nature of the interactions that lead to ferroquadrupole order, in this case, magnetoelastic coupling.

Ferroquadrupole order associated with local 4f atomic orbitals of rare-earth ions is a realization of electronic nematic order. However, there are relatively few examples of intermetallic materials which exhibit continuous ferroquadrupole phase transitions, motivating the search for additional materials that fall into this category. Furthermore, it is not clear a priori whether experimental approaches based on transport measurements which have been successfully used to probe the nematic susceptibility in materials such as the Fe-based superconductors will be as effective in the case of 4f intermetallic materials, for which the important electronic degrees of freedom are local rather than itinerant and are consequently less strongly coupled to the charge-carrying quasiparticles near the Fermi energy. In the present work, we demonstrate that the intermetallic compound YbRu2Ge2 exhibits a tetragonal-to-orthorhombic phase transition consistent with ferroquadrupole order of the Yb ions and go on to show that elastoresistivity measurements can indeed provide a clear window on the diverging nematic susceptibility in this system. This material provides an arena in which to study the causes and consequences of electronic nematicity.

Intertwined Magnetic and Nematic Orders in Semiconducting KFe_{0.8}Ag_{1.2}Te_{2}

Superconductivity in the iron pnictides emerges from metallic parent compounds exhibiting intertwined stripe-type magnetic order and nematic order, with itinerant electrons suggested to be essential for both. Here we use x-ray and neutron scattering to show that a similar intertwined state is realized in semiconducting KFe_{0.8}Ag_{1.2}Te_{2} (K_{5}Fe_{4}Ag_{6}Te_{10}) without itinerant electrons. We find that Fe atoms in KFe_{0.8}Ag_{1.2}Te_{2} form isolated 2×2 blocks, separated by nonmagnetic Ag atoms. Long-range magnetic order sets in below T_{N}≈35  K, with magnetic moments within the 2×2 Fe blocks ordering into the stripe-type configuration. A nematic order accompanies the magnetic transition, manifest as a structural distortion that breaks the fourfold rotational symmetry of the lattice. The nematic orders in KFe_{0.8}Ag_{1.2}Te_{2} and iron pnictide parent compounds are similar in magnitude and in how they relate to the magnetic order, indicating a common origin. Since KFe_{0.8}Ag_{1.2}Te_{2} is a semiconductor without itinerant electrons, this indicates that local-moment magnetic interactions are integral to its magnetic and nematic orders, and such interactions may play a key role in iron-based superconductivity.