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Timestamp: 2019-04-18 20:49:31+00:00

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Nematic order in iron superconductors – who is in the driver’s seat?
Although the existence of nematic order in iron-based superconductors is now a well-established experimental fact, its origin remains controversial. Nematic order breaks the discrete lattice rotational symmetry by making the x and y directions in the Fe plane non-equivalent. This can happen because of (i) a tetragonal to orthorhombic structural transition, (ii) a spontaneous breaking of an orbital symmetry, or (iii) a spontaneous development of an Ising-type spin-nematic order – a magnetic state that breaks rotational symmetry but preserves time-reversal symmetry. The Landau theory of phase transitions dictates that the development of one of these orders should immediately induce the other two, making the origin of nematicity a physics realization of a “chicken and egg problem”. The three scenarios are, however, quite different from a microscopic perspective. While in the structural scenario lattice vibrations (phonons) play the dominant role, in the other two scenarios electronic correlations are responsible for the nematic order. In this review, we argue that experimental and theoretical evidence strongly points to the electronic rather than phononic mechanism, placing the nematic order in the class of correlation-driven electronic instabilities, like superconductivity and density-wave transitions. We discuss different microscopic models for nematicity in the iron pnictides, and link nematicity to other ordered states of the global phase diagram of these materials – magnetism and superconductivity. In the magnetic model nematic order pre-empts stripe-type magnetic order, and the same interaction which favors nematicity also gives rise to an unconventional s+− superconductivity. In the charge/orbital model magnetism appears as a secondary effect of ferro-orbital order, and the interaction which favors nematicity gives rise to a conventional s++ superconductivity. We explain the existing data in terms of the magnetic scenario, for which quantitative results have been obtained theoretically, including the phase diagram, transport properties of the nematic phase, scaling of nematic fluctuations, and the feedback of the nematic order on magnetic and electronic spectra.
The discovery of iron-based superconductors (FeSCs) with transition temperatures Tc as high as 65K has signaled the beginning of a new era in the investigation of unconventional superconductivity (for a review, see (1) ). The key first step to unveil the nature of the superconducting phase is to understand the normal state from which superconductivity arises. In most FeSCs, superconductivity is found in proximity to a magnetically ordered state (transition temperature Tmag), which led early on to the proposal that magnetic fluctuations play the key role in promoting the superconducting pairing (2); (3) . A more careful examination of the phase diagram, however, revealed that there is another non-superconducting ordered state besides magnetism. Namely, at a certain temperature Tnem, the system spontaneously breaks the symmetry between the x and y directions in the Fe plane, reducing the rotational point group symmetry of the lattice from tetragonal to orthorhombic, while time-reversal symmetry is preserved. In some materials, such as hole-doped (Ba1−xKx)Fe2As2, the tetragonal-to-orthorhombic and magnetic transitions are simultaneous and first-order (Tnem=Tmag), whereas in electron-doped Ba(Fe1−xCox)2As2 and isovalent-doped BaFe2(As1−xPx)2, they are split (Tnem>Tmag) and second order (5); (6); (7) (see Fig. 1). As doping increases, the Tnem line tracks the Tmag line across the phase diagram, approaching the superconducting dome. It is therefore essential to understand the origin of this new order as it may support or act detrimentally to superconductivity.
Figure 1: Schematic phase diagram of hole-doped and electron-doped iron pnictides of the BaFe2As2 family. The blue area denotes stripe-type orthorhombic magnetism, the red area denotes nematic/orthorhombic paramagnetic order, and the yellow area, superconductivity. The green area corresponds to a magnetically-ordered state that preserves tetragonal (C4) symmetry, as observed recently (45) . The shaded red region denotes a regime with strong nematic fluctuations. Bent-back dotted lines illustrate the magnetic and nematic transition lines inside the superconducting dome. Second-order (first-order) transitions are denoted by solid (dashe) lines. The insets show the temperature-dependence of the nematic (φ) and magnetic (M) order parameters in different regions of the phase diagram: region (I) corresponds to simultaneous first-order magnetic and nematic transitions; region (II), to split second-order nematic and first-order magnetic transitions; and region (III) to split second-order transitions.
The order parameter for a transition in which a rotational symmetry is broken but time-reversal symmetry is preserved is a director (i.e. a vector without an arrow), similar to the order parameter in the nematic phase of liquid crystals (8) . By analogy, the orthorhombic state in FeSCs has been called a “nematic state”. Unlike isotropic liquid crystals, however, in FeSCs the lattice symmetry forces the director to point only either along x or y directions, what makes the nematic order parameter Ising-type (Ising-nematic).
At first sight, one might view this tetragonal-to-orthorhombic transition as a regular structural transition driven by lattice vibrations (phonons). However, experiments find anisotropies in several electronic properties, such as the dc resistivity (9); (10) , to be much larger than the anisotropy of the lattice parameters. This led to the idea that the tetragonal-to-orthorhombic transition may be driven by electronic rather than lattice degrees of freedom. If this is the case, then the transition into the nematic phase is driven by the same fluctuations that give rise to superconductivity and magnetic order, and therefore is an integral part of a global phase diagram of FeSCs. Electronic nematic phases have been recently proposed in other unconventional superconductors, such as high-Tc cuprates and heavy-fermion materials (8) . An electronically driven nematic state in FeSCs would be in line with a generic reasoning that the pairing in all these correlated electron systems has the same origin.
The discussion on the “nematicity” in FeSCs has been largely focused on two key issues: (i) Can the experiments distinguish “beyond reasonable doubt” between phonon-driven and electron-driven tetragonal symmetry breaking? (ii) If this transition is driven by electrons, which of their collective degrees of freedom are driving it - charge/orbital fluctuations or spin fluctuations? Answering the last question is crucial for the understanding of superconductivity in FeSCs because we argue below that charge/orbital fluctuations favor a sign-preserving s-wave state (s++) whereas spin fluctuations favor a sign-changing s-wave (s+−) or a d-wave state. Here we give our perspective on these issues, discuss the phenomenology of the nematic state, its experimental manifestations, and the underlying microscopic models.
Spin-nematic order – the static spin susceptibility χmag(q) becomes different along the qx and qy directions of the Brillouin zone before a conventional SDW state develops (7) . The appearance of such an order is normally associated with divergent quadrupole magnetic fluctuations.
Figure 2: Manifestations of nematic order in the iron pnictides: (a) Structural distortion from a tetragonal (dashed line) to an orthorhombic (solid line) unit cell (5) . (b) Anisotropy in the uniform spin susceptibility χij=mi/hj, where mi denotes the magnetization along the i direction induced by a magnetic field hj applied along the j direction (7) . (c) Splitting of the dxz and dyz orbitals (orange and blue lines, respectively) (11) . The corresponding distortion of the Fermi surface is also shown (see also Fig. 5a).
Because the nematic transition is driven by ψ1, the coefficient χ1, which corresponds to the order parameter susceptibility in the disordered state, diverges at T=Tnem and becomes negative for T<Tnem, while χ2 and χ3 remain finite and positive (although fluctuations of ψ2 and ψ3 may shift slightly Tnem). For T<Tnem, ψ1 orders on its own: ⟨ψ1⟩=±(−χ−11/b)1/2. If λij in (1) were zero, the other two fields ψ2 and ψ3 would not order, but once λij are finite, a non-zero ⟨ψ1⟩ instantly induces finite values of the secondary order parameters ⟨ψ2⟩=−λ12χ2⟨ψ1⟩, ⟨ψ3⟩=−λ13χ3⟨ψ1⟩. As a consequence, there is only one nematic transition temperature at which all three ⟨ψi⟩ become non-zero (e.g., lattice symmetry is broken at the same temperature where electronic nematic order emerges), and it is not possible to determine who causes the instability by looking solely at equilibrium order parameters. An additional experimental complication is the presence of nematic twin domains below Tnem, what effectively averages ⟨ψ1⟩ to zero. This problem can be circumvented by applying a small detwinning uniaxial stress (9); (10) , which acts as a conjugate field to ψ1 and breaks the tetragonal symmetry at all temperatures, making Tnem an ill-defined quantity.
where χ1=⟨ψ21⟩ is the susceptibility of the primary field. The renormalized susceptibilities of the secondary fields do diverge at the nematic transition, however for small enough λ12 and λ13, ~χ2 and ~χ3 begin to grow only in the immediate vicinity of Tnem, where χ1 is already large. If one can measure the three susceptibilities independently, Eq. (2) in principle provides a criterion to decide which order parameter drives the instability. The implementation of this procedure is possible (see next section), but is complicated by two factors. First, it only works if λ12 and λ13 are relatively weak, what normally implies that the systems falls into the weak/moderate coupling category. If the coupling is large, all three order parameters become so inter-connected that the question “who is in the driver’s seat?” becomes meaningless. Second, in some FeSCs the nematic transition is first order, in which case all three susceptibilities jump from one finite value to another, even before the susceptibility of the primary field gets enhanced.
The first evidence for the electronic character of the tetragonal-to-orthorhombic transition came from resistivity measurements in detwinned samples (9); (10) , which revealed that resistivity anisotropies are significantly larger than relative lattice distortions and also display a nontrivial dependence on doping and disorder (15) . Other non-equilibrium quantities, such as thermopower (13) and optical conductivity (14); (15) , were also found to display large anisotropies, which in optical measurements were observed to extend to energies of several hundreds of meV. Anisotropies in observables related to charge and spin were also seen: angle-resolved photoemission spectroscopy (ARPES) found a splitting between the on-site energies of the dxz and dyz orbitals, indicative of ferro-orbital order (11) and torque magnetometry revealed different uniform magnetic susceptibilities along the x and y directions (7) . The onset of magnetic anisotropy coincides with the observation of a non-zero orthorhombic distortion, in agreement with the discussions of the previous section. Strong signatures of emerging magnetic anisotropy were also found in the behavior of the nuclear magnetic resonance (NMR) lines across Tnem (16) .
The direct observation of electronic anisotropy in the nematic state was made possible by scanning tunneling microscopy (STM). The first measurements, performed deep inside the magnetic phase, found that the local density of states around an impurity is characterized by a dimer-like structure extended along the magnetic ordering vector direction (18) . Subsequent measurements showed that these dimers persist above Tmag, in the temperature regime of the nematic state (19) . An additional piece of evidence in favor of the electronic character of the nematic transition came, ironically, from x-ray measurements of the orthorhombic distortion inside the SC phase. These measurements found a strong suppression of the distortion below Tc (20) , what is a characteristic signature of the competition for the same electronic states between two electronically-driven orders.
A few recent measurements focused on fluctuations in the tetragonal state, in particular, on the shear modulus Cs, which is the inverse susceptibility of the structural order parameter (21); (22); (23) . If the structural transition is driven not by the lattice but by some other electronic degree of freedom, Eq. (2) provides a natural way to connect the shear modulus to the electronic nematic susceptibility χ1. An experimentally observed softening of the shear modulus above Tnem was successfully fitted by Eq. (2) using both magnetic (21) and charge/orbital (22) phenomenological models for χ1, indicating that structural distortion is very likely not the primary order.
Perhaps the strongest evidence that the nematic transition is electronically-driven came from the recent measurements of the anisotropy of the resistivity (24) . Using a piezoelectric, the measurements were performed by using strain (the structural distortion) as the control parameter, rather than stress, as in previous setups. The strain δ is one of the order parameter fields in the free energy Eq. (1). Using the resistivity anisotropy ρanis=ρxx−ρyy as a proxy of the nematic order parameter, it was experimentally shown that the susceptibility ∂ρanis/∂δ diverges near the nematic transition. This is only possible if the structural distortion is a conjugate field to the primary order parameter, rather than the primary order parameter itself – otherwise ρanis would be simply proportional to the order parameter δ, with a constant prefactor.
A successful microscopic theory for electronic nematicity must describe the global phase diagram of FeSCs, i.e. not only the nematic order but also magnetism and superconductivity. A popular starting point is the multi-orbital Hubbard model, which describes hopping between all Fe-As orbitals and local interactions, such as intra-band and inter-band Hubbard repulsions and Hund’s exchange (2) . There is a general agreement among researches that this model does contain all information about the phase diagram. The model has been analyzed at both weak/intermediate coupling, when the system is a metal, and at strong coupling, when electrons on at least some orbitals were assumed to be localized or “almost localized”. The nematic order has been obtained in both limits, what is yet another indication that it is a generic property of FeSCs. We adopt the itinerant approach, since most FeSCs are metals. In this itinerant scenario, the low-energy electronic states lie around hole-like Fermi-surface pockets at the center of the Fe-square lattice Brillouin zone and electron-like Fermi-surface pockets at the borders of the Brillouin zone, see Fig. 5a. The microscopic reasoning for either magnetic or orbital scenarios of electronic nematicity follows from two different assumptions about the sign of the effective inter-pocket interaction U (26) , which is a combination of the Hubbard and Hund interactions dressed up by coherence factors associated with the transformation from the orbital to the band basis. As we will see, each scenario leads to a prediction of a particular superconducting pairing state.
Figure 3: Nematic order in both real and momentum space. In (a), we show the stripe magnetic configuration in real space, which can be interpreted as two inter-penetrating Neel sublattices (green and yellow) with staggered magnetization M1 and M2. In terms of the two magnetic order parameters MX and MY defined in momentum space and used throughout the text, we have M1,2=MX±MY. In (b), we show the onset of nematic order in the paramagnetic phase (⟨Mi⟩=0), in terms of the magnetic susceptibility χ(q) across the first Brillouin zone. For T>Tnem, the two inelastic peaks at QX=(π,0) and QY=(0,π) have equal amplitudes, i.e. ⟨M2X−M2Y⟩≡⟨M1⋅M2⟩=0. For Tmag<T<Tnem, one of the peaks becomes stronger than the other, i.e. ⟨M2X−M2Y⟩≡⟨M1⋅M2⟩≠0, which breaks the equivalence between the x and y directions.
The magnetic mechanism for the nematic order follows from the observation that in most FeSCs the observed magnetic order on the Fe atoms is of stripe type, with ordering vectors QX=(π,0) or QY=(0,π) i.e. spins are parallel to each other along one direction and anti-parallel along the other (27) (see Fig. 3a). This order breaks not only the O(3) spin-rotational symmetry (and time-reversal symmetry), but it also breaks the 90∘ lattice rotational symmetry down to 180∘ by choosing the ordering vectors to be either QX or QY. This additional tetragonal symmetry breaking enhances the order parameter manifold to O(3)×Z2 (29); (28) . In terms of the two magnetic order parameters MX=∑kc†k+QX,ασαβck,β and MY=∑kc†k+QY,ασαβck,β, associated with the ordering vectors QX and QY, the breaking of the O(3) symmetry implies ⟨Mi⟩≠0 while the breaking of the Z2 symmetry implies ⟨M2X⟩≠⟨M2Y⟩ (30) . In a mean-field approach both O(3) and Z2 symmetries are broken simultaneously at Tmag. However, fluctuations split the two transitions and give rise to an intermediate phase at Tmag<T<Tnem where tetragonal symmetry is broken but the spin-rotational O(3) symmetry is not, i.e. ⟨M2X⟩≠⟨M2Y⟩ while ⟨Mi⟩=0. This is by definition a nematic order, which, viewed this way, is an unconventional magnetic order which preserves time-reversal symmetry (a spin nematic). In real space, the stripe magnetic state can be viewed as two inter-penetrating Neel sublattices with staggered magnetizations M1=MX+MY and M2=MX−MY. In terms of these quantities, the nematic state is characterized by ⟨M1⋅M2⟩≠0 while ⟨Mi⟩=0 (see Fig. 3).
where T is the temperature, and g∝U2 is the composite coupling which, when positive, sets the magnetic order to be of stripe type. In dimensions d<4, ∑qχ2mag(q) diverges at Tmag (assuming that the magnetic transition is second order). Eq. (3) then shows that the nematic susceptibility diverges at a higher Tnem>Tmag, when g∑qχ2mag(q)=1, i.e. at sufficiently large but still finite magnetic correlation length. This mechanism naturally ties the nematic and magnetic ordering temperatures to each other over the entire phase diagram. In between Tnem and Tmag, the Z2 symmetry is broken but O(3) is not, i.e., ⟨M2X⟩≠⟨M2Y⟩ but ⟨Mi⟩=0. The difference between Tnem and Tmag is stronger in quasi-2D systems where Tmag is further decreased by thermal fluctuations, while Tnem remains unaffected (28); (31) .
More detailed microscopic calculations show that for some system parameters the nematic transition is second order, but for other input parameters it becomes first-order (30) . In the latter case, a jump in the nematic order parameter induces a jump in the magnetic correlation length, which may instantaneously trigger a first-order magnetic transition. In any case, when the Fermi pockets are decomposed into their orbital characters, one finds within the same microscopic model that the emergence of spin-nematic order gives rise to orbital order Δn=nxz−nyz, since the electron pocket at QX has mostly dyz character, whereas the electron pocket at QY has mostly dxz character. Similarly, a spin-nematic order induces a structural distortion a≠b (32); (33) .
We see therefore that the repulsive inter-pocket interaction U>0 enhances spin fluctuations, which gives rise to both magnetism and nematicity. To describe the global phase diagram of FeSCs, one needs also to investigate superconductivity. Spin fluctuations peaked at QX and QY strongly enhance inter-pocket repulsion, which becomes larger than intra-pocket repulsion. In this situation, the system is known to develop either an unconventional s+− superconductivity, in which the gap functions have different signs in the hole and in the electron pockets, or a dx2−y2 superconductivity (2); (3) . We emphasize that spin-nematic order and s+− superconductivity are both intrinsic consequences of the same magnetic scenario.
Other microscopic models also find nematic order in proximity to a magnetic instability. For instance, explicit evaluation of the ferro-orbital susceptibility using the multi-orbital Hubbard model finds that it is enhanced only in the presence of spin fluctuations (34) , similarly to what is described by Eq. (3). Studies of models with both localized and itinerant orbitals also found (37); (38); (39) that the proximity to magnetism is an important ingredient for orbital order. In purely localized-spin models the interplay between magnetism and ferro-orbital order is blurred by the complicated form of the effective Hamiltonian, which deviates from a simpler Kugel-Khomskii type (35); (36) .
In its simplest form, the charge/orbital scenario for the nematic order parallels the magnetic scenario, the only difference being the sign of the interaction U between electron and hole pockets. If this interaction turns out to be negative, it is the charge/orbital susceptibility rather than the spin susceptibility that is enhanced as χorb(Q)=χ0(Q)/(1+Uχ0(Q)), diverging at QX and QY at a certain Torb. This divergence would signal the onset of a charge density-wave state with ordering vectors QX or QY (or both) and order parameters WX=∑kc†k+QX,αδαβck,β and WY=∑kc†k+QY,αδαβck,β. This order breaks translational symmetry and, like in the magnetic scenario, breaks also an additional Z2 symmetry if only one order parameter becomes non-zero. It is natural to expect, although no explicit calculations have been done to the best of our knowledge, that fluctuations split the temperatures at which the translational and the Z2 symmetries are broken, in a manner similar to Eq. (3). Then, in the intermediate temperature range Torb<T<Tnem, the system spontaneously develops ferro-orbital order in which ⟨W2X⟩≠⟨W2Y⟩ while ⟨WX⟩=⟨WY⟩=0. A structural distortion and the difference between ⟨M2X⟩ and ⟨M2Y⟩ appear instantly once ferro-orbital order sets in. However, magnetic order only appears at a smaller temperature, presumably via changes in the magnetic correlation length induced by the ferro-orbital order.
For the Cooper pairing, the orbital scenario implies that the inter-pocket interaction is attractive and enhanced. Once this interaction exceeds the intra-pocket repulsion, the system develops a superconducting instability towards an s++ state – a conventional pairing state where the gap functions have the same sign in all pockets (3) .
What we described above is the simplest scenario for orbital order. More complex models have been also proposed to account for the nematic transition without involving magnetic degrees of freedom. In Ref. (40) , it was proposed that nematicity could arise as an unequal hybridization between localized dxy orbitals and itinerant dxz/dyz orbitals. In Ref. (41) it was suggested that both spin and charge interactions are present and that the larger interaction in the spin channel gives rise to magnetic order at, say, QX. However, before this happens, a weaker charge interaction gives rise to charge order at the other momentum QY (a pocket density-wave state), which would break the tetragonal symmetry of the system. Whether such a pocket density-wave is experimentally realized in FeSCs remains to be seen.
Figure 4: Schematic representation of the evolution of the magnetic and nematic transitions as function of the inverse nematic coupling g, according to the microscopic itinerant spin-nematic model. Second-order (first-order) lines are denotes by solid (dashed) lines. Regions (I)-(III) correspond to those of the phase diagram in Fig. 1. The arrows show how the nematic order parameter g is expected to change as function of various control parameters.
Although the experimental evidence presented in Section III favors an electronic nematic instability, disentangling the orbital and magnetic scenarios is difficult on a qualitative level, what begs for a more direct comparison between microscopic models and experimental results. In this regard, the doping evolution of the magnetic and structural transitions is an important benchmark. BaFe2As2, one of the compounds most extensively investigated, displays a second-order nematic transition at Tnem followed by a a first-order “meta-nematic transition” at a lower T, where the system simultaneously undergoes a first-order magnetic transition. The meta-nematic transition has been observed by x-ray (5); (6) and torque magnetometry (7) , although the data disagree on the precise value of Tnem. As charge carriers are introduced in the system via Co substitution in the Fe sites, the splitting between the two transitions increases, and eventually the meta-nematic transition disappers and the magnetic transition becomes second-order.
How does this compare to theory? For the magnetic scenario, a detailed theoretical analysis (30) shows that three types of system behavior are possible in systems that are moderately anisotropic, depending on the value of the nematic coupling g (see Fig. 4). At large g, nematic and magnetic transitions are simultaneous and first order. At intermediate g, nematic order develops via a second-order transition, and there is a meta-nematic transition at a lower T, where magnetic order also develops discontinuously. At smaller g, nematic and magnetic transitions are separate and second-order, with an intermediate spin-nematic phase between Tnem and Tmag. The microscopic calculations found that g decreases with electron doping, and the theoretical phase diagram in Fig. 4 is fully consistent with the one for the electron-doped Ba(Fe1−xCox)2As2 if we place the x=0 point in the region II in Fig. 4. No calculations of how the nematic and magnetic transitions evolve with carrier concentration have been done within the charge/orbital scenario.
One can take the comparison with the data even further and compare the two versions of the magnetic scenario – for itinerant and for localized spins. Both predict stripe magnetic order and pre-emptive Z2 symmetry-breaking but differ in the details. In particular, in localized models g is generally small and is unaffected by carrier concentration (42); (28) . This makes the description of the doping dependence in the localized spin approach somewhat problematic, although not impossible (31) . A more essential difference is that in localized models the coupling g is always positive, while in itinerant models g may become negative at large enough hole doping (43); (44) . For negative g, there is no tetragonal symmetry breaking either above or below the magnetic transition as the system selects a tetragonally-symmetric combination of both QX and QY orders. A symmetry-preserving magnetic state with orders at QX and QY has been recently observed in Ba1−xNaxFe2As2 (45) and Ba(Fe1−xMnx)2As2 (46) at large enough doping – a strong argument in favor of the itinerant magnetic scenario.
Another key quantity to compare experiment and theory is the resistivity anisotropy. Deep in the magnetically ordered state, the anisotropic folding of the Fermi surface plays the major role in determining the resistivity anisotropy (10); (47) . In the nematic state, T>Tmag, orbital order and spin-nematic order have different effects on the dc resistivity: while the former causes an anisotropy in the Drude weight (48); (37) , the latter gives rise to anisotropy in the scattering rate (49) . The calculated anisotropy in the Drude weight has the opposite sign to the one observed experimentally (48); (37) , whereas the calculated anisotropy in the magnetic scattering rate was shown to agree with experiments, including a sign-change of the anisotropy between electron-doped and hole-doped materials (50) .
Figure 5: (a) The minimal Fermi surface with hole pockets (green lines) at the center Γ of the Fe-square lattice Brillouin zone and electron pockets (blue lines) centered at the X=(π,0) and Y=(0,π) points of the Brillouin zone. U is the inter-pocket interaction discussed in the main text. (b) Depending on the sign of U, either spin fluctuations (U>0, repulsion) or charge fluctuations (U<0, attraction) dominate. In the former, a stripe-type spin density-wave state is pre-empted by a spin-nematic phase, and the superconducting state is s+− (opposite-sign gaps around the hole and electron pockets). In the latter, a stripe-type charge density-wave is pre-empted by a charge-nematic phase, and the superconducting state is s++ (same-sign gaps around the hole and electron pockets). In this scenario, magnetic order only appears as a secondary consequence of ferro-orbital order.
One can also compare theoretical and experimental results for the feedback effects from the nematic order on the electronic and the magnetic spectrum (37); (30); (51) . In the magnetic scenario, nematic order enhances the magnetic correlation length, what gives rise to strong magnetic fluctuations and a possible pseudogap in the electronic spectrum. A significant increase of magnetic fluctuations below Tnem has been observed via NMR in compounds where Tnem and Tmag are well separated (52) . Also, recent ARPES experiments found the pseudogap behavior (a suppression in the density of states at low energies) whose onset coincides with the nematic transition (53) . Within the orbital scenario, the key feedback from the orbital order is a Pomeranchuk distortion of the Fermi surface induced by orbital order (37) .
Nematic fluctuations above Tnem have also been used to compare experiment and theory. Orbital fluctuations have been argued to affect the density of states at the Fermi level (54) and leave signatures in point-contact spectroscopy consistent with the data (55) . Alternatively, one can employ Eq. (2) to compare the renormalized lattice susceptibility (the shear modulus Cs), assumed to be non-critical, with the susceptibility χ1 associated with either the orbital or the spin-nematic order parameter. Eq. (2) must be satisfied if the corresponding electronic order drives the nematic instability. In Ref. (56) , a quasi-elastic peak in the Raman response was attributed to charge/orbital fluctuations and used to extract the corresponding orbital susceptibility. On the other hand, the spin-nematic susceptibility, being proportional to ∑qχ2mag(q) (see Eq. 3), can be measured via the NMR spin-lattice relaxation rate 1/T1. Comparison with shear modulus data for a family of electron-doped FeSCs found that there is a robust scaling between Cs and 1/T1 data (57) . This provides strong support to the idea that the nematic transition is magnetically-driven.
The bulk of experimental and theoretical results which we presented in this mini-review supports the idea that nematic order in FeSCs is of electronic origin, what places it at par with other known electronic instabilities such as superconductivity or density-wave orders. It is likely that magnetic fluctuations drive the nematic instability. In any case, all three orders – spin-nematic, orbital, and structural, appear simultaneously below Tnem. The important question not addressed until very recently is the role of nematicity for high-temperature superconductivity. It is unlikely that nematic fluctuations can mediate superconductivity as spin or charge fluctuations do, but nematic fluctuations may nevertheless enhance Tc by reducing the bare intra-pocket repulsion. Below Tc, however, nematic order has been found to compete with superconductivity (20); (58) , like density-wave orders do. A special case in which nematicity strongly affects Tc is when s-wave and d-wave superconducting instabilities are nearly degenerate, what was suggested to be the case for strongly hole-doped and strongly electron-doped FeSCs (59) . In this situation nematic order leads to a sizeable enhancement of Tc by lifting the frustration associated with the competing pairing states (61); (60); (62) . These results clearly point to the need of additional investigations of the interplay between nematicity and superconductivity.
We acknowledge useful discussions with E. Abrahams, J. Analytis, E. Bascones, A. Böhmer, J. van den Brink, P. Brydon, S. Bud’ko, P. Canfield, P. Chandra, P. Dai, M. Daghofer, L. Degiorgi, I. Eremin, I. Fisher, Y. Gallais, A. Goldman, A. Kaminski, V. Keppens, D. Khalyavin, M. Khodas, S. Kivelson, J. Knolle, H. Kontani, A. Kreyssig, F. Krüger, W. Ku, W.C. Lee, J. Lorenzana, W. Lv, S. Maiti, D. Mandrus, R. McQueeney, Y. Matsuda, I. Mazin, C. Meingast, A. Millis, R. Osborn, A. Pasupathy, I. Paul, P. Phillips, R. Prozorov, S. Sachdev, Q. Si, T. Shibauchi, L. Taillefer, M. Takigawa, M. Tanatar, M. Vavilov, P. Wölfle, and M. Yoshizawa. The authors benefited a lot from the discussions with our great colleague Z. Tesanovic who unexpectedly passed away last year. A.V.C. is supported by the Office of Basic Energy Sciences U.S. Department of Energy under the grant #DE-FG02-ER46900.
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