Emergent XY electronic nematicity in iron-based superconductors

TL;DR

This paper reports the discovery of nearly isotropic XY-nematic fluctuations in hole-doped iron-based superconductors, revealing a new quantum liquid-crystalline state with potential for unique physical phenomena.

Contribution

It provides experimental evidence of XY-nematic fluctuations in iron-based superconductors, a state previously considered elusive, using elastoresistance measurements across different doping levels.

Findings

Observation of nearly isotropic XY-nematic fluctuations.

Doping-dependent orientation of the nematic director.

Comparable Curie-Weiss behavior in two nematic channels.

Abstract

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$_{x}$Fe$_{2}$As$_{2}$ iron-based superconductors. While for $x=0$ the nematic director points along the in-plane diagonals of the tetragonal lattice, for $x=1$ 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 new route to assess this elusive electronic quantum liquid-crystalline state, which is a candidate to host unique phenomena not present in the Ising-nematic case.

Equations

$$F=\frac{\gamma }{2}(N_1^2-N_2^2)+\frac{a}{2}(N_1^2+N_2^2)+\frac{u}{4}(N_1^2+N_2^2{)}^2.$$

$${\chi }_{\mathrm{nem}}={\chi }_0+\frac{C}{T-T_0}.$$

Full text

I INTRODUCTION

Liquid crystals are composed of molecules with anisotropic shapes, which possess a degree of orientational order. As a result, they can form nematic phases, in which translational invariance is preserved, but rotational symmetry is spontaneously broken by the selection of a director along which the molecules align. Strongly correlated electron systems often show exotic states of matter analogous to liquid-crystalline states, originating from nontrivial quantum many-body interactions. In particular, electronic nematic phases have been found in a number of quantum materials and have generated a lot of attention Fradkin2010 ; Fernandes2014 . After its original experimental identification in quantum Hall systems Lilly1999 , electronic nematicity has increasingly been recognized as a ubiquitous feature of unconventional superconductors. In iron-based superconductors, electronic nematic order and fluctuations span a wide region of the phase diagram Chu2010 ; Kasahara2012 , and likely impact superconductivity Lederer2016 . Electronically-driven rotational symmetry breaking has also been reported inside the mysterious pseudogap phase of high-$T_{{\rm c}}$ cuprates Hinkov2008 ; Daou2010 ; Sato2017 as well as in the elusive hidden-order phase of the heavy-fermion material URu2Si2 Okazaki2011 .

A crucial difference between electronic and liquid-crystalline nematic orders is that the former develops in the presence of a crystal, which by itself does break the rotational symmetry of the system – for instance, by restricting the possible paths along which the electrons can hop. As a result, while in liquid crystals the nematic director can point anywhere in space (i.e., it has a continuous symmetry), in electronic systems it is generally restricted to a few high-symmetry directions of the crystal (i.e., it has a discrete symmetry). While such a discrete electronic nematic state still displays interesting properties Fradkin2010 ; Fernandes2014 , a hypothetical continuous electronic nematic state would host several unique quantum many-body phenomena Oganesyan2001 . For instance, the transverse and longitudinal fluctuations in the disordered phase have completely different dynamics, which may lead to multiscale critical behavior Zacharias2009 . Moreover, the ordered phase has a Goldstone mode associated with the breaking of the continuous symmetry by the director. Remarkably, unlike other Goldstone modes such as phonons or magnons, the nematic mode couples directly to the electronic density rather than to gradients of the density Watanabe2014 . This can lead to strange metallic behavior Oganesyan2001 and strongly impact superconductivity Kim2004 .

At first sight, the inevitable presence of the crystal may be taken as an insurmountable obstacle to realize a continuous electronic nematic state in actual materials. However, as we argue now, and demonstrate below experimentally in the hole-doped iron pnictide Ba1-xRbxFe2As2, a nearly-continuous nematic phase can be observed under certain conditions. To make the discussion concrete, we consider layered crystals with tetragonal symmetry, i.e., the symmetry of a square. Indeed, many of the materials where nematicity has been observed do possess this symmetry. We start our discussion by first assuming that there was no underlying crystal. Then the electronic nematic director could point anywhere in the layer plane. In this case, in analogy to a two-dimensional nematic liquid crystal, the nematic order parameter is described in terms of a traceless symmetric $2\times 2$ matrix Oganesyan2001 given by: \bm{\mathcal{Q}}=\left(\begin{array}[]{rr}N_{1}&N_{2}\\ N_{2}&-N_{1}\end{array}\right). Here, the order parameter $N_{1}$ corresponds to a quadrupolar charge density that makes the horizontal ($x$) and vertical ($y$) directions inequivalent ($d_{x^{2}-y^{2}}$-wave form factor), whereas $N_{2}$ corresponds to a quadrupolar charge density that makes the two diagonal directions inequivalent ($d_{xy}$-wave form factor). A linear combination of $N_{1}$ and $N_{2}$ can be constructed such that the quadrupolar charge density can be rotated along any in-plane direction. The static and uniform free energy depends only on traces of even powers of $\bm{\mathcal{Q}}$, i.e., only on the combination $\left(N_{1}^{2}+N_{2}^{2}\right)$, reflecting the continuous symmetry of the nematic director. Therefore, $N_{1}$ and $N_{2}$ can be interpreted as two orthogonal components of the continuous nematic order parameter. Borrowing the nomenclature of magnetic phases, the resulting nematic phase is called an XY-nematic state.

The existence in reality of the tetragonal crystal, of course, dramatically changes this scenario. Now, $N_{1}$ and $N_{2}$ transform as two different irreducible representations of the tetragonal point group, denoted respectively by $B_{1g}$ and $B_{2g}$. Phenomenologically, this leads to a term in the free energy that explicitly lifts the degeneracy between $N_{1}$ and $N_{2}$, changing the symmetry of the nematic state from XY (continuous) to Ising (discrete). More formally, the free energy expansion becomes

[TABLE]

Here, $a$ and $u$ are Landau coefficients that depend on the microscopic model. The anisotropic term $\gamma$ selects one of the two components of the XY nematic order parameter: $B_{1g}$ nematic order when $\gamma<0$, and $B_{2g}$ nematic order when $\gamma>0$. The situation is analogous to an XY (i.e., planar) ferromagnet with an easy-axis anisotropy. Indeed, the nematic order parameter can now be written in terms of Pauli matrices as $\bm{\mathcal{Q}}=N_{1}\bm{\sigma}_{z}+N_{2}\bm{\sigma}_{x}$, expressing the analogy between $N_{i}$ and magnetic moment components. In the magnetic analogue, the $\gamma$ term corresponds to the single-ion anisotropy that defines the magnetization easy axis. This analogy offers a fruitful insight: in ferromagnets, when the single-ion anisotropy is much smaller than the exchange coupling, although the magnetic transition is formally Ising-like, the fluctuation spectrum is very similar to that of the XY ferromagnet, except at very low energies. Moreover, once long-range order sets in, the collective modes correspond essentially to a Goldstone mode with a very small gap.

Thus, going back to the nematic case, it is expected that many of the properties of the XY-nematic state will be relevant to describe the system’s behavior over a wide energy range as long as the anisotropy coefficient $\left|\gamma\right|$ is much smaller than the nematic energy scale, which we can set to be the nematic transition temperature $T_{{\rm nem}}$. The remaining task is to find a concrete system with $|\gamma|\ll T_{{\rm nem}}$. A promising strategy is to interpolate between two different compounds that display $B_{1g}$ nematic order ($\gamma<0$) and $B_{2g}$ nematic order ($\gamma>0$), forcing $\gamma$ to cross zero. In the remainder of this paper, we show that such a situation is realized in the hole-doped iron pnictide Ba1-xRbxFe2As2.

II METHODS

The parent compound BaFe2As2 ($x=0$) displays an antiferromagnetic transition that is preceded by a tetragonal to orthorhombic structural transition, which makes the diagonals $[110]$ and $[1\bar{1}0]$ of the tetragonal crystallographic unit cell inequivalent. It is well established that the structural transition is not driven by lattice degrees of freedom Chu2012 , but is the result of an electronic nematic instability in the $B_{2g}$ channel along the Fe-Fe direction [Fig. 1(a)]. Note that our notation refers to the crystallographic 2-Fe unit cell; in the 1-Fe unit cell notation sometimes used in the literature, $B_{1g}$ and $B_{2g}$ are exchanged. Upon increasing doping, the structural transition temperature is suppressed, approaching a putative quantum critical point near optimal doping, where the superconducting dome is peaked. The superconducting dome extends all the way up to $x=1$. Interestingly, in all hole-doped systems Ba1-x$A_{x}$Fe2As2, with alkali metal $A=$ K, Rb, Cs, the effective electronic mass is observed to be strongly enhanced near $x=1$ Hardy13 ; Eilers2016 ; Hardy2016 ; Mizukami2016 . This suggests a progressively more important role of electronic correlations in the $3d^{5.5}$ electronic configuration ($x=1$) as compared to the $3d^{6}$ configuration ($x=0$), which may be related to proximity to a hypothetical $3d^{5}$ Mott insulating state Medici2014 ; Valenti15 . This motivates a careful investigation of nematic tendencies in extremely hole-doped pnictides by using high-quality single crystals (see Appendix A), with the prospect of finding a suitable alloy series with a crossover to $B_{1g}$ dominance, as the stronger correlations may promote a mechanism for electronic nematicity different from that of the undoped compound Kivelson1998 .

To study nematicity in the extremely hole-doped regime, we use elastoresistance measurements, which have been shown to probe the bare static nematic susceptibility Chu2012 ; Kuo2016 ; Hosoi2016 . In this technique, we measure the relative change in resistance $\Delta R/R$ in response to an externally applied strain $\epsilon$, which is controlled by a piezoelectric device. Because uniaxial strain couples linearly to the nematic order parameters, in the linear response regime the quantity $\chi_{{\rm nem}}\sim\frac{1}{R}\frac{\partial\Delta R}{\partial\epsilon}$ is proportional to the nematic susceptibility (see Appendix B). Specifically, in the iron pnictides, the nematic susceptibilities in the $B_{1g}$ and $B_{2g}$ symmetry channels (denoted here by $\chi_{{\rm nem}}^{[100]}$ and $\chi_{{\rm nem}}^{[110]}$, respectively, with the superscript denoting the strain direction) can be obtained by applying strain along the nearest neighbor Fe-As and Fe-Fe bond directions, corresponding to $[100]$ and $[110]$, respectively [Fig. 1(a)].

III RESULTS AND DISCUSSION

Figure 1(b)-(e) shows representative data of $\Delta R/R$ as a function of applied strain $\epsilon$ along the $[100]$ and $[110]$ directions. In BaFe2As2, which displays the $3d^{6}$ electronic configuration, the elastoresistances in the two channels show a striking anisotropy, with the response to the strain along the $[110]$ direction being much larger than that along $[100]$ [Fig. 1(b),(c)]. This is consistent with previous reports Chu2012 , and reflects the Ising-like character of the $B_{2g}$ nematic state, which sets in when $\chi_{{\rm nem}}^{[110]}$ peaks. In sharp contrast, in CsFe2As2 with $3d^{5.5}$ electronic configuration, the elastoresistance for strain applied along $[100]$ becomes much larger than that along $[110]$, indicating that Ising-like $B_{1g}$ nematic fluctuations are dominant. Such a difference can be clearly seen in the temperature dependence of the nematic susceptibilities obtained from the slopes of $\Delta R/R$ as function of $\epsilon$, as shown in Fig. 2(a),(b). Although the nematic susceptibilities presented here were evaluated from longitudinal elastoresistance measurements, we confirmed that they are identical to those determined by using a modified Montgomery method, indicating negligible contaminations of $A_{1g}$-symmetry strain (see Appendix:B).

In the case of BaFe2As2 [Fig. 2(a)], $\chi_{{\rm nem}}^{[110]}$ is much larger and has a more pronounced temperature dependence as compared with $\chi_{{\rm nem}}^{[100]}$. Moreover, as demonstrated previously Chu2012 , $\chi_{{\rm nem}}^{[110]}$ has a peak at the nematic transition temperature $T_{\mathrm{nem}}$. Above this temperature, the nematic susceptibility can be described by a Curie-Weiss expression

[TABLE]

Here, $C$ is a constant, and $\chi_{0}$ is a temperature-independent term not related to the nematic fluctuations. The Curie-Weiss temperature $T_{0}$ represents the “bare” nematic transition temperature without nemato-lattice coupling effects. Such a $B_{2g}$ Ising-nematic order has been observed in many iron-based superconductors Kuo2016 ; Hosoi2016 . However, in CsFe2As2 [Fig. 2(b)], we find that the dominant nematic fluctuations appear in the $B_{1g}$ channel, since $\chi_{{\rm nem}}^{[100]}$ displays a considerably larger magnitude and stronger temperature dependence than $\chi_{{\rm nem}}^{[110]}$. We also note that the sign of $\chi_{{\rm nem}}^{[100]}$ is positive for CsFe2As2, in contrast to the negative sign of $\chi_{{\rm nem}}^{[110]}$ in BaFe2As2. Such a sign reversal has also been reported for the in-plane resistivity anisotropy in Ba1-xKxFe2As2, where the hole doping changes sign of $\rho_{b}/\rho_{a}-1$ from positive to negative Blomberg2013 .

As shown in Fig. 2(c), similar $B_{1g}$ Ising-nematic behavior as in CsFe2As2 is also observed in RbFe2As2, suggesting that nematicity along the Fe-As direction, 45*∘* tilted with respect to the Fe-Fe direction, is a generic feature of the $3d^{5.5}$ electronic configuration. Indeed, recent nuclear magnetic resonance and scanning tunneling microscopy studies suggested $B_{1g}$ nematicity in these compounds Li2016 ; Liu2018 , although the presence of long-range nematic order has not been settled. Here, we note that not only do the $\chi_{{\rm nem}}^{[100]}(T)$ curves for both CsFe2As2 and RbFe2As2 follow Curie-Weiss behavior over a wide temperature range, but also does the nematic susceptibility of RbFe2As2 exhibit a distinct peak at $T_{{\rm nem}}\sim 38$ K, indicative of the onset of long-range nematic order in the $B_{1g}$ channel. Recent specific-heat measurements under in-plane field rotation indeed show two-fold oscillations near $T_{c}$, supporting the $B_{1g}$ nematic order in RbFe2As2 Tanaka . This is consistent with a positive Curie-Weiss temperature $T_{0}\approx 25$ K obtained for RbFe2As2 by fitting Eq. (2) to $\chi_{{\rm nem}}^{[100]}(T)$. In contrast, we have found $T_{0}\approx-4$ K for CsFe2As2, from which we conclude that CsFe2As2 is close to a possible $B_{1g}$ nematic quantum critical point (QCP), which would correspond to $T_{0}=0$. The largely enhanced magnitude of $\chi_{{\rm nem}}^{[100]}$ is also consistent with quantum-critical fluctuations of $B_{1g}$ nematicity Hosoi2016 . We note in passing that, at low temperatures, $\chi_{{\rm nem}}^{[100]}$ in CsFe2As2 shows a noticeable deviation from the Curie-Weiss behavior. Similar deviations have been seen in several iron-based materials in the vicinity of their putative $B_{2g}$ nematic QCP, when the Curie-Weiss temperature extrapolates to zero Kuo2016 ; Hosoi2016 . Although disorder effects have been discussed as a possible origin of this behavior, the observation of a similar deviation in the very clean CsFe2As2 compound, where sharp quantum oscillations are clearly observed Eilers2016 , suggests that mechanisms other than disorder may be important for the behavior close to a nematic QCP.

Our elastoresistance measurements thus establish two parent iron-pnictide compounds, the $3d^{6}$ BaFe2As2 and the $3d^{5.5}$ RbFe2As2, that display Ising-nematic orders in two orthogonal channels, namely, $B_{2g}$ (corresponding to $\gamma>0$ in Eq. (1)) and $B_{1g}$ (corresponding to $\gamma<0$ in Eq. (1)), respectively. Following the strategy outlined above, we thus search for signatures of XY nematicity in the doped compound Ba1-xRbxFe2As2. Note that just because Ba1-xRbxFe2As2 interpolates between $B_{1g}$ and $B_{2g}$ nematic orders is not enough to ensure XY nematicity. On the contrary, if these ordered states had completely different origins, one would expect quite generically the $B_{1g}$ fluctuations to be suppressed before $B_{2g}$ fluctuations become sizable (and vice-versa). However, if these two states were manifestations of a common nematic state, differing from each other only in the direction of the nematic director, one would expect XY nematicity to emerge in the regime where the director switches from one direction to the other.

The doping evolution of the elastoresistance is depicted in Fig. 3(a)-(d). For Rb concentrations up to $x=0.65$, $\chi_{{\rm nem}}^{[110]}$ remains considerably larger than $\chi_{{\rm nem}}^{[100]}$. For $x=0.65$, the temperature dependence of $\chi_{{\rm nem}}^{[110]}$ continues to follow a Curie-Weiss behavior with a negative $T_{0}$, indicating the dominance of $B_{2g}$ nematic fluctuations, but there is no evidence for long-range nematic order [Fig. 3(a)]. As the Rb concentration further increases ($x=0.75,0.80,$ and $0.86$), the magnitude of $\chi_{{\rm nem}}^{[100]}$ gradually grows at the expense of $\chi_{{\rm nem}}^{[110]}$, as shown in Fig. 3(b)-(d), reflecting the development of a $B_{1g}$ nematic instability near $x=1$. Remarkably, for $x=0.80$ and $0.86$, the nematic susceptibilities in the two channels have comparable magnitudes and similar temperature dependencies at high temperatures. In this doping range, the system displays long-range $B_{1g}$ nematic order, as signaled by the positive $T_{0}$ and, more importantly, by the peak in $\chi_{{\rm nem}}^{[100]}(T)$, which defines $T_{{\rm nem}}$. Interestingly, despite the onset of long-range order in the $B_{1g}$ channel, $\chi_{{\rm nem}}^{[110]}$ continues to increase even below $T_{{\rm nem}}$, indicating the importance of both nematic modes for all temperatures. In the $x$ range between $B_{1g}$ and $B_{2g}$ nematic orders, i.e., for $x=0.75$, the absence of peaks in the nematic susceptibilities indicates the absence of long-range nematic order. In addition, the two nematic susceptibilities show almost identical temperature dependencies, with very similar Curie-Weiss temperatures. The observed comparable Curie-Weiss behavior in these two symmetry-irreducible channels immediately indicates that the nematic fluctuations are isotropic, in contrast to the strongly anisotropic fluctuations found for $x=0$ and $x=1$. Note that to establish that the nematic fluctuations are nearly isotropic (i.e. XY-like), it is sufficient to show that the susceptibilities in the two orthogonal channels, $B_{1g}$ and $B_{2g}$, are very similar. This is because a director pointing along an arbitrary in-plane direction can always be decomposed as a linear combination of the $B_{1g}$ director and the $B_{2g}$ director. The situations is analogous to the case of a nearly-XY ferromagnet, which displays comparable susceptibilities $\chi_{\mathrm{FM}}^{xx}$ and $\chi_{\mathrm{FM}}^{yy}$. Thus, our results provide direct evidence that the $x=0.75$ concentration is very close to an XY-nematic state.

Our main findings are summarized in the Ba1-xRbxFe2As2 phase diagram shown in Fig. 4. The electronic nematic director rotates by $45^{\circ}$ with increasing Rb doping, changing from $B_{2g}$ Ising-nematic order near $x=0$ (blue region) to $B_{1g}$ Ising-nematic order at $x=1$ (red region). In an intermediate doping range, the nematic susceptibilities in the two channels become very similar, thus revealing an emergent nearly-XY nematic fluctuating regime (purple region). The Curie-Weiss temperature extracted from the $B_{1g}$ nematic susceptibility shows an abrupt change near the $x=0.80$ composition, below which long-range $B_{1g}$ nematic order is no longer detected. It is noteworthy that in the related compound Ba1-xKxFe2As2, the Fermi pockets near the Brillouin zone corner change from electron-like to hole-like near a similar doping composition Xu2013 .

Our results offer important new insights about the microscopic mechanism of nematicity in iron-based superconductors. More specifically, one can envision two possible general theoretical scenarios to explain our results. In the first scenario, both the $B_{2g}$ and $B_{1g}$ nematic states have the same origin. Given the evidence in favor of a magnetically-driven $B_{2g}$ nematic state for the $x=0$ compound Fernandes13 , the question is whether magnetic fluctuations can also promote $B_{1g}$ nematicity. In this regard, recent works have proposed that the wave-vector $\mathbf{Q}$ of the dominant magnetic fluctuations change from being parallel to the Fe-Fe direction for $x=0$ to being parallel to the Fe-As direction for $x=1$ Borisov2019 ; Wang19 . This would naturally lead to a change in the vestigial nematic order from $B_{2g}$ to $B_{1g}$ Zhang17 ; Bishop17 . Indeed, first-principle calculations Borisov2019 predict that the magnetic ground state changes from single-stripe, $\mathbf{Q}=(\pi,\pi)$, for $x=0$, to double-stripe, $\mathbf{Q}=(\pi,0)$, for $x=1$ (recall that we are referring to the crystallographic 2-Fe unit cell). A related theoretical proposal also attributes the change in nematicity to a change in the magnetic spectrum, which would become incommensurate due to the occurrence of a Lifshitz transition as $x=1$ is approached Onari2018 . While long-range magnetic order has not been observed in RbFe2As2, inelastic neutron scattering measurements, which probe directly the magnetic fluctuation spectrum, could be used to verify the applicability of this type of scenarios.

An alternative scenario is that the two nematic states have different origins. In this context, recent studies of possible charge order in RbFe2As2 Civardi2016 and in KFe2As2 under high pressure Wang2016 suggest that charge degrees of freedom could be important. Moreover, a distinguishing feature of the $x=1$ compound, and of all $3d^{5.5}$ stoichiometric compounds, is the proposed enhanced strength of electronic correlations Hardy13 ; Eilers2016 ; Hardy2016 ; Mizukami2016 ; Medici2014 ; Valenti15 . One interesting possibility then is that the $B_{1g}$ nematic state in RbFe2As2 could be the result of a symmetry-breaking Kondo-like hybridization gap between the more localized $d_{xy}$ orbital and the more itinerant $d_{xz}$, $d_{yz}$ orbitals Schmalian . Another possibility would be ferro-orbital order involving the $d_{xz}$, $d_{yz}$ orbitals only, although a microscopic mechanism that does not involve magnetic fluctuations has yet to be proposed.

IV CONCLUSIONS

From the elastoresistance measurements in Ba1-xRbxFe2As2, it is found that the nematic director rotates from Fe-Fe to Fe-As directions with hole doping. In the intermediate doping regimes, the nematic susceptibilities in the two symmetry-irreducible ($B_{1g}$ and $B_{2g}$) channels show very similar Curie-Weiss behavior, indicating nearly isotropic nematic fluctuations that are different from the Ising nematicity found in other iron-based superconductors.

The present observation of electronic XY-nematic fluctuations in overdoped Ba1-xRbxFe2As2 offers a new route to realizing this novel quantum liquid-crystalline state in iron-based superconductors. A promising direction is attempting to stabilize long-range XY-nematic order, for instance by combining chemical substitution and application of pressure Borisov2019 . One of the unique properties of such an XY-nematic state is that its Goldstone mode couples directly to the electronic density, as discussed above. Of course, the system will inevitably have a small anisotropy favoring one of the two nematic components, which will gap the Goldstone mode. Yet, for a small anisotropy, as implied by our elastoresistance data for the $x=0.75$ composition, much of the system’s behavior (except at very low energies) is similar to that of a true XY-nematic state.

Acknowledgements

We thank fruitful discussion with A. E. Böhmer, V. Borisov, A. Chubukov, A. Fujimori, Y. Gallais, H. Kontani, C. Meingast, S. Onari, J. C. Palmstrom, I. Paul, J. Schmalian, Q. Si, and R. Valenti. This work was supported by “Kakehashi” collaborative research program of Tsukuba Innovation Arena and by Grants-in-Aid for Scientific Research (Nos. 15H02106, 18H05227, 18K13492) from Japan Society for the Promotion of Science. X-ray diffraction measurements were partly supported by the joint research in the Institute for Solid State Physics, the University of Tokyo. Theory work (R.M.F.) was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0012336.

Appendix A: Single crystals

Single crystals of RbFe2As2 and CsFe2As2 were grown from arsenic-rich flux as described in Ref. Eilers2016, . The crystals grown by this method have been characterized by several measurements including X-ray, thermal expansion, ac susceptibility, and specific heat Eilers2016 ; Hardy2016 ; Mizukami2016 , and quantum oscillations have been observed in the magnetostriction Eilers2016 indicating the cleanness of these samples. Single crystals of Ba1-xRbxFe2As2 with various Rb concentrations $x$ were grown by the FeAs self-flux method Tsujii . $x$ was determined by energy dispersive X-ray spectrometry and also checked from the $c$-axis lattice constant measured by X-ray diffraction. The temperature dependence of the in-plane resistivity is measured by the standard four-probe method, and the results of Ba1-xRbxFe2As2 single crystals are shown in Fig. 5. The superconducting transition temperature $T_{{\rm c}}$ of the crystals was determined by the onset of superconducting transition (black arrows). The characteristic S-shaped temperature dependence as well as overall doping dependence of $T_{c}$ are similar to that reported in Ba1-xKxFe2As2 single crystals Liu2014 .

Appendix B: Nematic susceptibility measurements

We have systematically evaluated the $B_{1g}$ and $B_{2g}$ nematic susceptibility in $A$Fe2As2 ($A=$ Ba, Cs and Rb) and Ba1-xRbxFe2As2 (Figs. 2 and 3) from longitudinal elastoresistance measurements, where the current direction is parallel to the applied strain $\epsilon_{xx}$. At present, we have not obtained reliable data for KFe2As2, whose crystals are more sensitive to air.

For the longitudinal measurements we use rectangular crystals with typical dimensions of $\sim 1.0\times 0.4\times 0.02$ mm3. The resistance of the sample ($R_{xx}$) was measured by the conventional four-probe method [Fig. 6(a)]. For the $B_{1g}$ and $B_{2g}$ nematic susceptibility measurements, the edges of the samples are aligned along the tetragonal [100] and [110] directions, respectively.

It should be noted that, strictly speaking, the strain transmitted to the sample via the orthorhombic deformation of the piezoelectric stack has not uniaxial but highly anisotropic biaxial character, which can be decomposed into in-plane $A_{1g}$-symmetry strain ($\frac{1}{2}(\epsilon_{xx}+\epsilon_{yy})$) and $B_{1g/2g}$-symmetry strain ($\frac{1}{2}(\epsilon_{xx}-\epsilon_{yy})$). However, since $\epsilon_{xx}$ and $\epsilon_{yy}$ have opposite signs, and their ratio is characterized by the in-plane Poisson’s ratio $\nu_{p}$ of the piezoelectric device ($\epsilon_{yy}=-\nu_{p}\epsilon_{xx}$), the $A_{1g}$-symmetry strain is small compared to the $B_{1g/2g}$-symmetry strain Kuo2013 . To see the effect of $A_{1g}$-symmetry strain on the elastoresistance, we also conducted additional elastoresistance measurements using a modified Montgomery method Kuo2016 . For this type of measurement, samples with square shape were used ($\sim 0.6\times 0.6\times 0.03$ mm3), and the resistances along both $x$ and $y$ directions ($R_{xx}$ and $R_{yy}$) are measured by switching the current and voltage configurations [Fig. 6(a)]. In this method, the nematic susceptibility $\chi_{{\rm nem}}$ is given by $\chi_{{\rm nem}}\sim(1/2((\Delta R/R)_{xx}-(\Delta R/R)_{yy}))/\epsilon_{xx}$, in which $A_{1g}$-symmetry strain effect can be removed.

As shown in Fig. 6(b), the data of $\chi_{{\rm nem}}^{[110]}(T)$ in Ba1-xRbxFe2As2 with $x=0.65$ determined by two methods are almost identical, revealing that the $A_{1g}$-symmetry strain effect is negligibly small in the present measurements, and the longitudinal elastoresistance measurements can be used to extract nematic susceptibility in our study.