Diverse Exotic Orders and Fermiology in Fe-Based Superconductors: A Unified Mechanism for B_1g/B_2g Nematicity in FeSe/(Cs,Rb)Fe₂As₂ and Smectic Order in BaFe₂As₂

Abstract

A rich variety of nematic/smectic orders in Fe-based superconductors is an important unsolved problem in strongly correlated electron systems. A unified understanding of these orders has been investigated for the last decade. In this study, we explain the B_1g symmetry nematic transition in FeSe_1−xTe_x, the B_2g symmetry nematicity in AFe₂As₂ (A = Cs, Rb), and the smectic state in BaFe₂As₂ based on the same framework. We investigate the quantum interference mechanism between spin fluctuations by developing the density wave equation. The observed rich variety of nematic/smectic orders is naturally understood in this mechanism. The nematic/smectic orders depend on the characteristic shape and topology of the Fermi surface (FS) of each compound. 1) In FeSe_1−xTe_x (n_d = 6.0), each FS is very small and the d_xy-orbital hole pocket is below the Fermi level. In this case, the small spin fluctuations on three d_xz, d_yz, and d_xy orbitals cooperatively lead to the B_1g nematic (q = 0) order without magnetization. The experimental Lifshitz transition below the nematic transition temperature (T_S) is naturally reproduced. 2) In BaFe₂As₂ (n_d = 6.0), the d_xy-orbital hole pocket emerges around the M point, and each FS is relatively large. The strong spin fluctuations due to the d_xy-orbital nesting give rise to the B_1g nematic (q = 0) order and the smectic [q = (0, π)] order, and the latter transition temperature (T* ∼ 170K) exceeds the former one (T_S ∼ 140K). 3) In heavily hole-doped AFe₂As₂ (n_d = 5.5), the large d_xy-orbital hole pocket and the four tiny Dirac pockets appear due to the hole-doping. The B_2g nematic bond order emerges on the d_xy-orbital hole pocket because of the same interference mechanism. The present paramagnon interference mechanism provides a unified explanation of why the variety of nematic/smectic orders in Fe-based superconductors is so rich, based on the well-established fermiology of Fe-based superconductors.

1 Introduction

The emergence of an electron nematic (q = 0) state is one of the most important unsolved problems in Fe-based superconductors [1]. In LaFeAsO and Ba122 compounds, the antiferro (AF) magnetic state appears at the Néel temperature T_N, which is lower than the nematic transition temperature T_S. Since the superconducting phase with a high transition temperature (T_c) appears near the nematic phase and the AF magnetic phase, it is expected that the nematic fluctuations and the spin fluctuations are related to the mechanism of the high-T_c superconductivity. However, the questions appear before discussing the superconductivity: 1) what is the order parameter of the nematic state? 2) What is the driving force of the nematic state? 3) Why do the diverse nematic states emerge in various compounds?

It is known that the nematic order cannot be derived from the mean-field theory since the spin-channel order always dominates over the nematic order unless unphysical parameters (such as negative Hund’s coupling) are assumed. Previously, to explain the nematic state [2], the vestigial order (spin nematic) scenario [3–9] and the orbital order scenario [10–22] have been proposed.

To investigate the nematic state, the FeSe family is an ideal platform since the AF magnetic state is absent [23–26]. This family is also ideal from the aspect of superconductivity since the highest T_c ≳ 65K in Fe-based superconductors has been reported in electron-doped FeSe [27–31]. In FeSe, the orbital polarization between d_xz and d_yz orbitals in the nematic state has been observed by angle-resolved-photoemission spectroscopy (ARPES) [32–36]. To be more precise, the orbital polarization energy E_xz − E_yz has k dependence and changes the sign between the Γ point and the X (Y) point. This sign reversal orbital polarization has been explained by the orbital order scenario [16, 17, 19, 20] based on the paramagnon interference mechanism and by the renormalization group (RG) theory [37, 38]. In both theories, the vertex correction (VC) of the Coulomb interaction, which corresponds to the higher-order many-body effect, plays an essential role. Since the AF magnetic correlation is weak in FeSe, it is difficult to explain the nematic state by the vestigial order (spin nematic) scenario. Based on the paramagnon interference mechanism, the B_1g nematic orders in LaFeAsO and FeSe [14–18] and the nematic orders in cuprate superconductors [39–41] and magic-angle twisted bilayer graphene [42] have been explained as the orbital/bond orders. CDW orders in the transition metal dichalcogenide [43] and kagome metal [44] have also been explained by the paramagnon interference mechanism.

The rich variety of nematicity in the FeSe family remains a significant open problem. In FeSe_1−xS_x, T_S disappears at x ∼ 0.17, where the emergence of the nematic quantum critical point (QCP) has been suggested by experiments [45–48]. Recently, the whole x dependent phase diagram for FeSe_1−xTe_x (x ≲ 0.6) has been reported [49–51]. In the phase diagram shown in Figure 1A, T_S decreases with Te doping x, and T_S disappears at x ∼ 0.5. T_c becomes maximum K at x ∼ 0.6, which indicates that the nematic fluctuations enlarge the superconducting pairing interaction near the nematic QCP. Thus, it is essential to clarify the mechanism of x dependence of T_S to understand the mechanism of superconductivity in the FeSe family.

FIGURE 1

In addition, a significant open issue in nematicity is the emergence of another type of nematicity in various Ba122 compounds below T = T*, which is higher than T_S by tens of Kelvin, as shown in Figure 1B. An actual bulk nematic transition at T = T* has been reported in many experimental studies, such as a magnetic torque study [52], an X-ray study [53], an optical measurement study [54], and a laser photoemission electron microscope (PEEM) study [55]. Since the orthorhombicity (a − b)/(a + b) ≪ 0.1% below T* is tiny, an extrinsic origin such as inhomogeneity of the nematic transition temperature T_S due to local uniaxial pressure and randomness was proposed [4, 56–60]. On the other hand, T* seems not to be sensitive to the sample quality and the local strain, and the domain structure of nematicity observed above T_S is homogeneous [54, 55]. It is noteworthy that bulk orbital polarization starts to emerge at T = T*(>T_S) in Ba122 compounds, according to the recent PEEM study [55]. In this study, we will explain the multistage smectic/nematic transitions: the smectic order (q ≠ 0) at T = T* and the nematic order (q = 0) at T_S. In this scenario, T* is given by the intrinsic smectic order free from randomness.

In contrast to B_1g nematicity in typical Fe-based superconductors, the emergence of 45°-rotated B_2g nematicity in heavily hole-doped AFe₂As₂ (A = Cs, Rb) has been reported in Refs. [61–64], while Refs. [65, 66] have reported the absence of the nematic order. As shown in Figure 1C, the dominant B_1g nematicity changes to B_2g nematicity with doping x in Ba_1−xRb_xFe₂As₂. As for the mechanism of B_2g nematicity, vestigial nematic order by using the double-stripe magnetic configuration was suggested [67]. However, no SDW transition has been observed [64, 68] in AFe₂As₂, and the spin fluctuations are weak around T_S in RbFe₂As₂ [69]. In this study, we reveal the emergence of B_2g-symmetry bond order in AFe₂As₂.

As described previously, the variety of nematicity in Fe-based superconductors is very rich. To understand the mechanism of nematic/smectic states and superconductivity, it is important to explain these nematic/smectic states in the same theoretical framework.

In this study, we study B_1g nematicity in FeSe_1−xTe_x (n_d = 6.0), the tiny nematicity below T* in BaFe₂As₂ (n_d = 6.0), and B_2g nematicity in AFe₂As₂ (A = Cs, Rb) (n_d = 5.5) by developing the density wave (DW) equation theory. In this theory, the paramagnon interference mechanism due to the Aslamazov–Larkin (AL) type VCs shown in Figure 1D is taken into account. We also take into account of the self-energy effect shown in Figure 1E. In this mechanism, the rich variety of nematicity is naturally understood. The obtained nematicity depends on the shape and topology of FSs, as shown in Figures 2A–C. 1) In FeSe_1−xTe_x, all FSs are very small, and d_xy-orbital hole pocket is absent. The small spin fluctuations on the three d_xz, d_yz, and d_xy orbitals cooperatively lead to the B_1g nematic order, where the orbital order for d_xz and d_yz orbitals coexists with the bond order for the d_xy orbital. The experimental Lifshitz transition below T_S is naturally explained by the nematic order. 2) In BaFe₂As₂, the d_xy hole pocket emerges, and each FS is relatively large. The smectic order at T = T*(>T_S) and the nematic order at T = T_S emerge due to the strong d_xy-orbital nesting. The smectic order explains the tiny nematicity below T*, and the multistage transitions are explained by the smectic and nematic orders. 3) In heavily hole-doped AFe₂As₂, the large d_xy-orbital hole pocket and the four tiny Dirac pockets appear. The B_2g nematic bond order emerges due to the d_xy-orbital paramagnon interference mechanism, where the nesting between the Dirac pockets and the large d_xy-orbital hole pocket plays an important role. By considering the fermiology of each compound, these various nematic/smectic states are explained by the same theoretical framework based on the paramagnon interference mechanism.

FIGURE 2

In the present study, we intensively study the effect of self-energy on the nematic/smectic orders. It has been dropped in many previous studies, despite the fact that self-energy is necessary to satisfy the criteria of Baym–Kadanoff’s conserving approximation [70, 71]. We revealed that 1) the nematic/smectic order is stabilized by the AL-type VCs, while 2) T_S is reduced to become realistic (K) by introducing self-energy. These results validate the idea of the “nematic/smectic state due to the paramagnon interference mechanism” proposed in our previous studies [14–20, 39–44]. In addition, 3) the phase diagram of FeSe_1−xTe_x [49–51] is understood by using a fixed Coulomb interaction because of self-energy (in the absence of self-energy, add-hoc doping x dependence of the Coulomb interaction has to be introduced). The main merits (1)–(3) in the present study strongly indicate that the nematic/smectic states originate from the paramagnon interference mechanism [14–20, 39–44].

2 Multiorbital Models and Formulation

2.1 Multiorbital Models

Here, we introduce multiorbital models based on the first principle’s calculations. We analyze the following two-dimensional d-p Hubbard model with a unique parameter r, which controls the strength of the Coulomb interaction [16]:where is the first-principle’s model, and H^U is the Coulomb interaction for d-orbitals. We neglect the Coulomb interaction for p-orbitals. We denote the five Fe d-orbitals , d_xz, d_yz, and d_xy, as l = 1, 2, 3, 4, and 5, and three Se(As) p-orbitals p_x, p_y, and p_z as l = 6, 7, and 8. To obtain the model, we first use the WIEN2k [72] and Wannier90 [73] codes. Next, to reproduce the experimentally observed FSs, we introduce the k-dependent shifts for orbital l, δE_l, by modifying the intra-orbital hopping parameters, as explained in Ref. [17]. In the FeSe_1−xTe_x model, we shift the d_xy-orbital band and the d_xz/yz-orbital band at [Γ, M, X] points by [0eV, − 0.27 eV, + 0.40 eV] and [ − 0.24 eV, 0 eV, + 0.13 eV], respectively. In the BaFe₂As₂ model, the shifts are absent. In the CsFe₂As₂ model, we shift the d_xy-orbital band and the d_xz/yz-orbital band at [Γ, M, X] points by [0 eV, + 0.40 eV, 0 eV] and [ − 0.40 eV, 0 eV, + 0.10 eV], respectively.

We use the d-orbital Coulomb interaction introduced by the constraint random phase approximation (RPA) method in Ref. [74]. The Coulomb interactions for the spin and charge channels are generally given as

The Hamiltonian of the Coulomb interaction is given aswhere σ, σ′ = ±1 denote spin.

By using the multiorbital Coulomb interaction, the spin (charge) susceptibility for q = (q, ω_m = 2mπT) is given bywhere irreducible susceptibility is is the multiorbital Green function with the self-energy and given as for = [k, ϵ_n = (2n + 1)πT]. Here, is the matrix expression of H⁰, and μ is the chemical potential. The spin (charge) Stoner factor α_s(c) is defined as the maximum eigenvalue of . Since holds, spin (charge) fluctuations develop with increasing α_s(c), and α_s(c) = 1 corresponds to the spin- (charge)-channel ordered state.

2.2 FLEX Approximation

Here, we introduce the multiorbital fluctuation exchange (FLEX) approximation [15, 75]. The FLEX approximation satisfies the conserving approximation formalism of Baym and Kadanoff [70, 71]. In the FLEX approximation, self-energy is given aswhich is shown by the Feynman diagram in Figure 1E. The effective interaction for self-energy in the FLEX approximation is given aswhere and are denoted. We set μ = 0. , , and are calculated self-consistently. In multiband systems, the FSs are modified from the original FSs because of the self-energy correction. To escape from this difficulty, we subtract the Hermite term from the original self-energy, which corresponds to the elimination of double-counting terms between the LDA and FLEX.

2.3 Density-Wave Equation

We derive the strongest charge-channel density-wave (DW) instability without assuming the order parameter and wave vector. For this purpose, we use the DW equation method developed in Refs. [16, 19, 76]. We obtain the optimized non-local form factor with the momentum and orbital dependences by solving the following linearized DW equation shown in Figure 3A:where λ_q is the eigenvalue of the form factor , , and is the charge-channel irreducible four-point vertex shown in Figure 3B. The four-point vertex interaction in the DW Eq. 10 [16, 19] is given bywhere a^s = 3, a^c = 1, p = (p, ω_l), and .

FIGURE 3

In Eq. 11, the first line corresponds to the Maki-Thompson (MT) term, and the second and third lines give the AL terms, respectively. Feynman diagrams of the MT terms and AL terms are shown in Figure 3B.

The AL terms are enhanced by the paramagnon interference shown in Figure 1D. Thus, q = Q + Q′ = 0 nematic order is naturally induced by the paramagnon interference at the same nesting vector (Q′ = −Q). In the MT term, the first-order term with respect to gives the Hartree–Fock (HF) term in the mean-field theory. The charge-channel DW with wave vector q is established when the largest λ_q = 1. Thus, the smaller λ_q corresponds to the lower T_S. DW susceptibility is proportional to 1/(1 − λ_q) as explained in Ref. [20]. Therefore, λ_q represents the strength of the DW instability.

3 Results and Discussions

3.1 Results of FeSe_1−xTe_x

In this section, we show that 1) the B_1g nematic orbital + bond order originates from the paramagnon interference, and 2) the effect of self-energy is essential to reproduce the x dependence of T_S as shown in Figure 1A in FeSe_1−xTe_x. The effect of self-energy on the nematic/smectic order caused by the VCs is systematically studied in the present work. Because of self-energy, T_S is reduced to become realistic (K), while the symmetry of the nematic/smectic order is unchanged. Thus, the idea of electronic nematicity due to “the paramagnon-interference mechanism” proposed in Refs. [14–20, 39–44] has been confirmed by the present study.

Hereafter, we fix r = 0.35, T = 15 meV in calculations with self-energy and r = 0.15, T = 15 meV in calculations without the self-energy, unless otherwise noted.

Figures 4A,B show x dependent FSs and band structures, respectively. The FSs are small compared to other Fe-based superconductors. The d_xy orbital level at the M point increases with increasing x, as shown in Figure 4C. This behavior is consistent with ARPES measurements [77, 78]. On the other hand, the d_xy orbital level at the Γ point decreases with increasing x. becomes lower than the d_xz(yz) orbital level for x ≳ 0.3, and the topology of band changes. The change in topology has been observed between Γ and Z points in ARPES measurements of FeSe_0.5Te_0.5 [79, 80]. Figure 4D shows the density of state (DOS) of orbitals 3 and 4 for x = 0, 0.5. The DOS near the Fermi level for x = 0.5 is larger than that for x = 0 since the bandwidth decreases, and comes close to the Fermi level with increasing x. In addition, the dispersion of orbitals 2 and 3 at the Γ point becomes flat as decreases with increasing x, which also enlarges the DOS for orbitals 2 and 3 near the Fermi level.

FIGURE 4

To discuss the self-energy effect, we calculate the mass enhancement factors. Figure 5 shows the obtained x dependence of the mass enhancement factors for orbital l = 3, 4, which are given by in the FLEX approximation. The value of increases with increasing x since the electron correlation increases due to the reduction in the bandwidth and the increase in the DOS as shown in Figure 4D. Particularly, is enhanced by the d_xy orbital electron correlation between the electron pockets and the band around the M point since comes close to the Fermi level, as shown in Figure 4C. The behaviors of are similar to those given by the dynamical mean-field theory [81] and experiment [82].

FIGURE 5

Figure 6 shows x dependences of and in the FLEX approximation and the RPA. is almost independent of doping x, which means that change in topology or the number of FS around Γ comprising d_xz and d_yz orbitals does not strongly affect the spin fluctuation for the d_xz(yz) orbital. On the other hand, in the RPA without self-energy is strongly enhanced with increasing x since the electron correlation for the d_xy orbital between electron pockets and the band around the M point is significant for the enhancement of . The strong enhancement of is suppressed by the self-energy in the FLEX approximation. This suppression is necessary to reproduce the x dependence of T_S in the phase diagram.

FIGURE 6

Hereafter, we discuss the DW instability given by the DW Eq. 9. Figure 7A shows x dependences of λ₀ for the B_1g nematic state with and without the self-energy. λ₀ without the self-energy rapidly increases with doping x due to the paramagnon interference shown in Figure 1D. λ₀ is enlarged by the interference between strongly enhanced in the RPA, as shown in Figure 6. Since this result means T_S increases with x, the phase diagram in Figure 1A cannot be explained when the self-energy is absent. However, λ₀ including the self-energy decreases with doping x since the enhancement of in the FLEX approximation is moderate and the self-energy suppresses the and in the DW Eq. 10. The value of λ₀ increases with decreasing T, as shown in Figure 7B, and T = T_S is given when λ₀ = 1 is satisfied. Thus, T_S at x = 0 is higher than that at x = 0.5, and T_S at x = 0.65 cannot be obtained for T > 6 meV. The x dependence of T_S obtained by the paramagnon interference mechanism is consistent with the phase diagram in Figure 1A [49]. We see that T dependences of the strength of nematic fluctuations 1/(1 − λ₀) satisfy the Curie–Weiss law at low temperatures, as shown in Figure 7C. We note that the B_1g nematic state is realized because of the small FSs even for the weak spin fluctuations [16, 17].

FIGURE 7

Here, we analytically explain that T_S is reduced by self-energy by focusing on the mass renormalization factor z. As discussed in Ref. [17], α_s(c) is independent of z under the scaling T → zT and r → r/z. Under this scaling, the eigenvalue of the DW equation is unchanged [17]. Thus, T_S obtained by the DW equation without the self-energy is reduced to zT_S because of the self-energy. As a result, realistic T_S is obtained by taking self-energy into account.

Figure 8A shows q dependences of λ_q with the self-energy at x = 0, 0.5. λ_q has peak at q = 0, which means that the ferro nematic order is favored. Figure 8B shows k dependences of the static form factors and , where is given by the analytic continuation of . represents B_1g orbital order between orbitals 2 and 3. From the k dependence of , the sign-reversing orbital order is confirmed along the k_x(k_y) axis. As shown in Figure 8C, k dependence of causes the B_1g nearest-neighbor bond order, which is the modulation of correlated hopping. Based on the paramagnon interference mechanism, we find that the small spin fluctuations on the three d_xz, d_yz, and d_xy orbitals cooperatively cause the B_1g nematic orbital + bond order. The FSs and the band structure under the nematic order with the maximum value of the form factor meV are shown in Figures 8D,E. meV with the mass enhancement factor is consistent with ARPES measurements [35, 36]. The Lifshitz transition, where the FS around the Y point is missing, has been reported in recent experiments [83–86]. The Lifshitz transition is naturally explained by the increase of the d_xy level around the Y point induced by . We note that the obtained coexistence of the bond order on the d_xy orbital and the orbital order on the (d_xz, d_yz) orbitals has already been shown in the supplementary material of Refs. [19, 20]. In Figure 8D, we derived the Lifshitz transition by setting meV by hand. It is noteworthy that the same result is recently obtained by solving the full DW equation in Ref. [87]. The full DW equation enables us to study the electronic states below T_S without introducing additional fitting parameters.

FIGURE 8

Here, we confirm that the d_xy orbital levels at Γ and M points are important for the x dependence of λ₀. We use the simple model, where only the shift of or is introduced for the x = 0 model. Figure 9A shows dependences of and λ₀, respectively. is almost independent of the value of . λ₀ decreases with decreasing , which is consistent with the result shown in Figure 7A. The topology of the band structure changes at the Γ point with decreasing , which plays an important role in decreasing λ₀. Figure 9B shows dependences of and λ₀, respectively. The behaviors of and λ₀ are similar to the results shown in Figure 5, Figure 7A. The x dependences of and λ₀ are explained by the electron correlation between the electron pockets and the d_xy band around the M point. λ₀ is suppressed by self-energy for the d_xy orbital. The suppression becomes strong with increasing due to the feedback effect of the self-energy. To summarize, the B_1g nematic orbital + bond order is explained by the paramagnon interference mechanism in FeSe_1−xTe_x, and x dependence of T_S is well reproduced by the self-energy effect for the d_xy orbital.

FIGURE 9

3.2 Results of BaFe₂As₂

In this section, we discuss the multi-nematicity in BaFe₂As₂ [20].

The effect of self-energy on the nematic/smectic orders caused by the VCs is studied in the present work. Transition temperatures are reduced to become realistic because of the self-energy, while the symmetries of the nematic/smectic orders are unchanged. We reveal the origin of the tiny nematicity below T = T* and explain the multistage transitions at T = T* and T_S in the phase diagram shown in Figure 1B. As shown in Figure 2B, the size of the hole FS around the M point comprising the d_xy orbital is similar to that of electron FSs around the X and Y points, which causes good intra- and inter-orbital nestings. As explained later, inter-orbital nesting is important to realize the smectic state at T = T*.

Figure 10A shows the q-dependence of λ_q with and without self-energy. The q = (0, π) smectic bond order is dominant over the q = 0 nematic orbital + bond order because of the relation λ_(0,π) > λ₀, which is robust in the presence of moderate spin fluctuations α_s ≳ 0.85. Thus, the nematic orbital + bond transition temperature T_S is lower than T*, where the smectic bond order appears. Figure 10B shows the dominant component of the static form factor, , for q = (0, π). Focusing on the X and M points, is proportional to − cos(k_y), which corresponds to the inter-orbital smectic bond order, where the y-direction hoppings between orbitals 3 and 4 are modulated by the correlated hopping δt_3,4(y; y ± 1) = −δt_4,3(y; y ± 1) = δt(−1)^y. It is to be noted that δt_l,m(y; y′) is real and equal to δt_m,l(y′; y).

FIGURE 10

As shown in Figure 1D, the origin of the smectic bond order is the quantum interference between the spin fluctuations χ^s(Q) for Q ≈ (0, π) and χ^s(0) due to the AL terms. In this case, q = (0, π) (= Q + Q′) is given by Q′ = 0. χ^s(Q) is enhanced when the FS appears around the M point since nesting between FSs around the X and M points becomes good, while the moderate χ^s(0) is caused by forward scattering. We find that is significantly enlarged by inter-orbital nesting between the d_xy-orbital FS around the M point and the d_yz-orbital FS around the X point. In addition to the quantum interference due to the AL terms, the MT terms strengthen the sign change of between the X and M points, as reported previously [16, 19, 37]. Thus, the smectic bond order originates from the cooperation between the AL and MT terms due to good inter-orbital nesting between FSs around the X and M points. In contrast, the B_1g nematic orbital + bond order shown in Figures 10C,D originates from the interference between χ^s(Q) and χ^s(−Q). This nematic orbital + bond order is similar to that in FeSe and FeSe_1−xTe_x.

Here, we examine the DOS under the smectic bond order to verify the present theory. For T < T* = 32.4 meV without self-energy, we introduce the mean-field-like T-dependent form factor , where is the obtained form factor for q = (0, π) normalized as max_k| f^q(k)| = 1. We put f^max = 60 meV. Figure 11A shows the DOS at T = T* and 28 meV(<T*). For T < T*, a pseudogap appears due to the smectic bond order, which is consistent with the experiments [88, 89]. Since the smectic bond order is an antiferroic order, the folded band structure emerges below T*, which is also consistent with the experiment [90].

FIGURE 11

Next, we focus on another mystery, the T-linear behavior of tiny nematicity ψ in Ba122 [52] below T*. To solve this mystery, we calculate the T dependence of uniform nematicity ψ = (n₂ − n₃)/(n₂ + n₃) in Figure 11B, where both for T < T* and the ferro nematic orbital + bond order for meV) are introduced. For T < T_S, we assume , where is the obtained form factor normalized as max_k|f⁰(k)| = 1. We use f^max = 60 meV, which corresponds to the d_xz(yz) orbital energy split meV in the ARPES measurements [91] by considering the mass enhancement factor for l = 2, 3. The T-linear behavior ψ ∝ (T* − T) for T_S < T < T* is a consequence of the relation because the f^(0,π) term cannot contribute to any q = 0 linear response. It is to be noted that the form factor for q = (π, 0) gives ψ < 0. Thus, the T-linear behavior of ψ below T* is also naturally explained by the smectic bond order. On the other hand, for T < T_S is induced by the nematic orbital + bond order. To summarize, the multistage transitions at T = T* and T_S, and the T-linear ψ below T*, are naturally explained by the smectic bond order and nematic orbital + bond order. The hole pocket around the M point is necessary to realize the smectic bond order by the paramagnon interference mechanism.

We stress that the present mechanism of the bulk nematicity for T_S < T < T* is intrinsic and free from the strength of the disorder and local strain in the system. The present smectic order originates from the AL–VC and the FS nesting between the d_xy-orbital hole pocket and the electron pockets [20]. We stress that the present theory explains the absence of the smectic order in bulk FeSe [55] because the d_xy-orbital hole pocket, which is necessary for smectic order formation, is below the Fermi level in FeSe.

Here, we explain the details of the recent microscopic measurements in P-doped Ba122 [54, 55] that support the present intrinsic scenario. These are bulk and real-space measurements. In the PEEM measurement [55], very uniform bulk nematic domains have been observed for T_S < T < T*. The width of each nematic domain is about 500 nm. The structure of the nematic domains is unchanged for T < T_S. In addition, once the nematic domain completely disappears by increasing T, it never appears at the same location if the temperature is lowered again. These results are consistent with the present intrinsic smectic order scenario for T_S < T < T* in P-doped Ba122. In the photo-modulation measurement [54], uniform nematic domains have also been observed. The observed nematicity becomes small near the nematic domain boundary, irrespective of the fact that large local strain anisotropy is observed at the domain boundary. The observed anticorrelation between the nematicity and the local strain anisotropy may conflict with the assumption of the extrinsic scenario of the nematicity above T_S.

In contrast, the extrinsic scenario has been proposed by other groups [4, 56–60]. In the extrinsic mechanism, nematicity for T > T_S in Co-doped Ba122, which exhibits large residual resistivity (>100 μΩcm), has been explained by the inhomogeneity of T_S induced by the disorder and local strain. However, it is not easy to explain the nematicity above T_S in clean P-doped (non-doped) Ba122 on the same footing in the extrinsic scenario.

We note that the multistage smectic/nematic transitions observed in NaFeAs [92] are also explained by the present intrinsic mechanism [20].

3.3 Results of Ba_1−xCs_xFe₂As₂

In this section, we discuss B_2g nematicity in heavily hole-doped compound AFe₂As₂ (A = Cs, Rb) [19]. The effect of self-energy on the nematic order caused by the VCs is studied in the present work. Because of self-energy, T_S is reduced to become realistic, whereas the symmetry of the nematic order is unchanged. The direction of B_2g nematicity is rotated by 45° from that of the conventional B_1g nematicity. Figure 2C shows FSs of CsFe₂As₂: the hole FS around the M point comprising the d_xy-orbital is large, whereas the Dirac pockets near the X and Y points are small. In this system, the d_xy-orbital spin fluctuations are dominant.

Figure 12A shows the q dependence of the largest eigenvalue λ_q with self-energy for r = 0.96 at T = 5 meV and that without self-energy for r = 0.30 at T = 20 meV λ_q becomes maximum at q = 0 and the dominant form factor at q = 0 is shown in Figure 12B. As shown in Figure 12C, this form factor corresponds to the B_2g next-nearest-neighbor bond order for the d_xy orbital, which is consistent with the experimentally observed B_2g nematicity [61–64]. By analyzing the irreducible four-point vertex in the DW Eq. 10, we find that the attractive (repulsive) interactions originate from the AL (MT) terms, as shown in Figure 12D. The obtained q = 0B_2g bond order is derived from these interactions. Since the AL terms are enhanced by the quantum interference between the spin fluctuations with Q and Q′(= −Q), as shown in Figure 1D, the q = 0 nematic bond order is realized. The value of λ₀ is strongly enhanced by the attractive interactions for the d_xy orbital due to the AL terms. In this system, the nesting vector is short Q ∼ (0.5π, 0), as shown in Figure 12B. Because of repulsive interaction by the MT terms, changes sign between the k points on the FSs connected by Q, as shown in Figure 12D. To summarize, the AL terms strongly enlarge λ₀ due to the paramagnon interference mechanism, and the MT terms favor B_2g symmetry. Cooperation of the AL and MT terms is important to realize the B_2g bond order.

FIGURE 12

We comment on the recent experiments on RbFe₂As₂. The specific heat jump at T_S = 40K (ΔC/T_S) is very small [64]. However, it is naturally understood based on the recent theoretical scaling relation with b ∼ 3 derived in Ref. [87]. Although the smallness of B_2g nematic susceptibility in RbFe₂As₂ was recently reported in Refs. [65, 66], the field angle-dependent specific heat measurement has shown finite B_2g nematicity above T_c [93]. Further experimental and theoretical studies are necessary to clarify the nematicity in AFe₂As₂ (A = Cs, Rb).

Finally, we discuss the x dependence of nematicity in Ba_1−xA_xFe₂As₂ (A = Cs, Rb). The schematic phase diagram of Ba_1−xRb_xFe₂As₂ given by the experiment [64] is shown in Figure 1C. We introduce the model Hamiltonian for Ba_1−xCs_xFe₂As₂, by interpolating between the BaFe₂As₂ model and the CsFe₂As₂ model with the ratio 1 − x: x. Figure 13A shows x dependences of λ_q=0 without self-energy for the B_2g and the B_1g symmetries by fixing T = 30 meV and r = 0.30. Below x = x_c ∼ 0.5, the B_1g nematic orbital order is dominant as discussed in the previous section, while the B_2g nematic bond order dominates over the B_1g nematic orbital order for x > x_c. As shown in Figure 13B, the Lifshitz transition occurs at x ∼ x_c, where the electron pockets split into the four tiny Dirac pockets. Thus, the B_2g nematic bond order appears when the nesting vector Q between the electron pockets and hole pocket around the M point becomes short Q ∼ (0.5π, 0). By taking account of the Lifshitz transition at x ∼ x_c, the schematic phase diagram in Figure 1C is also well reproduced by the orbital/bond order because of the paramagnon interference mechanism. We note that the q = (0, π) smectic order is dominant over the q = 0B_1g nematic order at x = 0, as shown in the previous section.

FIGURE 13

4 Conclusion

We discussed the rich variety of nematic/smectic states in Fe-based superconductors in the same theoretical framework based on the paramagnon interference mechanism. In this mechanism, the charge-channel order is induced by the quantum interference between the spin fluctuations, as shown in Figure 1D. The form factor and wave vector of the DW instability are derived from the DW equation based on the paramagnon interference mechanism. Recently, a rigorous formalism of the DW equation has been constructed based on the Luttinger–Ward (LW) theory in Ref. [87]. According to Ref. [87], the solution of the DW equation gives the minimum of the grand potential in the LW theory. Thus, the nematic/smectic order discussed in the present study is thermodynamically stable in the framework of the conserving approximation.

By considering the characteristic fermiology of each compound, the paramagnon interference mechanism explains the rich variety of the nematic/smectic states. In Figures 14A–C, we summarized the nematic/smectic orders revealed by the mechanism in the present study. 1) In FeSe_1−xTe_x, each FS is very small and the d_xy-orbital hole pocket is absent. In this case, the small spin fluctuations on the three orbitals cooperatively lead to the B_1g orbital order for the d_xz and d_yz orbitals coexisting with the d_xy-orbital bond order, as shown in Figure 14A. The nematic orbital + bond order causes the Lifshitz transition, where the FS around the Y point disappears, which is consistent with the recent experiments. The x dependence of T_S in the phase diagram is reproduced by introducing self-energy. 2) In BaFe₂As₂, the d_xy-orbital hole pocket emerges. Since each electron and hole pocket is relatively large and similar in size, the strong d_xy-orbital spin fluctuations due to good nesting give rise to the smectic order shown in Figure 14B and the B_1g nematic order. The smectic order explains the tiny T-linear nematicity below T = T*(>T_S). We predict the multistage transitions with the smectic order at T = T* and the nematic order at T_S. 3) In heavily hole-doped AFe₂As₂ (A = Cs, Rb), the tiny Dirac pockets around the X(Y) point and the large d_xy-orbital hole pocket appear due to hole-doping. The B_2g bond order for the d_xy orbital shown in Figure 14C emerges due to the d_xy-orbital paramagnon interference mechanism. The B_2g bond order is triggered by the Lifshitz transition of the electron FSs by hole-doping.

FIGURE 14

The limitation of this theory is that the calculated VCs are reduced to an infinite series of the MT and AL terms. To verify the validity of the present theory, we performed the functional renormalization group (fRG) analysis for the single-orbital Hubbard model for cuprates [41] and the two-orbital Hubbard model for ruthenates [94], and obtained the bond-order (orbital order) in the former (latter) model. These results are consistent with previous experiments, and they are also obtained by the DW equation analysis. In the fRG theory, a huge number of higher-order VCs are generated in an unbiased manner by solving the RG equation. Thus, the significance of the MT and AL terms in the present theory has been confirmed by the different and excellent theoretical frameworks.

In future, it is to clarify the mechanism of superconductivity and non-Fermi-liquid behaviors of transport phenomena in the FeSe family by considering the nematic fluctuations enlarged near the nematic QCP. This issue will be discussed in future studies [95].

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