DOI https://doi.org/10.1080/00018732.2022.2144590

Unconventional density waves and superconductivities in Fe-based
superconductors and other strongly correlated electron systems

_Hiroshi Kontani 1, Rina Tazai2, Youichi Yamakawa1, Seiichiro Onari1 _

1 Department of Physics, Nagoya University, Furo-cho, Nagoya 464-8602, Japan.

2 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502,
Japan

<EMAIL_ADDRESS>
Abstract In this article, we review the recent significant progress in the
theoretical studies of the electronic states by mainly focusing on Fe-based
and cuprate superconductors. These superconductors are “unconventional” in
that strong electron-electron correlation mediates the pairing;  they are
different from conventional phonon-mediated BCS superconductors. To seek the
high-$T_{\rm c}$ pairing mechanism, many scientists have focused on the
mysterious spontaneous rotational symmetry breaking above $T_{\rm c}$, such as
nematic order at ${{\bm{q}}}={\bm{0}}$ and smectic order at
${{\bm{q}}}\neq{\bm{0}}$. Such exotic correlation-driven symmetry breaking in
metals has become a central issue in condensed matter physics. We demonstrate
the emergence of the nematic and smectic orders due to orbital polarization
($n_{xz}\neq n_{yz}$) and the symmetry breaking in the correlated intersite
hopping (= bond order $\delta t_{i,j}$) in Fe-based and cuprate
superconductors. In addition, we discuss exotic spontaneous loop current
orders driven by the pure imaginary $\delta t_{i,j}$. These interesting
“unconventional density-waves” originate from the quantum interference between
different spin fluctuations that is described by the vertex correction (VC) in
the field theory. In the next stage, we discuss electron-correlation driven
superconductivity due to the fluctuations of unconventional density-waves. For
this purpose, we suggest the beyond-Migdal-Eliashberg gap equation by
including the VCs into the equation. In Fe-based superconductors, high-$T_{\rm
c}$ $s$-wave superconductivity can be mediated by nematic and smectic
fluctuations because the pairing interaction is magnified by the VCs. We also
discuss the multipolar fluctuation pairing mechanism in heavy fermion systems,
owing to the cooperation between the strong spin-orbit interaction and the
strong electron correlation. To summarize, we suggest that the quantum
interference mechanism described by the VCs plays essential roles in not only
various unconventional density-waves, but also exotic superconducting states
in many strongly correlated metals. We finally discuss some interesting future
issues with respect to the quantum interference mechanism.

Keywords: unconventional superconductivity, electronic nematic order, current
order, Fe-based superconductors, cuprate superconductors

###### Contents

  1. 1 Introduction
    1. 1.1 Unconventional superconductivity due to electron correlation
    2. 1.2 Fe-based superconductors
    3. 1.3 cuprate superconductors
    4. 1.4 Other exotic strongly correlated electron systems
  2. 2 Form factors of unconventional orders
    1. 2.1 Form factors
    2. 2.2 Difficulties in the mean-field approximation
    3. 2.3 Spin fluctuation theory
    4. 2.4 Vertex corrections for susceptibilities: Self-consisting vertex correction (SC-VC) theory
  3. 3 Why vertex corrections cause unconventional orders?
    1. 3.1 Orbital order due to vertex corrections
    2. 3.2 Orbital order owing to VCs: fRG theory
    3. 3.3 Density-wave equation: Optimization of ${{\bm{k}}}$-dependent form factors
  4. 4 Unconventional orders in single-orbital models: with a focus on cuprate superconductors
    1. 4.1 Motivation
    2. 4.2 Analysis by DW equation for cuprates
    3. 4.3 Renormalization group study for cuprates
    4. 4.4 Odd parity current orders: spin loop current
    5. 4.5 Odd parity current orders: charge loop current in quasi 1D systems
  5. 5 Unconventional orders in Fe-based superconductors
    1. 5.1 Motivation
    2. 5.2 Nematic order and fluctuations due to paramagnon-interference: the SC-VC theory
    3. 5.3  Comparison between 1111 and 11 systems: the SC-VC theory
    4. 5.4 Nematic electronic states in FeSe below $T_{S}$: DW equation analysis
    5. 5.5 FeSe phase diagram under pressure: DW equation analysis
    6. 5.6 $B_{2g}$ nematic order in heavily hole-doped $A$Fe2As2 ($A$=Cs,Rb)
    7. 5.7 Hidden nematic order in Ba122 family
  6. 6 Superconductivity in Fe-based superconductors
    1. 6.1 Spin and orbital fluctuation theories
      1. 6.1.1 Basic idea
      2. 6.1.2 Charge quadrupole interaction
    2. 6.2 Phase sensitive experiments
      1. 6.2.1 Impurity effect
      2. 6.2.2 Resonance in inelastic neutron scattering
      3. 6.2.3 Nuclear relaxation rate $1/T_{1}$
    3. 6.3 Theory of superconductivity beyond-Migdal approximation
      1. 6.3.1 beyond-Migdal pairing interactions
    4. 6.4 Ba122: Cooperation and Competition between spin and orbital fluctuations
    5. 6.5 LiFeAs: Cooperation of $e$-ph and Coulomb interaction
    6. 6.6 La1111: Double-dome superconducting phase
    7. 6.7 FeSe: High-$T_{\rm c}$ superconductivity without hole-pocket
  7. 7 Unconventional superconductivity and multipole orders in heavy fermions
    1. 7.1 Introduction of heavy fermions
    2. 7.2 Multipole degrees of freedom by spin-orbit coupling
    3. 7.3 $S$-wave superconductivity in $\rm{CeCu}_{2}\rm{Si}_{2}$
    4. 7.4 Quadrupole order in CeB6
  8. 8 Summary

##  1 Introduction

###  1.1 Unconventional superconductivity due to electron correlation

In conventional BCS-type superconductors, the electron-phonon interaction
mediates the pairing interaction. Since the phonon-mediated interaction is
attractive, $s$-wave superconductivity is realized. Fundamental
superconducting (SC) electronic properties, such as transition temperature
$T_{\rm c}$ and thermodynamic quantities, are well explained based on the BCS
theory [1]. In general, the realized $T_{\rm c}$ is low since the phonon-
mediated pairing interaction is weak and the Debye temperature is only on the
order of $100$K. However, there are several exceptional high-$T_{c}$ phonon-
mediated superconductors, such as MgB2 and hydrogen-based compounds. The
$T_{\rm c}$ of the latter compounds exceeds 200K under ultrahigh pressure
(over $200$GPa) thanks to high Debye temperature of H-ion oscillation [2].

In contrast, in unconventional superconductors, the pairing interactions
originate from strong electron-electron correlations. Since the correlation-
driven pairing interaction strongly depends on systems, unconventional
superconductivity exhibits amazing variety in various strongly correlated
electron systems. For example, nodal $d$-wave superconductivity is realized in
cuprate high-$T_{\rm c}$ superconductors and heavy fermion compound Ce$M$In5
($M$=Co,Rh,Ir). In contrast, fully-gapped or nodal $s$-wave superconductivity
is realized by electron-electron correlation in Fe-based superconductors.
Also, spin-triplet superconductivity is expected to be realized in several
U-based heavy fermions.

The gap function is obtained by solving the gap equation. A simplified
expression of the linearized gap equation is [1, 3, 4],

$\displaystyle\lambda\Delta_{{\bm{k}}}=-\frac{1}{N}\sum_{{\bm{p}}}V^{\rm
SC}({{\bm{k}}}-{{\bm{p}}})\delta({\epsilon}_{{\bm{p}}}-\mu)\Delta_{{\bm{p}}}\ln(\omega_{c}/T)$
(1)

where $T$ is the temperature, $\mu$ is the chemical potential, and
${\epsilon}_{{\bm{p}}}$ is the conduction electron energy. $V^{\rm
SC}({{\bm{k}}}-{{\bm{p}}})$ is the pairing interaction due to the collective
bosonic fluctuations, and $\omega_{c}$ is the fluctuation energy-scale. The
gap function $\Delta_{{\bm{k}}}$ is uniquely determined as the eigenfunction
of the largest eigenvalue $\lambda$ in Eq. (1), and $T_{\rm c}$ is given by
the condition $\lambda=1$. Therefore, to reveal the unconventional SC state,
we have to know accurate $V^{\rm SC}({{\bm{q}}})$ in the normal state. For
this purpose, we study the normal state electronic properties before analyzing
the SC state. One of the main aim of this article is to explain the “phase
diagram in the normal state” that is the parent state of the
superconductivity.

The most famous correlation-driven superconductivity would be the spin-
fluctuation-mediated singlet pairing state. A schematic phase diagram of the
spin fluctuation scenario is shown in Fig. 1 (a). Here, the antiferromagnetic
(AFM) second-order transition temperature $T_{N}$ decreases with $x$, and the
AFM order disappears ($T_{N}=0$) at the quantum critical point (QCP)
$x=x_{c}$. Near the QCP, strong spin fluctuations develop at the nesting
vector ${{\bm{Q}}}_{s}$, and the spin susceptibility $\chi^{s}({{\bm{q}}})$
shows large peak at ${{\bm{q}}}={{\bm{Q}}}_{s}$. For the spin-singlet Cooper
pair, spin fluctuations give repulsive (=positive) interaction as $V^{\rm
SC}({{\bm{q}}})\propto\chi^{s}({{\bm{q}}})$. Therefore, in general, the
$d$-wave gap function $\Delta({{\bm{k}}})\propto\cos k_{x}-\cos k_{y}$ is
mediated by the staggered spin fluctuations with
${{\bm{Q}}}_{s}\approx(\pi,\pi)$, which is expected to be realized in Ce$M$In5
($M$=Co,Rh,Ir). Near the ferromagnetic (${{\bm{Q}}}_{c}={\bm{0}}$) QCP, the
spin triplet superconductivity is expected to appear. This “spin-fluctuation
pairing mechanism” has been established by many theorists. [4].

Figure 1:  (a) A schematic phase diagram due to the spin fluctuation mechanism
with antiferromagnetic (AFM) state and $d$-wave superconductivity. Strong spin
fluctuations appear near the AFM quantum critical point (QCP). (b) Paramagnon
interference mechanism that induces various nonmagnetic nematic or bond orders
in metals. (c) The expected phase diagram with exotic bond/orbital order that
is the unconventional (= non-$A_{1g}$ symmetry) order parameters. It
originates from the paramagnon interference mechanism. Exotic superconducting
state is expected to emerge near the bond/orbital order QCP

Here, we consider strongly correlated metals with large on-site Coulomb
interaction

$\displaystyle
H_{U}=\sum_{i}Un_{i\uparrow}n_{i\downarrow}=\sum_{i}U(n_{i}^{2}-m_{i}^{2})/4,$
(2)

where $i$ is the site index,
$n_{i{\sigma}}=c_{i{\sigma}}^{\dagger}c_{i{\sigma}}$,
$n_{i}=n_{i\uparrow}+n_{i\downarrow}$, and
$m_{i}=n_{i\uparrow}-n_{i\downarrow}$. In strongly correlated metals, strong
AFM fluctuations frequently appear since $U$ tends to induce on-site spin
polarization $\langle m_{i}\rangle\neq 0$, while on-site charge polarization
is suppressed by $U$. This consideration leads to the phase diagram in Fig. 1
(a). Nonetheless of this commonsense, the phase diagrams of Fe-based and
cuprate high-$T_{\rm c}$ superconductors are very different from Fig. 1 (a)
because of the presence of bond-order and orbital-order. These orders are
called nematic at wavevector ${{\bm{q}}}={\bm{0}}$, and smectic at
${{\bm{q}}}\neq{\bm{0}}$. The main origin of these nonmagnetic orders, which
cannot be explained by mean-field-level approximations, is the quantum
interference between spin fluctuations exhibited in Fig. 1 (b) [5]. This
paramagnon interference mechanism totally modifies the phase diagram in
strongly correlated metals: Figure 1 (c) is an example of expected phase
diagrams due to the paramagnon interference mechanism. The fluctuations near
the nematic or smectic QCP would induce non $d$-wave SC state, which will be
explained in this article.

In the last decade, interesting quantum phase transitions have been discovered
in many strongly-correlated superconductors. In Fe-based superconductors, we
encounter the electronic nematic states, which are the uniform
(${{\bm{q}}}={\bm{0}}$) rotational symmetry breaking states due to electron
correlations. In $R$1111 families ($R$=La,Ce,Pr,Nd,Sm) [6, 7], A122 families
($A$=Ba,Ca,Sr) [8] and FeSe families [9, 10, 11], $B_{1g}$ symmetry nematic
order with sizable orbital polarization ($n_{xz}\neq n_{yz}$) appears at the
structural transition temperature $T_{S}$. Interestingly, $B_{2g}$ symmetry
nematic order emerges in heavily $e$-doped system AFe2As2 (A=Cs,Rb) [12, 13,
14].

In cuprate superconductors, stripe order formation was reported by x-ray [15]
and neutron [16] scattering studies. Recently, the emergence of charge-
density-wave (CDW) order with finite wavevector ${{\bm{Q}}}=(\pi/2,0)$ or
$(0,\pi/2)$ has been confirmed by resonant X-ray scattering measurements [17,
18, 19, 20, 21, 22], the Scanning tunneling spectroscopy/Scanning unneling
microscopy (STM/STS) measurements [23, 24, 25], and the Nuclear magnetic
resonance (NMR) studies [26, 27]. These experimental findings manifest that
the spin, charge, and orbital degrees of freedom are all active and strongly
coupled in these high-$T_{\rm c}$ superconductors. Then, strong fluctuations
of orbital/charge degrees of freedoms may be significant for high-$T_{\rm c}$
superconductivity. Note that the CDW order below $T_{\rm CDW}\sim 200$K is
“short range” with the correlation length $\xi\ll 10$nm. It becomes the true
long-range order under high magnetic field ($\sim 20$T) [28, 29] or applying
the uniaxial strain ($|{\epsilon}_{xx}|\lesssim 0.01$) [30] below $\sim 100$K.

In this article, we review the recent progress on the study of the mechanisms
of unconventional (= non-$A_{1g}$ symmetry) order parameter and
superconductivity, mainly in Fe-based and cuprate superconductors and related
compounds. We explain that the strong coupling between spin, charge, and
orbital degrees of freedom are induced by the inter-mode coupling due to
vertex corrections (VCs), which are neglected in previous mean-field-type
approximations. The VCs are important key ingredients to understand the
nematicity, CDW and superconductivity in many strongly correlated electron
systems.

In later subsections of Sect. 1, we review the phase diagrams in Fe-based and
cuprate superconductors. In Sect. 2 - Sect. 5, we discuss the origin of
unconventional ordered states, such as the orbital order and $d$-wave bond
order, by considering the spin-fluctuation-driven VCs. In Sect. 6, we discuss
the pairing mechanism in Fe-based superconductors, by focusing on both orbital
and spin fluctuations. In Sect. 7, we review the multipole order and multipole
fluctuation pairing mechanism in $f$-electron systems with strong spin-orbit
interaction (SOI). The discussions of this article are summarized in Sect. 8.

###  1.2 Fe-based superconductors

The discovery of Fe-based high-$T_{\rm c}$ superconductors by Kamihara and
Hosono is an epoch-making event of in condensed matter physics. $T_{\rm
c}=26$K in the first discovered 1111 compound LaFeAsO [31] had immediately
increased to 56K by replacing La with other rare-earth elements [32, 33].
Also, $T_{\rm c}=30\sim 50$K in 122 compounds $A$Fe2As2 ($A$=Ba,Ca,Sr, etc)
[34, 35, 36]. In 11 compounds, $T_{\rm c}\sim 40$K is realized in Li-
intercalated FeSe [37, 38, 39, 40, 41, 42, 43], and $T_{\rm c}>60$K is
reported in single layer FeSe grown on SrTiO3 substrates [44, 45, 46, 47, 48].
In all families, the metallicity and superconductivity occur in two-
dimensional FePn (Pn=As,Se) plane, which is shown in Figs. 2 (a) and (b). In
usual compounds, the $d$-electron filling of each Fe ion is $n_{d}\sim 6$.

In this article, we set $x$ and $y$ axes parallel to the nearest Fe-Fe bonds,
and represent the $z^{2}$, $xz$, $yz$, $xy$, and $x^{2}-y^{2}$ $d$-orbitals as
1,2,3,4, and 5, respectively. The Fermi surfaces (FSs) are mainly composed of
$t_{2g}$ orbitals ($l=2-4$), although $z^{2}$ orbitals hole-pocket exists in
122 families. Since the Pn-A (Pn-B) ions form the upper- (lower-) plane, the
unit cell contains Fe-A and Fe-B. The FSs of this original ten-orbital tight-
binding model are shown in Fig. 2 (c). The orbital character is shown by green
($d_{xz}$), red ($d_{yz}$), and blue ($d_{xy}$) colors. The good nesting
between hole-pockets and electron-pockets with ${{\bm{Q}}}\sim(\pi,0),(0,\pi)$
causes the stripe AFM order. In Refs. [49, 50], the authors introduced the
“unfold-gauge transformation” $|l\rangle\rightarrow-|l\rangle$ ($l=1,4,5$)
only for Fe-B sites. Due to this gauge transformation, the unit cell is halved
to become the single-iron unit cell. The FSs of the unfolded five-orbital
tight-binding model are shown in Fig. 2 (d). This unfolded model is very
convenient for theoretical analyses.

In almost all Fe-based superconductors, the $s$-wave superconductivity is
realized. However, basic properties of the gap function, such as the momentum
and orbital dependences and the presence or absence of the sign-reversal in
the gap function, are still under debate in many compounds. To uncover the
pairing mechanism, we should first understand the basic many-body electronic
properties in the normal state.

Figure 2:  (a)(b) Crystal structure of FePn-layer (Pn=As,Se), in which the
unit cell is given by the two-iron unit cell composed of Fe-A, Fe-B, Pn-A, and
Pn-B. The single-iron unit cell is realized by applying the “unfold-gauge
transformation”. (c) Original folded and (d) unfolded FSs in 1111 and 122
families with $n_{d}=6$.

First, we explain the diverse phase diagrams of Fe-based superconductors.
Figure 3 shows the phase diagrams for (a) 1111 family [36] and (b) 122 family
[51] with $n_{d}\sim 6$. $T_{S}$ is the second-order structural transition
temperature from tetragonal state ($C_{4}$, $a=b$) to orthorhombic ($C_{2}$,
$a>b$) state. The ratio $(a-b)/(a+b)$ below $T_{S}$ is at most $0.3\%$. As
decreasing the temperature, stripe magnetic order with
${{\bm{Q}}}_{s}\approx(\pi,0)$ or $(0,\pi)$ appears at $T_{N}\ (\leq T_{S})$
in both 1111 and 122 compounds. These phase diagrams indicate the close
relationship between the structural transition and magnetism.

The origin of the orthorhombic transition is the spontaneous rotational
symmetry breaking due to electron correlation, so it is called the “electronic
nematic state”. The nematic states in Figs. 3 (a) and (b) possess $B_{1g}$
($d_{x^{2}-y^{2}}$) symmetry, where $x$ and $y$ axes are along the nearest Fe-
Fe direction shown in Fig. 2 (b). To explain ubiquitous electronic nematicity,
up to now, the spin nematic order, the orbital order, and the bond order have
been proposed as the real order parameters. The microscopic mechanism for each
order parameter has been developed intensively in the last decade. The origin
and the mechanism of nematicity is one of the main topics of this review
article.

In addition, almost hidden nematic transition with tiny nematicity
($(a-b)/(a+b)\sim 10^{-4}$) occurs at $T=T^{*}\ (>T_{S})$ in several FeAs
families. For example, it is recognized in the phase diagram of
BaFe2(As1-xPx)2 in Fig. 3 (b). At present, the origin of this “slight nematic
state” is unknown. Some extrinsic origins (such as surface induced nematicity)
have been discussed so far. As the intrinsic order, we propose the smectic
orbital order at ${{\bm{q}}}=(\pi,0)$ based on a microscopic theory, as we
will explain in Sect. 5.7. We stress that similar smectic order has been
recently observed by the Angle-resolved photoemission spectroscopy (ARPES)
studies in (Ba,K)Fe2As2 [52], CaKFe4As4 and KCa2Fe4As4F2 [53], and electron-
doped thin layer FeSe [54].

Figure 3:  Experimental phase diagrams of FeAs families with $n_{d}\sim 6$:
(a) LaFeAsO1-xHx at ambient pressure ($P=0$) and at $P=2$GPa. Reprinted from
Ref. [36]. (b) BaFe2(As1-xPx)2 [51]. The electronic nematic transition occurs
at $T=T_{S}$, below which small orthorhombic structural transition
($\delta=(a-b)/(a+b)\sim 3\times 10^{-3}$) is accompanied. In (b), hidden
nematicity with tiny otrhorhombicity ($\delta\sim 10^{-4}$) appears at
$T=T^{*}$, which will be discussed in Sect. 5. Reprinted by permission from
Springer Nature: Nature [51], copyright 2012.

Next, we discuss the FeSe family without carrier doping ($n_{d}=6$): The phase
diagram of FeSe1-xSx [55] and that of FeSe1-xTex [11] are shown in Fig. 4 (a).
In both compounds, no magnetism appears in the nematic phase below $T_{S}$,
and superconductivity emerges in the nematic phase. The symmetry of the
nematic state in FeSe1-xTex and FeSe1-xSx is $B_{1g}$ ($d_{x^{2}-y^{2}}$).
Similar phase diagram is realized in FeSe1-xSx. Above $T_{S}$, observed low-
energy spin fluctuations are very small in both S- and Te-doped FeSe. The
discovery of “nematicity without magnetism in FeSe families” gives us
significant information on the origin of nematicity. The unfolded FSs in the
tetragonal state ($T>T_{S}$) is presented in Fig. 4 (c). The $C_{4}$ symmetry
of the FS shape is spontaneous violated below $T_{S}$. Figure 4 (d) shows the
FS of detwined FeSe below $T_{S}$ observed by ARPES [56, 57]. Due to the
smallness of the Fermi energy $E_{F}$, the SC coherence length $\xi$ becomes
comparable to the Fermi momentum $k_{\rm F}$, so the BCS-BEC crossover is
expected to be realized [58].

Figure 4:  (a) Experimental phase diagram of FeSe1-xSx. Reprinted by
permission from Springer Nature: Nature [55], copyright 2019. (b) Experimental
phase diagram of FeSe1-xTex. Reprinted from Ref. [11]. (c) FS of the unfolded
FeSe model in the tetragonal state. (d) FS of the nematic state in FeSe
observed by ARPES. Reprinted with permission from [56]. Copyright (2015) by
the American Physical Society.

The pressure-temperature ($P$-$T$) phase diagram of FeSe is also remarkably
interesting: As shown in Fig. 5, the nematic order is quickly suppressed under
pressure, and instead, the magnetic order appears [9, 10]. $T_{\rm c}$
gradually increases with $P$, and the maximum $T_{\rm c}\sim 40$K is realized
at $P\sim 6$GPa. In the $P$-$T$ phase diagram of FeSe1-xSx with $x\sim 0.1$,
the nematic phase at $P\sim 0$GPa is completely separated from the pressure
induced magnetic order phase, which appears just around $P\sim 5$GPa [10]. We
will explain the $P$-$T$ phase diagram of FeSe in terms of the quantum
interference mechanism in later section.

The thin atomic-layer FeSe attracts great attention because this system
exhibits the highest $T_{\rm c}$ ($\gtrsim 70$K) among Fe-based
superconductors [44, 45, 46, 47, 48]. In mono atomic-layer FeSe, all hole-
pockets disappear because $10\sim 20$% electron carrier is naturally
introduced. In this e-doped FeSe, electronic nematic order disappears, and
instead high-$T_{\rm c}$ SC state emerges on two electron-pockets. Similar
e-doped high-$T_{\rm c}$ SC state in FeSe is also realized by
$Li$-intercalation [37, 38, 39, 40, 41, 42, 43], $K$-deposition [59, 60], and
ionic-liquid gating [61, 62]. In e-doped FeSe, the observed low-energy spin
fluctuations are tiny since the hole-pocket is absent. Thus, the Fermi pocket
nesting is inessential for realizing high-$T_{\rm c}$ state in electron-doped
FeSe.

Recently, nonmagnetic smectic order at ${{\bm{q}}}\sim(\pi/4,0)$ is discovered
in atomic layer FeSe [54], in the vicinity of the high-$T_{\rm c}$ SC phase.
This experimental finding indicates that the high-$T_{\rm c}$ SC state is
mediated by the smectic order fluctuations. In this article, we explain the
emergence of the smectic order in e-doped FeSe, and propose the mechanism of
the smectic fluctuation mediated high-$T_{\rm c}$ $s$-wave SC state.

Figure 5:  $T$-$P$ phase diagram of bulk FeSe. Reprinted from Ref. [10].

Finally, we discuss RbFe2As2 and CsFe2As2, which are heavily hole-doped
($n_{d}=5.5$) 122 compounds. Here, the symmetry of nematic state is $B_{2g}$
($d_{xy}$), which is 45 degree rotated from the $B_{1g}$ nematicity in usual
compounds. This $B_{2g}$ nematicity in these compounds has been reported by
the NMR [12] and STM [14], and the nematic susceptibility [13] measurements.
The discovered $B_{2g}$ nematicity provides a very severe constraint on the
mechanism of nematicity. In this article, we try to explain rich variety of
nematicity in Fe-based superconductors in terms of the paramagnon interference
mechanism.

In summary, the nematic states universally appear in Fe-based superconductors,
while the symmetry of nematicity and the relation between nematicity and
magnetism exhibit amazing system dependence. These facts strongly indicate
that the electronic properties, especially the SC states, are strongly
compound-dependent. On the other hand, one may expect that the rich variety of
nematicity can be understood based on the same mechanism, from the perspective
of universality. Thus, diverse nematicity in Fe-based superconductors is a
significant test for constructing the reliable theory of nematicity. The
knowledge of nematicity is significant to establish the mechanism of
superconductivity, which is another significant open problem in Fe-based
superconductors. It is widely believed that the nematic fluctuations mediate
the superconductivity, since high-$T_{\rm c}$ state is frequently realized
next to the nematic phase. The aim of the present study is to discuss the
origin of nematicity and the pairing mechanism from a unified viewpoint.

###  1.3 cuprate superconductors

The discovery of cuprate high-$T_{\rm c}$ superconductors in 1986 has
triggered significant progress in the study of strongly correlated electron
systems [63, 64, 65, 66, 67]. The superconductivity is realized by introducing
hole-carrier or electron-carrier into the parent antiferromagnetic
(${{\bm{Q}}}=(\pi,\pi)$) insulators. In the hole-doping systems, the maximum
$T_{\rm c}$ at ambient pressure is about $40$K in La-compounds (e.g.,
La2-xSrxCuO4, La2-xNdxCuO4 ), and around $100$K in Y-compounds (e.g.,
Ba2Cu3O7-x ), and Bi, Ta, and Hg-compounds. In the electron-doping compounds,
the maximum $T_{\rm c}$ is around $30$K in $R_{2-x}$CexCuO4 ($R$=Nd,Pr). In
all compounds, the metallicity and superconductivity occur in two-dimensional
CuO2 plane, which is shown in Fig. 6 (a). It is widely believed that the spin-
fluctuation-mediated $d$-wave superconductivity is realized. The FS and
$\chi^{s}({{\bm{q}}})$ in the RPA for YBCO model are shown in Fig. 6 (b) and
(c), respectively.

Figure 6:  (a) Schematic CuO2 plane of cuprate superconductors. (b) FSs of
hole-doped cuprate (YBCO) and main nesting vectors. (c) Spin susceptibility
$\chi^{s}({{\bm{q}}})$ obtained by the RPA.

In real cuprate superconductors, however, exotic orders composed of charge and
spin degrees of freedom emerge in a cooperative fashion [68]. These
unconventional density-wave states cannot be explained by simple mean-field-
type approximations. For example, in Y-based, Bi-based, and Hg-based cuprates,
the charge-density-wave (CDW) order with finite wavevector
${{\bm{Q}}}=(\pi/2,0)$ or $(0,\pi/2)$ appears inside the pseudogap region in
the presence of spin fluctuations [17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27]. Thus, spin and charge degrees of freedom are strongly coupled in
high-$T_{\rm c}$ cuprates. Then, strong fluctuations of charge degrees of
freedoms may be significant for high-$T_{\rm c}$ superconductivity.

Figure 7 shows the phase diagram of YBCO compound [69]. Above the pseudogap
temperature $T^{*}$, the antiferromagnetic fluctuations develop monotonically
as decreasing the temperature. The observed non-Fermi liquid behaviors, such
as $1/T_{1}T\propto T^{-1}$, and $\rho\propto T$, and $R_{H}\propto T^{-1}$,
are satisfactorily explained based on the spin fluctuation theories. Below
$T^{*}$, the pseudogap appears in the electron density-of-states (DOS),
$N(0)$, so the Knight shift starts to decrease. Below $T^{*}$, the increment
of spin fluctuations becomes moderate, so Fermi liquid behavior tends to be
recovered. Until now, a number of theoretical studies have been performed [70,
71, 72, 73, 74, 75, 66, 76, 77, 67]. However, the origin and nature of the
pseudogap have been significant open problems.

Figure 7:  Experimental phase diagram of cuprate high-$T_{\rm c}$
superconductor YBCO. Reprinted by permission from Springer Nature: Nat. Phys.
[69], copyright 2017.

The CDW phase around $x\sim 0.1\sim 0,15$ in Fig. 7 has been discovered in the
last decade by $X$-ray measurements [17, 18, 26, 19, 20, 21, 22] STM/STS
measurements [23, 24, 25], and NMR studies [26, 27]. The wavevector of the CDW
state is ${{\bm{q}}}\approx(0.5\pi,0),(0,0.5\pi)$, which corresponds to the
nesting vector between the adjacent hot spots; see Fig. 6. The CDW phase
emerges universally in all hole-doped compounds, and its high transition
temperature ($T_{\rm CDW}\sim 200$K) indicates that the CDW is driven by the
electron correlation. However, it is highly nontrivial why the CDW is driven
by the electron correlation. It cannot be explained in the mean-field
approximations. To explain the CDW phase, various order parameters have been
proposed, such as the pair-density-wave (PDW) [78, 79] and the bond order [80,
81, 25, 82, 83, 84, 85, 86, 87]. The latter is the modulation of the hopping
integrals given by the symmetry-breaking in the self-energy.

Recently, experimental evidences for the phase transition at $T^{*}$ have been
accumulated. The nematic transition has been observed at $T^{*}$ in Y-, Hg-,
and Bi-based compounds [69, 88, 89]. As the candidates of the pseudogap order
parameters, the uniform (${{\bm{q}}}=\bm{0}$) bond order [83, 84], charge loop
current [90], spin loop current [91], and various composite orders [92] have
been discussed intensively.

In this article, we discuss the origin of the unconventional density waves by
considering the electron correlation effects. We explain the importance of the
quantum interference as the origin of the unconventional density waves. It is
important to uncover the origin of the CDW and the pseudogap order parameter,
since the QCPs of these orders might be significant for the pairing mechanism,
as indicated by the phase diagram in Fig. 7.

###  1.4 Other exotic strongly correlated electron systems

In previous subsections, we explained the phase diagrams of two high-$T_{\rm
c}$ superconductors. Other candidates of superconductors due to electron
correlation would be Ru-based superconductor Sr2RuO4 [93], Cr-based
superconductors $A_{2}$Cr3As3 and $A$Cr3As3 ($A$=K, Rb, Cs, Na) [94], V-based
superconductors [95, 96], and organic superconductors $\kappa$-(BEDT-TTF)2X
[97, 98] and (TMTSF)2PF6. [99].

In addition, there are many exotic superconductors in $f$-electron systems.
The $d$-wave superconductivity in Ce$M$In5 ($M=$Rh, Co, Ir) has been
established by many experiments [100]. Also, spin triplet superconductivity is
expected in several U-based compounds; UPt3, UGe2, UCoGe, UTe2, etc [100].
Among these compounds, CeCu2Si2 is one of the most famous $f$-electron
superconductor, since it is the first discovered unconventional superconductor
in 1979 [101]. For long time, CeCu2Si2 is believed to be a $d$-wave
superconductor since strong AFM fluctuations are observed above $T_{\rm c}$.
However, recent experimental study has revealed the realization of the fully-
gapped $s$-wave gap state [102, 103, 104], based on the penetration depth,
impurity effect on $T_{\rm c}$, and specific heat measurements. These
surprising results mean that very strong (charge-channel) attractive pairing
interaction does exist in heavy fermion systems with large on-site $U$.

Figure 8:  Impurity effect on $T_{\rm c}$ in CeCu2Si2 measured by electron
irradiation study. Reprinted from Ref. [102].

Figure 8 shows the impurity effect on $T_{\rm c}$ in CeCu2Si2 and other
superconductors obtained by the irradiation experiments [102]. The horizontal
axis represents the normalized impurity scattering strength. The $T_{\rm c}$
of $d$-wave superconductors, YBCO and CeCoIn5, are quickly suppressed by
impurity scattering, by following the Abrikosov-Gorkov (AG) theory. This is
because the gap function with sign reversal is easily destroyed by the
impurity scattering when the mean-free-path of electrons $l_{\rm m.f.p}$ is
comparable to the coherence length. In sharp contrast, $T_{\rm c}$ in CeCu2Si2
is extremely strong against impurity scattering, much stronger than the
conventional (anisotropic) $s$-wave superconductor MgB2 and YNi2B2C. This
experiment clearly indicates that CeCu2Si2 exhibits the $s$-wave gap function
without sign reversal.

To understand the origin of $s$-wave pairing glue in CeCu2Si2, in Sect. 7, we
focus on the higher-rank multipole operators, which is active thanks to the
strong spin-orbit interaction (SOI) in heavy fermion systems. We find that
charge-channel (quadrupole or hexadecapole) multipole fluctuations are induced
by the quantum interference among magnetic fluctuations near the magnetic QCP.
In addition, the obtained multipole fluctuations are magnified with the aid of
moderate electron-phonon interaction. We reveal that the derived strong
charge-channel multipole fluctuations give sizable attractive pairing
interaction, so the $s$-wave superconductivity in CeCu2Si2 is naturally
understood in terms of the multipole fluctuations theory.

The present theory is also useful for understanding the mechanism of rich
multipolar order physics in heavy fermion systems, many of which have been
unsolved at present. We successfully applied the present theory to the
multipole order in CeB6 [105], which is a famous multipolar heavy fermion
system.

##  2 Form factors of unconventional orders

In strongly correlated metals, various symmetry breakings occur in the normal
state. In this section, we discuss the variety of symmetry breaking states and
their microscopic origin based on the recently developed theories.

###  2.1 Form factors

Here, we introduce the unconventional density orders in a square lattice
single orbital tight-binding Hubbard Hamiltonian, $H=H_{0}+H_{I}$. Here,
$H_{I}$ is the Coulomb interaction term, and $H_{0}$ is the kinetic term given
as

$\displaystyle
H_{0}=\sum_{i,j,{\sigma}}t_{i,j}c_{i{\sigma}}^{\dagger}c_{j{\sigma}}=\sum_{{{\bm{k}}},{\sigma}}{\epsilon}_{{\bm{k}}}c_{{{\bm{k}}}{\sigma}}^{\dagger}c_{{{\bm{k}}}{\sigma}}$
(3)

where $i,j$ are the site indices in real space, ${{\bm{k}}}$ is the momentum,
and ${\sigma}\ (=\pm 1)$ is the spin index. Both $t_{i,j}=t_{i-j}$ and its
Fourier transformation ${\epsilon}_{{\bm{k}}}$ belong to $A_{1g}$
representation of the square lattice.

Due to the Coulomb interaction $H_{I}$, the kinetic term is modified by the
self-energy correction $\Sigma$. In this subsection, we neglect its energy-
dependence to simplify the discussion. The self-energy without symmetry
breaking, $\Sigma_{i,j}^{0}$, belongs $A_{1g}$ representation. In the field
theory, the quantum phase transition is given by the spontaneous symmetry
breaking in the self-energy. The self-energy after the phase transition is
expressed as
$\Sigma_{i,j}^{\sigma}=\Sigma_{i,j}^{0}+\Delta\Sigma_{i,j}^{\sigma}$, where
$\Delta\Sigma_{i,j}^{\sigma}$ is the order parameter that belongs to a
non-$A_{1g}$ representation.

Its Fourier transformation is given as

$\displaystyle\Delta\Sigma_{{{\bm{q}}}}^{\sigma}({{\bm{k}}})=\sum_{i,j}\Delta\Sigma_{i,j}^{\sigma}e^{i{{\bm{k}}}\cdot({\bm{r}}_{i}-{\bm{r}}_{j})}e^{-i{{\bm{q}}}\cdot{\bm{r}}_{j}}$
(4)

where ${{\bm{q}}}$ is the wavenumber of the order parameter. Hereafter, we put
$\Delta\Sigma_{{{\bm{q}}}}^{\sigma}({{\bm{k}}})\equiv\Delta E\cdot
f_{{{\bm{q}}}}^{\sigma}({{\bm{k}}})$, where $f_{{{\bm{q}}}}^{s}({{\bm{k}}})$
is the normalized
($\frac{1}{N}\sum_{{\bm{k}}}|f_{{{\bm{q}}}}^{\sigma}({{\bm{k}}})|^{2}=1$)
dimensionless order parameter. We also introduce the charge (spin channel form
factor $f_{{{\bm{q}}}}^{c(s)}({{\bm{k}}})\equiv
f_{{{\bm{q}}}}^{\uparrow}({{\bm{k}}})+(-)f_{{{\bm{q}}}}^{\downarrow}({{\bm{k}}})$.

In the present article, we call $f_{{{\bm{q}}}}^{x}({{\bm{k}}})$ ($x=s,c$) or
its Fourier transformation $f_{i,j}^{x}$ the “form factor” of the density
wave. The form factor is a central and essential concept of the unconventional
density-wave state. We stress that the Hermite condition
$f_{i,j}^{x}=(f_{j,i}^{x})^{*}$ should be satisfied because we consider the
thermal equilibrium phase transitions.

Figure 9:  (a) Local density order (CDW or SDW) with the wavevector
${{\bm{q}}}=(\pi,pi)$. The form factor is $f=1$. (b)(c) Non-local $d$-wave
bond order with the wavevector ${{\bm{q}}}={\bm{0}}$ and
${{\bm{q}}}=(\pi/2,0)$. The form factor is $f=\cos k_{x}-\cos k_{y}$. (d) Non-
local current order with the wavevector ${{\bm{q}}}=(\pi/2,\pi/2)$. The form
factor is $f=\sin k_{x}+\sin k_{y}$.

Here, we introduce typical examples of the density-wave states, which are
actually expected to emerge in several strongly correlated metals.

(i) Conventional local order $f_{i,i}^{x}={\rm real}$: Figure 9 (a) exhibits
the conventional CDW or SDW at wavevector ${{\bm{q}}}=(\pi,\pi)$. Here,
$f_{i,i}^{x}=e^{i{\bm{r}}_{i}\cdot{{\bm{q}}}}$ for general ${{\bm{q}}}$. (Note
that the ferro-CDW is prohibited by the charge conservation law.)

(ii) Nonlocal bond order $f_{i,j}^{x}=f_{j,i}^{x}={\rm real}$: Figures 9 (b)
and (c) exhibits the $d$-wave bond orders at wavevectors ${{\bm{q}}}={\bm{0}}$
and ${{\bm{q}}}=(\pi/2,0)$, respectively. Its form factor at
${{\bm{q}}}={\bm{0}}$ is $f_{x^{2}-y^{2}}^{x}({{\bm{k}}})=\cos k_{x}-\cos
k_{y}$ in ${{\bm{k}}}$-space, and its real-space expression is
$f_{i,j}^{x}=(\delta_{i_{x},j_{x}\pm 1}\delta_{i_{y},j_{y}}-(x\rightarrow
y))$, where $(i_{x},i_{y})$ is the coordinate of site $i$. Here, the relation
$f_{i,j}^{x}=f_{j,i}^{x}$ holds. The $d$-wave bond order is observed in the
CDW phase in cuprates and in the nematic state in FeSe (within the $d_{xy}$
orbital).

(iii) Nonlocal current order $f_{i,j}^{x}=-f_{j,i}^{x}={\rm imaginary}$:
Figure 9 (d) exhibits the current order at wavevector
${{\bm{q}}}=(\pi/2,\pi/2)$. Its form factor at ${{\bm{q}}}={\bm{0}}$ is
$f_{x+y}^{x}({{\bm{k}}})=\sin k_{x}+\sin k_{y}$ in ${{\bm{k}}}$-space, and its
real-space expression is
$f_{i,j}^{x}=i(\delta_{i_{x},j_{x}+1}-\delta_{i_{x},j_{x}-1})\delta_{i_{y},j_{y}}+(x\rightarrow
y))e^{i({\bm{r}}_{i}+{\bm{r}}_{j})\cdot{{\bm{q}}}/2}$. Here, the relation
$f_{i,j}^{x}=-f_{j,i}^{x}$ holds. $f^{c(s)}$ induces the loop charge (spin)
current order, and its existence is hotly discussed in cuprates and irritates.
(Note that the macroscopic current is prohibited by the translational symmetry
even in ferro-current orders [91].

Both bond and current orders are difficult to be explained within the mean-
field-level approximation, as we will explain in Sect. 2.2. However, these
unconventional orders are caused by the non-local vertex corrections (VCs),
which is the main issue of the present article.

In Fig. 10, we summarize the classification of the non-local form factor
$f_{i,j}^{\sigma}$. Here, ${\cal P}_{\rm bond}$ is the parity (=eigenvalue
$\pm 1$) of the site exchange operator ${\hat{P}}_{\rm
bond}\\{f_{i,j}^{\sigma}\\}\equiv f_{j,i}^{\sigma}$. ${\cal P}_{\rm spin}$ is
the parity of the spin-flip operator ${\hat{P}}_{\rm
spin}\\{f_{i,j}^{\sigma}\\}\equiv f_{i,j}^{-{\sigma}}$. ${\cal T}$ is the
parity of the time-reversing operator
${\hat{T}}\\{f_{i,j}^{\sigma}\\}\equiv(f_{i,j}^{-{\sigma}})^{*}$.

Figure 10:  Classification of the non-local form factor $f_{i,j}^{\sigma}$.
${\cal P}_{\rm bond}$ is the parity of the site exchange ($i\leftrightarrow
j$), and ${\cal T}$ represents the time-reversal symmetry. The hermitian
condition of form factor is ${\cal S}\cdot{\cal T}\cdot{\cal P}_{\rm bond}=1$,
where ${\cal T}$ is the parity of spin exchange.

Finally, we introduce the orbital order in a square lattice two-orbital tight-
binding Hubbard model. Its kinetic term is

$\displaystyle
H_{0}=\sum_{{\sigma},i,j,l,m}t_{il,jm}c_{il{\sigma}}^{\dagger}c_{jm{\sigma}},$
(5)

where $l,m=1,2$ represents the $d$-orbital ($1=xz$, $2=yz$). $t_{il,jm}$ is
the hopping integral between the orbital $l$ at site $i$ and the orbital $m$
at site $j$.

Figure 11:  (a) Orbital nematic (${{\bm{q}}}={\bm{0}}$) order due to the
orbital polarization $n_{i,xz}>n_{i,yz}$. The form factor is
${\hat{f}}={\hat{\tau}}_{z}$ in the ($d_{xz}$, $d_{yz}$) orbital basis. The
corresponding energy splitting is shown. (b) Tetragonal ($C_{4}$) state and
(d) nematic ($C_{2}$) state in square lattice model with spin degrees of
freedom. $\phi_{A}$ ($\phi_{B}$) is the spin direction of sublattice A (B). In
the paramagnetic state, $\langle\phi_{A}\rangle=\langle\phi_{B}\rangle=0$. The
spin nematic ($C_{2}$) state is realized when
$\langle\phi_{A}\phi_{B}\rangle\neq 0$, depicted in (d).

In $2$-orbital Hubbard models, the form factor is expressed as the $2\times 2$
matrix form, ${\hat{f}}_{{{\bm{q}}}}^{c,s}({{\bm{k}}})$. In this subsection,
we consider a simple ${{\bm{k}}}$-independent form factor
${\hat{f}}^{c}={\hat{\tau}}_{z}$, where ${\hat{\tau}}_{\mu}$ is the Pauli
matrix in the $d$-orbital basis. The induced orbital polarization
$n_{i,xy}\neq n_{i,yz}$ at ${{\bm{q}}}={\bm{0}}$ is depicted in Fig. 11 (a).
The orbital order is realized in the nematic phase in many Fe-based
superconductors. The orbital order is “non-local” and “non-$A_{1g}$” in the
orbital space, and therefore “unconventional”. This unconventional order is
difficult to be explained within the mean-field-level approximation. However,
it is caused by the VCs, which is the main issue of the present article.

Finally, we explain the scenario of spin nematic order for the nematic
transition proposed in Fe-based superconductors [106, 107, 108, 109]. Figure
11 (b) shows the square lattice quantum spin model. When the magnetic
frustration is large, the paramagnetic state
$\langle\phi_{A}\rangle=\langle\phi_{B}\rangle=0$ is realized, where
$\phi_{A}$ ($\phi_{B}$) is the spin direction of sublattice A (B). Then,
nonmagnetic spin nematic ($C_{2}$) state can be established by emerging the
order parameter $\langle\phi_{A}-\phi_{B}\rangle\neq 0$, which is depicted in
Fig. 11 (b). (Mathematically, this is the non-local spin quadrupole order.) In
the spin nematic scenario, only the rotational symmetry breaks
$(C_{4}\rightarrow C_{2})$ at $T=T_{S}$, and the time-reversal symmetry breaks
at $T=T_{N}$ successively. In this sense, the spin nematic order is sometimes
called the “vestigial order”. The expected phase diagram with
$T_{S}>T_{N}(>0)$ would be consistent with 1111 and 122 phase diagrams in
Figs. 3 (a) and (b), while the origin of the nematicity without magnetism in
Fe(Se,Te) and Fe(Se,S), shown in Figs. 4 (a) and (b) respectively, may not be
trivial.

###  2.2 Difficulties in the mean-field approximation

Here, we discuss the microscopic mechanism of the unconventional orders
introduced in the previous subsections.

As the first step, we perform the mean-field analysis for the single orbital
Hubbard model. We introduce the on-site Coulomb interaction $U$ in addition to
the off-site Coulomb interaction $V_{i,j}$ in the interaction term:

$\displaystyle H_{I}$ $\displaystyle=$
$\displaystyle\frac{U}{2}\sum_{i}n_{i{\sigma}}n_{i-{\sigma}}+\sum_{i,\neq
j}V_{i-j}n_{i}n_{j}$ (6)

where $n_{i{\sigma}}=c_{i{\sigma}}^{\dagger}c_{i{\sigma}}$ and
$n_{i}=\sum_{\sigma}n_{i{\sigma}}$, and ${\sigma}=\pm 1$ is the spin index.
For simplicity, here we consider $V_{i-j}$ only for the nearest sites. As a
consequence of $SU(2)$ symmetry in spin space, $H_{I}$ is written as

$\displaystyle H_{I}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{{{\bm{k}}}{{\bm{k}}}^{\prime}{{\bm{q}}},{\sigma}{\sigma}^{\prime}}I_{{\sigma}{\sigma}^{\prime}}^{(0)}({{\bm{k}}},{{\bm{k}}}^{\prime},{{\bm{q}}})c_{{{\bm{k}}}+{{\bm{q}}},{\sigma}}^{\dagger}c_{{{\bm{k}}}^{\prime},{\sigma}^{\prime}}^{\dagger}c_{{{\bm{k}}}^{\prime}+{{\bm{q}}},{\sigma}^{\prime}}c_{{{\bm{k}}},{\sigma}}$
(7) $\displaystyle
I_{{\sigma}{\sigma}^{\prime}}^{(0)}({{\bm{k}}},{{\bm{k}}}^{\prime},{{\bm{q}}})$
$\displaystyle=$
$\displaystyle(I^{c(0)}({{\bm{k}}},{{\bm{k}}}^{\prime},{{\bm{q}}})+I^{s(0)}({{\bm{k}}},{{\bm{k}}}^{\prime},{{\bm{q}}}){\sigma}{\sigma}^{\prime})/2$
(8)

and $I^{c(0)}$ and $I^{s(0)}$ are explicitly given as

$\displaystyle I^{c(0)}({{\bm{k}}},{{\bm{k}}}^{\prime},{{\bm{q}}})$
$\displaystyle=$
$\displaystyle-U-2V\gamma_{{\bm{q}}}+V\gamma_{{{\bm{k}}}-{{\bm{k}}}^{\prime}}$
(9) $\displaystyle I^{s(0)}({{\bm{k}}},{{\bm{k}}}^{\prime},{{\bm{q}}})$
$\displaystyle=$ $\displaystyle U+V\gamma_{{{\bm{k}}}-{{\bm{k}}}^{\prime}}$
(10)

where $\displaystyle\gamma_{{{\bm{k}}}}=2(\cos k_{x}+\cos k_{y})$ originates
from the Fourier transformation of $V_{i,j}$. The diagrammatic expressions of
$I^{c(0)}$ and $I^{s(0)}$ are shown in Fig. 12. The following relationship is
important:
$\displaystyle\gamma_{{{\bm{k}}}-{{\bm{k}}}^{\prime}}=\sum_{\Gamma}^{x\pm
y,x^{2}\pm y^{2}}f_{\Gamma}({{\bm{k}}})f_{\Gamma}({{\bm{k}}}^{\prime})$, where
$\displaystyle f_{x^{2}\pm y^{2}}({{\bm{k}}})=\cos k_{x}\pm\cos k_{y}$ and
$\displaystyle f_{x\pm y}({{\bm{k}}})=\sin k_{x}\pm\sin k_{y}$. Note that
$f_{\Gamma}$ is normalized as
$\displaystyle\frac{1}{N}\sum_{{\bm{k}}}|f_{\Gamma}({{\bm{k}}})|^{2}=1$.

Figure 12:  Bare charge- and spin-channel interactions.
$I^{c(0)}=I^{(0)}_{{\sigma},{\sigma}}+I^{(0)}_{{\sigma},{\bar{\sigma}}}$ and
$I^{s(0)}=I^{(0)}_{{\sigma},{\sigma}}-I^{(0)}_{{\sigma},{\bar{\sigma}}}$.
Here, the Fock term $\gamma_{{\bm{k}}}=2(\cos k_{x}+\cos k_{y})$ originates
from non-local interaction $V$.

In the mean-field theory, the Stoner criterion determines the condition of the
phase transition at ${{\bm{q}}}$. The spin ($x=s$) or charge ($x=c$) channel
Stoner factor at ${{\bm{q}}}$ with respect to the form factor $f_{\Gamma}$ is
${\alpha}_{x}^{\Gamma}({{\bm{q}}})=g_{x}^{\Gamma}({{\bm{q}}})\chi^{(0)}_{\Gamma}({{\bm{q}}})$,
where

$\displaystyle g_{x}^{\Gamma}({{\bm{q}}})$ $\displaystyle\equiv$
$\displaystyle\frac{1}{N^{2}}\sum_{{{\bm{k}}},{{\bm{k}}}^{\prime}}I^{x(0)}({{\bm{k}}},{{\bm{k}}}^{\prime},{{\bm{q}}})f_{\Gamma}({{\bm{k}}}+{{\bm{q}}}/2)f_{\Gamma}({{\bm{k}}}^{\prime}+{{\bm{q}}}/2)$
(11) $\displaystyle\chi^{(0)}_{\Gamma}({{\bm{q}}})$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{{\bm{k}}}\frac{n({\epsilon}_{{{\bm{k}}}+{{\bm{q}}}})-n({\epsilon}_{{\bm{k}}})}{{\epsilon}_{{\bm{k}}}-{\epsilon}_{{{\bm{k}}}+{{\bm{q}}}}}|f_{\Gamma}({{\bm{k}}}+{{\bm{q}}}/2)|^{2}$
(12)

and we denote
${\alpha}_{x}^{\Gamma}\equiv\max_{{\bm{q}}}{\alpha}_{x}^{\Gamma}({{\bm{q}}})$.
Note that $\chi^{(0)}_{\Gamma=1}({{\bm{q}}})$ is equal to the density-of-
states (DOS) $N(0)$ at ${{\bm{q}}}={\bm{0}}$. Here,
$g_{x}^{\Gamma}({{\bm{q}}})$ for each form factor is given as

(i) Local order ($\Gamma=1$):

$\displaystyle g_{c}^{1}({{\bm{q}}})=-U-2V\gamma_{{\bm{q}}},\
g_{s}^{1}({{\bm{q}}})=U,$ (13)

(ii) $d$-wave bond order ($\Gamma=x^{2}-y^{2}$):

$\displaystyle
g_{c}^{x^{2}-y^{2}}({{\bm{q}}})=g_{s}^{x^{2}-y^{2}}({{\bm{q}}})=V,$ (14)

(iii) Current order ($\Gamma=x\pm y$):

$\displaystyle g_{c}^{x\pm y}({{\bm{q}}})=g_{s}^{x\pm y}({{\bm{q}}})=V,$ (15)

We note that only the Fock term ($=V\gamma_{{{\bm{k}}}-{{\bm{k}}}^{\prime}}$)
in Eqs. (9) and (10) contribute to the non-local orders (ii) and (iii). In
contrast, the Hartree term ($=U,2V\gamma_{{\bm{q}}}$) contribute to the local
order (i). Note that we dropped the Fock term for the local order (i) because
it is very small.

As for the local order (i), at ${{\bm{q}}}=(\pi,\pi)$, the SDW Stoner factor
${\alpha}_{S}^{1}$ is larger than the CDW Stoner factor ${\alpha}_{C}^{1}$ for
$U>4V$. Therefore, the SDW (CDW) order occurs in the case of $U>4V$ ($U<4V$)
when the interaction is strong.

As for the non-local orders (ii) and (iii), ${\alpha}_{x}^{\Gamma}$
($\Gamma=x^{2}-y^{2},x\pm y$) will exceeds SDW Stoner factor
${\alpha}_{S}^{\Gamma=1}$ under the condition $4V\gtrsim U$, because
$\chi^{(0)}_{\Gamma}({{\bm{q}}})<4\chi^{(0)}_{\Gamma=1}({{\bm{q}}})$ due to
the relation $|f_{\Gamma}({{\bm{k}}})|\leq 2$. Thus, in the present $U$-$V$
Hubbard model, $\max\\{{\alpha}_{s}^{1},{\alpha}_{C}^{1}\\}$ is larger than
${\alpha}_{s,c}^{\Gamma\neq 1}$, so SDW/CDW orders always dominates over the
bond and current orders. Therefore, additional non-local interactions, such as
the RKKY interaction, must be introduced to realize the bond/current order
[110] within the mean-field approximation. This difficulty will be resolved by
developing the many-body theory beyond the RPA, as we will explain in Sect. 3

Finally, we briefly discuss the Stoner factors in the two-orbital Hubbard
model $H=H_{0}+H_{I}$. The local multiorbital Coulomb interaction $H_{I}$ is
composed of intra- and inter-orbital repulsions, $U$ and $U^{\prime}$, in
addition to the Hund’s coupling, $J$. The relation $U=U^{\prime}+2J$ holds
with respect to $t_{2g}$ (=$xz,yz,xy$) orbitals. In the mean-field
approximation, the spin-channel Stoner factor is always larger (smaller) than
the charge channel Stoner factor when $J$ is positive (negative). [111, 112].
Therefore, under the realistic condition $J>0$, the SDW always dominates over
the orbital order within the mean-field approximation. In later sections, we
will explain that the orbital order can emerge even for $J>0$ by introducing
the VCs into the theory.

###  2.3 Spin fluctuation theory

In the previous subsection, we see that the non-local (bond or current) order
can be induced by off-site interactions. However, its realization condition is
very severe within the mean-field approximation. Hereafter, we study the non-
local orders based on the Hubbard model with local Coulomb interaction $U$
($V_{i-j}\equiv 0$ in Eq. (6)) by constructing “beyond mean-field
approximation”. In other words, we develop the mechanism of non-local orders
driven by the off-site many-body interaction described by higher-order VCs.

To prepare for that, we briefly review the random-phase-approximation (RPA)
and fluctuation-exchange (FLEX) approximation. In these approximations, the
irreducible susceptibility without VC is given as

$\displaystyle\chi^{(0)}(q)$ $\displaystyle=$
$\displaystyle-T\sum_{k}G(k+q)G(k),$ (16)

where $q=({{\bm{q}}},{\omega}_{l}=2\pi Tl)$ and
$k=({{\bm{k}}},{\epsilon}_{n}=\pi T(2n+1))$. $G$ is the electron Green
function given as

$\displaystyle G(k)$ $\displaystyle=$
$\displaystyle\frac{1}{i{\epsilon}_{n}+\mu-{\epsilon}_{{\bm{k}}}-\Sigma(k)}$
(17)

where $\Sigma$ is the self-energy. Note that Eq. (16) at ${\omega}_{l}=0$
becomes $\chi_{\Gamma=1}^{(0)}({{\bm{q}}})$ in Eq. (12) in the case of
$\Sigma=0$.

Below, we consider the on-site Coulomb Hubbard model in Eqs. (6)-(10) with
$V=0$. As we discussed in the previous subsection, the spin (charge)
susceptibility is magnified by the factor
$(1-{\alpha}_{s(c)}^{\Gamma=1}({{\bm{q}}}))^{-1}$, and therefore

$\displaystyle\chi^{s(c)}(q)=\chi^{(0)}(q)/(1-U^{s(c)}\chi^{(0)}(q))$ (18)

where $U^{s}=U$ and $U^{c}=-U$. Thus, $\chi^{s}({{\bm{Q}}},0)$ diverges when
${\alpha}_{S}$ approaches to unity. In cuprate superconductors,
$\chi^{(0)}({{\bm{Q}}},0)\sim N(0)\sim 1{\rm[eV^{-1}]}$, whereas
$\chi^{s}({{\bm{Q}}},0)$ exceeds $100{\rm[eV^{-1}]}$ in optimally or slightly
under-doped cuprates above $T_{\rm c}$ [113]. This fact indicates that
$(1-{\alpha}_{S})^{-1}\gg 100$ in cuprate superconductors.

The self-energy due to the spin fluctuations is given as

$\displaystyle\Sigma(k)$ $\displaystyle=$ $\displaystyle T\sum_{q}V^{\rm
FLEX}(q)G(k+q),$ (19) $\displaystyle V^{\Sigma}(q)$ $\displaystyle=$
$\displaystyle\frac{3}{2}(U^{s})^{2}\chi^{s}(q)+\frac{1}{2}(U^{c})^{2}\chi^{c}(q)-(U^{s})^{2}\chi^{(0)}(q)$
(20)

which becomes important near the magnetic crucial point;
${\alpha}_{S}^{\Gamma=1}\lesssim 1$. In the FLEX approximation, both the
susceptibilities and the self-energy are calculated self-consistently.

For convenience, we introduce a phenomenological form of spin susceptibility
[114, 115, 116, 117, 118]:

$\displaystyle\chi^{s}({{\bm{q}}},{\omega}+i0)=\sum_{{{\bm{Q}}}}\frac{\chi_{Q}}{1+\xi_{s}^{2}({{\bm{q}}}-{{\bm{Q}}})^{2}-i{\omega}/{\omega}_{s}},$
(21)

where ${{\bm{Q}}}$ is the AF wavevector and $\xi_{s}$ is the AF correlation
length. In cuprates for $T>T^{*}$, both $\chi_{Q}$ and $1/{\omega}_{s}$ are
scaled by $\xi_{s}^{2}$ as follows [119]:

$\displaystyle\xi_{s}^{2}\approx c_{0}/(T-\theta_{s}),$ (22)
$\displaystyle\chi_{Q}\approx c_{1}\cdot\xi_{s}^{2},\ \ \
1/{\omega}_{s}\approx c_{2}\cdot\xi_{s}^{2},$ (23)

where $\theta_{S}$, $c_{0}$, $c_{1}$ and $c_{2}$ are constants. Since
$\chi_{Q}{\omega}_{s}\propto\xi_{s}^{0}$ in eq. (23), the dynamical exponent
is $z=2$. The relationship ${\omega}_{s}\gtrsim T$ is satisfied in optimally-
doped cuprates.

The relationships in Eqs. (22) and (23) are explained by the self-consistent-
renormalization (SCR) theory, in which the “mode-mode coupling effect” is
calculated self-consistently[120]. Relationships (22) and (23) are also
satisfied by the FLEX approximation [121, 122, 111, 123, 124, 125, 126]. In
both theories, long-range magnetic order does not occur in pure 2D systems,
because both theories satisfy the Mermin-Wagner theorem [127].

We stress that the charge susceptibility in Eq. (18) remains small. Therefore,
theoretical methods beyond the FLEX approximation should be established to
explain the strong charge-channel fluctuations observed in Fe-based and
cuprate superconductors.

###  2.4 Vertex corrections for susceptibilities: Self-consisting vertex
correction (SC-VC) theory

Now, we analyze SDW and CDW susceptibilities with constant form factor
($f_{\Gamma=1}=1$) by including the VCs:

$\displaystyle\chi^{x}(q)$ $\displaystyle=$
$\displaystyle(1-U^{x}\Phi^{x}(q))^{-1}\Phi^{x}(q),$ (24)
$\displaystyle\Phi^{x}(q)$ $\displaystyle=$
$\displaystyle\chi^{(0)}(q)+X^{x}(q),$ (25) $\displaystyle X^{x}(q)$
$\displaystyle=$ $\displaystyle
T^{2}\sum_{kk^{\prime}}G(k+q)G(k)\Gamma^{x}_{U-{\rm
irr}}(k,k^{\prime};q)G(k^{\prime}+q)G(k^{\prime})$ (26)

where $x=c,s$. Here, $\Phi^{x}(q)$ and $\Gamma^{x}_{U-{\rm
irr}}(k,k^{\prime};q)$ are irreducible with respect to single $U^{x}$.
$\Gamma^{x}$ is expressed by using the irreducible vertex with respect to the
particle-hole (p-h) channel $I^{x}$ by using the following Bethe-Salpeter
equation.

$\displaystyle\Gamma^{x}(k,k^{\prime};q)=I^{x}(k,p;q)-T\sum_{p}I^{x}(k,p;q)G(p+q)G(p)\Gamma^{x}(p,k^{\prime};q)$
(27)

Note that $\Gamma^{x}_{U-{\rm irr}}$ is given by Eq. (27) by replacing $I^{x}$
with $I^{x}-U^{x}$

To avoid unphysical results, it is important to satisfy the following Ward
identity:

$\displaystyle\Sigma_{{\sigma}}(x_{i},x_{j})$ $\displaystyle=$
$\displaystyle\frac{\delta\Phi_{\rm LW}}{\delta G(x_{i},x_{j})}$ (28)
$\displaystyle
I_{{\sigma},{\sigma};{\sigma}^{\prime}{\sigma}^{\prime}}(x_{i},x_{j};x_{i}^{\prime},x_{j}^{\prime})$
$\displaystyle=$ $\displaystyle-\frac{\delta^{2}\Phi_{\rm LW}}{\delta
G(x_{i},x_{j})\delta G(x_{i}^{\prime},x_{j}^{\prime})}$ (29)

where $x_{i}=({\bm{r}}_{i},{\epsilon}_{n})$, and $\Phi_{\rm LW}$ is the
Luttinger-Ward function. In the ${{\bm{k}}}$-representation, the Luttinger-
Ward function in the FLEX scheme is given as

$\displaystyle\Phi_{\rm LW}={\rm Tr}[\frac{3}{2}{\rm
ln}(1-U\chi^{0}(q))+\frac{1}{2}{\rm
ln}(1+U\chi^{0}(q))+\frac{U^{2}}{2}\\{\chi^{0}(q)\\}^{2}+U\chi^{0}(q)]$ (30)

Then, $\Gamma_{{\sigma},{\sigma};{\sigma}^{\prime}{\sigma}^{\prime}}^{I}$ is
given by three terms: one Maki-Thompson (MT) and two Aslamazov-Larkin (AL)
terms.

$\displaystyle I^{x}$ $\displaystyle=$ $\displaystyle I^{x,{\rm MT}}+I^{x,{\rm
AL1}}+I^{x,{\rm AL2}}$ (31)

which are expressed in Fig. 13, and their analytic expressions are given in
Ref. [91].

Figure 13:  Hartree-Fock (HF) term, MT term, and two AL terms included in the
irreducible four-point vertex $I$. Only the HF term is included in the RPA.
The nonlocal interaction by MT and AL terms cause exotic bond and current
orders [128, 129, 91, 130].

For general wavevector ${{\bm{q}}}$ of the density-wave, the kernel in Eq.
(31) in the momentum representation is given as

$\displaystyle I^{c,{\rm
MT}}({{\bm{q}}};k,k^{\prime})=-\left(\frac{3}{2}V_{0}^{s}(k-k^{\prime})+\frac{1}{2}V_{0}^{c}(k-k^{\prime})\right)$
(32) $\displaystyle I^{c,{\rm
AL}}({{\bm{q}}};k,k^{\prime})=T\sum_{p}\left(\frac{3}{2}V_{0}^{s}(p+{{{\bm{q}}}})V_{0}^{s}(p)+\frac{1}{2}V_{0}^{c}(p+{{{\bm{q}}}})V_{0}^{c}(p)\right)G(k-p)[G(k^{\prime}-p)+G(k^{\prime}-p+{{\bm{q}}})]$
(33)

where double-counting terms in the $U^{2}$-order should be subtracted. Near
the magnetic QCP, the relationship
$\Phi^{c}({{\bm{q}}})\gg\chi^{(0)}({{\bm{q}}})$ is realized for
$|{{\bm{q}}}|\ll 1$ since $I^{c,{\rm AL1,2}}$ is proportional to the
convolution,
$C_{q}\equiv\sum_{{\bm{k}}}\chi^{s}({{\bm{k}}}+{{\bm{q}}})\chi^{s}({{\bm{k}}})\propto\xi_{s}^{2}$.
(In contrast, $\Phi^{s}({{\bm{q}}})\sim\chi^{(0)}({{\bm{q}}})$ even for
$\xi_{s}\gg 1$ [131].) In a single orbital Hubbard model, the charge
susceptibility
$\chi^{c}({{\bm{q}}})=(1+U\Phi^{c}({{\bm{q}}}))^{-1}\Phi^{c}({{\bm{q}}})$ is
always smaller than $U^{-1}$ due to the suppression by the Hartree term. In
multiorbital Hubbard models, in contrast, the orbital order is realized when
$\Phi^{c}({{\bm{q}}})\gg U^{-1}$, as we will explain in the next section.

Hereafter, we calculate the susceptibilities on the basis of Eqs. (24)-(26)
self-consistently by setting $\Gamma^{c}_{U-{\rm irr}}=I^{c,{\rm
MT}}+I^{c,{\rm AL}}$ for simplicity. We call this theoretical framework the
self-consistent vertex correction (SC-VC) method [129]. The nematic order in
Fe-based superconductors is satisfactorily derived from the SC-VC method.

##  3 Why vertex corrections cause unconventional orders?

###  3.1 Orbital order due to vertex corrections

In order to understand why exotic density waves are caused by the VCs, we
first explain the orbital order in a simple two-orbital Hubbard model. We
demonstrate that $\Phi^{c}({\bm{0}})\gg\chi^{(0)}({\bm{0}})$ due to AL terms
give orbital order in multiorbital Hubbard models with $U$. Here, we consider
the following simple two-orbital model $H=H_{0}+H_{I}$:

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\sum_{{{\bm{k}}};{\sigma}=\uparrow,\downarrow;l,m=1,2}\xi_{{\bm{k}}}^{lm}c_{{{\bm{k}}}l{\sigma}}^{\dagger}c_{{{\bm{k}}}m{\sigma}},$
(34) $\displaystyle H_{I}$ $\displaystyle=$ $\displaystyle
U\sum_{i,l}n_{il\uparrow}n_{il\downarrow}+U^{\prime}\sum_{i,l<m}n_{il}n_{im}$
(35)
$\displaystyle+J\sum_{i,l<m,{\sigma}{\sigma}^{\prime}}c_{im{\sigma}}^{\dagger}c_{il{\sigma}^{\prime}}^{\dagger}c_{im{\sigma}^{\prime}}c_{il{\sigma}}+J^{\prime}\sum_{i,l\neq
m}c_{il\uparrow}^{\dagger}c_{il\downarrow}^{\dagger}c_{im\downarrow}c_{im\uparrow}$

where $n_{il{\sigma}}=c_{il{\sigma}}^{\dagger}c_{il{\sigma}}$ and
$n_{il}=\sum_{\sigma}n_{il{\sigma}}$, and $l,m=1,2$ represents the
$d$-orbital; $1=xz$ and $2=yz$. This model describes the ${\alpha}$-FS and
${\beta}$-FS of Ru-oxides, and it has been used in the study of anomalous and
spin Hall effects [132]. Here, we introduce the nearest intra-orbital ($t$)
and the next-nearest inter-orbital ($t^{\prime}$) hopping integrals;
$\xi_{{\bm{k}}}^{11}=-2t\cos k_{x}$, $\xi_{{\bm{k}}}^{22}=-2t\cos k_{y}$, and
$\xi_{{\bm{k}}}^{12}=4t^{\prime}\sin k_{x}\sin k_{y}$. The bandstructure and
the FSs for the electron filling $n=3.3$ are shown in Figs. 14 (a) and (b),
respectively. $H_{I}$ represents the multiorbital Coulomb interaction composed
of intra (inter) orbital interaction $U$ ($U^{\prime}$) and the exchange
interaction $J$.

Figure 14:  (a) Band structure of the two-orbital model. The low-energy
excitations of electrons
($|\xi_{\bm{k},\nu}^{\mathrm{linear}}|\leq\Lambda_{0}$) are denoted by the
shaded area. (b) FSs of the two-orbital model for $n=2.7$. The colors
correspond to $1=xz$ (red) and $2=yz$ (blue). The arrow represents the major
nesting vector. The patch index ($1\sim 24$) on the FSs is shown. (c) AL term
$X^{c}_{11,11}({{\bm{q}}})$ and (d) orbital susceptibility
$\chi^{c}_{x^{2}-y^{2}}({{\bm{q}}})$ at $n=3.3$ obtained by the SC-VC method.
Strong ferro-orbital susceptibility is induced by large
$X^{c}_{11,11}({\bm{0}})$. Cited from Ref. [133].

Hereafter, we calculate the susceptibilities by using the RPA. The local
quadruple (order) order parameters in the present model is [112, 133]:

$\displaystyle{\hat{O}}_{x^{2}-y^{2}}^{j}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\sum_{\sigma}(c_{j,1,\sigma}^{\dagger}c_{j,1,\sigma}-c_{j,2,\sigma}^{\dagger}c_{j,2,\sigma})=n_{j,1}-n_{j,2},\,\,\,$
(36) $\displaystyle{\hat{O}}_{xy}^{j}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\sum_{\sigma}(c_{j,1,\sigma}^{\dagger}c_{j,2,\sigma}+c_{j,2,\sigma}^{\dagger}c_{j,1,\sigma}),$
(37)

where $j$ is the site index. The quadrupole susceptibility per spin is given
by ($\gamma=x^{2}-y^{2}$ or $xy$)
$\chi_{\gamma}^{c}(\bm{q})=\frac{1}{2}\int_{0}^{\beta}d\tau\left\langle
T_{\tau}\,{\hat{O}}_{\gamma}(\bm{q},\tau)\,{\hat{O}}_{\gamma}(-\bm{q},0)\right\rangle$,
where $\tau$ is the imaginary time and $\beta=1/(k_{B}T)$. The divergence of
$\chi^{c}_{x^{2}-y^{2}}(\bm{q}=\bm{0})$ reflects the emergence of the orbital
nematic state ($\langle n_{xz}\rangle\neq\langle n_{yz}\rangle$). In addition,
we analyze the charge susceptibility
$\chi^{\mathrm{c}}_{\Gamma=1}(\bm{q})=\frac{1}{2}\int_{0}^{\beta}d\tau\langle
T_{\tau}n(\bm{q},\tau)n(-\bm{q},0)\rangle$ where $n(\bm{q},\tau)$ is the
charge density operator.

In this model, the $2\times 2$ matrix Green function in the $d$-orbital basis
is

$\displaystyle{\hat{G}}(k)$ $\displaystyle=$
$\displaystyle((i{\epsilon}_{n}+\mu){\hat{1}}-\xi_{{\bm{k}}}-\hat{\Sigma})^{-1}$
(38)

which is approximately diagonal when $|t^{\prime}/t|\ll 1$. Then, the
irreducible susceptibility is also diagonal in the orbital basis
approximately:

$\displaystyle{\hat{\Phi}}^{x}(q)$ $\displaystyle\approx$
$\displaystyle\left(\begin{array}[]{cc}\Phi_{1}^{x}(q)&0\\\
0&\Phi_{2}^{x}(q)\end{array}\right)$ (41)

Here, we consider a simplified Coulomb interaction with $J=0$ and $U\neq
U^{\prime}$ in Eq. (35). According to Fig. 12, the spin-channel and charge-
channel Hartree terms are given as ${\hat{U}}^{s}=U{\hat{\tau}_{0}}$ and
${\hat{U}}^{c}=-U{\hat{\tau}_{0}}+2U^{\prime}{\hat{\tau}}_{x}$, respectively.
By following Eq. (24), we obtain
${\hat{\chi}}^{x}(q)=(1-{\hat{U}}^{x}{\hat{\Phi}}^{x}(q))^{-1}{\hat{\Phi}}^{x}(q)$.
for $x=s,c$.

Then, the SDW susceptibility at $q=0$ is
$\chi^{s}(0)=\Phi^{s}_{1}(1-{\alpha}_{S})^{-1}$, where
${\alpha}_{S}=U\Phi^{s}_{1}$. Also, the orbital susceptibility with respect to
${\hat{O}}_{x^{2}-y^{2}}$ at $q=0$ is
$\chi_{x^{2}-y^{2}}^{c}(0)=\Phi^{c}_{1}(1-{\alpha}_{C})^{-1}$, where
${\alpha}_{C}=(2U^{\prime}-U)\Phi^{c}_{1}$. In the RPA,
$\Phi^{c}_{1}=\chi^{(0)}(0)$, and therefore ${\alpha}_{S}>{\alpha}_{C}$ for
$U>U^{\prime}$. Since the relation $U>U^{\prime}$ holds in transition metals,
the SDW instability dominates over the orbital order instability within the
RPA. However, when $\Phi^{c}_{1}$ is larger than $\Phi^{s}_{1}$ owing to the
VCs, orbital order can emerge in the paramagnetic phase. This novel orbital
order mechanism is important for the nematic state of Fe-based
superconductors. The divergence of $\chi_{x^{2}-y^{2}}^{c}$ immediately leads
to the ferro-quadrupole order $n_{1}\neq n_{2}$, resulting in the “nematic”
deformation of the FSs.

Finally, we perform the numerical study for the two-orbital model. Here, we
set $J/U=0.1$ under the relation $U=U^{\prime}+2J$ [133]. In the RPA,
$\chi^{s}({{\bm{q}}})$ have a peak at the nesting vector
${{\bm{q}}}\sim(0.3\pi,0.3\pi)$. In contrast,
$\chi^{c}_{x^{2}-y^{2}}({{\bm{q}}})\sim O(1)$ in energy unit $|t|$, so the
orbital order is not realized in the RPA [133]. Now, we calculate the VCs on
the charge-channel susceptibilities based on the SC-VC theory, which was
introduced in Sect. 2.4. Figure 14 (c) exhibits the charge-channel AL-type VC
for the irreducible susceptibility, $X^{c}_{1}(q)$, on orbital 1. (Note that
$X^{c}_{2}(q_{x},q_{y},{\omega}_{l})=X^{c}_{1}(q_{y},q_{y},{\omega}_{l})$.) We
find that $X_{1}^{c}(q)\sim 0.5$ is comparable to the DOS per orbital $N(0)$,
while $X_{1}^{s}(q)$ exhibits small negative value [133]. In this case, the
relation $\Phi^{c}_{1}\sim 2\Phi^{s}_{1}$ holds, and therefore the orbital
order appears in the paramagnetic state. Figure 14 (d) shows the obtained
$\chi^{c}_{x^{2}-y^{2}}({{\bm{q}}})$, derived from the self-consistent
calculation of $\chi^{s,c}(q)$ and $X^{s,c}(q)$ [133]. In this numerical
study, the nematic orbital order is realized when ${\alpha}_{S}\sim 0.9$. To
summarize, the interference between spin fluctuations, which is described by
the AL diagram, gives rise to the ferro-orbital order.

###  3.2 Orbital order owing to VCs: fRG theory

We have shown that the spin-fluctuation-driven VCs induces the orbital order
in a two-orbital Hubbard model with on-site Coulomb interaction. However,
limited numbers of VCs were considered in the previous subsection. Thus, the
importance of VCs should be clarified by other unbiased theoretical
techniques. For this purpose, the functional-renormalization-group (fRG)
theory is quite suitable because the RG method enable us to perform the
systematic calculations of VCs. The energy band structure and the FSs obtained
from $H_{0}$ [Eq. (34)] are shown in Figs. 14 (a) and (b). The $\alpha$ band
forms a hole-like FS centered at ${{\bm{k}}}=(\pi,\pi)$ while the $\beta$ band
forms an electron-like FS centered at ${{\bm{k}}}=(0,0)$.

We apply the one-loop RG method, by which the VCs given by the parquet
equations are calculated. The RG equations are shown in Fig. 15, where
$\chi(\bm{q})$, $R(\bm{q};k_{1},k_{2})$, and $\Gamma(k_{1},k_{2};k_{3},k_{4})$
are the susceptibility, the three-point and four-point vertices, respectively
[134]. The scattering processes of electrons having energies less than a
cutoff $\Lambda_{0}$ are integrated within the RG scheme. Here, the band
dispersion near the Fermi level is divided into $N_{p}\ (=24)$ patches, as
shown in Fig. 14 (b). Here, the vertex functions $R$ and $\Gamma$ are
functions of the patch indices $k_{i}$, and their frequency dependences are
ignored [135, 136, 81]. In contrast to conventional patch fRG [137], in Ref.
[138], we introduced the initial cutoff $\Lambda_{0}$ ($T\ll\Lambda_{0}\gg
W_{\rm band}$) shown in Fig. 14 (a): The VCs due to lower energy contributions
($<\Lambda_{0}$) are calculated accurately by solving the RG equation, while
higher-energy contributions ($>\Lambda_{0}$) are calculated within the RPA
since the VCs are less important [139].

Figure 15:  Schematic explanation for the RG+cRPA method [138]. The higher-
energy process ($|\xi_{{{\bm{k}}},\nu}|>\Lambda_{0}$) is calculated by the
constrained RPA. Next, we calculate the lower-energy process
($|\xi_{{{\bm{k}}},\nu}|<\Lambda_{0}$) using the fRG method. The slashed
(crossed) line represents on-shell Green function with
$\Lambda_{l+dl}<|\xi_{{{\bm{k}}},\nu}|<\Lambda_{l}$
($\xi_{{{\bm{k}}},\nu}|<\Lambda_{l}$), where $\Lambda_{l}=\Lambda_{0}e^{-}l$.
The RG equations for four-point vertex, three-point vertex, and susceptibility
are shown.

In a conventional patch RG scheme, the higher-energy contributions are treated
less accurately because of the projection of momenta on the FS. In the RG+cRPA
method [138], in contrast, the higher-energy contributions are accurately
calculated numerically by using fine $\bm{k}$ meshes. This “RG+cRPA method” is
a natural combination of the merits of the RG (for lower energy) and RPA (for
higher energy), and enables us to obtain accurate results. In the RG+cRPA
theory, the initial values of $\Gamma$, $R$, and $\chi$ in the RG equations
are given by the “constrained” RPA (cRPA), shown in Fig. 15. The cRPA
susceptibility $\chi_{\mathrm{cRPA}}(\bm{q})$ is given by omitting the low-
energy particle-hole (p-h) contributions ($|E|<\Lambda_{0}$) from the RPA
susceptibility. Then, the contribution from the RG and that from the cRPA are
not over-counted. We have verified that the numerical results depend on
$\Lambda_{0}$ only weakly.

Figure 16:  (a) Temperature dependences of $\chi^{c}_{x^{2}-y^{2}}(\bm{0})$
and $\chi^{c}_{xy}(\bm{0})$ for $n=3.3$, $U=2.13$, and $U^{\prime}/U=0.9$. The
solid (dashed) lines represent the RG (RPA) results. In the inset, the same
data of $\chi^{\mathrm{s}}(\bm{Q})$ and $\chi^{c}_{x^{2}-y^{2}}(\bm{0})$ are
plotted on a different vertical scale. (b) Momentum dependences of
$\chi^{c}_{x^{2}-y^{2}}(q)$ for $T=0.06$. The solid (dashed) lines represent
the RG (RPA) results. Cited from Ref. [138].

As we already explained, only $\chi^{\mathrm{s}}(\bm{q})$ is strongly enhanced
for $U>U^{\prime}$ in the RPA without VCs [111, 133]. That is, orbital
fluctuations remain small in the RPA.

Next, we perform the RG+cRPA study [138]. The obtained $T$ dependences of the
spin and quadrupole susceptibilities are shown in Fig. 16 (a). In the high
temperature ($T\gtrsim 0.3$) region, all the susceptibilities exhibit similar
behavior to the RPA results [139]. Even at low temperatures,
$\chi^{c}_{xy}(\bm{0})$ show the same $T$ dependences as in RPA. The effect of
VCs suppresses $\chi^{\mathrm{s}}(\bm{Q})$ at low temperatures. The most
striking feature of Figs. 16 (a) and (b) are the critical enhancement of
$\chi_{x^{2}-y^{2}}^{c}({{\bm{q}}})$ around ${{\bm{q}}}\approx{\bm{0}}$ at low
temperatures, which cannot be derived from RPA. Thus, spin-fluctuation-driven
ferro-orbital fluctuations due to the AL process shown in the previous
subsection is well confirmed by the present fRG study. Thus, the importance of
the VCs on the charge-channel fluctuations has been confirmed by the fRG
study. The present mechanisms of the orbital nematic phase would be realized
in various multiorbital systems.

###  3.3 Density-wave equation: Optimization of ${{\bm{k}}}$-dependent form
factors

In previous subsections, we studied the mechanism of the orbital order in
multiorbital Hubbard models, by assuming that orbital order parameters are
local. However, as we discussed in Sect. 2, nonlocal p-h interaction (due to
VCs) can induce various non-local order parameters described by the form
factor, such as the bond order and current order. In order to study symmetry
breaking phenomena with non-local order parameters, we have to develop the
theory of unconventional density waves with nontrivial form factors. For this
purpose, we introduce generalized density-wave (DW) states for both charge-
and spin-channels. They are expressed in real space as follows [110]:

$\displaystyle D_{i,j}^{{\sigma},\rho}$ $\displaystyle\equiv$
$\displaystyle(1-{\cal P}_{\rm A_{1g}})\langle
c_{i{\sigma}}^{\dagger}c_{j\rho}\rangle$ (42) $\displaystyle=$ $\displaystyle
d^{c}_{i,j}\delta_{{\sigma},\rho}+\bm{d}^{s}_{i,j}\cdot\bm{{\sigma}}_{{\sigma},\rho},$

where $D_{i,j}^{{\sigma},\rho}=\\{D_{j,i}^{\rho,{\sigma}}\\}^{*}$, and
$d^{c}_{i,j}$ (${\bm{d}}^{s}_{i,j}$) is the charge (spin) channel order
parameter. ${\cal P}_{\rm A_{1g}}$ is the projection operator on the $A_{1g}$
symmetry space. General DW states with various symmetries (such as SDW, CDW,
orbital order, bond order, current order) can be expressed by Eq. (42).

The DW formation given by Eq. (42) directly drives the “symmetry breaking in
the self-energy”:
$\Sigma_{0}(i-j)\rightarrow\Sigma_{0}(i-j)+\Delta\Sigma(i,j)$. Here,
$\Sigma_{0}(i-j)$ is the without symmetry breaking, and it possesses $A_{1g}$
symmetry. $\Delta\Sigma(i,j)$ is the symmetry-breaking component, which breaks
the $A_{1g}$ symmetry. It is expressed as

$\displaystyle\Delta\Sigma_{i,j}^{{\sigma}\rho}$ $\displaystyle=$
$\displaystyle
f_{i,j}\delta_{{\sigma},\rho}+\bm{g}_{i,j}\cdot\bm{{\sigma}}_{{\sigma},\rho}$
(43)

which we call the form factors in this article. For example, the bond order is
given by real even-parity function $f_{i,j}=f_{j,i}$, and the spin current
order is given by pure imaginary odd-parity vector
${\bm{g}}_{i,j}=-{\bm{g}}_{j,i}$, as we will discuss below. After the Fourier
transformation in Eq. (4),

$\displaystyle\Delta\Sigma_{{\bm{q}}}^{{\sigma}\rho}({{\bm{k}}})$
$\displaystyle=$ $\displaystyle
f_{{\bm{q}}}({{\bm{k}}})\delta_{{\sigma},\rho}+\bm{g}_{{{\bm{q}}}}({{\bm{k}}})\cdot\bm{{\sigma}}_{{\sigma},\rho}$
(44)

which we call the form factors in this paper. Below, we assume
$\bm{g}_{{{\bm{q}}}}({{\bm{k}}})=g_{{{\bm{q}}}}({{\bm{k}}}){\bm{e}}_{z}$
without losing generality. The DW is interpreted as the electron-hole pairing
condensation [110].

Now, we consider the symmetry breaking in the self-energy by following Ref.
[140]. The self-energy is functional of the Green function:
$\Sigma^{[G]}_{i,j}$. It is given as $\Sigma^{[G]}_{i,j}=\delta\Phi[G]/\delta
G_{i,j}$, where $\Phi[G]$ is Luttinger Ward function composed of $G$ and $U$.
Here, we denote the self-energy and Green function without symmetry breaking
as $\Sigma^{0}=\Sigma^{[G^{0}]}$ and $G^{0}=((G^{\rm
fr})^{-1}-\Sigma^{0})^{-1}$. Then, the symmetry breaking in the self-energy is
given as

$\displaystyle\Delta\Sigma_{i,j}$ $\displaystyle=$ $\displaystyle(1-{\cal
P}_{\rm A_{1g}})\Sigma^{[G]}_{i,j}$ (45)

where $i=({\bm{r}},{\epsilon}_{n},{\sigma})$ and
$G=((G^{0})^{-1}-\Delta\Sigma)^{-1}$. Here, $\Sigma^{[G]}$ in the right-hand-
side is composed of the full Green function with $\Delta\Sigma\neq 0$:
$G=((G^{0})^{-1}-\Delta\Sigma)^{-1}$. Based on this “full DW equation” in Eq.
(45), we can obtain the self-consistent solution of $\Delta\Sigma$.

Next, we expand the right-hand-side of Eq. (45) with respect to $\Delta\Sigma$
by using the Dyson equation $G=G^{0}+G^{0}\Delta\Sigma G$. The obtained
linearized self-consistent equation is

$\displaystyle\lambda\Delta\Sigma_{i,j}$ $\displaystyle=$ $\displaystyle
T\sum_{l,m}I(i,j;l,m)[G^{0}\cdot\Delta\Sigma\cdot G^{0}]_{l,m}$ (46)

where $\lambda$ is the eigenvalue and $\Delta\Sigma$ is the eigenfunction.
$I(i,j;l,m)=\delta\Sigma_{i,j}^{[G^{0}]}/\delta G^{0}_{l,m}$ is the four-point
vertex which is irreducible with respect to the p-h channel.

After the Fourier transformation of Eq. (46), we obtain the “linearized DW
equation” for both spin/charge channels [83]:

$\displaystyle\\!\\!\\!\\!\lambda_{{{\bm{q}}}}f_{{\bm{q}}}(k)=-\frac{T}{N}\sum_{p}I_{{\bm{q}}}^{c}(k,p)G(p)G(p+{{\bm{q}}})f_{{\bm{q}}}(p),$
(47)
$\displaystyle\\!\\!\\!\\!\eta_{{{\bm{q}}}}g_{{\bm{q}}}(k)=-\frac{T}{N}\sum_{p}I_{{\bm{q}}}^{s}(k,p)G(p)G(p+{{\bm{q}}})g_{{\bm{q}}}(p),$
(48)

where $\lambda_{{{\bm{q}}}}$ ($\eta_{{\bm{q}}}$) is the eigenvalue that
represents the charge (spin) channel DW instability,
$k\equiv({{\bm{k}}},{\epsilon}_{n})$, $p\equiv({{\bm{p}}},{\epsilon}_{m})$,
and ${\epsilon}_{n}$, ${\epsilon}_{m}$ are fermion Matsubara frequencies. The
charge (spin) channel kernel function is
$I_{{\bm{q}}}^{c(s)}=I_{{\bm{q}}}^{\uparrow,\uparrow}+(-)I_{{\bm{q}}}^{\uparrow,\downarrow}$.
These DW equations are interpreted as the “spin/charge channel electron-hole
pairing equations”. This DW equations is easily extended to the multiorbital
Hubbard models.

In the RPA, $I_{q}$ is given by the first HF term in Fig. 13:
$I_{{\bm{q}}}^{c}=-U$ and $I_{{\bm{q}}}^{s}=U$. Then, the form factors
$f_{{\bm{q}}}$ and $g_{{\bm{q}}}$ become constants, which represents
conventional CDW and SDW order parameters. Their eigenvalues are
$\lambda_{q}=-U\chi^{(0)}(q)$ and $\eta_{q}=U\chi^{(0)}(q)$, respectively.

However, $I_{{\bm{q}}}^{c}(k,p)$ by MT and AL terms possesses
$k,p$-dependence. In Sect. 5 and Sect. 4, we discuss that various non-local DW
orders in strongly correlated electron systems can be understood using the
irreducible four-point vertices in Eq. (31). They are composed of one single-
magnon exchange (MT) term and two double-magnon exchange (AL) terms: The
significance of these VCs have been revealed by the functional-
renormalization-group (fRG) study, in which higher-order VCs are produced in
an unbiased way [84, 138, 141, 142]. The AL terms frequently cause the
${{\bm{q}}}={\bm{0}}$ CDW instability since its functional form
$\propto\sum_{{\bm{k}}}\chi^{s}({{\bm{k}}}+{{\bm{q}}})\chi^{s}({{\bm{k}}})$ is
large for ${{\bm{q}}}\approx{\bm{0}}$ in two-dimensional systems [129].

In principle, the DW order parameter is given as the “symmetry breaking in the
self-energy $\Delta\Sigma$”, similarly to the superconductivity given as the
symmetry-breaking in the anomalous self-energy. The solution $\Delta\Sigma$
obtained by the DW equation naturally explains the nematic symmetry breaking
in Fe-based [143] and cuprate [83] superconductors. By solving the linearized
equation, the higher-order diagrams with respect to these terms are generated.
Examples of generated higher-order AL+MT processes are given in Fig. 17 (b).

Figure 17:  (a) Schematic linearized DW equation for general wavenumber
${{\bm{q}}}$ [83, 91]. $f_{{\bm{q}}}({{\bm{k}}})$ is the charge-channel form
factor. The irreducible vertex $I^{c}$ includes the HF, MT, AL1, AL2 terms
shown in Fig. 13. (b) Full vertex $\Gamma^{c}$ composed of infinite series of
$I^{c}$’s. ($\Gamma^{c}$ is the solution of the Bethe-Salpeter equation (27).)
$\Gamma^{c}$ is also generated by solving the DW equation.

Here, we stressed the importance of the AL terms for various DW states. In
contrast, the MT term is significant for the non-Fermi liquid transport
phenomena [144] and the charge current order [130]. The MT term is also
important in the SC gap equation that represents the particle-particle (p-p)
condensation. Mathematically, the gap equation is given by replacing the p-h
propagator [$G(p+{{\bm{q}}})G(p)$] in the DW equation with the p-h propagator
at ${{\bm{q}}}={\bm{0}}$ [$G(p)G(-p)$]. In this replacement, the AL terms
disappears because of the particle conservation laws in the three-point
vertex. Since the AL terms are larger than the MT term near the magnetic QCP,
the DW order transition can occur above $T_{\rm c}$.

##  4 Unconventional orders in single-orbital models: with a focus on cuprate
superconductors

###  4.1 Motivation

In cuprate superconductors, the normal electronic states exhibit amazing
variety as we explained in Sect. 1.3. Exotic orders composed of charge and
spin degrees of freedom emerge in a cooperative fashion; see Ref. [68]. In
Y-based, Bi-based, and Hg-based cuprates, the charge-density-wave (CDW) order
with finite wavevector ${{\bm{Q}}}=(\pi/2,0)$ or $(0,\pi/2)$ appears in the
presence of strong spin fluctuations [17, 18, 26, 19, 20, 21, 22, 23, 24, 25,
26, 27]. The discovery of these unconventional density-wave (DW) states in
high-$T_{\rm c}$ cuprates have triggered significant progress in the field of
strongly correlated electron systems. This significant open issue sets a very
severe constraint on the theory, and the theoretical study will serves to
understand the high-$T_{\rm c}$ pairing mechanism. The rich variety of
correlation-driven spontaneous symmetry breaking in metals becomes a central
issue in condensed matter physics.

The aim of this section is to discuss the mechanisms of unconventional DW
states in cuprate superconductors near around the optimum doping, with the
electron filling $n\sim 0.85$. For this purpose, we analyze a simple single-
orbital square-lattice Hubbard model, which describes an effective model of
cuprate superconductors. It is expressed as

$\displaystyle
H=\sum_{{{\bm{k}}},{\sigma}}{\epsilon}_{{\bm{k}}}c_{{{\bm{k}}}{\sigma}}^{\dagger}c_{{{\bm{k}}}{\sigma}}+U\sum_{i}n_{i\uparrow}n_{i\downarrow}.$
(49)

We denote the hopping integrals $(t_{1},t_{2},t_{3})=(-1,1/6,-1/5)$, where
$t_{l}$ is the $l$-th nearest hopping integral [144, 145]. Hereafter, we set
the unit of energy as $|t_{1}|=1$, which corresponds to $\sim 4000$ [K] in
cuprates, and fix the temperature $T=0.05\ (\sim 200{\rm K})$. The FS at
filling $n=0.85$ is given in Fig. 6 (b).

As for the value of $U$, early constrained LDA (cLDA) studies lead $U=7\sim
10$ eV in La2CuO4 [146, 147, 148]. On the other hand, recent cRPA analysis
predicts $U\approx 3.2$ eV ($U/|t|\approx 6.5$) in La2CuO4, and $U\approx 2.2$
eV ($U/|t|\approx 4.5$) in Hg-based cuprates [149]. Hereafter, we use smaller
$U$ when the self-energy is neglected in electron Green functions in order to
satisfy the paramagnetic condition ${\alpha}_{S}<1$.

The spin susceptibility in the random-phase-approximation (RPA) is
$\chi^{s}(q)=\chi^{0}(q)/(1-U\chi^{0}(q))$, where $\chi^{0}(q)$ is the
irreducible susceptibility without $U$ and $q\equiv({{\bm{q}}},{\omega}_{l})$.
The spin Stoner factor is defined as
${\alpha}_{S}\equiv\max_{q}\\{U\chi^{0}(q)\\}=U\chi^{0}({{\bm{Q}}}_{s},0)$.
Figure 6 (c) shows the obtained $\chi^{s}(q)$ at ${\alpha}_{S}=0.99$
($U=3.27$). Here, $\chi^{s}({{\bm{Q}}}_{s},0)\sim 30$ [$1/{t_{1}}$] $\sim 80$
[$\mu_{\rm B}^{2}/{\rm eV}$], which is still smaller than
Im$\chi^{s}({{\bm{Q}}}_{s},E=31{\rm meV})\sim 200$ [$\mu_{\rm B}^{2}/{\rm
eV}$] at $T\sim 200$K in 60K YBCO [113]. Thus, ${\alpha}_{S}>0.99$ in real
compounds. In the FLEX approximation, the relation ${\alpha}_{S}\lesssim 1$ is
satisfied for $U/|t|\gg 3.3$ without fine tuning of $U$, because of the large
negative feedback of spin-fluctuation-induced self-energy on ${\alpha}_{S}$ in
two-dimensional systems. In Ref. [144], non-Fermi liquid transport phenomena
are satisfactorily for $U/|t|=5\sim 8$ based on the FLEX + current VC study.
It is notable that the FLEX approximation satisfies the Mermin-Wagner theorem
[127], so ${\alpha}_{S}$ never exceeds unity in two-dimensional systems.

In strongly correlated metals, the “correlation-driven density-wave (DW)” has
been studied intensively [129, 138, 150, 151, 152, 92, 25, 80]. Various
beyond-mean-field approximations have been developed to explain the nematic
and smectic orders in cuprates. Here, we focus on the impact of th vertex
corrections (VCs) that describe the paramagnon interference process in Fig. 1
(b). In the Fermi liquid theory, The irreducible VC $I$ is derived from the
Ward-identity ($\delta{\hat{\Sigma}}/\delta{\hat{G}}$). In the one-loop
approximation, $I$ is composed of the Maki-Thompson (MT) and Aslamazov-Larkin
(AL) VCs as we depicted in Fig. 13. As studied in Ref. [150], the higher-order
MT processes give the diagonal bond order with
${{\bm{q}}}={{\bm{Q}}}_{d}=(\delta,\delta)$. However, this wavevector is
inconsistent with experiments. The axial bond order is given by the lowest-
order AL process if small inter-site Coulomb interaction exists [82], while
the uniform nematic order that is observed at $T=T^{*}$ [69] is bot explained.
Therefore, new theoretical method should be developed.

Hereafter, we analyze the DW instabilities based on the DW equation as well as
the functional renormalization group (fRG) theory. In both theories, we obtain
the uniform (${{\bm{q}}}={\bm{0}}$) $d$-wave bond order, which is
schematically shown in Fig. 9 (b). This uniform bond order strongly enlarges
the axial nematic bond order instability at ${{\bm{q}}}={{\bm{Q}}}_{a}$ shown
in Fig. 9 (c). These studies lead to the prediction that the uniform bond
order occurs at $T^{*}$, and axial ${{\bm{q}}}={{\bm{Q}}}_{a}$ CDW is induced
at $T_{\rm CDW}<T^{*}$. The higher-order AL processes are significant for the
rich variety of bond orders.

###  4.2 Analysis by DW equation for cuprates

Here, we analyze the DW instabilities in a simple Hubbard model (49) by using
the DW equation, without assuming any ${{\bm{q}}}$-dependence and the form
factor. Figure 18 (a) shows the ${{\bm{q}}}$-dependence of the eigenvalue
$\lambda_{{\bm{q}}}$ obtained for ${\alpha}_{S}=0.995$ at $T=50$meV. (In FIg.
18, we introduce the imaginary part of the self-energy $\gamma=0.3$ eV.) Here,
$\lambda_{{\bm{q}}}$ is the largest at ${{\bm{q}}}={\bm{0}}$, meaning that the
uniform DW emerges at the highest temperature. As shown in Fig. 18 (b), the
corresponding form factor $f_{{\bm{0}}}({{\bm{k}}})$ has the $d$-wave
symmetry. The second largest peak is Fig. 18 (a) locates
${{\bm{q}}}={{\bm{Q}}}_{a}=(\delta,0)$, which corresponds to the axial CDW.
Since these form factors have sign reversal in ${{\bm{k}}}$-space, the
contribution from the Hartree term ($U$) is absent. To summarize, the present
theory predicts the emergence of the nematic (${{\bm{q}}}={\bm{0}}$) and
smectic ${{\bm{Q}}}={{\bm{Q}}}_{a}$ bond orders shown in Figs. 9 (b) and (c),
respectively. Thus, the present theory predicts that uniform bond order
transition emerges at a high temperature, and it stabilize the axial
${{\bm{q}}}={{\bm{Q}}}_{a}$ bond order at $T=T_{\rm CDW}$.

Figure 18:  (a) Charge-channel eigenvalue $\lambda_{{\bm{q}}}$ of DW equation
due to the MT+AL processes. Cited from [83]. The eigenvalue due to AL
processes are also shown. (b) Form factor for ${{\bm{q}}}={\bm{0}}$
($d$-wave), $f_{{\bm{q}}}({{\bm{k}}})$, normalized as
$\max_{{\bm{k}}}\\{f_{{\bm{q}}}({{\bm{k}}})\\}=1$. (c)Charge-channel kernel
functions on the FS, $I_{{{\bm{q}}}={\bm{0}}}^{c}(\theta,\theta^{\prime})$,
where $\theta$ represents the position of ${{\bm{k}}}$. We see that
$I_{{\bm{0}}}^{c}$ has large positive values due to the AL1 and AL2 terms
shown in Fig. 13 (a). (d) Origin of $d$-wave bond order. Red (blue) color
arrows represent the attractive (repulsive) interaction.

To find the origin of the DW instability, we solve the linearized DW equation
by including only AL terms. The obtained “charge-channel” eigenvalue
$\lambda^{\rm AL}_{{\bm{q}}}$ is shown in Fig. 18 (a). The similarity between
$\lambda_{{\bm{q}}}$ and $\lambda^{\rm AL}_{{\bm{q}}}$ means that the DW
instabilities at ${{\bm{q}}}={\bm{0}}$ and ${{\bm{Q}}}_{a}$ originate from the
AL processes, whereas the instability at ${{\bm{q}}}={{\bm{Q}}}_{d}$ is mainly
derived from the MT processes. Thus, it is confirmed that the higher-order AL
processes cause the multi bond order transition at both ${{\bm{q}}}={\bm{0}}$
and ${{\bm{Q}}}_{a}$.

We stress that the large eigenvalues in Fig. 18 are strongly suppressed to
$O(1)$ by considering the small quasiparticle weight $z=m/m^{*}\sim
O(10^{-1})$ and large quasiparticle damping $\gamma\gg T$ in cuprates [83,
153].

We discuss the reason why $d$-wave bond order is obtained based on a
simplified DW equation (60): The charge-channel “electron-hole pairing
interaction” $I^{c}_{{{\bm{q}}}={\bm{0}}}({{\bm{k}}},{{\bm{k}}}^{\prime})$ on
the FS is shown in Fig. 18 (c). Here, $\theta$ represents the position of
${{\bm{k}}}$ shown in Fig. 18 (d).
$I_{{\bm{q}}}^{c}({{\bm{k}}},{{\bm{k}}}^{\prime})$ in Fig. 18 (c) gives large
attractive interaction for A: ${{\bm{k}}}\approx{{\bm{k}}}^{\prime}$ and B:
${{\bm{k}}}\approx-{{\bm{k}}}^{\prime}$, and weak repulsive interaction for C.
Then, we naturally obtain the $d$-wave form factor $g_{{\bm{q}}}({{\bm{k}}})$
shown in Fig. 18 (d). Here, red (blue) arrows represent the attractive
(repulsive) interaction by A and B (C). The large positive $I_{{\bm{0}}}^{c}$
around A originates from the p-h channel in AL1 term, and that around B
originates from the p-p channel in AL2 term [91]. The moderate repulsive
$I_{{\bm{0}}}^{c}$ around C originates from the MT term.

Figure 19:  Diagrammatic expression for the even/odd parity magnon pair
condensation, which is the physical origin of the sLC and bond order. Cited
from Ref. [91].

In the DW equation formalism, the order parameter of bond order is the form
factor that represents the electron-hole pairing. Another physical
interpretation of the bond order is the “condensation of even (odd) parity
magnon-pairs”, which is the origin of the nematic order in quantum spin
systems [154, 155, 156]. In fact, the two-magnon propagator shown in Fig. 19
diverges when the eigenvalue of DW equation reaches unity [91]. The
condensation of magnon-pairs and the p-h pair condensation occur
simultaneously. Thus, the bond order discussed here and the spin nematic order
in quantum spin systems are the same phenomenon. The predicted multistage
symmetry breaking will be a key ingredient in understanding pseudogap phase
and electronic nematicity in cuprates and other strongly correlated electron
systems.

###  4.3 Renormalization group study for cuprates

In this subsection, we analyze the unconventional DW formation based on the
functional renormalization group (fRG) theory. One of the great merit of the
fRG method is that huge numbers of higher-order VCs are calculated in unbiased
way.

Here, we study a standard three-orbital $d$-$p$ Hubbard model [82, 157, 158,
84] expressed as
$H=\sum_{\bm{k},\sigma}\bm{c}_{\bm{k},\sigma}^{\dagger}\,\hat{h}_{0}(\bm{k})\,\bm{c}_{\bm{k},\sigma}+U\sum_{\bm{j}}n_{d,\bm{j},\uparrow}n_{d,\bm{j},\downarrow}$,
where
$\bm{c}_{\bm{k},\sigma}^{\dagger}=(d_{\bm{k},\sigma}^{\dagger},p_{x,\bm{k},\sigma}^{\dagger},p_{y,\bm{k},\sigma}^{\dagger})$
is the creation operator for the electron on $d$, $p_{x}$, and $p_{y}$
orbitals, and $\hat{h}_{0}(\bm{k})$ is the kinetic term. $U$ is the Hubbard-
type on-site Coulomb interaction for the $d$ orbital, and
$n_{d,\bm{j},\sigma}=d^{\dagger}_{\bm{j},\sigma}d_{\bm{j},\sigma}$ at site
$\bm{j}$. Hereafter, we study the $10\%$ hole doping case.

Figure 20:  (a) RG+cRPA result of the $p$O-CDW susceptibility
$\chi^{p\mbox{-}\mathrm{orb}}_{d}(\bm{q})$ obtained for $U=4.32$ eV. The axial
wavevector is $\bm{Q}_{\mathrm{a}}\approx(0.37\pi,0)$ and the diagonal
wavevector is $\bm{Q}_{\mathrm{d}}\approx(0.40\pi,0.40\pi)$. Both
$\bm{Q}_{\mathrm{a}}$ and $\bm{Q}_{\mathrm{d}}$ correspond to the wavevector
connecting the hot spots. (b) RG+cRPA result of
$\chi^{p\mbox{-}\mathrm{orb}}_{d}(\bm{q})$ at $\bm{q}={\bm{0}}$,
$\bm{Q}_{\mathrm{a}}^{x,y}$ and ${{\bm{Q}}}_{\mathrm{d}}$ as a function of
$\chi^{\mathrm{spin}}_{\mathrm{max}}$ for $\Delta E_{p}=0.3$ eV. The inset
shows the FS. (c) Schematic phase diagram of cuprate superconductor predicted
by the DW-equation and RG+cRPA theory. Nematic and smectic bond order are
shown in Figs. 9 (b) and (c). Cited from Ref. [157].

By using the RG+cRPA theory in Ref. [157], we find that the spin
susceptibility for $d$-electrons,

$\chi^{\mathrm{spin}}(\bm{q})=\frac{1}{2}\int_{0}^{1/T}d\tau\,\left\langle
S_{d}(\bm{q},\tau)S_{d}(-\bm{q},0)\right\rangle,$ (50)

and the $B_{1g}$-symmetry ($d$-symmetry) charge-channel susceptibility for
$p$-electrons,

$\chi^{p\mbox{-}\mathrm{orb}}_{d}(\bm{q})=\frac{1}{2}\int_{0}^{1/T}\\!d\tau\,\left\langle
n^{p\mbox{-}\mathrm{orb}}_{d}(\bm{q},\tau)n^{p\mbox{-}\mathrm{orb}}_{d}(-\bm{q},0)\right\rangle,$
(51)

are the most enhanced susceptibilities [84]. Here, $S_{d}(\bm{q},\tau)$ is the
$d$-electron spin operator, and $n^{p\mbox{-}\mathrm{orb}}_{d}(\bm{q})\equiv
n_{x}(\bm{q})-n_{y}(\bm{q})$
($n_{x(y)}(\bm{q})=\sum_{\bm{k},\sigma}p_{x(y),\bm{k},\sigma}^{\dagger}p_{x(y),\bm{k+q},\sigma}$)
is the $p$-orbital charge-density-wave ($p$O-CDW) operator with $B_{1g}$
symmetry. If $\chi^{p\mbox{-}\mathrm{orb}}_{d}(\bm{q})$ diverges at
$\bm{q}=\bm{Q}_{\rm a}$ [$\bm{q}=\bm{0}$], the bond order similar to Fig. 9
(c) [Fig. 9(b)] is realized.

Hereafter, we perform the RG+cRPA analysis by putting $N_{p}=128$ and
$\Lambda_{0}=0.5$ eV. In the RG+cRPA method, the numerical accuracy of the
susceptibilities is drastically improved by applying the cRPA for the higher-
energy processes, because the $N_{p}$-patch RG scheme is less accurate. We
verified that the numerical results are essentially independent of the choice
of $\Lambda_{0}$ when $E_{\rm F}\gtrsim\Lambda_{0}\gg T$. In Fig. 20(a), we
show the $p$O-CDW susceptibility $\chi^{p\mbox{-}\mathrm{orb}}_{d}(\bm{q})$
given by the RG+cRPA method for $U=4.32$ eV at $T=0.1$ eV. The obtained large
peaks at $\bm{q}=\bm{0}$, $\bm{Q}_{\mathrm{a}}$, and $\bm{Q}_{\mathrm{d}}$
originate from the VCs, since the RPA result is less singular. The obtained
highest peak at $\bm{q}=\bm{0}$ is consistent with the experimental uniform
nematic transition at $T^{*}\ (>T_{\rm CDW})$ [69]. We also obtain the peak
structures at $\bm{q}=\bm{Q}_{\mathrm{a}}$ and $\bm{Q}_{\mathrm{d}}$. Note
that the temperature $T=0.1$ eV is comparable to $T^{*}\sim 300$ K if the
mass-enhancement factor $m^{*}/m_{\rm band}\sim 3$ is considered.

In order to discuss the CDW instabilities inside the nematic phase, we perform
the RG+cRPA analysis in the presence of the uniform $p$O-CDW order
$H^{\prime}=-\frac{1}{2}\Delta E_{p}[n_{x}(\bm{0})-n_{y}(\bm{0})]$. In Fig. 20
(b), we plot the peak values of $\chi^{p\mbox{-}\mathrm{orb}}_{d}(\bm{q})$ in
the uniform $p$O-CDW state with $\Delta E_{p}=0.3$ eV. Due to small $\Delta
E_{p}>0$, $\chi^{p\mbox{-}\mathrm{orb}}_{d}(\bm{q})$ at
$\bm{q}=\bm{Q}_{\mathrm{a}}^{x}=(\delta,0)$ strongly increases whereas that at
$\bm{q}=\bm{Q}_{\mathrm{a}}^{y}=(0,\delta)$ decreases. Thus, the $p$O-CDW at
$\bm{q}=\bm{Q}_{\mathrm{a}}^{x}$ is expected to emerge below $T_{\mathrm{\rm
CDW}}$, consistently with the phase diagram in Fig. 7.

Because $p$-electrons are non-interacting in this $d$-$p$ Hubbard model, the
enhancement of $\chi^{p\mbox{-}\mathrm{orb}}_{d}(\bm{q})$ originates from the
$d$-orbital Coulomb interaction. The obtained $p$O-CDW susceptibility is
equivalent to the $d$-wave bond order between $d$-orbitals, which is obtained
in Ref. [84] based on the DW equation. Figure 20(c) represents the schematic
phase diagram of cuprate superconductor predicted by the DW-equation and
RG+cRPA theory. Nematic and smectic bond orders are shown in Figs. 9 (b) and
(c), respectively.

###  4.4 Odd parity current orders: spin loop current

In previous subsections, we discussed the spin-fluctuation-mediated $d$-wave
orbital/bond formation in Fe-based and cuprate superconductors. They are
expressed by the “charge-channel, even-parity (${\cal P}_{\rm bond}=+1$)” form
factor $f^{c}_{{\bm{k}}}=f^{c}_{-{{\bm{k}}}}$. Here, we explain that the “odd-
parity (${\cal P}_{\rm bond}=-1$)” form factor can be caused by the same spin
fluctuation mechanism. We derive the $p$-wave spin-channel form factor
$f^{s}\propto(\sin k_{x},\sin k_{y})$ that accompanies the spontaneous spin
loop current (sLC). The obtained sLC may be the origin of the pseudogap
behaviors in cuprate superconductors. In the sLC state, the time reversal
symmetry is preserved $({\cal T}=+1)$.

Figure 21:  (a) Spin-channel eigenvalue $\eta_{{\bm{q}}}$ of DW equation
discussed in Ref. [91]. (b) Form factor for ${{\bm{q}}}={\bm{0}}$ ($p$-wave)
normalized by its maximum value. (c) Spin-channel kernel functions on the FS,
$I_{{{\bm{q}}}={\bm{0}}}^{s}(\theta,\theta^{\prime})$, where $\theta$
represents the position of ${{\bm{k}}}$. We see that $I_{{\bm{0}}}^{s}$ has
large positive and negative values for A $\theta^{\prime}\approx\theta$ due to
AL1 and B $\theta^{\prime}\approx\theta+\pi$ due to AL2, respectively. AL1 and
AL2 are shown in Fig. 13 (a). (d) Origin of $d$-wave bond order. Red (blue)
color arrows represent the attractive (repulsive) interaction due to A (B).
Cited from Ref. [91].

Here, we discuss the spin-fluctuation-driven sLC order. Figure 21 (a) exhibits
the spin-channel eigenvalue $\eta_{{{\bm{q}}}}$ derived from the DW eq. (48).
Peaks of $\eta_{{{\bm{q}}}}$ are located at the nesting vectors
${{\bm{q}}}={{\bm{Q}}}_{\rm d}$ (diagonal) and ${{\bm{q}}}={{\bm{Q}}}_{\rm a}$
(axial). The obtained form factor $g_{{\bm{q}}}({{\bm{k}}})$ at
${{\bm{q}}}={{\bm{Q}}}_{\rm d}$ (diagonal sLC) is shown in Fig. 21 (b). The
obtained odd-parity solution
$g_{{\bm{q}}}({{\bm{k}}})=-g_{{\bm{q}}}(-{{\bm{k}}})$ corresponds to the sLC
order.

To understand why sLC state is obtained, we simplify Eq. (48) by taking the
Matsubara summation analytically by approximating that $I_{{\bm{q}}}^{s}$ and
$g_{{\bm{q}}}(k)$ are static:

$\displaystyle\eta_{{{\bm{q}}}}g_{{\bm{q}}}({{\bm{k}}})=\frac{1}{N}\sum_{{{\bm{p}}}}I_{{\bm{q}}}^{s}({{\bm{k}}},{{\bm{p}}})F_{{\bm{q}}}({{\bm{p}}})g_{{\bm{q}}}({{\bm{p}}}),$
(52)

where $\displaystyle
F_{{\bm{q}}}({{\bm{p}}})\equiv-T\sum_{m}G(p+{{\bm{q}}})G(p)=\frac{n({\epsilon}_{{{\bm{p}}}+{{\bm{q}}}})-n({\epsilon}_{{{\bm{p}}}})}{{\epsilon}_{{{\bm{p}}}}-{\epsilon}_{{{\bm{p}}}+{{\bm{q}}}}}$
is a positive function. We exhibit the spin-channel “electron-hole pairing
interaction” $I^{s}_{{{\bm{q}}}={\bm{0}}}({{\bm{k}}},{{\bm{k}}}^{\prime})$ on
the FS in Fig. 21 (c), where $\theta$ is the position of ${{\bm{k}}}$ shown in
Figs. 21 (d). We see that $I_{{\bm{q}}}^{s}({{\bm{k}}},{{\bm{k}}}^{\prime})$
gives large attractive and repulsive interactions at A
(${{\bm{k}}}\approx{{\bm{k}}}^{\prime}$) and B
(${{\bm{k}}}\approx-{{\bm{k}}}^{\prime}$), respectively. Thus, the $p$-wave
form factor $g_{{\bm{q}}}({{\bm{k}}})$ is naturally obtained as we explain in
Fig. 21 (d). Here, red (blue) arrows represent the attractive (repulsive)
interaction.

As shown in Ref. [91], large $I^{s}$ originates from the convolution of
transverse spin fluctuations $C_{{\bm{q}}}^{\rm
tr}=\sum_{{\bm{k}}}\chi^{s}_{\pm}({{\bm{k}}}+{{\bm{q}}})\chi^{s}_{\mp}({{\bm{k}}})$.
As we found in Ref. [91], $I^{s}\approx\mbox{[AL1]}-\mbox{[AL2]}$, and the
expressions of AL1 and AL2 are shown in Fig. 13. Since [AL1] and [AL2] take
large positive value for ${{\bm{k}}}\approx{{\bm{k}}}^{\prime}$ and
${{\bm{k}}}\approx-{{\bm{k}}}^{\prime}$, respectively, the numerical results
in Fig. 21 (c) is naturally understood. In contrast, the charge channel kernel
$I_{{\bm{q}}}^{c}({{\bm{k}}},{{\bm{k}}}^{\prime})$ gives an attractive
interaction for both ${{\bm{k}}}\approx\pm{{\bm{k}}}^{\prime}$ as shown in
Fig. 18 (c), because $I^{c}=3(\mbox{[AL1]}+\mbox{[AL2]})/2$. Therefore, the
$d$-wave change bond order is obtained [91].

In the present sLC mechanism due to the spin-flipping AL process
$C_{{\bm{q}}}^{\rm tr}$ [91], the $\bm{g}$-vector will be parallel to
$z$-direction when
$\chi^{s}_{x(y)}({{\bm{Q}}}_{s})>\chi^{s}_{z}({{\bm{Q}}}_{s})$ (XY-anisotropy)
due to the spin-orbit interaction (SOI). When the XY-anisotropy of
$\chi^{s}_{\mu}({{\bm{Q}}}_{s})$ is very large, $I^{c}$ due to AL terms is
multiplied by $2/3$ whereas $I^{s}$ is unchanged, so it is suitable condition
for the sLC order.

Next, we investigate the spin current in real space, which is driven by a
fictitious Peierls phase due to the “spin-dependent self-energy” $\delta
t_{i,j}^{\sigma}={\sigma}g_{i,j}$. In the current order, $\delta t_{i,j}$ is
purely imaginary and odd with respect to $i\leftrightarrow j$. The
conservation law $\dot{n}_{i}^{\sigma}=\sum_{j}j_{i,j}^{\sigma}$ directly
leads to the definition the spin current operator from site $j$ to site $i$ as
$j_{i,j}^{\sigma}=-i\sum_{{\sigma}}{\sigma}(h_{i,j}^{\sigma}c_{i{\sigma}}^{\dagger}c_{j{\sigma}}-(i\leftrightarrow
j))$, where $h_{i,j}^{\sigma}=t_{i,j}+\delta t_{i,j}^{\sigma}$. Then, the
spontaneous spin current from $j$ to $i$ is $J_{i,j}^{s}=\langle
j_{i,j}^{s}\rangle_{\hat{h}^{\sigma}}$. The spin current for the commensurate
sLC order at ${{\bm{q}}}_{\rm sLC}=(\pi/2,\pi/2)$ is shown in Fig. 9 (d) [91].

###  4.5 Odd parity current orders: charge loop current in quasi 1D systems

Spontaneous current orders due to odd-parity order parameters attract
increasing attention in various strongly correlated metals. Here, we propose a
novel spin-fluctuation-driven charge loop current (cLC) mechanism based on the
functional renormalization group (fRG) theory. By analyzing a simple
frustrated Hubbard model, we find that the ferro cLC appears between the
antiferromagnetic (AFM) and $d$-wave SC ($d$SC) phases. The key ingredients of
the present cLC mechanism are the geometrical frustration and magnetic
criticality. The cLC is expressed as the $p$-wave charge-channel form factor
without time-reversal symmetry (${\cal T}=-1$). The present study indicates
that the cLC is ubiquitous in metals near the magnetic criticality with
geometrical frustration.

Figure 22:  (a) Geometrically frustrated coupled chain model. $t$ and
$t^{\perp}$ represent the intra- and inter-chain hopping integrals,
respectively. Intra-unit-cell cLC pattern obtained in Ref. [130] is
illustrated. (b) Obtained transition temperatures as function of $t^{\perp}$.
The cLC state is realized for $t^{\perp}=0.1\sim 0.2$. The AFM and d-wave SC
appear for $t^{\perp}\lesssim 0.1$ and $t^{\perp}\gtrsim 0.2$, respectively.
The cLC appears in the Fermi liquid (FL) regime. (c) Obtained charge-channel
odd-parity form factor $f^{{{\bm{q}}}={\bm{0}}}({{\bm{k}}})$ ($\propto\sin
k_{x}+b\sin 3k_{x}$). (d) (upper) The cLC order induced by inter-branch
repulsion and intra-branch attraction. (lower) Inter-branch repulsion [intra-
branch attraction] due to odd-number [even-number] spin fluctuation exchange
processes. Cited from Ref. [130].

Here, we calculate the spin and charge susceptibilities with nonlocal form
factor $\chi_{f}^{s,c}({{\bm{k}}})$ in the geometrically frustrated coupled
chain model shown in Fig. 22 (a). The kinetic energy is given as
${\epsilon}_{{\bm{k}}}=2t\cos k_{x}+2t_{\perp}(\cos k_{y}+\cos(k_{x}+k_{y}))$.
Here, we optimize the form factor $f^{{{\bm{q}}}}({{\bm{k}}})$ so as to
maximize $\chi_{f}^{s,c}({{\bm{k}}})$ in the RG theory by following Ref.
[130], under the constraint
$\sum_{{{\bm{k}}}}|f^{{{\bm{q}}}}_{{{\bm{k}}}}|^{2}=1$ at each
${{\bm{q}}}$-point. For this purpose, we introduce the Fourier expansion form
of $f^{{{\bm{q}}}}({{\bm{k}}})$ as

$\displaystyle
f^{{{\bm{q}}}}({{\bm{k}}})=\sum_{n,m=1}^{7}a_{nm}^{{{\bm{q}}}}h_{n}(k_{x})h_{m}(k_{y}),$
(53)

where $\frac{h_{n}(k)}{\sqrt{2}}=\\{\frac{1}{\sqrt{2}},\cos k,\cos 2k,\cos
3k,\sin k,\sin 2k,\sin 3k\\}$ for $n=1,2,3,4,5,6,7$, respectively. More
detailed explanations are presented in Ref. [159]. Hereafter, we study the
half-filling case at $U=2.0$.

Figure 22 (b) shows the transition temperatures obtained by the fRG method as
function of $t^{\perp}$. The charge-channel odd-parity solution at wavevector
${{\bm{q}}}={\bm{0}}$, which corresponds to the cLC state, is obtained for
$t^{\perp}=0.1\sim 0.2$. The AFM and d-wave SC appear for $t^{\perp}\lesssim
0.1$ and $t^{\perp}\gtrsim 0.2$, respectively. The cLC appears in the Fermi
liquid (FL) regime, where inter-chain coherence is established. The optimized
charge-channel form factor at ${{\bm{q}}}=\bm{0}$ is depicted in Fig. 22 (c).
The relation $f^{\bm{0}}({{\bm{k}}})\simeq-f^{\bm{0}}(-{{\bm{k}}})(\propto\sin
k_{x}+b\sin 3k_{x})$ holds. Then, the real-space order parameter is $\delta
t_{ij}=-\delta t_{ji}$ that leads to the emergence of ferro-cLC order. The
schematic picture of the cLC in real space is depicted in Fig. 22 (a), which
is a magnetic-octupole-toroidal order. Thus, the intra-unit-cell cLC order is
obtained without any bias in a simple frustrated chain Hubbard model.

Here, the $p$-wave form factor
$f^{\bm{0}}({{\bm{k}}}_{L})=-f^{\bm{0}}({{\bm{k}}}_{R})$ is induced by the
inter-branch repulsion and the intra-branch attraction in Fig. 22 (d). By
means of the $g$-ology theory, it originates from the enhancement of the
forward scatterings ($g_{2}$,$g_{4}$) [130]. The origin of the $p$-wave form
factor is also understood based on the ”particle-hole (ph) gap equation”:
Inter-branch repulsion [intra-branch attraction] due to odd-number [even-
number] spin fluctuation exchange processes, as shown in Fig. 22 (d). Thus,
the cLC is explained by the spin-fluctuation-driven mechanism based on 2D FL
concept [144, 63, 64, 65]. Therefore, the cLC order emerges next to the AFM
phase in Fig. 22 (b). We stress that the cLC phase in the FL regions is
replaced with the AFM phase if we remove the geometrical frustration. The
present theory provides important guidelines for generating the cLC in
materials; the geometrical frustration and the magnetic criticality. It is an
important issue to understand the cLC order recently observed in ladder [160]
and square lattice [161, 162, 163, 164, 165] systems based on the present cLC
mechanism.

##  5 Unconventional orders in Fe-based superconductors

###  5.1 Motivation

The normal state electronic states in Fe-based superconductors exhibit amazing
variety as we explained in Sect. 1.2. The discovery of the “electronic nematic
states” in Fe-based superconductors has triggered significant progress in the
field of strongly correlated electron systems. For example, the $B_{1g}$
nematic states with (without) magnetization emerge in BaFe2As2 (FeSe)
families, and $B_{2g}$ nematicity appears in RbFe2As2. Such rich variety of
the electronic states set a very severe constraint on the theory of Fe-based
superconductors, and the theoretical study will serve to understand the
pairing mechanism of high-$T_{\rm c}$ superconductivity. Now, it is widely
accepted that the correlation-driven spontaneous symmetry breaking in metals
exhibits richer variety than we had expected before the discovery of LaFeAsO
in 2008 [31], and this issue has become a central open problem in condensed
matter physics.

In strongly correlated metals, the “correlation-driven density-wave (DW)” has
been studied intensively [129, 138, 150, 151, 152, 92, 25, 80]. Various
beyond-mean-field approximations have been developed to explain the electronic
nematic phases in Fe-based superconductors, such as spin nematic scenarios
[166, 106, 167, 168, 169, 108] and the orbital/charge-order scenarios [128,
129, 170, 143, 131, 171, 172, 173, 109] These scenarios were successfully
applied to the nematicity in BaTi2Sb2O [174] and 1T-TaS2 [175].

In this section, we try to explain various key experiments of many Fe-based
superconductors based on the itinerant picture, by focusing on the “paramagnon
interference mechanism” shown in Fig. 1 (b). First, we introduce the unfolded
five-orbital Hubbard model for Fe-based superconductors, $H=H_{0}+H_{I}$. The
kinetic term is given as

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\sum_{{{\bm{k}}};{\sigma}=;l,m}h_{0}^{lm}({{\bm{k}}})c_{{{\bm{k}}},l,{\sigma}}^{\dagger}c_{{{\bm{k}}},m,{\sigma}},$
(54)

where $l,m$ are the $d$-orbital indices. Hereafter, we denote
$d_{3z^{2}-r^{2}}$, $d_{xz}$, $d_{yz}$, $d_{xy}$, $d_{x^{2}-y^{2}}$ as
$l=1,2,3,4,5$. The unfolded Fermi surfaces (FSs) for Ba122 and FeSe are shown
in Fig. 2 (c) and 4 (d), respectively. In both compounds, the FSs are mainly
composed of $xz$, $yz$, and $xy$ orbitals. The multiorbital Coulomb
interaction term is given in Eq. (35). In transition metals, the relations
$U=U^{\prime}+2J$ and $J=J^{\prime}$ holds approximately. In addition,
$J/U=0.1\sim 0.2$, which means that $U^{\prime}/U=0.8\sim 0.6$ according to
first principles study [50]. $H_{I}$ in Eq. (35) is compactly expressed as

$\displaystyle H_{I}=\frac{1}{4}\sum_{i,1\sim
4}({\hat{U}})_{1,2,3,4}c_{1}^{\dagger}c_{2}c_{3}c_{4}^{\dagger},$ (55)

where $1\equiv(l_{1},{\sigma}_{1})$ and so on. Reflecting the $SU(2)$
symmetry, ${\hat{U}}$ in Eq. (55) is expressed as

$\displaystyle{\hat{U}}=\frac{1}{2}{\hat{U}}^{c}\delta_{{\sigma}_{1},{\sigma}_{2}}\delta_{{\sigma}_{4},{\sigma}_{3}}+\frac{1}{2}{\hat{U}}^{s}{\bm{{\sigma}}}_{{\sigma}_{1},{\sigma}_{2}}\cdot{\bm{{\sigma}}}_{{\sigma}_{4},{\sigma}_{3}}$
(56)

where ${\hat{U}}^{c(s)}$ is the charge (spin) channel Coulomb interaction in
$5^{2}\times 5^{2}$ matrix form; see Refs. [128, 131]. (Note that $U^{s}=U$
and $U^{c}=-U$ in single-orbital models.)

According to the constrained RPA (cRPA) study [50], the averaged intra-orbital
Coulomb interaction on Fe-ion ${\bar{U}}$ in the $d$-$p$ orbital (8 orbital)
model ranges from $4$eV to $7$eV for 1111, 122, 111, and 11 compounds. In
contrast, ${\bar{U}}$ is reduced to $2.5$eV to $4$eV in the $d$ orbital (5
orbital) model, due to the screening effect by $p$-orbitals. In later
sections, we set the Coulomb interaction in Eq. (55) as ${\hat{U}}\equiv
r{\hat{U}}_{\rm X}^{d-p}$ (X=LaFeAsO, FeSe, and BeFe2As2), which is given by
the cRPA result for the $d$-$p$ orbital model [50] multiplied by the reduction
factor $r$. Here, $r$ is the solo model parameter in the majority of the
present study. Here, we set $U\sim 1.7$eV ($r\sim 0.4$ in LaFeAsO and $r\sim
0.25$ in FeSe) when the self-energy is neglected to satisfy the paramagnetic
condition ${\alpha}_{S}<1$. If FLEX self-energy is included in the electron
Green functions, we use larger $U$ ($U\sim 2.8$eV and $r\sim 0.4$) for FeSe
model because $\alpha_{S}$ is reduced by the self-energy.

###  5.2 Nematic order and fluctuations due to paramagnon-interference: the
SC-VC theory

In this subsection, we analyze the orbital fluctuations driven by the VCs
based on the self-consistent vertex correction (SC-VC) method in Sect. 2.4.
The formalism in Sect. 2.4 can be naturally extended to multiorbital Hubbard
models [129].

Here, study the five-orbital Hubbard model derived from the Density-functional
theory (DFT) band calculation for La1111 compound [49, 129]. Its unfolded FS
is similar to Fig. 2 (d), whereas the $xy$-orbital hole pocket around
$(\pi,\pi)$ point is absent; see Fig. 23 (a). In the RPA, in which any VCs are
neglected, strong spin fluctuations develop at
${{\bm{q}}}\approx(\pi,0),(0,\pi)$. In contrast, no charge-channel
fluctuations develop at all. Thus, the nematic phase transition without
magnetism at $T_{S}\ (>T_{N})$ cannot be explained by the RPA.

For convenience, we introduce the matrix expressions of the quadrupolar
operators with respect to orbitals 2-4 as:

$\displaystyle({\hat{O}}_{x^{2}-y^{2}})_{lm}=(-1)^{l}\ \ {\rm for}\
(lm)=(22),(33),$ (57) $\displaystyle({\hat{O}}_{xz})_{lm}=+1\ \ {\rm for}\
(lm)=(3,4),(4,3),$ (58)

and other elements are zero. Note that
$({\hat{O}}_{\mu\nu})_{lm}\propto\langle
l|(l_{\mu}l_{\nu}+l_{\nu}l_{\mu})|m\rangle$ with $\mu,\nu=x,y,z$. Then,
$\Gamma$-channel susceptibility is given as $\chi_{\Gamma}^{x}(q)={\rm
Tr}\\{{\hat{O}}_{\Gamma}{\hat{\chi}}^{x}(q){\hat{O}}_{\Gamma}\\}\
(=\sum_{ll^{\prime}mm^{\prime}}({\hat{O}}_{\Gamma})_{ll^{\prime}}\chi_{ll^{\prime},mm^{\prime}}^{x}(q)({\hat{O}}_{\Gamma})_{m^{\prime}m})$

In the five-orbital model, the Green function ${\hat{G}}(k)$ is expressed as
$5\times 5$ matrix, and the $25\times 25$ matrix of the irreducible
susceptibility in the RPA is

$\displaystyle[{\hat{\chi}}^{(0)}(q)]_{ll^{\prime},mm^{\prime}}=-T\sum_{k}G_{lm}(k+q)G_{m^{\prime}l^{\prime}}(k).$
(59)

The bare Coulomb interaction ${\hat{U}}^{x}$ ($x=c,s$) is expressed as
$25\times 25$ matrix. Note that $U_{ll,ll}^{s}=-U_{ll,ll}^{c}=U$. In the
present model, $\chi^{s}_{22,22}({{\bm{q}}})$ [$\chi^{s}_{33,33}({{\bm{q}}})$]
possesses broad peak around ${{\bm{q}}}=(0,\pi)$ [${{\bm{q}}}=(\pi,0)$] due to
the intra-orbital nesting between electron- and hole-pockets; see Fig. 23 (a).
In contrast, $\chi^{s}_{44,44}({{\bm{q}}})$ exhibits small peak around
${{\bm{q}}}=(\pi,\pi/2),(\pi/2,\pi)$ due to the weak nesting between electron-
pockets. In Ba122 model, the peak of $\chi^{s}_{44,44}({{\bm{q}}})$ shifts to
${{\bm{q}}}=(0,\pi),(\pi,0)$ because of the additional $d_{xy}$-orbital hole-
pocket around M point; h3 in Fig. 2 (d).

Next, we calculate the charge-channel susceptibility ${\hat{\chi}}^{c}(q)$ by
including both the MT and AL terms in Fig. 13, by applying the SC-VC theory
given as Eqs. (24)-(26). In the presence of moderate spin fluctuations,
${\hat{\chi}}^{c}(q)$ is strongly enhanced by the charge-channel AL-VC
$X^{{\rm AL},c}\sim T\sum\chi^{s}\chi^{s}$, which is shown in Fig. 23 (b)
[129]. On the other hand, it is verified that ${\hat{X}}^{{\rm AL},s}\sim
T\sum\chi^{s}\chi^{c}$ is less important in Fe-based superconductors [131].
Figure 23 (c) shows the charge-channel susceptibilities given by the self-
consistent calculation of the AL and MT terms (SC-VC method) for $n=6.1$,
$J/U=0.088$ and $U=1.53$eV. Both orbital susceptibilities
$\chi^{c}_{x^{2}-y^{2}}({{\bm{q}}})=2(\chi^{c}_{22,22}-\chi^{c}_{22,33})$ and
$\chi^{c}_{xz}({{\bm{q}}})=2(\chi^{c}_{34,34}+\chi^{c}_{34,43})$ are strongly
enhanced by the charge AL term, ${\hat{X}}^{{\rm AL},c}$. (The obtained
results are essentially unchanged even if MT term is dropped.) The
enhancements of other orbital susceptibilities are small. We well discuss in
Sect. 5.7 that the relation
$\chi^{c}_{xz}({{\bm{Q}}})>\chi^{c}_{x^{2}-y^{2}}({\bm{0}})$ is realized in
Ba122 model, in which large $xy$-orbital hole pockets exists.

Figure 23:  (a) FSs of La1111 model at $n=6.1$ [129]. (b) VC for the charge-
channel susceptibility in the SC-VC theory. In this theory,
$I_{{\bm{q}}}^{{\sigma},\rho}$ is given in Fig. 13. (c) Obtained enhancements
of orbital susceptibilities $\chi_{xz}({{\bm{q}}})$ and
$\chi_{x^{2}-y^{2}}({{\bm{q}}})$ in La1111 Hubbard model [129]. They exhibit
peaks at ${{\bm{q}}}\approx(\pi,0),(0,\pi)$ and at
${{\bm{q}}}\approx{\bm{0}}$, respectively. Both susceptibilities in the RPA
shown by broken lines are very small. Cited from Ref. [129].

In summary, we developed the SC-VC method for multiorbital Hubbard models, and
obtained strong nematic and smectic orbital fluctuations in Fe-based
superconductors due to the AL process that describes the paramagnon
interference effect [128]. The AL process is dropped in the RPA. In the SC-VC
theory, the structure transition (${\alpha}^{c}\approx 1$) occurs prior to the
magnetic transition (${\alpha}^{s}\approx 1$) for $J/U\lesssim 0.2$,
consistently with experiments. (In the DW equation explained in Sect. 3.3
nonmagnetic orbital order is realized even for $J/U>0.2$.) When
${\alpha}^{s}\sim{\alpha}^{c}$, both $s_{++}$\- and $s_{\pm}$-states could be
realized, depending on model parameters like the impurity concentration [112,
176]. Thus, the orbital-fluctuation-mediated superconductivity is expected
near the nematic QCP.

###  5.3  Comparison between 1111 and 11 systems: the SC-VC theory

Near the quantum critical point (QCP) of the nematic order, prominent nematic
criticality has been observed in various compounds by many experimental
techniques, such as the shear modulus measurement [177, 178], in-plane
resistivity anisotropy [179, 13], Raman spectroscopy [180, 181, 182], and the
phonon softening measurement by x-ray scattering [183]. In these measurements,
one can derive the “electronic nematic susceptibility” driven by the electron
correlation, free from the electron-phonon interaction. For example, Raman
nematic spectroscopy is free from the acoustic phonon contribution because of
the relationship $({\omega}_{k}/k)_{\rm photon}\gg({\omega}_{k}/k)_{\rm
phonon}$ [184]. It has been established that the nematicity of Fe-based
superconductors originates from the electron correlation, while its transition
temperature is raised by just several tens of Kelvins, which we call the Jahn-
Teller energy $T_{\rm JT}$. That is, the relationship $T_{\rm S}=T_{\rm
nem}+T_{\rm JT}$ holds, where $T_{\rm S}$ is the observed structural
transition temperature, and $T_{\rm nem}$ is the correlation-driven nematic
transition temperature. In many Fe-based superconductors, $T_{\rm JT}$ is just
$20\sim 30$K. For simplicity, we set $T_{S}=T_{nem}$ by neglecting the Jahn-
Teller energy. in later discussions.

Experimentally, both AFM susceptibility and the electronic nematic
susceptibility follows the Curie-Weiss behaviors in the “tetragonal phase
above $T_{S}$”: $\chi^{s}({{\bm{Q}}})\propto(T-T_{\rm N})^{-1}$ and $\chi_{\rm
nem}\propto(T-T_{\rm S})^{-1}$, respectively. Interestingly, the relationship
between $T_{\rm N}$ and $T_{\rm S}$ is strongly depends on compounds. In many
Ba122 families, the relation $T_{\rm S}\gtrsim T_{\rm N}$ holds, which is
naively expected as the spin-fluctuation-driven nematicity. In 1111 families
and NaFeAs ($T_{S}=60$K), in contrast, $T_{\rm S}$ is clearly larger than
$T_{\rm N}$. The AFM order appears inside the nematic phase because the AFM
correlation is magnified by the nematic order. In FeSe with $T_{S}=90$K, no
AFM order appears down to zero temperature. In fact, $T_{\rm N}$ derived from
the Weiss temperature of $\chi^{s}({{\bm{Q}}})$ above $T_{S}$ takes large
negative value. Therefore, $\chi_{\rm nem}$ is strongly enlarged in the
$C_{4}$ phase near the nematic phase, whereas $\chi^{s}({{\bm{q}}})$ at
$T=T_{S}$ drastically depends on materials. Such drastic material dependences
of $\chi^{s}({{\bm{Q}}})$ and $\chi_{\rm nem}$ put a strong constraint on the
theory of nematicity.

Figure 24:  Enhancement factor of the nematic susceptibility
($S_{c}=1/(1-{\alpha}_{C})$) and that of the spin susceptibility
($S_{s}=1/(1-{\alpha}_{S})$) obtaind for (a) LaFeAsO model and (b) FeSe model
[131]. In FeSe model, $S_{c}$ stronlgy increases due to the AL-process, while
$S_{s}$ is nearly constant. Cited from Ref. [131].

Now, we analyze the LaFeAsO model based on the SC-VC theory. Figure 24 (a)
shows the spin-channel Stoner enhancement factor
$S_{s}\equiv(1-{\alpha}_{S})^{-1}$ in LaFeAsO model with $U=1.74$ eV. We also
show the charge-channel Stoner enhancement factor
$S_{c}\equiv(1-{\alpha}_{C})^{-1}$ that is proportional to the nematic orbital
susceptibility. Both $S_{c}$ and $S_{s}$ follow the Curie-Weiss behaviors with
$\theta_{C}=T_{\rm S}=48$ meV and $\theta_{S}=T_{\rm N}=40$ meV, respectively.
Similar results are obtained by analyzing Ba122 Hubbard model [131]. The
obtained relation $T_{\rm S}\gtrsim T_{\rm N}$, which is naively expected as
the spin-fluctuation-driven mechanism, is consistent with experimental reports
in Ba122 families [177, 178, 182, 180, 184, 179].

Next, we analyze the FeSe model, in which the particle-hole asymmetry of the
bandstructure is very large, and the ratio $\bar{J}/\bar{U}$ is just $\sim
0.1$ [50]. These situations are favorable for the nematic ordering due to
paramagnon interference [131]. Figure 24 (b) shows the enhancement factors in
FeSe model, in which the renormalized Coulomb is $U^{*}=zU=1.75$eV and
$z=0.25$ (i.e., $U=7$eV). We stress that $S_{C}$ approximately follow the
Curie-Weiss behavior with the Weiss temperature $\theta_{C}=12$ meV, which is
consistent with the experimental positive $\theta_{C}$ in FeSe [185]. In
contrast, the spin Weiss temperature takes a large negative value
($\theta_{S}\sim-7.5$ meV), so the nematicity without magnetization in FeSe
families is naturally explained by the paramagnon interference mechanism.

Here, we discuss the origin of the relation $S_{C}\gg S_{S}$ in FeSe. The
nematic susceptibility is enhanced not only by $\chi^{s}({{\bm{Q}}})$, but
also by the “significant $T$-dependence of the spin-charge coupling term
$\Lambda\sim T\sum_{k}(G(k))^{2}G(k+{{\bm{Q}}})$” in Fig. 1 (b) [109, 184].
According to Ref. [131], the relation $\Lambda^{2}\propto T^{-a}$ with
$a\approx 1$ is realized in FeSe models, by reflecting prominent particle-hole
asymmetric bandstructure. Since ${\hat{X}}^{{\rm
AL},c}\sim|\Lambda|^{2}(T\xi_{s}^{2})$ given in Fig. 23 (b) increases at low
temperatures, and therefore $\chi^{c}_{x^{2}-y^{2}}({\bm{0}})$ is enlarged by
following the Curie-Weiss law. Another favorable condition for the nonmagnetic
nematicity in FeSe is the absence of h3: Its existence magnifies the
$d_{xy}$-orbital spin fluctuations while they are unimportant for the AL-
process driven orbital order. In this case, spin fluctuations develop only in
($d_{xz},d_{yz}$)-orbitals. This “orbital selective spin fluctuations” is
favorable for the orbital polarization $n_{xz}\neq n_{yz}$ [131].

To summarize, we studied the origin of the nematicity in Fe-based
superconductors, by paying the special attention to the nonmagnetic nematic
order in FeSe. Based on the SC-VC theory, we succeeded in explaining the rich
variety of the phase diagrams in Fe-based superconductors, such as the
nonmagnetic/magnetic nematic order in FeSe/LaFeAsO. In the present theory, the
ratio $\theta_{S}/\theta_{C}$ decreases when the size of the FS and the ratio
$J/U$ are small. The present results are verified by the DW equation analysis
[143, 153, 159, 186] as we explain in the next subsection.

###  5.4 Nematic electronic states in FeSe below $T_{S}$: DW equation analysis

FeSe provides a very suitable platform to understand the central issues of Fe-
based superconductors, i.e., the relationship between nematicity and
magnetism, and the origin of high-$T_{\rm c}$ superconductivity above 60K. As
we discussed in previous subsections, the nonmagnetic nematicity in bulk FeSe
at ambient pressure is naturally explained by the paramagnon interference
mechanism.

Recently, the nematic electronic states below $T_{S}=90$K have been precisely
determined by experimentally. The two main characteristics of the nematic
state in FeSe would be (i) Sign reversing $xz,yz$ orbital order [56, 57], and
(ii) Disappearance of the e-pocket around Y point [187, 188, 189]. Here, we
reproduce these two characteristics theoretically, by calculating the momentum
and orbital dependences of the form factor self-consistently based on the DW
equation (47) introduced in Sect. 3.3.

In the DW equation method, macroscopic conservation laws are rigorously
satisfied if the kernel of the DW equation and the self-energy are derived
from the same Luttinger-Ward function $\Phi_{\rm LW}$ [190]. To satisfy the
conservation laws, here we calculate the $5\times 5$ self-energy
${\hat{\Sigma}}(k)$ in the FLEX approximation shown in Fig. 25 (a). The
obtained mass-enhancement factor for orbital $l$ is $z_{l}^{-1}=1-{\rm
Re}{\partial}\Sigma_{l,l}/{\partial}{\epsilon}|_{{\epsilon}=0}$. In the
present study, we obtain $z_{xz}^{-1}\approx 3.5$ and $z_{xy}^{-1}\approx
6.5$, which are consistent with experimental values. Both band-dispersion and
form factor are renormalized by the factor $z$. Here, we set ${\hat{U}}$ in
Eq. (55) as $r{\hat{U}}_{\rm FeSe}^{d-p}$, which is the cRPA Coulomb
interaction for $d$-$p$ model of FeSe [50] multiplied by the reduction factor
$r$. Now, we set $r=0.37$ that corresponds to $U\sim 2.6$eV.

Now, we solve the linearized DW equation (47) by using the full Green function
with ${\hat{\Sigma}}(k)$ in order to satisfy the conservation laws. Figures 25
(b) and (c) represents the obtained eigenvalue $\lambda_{{\bm{q}}}$ at $T=5$me
and $r=0.40$. The ferro (${{\bm{q}}}={\rm 0}$) DW instability strongly
develops due to the interference between small spin fluctuations
(${\alpha}_{S}\lesssim 0.9$). In addition, the eigenvalue
$\lambda_{{{\bm{q}}}={\bm{0}}}$ reaches unity at $T\approx 10$meV for $r\geq
0.36$, which is consistent with the experimental transition temperature
$T_{S}=90$K. The obtained form factor
${\hat{f}}^{{{\bm{q}}}={\bm{0}}}({{\bm{k}}})$ induces $B_{1g}$ nematic orbital
order ($n_{xz}\neq n_{yz}$), consistently with the enlarged orbital
susceptibility $\chi^{c}_{x^{2}-y^{2}}({{\bm{q}}})$ at ${{\bm{q}}}={\bm{0}}$
obtained by the SC-VC theory in Fig. 23 (c).

Figure 25:  (a) Self-energy $\Sigma(k)=\delta\Phi_{\rm LW}/\delta G(k)$ in the
one-loop approximation. (b) Obtained ${{\bm{q}}}$-dependence of
$\lambda_{{\bm{q}}}$ and (c) $T$-dependence of $\lambda_{{{\bm{q}}}={\bm{0}}}$
given by the “linearized DW equation” with the self-energy in the framework of
the conserving approximation [140]. (d) Renormalized form factors
${\bar{f}}_{l,l}^{{{\bm{q}}}={\bm{0}}}({{\bm{k}}})$ for $l=xz$ ,$yz$, and
$xy$, derived from the full DW equation” in the nematic state [140]. The
e-pocket around Y point disappears under the nematic order parameters. The
obtained “one e-pocket + one h-pocket” in the nematic phase of FeSe is
consistent with recent experimental reports [57, 187, 188, 189]. Both
${\bar{f}}_{2,2}^{\bm{0}}({{\bm{k}}})$ along $k_{y}$-axis and
${\bar{f}}_{3,3}^{\bm{0}}({{\bm{k}}})$ along $k_{x}$-axis exhibit sign
reversal. (e) Renormalized form factors as functions of $T$, and (f)
Bandstructure in the nematic phase [140]. Here, two characteristics (i) and
(ii) in the nematic state in FeSe are well reproduced. Sign reversal in the
bandshift along $\Gamma$-X,Y axis is observed experimentally. Cited from Ref.
[140].

Next, we solve the “full DW equation” in Eq. (45) self-consistently, in order
to analyze the electronic nematic states under $T_{S}$ [140, 186]. Figure 25
(d) shows the renormalized form factors ${\bar{f}}_{l}({{\bm{k}}})\equiv
z_{l}{f}_{l,l}^{{\bm{0}}}({{\bm{k}}})$ for $l=3\ (yz)$ and $l=4\ (xy)$. In the
obtained $B_{1g}$ symmetry solution satisfies the relations
${\bar{f}}_{xz}(k_{x},k_{y})=-{\bar{f}}_{yz}(k_{y},k_{x})$ and ${\bf
f}_{xy}\sim\cos k_{x}-\cos k_{y}$. The obtained nematic FS is shown in each
panel. Figure 25 (e) exhibits the $T$-dependence of the form factor, and the
band dispersion in the nematic state is shown in Fig. 25 (f). We see that the
e-pocket around Y-point is lifted by the $d_{x^{2}-y^{2}}$-wave bond order
${\bar{f}}_{xy}(0,\pi)>0$. Thus, the obtained coexistence of the bond order on
$xy$-orbital the orbital order on $xz,yz$-orbital with sign reversal are
consistent with the two characteristics (i) and (ii) in the nematic states in
FeSe [56, 57, 57, 187, 188, 189].

In Figs. 25 (b)-(g), we performed the conserving approximation by taking
account of the FLEX self-energy $\Sigma_{\rm FLEX}$. It is found that the
effect of $\Sigma_{\rm FLEX}$ is just to reduce the DW instability, while the
$q$-dependence of $\lambda_{{\bm{q}}}$ and the form factor are essentially
unchanged [143]. Therefore, in later subsections, we sometimes drop
$\Sigma_{\rm FLEX}$ in order to simplify the analysis.

###  5.5 FeSe phase diagram under pressure: DW equation analysis

Next, we discuss the rich $P$-$T$ phase diagram in FeSe exhibited in Fig. 5,
which vividly demonstrates the strong interplay between the nematicity,
magnetism and superconductivity. For this purpose, we construct the first
principles model for FeSe as function of $P$, and discovered theoretically the
pressure-induced emergence of $d_{xy}$-orbital hole-pocket. The obtained
$P$-dependent bandstructure is shown in Figs. 26 (a). The $xy$-orbital level
around M point prominently shifts upwards because the Se-height increases
under pressure. For this reason, $xy$-orbital pocket appears for $P\gtrsim
2$GPa. This pressure-induced Lifshitz transition has been confirmed by the
sign change in $R_{\rm H}$ [191] and the abrupt increment of the knight shift
[192].

Figure 26:  (a) Pressure-induced change in the bandstructure in FeSe. (b)
Obtained eigenvalue of the nematic order $\lambda_{\bm{0}}$ and the spin
Stoner factor ${\alpha}_{S}$, and (c) Obtained mass-enhancement factor, as
function of $P$ at $T=5{\rm meV}$. (Y. Yamakawa et al., unpublished.)

Now, we analyze this FeSe model using the linearized DW equation (47) [193].
We calculate the self-energy ${\hat{\Sigma}}(k)$ based on the FLEX
approximation, and incorporate it into the DW equation. Figure 26 (b) shows
the obtained eigenvalue of the nematic order $\lambda$ and the spin Stoner
factor ${\alpha}_{S}$ as function of $P$ at $T=5{\rm meV}$. With increasing
$P$, $\lambda$ monotonically decreases because of the slight increment of the
bandwidth. Around $P=2$GPa, $xy$-orbital e-pocket emerges. Due to this
pressure-induced Lifshitz transition, the spin susceptibility on the $d_{xy}$
orbital is enhanced, whereas that on $d_{xz},d_{yz}$ orbitals is gradually
reduced. Since the nematicity is mainly driven by the spin fluctuations on
$d_{xz},d_{yz}$ orbitals through the intra-orbital VCs, the nematic eigenvalue
$\lambda$ in Fig. 26 (b) remains small for $P>2$GPa. Thus, rich $T$-$P$ phase
diagram in the normal state of FeSe is naturally understood based on the
paramagnon interference mechanism.

Figure 26 (c) show the mass-enhancement factors $z_{l}^{-1}=(m^{*}/m)_{l}$ for
the orbitals $l=xz(yz)$ and $l=xy$ obtained by the FLEX approximation. The
obtained relations $z_{xy}^{-1}\approx 5$ and $z_{xz}^{-1}\approx 3$ are
consistent with the LDA+DMFT analysis.

###  5.6 $B_{2g}$ nematic order in heavily hole-doped $A$Fe2As2 ($A$=Cs,Rb)

In previous subsections, we explain the $B_{1g}$ nematic order in typical Fe-
based superconductors based on the paramagnon interference mechanism.
Surprisingly, a new type of nematic order was recently discovered in heavily
hole-doped ($n_{d}=5.5$) compound AFe2As2 (A=Cs, Rb) [13, 12, 14]. The
discovered nematicity has $B_{2g}$ (=$d_{xy}$) symmetry, rotated by
$45^{\circ}$ from the $B_{1g}$ (=$d_{x^{2}-y^{2}}$) nematicity in usual
compounds with $n_{d}\approx 6$. The discovery of $B_{2g}$ nematicity provides
a very useful information to figure out the unique mechanism of nematicity for
all Fe-based superconductors. To reveal the origin of the $B_{2g}$ nematicity,
spin nematic (or vestigial order) scenario has been proposed in Refs. [194,
195]. In this article, we investigate the symmetry-breaking in the self-energy
based on the DW equation (47).

The FSs of heavily hole-doped system AFe2As2 ($n_{d}=5.5$) derived from the
first principles study are shown in Fig. 27 (a). The hole FS around M point
composed of $d_{xy}$-orbital is large, while the Dirac pockets near X and Y
points are small. The arrows ${{\bm{Q}}}$ denote the most important