# Anisotropy of the dc conductivity due to orbital-selective spin fluctuations
in the nematic phase of iron superconductors

Raquel Fernández-Martín  Materials Science Factory, Instituto de Ciencia de
Materiales de Madrid, ICMM-CSIC, Cantoblanco, E-28049 Madrid, Spain  Laura
Fanfarillo<EMAIL_ADDRESS>CNR-IOM and International School for
Advanced Studies (SISSA), Via Bonomea 265, I-34136, Trieste, Italy  Lara
Benfatto  ISC-CNR and Dep. of Physics, “Sapienza” University of Rome, P.le A.
Moro 5, 00185, Rome, Italy  Belén Valenzuela<EMAIL_ADDRESS>Materials
Science Factory, Instituto de Ciencia de Materiales de Madrid, ICMM-CSIC,
Cantoblanco, E-28049 Madrid, Spain

###### Abstract

We study the DC conductivity of iron-based superconductors within the orbital-
selective spin fluctuation scenario. Within this approach, the anisotropy of
spin fluctuations below the spin-nematic transition at $T_{s}$ is also
responsible for the orbital ordering, induced by nematic self-energy
corrections to the quasiparticle dispersion. As a consequence, the anisotropy
of the DC conductivity below $T_{s}$ is determined not only by the anisotropy
of the scattering rates as expected within a spin-nematic scenario, but also
by the modification of the Fermi velocity due to the orbital reconstruction.
More interestingly, it turns out that these two effects contribute to the DC-
conductivity anisotropy with opposite signs. By using realistic band-structure
parameters we compute the conductivity anisotropy for both 122 and FeSe
compounds, discussing the possible origin of the different dc-conductivity
anisotropy observed experimentally in these two families of iron-based
superconductors.

The driving force of electronic nematicity is one of the most intriguing
puzzles in iron-based superconductors (IBS). The structural transition from
tetragonal to orthorhombic at $T_{S}$ comprised the nematic phase
characterized by a marked electronic anisotropy, much larger than the one
expected by the structural transition itselfGallais and Paul (2016). In most
iron pnictides the structural transition precedes or coincides with the
magnetic transition at $T_{N}$. The proximity of the magnetic phase led to the
proposal of the band spin-nematic scenarioFernandes et al. (2012, 2014) where
the spin fluctuations (SF) at $\mathbb{Q}_{X}=(\pi,0)$ and at
$\mathbb{Q}_{Y}=(0,\pi)$ become anisotropic below $T_{S}$. The lack of long-
range magnetic order in FeSe has cast some doubts on the validity of the spin-
nematic scenario in this compound. FeSe presents a nematic phase below the
structural transition at $T_{S}=90K$ down to the critical superconducting
temperature $T_{c}\sim 9K$. Even though a magnetic phase is not stabilized in
temperature, sizeable SF have been detected also in FeSe Rahn et al. (2015);
Wang et al. (2016); He et al. (2018); Wiecki et al. (2017, 2018). ARPES
experiments in the nematic phase report a momentum-modulated orbital splitting
between the $\Gamma$ and M point of the Brillouin zone Coldea and Watson
(2018) that has been interpreted via both the orbital-ordering scenario Baek
et al. (2014); Onari et al. (2016); Su et al. (2015); Mukherjee et al. (2015);
Jiang et al. (2016); Xing et al. (2017) and the spin-nematic scenario
Fanfarillo et al. (2016). In this situation two related questions arise: what
is the role of the spin-orbital interplay and whether the origin of nematicity
is universal in IBS or material-dependentChubukov et al. (2016); Yamakawa et
al. (2016); Fanfarillo et al. (2018).

Resistivity anisotropy is a hallmark of nematicity in IBS. In detwinned
electron(e)-doped 122 compounds $\Delta\rho=\rho_{x}-\rho_{y}<0$ is found
below the structural transitionTanatar et al. (2010); Dusza et al. (2011);
Fisher et al. (2011); Dusza et al. (2012); Mirri et al. (2014) while detwinned
hole(h)-doped compounds present the opposite anisotropyBlomberg et al. (2013).
There is an on-going debate in the literature on whether the observed DC
anisotropy is due to the anisotropy in the scattering rate or to the
anisotropy in Fermi Surface (FS) parametersChuang et al. (2010); Nakajima et
al. (2012); Ishida et al. (2013); Allan et al. (2013); Kuo and Fisher (2014);
Mirri et al. (2015, 2016); Schütt et al. (2016); Chinotti et al. (2017). In
principle, within an orbital-ordering scenario the different occupation of the
various orbitals affects mainly the FSLee et al. (2009); Lv et al. (2010);
Kontani et al. (2011), while within a spin-driven scenario the largest effect
is expected to come from an anisotropy in the inelastic scattering
rateFernandes et al. (2010); Chuang et al. (2010); Breitkreiz et al. (2014);
Gastiasoro et al. (2014); Wang et al. (2015) . Specifically, in the band spin-
nematic scenario, depending on the FS shape and size, the band nesting is
active at the so called hot spots on the FS, where the scattering rate is
maximum. It has been argued that the location of the hot spots could explain
the different signs between e-doped compounds and h-doped compoundsBlomberg et
al. (2013) in pnictides. Besides the spin-nematic or orbital order scenario,
further attempts to explain the DC anisotropy in pnictides taking into account
the spin-orbital interplay has been performed using effective spin-fermion
modelLiang et al. (2012) or multiorbital microscopic model in the magnetic
phaseValenzuela et al. (2010); Bascones et al. (2016); Sugimoto et al. (2014).

Recent experiments in FeSe have found the opposite anisotropy with respect to
the e-doped 122 compoundsTanatar et al. (2016), i.e.
$\Delta\rho=\rho_{x}-\rho_{y}>0$. Given the significant FS reconstruction
observed in the nematic phase of FeSeColdea and Watson (2018), we need to
revise the role of the scattering rate and velocity anisotropies taking into
account the spin-orbital interplay in order to theoretically address both
pnictides and FeSe. Within an orbital-ordering scenario, the opposite
anisotropy of the resistivity of 122 and FeSe compounds in the nematic phase
has been ascribed to the orbital-dependent inelastic quasiparticle scattering
by orbital-dependent SFOnari and Kontani (2017). However, an analogous study
of the DC conductivity anisotropy within a spin-nematic scenario accounting
for the spin-orbital interplay and able to address pnictides and FeSe, is
still missing.

The aim of this work is then to provide an interpretation for the observed
differences, using as a starting point the orbital-selective spin fluctuation
(OSSF) model. The model, derived in the itinerant approachFanfarillo et al.
(2018), exploits the original idea of the orbital selective character of the
SF in IBS discussed in Ref. [Fanfarillo et al., 2015]. Within the OSSF model,
due to the orbital composition of the FS the SF peaked at
$\mathbb{Q}_{X}=(\pi,0)$ involve only the $yz$ orbital, while the SF at
$\mathbb{Q}_{Y}=(0,\pi)$ involve only the $xz$ orbital, Fig. 1. As in the
spin-nematic scenario, the nematic phase emerges when SF at
$\mathbb{Q}_{X}=(\pi,0)$ and at $\mathbb{Q}_{Y}=(0,\pi)$ become anisotropic,
however in the OSSF model such anisotropy directly affects the $yz/xz$ orbital
symmetry. The OSSF is a minimal model that explains successfully the enhanced
nematic tendency of FeSe as compared to 122 systemsFanfarillo et al. (2018)
and clarifies controversial experimental issues in FeSe such as the
temperature evolution of the FS of FeSe and the odd orbital ordering observed
by ARPES experimentsFanfarillo et al. (2016), the decrease of the nematic
critical temperature and the emergence of magnetism in FeSe with
pressureFanfarillo et al. (2018) as recently observed inSun et al. (2016);
Kothapalli et al. (2016); Coldea and Watson (2018). Moreover the analysis of
the superconductivity mediated by anisotropic OSSFBenfatto et al. (2018)
successfully account for the enigmatic anisotropy of the superconducting gap
revealed by STMSprau et al. (2017) and ARPESKushnirenko et al. (2018); Rhodes
et al. (2018); Liu et al. (2018) experiments in FeSe.

In this work we analyze, within the OSSF model, the effect of anisotropic
self-energy corrections on the conductivity anisotropy in the nematic phase of
IBS. In contrast to the band spin-nematic scenarioFernandes et al. (2012,
2014), where just the scattering rate contributes to the conductivity
anisotropy, we found that also the velocity contributes. The contribution of
the scattering rate to the resistivity anisotropy is dominated by the location
of the cold spots where the scattering rate is minimum (see Fig. 1), which,
within our model, is determined by the orbital composition of the FS and by
the spin-orbital interplay of the OSSF. The contribution of the velocity to
the resistivity anisotropy is counter-intuitive and opposite to the one of the
scattering rate. We find indeed that the conductivity is larger in the
direction where the self-energy is also larger. This interesting new effect is
due to an orbital character exchange in the pockets arising from the OSSF
self-energy in the nematic phase. Our study shows that the sign of the
anisotropy of the DC conductivity depends on whether scattering rate or
velocity anisotropy dominates on each pocket, as well as other parameters such
as the ellipticity and the quasiparticle renormalization due to local
interactionsde’ Medici (2015). Thus, different experimental results among the
various families of IBS can be explained within the same OSSF scenario.

Figure 1: General sketch of the orbital content of the Fermi surface of the
4-pocket model for iron superconductors. The green and red arrows show the
OSSF with $yz$ content in the x-direction and $xz$ content in the y-direction.
Cold spots, where the scattering rate is minimum, are shown by a circle and
they are found on the $xy$ and $xz$ orbitals in the nematic phase due to
anisotropic self-energy corrections. See text.

The structure of the paper is the following. In sec. I we introduce the OSSF
model. In sec. II we outline the calculation of the DC conductivity: in sec.
II A we derive analytical expressions for the DC anisotropy within the
perfectly-nested parabolic-band approximation; in sec. II B we discuss
numerical results obtained using realistic parameters for 122 compounds and
FeSe i.e. accounting for the effects of spin-orbit coupling and elliptical
e-pockets; in sec. II C we discuss our results in connection to experiments in
IBS. In sec. III we summarize our results and draw the conclusions of our
work.

##  I Model

The OSSF low–energy model has been derived in detailed in Ref. [Fanfarillo et
al., 2018]. Here we summarize the main features of the model, further details
can be found in App. A. The starting point is a general 4-pocket model with
two h-pockets at $\Gamma$, denoted as $\Gamma^{\pm}$, and two e-pockets at $X$
and $Y$. The model can be easily adapted to describe different compounds among
the 122 and 11 families. The kinetic part of the Hamiltonian is derived
following the low-energy approach considered in Ref. [Cvetkovic and Vafek,
2013], where each pocket is described using a spinor representation in the
orbital space.

$H_{0}^{l}=\sum_{{\bf k},\sigma}\psi^{\dagger l}_{{\bf k}\sigma}\hat{H}_{0{\bf
k}}^{l}\psi^{l}_{{\bf k}\sigma},\;\;\hat{H}^{l}_{0{\bf k}}=h_{0{\bf
k}}^{l}\hat{\tau_{0}}+\vec{h}^{l}_{{\bf k}}\cdot\vec{\hat{\tau}}$ (1)

with $l=\Gamma,X,Y$ and $\hat{\tau}$ the Pauli matrices representing the
orbital pseudo-spin. The spinors are defined as $\psi^{\Gamma}_{{\bf
k}\sigma}=(c^{yz}_{{\bf k},\sigma},c^{xz}_{{\bf k}\sigma})$ and
$\psi^{X/Y}_{{\bf k}\sigma}=(c^{yz/xz}_{{\bf k}\sigma},c^{xy}_{{\bf
k}\sigma})$. Rotating the Hamiltonian into the band basis we have

$H_{0}^{l}=\sum_{{\bf k},\sigma}E^{l\pm}_{{\bf k}}c^{\dagger l\pm}_{{\bf
k}\sigma}c^{l\pm}_{{\bf k}\sigma},$ (2)

where $E^{l\pm}_{{\bf k}}=h_{0{\bf k}}^{l}\pm h^{l}_{{\bf k}},\quad
h^{l}_{{\bf k}}=|\vec{h}_{{\bf k}}^{l}|$ are the band dispersions. The
fermionic band operators $c^{l\pm}$ are obtained rotating the orbital spinors
via an unitary matrix ${\hat{\cal U}}^{l}$. Explicitly for the h-pockets at
$\Gamma$, $c^{\Gamma\pm}\equiv h^{\pm}$ we have

$\begin{pmatrix}h^{+}\\\ h^{-}\\\
\end{pmatrix}=\begin{pmatrix}u^{\Gamma}&-v^{\Gamma}\\\
v^{*\Gamma}&u^{*\Gamma}\\\ \end{pmatrix}\begin{pmatrix}c^{yz}\\\ c^{xz}\\\
\end{pmatrix}={\hat{\cal U}}^{\Gamma}\begin{pmatrix}c^{yz}\\\ c^{xz}\\\
\end{pmatrix}$ (3)

where we have dropped the momentum and spin indices for simplicity. Analogous
expressions hold for the $X/Y$ e-pockets fermionic operators $c^{X/Y\pm}\equiv
e^{X/Y\pm}$. Since only the $E^{X/Y+}$ band crosses the Fermi level at $X/Y$,
in the following we will drop the $+$ subscript from $e^{X/Y}$.

The interacting Hamiltonian simplifies substantially once the spin-exchange
interaction is projected at low-energy Fanfarillo et al. (2015, 2018). The
generic intraorbital spin operator reads ${\bf S}^{\eta}_{{\bf q}}=\sum_{{\bf
k}\alpha\beta}(c^{\eta\dagger}_{{\bf
k}\alpha}\vec{\sigma}_{\alpha\beta}c^{\eta}_{{\bf k}+{\bf q}\beta})$, with
$\eta$ the orbital index and $\vec{\sigma}_{\alpha\beta}$ the Pauli matrices
for the spin. Thus, taking into account the orbital composition of the FS
shown in Fig. 1, the relevant intraorbital spin operator occur at momenta
${\bf q}$ near ${\bf Q}_{X}=(\pi,0)$ and ${\bf Q}_{Y}=(0,\pi)$, which connect
a h-pocket with the $X$ and $Y$ e-pockets respectively. Since the only common
orbital is the $yz/xz$ along the $x/y$ direction the spin interaction reduces
to

$H_{int}=-\frac{\tilde{U}}{2}\sum_{{\bf q}\\\ ^{\prime}}{\bf
S}^{yz/xz}_{X/Y}\cdot{\bf S}^{yz/xz}_{X/Y}.$ (4)

Here, $\tilde{U}$ is the intraorbital interaction renormalized at low energy
and ${\bf S}_{X/Y}^{\eta}\equiv\vec{S}_{{\bf q}={\bf Q}_{X/Y}}^{\eta}$. The
relevant magnetic fluctuations peaked at ${\bf Q}_{X}$/${\bf Q}_{Y}$ are
orbital selective, as sketched in Fig. 1, having only $yz$/$xz$ orbital
character:

$\displaystyle\langle{\bf S}\cdot{\bf S}\rangle({\bf Q}_{X})$
$\displaystyle\Rightarrow$ $\displaystyle\langle{\bf S}^{yz}_{X}\cdot{\bf
S}^{yz}_{X}\rangle$ (5) $\displaystyle\langle{\bf S}\cdot{\bf S}\rangle({\bf
Q}_{Y})$ $\displaystyle\Rightarrow$ $\displaystyle\langle{\bf
S}^{xz}_{Y}\cdot{\bf S}_{Y}^{xz}\rangle$ (6)

The SF exchange between h- and e-like pockets renormalizes the quasiparticles
via single-particle self-energy corrections. Within the OSSF model, due to the
orbital-selective nature of SF, this mechanism is also orbital dependent. The
self-energy corrections in the orbital basis can be computed within an
Eliashberg-like treatmentFanfarillo et al. (2016) for each pockets. For the
h-pockets at $\Gamma$ we find

$\hat{\Sigma}^{\Gamma}(\omega)=\begin{pmatrix}\Sigma^{\Gamma}_{yz}(\omega)&0\\\
0&\Sigma^{\Gamma}_{xz}(\omega)\\\
\end{pmatrix}=\Sigma^{\Gamma}_{0}(\omega)\hat{\tau}_{0}+\Sigma_{3}^{\Gamma}(\omega)\hat{\tau}_{3},$
(7)

with
$\Sigma^{\Gamma}_{0}(\omega)=(\Sigma^{\Gamma}_{yz}(\omega)+\Sigma^{\Gamma}_{xz}(\omega))/2$,
$\Sigma_{3}^{\Gamma}(\omega)=(\Sigma^{\Gamma}_{yz}(\omega)-\Sigma^{\Gamma}_{xz}(\omega))/2$,
while for the e-pockets we have

$\hat{\Sigma}^{X/Y}(\omega)=\begin{pmatrix}\Sigma^{X/Y}_{yz/xz}(\omega)&0\\\
0&0\\\
\end{pmatrix}=\Sigma^{X/Y}_{0}(\omega)\hat{\tau_{0}}+\Sigma_{3}^{X/Y}(\omega)\hat{\tau_{3}},$
(8)

with
$\Sigma^{X/Y}_{0}(\omega)=\Sigma_{3}^{X/Y}(\omega)=\Sigma^{X/Y}_{yz/xz}(\omega)/2$.
In the tetragonal state, above the structural transition, the isotropic SF
lead to equivalent self-energies for the $C_{4}$ symmetric $xz/yz$ orbitals,
i.e. $\Sigma^{\Gamma}_{3}=0$, and $\Sigma^{X}_{3}=\Sigma^{Y}_{3}$. In the
nematic phase the anisotropy of the OSSF below $T_{S}$ generates a
differentiation of the $xz/yz$ self-energy corrections

$\displaystyle\Sigma^{\Gamma}_{3}(\omega)$ $\displaystyle\neq$ $\displaystyle
0$ $\displaystyle\Sigma^{X}_{3}(\omega)$ $\displaystyle\neq$
$\displaystyle\Sigma^{Y}_{3}(\omega),$ (9)

and gives rise to an effective orbital nematicity. Eq.s (7)-(8) are generic
but their specific values are material dependent and for FeSe were calculated
at the RPA level in Ref. [Fanfarillo et al., 2016]. The dressed Green’s
functions are obtained via the Dyson equation $\hat{G}^{l^{-1}}({\bf
k},\omega)=\hat{G}_{0}^{l^{-1}}({\bf k},\omega)-\hat{\Sigma}^{l}(\omega)$
where $\hat{G}_{0}^{l^{-1}}({\bf
k},\omega)=\omega\hat{\mathbb{1}}-\hat{H}_{0}^{l}({\bf k})$. We diagonalize
the renormalized Green’s function via the unitary transformation ${\hat{\cal
U}}_{R}^{l}({\bf k},\omega)$ defined as

$\hat{G^{l}}({\bf k},\omega)={\hat{\cal U}_{R}^{l}}({\bf
k},\omega)(\omega\hat{\mathbb{1}}-\hat{\Lambda}_{R}^{l}({\bf
k},\omega))^{-1}{\hat{\cal U}}_{R}^{l^{-1}}({\bf k},\omega).$ (10)

where $\hat{\Lambda}_{R}^{l}=diag(E^{l+}_{R},E^{l-}_{R})$ and

$\displaystyle E_{R}^{l_{\pm}}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle h_{{\bf k}0}^{l}+\Sigma_{0}^{l}(\omega)\pm h^{l}_{R{\bf
k},\omega}$ $\displaystyle h^{l}_{R{\bf k},\omega}$ $\displaystyle=$
$\displaystyle\sqrt{(h_{3}^{l}+\Sigma_{3}^{l}(\omega))^{2}+(h^{l}_{1})^{2}+(h^{l}_{2})^{2}}.$
(11)

Since the self-energy is a complex function, Eq. (I) accounts both for the
renormalization of the band dispersion

$\epsilon_{R}^{l_{\pm}}({\bf k},\omega)=ReE_{R}^{l_{\pm}}({\bf k},\omega)$
(12)

and for the renormalized scattering rate

$\Gamma_{R}^{l_{\pm}}({\bf k},\omega)=\delta\Gamma+|ImE_{R}^{l_{\pm}}({\bf
k},\omega)|$ (13)

where we also added a residual constant broadening term $\delta\Gamma$. Eq.
(13) establishes a connection between the scattering rate and the self-energy
renormalizations contained in $E_{R}^{l_{\pm}}$. The qualitative behavior of
the self-energies in the nematic phase allows us to easily localize the
minimum value of $\Gamma_{R}^{l_{\pm}}$ on the FS, i.e. the cold spots shown
in Fig. 1. As we discussed in Ref. [Fanfarillo et al., 2016], the
reconstruction of the FS below $T_{S}$ is consistent with the Ising-nematic
spin-fluctuations being bigger at $\mathbb{Q}_{X}$ than at $\mathbb{Q}_{Y}$.
This implies that self-energy corrections are stronger on the $yz$ orbital
than on the $xz$ one. As a consequence on the h-pockets the smaller scattering
rate corresponds to the $xz$ orbital. On the e-pockets instead, the smaller
scattering rate is found for the $xy$ orbital, given the absence of $xy$-SF
within our model. The result is an example of the spin-orbital interplay
retained by the OSSF approach that allows us to directly link the cold spots
position with the FS orbital character and is not present in the band-based
spin-nematic scenarioFernandes et al. (2012).

The rotation matrix ${\hat{\cal U}_{R}^{l}}({\bf k},\omega)$ in Eq. (10), has
a structure analogous to Eq. (3). The renormalized coherence factors,
$u^{l}_{R}$ and $v^{l}_{R}$, whose exact expressions are given in App. A,
depend now on the self-energies Eq.s (7)-(8). The approximated expressions for
$u^{l}_{R{\bf k}}(\omega)$, $v^{l}_{R{\bf k}}(\omega)$ obtained at low energy
$\omega$ by expanding in $\Sigma^{l}_{3}$ at the first order read:

$\displaystyle\begin{split}|u^{l}_{R}(\omega)|^{2}=&|u^{l}|^{2}\left(1+\frac{2{Re}\Sigma_{3}^{l}(\omega)}{h^{l}}|v^{l}|^{2}\right),\\\
|v^{l}_{R}(\omega)|^{2}=&|v^{l}|^{2}\left(1-\frac{2{Re}\Sigma_{3}^{l}(\omega)}{h^{l}}|u^{l}|^{2}\right)\end{split}$
(14)

where we neglected the imaginary part of the self-energy which approaches to
zero at low energy and low temperature. $u^{l}$ and $v^{l}$ are the bare
coherence factors appearing in Eq.(3) and detailed in App. A. From Eq. (14)
one sees that the correction term $\sim{Re}\Sigma_{3}^{l}(\omega)$ mixes the
orbital character in each pocket, i.e. contribute to $u^{l}_{R}$ with a term
proportional to $v^{l}$ and viceversa. This effect of the OSSF self-energy in
the coherence factors will have important consequences for the renormalized
velocities as we will see in the following section.

##  II DC conductivity

The DC conductivity is the $\Omega\to 0$ limit of the longitudinal optical
conductivity given by

$\text{Re}\,\sigma_{\alpha}(\Omega)=-\frac{e^{2}}{V}\frac{\text{Im}\,\Pi_{\alpha}({\bf
q}=0,i\Omega_{m}\rightarrow\Omega)}{\Omega},$ (15)

where $\alpha=x,y$, $V$ is the unit-cell volume and $\Pi_{\alpha}$ is the
current-current correlation function that in the bubble approximation reads

$\displaystyle\Pi_{\alpha}({\bf q},{i\Omega_{m}})$ $\displaystyle=$
$\displaystyle 2T\sum_{l{\bf k}n}\text{Tr}\Big{[}\hat{G^{l}}({\bf k}-{\bf
q}/2,i\omega_{n})\hat{V}_{{\bf k}_{\alpha}}^{l}$ (16)
$\displaystyle\hat{G}^{l}({\bf k}+{\bf
q}/2,i\omega_{n}+i\Omega_{m})\hat{V}_{{\bf k}_{\alpha}}^{l}\Big{]}.$

$\hat{V}_{{\bf k}_{\alpha}}^{l}=\partial_{{\bf k}_{\alpha}}\hat{H}_{0}^{l}$ is
the bare velocity operator and $\hat{G^{l}}({\bf k},i\omega_{n})$ is the
renormalized Green’s function defined in Eq. (10). We rotate Eq.(16) in the
band basis using Eq. (10), perform the trace and take the $\Omega\rightarrow
0$ limit. The $\alpha=x,y$ component of the DC conductivity in the band basis,
${\sigma_{DC}}_{\alpha}=\sum_{l_{\pm}}\sigma^{l_{\pm}}_{\alpha}$, is obtained
from the sum over all the pockets $l_{\pm}=\Gamma_{+},\Gamma_{-},X,Y$. The
pocket conductivity reads (see App. B for further details)

$\sigma^{l_{\pm}}_{\alpha}=\frac{2\pi e^{2}}{N}\sum_{{\bf
k}}\int_{-\infty}^{\infty}d\omega(-\frac{\partial
f(\omega)}{\partial\omega})\big{(}{V}_{R{\bf
k}_{\alpha}}^{l_{\pm}}(\omega)\big{)}^{2}\big{(}A^{l_{\pm}}_{\bf
k}(\omega)\big{)}^{2}.$ (17)

In this basis the spectral function is diagonal

$A^{l_{\pm}}_{\bf
k}(\omega)=\frac{1}{\pi}\frac{\Gamma_{R}^{l_{\pm}}(\omega)}{(\Gamma_{R}^{l_{\pm}}(\omega))^{2}+(\omega-\epsilon_{R{\bf
k}}^{l_{\pm}}(\omega))^{2}}$ (18)

with $\Gamma_{R}^{l_{\pm}}(\omega)$ and $\epsilon_{R{\bf
k}}^{l_{\pm}}(\omega)$ defined in Eq.s (12)-(13). $V^{l_{\pm}}_{R{\bf
k}\alpha}$ in Eq.s (17) is the bare velocity operator rotated into the band
basis. As a consequence of the orbital structure we have

$V^{l_{\pm}}_{R{\bf k}\alpha}=V_{{\bf k}\alpha}^{l_{11}}|u^{l}_{R}|^{2}\pm
V_{{\bf k}\alpha}^{l_{12}}u^{*l}_{R}v^{*l}_{R}\pm V_{{\bf
k}\alpha}^{l_{21}}u^{l}_{R}v^{l}_{R}+V_{{\bf
k}\alpha}^{l_{22}}|v^{l}_{R}|^{2}$ (19)

Hereafter we omit the dependence on $\omega$ for simplicity. ${V_{{\bf
k}\alpha}^{l_{\eta\eta^{\prime}}}}$ are the $\eta\eta^{\prime}$ component of
the velocity and $({u/v})^{l}_{R}$ are the renormalized coherence factors. Via
the coherence factors $V^{l_{\pm}}_{R{\bf k}\alpha}$ depends on the
$\hat{\tau}_{3}$ component of the self-energies, $\Sigma_{3}^{l}(\omega)$ that
mixes the orbital content of each pocket. This effect can be easily understood
considering the approximated expressions in Eq.s (14). By neglecting the
imaginary part of the self-energy in $({u/v})^{l}_{R}$, Eq. (19) can also be
written as

$V^{l_{\pm}}_{R{\bf k}\alpha}=\partial\epsilon_{R}^{l_{\pm}}({\bf k})/\partial
k_{\alpha}.$ (20)

In the $T\rightarrow 0$ limit we can approximate the Fermi function with a
$\delta(\omega)$ which selects only states at the Fermi level $\omega=0$. By
further assuming $\Gamma_{R}^{l_{\pm}}$ to be small we can also approximate
the spectral functions with a delta function and Eq. (17) reduces to

$\sigma^{l_{\pm}}_{\alpha}=\frac{e^{2}}{N}\sum_{{\bf k}}\frac{({V}_{R{\bf
k}_{\alpha}}^{l_{\pm}})^{2}}{\Gamma_{R{\bf k}}^{l_{\pm}}}\
\delta(\epsilon_{R\mathbb{k}}^{l_{\pm}}).$ (21)

###  II.1 Analytical calculation

To gain physical insight on the DC anisotropy and disentangle the effect of
the velocity and scattering rate in Eq. (21), we estimate analytically
$\sigma_{\alpha}$. We approximate the h- and e-bands with perfectly nested
parabolic bands and assume that the nematic order is small enough to allow one
for a perturbative expansion of the renormalized energy $E_{R}^{l\pm}$. We use
a symmetric nematic splitting around the isotropic value $\Sigma^{l}_{0}$ in
the tetragonal phase. By expanding Eq.(I) at first order in the $\tau_{3}$
self-energy component we can estimate analytically for each pocket ${V}_{R{\bf
k}_{\alpha}}^{l_{\pm}}$ and $\Gamma^{l_{\pm}}_{R{\bf k}}$ via Eq.s (12), (13)
and (20).

Deriving for example with respect $k_{x}$ the renormalized energy of the
pocket $\Gamma_{+}$ we find

$V^{\Gamma_{+}}_{R{\bf
k}x}=-\frac{k\cos\theta}{\,m^{\Gamma_{+}}}+4\,Re\Sigma^{\Gamma_{+}}_{3}\sin^{2}\theta\frac{k\cos\theta}{k^{2}}.$
(22)

where $m^{\Gamma_{+}}$ is the bare mass of the $\Gamma_{+}$ pocket whose
definition in terms of the Hamiltonian parameters is given in App. B. The
first term on the r.h.s. of Eq.(22), is the $x$-component of the bare
velocity, while the second term ${\cal O}(Re\Sigma^{\Gamma}_{3})$ is an
additional contribution due to the orbital mixing induced by the nematic order
as expected from the $(u,v)^{l}_{R}$ factors in Eq.(19). To compute the ${\bf
k}$ integration in Eq. (21) we use the delta function and evaluate
$V^{\Gamma_{+}}_{R{\bf k}x}$ at the renormalized FS. Notice that, in the
nematic phase $k^{\Gamma_{+}}_{F}(\theta)$ is no longer constant but gets
deformed because of the anisotropic self-energy renormalization. This effect
is also of order ${\cal O}(Re\Sigma^{\Gamma}_{3})$ and has to be taken into
account. We estimate the change in the Fermi wave vector at the first order in
the self-energy. Replacing the expression of $k^{\Gamma_{+}}_{F}(\theta)$ into
Eq. (22) we find

${V}^{\Gamma_{+}}_{Rx}={V_{0}}^{\Gamma_{+}}_{x}\bigg{(}1+\cos
2\theta\frac{Re\Sigma^{\Gamma}_{3}}{2\epsilon^{h}_{0}}-4\sin^{2}\theta\frac{Re\Sigma^{\Gamma}_{3}}{2\epsilon^{h}_{0}}\bigg{)},$
(23)

where ${V_{0}}^{\Gamma_{+}}_{x}=-{k_{0}}_{Fx}^{\Gamma_{+}}/m^{\Gamma_{+}}$ and
$\epsilon^{h}_{0}=\epsilon^{\Gamma}+Re\Sigma_{0}^{\Gamma}$ are the velocity
and the Fermi energy in the tetragonal phase respectively. From Eq. (23), one
sees that the bare Fermi velocity in the nematic phase has two contributions
${\cal O}(Re\Sigma^{\Gamma}_{3})$ opposite in sign: the first one is due to
the change in ${k}_{F}^{\Gamma_{+}}$, while the second one comes from the
orbital mixing produced by the nematic order. Analogous calculation of the
velocity contributions along $y$ for the $\Gamma_{+}$ as well as for the other
pockets lead to similar expressions (see Eq.s (APPENDIX B: Computation of the
DC conductivity and analytical results for perfectly nested circular FS) in
App. B) with the band velocity of the tetragonal phase renormalized by two
additional contributions ${\cal O}(Re\Sigma^{l}_{3})$ of opposite sign. The
scattering rate is analytically estimated from Eq.(13) using again the
expansion of $E^{\Gamma_{+}}_{R}$ at the first order in $\Sigma^{l}_{3}$

$\Gamma^{\Gamma_{+}}_{R}(\theta)=\Gamma^{h}_{0}+cos2\theta|Im\Sigma^{\Gamma}_{3}|.$
(24)

Here we separate the tetragonal phase scattering rate,
$\Gamma^{h}_{0}=\delta\Gamma+|Im\Sigma^{\Gamma}_{0}|$, from the the angular-
dependent correction due to the nematic effect $\sim Im\Sigma^{\Gamma}_{3}$.
The explicit expression of the $\Gamma_{+}$ pocket DC conductivity follows
from Eq. (21) using Eq.s (23)-(24):

$\displaystyle\sigma_{x}^{\Gamma_{+}}$ $\displaystyle=$
$\displaystyle\sigma^{h}\bigg{(}1+\frac{{Re}\Sigma_{3}^{\Gamma}}{2\epsilon^{h}_{0}}-\frac{{Re}\Sigma_{3}^{\Gamma}}{\epsilon^{h}_{0}}-\frac{|{Im}\Sigma^{\Gamma}_{3}|}{2\Gamma^{h}_{0}}\bigg{)}$
(25) $\displaystyle=$
$\displaystyle\sigma^{h}\bigg{(}1-\frac{\Phi^{h}}{2\epsilon^{h}_{0}}+\frac{\Phi^{h}}{\epsilon^{h}_{0}}-\frac{\Delta\Gamma^{h}}{2\Gamma^{h}_{0}}\bigg{)}$

$\epsilon^{h}_{0}$, $\Gamma^{h}_{0}$ and
$\sigma^{h}=e^{2}\epsilon^{h}_{0}/(2\pi\hbar)\Gamma_{0}^{h}$ are respectively
the Fermi energy, the scattering rate and the DC conductivity in the
tetragonal phase. We also defined the real and imaginary part of the nematic
order parameter for the h-pockets ($\Phi^{h},\Delta\Gamma^{h}$) as

$\displaystyle\Phi{{}^{h}}\equiv\displaystyle\frac{Re\Sigma^{\Gamma}_{xz}-Re\Sigma^{\Gamma}_{yz}}{2}=-Re\Sigma^{\Gamma}_{3},$
$\displaystyle\Delta\Gamma{{}^{h}}\equiv|Im\Sigma^{\Gamma}_{3}|,$ (26)

taking also into account that stronger spin-fluctuation at ${\bf Q}_{X}$
implies ${Re}\Sigma_{3}^{\Gamma}<0$, so that now the nematic order parameters
are all positive defined.

Performing analogous calculations (see App. B) we derive the DC conductivities
along $x$ and $y$ for each pocket. For the h-pockets we find:

$\displaystyle\sigma_{x/y}^{\Gamma_{+}}$ $\displaystyle=$
$\displaystyle\sigma^{h}\bigg{(}1\mp\frac{\Phi^{h}}{2\epsilon^{h}_{0}}\pm\frac{\Phi^{h}}{\epsilon^{h}_{0}}\mp\frac{\Delta\Gamma^{h}}{2\Gamma^{h}_{0}}\bigg{)},$
$\displaystyle\sigma_{x/y}^{\Gamma_{-}}$ $\displaystyle=$
$\displaystyle\sigma^{h}\bigg{(}1\pm\frac{\Phi^{h}}{2\epsilon^{h}_{0}}\mp\frac{\Phi^{h}}{\epsilon^{h}_{0}}\pm\frac{\Delta\Gamma^{h}}{2\Gamma^{h}_{0}}\bigg{)}.$
(27)

In the absence of spin-orbit interaction the h-pockets have the same
$\epsilon_{0}^{h}$, so they also have the same conductivity $\sigma^{h}$ in
the tetragonal phase. Additional terms proportional to $\Phi^{h}$ and
$\Delta\Gamma^{h}$ arise in the nematic phase and make the conductivity
different for the two h-pocket. As extensively discussed within the
calculation of the velocity operator for the $\Gamma_{+}$ pocket in Eq. (23),
the nematic order has two opposite effects ${\cal O}(\Phi^{h})$ in the
velocity and this is reflected into the pocket DC conductivity anisotropy as
one sees from Eq.s (27). The first correction comes directly from the
$k^{\Gamma_{\pm}}_{F}$ changes due to the nematic FS reconstruction, while the
second one, opposite in sign, is due to the orbital mixing. Notice that this
last term also determine the overall sign of the correction $\sim\Phi^{h}$ in
each pocket. Due to the $xz/yz$ orbital arrangement of the $\Gamma_{\pm}$ FS,
the two h-pockets contribute with opposite sign to the conductivity anisotropy
i.e. in Eq. (27) we find the same sign of the nematic terms in the
conductivity along $x$ of the $\Gamma_{+}$ pocket and in the conductivity
along $y$ of the $\Gamma_{-}$ one. In particular the opposite sign of the
contribution ${\cal O}(\Delta\Gamma^{h})$ giving negative/positive anisotropy
for the $\Gamma_{+/-}$ pocket is a direct consequence of the cold-spots
physics, Fig. 1, from where we can easily infer the sign of the anisotropic
contribution for $\Gamma_{\pm}$ having in mind that lower scattering implies a
bigger conductivity. By computing the h-DC conductivity anisotropy,
$\Delta\sigma^{h_{\pm}}\equiv\sigma^{\Gamma_{\pm}}_{x}-\sigma^{\Gamma_{\pm}}_{y}$,
we find:

$\displaystyle\Delta\sigma^{h+}$ $\displaystyle=$
$\displaystyle\sigma^{h}\left(\frac{\Phi^{h}}{\epsilon^{h}_{0}}-\frac{\Delta\Gamma^{h}}{\Gamma^{h}_{0}}\right),$
$\displaystyle\Delta\sigma^{h-}$ $\displaystyle=$
$\displaystyle\sigma^{h}\left(-\frac{\Phi^{h}}{\epsilon^{h}_{0}}+\frac{\Delta\Gamma^{h}}{\Gamma^{h}_{0}}\right).$
(28)

The DC conductivity components for the e-pocket at $X$ read

$\displaystyle\sigma_{x}^{X}$ $\displaystyle=$
$\displaystyle\sigma^{e}\bigg{(}1-\frac{Re\Sigma^{X}_{yx}}{4\epsilon^{e}_{0}}+\frac{|Im\Sigma^{X}_{yz}|}{4\Gamma^{e}_{0}}\bigg{)},$
$\displaystyle\sigma_{y}^{X}$ $\displaystyle=$
$\displaystyle\sigma^{e}\bigg{(}1+\frac{3Re\Sigma^{X}_{yx}}{4\epsilon^{e}_{0}}-\frac{|Im\Sigma^{X}_{yz}|}{4\Gamma^{e}_{0}}\bigg{)}.$
(29)

As already done in Eq. (LABEL:holeDC1) we defined the nematic correction with
respect to the tetragonal $x/y$ DC conductivities. $\sigma_{x/y}^{e}$ are both
equivalent to $\sigma^{e}=e^{2}\epsilon^{e}_{0}/(2\pi\hbar)\Gamma_{0}^{e}$
since within the parabolic band approximation we neglect the ellipticity of
the e-pockets. The same expressions of Eq.s (29) hold for the $Y$ pocket once
replaced $\Sigma^{X}_{yz}\rightarrow\Sigma^{Y}_{xz}$ and $k_{x}\rightarrow
k_{y}$. Thus also the $X/Y$ pockets contribute with the opposite sign to the
overall DC conductivity. By defining the real and imaginary part of the
e-pocket nematic order parameter $(\Phi^{e},\Delta\Gamma^{e}$):

$\displaystyle\Phi{{}^{e}}\equiv\frac{Re\Sigma^{X}_{yz}-Re\Sigma^{Y}_{xz}}{2},\quad\quad\Delta\Gamma{{}^{e}}=\frac{|Im\Sigma^{X}_{yz}|-|Im\Sigma^{Y}_{xz}|}{2}$
(30)

we can write the electronic DC conductivity anisotropy
$\Delta\sigma^{e}\equiv\Delta\sigma^{X}+\Delta\sigma^{Y}$ as

$\Delta\sigma^{e}=\sigma^{e}\left(-\frac{\Phi^{e}}{\epsilon^{e}_{0}}+\frac{\Delta\Gamma^{e}}{\Gamma^{e}_{0}}\right)$
(31)

Also for the e-pockets we find that the renormalized velocity and the
scattering rate contribute with opposite sign to the DC conductivity
anisotropy. The balance between the two effects is controlled by the nematic
order parameters normalized to the Fermi energy and isotropic scattering rate,
respectively, i.e. $\Phi^{e}/\epsilon^{e}_{0}$ vs
$\Delta\Gamma^{e}/\Gamma^{e}_{0}$.

Summarizing, we computed analytically the anisotropy of the DC conductivity of
the various pockets using the parabolic-band approximation. We find for all
the pockets that the anisotropy is given by a contribution ${\cal
O}(Re\Sigma^{l})$ and another ${\cal O}(Im\Sigma^{l})$, opposite in sign with
respect to each other, whose relevance is controlled by the values of
$\Phi^{h/e}/\epsilon^{h/e}_{0}$ vs $\Delta\Gamma^{h/e}/\Gamma^{h/e}_{0}$.
Summing up the h-and e-pockets $\Delta\sigma^{h/e}$ we find that the sign of
the anisotropy of the total DC conductivity depends on which pocket
contributes more to the total conductivity and on which effect, among the
scattering rate and velocity renormalization, dominates. Within the perfectly
nested parabolic band approximation, in which the two h-pockets are
equivalent, their anisotropic contributions, Eq.s (28), are opposite in sign
and cancel out, so that the DC anisotropy is determined only by the e-pockets.
In this situation the overall sign of $\Delta\sigma$ depends on which effect
dominates in $\Delta\sigma^{e}$, Eq. (31) i.e. the anisotropy of the velocity
or the one of the scattering rate.

In real IBS systems, however, we need to account for the presence of the spin-
orbit interaction that splits the h-pockets at $\Gamma$ and mixes their
orbital content at the FS already in the tetragonal phase. Moreover, the
parabolic band approximation is particularly inaccurate for the e-pockets that
are strongly elliptical in all IBS. Furthermore, especially for FeSe, the
nematic self-energy components $\Sigma^{l}_{3}$ are not smallFanfarillo et al.
(2016), thus the expansion of the renormalized energy in $\Sigma^{l}_{3}$
performed above is not longer justified. For realistic cases then, we cannot
use the analytical expressions, Eq.s (23)-(24), and we need to compute the DC
conductivity from Eq.(21) using a numerical estimate of the velocity and
scattering rate from Eq.s (I)-(13) and (20).

###  II.2 Beyond the analytical approach

We perform a numerical estimate of the conductivity anisotropy using realistic
parameters for 122 and FeSe systems in order to assess the limits of validity
of the analytical expressions Eq.s (28) and (31) and qualitatively discuss our
results in the context of the experimental outcomes found for 122 pnictides
and FeSe. We assume for both 122 and FeSe equivalent band-structure parameters
that results in the tetragonal FS shown in Fig.1. The FS topology of FeSe with
just the outer h-pocket crossing the Fermi level at $\Gamma$ already in the
tetragonal phase is achieved in the calculation using a larger value of the
spin-orbit interaction as well as a larger values of the real part of the
self-energy renormalizations in agreement with previous analysisFanfarillo et
al. (2016). The numerical values of the parameters used in the following are
detailed in App. C.

#### 122 pnictides

Figure 2: Numerical computation of FS wavevectors and velocity components for
122 system parameters in the tetragonal and nematic phase.
$\Phi_{h}=\Phi_{e}=4$ meV, the spin-orbit interaction is $5$ meV, other band
parameters are detailed in App. C. The $k_{F}$ are measured in units $1/a\sim
0.375$ Å, where $a=a_{FeFe}$ is the lattice constant of the 1-Fe unit cell.
The velocities are in eV. Figure 3: Renormalized scattering rate (in meV)
computed using realistic 122 system parameters. Here $\Gamma^{h}_{0}=3$ meV
and $\Gamma^{e}_{0}=2$ meV. We fix $\Delta\Gamma^{h}$ and $\Delta\Gamma^{e}$
considering the imaginary part of the self-energy for each pocket changing
proportionally to the real part in the nematic phase (App. C). Figure 4:
Numerical computation of the velocity contribution to the DC conductivity
anisotropy $\Delta\sigma_{V}$ and of the total DC conductivity anisotropy
$\Delta\sigma$ for realistic parameter for 122. In panels a-b we renormalized
the pocket contributions to their value in the tetragonal phase i.e.
$\Delta\sigma_{V}^{l_{\pm}}(\Phi^{h/e}=0)$ and analogously in panels d-e. In c
and f panels instead we renormalize the h- and e-contributions to the total
tetragonal values, i.e. $\sigma_{Vtot}=\sigma_{V}(\Phi^{h/e}=0)$ and
$\sigma_{tot}=\sigma(\Phi^{h/e}=0)$.

In Fig. 2 we show for each pockets the FS wavevectors and velocities along
$x/y$. To better appreciate the changes induced by the nematic order we plot
in the first row the results for the tetragonal phase and in the second ones
the results obtained in the nematic phase assuming $\Phi^{h}=\Phi^{e}=4$ meV.
The h-pockets, circular in the tetragonal phase, are weakly deformed in the
nematic phase due to the small nematic order that also makes the $X/Y$ pockets
slightly different (a-d panel). The changes in the velocities for the
$\Gamma_{\pm}$ pockets appear to be quite small and do not follow monotonously
the renormalization of the Fermi vectors as one could have expected (e-f
panels). This is in agreement with the analytical calculation outlined above,
where we found that the renormalization in the velocities due to the orbital
mixing and the one coming from the Fermi vector renormalization are opposite
in sign, reducing the overall anisotropic effect on the velocity (see
Eq.(23)). Due to the ellipticity of the FS, the e-pockets have anisotropic
velocities already in the tetragonal phase (b-c panels) with the $X/Y$ pockets
showing larger velocity along $y/x$. No qualitatively changes are visible in
the nematic phase (e-f panels). In Fig. 3 we show for each pocket the
scattering rate obtained from Eq. (13). In the three panels we show the
tetragonal value, $\Gamma_{0}$, the nematic one $\Gamma_{R}$ and their
difference. We find again a good agreement between the analytical calculations
and the numerical results for the h-pockets. As in Eq. (24) the angular
dependence of the correction
$\Gamma^{\Gamma_{\pm}}_{R}-\Gamma^{\Gamma_{\pm}}_{0}$ goes almost as a $\cos
2\theta$, even if the weak ellipticity of the h-FS induced by the nematic
order causes minor deviations, e.g. the correction vanishes for the
$\Gamma_{+}/\Gamma_{-}$ slightly before/after $\pi/4$. No renormalizations are
found along $x/y$ for the $X/Y$ pockets since, within our model, no scattering
is allowed in the $xy$ channel (Fig.1). The location of the so-called cold
spots i.e. the position of the minima of the scattering rate for both h- and
e-pockets, does not change once a realistic FS are considered and corresponds
to the ones shown in Fig.1.

We can disentangle the effect of the velocity and of the scattering rate on
the DC anisotropy by computing Eq.(21) using a constant scattering rate. This
result just account for the anisotropic effects coming from the velocity so we
will refer to it as $\Delta\sigma_{V}$. In Fig. 4 we show for each pockets
$\Delta\sigma_{V}$ (a-c panels) and the complete conductivity anisotropy
$\Delta\sigma$ (d-f panels) as a function of $\Phi^{h/e}$. To easily compare
the results of the numerics with the analytical estimate of Eq.s (28) and (31)
we renormalized the $h_{\pm}/e$ pocket anisotropy in panels a-b and d-e to
their value in the tetragonal phase. In panel c and f we renormalize instead
the h- and e-anisotropy to the total values of $\sigma_{V}$ and $\sigma$
obtained summing all the pockets contributions in the tetragonal phase. From
the analysis of $\Delta\sigma_{V}$ we find that the sign of the anisotropic
contribution proportional to $\Phi^{h/e}$ found in Eq.s (28) and (31) is
robust, with the $\Gamma_{+/-}$ and the $Y/X$ pockets contributing with
positive/negative terms to the DC-conductivity anisotropy (see panels a and b
and inset of b). The h-pockets anisotropy due to the velocity, panels a-c, are
opposite in sign and grows as $\Phi^{h}/\epsilon_{0}^{h}$ in agreement with
the analytical expectation. Even if the $\Gamma_{\pm}$ are not longer
equivalent due to a small spin-orbit interaction their anisotropic
contributions almost cancel out so that the negative anisotropy of the
e-pocket is the one that determines the final results. Once the effect of the
scattering rate is included in the calculation we see in Fig.4 d a reduction
of the conductivity anisotropy for the h-pocket that however still sum up to
an anisotropic conductivity terms close to zero (panel f). In contrast a
change of sign in the overall electronic term is observed due to the larger
positive contribution $\Delta\sigma^{Y}$ of the $Y$ pocket once that the
anisotropic scattering rate is correctly taken into account. For the set of
parameters used, thus, we find a final $\Delta\sigma>0$. The result comes from
the change in the relative weight of the contribution of the $X$ and $Y$
pockets in the e-term due to the different scattering rate $\Gamma^{X/Y}_{R}$.
The final outcome is thus particularly sensitive to the $\Gamma^{e}_{0}$ and
$\Delta\Gamma^{e}$ used and could be strongly affected by any mechanism
(temperature, disorder, interactions, ect.) affecting their absolute values.

#### FeSe

We repeat the numerical analysis considering the case of FeSe. In Fig. 6 we
show the pockets FS wavevectors and velocities both in the tetragonal and in
the nematic phase assuming $\Phi^{h}=\Phi^{e}=15$ meV.

Figure 5: Numerical computation of the tetragonal and nematic FS and Fermi
velocity components for FeSe parameters. $\Phi_{h}=\Phi_{e}=15$ meV, the spin-
orbit interaction is $20$ meV, the other band parameters are detailed in App.
C.

Figure 6: Renormalized scattering rate for FeSe. $\Gamma^{h}_{0}=5$ meV and
$\Gamma^{e}_{0}=2.5$ meV. $\Delta\Gamma^{h}$ and $\Delta\Gamma^{e}$ are
considered as proportional to the nematic variation of the real parts (App. C)
Figure 7: Numerical computation of the velocity contribution to the DC
conductivity anisotropy and of the total DC conductivity anisotropy for
realistic parameter for FeSe. All the pockets show consistent deviations of
the DC anisotropy with respect the analytical expectations Eq.s (28) and (31).

With respect to the previous case, here we clearly see that the outer
h-pocket, the only one crossing the Fermi level, becomes strongly elliptical
in the nematic phase due to the large nematic order which also makes the $X/Y$
pockets consistently different in size (a-d panel). The changes in the
velocities (e-f panels) are similar to the ones observed for the 122 case but
quantitatively more pronounced here due to the larger value of the nematic
order parameters. The scattering rates for all the pockets are shown in Fig.
6. For all the pockets we find a clear deviation of the renormalized
scattering rate from the analytical estimate. In particular, the angular
dependence of the $\Gamma_{+}$ scattering rate is very weak and does not
resemble the $\cos 2\theta$ predicted by Eq.(24). This is a consequence of the
FS nematic reconstruction of FeSe. In fact, the nematic order not only makes
the $\Gamma_{+}$ pocket elliptical but also affect its orbital content that
becomes almost completely $xz$ at the Fermi levelFanfarillo et al. (2016);
Rhodes et al. (2018); Liu et al. (2018). As a consequence, the cold spots of
the outer pocket shown in Fig.1 do not represent anymore a minimum of the
scattering since the $\Gamma_{+}$ FS is mostly $xz$ also at $\theta=0$.

We study also in this case for each pocket the behavior of $\Delta\sigma_{V}$
and $\Delta\sigma$ as a function of $\Phi^{h/e}$, Fig. 7. We use the same
renormalizations used in Fig. 4. The analysis of the velocity contribution
reveals that the sign of the $\Phi^{h/e}$ terms of Eq.s (28),(31) is robust
also in this case. We are no longer in the perturbative regime as one can see
from the non-linear behavior of $\Delta\sigma_{V}^{h_{+}}$, shown in panel a,
where the $\Gamma_{+}$ pocket anisotropy contribution grows much faster than
what expected from the linear dependence in Eq. (28). The final anisotropy of
$\Delta\sigma_{V}$ is the result of the competition between the h- and
e-terms. The inclusion of the scattering rate in the calculation strongly
affect the e-pockets contribution while leaving $\Delta\sigma^{h_{+}}$ almost
unchanged. As a matter of fact, the scattering rate of the outer h-pocket
$\Gamma^{\Gamma_{+}}_{R}$ is almost isotropic, see Fig.6, so that the
anisotropic velocity is the only factor which contributes to the DC anisotropy
of the $\Gamma_{+}$ pocket. The final result for the DC conductivity strongly
depends on the set of $\Gamma^{h/e}_{0}$ and $\Delta\Gamma^{h/e}$ used. In
Fig.7 d-f, we show a case in which the inclusion of the scattering rate
enhances the relative weight of the e-contribution with respect the h-term, so
that overall the negative anisotropy of the e-part, due to the $X$ pocket,
determines the final results shown in Fig.7 f.

#### High-energy renormalization and nesting

An effect neglected in the above calculation is the renormalization of the
quasiparticle due to local electronic interactions. It is well established
that in IBS the high-energy renormalizations of the quasiparticle $Z_{orb}$
coming from local interactions are quite strong and orbital-dependent. This
effect, usually named orbital-selective renormalization in the
literature,Sprau et al. (2017) should not be confused with the effects
discussed in the present work, where the orbital selectivity refers to the
spin fluctuations, which affect the various orbital in a different way in the
nematic phase. The high-energy renormalizations have noticeable effects on the
optical conductivity in the tetragonal phase of IBS, as discussed in Ref.
[Calderón et al., 2014] and should be included in the above calculation. We
then repeted the numerical analysis including phenomenologically the orbital
renormalizations into the coherence factors $({u/v})^{l}_{R}$ entering in Eq.
(19). This allows us to estimate the effects of the high-energy
renormalization on the velocity contributions to the DC conductivity
$\Delta\sigma_{V}$. As expected, the inclusion of a severe reduction of the
coherence of the $xy$ orbital ($Z_{xy}\sim 0.3$), that is the most correlated
orbital in all IBS, leads to the suppression of the $V^{x/y}_{X/Y}$
contributions enhancing the DC anisotropy in the e-pockets. Moreover, the
small differentiation ($\sim 10\%$) of the quasiparticle masses for the
$xz/yz$ orbitals in the nematic phaseFanfarillo et al. (2017) contributes to
enhance the differentiation of the $\Gamma_{\pm}$ and $X/Y$ pockets. However,
the sign of the velocity contribution to the DC anisotropy is robust within
the set of $Z_{orb}$ considered. The quasiparticle renormalizations affect
also the conductivity via the renormalized scattering rate, however their
relevance strongly depends on the set of parameters used, whose analysis goes
beyond the scope of the present work.

Finally, it is worth noting that the analysis presented here, as well as the
one carried out previouslyFanfarillo et al. (2016) within a momentum-
independent Eliashberg scheme, does not include the physics of the band-
nesting effects, which are the only ones to determine the location of the hot-
spots in the band-based description Fernandes et al. (2010); Blomberg et al.
(2013). In particular the ellipticity of the $X/Y$ pockets suggests that the
scattering rate is maximum at the location where the nesting with the
h-pockets is realized. Within the OSSF both the orbital character and the
degree of band nesting will contribute to the hot-spot location. Recent
multiorbital RPA calculation in the paramagnetic state support the idea that
the dominant effect in determining the scattering rate is still the orbital
character of the FSKemper et al. (2011); Korshunov (2017). How these results
evolve below $T_{S}$ within an orbital-selective spin-nematic scenario is
still an open question which certainly deserves further investigation.

###  II.3 Comparison with experiments

In the previous section we computed numerically the DC anisotropy for
realistic parameters of 122 pnictides and FeSe. The values of the band
structure parameters and self-energies used in the calculations quantitatively
reproduce the main features of the FS, including the FS shrinking and the
orbital FS reconstruction experimentally observed in the nematic phase of 122
and FeSe. In 122, where the nematic order parameters $\Phi^{h/e}$ are small,
the h-pockets contribution to the DC conductivity anisotropy is well
approximated by the analytical estimate Eq. (28), while we observe consistent
deviations in FeSe. Nonetheless, for both 122 and FeSe systems the sign of the
anisotropic contribution coming from the renormalized velocity,
$\Delta\sigma_{V}$, is robust. In both cases considered in sec. II.2 we
managed to match the experimental result, $\Delta\sigma_{DC}(FeSe)<0$ and
$\Delta\sigma_{DC}(122)>0$, once the renormalization of the scattering rate is
included in the calculation. As already mentioned, the final result is still
somehow sensitive to the set of parameters used. Thus, in this last section we
discuss in general which are the possibilities to match the experimental
results regardless the precise choice of parameters used in sec. II.2.

Concerning 122 systems, as long as the h-pockets contributions to the DC
anisotropy cancel out, the final result is controlled by the e-pockets. Since
they have a strong elliptical deformation, their overall contribution to the
DC anisotropy cannot be predicted from the analytical result Eq.(31), and the
final outcome depends on the relative weight of the $X$ and $Y$ pockets and on
the relevance of the scattering-rate anisotropy over the contribution
$\Delta\sigma^{e}_{V}$. Even in doped 122 compounds the h-pocket contributions
cancellation still occurs since the relatively small value of the spin-orbit
splitting at $\Gamma$ guarantees that the Fermi energy is the same for the
h-pockets. However, doping changes both the size of the pockets and the degree
of nesting between h- and e-pockets. Both effects contribute to change the
relative weight of the $X/Y$ e-pockets as well as the balance between the
velocity vs scattering-rate anisotropic contributions and can be at the origin
of the different sign of $\Delta\sigma_{DC}$ experimentally observed between
the h- and e-doped side of 122 phase diagram.

For what concerns FeSe, the presence of a single h-pocket and its strong
orbital reconstruction lead to rather different physics. In particular, since
the nematic FS reconstruction makes the whole $\Gamma_{+}$ FS mostly $xz$ even
at $\theta=0$, the expected anisotropy of the renormalized scattering rate is
absent, see Fig. 6. As a consequence the $\Gamma_{+}$ anisotropic contribution
is more likely controlled by the velocity anisotropy. This result should be
contrasted with the outcomes of Ref. [Onari and Kontani, 2017], where the
difference between 122 and FeSe is fully ascribed to a different behavior of
the scattering-rate anisotropy in the two compounds. In our picture the FeSe
DC-conductivity anisotropy emerges from a subtle interplay between the
competing effects coming from the scattering rate and the velocity, as it has
been recently suggested by optical conductivity experiments in FeSeChinotti et
al. (2018). It is worth noting that recent ARPES experiments reveal a strong
$k_{z}$ dependence of the orbital composition of the $\Gamma_{+}$
FSKushnirenko et al. (2018), with the FS a $k_{z}=\pi$ recovering
$yz$-character at $\theta=0$. As a consequence, also the scattering rate
anisotropy on $\Gamma_{+}$ is expected to be larger at $k_{z}=\pi$ and its
effect on the DC-conductivity anisotropy can possibly compete with the
velocity term at this $k_{z}$. This observation calls for a more complete
analysis of the DC anisotropy involving also the $k_{z}$-dependence of the FS.

##  III Conclusions

In conclusion we computed the DC-conductivity anisotropy in the nematic phase
of IBS using the orbital-selective spin-nematic scenario that accounts for the
orbital content of the FSFanfarillo et al. (2015, 2018). In this scenario, the
DC anisotropy of the nematic phase of IBS depends on the scattering rate and
velocity renormalizations due to self-energy corrections. Both scattering rate
and velocity are affected by the FS nematic reconstruction. The scattering
rate is strongly affected by the orbital content of the FS, and the location
of its minima on the FS is found in correspondence of the less renormalized
orbitals giving rise to cold spots. The velocity renormalization is sensitive
both to the orbital mixing and to the shrinking of the FS induced by the
nematic order, with the former effect dominating over the latter. Due to this
effect we find the unexpected result that the conductivity increases in the
direction in which the self-energy is larger and the shrinking is stronger.
For both h- and e-carriers the contribution of the velocity to the DC
anisotropy is opposite in sign to the one of the scattering rate. This is in
agreement with recent optical conductivity experiment in FeSeChinotti et al.
(2018) where it is shown that scattering rate and velocity contribute to the
conductivity anisotropy with opposite signs. Our results naturally follows
from the spin-orbital entanglement implicit in the OSSF model and are new
results in contrast to the band spin-nematic scenarioFernandes et al. (2012,
2014). In particular we demonstrated that the usual expectation of anisotropic
magnetic fluctuations giving rise only to an anisotropy in the inelastic
scattering rateFernandes et al. (2010) is not longer valid once the orbital
degree of freedom is taken into account in the theoretical description.

We performed numerical calculation for representative parameters for 122
pnictides and FeSe. We verified that for 122 system the analytical estimate
represents a good approximation of the numerical with the overall h-pockets
contribution vanishing even once a finite spin-orbit splitting at $\Gamma$ is
considered. Numerical results for FeSe instead deviate from the analytical
expectations due to the huge nematic FS reconstruction. We also discuss how
the conductivity anisotropy depends on the system parameters. It can be
dominated by either electron or hole pocket conductivity and depends on
ellipticity and high-energy renormalizations. The OSSF scenario provides then
a suitable framework where the same mechanism due to orbital-spin interplay
can reconcile the experimental observations reported in different families of
iron-based superconductors.

## ACKNOWLEDGMENTS

We acknowledge L. Degiorgi for useful discussions. L.B. acknowledges financial
support by Italian MAECI under the collaborative Italia-India project
SuperTop-PGR04879. B.V. acknowledges funding from MINECO (Spain) via Grants
No.FIS2014-53219-P and Fundación Ramón Areces. L.B., L.F. and B.V acknowledge
the cost action Nanocohybri CA16218.

## APPENDIX A: Green’s function and perfectly nested parabolic band
approximation.

In this appendix we calculate the dressed Green’s function to be used in the
DC conductivity. We also detail the perfect nested parabolic band
approximation to perform the analytical calculation of the conductivity.

### Green’s function

The bare Hamiltonian consists in a 4-pocket model with two h-pockets at
$\Gamma$, $\Gamma_{\pm}$ and two e-pockets at $X$ and $Y$ adapted from the
low-energy model considered in Ref. [Cvetkovic and Vafek, 2013]. Each pocket
is described using a spinor representation in the pseudo-orbital space.

$H_{0}^{l}=\sum_{{\bf k},\sigma}\psi^{\dagger l}_{{\bf k}\sigma}\hat{H}_{0{\bf
k}}^{l}\psi^{l}_{{\bf k}\sigma},$ (32)

where $l=\Gamma,X,Y$ and the spinors are defined as: $\psi^{\Gamma}_{{\bf
k}\sigma}=(c^{yz}_{{\bf k},\sigma},c^{xz}_{{\bf k}\sigma})$ and
$\psi^{X/Y}_{{\bf k}\sigma}=(c^{yz/xz}_{{\bf k}\sigma},c^{xy}_{{\bf
k}\sigma})$. The matrix $\hat{H}_{0{\bf k}}^{l}$ has the general form

$\hat{H}_{0{\bf
k}}^{l}=h_{0}^{l}{\color[rgb]{0,0,0}\hat{\tau_{0}}}+\vec{h}^{l}\cdot{\color[rgb]{0,0,0}\vec{\hat{\tau}}}=\begin{pmatrix}h_{0}^{l}+h_{3}^{l}\
\ &\ \ h_{1}^{l}-ih_{2}^{l}\\\ h_{1}^{l}+ih_{2}^{l}\ \ &\ \
h_{0}^{l}-h_{3}^{l}\\\ \end{pmatrix}$ (33)

with $\hat{\tau}$ matrices representing the pseudo-orbital spin. The
$h^{\Gamma}$ components read as

$\displaystyle h_{0}^{\Gamma}$ $\displaystyle=$
$\displaystyle\epsilon{{}^{\Gamma}}-a{{}^{\Gamma}}{\bf k}^{2},$ $\displaystyle
h_{1}^{\Gamma}$ $\displaystyle=$ $\displaystyle-2b{{}^{\Gamma}}k_{x}k_{y},$
$\displaystyle h_{3}^{\Gamma}$ $\displaystyle=$ $\displaystyle
b{{}^{\Gamma}}(k_{x}^{2}-k_{y}^{2}),$ (34)

and for the $X$ pocket,

$\displaystyle h_{0}^{X}$ $\displaystyle=$
$\displaystyle(h{{}^{yz}}+h^{xy})/2$ $\displaystyle h_{2}^{X}$
$\displaystyle=$ $\displaystyle vk_{y}$ $\displaystyle h_{3}^{X}$
$\displaystyle=$ $\displaystyle(h{{}^{yz}}-h^{xy})/2-b(k_{x}^{2}-k_{y}^{2})$
(35)

where $h{{}^{yz}}=-\epsilon{{}^{yz}}+a{{}^{yz}}{\bf k}^{2}$ and
$h{{}^{xy}}=-\epsilon{{}^{xy}}+a{{}^{xy}}{\bf k}^{2}$. Analogous expressions
hold for the $Y$ pocket provided that one exchange $k_{x}$ by $k_{y}$.
Rotating the Hamiltonian $\hat{H}_{0{\bf k}}^{l}$ into the band basis we have
$\hat{H}_{0{\bf k}}^{l}=\hat{\cal U}{{}^{l^{-1}}}\hat{\Lambda}^{l}\hat{\cal
U}^{l}$ where $\hat{\Lambda}{{}^{l}}=diag(E^{l+}_{{\bf k}},E^{l-}_{{\bf k}})$
is the eigenvalue matrix and $E^{l\pm}_{{\bf k}}$ is given by

$E^{l\pm}_{{\bf k}}=h_{0}^{l}\pm
h^{l}=h_{0}^{l}\pm\sqrt{(h^{l}_{1})^{2}+(h^{l}_{2})^{2}+(h^{l}_{3})^{2}},$
(36)

The fermionic band operators $c^{l{\color[rgb]{0,0,0}\pm}}$ from
$H_{0}^{l}=\sum_{{\bf k},\sigma}E^{l\pm}_{{\bf k}}{c}^{\dagger
l{\color[rgb]{0,0,0}\pm}}_{{\bf k}\sigma}c^{l{\color[rgb]{0,0,0}\pm}}_{{\bf
k}\sigma}$ are obtained rotating the orbital spinors via $\hat{\cal U}^{l}$.
The unitary $\hat{\cal U}^{l}$ matrix has the common form:

$\hat{\cal U}^{l}=\begin{pmatrix}u^{l}&-v^{l}\\\ v^{*l}&u^{*l}\\\
\end{pmatrix}$ (37)

where we drop the momentum and spin indices for simplicity. Explicitly for the
h-pockets at $\Gamma$, $c^{\Gamma{\color[rgb]{0,0,0}\pm}}\equiv
h^{{\color[rgb]{0,0,0}\pm}}$ we have

$\begin{pmatrix}h{{}^{+}}\\\ h{{}^{-}}\\\
\end{pmatrix}=\begin{pmatrix}u{{}^{\Gamma}}&-v{{}^{\Gamma}}\\\
v{{}^{*\Gamma}}&u{{}^{*\Gamma}}\\\ \end{pmatrix}\begin{pmatrix}c{{}^{yz}}\\\
c{{}^{xz}}\\\ \end{pmatrix}={\color[rgb]{0,0,0}{\hat{\cal
U}}^{\Gamma}}\begin{pmatrix}c{{}^{yz}}\\\ c{{}^{xz}}\\\ \end{pmatrix}$ (38)

Analogous expressions hold for the $X/Y$ e-pockets fermionic operators
$c^{X/Y{\color[rgb]{0,0,0}\pm}}\equiv e{{}^{X/Y\pm}}$, provided that the
corresponding orbital spinors $\psi^{X/Y}=(c{{}^{yz/xz}},c{{}^{xy}})$ are
used. The coherence factors are:

$\displaystyle u^{l}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\sqrt{1+\frac{h_{3}^{l}}{h^{l}}},$
$\displaystyle v^{*l}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\frac{h_{1}^{l}+ih_{2}^{l}}{\sqrt{(h_{1}^{l})^{2}+(h_{2}^{l})^{2}}}\sqrt{1-\frac{h_{3}^{l}}{h^{l}}}.$
(39)

Knowing the dispersion relation and the coherence factors inside the rotation
matrix ${\color[rgb]{0,0,0}\hat{\cal U}^{l}}$ we can obtain the bare pocket
Green’s functions. Since
$\hat{G}_{0}{{}^{l^{-1}}}=\omega{\color[rgb]{0,0,0}\hat{\mathbbm{1}}}-\hat{H}_{0{\bf
k}}^{l}$ simply $\hat{G}_{0}{{}^{l^{-1}}}={\color[rgb]{0,0,0}{\hat{\cal
U}}^{l^{-1}}}(\omega{\color[rgb]{0,0,0}\hat{\mathbbm{1}}}-{\color[rgb]{0,0,0}\hat{\Lambda}}^{l}){{\color[rgb]{0,0,0}\hat{\cal
U}}}^{l}$ with
${\color[rgb]{0,0,0}\hat{\Lambda}}^{l}=diag(E^{l_{+}},E^{l_{-}}).$

We now turn our attention to the OSSF self-energy. The low-energy interacting
Hamiltonian describing the spin exchange between h- and e-pockets is given in
the main text Eq. (4). The corresponding Dyson equation for each pocket is:

${\color[rgb]{0,0,0}\hat{G}^{l^{-1}}({\bf
k},\omega)}={\color[rgb]{0,0,0}\hat{G}_{0}^{l^{-1}}({\bf
k},\omega)}-\hat{\Sigma}^{l}(\omega),$ (40)

where $\hat{G}_{0}{{}^{l^{-1}}}({\bf
k},\omega)=\omega{\color[rgb]{0,0,0}\hat{\mathbbm{1}}}-\hat{H}_{0{\bf k}}^{l}$
and
$\hat{\Sigma}^{l}(\omega)=\Sigma^{l}_{0}(\omega)\hat{\tau}_{0}+\Sigma_{3}^{l}(\omega)\hat{\tau}_{3}$
is the self-energy due to the OSSF defined in the main text. From now on we do
not display the (${\bf k},\omega$) dependence. We can define the renormalize
Hamiltonian as $\hat{H}^{l}=\hat{H}^{l}_{0}+\hat{\Sigma}^{l}$. Rotating the
Hamiltonian $\hat{H}^{l}$ into the band basis we have
$\hat{H}^{l}={{\color[rgb]{0,0,0}\hat{\cal
U}_{R}}^{l^{-1}}}{\color[rgb]{0,0,0}\hat{\Lambda}}_{R}^{l}{{\color[rgb]{0,0,0}\hat{\cal
U}}_{R}}^{l}$ where
${\color[rgb]{0,0,0}\hat{\Lambda}}_{R}^{l}=diag(E_{R}^{l_{+}},E_{R}^{l_{-}})$
is the eigenvalue matrix and $E_{R}^{l_{\pm}}$ is given by

$\displaystyle E_{R}^{l_{\pm}}$ $\displaystyle=$ $\displaystyle
h_{0}^{l}+\Sigma^{l}_{0}\pm h_{R}^{l}=h_{0}^{l}+\Sigma^{l}_{0}+$ (41)
$\displaystyle\pm\sqrt{(h^{l}_{1})^{2}+(h^{l}_{2})^{2}+(h^{l}_{3}+\Sigma_{3}^{l})^{2}}.$

Since the self-energy given in Eq. (40) is complex, we define the renormalized
energy dispersion relation as $\epsilon_{R}^{l_{\pm}}=ReE_{R}^{l_{\pm}}$ and
the renormalized scattering rate as
$\Gamma_{R}^{l_{\pm}}={\color[rgb]{0,0,0}\delta\Gamma}+|ImE_{R}^{l_{\pm}}|$
with $\delta\Gamma$ some scattering background. The unitary $\hat{\cal
U}_{R}^{l}$ matrix has a structure analogous to Eq. (37) provided that the
coherence factors are given by

$\displaystyle u^{l}_{R}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\sqrt{1+\frac{h_{3}^{l}+\Sigma_{3}^{l}}{h_{R}^{l}}},$
$\displaystyle v^{*l}_{R}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\frac{h_{1}^{l}+ih_{2}^{l}}{\sqrt{(h_{1}^{l})^{2}+(h_{2}^{l})^{2}}}\sqrt{1-\frac{h_{3}^{l}+\Sigma_{3}^{l}}{h_{R}^{l}}}.$

We can take into account in our model the presence of a finite spin-orbit
interaction by replacing $h_{R}^{\Gamma}\to
h_{R,SO}^{\Gamma}=\sqrt{(h^{\Gamma}_{1})^{2}+(h^{\Gamma}_{3}+\Sigma_{3}^{\Gamma})^{2}+(\lambda^{2}/4)}$.
This affects both the $E_{R}^{\Gamma_{\pm}}$ and the coherence factors
$(u/v)^{\Gamma}_{R}$.

### Perfectly nested circular Fermi surfaces model

For the analytical approach, we write the bare pocket Hamiltonian Eq. (33) in
polar coordinates with $\theta=\arctan k_{y}/k_{x}$, for the hole

$\displaystyle h_{0}^{\Gamma}$ $\displaystyle=$
$\displaystyle\epsilon^{\Gamma}-ak^{2};$ $\displaystyle h_{1}^{\Gamma}$
$\displaystyle=$ $\displaystyle-bk^{2}\sin(2\theta);$ $\displaystyle
h_{3}^{\Gamma}$ $\displaystyle=$ $\displaystyle bk^{2}\cos(2\theta)$ (43)

and for the electron pockets

$\displaystyle h_{0}^{X/Y}$ $\displaystyle=$
$\displaystyle-\epsilon^{X/Y}+ak^{2};$ $\displaystyle h_{2}^{X/Y}$
$\displaystyle=$ $\displaystyle bk^{2}\sin(2\theta);$ $\displaystyle
h_{3}^{X/Y}$ $\displaystyle=$ $\displaystyle\mp bk^{2}\cos(2\theta)$ (44)

For simplicity let us further assume
$\epsilon^{X}\equiv\epsilon^{Y}\equiv\epsilon^{e}$. It is easy to check that
the orbital content of the 4-pocket model for IBS in Fig. 1 is reproduced in
the parabolic approximation. Using Eq. (Green’s function) the coherence
factors take the expression

$\displaystyle\begin{split}|u^{\Gamma}|^{2}&=|u^{Y}|^{2}=|v^{X}|^{2}=\cos\theta^{2},\\\
|v^{\Gamma}|^{2}&=|v^{Y}|^{2}=|u^{X}|^{2}=\sin\theta^{2}\end{split}$ (45)

To allow analytical treatment, we expand $E_{R}^{l_{\pm}}$ and $(u/v)^{l}_{R}$
Eq.(Green’s function) Eq.(41) up to first order in $\Sigma_{3}$ and we neglect
the imaginary part of the self-energy since goes to zero at low $\omega$ and T
= 0. The dressed coherence factors are given by:

$\displaystyle\begin{split}|u^{l}_{R}|^{2}=&|u^{l}|^{2}\left(1+\frac{2{Re}\Sigma_{3}^{l}}{h^{l}}|v^{l}|^{2}\right),\\\
|v^{l}_{R}|^{2}=&|v^{l}|^{2}\left(1-\frac{2{Re}\Sigma_{3}^{l}}{h^{l}}|u^{l}|^{2}\right)\end{split}$
(46)

that are the expressions quoted in the main text in Eq. (14). The dressed
dispersion relations $\epsilon_{R}^{l_{\pm}}=ReE_{R}^{l_{\pm}}$ become:

$\epsilon_{R}^{l_{\pm}}=\epsilon^{l_{\pm}}+Re\Sigma^{l}_{0}\pm\frac{h^{l}_{3}}{h^{l}}Re\Sigma_{3}^{l}.$
(47)

Replacing the values for the case of circular FS given in Eq. (43) and Eq.
(44) and calculate it at the Fermi surface we get:

$\displaystyle\epsilon_{R}^{\Gamma_{\pm}}$ $\displaystyle=\
\epsilon_{0}^{h}\pm\cos 2\theta Re\Sigma_{3}^{\Gamma},$ (48)
$\displaystyle\epsilon_{R}^{{X/Y}}$ $\displaystyle=\
{\color[rgb]{0,0,0}-}\epsilon_{0}^{e}\mp\cos 2\theta Re\Sigma_{3}^{X/Y},$

where we defined the tetragonal band energy for the h-pockets as
$\epsilon_{0}^{h}=\epsilon^{\Gamma}+Re\Sigma^{\Gamma}_{0}$ and
$\epsilon_{0}^{e}=\epsilon^{e}{\color[rgb]{0,0,0}-}Re\Sigma^{X/Y}_{0}$ for the
e-pockets. The scattering rate
$\Gamma_{R}^{l_{\pm}}=\delta\Gamma+|ImE_{R}^{l_{\pm}}|$ acquires the
expression:

$\Gamma_{R}^{l_{\pm}}\approx\delta\Gamma+|Im\Sigma^{l}_{0}|\pm\frac{h^{l}_{3}}{h^{l}}|Im\Sigma_{3}^{l}|,$
(49)

Using Eq. (43) and Eq. (44) we get:

$\displaystyle\Gamma^{\Gamma_{\pm}}_{R}$ $\displaystyle=\
\Gamma^{h}_{0}\pm\cos 2\theta|{Im}{\color[rgb]{0,0,0}\Sigma_{3}}^{\Gamma}|,$
(50) $\displaystyle\Gamma^{X/Y}_{R}$ $\displaystyle=\ \Gamma^{e}_{0}\mp\cos
2\theta|{Im}{\color[rgb]{0,0,0}\Sigma_{3}}^{X/Y}|,$

where we have separated the angular dependent renormalization $\sim
Im\Sigma^{l}_{3}$ from the tetragonal constant part,
$\Gamma^{h/e}_{0}=\delta\Gamma+|{Im}\Sigma_{0}^{\Gamma/e}|$. From Eq. (50) we
find analytically the locations of the cold spots on the FS where the minimum
value of the scattering rate is found:

$\displaystyle\Gamma^{\Gamma_{+}}_{R}(\theta=\frac{\pi}{2})$ $\displaystyle=$
$\displaystyle\delta\Gamma+|{Im}\Sigma^{\Gamma}_{xz}|,$
$\displaystyle\Gamma^{\Gamma_{-}}_{R}(\theta=0)$ $\displaystyle=$
$\displaystyle\delta\Gamma+|{Im}\Sigma^{\Gamma}_{xz}|,$
$\displaystyle\Gamma^{X/Y}_{R}(\theta=0)$ $\displaystyle=$
$\displaystyle\Gamma^{Y}_{R}(\theta=\frac{\pi}{2})=\delta\Gamma$ (51)

This result is easy to understand due to the OSSF shown in Fig. 1 where cold
spots are also shown. In the nematic phase, SF with momentum
${\color[rgb]{0,0,0}\mathbb{Q}}_{X}$ are bigger than the ones with
${\color[rgb]{0,0,0}\mathbb{Q}}_{Y}$. As a consequence, the largest scattering
is found in the $yz$ orbital due to the orbital selective nature of SF, while
the $xy$-orbital component is not renormalized given the absence of $xy$-OSSF.

## APPENDIX B: Computation of the DC conductivity and analytical results for
perfectly nested circular FS

We calculate the DC conductivity in the bubble approximation. The self-energy
corrections computed within an Eliashberg-like treatment are momentum
independent so that vertex corrections vanish identically. The DC conductivity
is given by

$\text{Re}\,\sigma_{\alpha}(\Omega)=-\frac{e^{2}}{V}\frac{\text{Im}\,\Pi_{\alpha}(\mathbb{q}=0,i\Omega_{m}\rightarrow\Omega)}{\Omega}$
(52)

where $\alpha=x,y$, V is the unit-cell volume and $\Pi_{\alpha}$ is the
current-current correlation function in the bubble approximation given by

$\Pi_{\alpha}({i\Omega_{m}})=2T\sum_{l{\bf
k}n}\text{Tr}\left[\hat{G{{}^{l}}}({\bf
k},i\omega_{n}){\color[rgb]{0,0,0}\hat{V}_{{\bf
k}_{\alpha}}^{l}}\hat{G}{{}^{l}}({\bf k},i\omega_{n}+i\Omega_{m})\hat{V}_{{\bf
k}_{\alpha}}^{l}\right].$ (53)

$\hat{V}_{{\bf k}_{\alpha}}^{l}=\partial_{{\bf
k}{{}_{\alpha}}}\hat{H}_{0}^{l}$ is the bare velocity operator and
$\hat{G{{}^{l}}}({\bf k},i\omega_{n})$ is the renormalized Green’s function
from the orbital-selective spin fluctuations. We work in the rotated basis and
replace ${\color[rgb]{0,0,0}\hat{G}^{l}}({\bf
k},i\omega_{n})={{\color[rgb]{0,0,0}\hat{\cal U}_{R}^{l}}}({\bf
k},i\omega_{n})(i\omega_{n}{\color[rgb]{0,0,0}\hat{\mathbb{1}}}-{\color[rgb]{0,0,0}\hat{\Lambda}}_{R}^{l}({\bf
k},\omega))^{-1}{\color[rgb]{0,0,0}{\hat{\cal U}}_{R}^{l^{-1}}}({\bf
k},i\omega_{n})$ in Eq. (53). Since we are interested in the DC conductivity
($\Omega\rightarrow 0$) the important term in the trace is

Tr $\displaystyle\big{[}$ $\displaystyle{\hat{\cal U}_{R}^{l}}({\bf
k},i\omega_{n})(i\omega_{n}\hat{\mathbb{1}}-\hat{\Lambda}_{R}^{l}({\bf
k},\omega))^{-1}{\hat{\cal U}_{R}^{l^{-1}}}({\bf k},i\omega_{n})\hat{V}_{{\bf
k}_{\alpha}}^{l}$ $\displaystyle{\hat{\cal U}_{R}^{l}}({\bf
k},i\omega_{n})(i\omega_{n}\hat{\mathbb{1}}-\hat{\Lambda}_{R}^{l}({\bf
k},\omega))^{-1}{\hat{\cal U}}_{R}^{l^{-1}}({\bf k},i\omega_{n})\hat{V}_{{\bf
k}_{\alpha}}^{l}\big{]}$

Using the cyclic property of the trace allows us to define the renormalized
velocity ${\color[rgb]{0,0,0}\hat{V}}_{R{\bf
k}_{\alpha}}^{l}={\color[rgb]{0,0,0}{\hat{\cal U}}_{R}^{l^{-1}}}({\bf
k},i\omega_{n}){\color[rgb]{0,0,0}\hat{V}_{{\bf
k}_{\alpha}}^{l}}{{\color[rgb]{0,0,0}\hat{\cal U}_{R}^{l}}}({\bf
k},i\omega_{n})$. Operating the trace and taking the limit $\Omega\to 0$ we
get that the trace in Eq. (APPENDIX B: Computation of the DC conductivity and
analytical results for perfectly nested circular FS) can be written as

$\displaystyle\begin{split}g_{+}^{2}({\bf
k},i\omega_{n}){\color[rgb]{0,0,0}(V_{R{\bf k}_{\alpha}}^{l_{+}})^{2}}\ +\
g_{-}^{2}(i\omega_{n})(V_{R{\bf k}_{\alpha}}^{l_{-}})^{2}\end{split}$ (54)

where $V_{R{\bf k}_{\alpha}}^{l_{\pm}}$ are given by

${\color[rgb]{0,0,0}V}^{l_{\pm}}_{R{\bf
k}_{\alpha}}={\color[rgb]{0,0,0}V_{\mathbb{k}\alpha}^{l_{11}}}|u^{l}_{R}|^{2}\pm{\color[rgb]{0,0,0}V_{\mathbb{k}\alpha}^{l_{12}}}u^{*l}_{R}v^{*l}_{R}\pm
V_{\mathbb{k}\alpha}^{l_{21}}u^{l}_{R}v^{l}_{R}+{\color[rgb]{0,0,0}V_{\mathbb{k}\alpha}^{l_{22}}}|v^{l}_{R}|^{2}$
(55)

Here $V_{\mathbb{k}\alpha}^{l_{\eta\eta^{\prime}}}$ are the
$\eta\eta^{\prime}$ component of $\hat{V}_{{\bf
k}_{\alpha}}^{l}=\partial_{{\bf k}{{}_{\alpha}}}\hat{H}_{0}^{l}$ and
$({u/v})^{l}_{R}$ the renormalized coherence factors defined in Eq. (Green’s
function). The multiorbital character of the problem gives rise to self-energy
effects in the velocities via the coherence factors especially in the nematic
phase. Expressing the Green’s function in terms of the spectral functions
$A^{l_{\pm}}({\bf k},\omega)$ we finally arrive to the results of Eq. (17)
quoted in main text:

${\color[rgb]{0,0,0}\sigma^{l_{\pm}}_{\alpha}}=\frac{2\pi e^{2}}{N}\sum_{{\bf
k}}\int_{-\infty}^{\infty}d\omega(-\frac{\partial
f(\omega)}{\partial\omega})\big{(}{V}_{R{\bf
k}_{\alpha}}^{l_{\pm}}(\omega)\big{)}^{2}\big{(}A^{l_{\pm}}_{\bf
k}(\omega)\big{)}^{2},$ (56)

where $f(\omega)$ the Fermi distribution function. Notice that the
renormalized velocities $V_{R{\bf k}_{\alpha}}^{l_{\pm}}(\omega)$ depend on
frequency and on the orbital self-energy through the dependence of the
coherence factors on $\Sigma^{l}_{3}(\omega)$. The renormalized spectral
function can be written as

$A^{l_{\pm}}_{\bf k}(\omega)=\frac{1}{\pi}\frac{\Gamma_{R{\bf
k}}^{l_{\pm}}(\omega)}{(\Gamma_{R{\bf
k}}^{l_{\pm}}(\omega))^{2}+(\omega-\epsilon_{R{\bf
k}}^{l_{\pm}}(\omega))^{2}}$ (57)

where $\epsilon_{R{\bf k}}^{l_{\pm}}(\omega)=ReE_{R}^{l_{\pm}}({\bf
k},\omega)$ and $\Gamma_{R{\bf
k}}^{l_{\pm}}(\omega)=\delta\Gamma+|ImE_{R}^{l_{\pm}}({\bf k},\omega)|$. At
low temperature $-\frac{\partial
f(\omega)}{\partial\omega}\rightarrow\delta(\omega)$ that selects only states
at the Fermi level

$\sigma^{l_{\pm}}_{\alpha}=\frac{2\pi e^{2}}{N}\sum_{{\bf k}}({V}_{R{\bf
k}_{\alpha}}^{l_{\pm}})^{2}\big{(}A^{l_{\pm}}_{\bf k}\big{)}^{2}$ (58)

Moreover, assuming $\Gamma_{R}^{l_{\pm}}\rightarrow 0$, the spectral function
$A^{l_{\pm}}_{\bf k}$ can be approximated as

$\big{(}A^{l_{\pm}}_{\bf
k}\big{)}^{2}\longrightarrow\frac{1}{2\pi\Gamma_{R{\bf k}}^{l_{\pm}}}\
\delta(\epsilon_{R{\bf k}}^{l_{\pm}})$ (59)

Replacing this expression in Eq. (58) we get that the conductivity is

$\displaystyle\sigma^{l_{\pm}}_{\alpha}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{N}\sum_{{\bf k}}\frac{({\color[rgb]{0,0,0}V}_{R{\bf
k}_{\alpha}}^{l_{\pm}})^{2}}{\Gamma_{R{\bf k}}^{l_{\pm}}}\
\delta(\epsilon_{R\mathbb{k}}^{l_{\pm}})=$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{N}\int\frac{d{\bf
k}^{2}}{(2\pi)^{2}}\frac{(V_{R{\bf k}_{\alpha}}^{l_{\pm}})^{2}}{\Gamma_{R{\bf
k}}^{l_{\pm}}}\ \frac{\delta({\bf k}-{\bf
k}_{F})}{|\bigtriangledown\epsilon_{R{\bf k}}^{l_{\pm}}|}$

Within the parabolic band approximation and using the expansion up to the
first order in $\Sigma^{l}_{3}$ of the renormalized energy, we can derive
analytically the conductivity given in Eq. (APPENDIX B: Computation of the DC
conductivity and analytical results for perfectly nested circular FS). We use
the expressions of the renormalized energies and scattering rates derived in
App. A. Using the expressions Eq. (46) for the coherence factors, it is easy
to check the velocity given in Eq.(55) can be expressed as the derivative of
the renormalized dispersion relation given in Eq. (47), so $V^{l\pm}_{R{\bf
k}\alpha}=\partial_{k_{\alpha}}\epsilon_{R}^{l_{\pm}}({\bf k})$. Explicitly
for the h-pockets at $\Gamma$ we have

$\displaystyle V^{\Gamma_{\pm}}_{R{\bf k}\alpha}$ $\displaystyle=$
$\displaystyle\partial_{k_{\alpha}}h_{0}^{\Gamma}\pm\partial_{k_{\alpha}}\,h_{1}^{\Gamma}\frac{h_{1}^{\Gamma}}{h^{\Gamma}}\pm\partial_{k_{\alpha}}\,h_{3}^{\Gamma}\frac{h_{3}^{\Gamma}}{h^{\Gamma}}$
(61) $\displaystyle\pm$
$\displaystyle{Re}\Sigma^{\Gamma}_{3}\frac{h_{1}^{\Gamma}}{(h^{\Gamma})^{2}}\Big{[}\partial_{k_{\alpha}}h_{3}^{\Gamma}\frac{h_{1}^{\Gamma}}{h^{\Gamma}}-\partial_{k_{\alpha}}h^{\Gamma}_{1}\frac{h_{3}^{\Gamma}}{h^{\Gamma}}\Big{]}$

and analogous expressions for $V^{X}_{R{\bf k}\alpha}$ and $V^{Y}_{R{\bf
k}\alpha}$. The first three terms in Eq. (61) corresponds to the bare
velocity, while the term multiplied by ${Re}\Sigma^{\Gamma}_{3}$ accounts for
the renormalization in the velocity due to OSSF self-energy corrections. Using
the explicit definition of $h^{l}_{0}$ and $\vec{h}^{l}$, the velocities for
the various pockets read:

$\displaystyle{\color[rgb]{0,0,0}V}^{\Gamma_{\pm}}_{Rx}$ $\displaystyle=$
$\displaystyle{\color[rgb]{0,0,0}-\frac{k\cos\theta}{m^{\Gamma_{\pm}}}\pm
4Re\Sigma^{\Gamma}_{3}\frac{k\cos\theta\sin\theta^{2}}{k^{2}}},$
$\displaystyle{\color[rgb]{0,0,0}V}^{\Gamma_{\pm}}_{Ry}$ $\displaystyle=$
$\displaystyle{\color[rgb]{0,0,0}-\frac{k\sin\theta}{m^{\Gamma_{\pm}}}\mp
4Re\Sigma^{\Gamma}_{3}\frac{k\sin\theta\cos\theta^{2}}{k^{2}}},$
$\displaystyle{\color[rgb]{0,0,0}V}^{X/Y}_{Rx}$ $\displaystyle=$
$\displaystyle{\color[rgb]{0,0,0}\frac{k\cos\theta}{m^{X/Y}}\mp
4Re\Sigma^{X/Y}_{3}\frac{k\cos\theta\sin\theta^{2}}{k^{2}}},$
$\displaystyle{\color[rgb]{0,0,0}V}^{X/Y}_{Ry}$ $\displaystyle=$
$\displaystyle{\color[rgb]{0,0,0}\frac{k\sin\theta}{m^{X/Y}}\pm
4Re\Sigma^{X/Y}_{3}\frac{k\sin\theta\cos\theta^{2}}{k^{2}}},$ (62)

where Eq. (43) and Eq. (44) have been used and $m^{l_{\pm}}$ is the bare mass
of the $l_{\pm}$ pocket whose definition in terms of the Hamiltonian
parameters is given by $m^{\Gamma_{\pm}}={2(a\mp b)}^{-1}$ and
$m^{X/Y}=m^{e}={2(a+b)}^{-1}$. To compute the k integration in the Eq.
(APPENDIX B: Computation of the DC conductivity and analytical results for
perfectly nested circular FS) we evaluate the velocity in the Eq. (APPENDIX B:
Computation of the DC conductivity and analytical results for perfectly nested
circular FS) at the renormalized FS. In the nematic phase $k_{F}(\theta)$ is
no longer constant. Using again the expansion up to the first order in
$\Sigma_{3}$ of the renormalized energy we can estimate $k_{F}(\theta)$ and
replace it in Eq. (APPENDIX B: Computation of the DC conductivity and
analytical results for perfectly nested circular FS). We find

$\displaystyle{V}^{\Gamma_{\pm}}_{Rx}$ $\displaystyle=$
$\displaystyle{\color[rgb]{0,0,0}{V_{0}}^{\Gamma_{\pm}}_{x}\bigg{(}1\pm\cos
2\theta\frac{Re\Sigma^{\Gamma}_{3}}{2\epsilon^{h}_{0}}\mp
4\sin^{2}\theta\frac{Re\Sigma^{\Gamma}_{3}}{2\epsilon^{h}_{0}}\bigg{)}},$
$\displaystyle V^{\Gamma_{\pm}}_{Ry}$ $\displaystyle=$
$\displaystyle{\color[rgb]{0,0,0}{V_{0}}^{\Gamma_{\pm}}_{y}\bigg{(}1\pm\cos
2\theta\frac{Re\Sigma^{\Gamma}_{3}}{2\epsilon^{h}_{0}}\pm
4\cos^{2}\theta\frac{Re\Sigma^{\Gamma}_{3}}{2\epsilon^{h}_{0}}\bigg{)}},$
$\displaystyle V^{X/Y}_{Rx}$ $\displaystyle=$
$\displaystyle{\color[rgb]{0,0,0}{V_{0}}^{X/Y}_{x}\bigg{(}1\pm\cos
2\theta\frac{Re\Sigma^{X/Y}_{3}}{2\epsilon^{X/Y}_{0}}\mp
4\sin^{2}\theta\frac{Re\Sigma^{X/Y}_{3}}{2\epsilon^{X/Y}_{0}}\bigg{)}},$
$\displaystyle V^{X/Y}_{Ry}$ $\displaystyle=$
$\displaystyle{\color[rgb]{0,0,0}{V_{0}}^{X/Y}_{y}\bigg{(}1\pm\cos
2\theta\frac{Re\Sigma^{X/Y}_{3}}{2\epsilon^{X/Y}_{0}}\pm
4\cos^{2}\theta\frac{Re\Sigma^{X/Y}_{3}}{2\epsilon^{X/Y}_{0}}\bigg{)}}.$

where $\epsilon^{h}_{0}$ and $\epsilon^{e}_{0}$ are the Fermi energy in the
tetragonal phase defined in Eq.s (48) and
${V_{0}}^{\Gamma_{\pm}}_{\alpha}=-{k_{0}}_{F\alpha}^{\Gamma_{\pm}}/m^{\Gamma_{\pm}}$,
${V_{0}}^{X/Y}_{\alpha}={k_{0}}_{F\alpha}^{e}/m^{e}$ is the $\alpha$ component
of the bare velocity with
${k_{0}}_{F}^{\Gamma_{\pm}}=\sqrt{\epsilon^{h}_{0}/(2m^{\Gamma_{\pm}})}$ and
${k_{0}}_{F}^{e}=\sqrt{\epsilon^{e}_{0}/(2m^{e}})$.

The last term we need to evaluate is the $|\bigtriangledown\epsilon_{R{\bf
k}}^{l_{\pm}}|$ (see Eq. (APPENDIX B: Computation of the DC conductivity and
analytical results for perfectly nested circular FS)). In the $\Gamma$ pocket
turns out to be $|\bigtriangledown\epsilon_{R{\bf
k}}^{l_{\pm}}|=k/m^{\Gamma_{+}}$ with similar results for the other pockets.
The important point is that the norm of the pocket velocity is independent of
the self-energy while the pocket velocity in a given direction $x$/$y$ depends
on the self-energy, Eq. (APPENDIX B: Computation of the DC conductivity and
analytical results for perfectly nested circular FS). Replacing all the
analytical expressions found for the velocities, Eq. (APPENDIX B: Computation
of the DC conductivity and analytical results for perfectly nested circular
FS), and the scattering rate, Eq. (50) in Eq. (APPENDIX B: Computation of the
DC conductivity and analytical results for perfectly nested circular FS) the
pockets DC conductivities read:

$\displaystyle\sigma_{x/y}^{\Gamma_{+}}$ $\displaystyle=$
$\displaystyle\sigma^{h}\bigg{(}1\pm\frac{{Re}\Sigma_{3}^{\Gamma}}{2\epsilon^{h}_{0}}\mp\frac{{Re}\Sigma_{3}^{\Gamma}}{\epsilon^{h}_{0}}\mp\frac{|{Im}\Sigma^{\Gamma}_{3}|}{2\Gamma^{h}_{0}}\bigg{)}$
$\displaystyle\sigma_{x/y}^{\Gamma_{-}}$ $\displaystyle=$
$\displaystyle\sigma^{h}\bigg{(}1\mp\frac{{Re}\Sigma_{3}^{\Gamma}}{2\epsilon^{h}_{0}}\pm\frac{{Re}\Sigma_{3}^{\Gamma}}{\epsilon^{h}_{0}}\pm\frac{|{Im}\Sigma^{\Gamma}_{3}|}{2\Gamma^{h}_{0}}\bigg{)}$
$\displaystyle\sigma_{x/y}^{X}$ $\displaystyle=$
$\displaystyle\sigma^{e}\bigg{(}1\pm\frac{{Re}\Sigma_{3}^{X}}{2\epsilon^{e}_{0}}\mp\frac{{Re}\Sigma_{3}^{X}}{\epsilon^{e}_{0}}\pm\frac{|{Im}\Sigma^{X}_{3}|}{2\Gamma^{e}_{0}}\bigg{)}$
$\displaystyle\sigma_{x/y}^{{\color[rgb]{0,0,0}Y}}$ $\displaystyle=$
$\displaystyle\sigma^{e}\bigg{(}1\mp\frac{{Re}\Sigma_{3}^{Y}}{2\epsilon^{e}_{0}}\pm\frac{{Re}\Sigma_{3}^{Y}}{\epsilon^{e}_{0}}\mp\frac{|{Im}\Sigma^{Y}_{3}|}{2\Gamma^{e}_{0}}\bigg{)}$
(64)

where $\sigma^{h/e}=e^{2}\epsilon^{h/e}_{0}/(2\pi\hbar)\Gamma_{0}^{h/e}$ is
the $x/y$-component of the DC conductivity in the tetragonal phase for the
h/e-pocket. The DC conductivity anisotropy for the h- and e-pockets can be
finally written as

$\displaystyle\Delta\sigma_{DC}^{h+}$ $\displaystyle=$
$\displaystyle\sigma^{\Gamma}\left(\frac{\Phi^{h}}{\epsilon^{h}_{0}}-\frac{\Delta\Gamma^{h}}{\Gamma^{h}_{0}}\right)$
$\displaystyle\Delta\sigma_{DC}^{h-}$ $\displaystyle=$
$\displaystyle\sigma^{\Gamma}\left(-\frac{\Phi^{h}}{\epsilon^{h}_{0}}+\frac{\Delta\Gamma^{h}}{\Gamma^{h}_{0}}\right)$
$\displaystyle\Delta\sigma_{DC}^{e}$ $\displaystyle=$
$\displaystyle\sigma^{{\color[rgb]{0,0,0}e}}\left(-\frac{\Phi^{e}}{\epsilon^{{\color[rgb]{0,0,0}e}}_{0}}+\frac{\Delta\Gamma^{e}}{\Gamma^{{\color[rgb]{0,0,0}e}}_{0}}\right)$
(65)

with the nematic order parameters for the h-pockets
($\Phi^{h},\Delta\Gamma^{h}$) and the e-pockets ($\Phi^{e},\Delta\Gamma^{e}$)
defined as

$\displaystyle\Phi{{}^{h}}\equiv\displaystyle\frac{Re\Sigma^{\Gamma}_{xz}-Re\Sigma^{\Gamma}_{yz}}{2}=-Re\Sigma^{\Gamma}_{3},\quad\Delta\Gamma{{}^{h}}\equiv|Im\Sigma^{\Gamma}_{3}|,$
$\displaystyle\Phi{{}^{e}}\equiv\frac{Re\Sigma^{X}_{yz}-Re\Sigma^{Y}_{xz}}{2},\quad\Delta\Gamma{{}^{e}}\equiv\frac{|Im\Sigma^{X}_{yz}|-|Im\Sigma^{Y}_{xz}|}{2}$

## APPENDIX C: Model parameters for FeSe and
122 systems

To perform the numerical analysis discussed in the main text we used a set of
band parameters which reproduce the experimental dispersions observed in 122
and FeSe systems. Using the band parameters listed in Table 1, a spin-orbit
interaction of $\lambda=5$ meV and
$|Re\Sigma_{yz/xz}^{\Gamma}|=Re\Sigma_{yz/xz}^{X/Y}=15$ meV, we obtain the FS
topology shown in Fig.1 that reproduce qualitatively well the FS of 122
systems in the tetragonal phase. The nematic phase is computed with a
symmetric nematic splitting of $\Phi^{h}=\Phi^{e}=4$ meV. For the FeSe case we
used the same set of band parameters listed in Table 1, with spin-orbit
interaction $\lambda=20$ meV, and $|Re\Sigma_{yz/xz}^{\Gamma}|=70/40$ meV and
$Re\Sigma_{yz/xz}^{X/Y}=45/15$ meV that result in a nematic order parameter
$\Phi^{h/e}=15$ meV. This set reproduce the FS and their orbital distribution
as experimentally observed by ARPESFanfarillo et al. (2016); Coldea and Watson
(2018); Rhodes et al. (2018) in the nematic phase.

|  | $\Gamma$ |  |  |  | X |  |
---|---|---|---|---|---|---|---|---
| $\epsilon_{\Gamma}$ |  | 46 |  |  $\epsilon_{xy}$ 72 |  |  $\epsilon_{yz}$ 55 |
| $a_{\Gamma}$ |  | 263 |  |  $a_{xy}$ 93 |  |  $a_{yz}$ 101 |
| $b_{\Gamma}$ |  | 182 |  |  $b$ 154 |  |  |
|  |  |  |  |  $v$ 144 |  |  |
Table 1: Low-energy model parameters used for FeSe and 122 system. All the
parameters are in meV, the $k$ vector is measured in units $1/a\sim 0.375$ Å,
where $a=a_{FeFe}$ is the lattice constant of the 1-Fe unit cell (so that
$\tilde{a}=\sqrt{2}a=3.77$ Å is the lattice constant of the 2Fe unit cell).

We fix the background scattering to $\delta\Gamma=1$ meV. The scattering rates
used in the tetragonal phase for 122 are
$Im\Sigma^{\Gamma}_{yz/xz}=Im\Sigma_{yz/xz}^{X/Y}=-2$ meV, while for the FeSe
case we used $Im\Sigma^{\Gamma}_{yz/xz}=-4$ meV and
$Im\Sigma_{yz/xz}^{X/Y}=-3$ meV. In both cases their variations of the
imaginary part of the self-energies in the nematic phase are assumed to be
proportional to the variation of their real parts, i.e. $\Delta\Gamma^{h}\sim
c_{h}\ (\Phi^{h}/Re\Sigma_{0}^{\Gamma})$ and $\Delta\Gamma^{e}\sim c_{e}\
(\Phi^{e}/Re\Sigma_{0}^{X/Y})$ with $c_{h/e}$ arbitrary coefficients.

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