# Order parameter fluctuation and ordering competition in
$\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$

Jing Wang  Department of Modern Physics, University of Science and Technology
of China, Hefei, Anhui 230026, P.R. China  Institute for Theoretical Solid
State Physics, IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany  Guo-Zhu
Liu  Department of Modern Physics, University of Science and Technology of
China, Hefei, Anhui 230026, P.R. China  Dmitry V. Efremov  Institute for
Theoretical Solid State Physics, IFW Dresden, Helmholtzstr. 20, 01069 Dresden,
Germany  Jeroen van den Brink  Institute for Theoretical Solid State Physics,
IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany  Institute for
Theoretical Physics, TU Dresden, 01069 Dresden, Germany

###### Abstract

The competition among superconductivity, stripe-type magnetic order, and a new
type of $C_{4}$ symmetric magnetic order in
$\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$ is theoretically studied, focusing on its
impact on the global phase diagram. By carrying out a renormalization group
analysis of an effective field theory, we obtain the energy-scale dependent
flows of all the model parameters, and then apply the results to understand
the observed phase diagram. On the basis of the renormalization group
analysis, we show that the critical line of nematic order has a negative slope
in the superconducting dome and superconductivity is suppressed near the
magnetic quantum critical point, which are both consistent with recent
experiments. Moreover, we find that, although the observed $C_{4}$ symmetric
magnetic state could be a charge-spin density wave or a spin-vortex crystal at
high temperatures, charge-spin density wave is the only stable $C_{4}$
magnetic state in the low-temperature regime. Therefore, ordering competition
provides a method to distinguish these two candidate $C_{4}$ magnetic states.

###### pacs:

74.70.Xa, 74.25.Dw, 74.40.Kb, 74.62.-c

##  I Introduction

A universal property shared by most known iron-based superconductors (FeSCs)
is the bulk coexistence of two or even more distinct long-range orders
Hinkov2009NPhys ; Paglione2010NPhys ; Stewart2011RMP ; Fisher2011RPP ;
Hirschfeld2011RPP ; Basov2011NPhys ; Dai2012NPhys ; Chubukov2012ARCMP ;
Dagotto2013RMP ; Fernandes2014NPhys ; Dai2015RMP ; Fernandes2017RPP , such as
superconductivity, stripe-type spin-density-wave (SDW) order, nematic order,
and other possible orders. The competition and coexistence of these orders
leads to a very complicated global phase diagram Hinkov2009NPhys ;
Paglione2010NPhys ; Stewart2011RMP ; Fisher2011RPP ; Hirschfeld2011RPP ;
Basov2011NPhys ; Dai2012NPhys ; Chubukov2012ARCMP ; Dagotto2013RMP ;
Fernandes2014NPhys ; Dai2015RMP ; Fernandes2017RPP . Acquiring a detailed
knowledge of the phase diagram is an important step towards a better
understanding of FeSCs.

Among the long-range orders competing with superconductivity, a particular
role is played by the nematic order, induced by an electronic state that
spontaneously breaks the $C_{4}$ (tetragonal) symmetry of the system down to a
$C_{2}$ (orthogonal) symmetry. Extensive experiments have confirmed that
nematic order exists in almost all FeSCs Paglione2010NPhys ; Stewart2011RMP ;
Fisher2011RPP ; Hirschfeld2011RPP ; Kim2011PRB ; Rotundu2011PRB ;
Kasahara2012Nature ; Avci2012PRB ; Zheng2013NComm . In most cases, the nematic
order sets in at a temperature $T_{\mathrm{n}}$ slightly higher than the
critical temperature of magnetic order $T_{\mathrm{m}}$ Basov2011NPhys ;
Fisher2011RPP ; Fernandes2014NPhys ; Fernandes2012PRB ; Fernandes2013PRL .
Usually, the magnetic order is generated by a stripe-type SDW, and possesses
two characteristic vectors $\mathbf{Q}_{X}=(\pi,0)$ and
$\mathbf{Q}_{Y}=(0,\pi)$ in the Brillouin zone of the iron square lattice,
which relate to the spin operator $S(\mathbf{r})$ in the form
$S(\mathbf{r})=M_{X,Y}e^{i\mathbf{Q}_{X,Y}\cdot\mathbf{r}}$ Fernandes2010PRB ;
Fernandes2017RPP . This stripe SDW breaks the discrete lattice rotational
symmetry by selecting out only one of the two characteristic vectors
$\mathbf{Q}_{X}$ and $\mathbf{Q}_{Y}$, preserving the $C_{2}$ symmetry
Basov2011NPhys ; Chubukov2012ARCMP ; Fernandes2014NPhys ; Fernandes2017RPP .
Because the nematic order and SDW order coexist over a large part of the
global phase diagram, it is widely believed Stewart2011RMP ; Fisher2011RPP ;
Hirschfeld2011RPP ; Basov2011NPhys ; Chubukov2012ARCMP ; Dagotto2013RMP ;
Fernandes2014NPhys ; Dai2015RMP ; Fernandes2017RPP  that the nematic order is
actually induced by the fluctuation of magnetic order.

It was unexpected that experiments had found a new type of $C_{4}$ symmetric
magnetic order that preserves the tetragonal symmetry in a number of hole-
doped FeSCs, including $\mathrm{Ba(Fe_{1-x}Mn_{x})_{2}As_{2}}$ Goldman2010PRB
, $\mathrm{Ba_{1-x}Na_{x}Fe_{2}As_{2}}$ Osborn2014NatureComm ; Wang2016PRB ,
$\mathrm{Sr_{1-x}K_{x}Fe_{2}As_{2}}$ Osborn2016NaturePhys , and
$\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$ Hassinger2012PRB ; Hardy2015NComm ;
Allred2015PRB ; Hassinger2016PRB . This $C_{4}$ magnetic state is
characterized by biaxial magnetic orders Lorenzana2008PRL ; Eremin2010PRB ;
Brydon2010PRB ; Giovannetti2011NComm ; Fernandes2016PRB , and the
corresponding spin operator is given by
$S(\mathbf{r})=M_{X}e^{i\mathbf{Q}_{X}\cdot\mathbf{r}}+M_{Y}e^{i\mathbf{Q}_{Y}\cdot\mathbf{r}}$
Fernandes2016PRB ; Schmalian2016PRB . It has been suggested that this
double-$\mathbf{Q}$ magnetic state has two possible realizations
Lorenzana2008PRL ; Knolle2010PRB ; Eremin2010PRB ; Brydon2010PRB ;
Giovannetti2011NComm ; Fernandes2016PRB : a charge-spin density wave (CSDW) in
which $M_{X}$ and $M_{Y}$ are collinear; a spin-vortex crystal (SVC) in which
$M_{X}$ and $M_{Y}$ are orthogonal. Since the largest value of $T_{c}$ of
FeSCs is observed at the proximity of tetragonal $C_{4}$ magnetic order
Osborn2014NatureComm ; Hardy2015NComm , there might exist a quantum critical
point (QCP) at certain doping $x_{c}$ in the superconducting (SC) dome
Fernandes2016PRB . After its discovery, the double-$\mathbf{Q}$ structured
$C_{4}$ SDW state has stimulated a variety of experimental Goldman2010PRB ;
Hassinger2012PRB ; Osborn2014NatureComm ; Hardy2015NComm ; Allred2015PRB ;
Osborn2016NaturePhys  and theoretical works Lorenzana2008PRL ; Eremin2010PRB ;
Brydon2010PRB ; Giovannetti2011NComm ; Kang2011PRB ; Anderson2014PRL ;
Fernandes2016PRB ; Schmalian2016PRB ; Anderson2015PRB ; Anderson2016PRB .

In this paper, we consider the effects caused by the competition of
superconductivity with both stripe-type $C_{2}$ symmetric and $C_{4}$
symmetric magnetic orders in a hole-doped FeSC
$\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$ Goldman2010PRB ; Hassinger2012PRB ;
Osborn2014NatureComm ; Hardy2015NComm ; Allred2015PRB ; Osborn2016NaturePhys .
Recently, Böhmer _et al._ Hardy2015NComm  have experimentally investigated the
global phase diagram of $\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$, and identified
five distinct thermodynamically stable ordered phases, which are schematically
shown in Fig. 1. One can see that the critical line for the nematic order
displays a rather complicated dependence on doping $x$ and temperature $T$: it
decreases with growing $x$ at high $T$, bends backwards to lower $x$ slightly
above $T_{c}$, and eventually exhibits a negative slope after penetrating into
the SC dome. In the narrow doping region in which the nematic critical line
has a positive slope, $T_{c}$ is moderately suppressed. Close to the putative
magnetic QCP, represented by $x_{2}$ in Fig. 1, there appears on the phase
diagram a region that manifests $C_{4}$ symmetric SDW state, which occupies
part of the usual $C_{2}$ symmetric SDW phase and coexists with
superconductivity below $T_{c}$. In principle, the experimentally observed
$C_{4}$ SDW state might be a CSDW or SVC type state, which needs to be
clarified theoretically.

Instead of trying to explain the entire phase diagram observed in Ref.
Hardy2015NComm , we perform a more moderate task in this work. In particular,
we will concentrate on the narrow doping region surrounding the magnetic QCP
$x_{2}$ inside the SC dome and endeavor to answer the following questions. How
to determine whether the observed $C_{4}$ magnetic state is of CSDW or SVC
type? What is the scenario that leads to suppresses superconductivity near the
magnetic QCP? Why does the nematic critical line display a negative, rather
than positive, slope in the SC dome? We will address these issues by
investigating the impact of order competition in the low-$T$ regime.

Figure 1: (Color online) Schematic global phase diagram of
$\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$ on $(x,T)$ plane Hardy2015NComm , with
$x_{1}$ and $x_{2}$ being two QCPs. $T_{M2}$ is the transition line between
paramagnetic (PM) and $C_{2}$-symmetric magnetic phases, and $T_{c}$ is the SC
transition line. $T_{M4}$ is the transition line between $C_{2}$\- and
$C_{4}$-symmetric magnetic phases above $T_{c}$, whereas $T_{m4}$ denotes such
a transition line inside the SC dome. The observed $C_{4}$ symmetric SDW state
could be either CSDW or SVC. Distinct phases are distinguished by different
values of the model parameters Fernandes2016PRB ; Schmalian2016PRB  defined in
Eq. (4): (i) $g<0$ and $w=0$, the system is in PM phase; (ii) $g>\max(0,-w)$
represents the $C_{2}$ symmetric SDW phase (with nematic order); (iii) $C_{4}$
symmetric magnetic state is of SVC-type if $g<0$ and $w>0$, and CSDW-type if
$g<-w$ and $w<0$.

We study an effective field theory that can be used to describe the low-energy
physics of $\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$ and other analogous FeSCs
Fernandes2010PRB ; Knolle2010PRB ; Eremin2010PRB ; Fernandes2012PRB ;
Fernandes2013PRL ; Fernandes2017RPP . When the parameters used in this model
take different values, the system might be in the paramagnetic state, $C_{4}$
tetragonal SDW state, or $C_{2}$ symmetric magnetic state. The fluctuations of
the associated order parameters and the competition between distinct orders
can qualitatively alter the magnitudes and even the sign of the model
parameters, which would drive phase transitions and reshape the global phase
diagram. After analyzing the competition between nematic and SC orders by
means of renormalization group (RG) method, we find that the slope of nematic
critical line is always negative in the SC dome. We also extract the
$T$-dependence of superfluid density from RG results, which clearly shows that
superconductivity is suppressed near magnetic QCP. Moreover, we infer from the
RG results that, although CSDW and SVC state are both possible in the high-$T$
regime, the CSDW state is the only stable one in the low-$T$ regime, which
provides a promising way to specify the true nature of the observed $C_{4}$
symmetric SDW state.

The rest of paper is organized as follows. In Sec. II, we present the
effective field theory for the ordering competition and then derive the
coupled flow equations of all the model parameters by performing perturbative
RG calculations. In Sec. III, we numerically solve the equations and apply the
RG solutions to understand several important features of the global phase
diagram of $\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$ observed in recent
experiments. The Sec. IV is followed to present some discussions. In Sec. V,
we present a brief summary of the results.

##  II Effective theory and RG analysis

Many of the basic properties of $\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$ can be
described by a three-band model that contains one hole pocket at the center of
Brillouin zone $\mathbf{Q}_{\Gamma}=(0,0)$ and two electron pockets centered
at two specific momenta $\mathbf{Q}_{X}=(\pi,0)$ and $\mathbf{Q}_{Y}=(0,\pi)$
Knolle2010PRB ; Eremin2010PRB ; Fernandes2010PRB ; Fernandes2012PRB ;
Fernandes2013PRL ; Fernandes2016PRB ; Schmalian2016PRB ; Fernandes2017RPP .
The microscopic model is written as Fernandes2013PRL ; Fernandes2017RPP

$\displaystyle
H=\sum_{\mathbf{k},i\in(X,Y,\Gamma)}\varepsilon_{\mathbf{k},i}c^{\dagger}_{\mathbf{k}\sigma,i}c_{\mathbf{k}\sigma,i}+H_{4},$
(1)

with the interacting term $H_{4}$ is given by

$\displaystyle H_{4}$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{k},i\in(X,Y)}\frac{U_{3}}{2}\left(c^{\dagger}_{\mathbf{k}\alpha,\Gamma}c^{\dagger}_{\mathbf{k}\gamma,\Gamma}c_{\mathbf{k}\delta,i}c_{\mathbf{k}\beta,i}+\mathrm{h.c.}\right)\delta_{\alpha\beta}\delta_{\gamma\delta}$
(2)
$\displaystyle+\sum_{\mathbf{k},i\in(X,Y)}U_{1}c^{\dagger}_{\mathbf{k}\alpha,\Gamma}c^{\dagger}_{\mathbf{k}\gamma,i}c_{\mathbf{k}\delta,i}c_{\mathbf{k}\beta,\Gamma}\delta_{\alpha\beta}\delta_{\gamma\delta}.$

Here, $U_{1}$ and $U_{3}$ represent density-density interaction and the pair
hoping interaction, respectively. They are responsible for the formation of
superconductivity and SDW state Fernandes2013PRL ; Chubukov2008PRB ;
Maiti2010PRB . The magnetic structure can be described by two order parameters
$\mathbf{M}_{X}$ and $\mathbf{M}_{Y}$, corresponding to the ordering vectors
$\mathbf{Q}_{X}=(\pi,0)$ and $\mathbf{Q}_{Y}=(0,\pi)$, which are defined as
$\mathbf{M}_{j}=\sum_{\mathbf{k}}c^{\dagger}_{\Gamma,\mathbf{k}\alpha}\vec{\sigma}_{\alpha\beta}c_{j,\mathbf{k}+\mathbf{Q}_{j}\beta}$
with $j=X,Y$ Fang2008PRB ; Xu2008PRB ; Fernandes_Schmalian2010PRL ;
Fernandes2012PRB ; Fernandes2013PRL . Both the $C_{2}$ and $C_{4}$ symmetric
magnetic orders are modeled by the following Ginzburg-Landau free energy
Fernandes2016PRB ; Schmalian2016PRB

$\displaystyle f[\mathbf{M_{X}},\mathbf{M_{Y}}]$ $\displaystyle=$
$\displaystyle
a_{m}(\mathbf{M}^{2}_{X}+\mathbf{M}^{2}_{Y})+\frac{u}{2}(\mathbf{M}^{2}_{X}+\mathbf{M}^{2}_{Y})^{2}$
$\displaystyle-\frac{g}{2}(\mathbf{M}^{2}_{X}-\mathbf{M}^{2}_{Y})^{2}+2w(\mathbf{M}_{X}\cdot\mathbf{M}_{Y})^{2}.$

As illustrated in Ref. Fernandes2016PRB , the term
$2w(\mathbf{M}_{X}\cdot\mathbf{M}_{Y})^{2}$ can be rewritten by using an
identity:

$\displaystyle(\mathbf{M}_{X}\cdot\mathbf{M}_{Y})^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(\mathbf{M}^{2}_{X}+\mathbf{M}^{2}_{Y})^{2}-\frac{1}{4}(\mathbf{M}^{2}_{X}-\mathbf{M}^{2}_{Y})^{2}$
(3) $\displaystyle-(\mathbf{M}_{X}\times\mathbf{M}_{Y})^{2}.$

Upon carrying out a Hubbard-Stratonovich transformation followed by an
integration over all the fermionic degrees of freedom, one can obtain an
effective field theory Fernandes2013PRL ; Fernandes2016PRB  for the interplay
of SDW magnetic and SC orders in the vicinity of magnetic QCP:

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\partial_{\mu}M_{X})^{2}+\frac{1}{2}(\partial_{\mu}M_{Y})^{2}+a_{m}\left(M_{X}^{2}+M_{Y}^{2}\right)$
(4)
$\displaystyle+\frac{(u+w)}{2}\left(M_{X}^{2}+M_{Y}^{2}\right)^{2}-\frac{\left(g+w\right)}{2}\left(M_{X}^{2}-M_{Y}^{2}\right)^{2}$
$\displaystyle+\partial_{\mu}\Delta^{\dagger}\partial_{\mu}\Delta+a_{s}\Delta^{2}(k)+\frac{u_{s}}{2}\Delta^{4}(k)+\frac{\varphi^{2}}{2w}+\mathcal{L}_{A}$
$\displaystyle-2\varphi
M_{X}M_{Y}+\lambda(M^{2}_{X}+M^{2}_{Y})\Delta^{2}+\lambda_{\Delta
A}\Delta^{2}A^{2},$

where $\Delta$ is the SC order parameter. Here we use a positive parameter
$\lambda$ to characterize the repulsive interaction (competition) between SC
and magnetic orders. In order to evaluate the superfluid density, we have
introduced a gauge potential $\mathbf{A}$ via the standard minimal coupling
Halperin1974PRL  with
$\mathcal{L}_{A}=-\frac{1}{4}(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})^{2}$.
This model contains eight fundamental parameters $a_{m}$, $a_{s}$, $u_{s}$,
$w$, $u$, $g$, $\lambda$, and $\lambda_{\Delta A}$, which are constants at the
mean-field level, but all become cutoff dependent due to interactions.

The transition lines for SDW and SC orders are determined by taking $a_{m}=0$
and $a_{s}=0$ respectively. For $s^{+-}$-wave superconductors, we employ the
relationship $\Delta_{\Gamma}=-\sqrt{2}\Delta_{X,Y}=\Delta$ Fernandes2013PRL ;
Chubukov2012ARCMP ; Fernandes2017RPP . An Ising-type nematic order is induced
by the magnetic order, and represented by a term of the form
$M^{2}_{X}-M^{2}_{Y}$ Fernandes2012PRB ; Chubukov2012ARCMP ; Fernandes2017RPP
. The property of $C_{4}$ magnetic order is determined by the parameter $w$
Fernandes2016PRB ; Schmalian2016PRB . In the SC dome, the SC order parameter
develops a nonzero mean value, i.e.,
$\langle\Delta\rangle=V_{0}=\sqrt{-a_{s}/u_{s}}$ near the magnetic QCP
$x_{2}$.

The effective model (4) displays different states when the model parameters
take various values Fernandes2016PRB ; Schmalian2016PRB . (i) If $g<0$ and
$w=0$, the effective model is in paramagnetic (PM) phase; (ii) The case of
$g>\max(0,-w)$ corresponds to the $C_{2}$ SDW phase (with nematic order);
(iii) The $C_{4}$ symmetric magnetic state is of SVC-type if $g<0$ and $w>0$,
and CSDW-type if $g<-w$ and $w<0$. Once some of these parameters are altered
by external forces, such as doping, magnetic field, and pressure, the system
would undergo transitions between distinct phases. However, the quantum
fluctuations of order parameters and the interaction between different order
parameters can also lead to remarkable changes of model parameters, and as
such drive phase transitions. In the next section, we will study the RG flows
of these parameters and examine how they are influenced by order parameter
fluctuation and ordering competition. The main results are schematically
illuminated in Fig. 1 and the detailed derivations and discussions are given
in the following.

To proceed, we perform a RG analysis of the effective theory (4). Our focus is
on the behavior of the system at low $T$ and in the vicinity of magnetic QCP.
Within this region, the quantum fluctuations of SC order parameter can result
in drastic effects even in the SC phase. For the complex SC order parameter
$\Delta(\mathbf{r})$, there are two sorts of fluctuations Varma2002JLTP ;
Zwerger2004PRL ; Podolsky2011PRB ; Podolsky2012PRB ; Pollet2012PRL ;
Bloch2012Nature ; Varma2013PRB ; Pekker2015ARCMP , namely the phase
fluctuation and amplitude fluctuation. The former fluctuation is gapless and
corresponds to the Nambu-Goldstone mode induced by continuous gauge symmetry
breaking. This mode does not play any role in the SC state because it is
absorbed by the vector gauge boson via the Anderson-Higgs mechanism. The
latter one, known as Higgs mode in a locally gauge invariant superconductor,
is found by both theoretical and experimental works to result in observable
effects Varma2002JLTP ; Zwerger2004PRL ; Podolsky2011PRB ; Podolsky2012PRB ;
Pollet2012PRL ; Bloch2012Nature ; Varma2013PRB ; Pekker2015ARCMP , and hence
should be seriously considered Kleinert2003NPB ; Wang2014PRD . In order to
capture the quantum fluctuation of SC order parameter around its mean value
$\langle\Delta\rangle$, we define two new fields $h$ and $\eta$ by
Kleinert2003NPB ; Wang2014PRD

$\displaystyle\Delta=V_{0}+\frac{1}{\sqrt{2}}(h+i\eta),$ (5)

where $\langle h\rangle=\langle\eta\rangle=0$. The fields $h$ and $\eta$ stand
for the Higgs mode and Nambu-Goldstone mode, respectively. We substitute Eq.
(5) into the effective Lagrangian density $\mathcal{L}$ (4), and obtain the
following new effective Lagrangian density:

$\displaystyle\mathcal{L}_{\mathrm{eff}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\partial_{\mu}M_{X})^{2}+\alpha_{X}M^{2}_{X}+\frac{\beta_{X}}{2}M_{X}^{4}+\frac{1}{2}(\partial_{\mu}M_{Y})^{2}+\alpha_{Y}M^{2}_{Y}+\frac{\beta_{Y}}{2}M_{Y}^{4}+\frac{1}{2}(\partial_{\mu}h)^{2}+\alpha_{h}h^{2}+\frac{\beta_{h}}{2}h^{4}+\gamma_{h}h^{3}$
(6)
$\displaystyle+\left(\mathcal{L}_{A}+\frac{\alpha_{A}}{2}A^{2}\right)+\alpha_{\varphi}\varphi^{2}+\gamma_{\varphi
XY}\varphi
M_{X}M_{Y}+\gamma_{X^{2}h}M^{2}_{X}h+\gamma_{Y^{2}h}M^{2}_{Y}h+\gamma_{hA^{2}}hA^{2}+\lambda_{Xh}M^{2}_{X}h^{2}$
$\displaystyle+\lambda_{Yh}M^{2}_{Y}h^{2}+\lambda_{XY}M_{X}^{2}M_{Y}^{2}+\lambda_{hA}h^{2}A^{2}.$

The gapless Nambu-Goldstone model $\eta$ naturally disappears after invoking
the Anderson-Higgs mechanism. However, the Higgs mode $h$ remains in the above
effective model, and couple directly to the magnetic order parameters
$M_{X,Y}$ and also to vector potential $A$. The originally massless gauge
field $A$ acquires a finite mass $\alpha_{A}$ after absorbing $\eta$.
Moreover, in the above Lagrangian density we have introduced a number of new
parameters that are related to the model parameters defined in (4) by the
following relations:

$\displaystyle\left.\begin{array}[]{ll}\alpha_{X}=\alpha_{Y}\equiv
a_{m}-\frac{\lambda
a_{s}}{u_{s}},\,\,\,\,\,\alpha_{A}\equiv\frac{-2\lambda_{\Delta
A}a_{s}}{u_{s}},\,\,\,\,\,\alpha_{h}\equiv-
a_{s},\,\,\,\,\,\alpha_{\varphi}\equiv\frac{1}{2w},\,\,\,\,\,\beta_{X}=\beta_{Y}\equiv
u-g,\,\,\,\,\,\beta_{h}\equiv\frac{u_{s}}{4},\\\
\gamma_{h}\equiv\frac{\sqrt{-2a_{s}u_{s}}}{2},\,\,\,\,\,\,\,\gamma_{hA^{2}}\equiv\sqrt{\frac{-2\lambda^{2}_{\Delta
A}a_{s}}{u_{s}}},\,\,\,\,\,\,\,\gamma_{\varphi
XY}\equiv-2,\,\,\,\,\,\,\,\gamma_{X^{2}h}=\gamma_{Y^{2}h}\equiv\sqrt{\frac{-2\lambda^{2}a_{s}}{u_{s}}},\\\
\lambda_{XY}\equiv
u+g+2w,\,\,\,\,\,\,\,\lambda_{hA}\equiv\frac{\lambda_{\Delta
A}}{2},\,\,\,\,\,\,\,\lambda_{Xh}=\lambda_{Yh}\equiv\frac{\lambda}{2}.\end{array}\right.$
(10)

Using these relations, we can derive the flow equations of fundamental
parameters by calculating the effective parameters $\alpha_{X}$, $\alpha_{h}$,
$\alpha_{\varphi}$, $\beta_{X}$, $\beta_{h}$, $\lambda_{XY}$, $\lambda_{Xh}$,
and $\lambda_{hA}$. By performing perturbative expansion in powers of small
coupling parameters Shankar1994RMP  and utilizing $\dot{u}$ to denote the
derivative of $u$ with respect to the varying length scale $l$, we arrive at
the following flow equations with the help of the identifies given by Eq. (10)
Wang2014PRD :

$\displaystyle\left.\begin{array}[]{ll}\dot{a}_{m}=2\left(a_{m}-\frac{\lambda
a_{s}}{u_{s}}\right)+\frac{1}{4\pi^{2}}\left\\{\frac{\lambda}{2}(1+2a_{s})+\frac{8\lambda^{2}a^{2}_{s}}{u_{s}}+\left[1-2\left(a_{m}-\frac{\lambda
a_{s}}{u_{s}}\right)\right]\left[2(2u-g)+\frac{4\lambda^{2}a_{s}}{u_{s}}\right]\right\\}\vspace{0.2cm}\\\
\hskip
21.90874pt+\left(\frac{\lambda}{u_{s}}\dot{a}_{s}+\frac{a_{s}}{u_{s}}\dot{\lambda}-\frac{a_{s}\lambda}{u_{s}^{2}}\dot{u}_{s}\right),\vspace{0.35cm}\\\
\dot{a}_{s}=2a_{s}-\frac{1}{12\pi^{2}}\left\\{\frac{27a_{s}u_{s}}{2}(1+4a_{s})+\frac{12a_{s}\lambda^{2}}{u_{s}}\left[1-4\left(a_{m}-\frac{a_{s}\lambda}{u_{s}}\right)\right]+3\lambda\left[1-2\left(a_{m}-\frac{a_{s}\lambda}{u_{s}}\right)\right]\right.\vspace{0.2cm}\\\
\hskip 21.90874pt\left.+\frac{9u_{s}}{4}(1+2a_{s})+3\lambda_{\Delta
A}\left(1+\frac{2a_{s}\lambda_{\Delta
A}}{u_{s}}\right)+\frac{32a_{s}\lambda^{2}_{\Delta
A}}{u_{s}}\left(1+\frac{4a_{s}\lambda_{\Delta
A}}{u_{s}}\right)\right\\},\vspace{0.35cm}\\\
\dot{u}_{s}=u_{s}+\frac{1}{\pi^{2}}\left\\{-\frac{9u^{2}_{s}}{4}(4a_{s}+1)+2\lambda^{2}\left[4\left(a_{m}-\frac{a_{s}\lambda}{u_{s}}\right)-1\right]+54a_{s}u^{2}_{s}(1+6a_{s})-\frac{4\lambda^{2}_{\Delta
A}}{3}\left(\frac{4a_{s}\lambda_{\Delta
A}}{u_{s}}+1\right)\right.\vspace{0.2cm}\\\ \hskip
21.90874pt\left.+\frac{32a_{s}\lambda^{3}}{u_{s}}\left(1+\frac{6a_{s}\lambda}{u_{s}}\right)+\frac{11072a_{s}\lambda^{3}_{\Delta
A}}{35u_{s}}\left(1+\frac{6a_{s}\lambda_{\Delta
A}}{u_{s}}\right)\right\\},\vspace{0.35cm}\\\
\dot{u}=u+w+\frac{1}{2\pi^{2}}\left\\{\left[9(u-g)^{2}+3(u+g+2w)(u-g)+5(u+g+2w)^{2}+12w(u-g)\right]\left[4\left(a_{m}-\frac{a_{s}\lambda}{u_{s}}\right)-1\right]\right.\vspace{0.2cm}\\\
\hskip
21.90874pt-\frac{3\lambda^{2}}{8}(4a_{s}+1)+\frac{24a_{s}\lambda^{2}(u-g)}{u_{s}}\left[1-2\left(2\left(a_{m}-\frac{a_{s}\lambda}{u_{s}}\right)-a_{s}\right)\right]+4w(u+g+2w)\left[4\left(a_{m}-\frac{a_{s}\lambda}{u_{s}}\right)-1\right]\vspace{0.2cm}\\\
\hskip
21.90874pt\left.+\frac{8a_{s}\lambda^{2}(u+g+2w)}{u_{s}}\left[1-4\left(a_{m}-\frac{a_{s}\lambda}{u_{s}}\right)+2a_{s}\right]+\frac{4a_{s}\lambda^{3}}{u_{s}}\left[1-2\left(a_{m}-\frac{a_{s}\lambda}{u_{s}}\right)+4a_{s}\right]\right\\}-\dot{w},\vspace{0.35cm}\\\
\dot{g}=g+w-\frac{1}{2\pi^{2}}\left\\{\left[9(u-g)^{2}-3(u+g+2w)(u-g)-3(u+g+2w)^{2}-12w(u-g)\right]\left[4\left(a_{m}-\frac{a_{s}\lambda}{u_{s}}\right)-1\right]\right.\vspace{0.2cm}\\\
\hskip
21.90874pt-\frac{\lambda^{2}}{8}(4a_{s}+1)+\frac{24a_{s}\lambda^{2}(u-g)}{u_{s}}\left[1-2\left(2\left(a_{m}-\frac{a_{s}\lambda}{u_{s}}\right)-a_{s}\right)\right]+4w(u+g+2w)\left[4\left(a_{m}-\frac{a_{s}\lambda}{u_{s}}\right)-1\right]\vspace{0.2cm}\\\
\hskip
21.90874pt\left.-\frac{8a_{s}\lambda^{2}(u+g+2w)}{u_{s}}\left[1-4\left(a_{m}-\frac{a_{s}\lambda}{u_{s}}\right)+2a_{s}\right]+\frac{4a_{s}\lambda^{3}}{u_{s}}\left[1-2\left(a_{m}-\frac{a_{s}\lambda}{u_{s}}\right)+4a_{s}\right]\right\\}-\dot{w},\vspace{0.35cm}\\\
\dot{\lambda}=\lambda+\frac{1}{\pi^{2}}\left\\{\frac{4a_{s}\lambda^{3}}{u_{s}}\left[1-4\left(a_{m}-\frac{a_{s}\lambda}{u_{s}}\right)+2a_{s}\right]+\lambda(2u-g+3w)\left[4\left(a_{m}-\frac{a_{s}\lambda}{u_{s}}\right)-1\right]\right.\vspace{0.2cm}\\\
\hskip
21.90874pt+\frac{16a_{s}\lambda^{2}(2u-g+w)}{u_{s}}\left[1-6\left(a_{m}-\frac{a_{s}\lambda}{u_{s}}\right)\right]+6a_{s}\lambda^{2}\left[1-2\left(a_{m}-\frac{a_{s}\lambda}{u_{s}}\right)+4\alpha_{s}\right]\vspace{0.2cm}\\\
\hskip
21.90874pt\left.-\frac{3u_{s}\lambda}{8}(4a_{s}+1)+2\lambda^{2}\left[2\left(a_{m}-\frac{a_{s}\lambda}{u_{s}}\right)-2a_{s}-1\right]+9a_{s}u_{s}\lambda(1+6a_{s})\right\\},\vspace{0.35cm}\\\
\dot{\lambda}_{\Delta A}=\lambda_{\Delta
A}+\frac{1}{3\pi^{2}}\left\\{27a_{s}u_{s}\lambda_{\Delta
A}(1+6a_{s})+\frac{4\lambda^{2}_{\Delta A}(16a_{s}\lambda_{\Delta
A}-3u_{s})}{u_{s}}\left[1+2a_{s}\left(1+\frac{2\lambda_{\Delta
A}}{u_{s}}\right)\right]\right.\vspace{0.2cm}\\\ \hskip
21.90874pt\left.-\frac{9u_{s}\lambda_{\Delta
A}}{8}(4a_{s}+1)+36a_{s}\lambda^{2}_{\Delta
A}\left[1+2a_{s}\left(2+\frac{\lambda_{\Delta
A}}{u_{s}}\right)\right]\right\\},\vspace{0.35cm}\\\
\dot{w}=\frac{w^{2}}{\pi^{2}}\left[1-4\left(a_{m}-\frac{\lambda
a_{s}}{u_{s}}\right)\right].\end{array}\right.$ (29)

##  III Comparison with experiments

In this section, we will compare the RG results with recent experiments. We
first numerically solve the self-consistently coupled RG equations, and then
manage to understand a number of important features observed by Böhmer _et
al._ in $\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$ Hardy2015NComm . We are
particularly interested in the doping dependence of nematic critical line in
the SC dome, the suppression of superconductivity observed at the magnetic
QCP, and the nature of the observed $C_{4}$ symmetric magnetic order, which
will be studied one by one based on the RG solutions.

As can be seen from the phase diagram presented in Fig. 1, the magnetic and SC
orders are assumed to coexist over a finite region, with $x_{2}$ being the
magnetic QCP. Such a coexistence can be realized if the bare values of model
parameters satisfy the constraint She2010PRB ; Vorontsov ; Fernandes2010PRB ;
Schmalian2010PRB ; Fernandes2013PRL ; Wang2013NJP
$\lambda<\sqrt{u_{s}[(u+w)-(g+w)]}=\sqrt{u_{s}(u-g)}$. For simplicity, we will
only consider the low-$T$ region in the close vicinity of the magnetic QCP
inside the SC dome. In addition, the external field $A$ is assumed to be weak,
but the basic conclusion does not depend on this assumption.

###  III.1 Slope of nematic critical line in SC dome

The nematic line is not shown apparently in Fig. 1. However, the $C_{2}$
magnetic phase is always accompanied (even preempted) by a nematic phase with
a critical temperature $T_{n}$ higher than that of $C_{2}$ magnetic order
Basov2011NPhys ; Chubukov2012ARCMP ; Fernandes2013PRL ; Fernandes2014NPhys .
Hence $T_{m4}$ is also a nematic critical line.

A known fact is that a long-range order can always be destroyed by thermal
fluctuation at sufficiently high $T$. In a system containing two or more
distinct orders, the competition between these orders might destroy some
specific order at very low $T$. As a result, the critical line on the $(x,T)$
phase diagram for this specific order has a positive slope in the low-$T$
region, which is often called back-bending behavior. Interestingly, such back-
bending behavior has been observed in some high-$T_{c}$ cuprate superconductor
Vishik ; Hashimotonp2014 ; Hashimotonm2015  and FeSCs Nandi2010PRL . In
cuprate Bi2Sr2CaCu2O8+δ, a pseudogap exists above $T_{c}$ on the $(x,T)$ phase
diagram. This pseudogap decreases rapidly with growing doping $x$, so its
critical line exhibits a negative slope above $T_{c}$. However, after entering
into the SC dome, the critical line for pseudogap was found to bend backwards
to lower doping, and thus displays a positive slope in the low-$T$ region
Vishik ; Hashimotonp2014 ; Hashimotonm2015 . A simimar behavior was also
observed in $\mathrm{Ba(Fe_{1-x}Co_{x})_{2}As_{2}}$ by Nandi _et al._
Nandi2010PRL . In this case, it is the nematic order that is in strong
competition with SC order. The nematic critical line has a negative slope on
$(x,T)$ phase diagram above $T_{c}$, but displays a positive slope below
$T_{c}$ Nandi2010PRL . A common feature observed in these two compounds is
that the critical line for the order competing with superconductivity has a
positive slope in the low-$T$ region. While a convincing theoretic explanation
for the back-bending of pseudogap critical line in Bi2Sr2CaCu2O8+δ is lacking,
a recent RG work reproduced the back-bending of nematic critical line by
studying the competition between nematic and SC orders in
$\mathrm{Ba(Fe_{1-x}Co_{x})_{2}As_{2}}$ Wang2015PRB . Different from
$\mathrm{Ba(Fe_{1-x}Co_{x})_{2}As_{2}}$, the nematic critical line has a
negative slope in the SC dome of $\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$
Hardy2015NComm  despite the presence of ordering competition.

Figure 2: (Color online) (a) $T$-dependence of parameter $g$ at several
different values of $w_{0}$. Calculations confirm that the positive parameter
$g$ never becomes negative as $T$ decreases. (b) $T$-dependence of
$\rho^{A}_{s}(T)/\rho^{A}_{s}(0)$. The quantity $\rho^{A}_{s}(T=0)$ is
temperature independent at sufficiently low temperatures in the absence of
ordering competition. It is apparent that $\rho^{A}_{s}(T)$ vanishes at
certain point $C$, where $T\approx 0.8T_{0}$.

In $\mathrm{Ba(Fe_{1-x}Co_{x})_{2}As_{2}}$, there is only $C_{2}$ symmetric
magnetic order, induced by a stripe-type SDW state. As discussed in Ref.
Fernandes2013PRL , the existence of nematic order is tuned by the quadratic
term $-g(M^{2}_{X}-M^{2}_{Y})^{2}$. The system can stay either in the PM
phase, or in one of the $M_{X}$ and $M_{Y}$ magnetically ordered phases. The
former case corresponds to a state in which $g<0$ and no nematic order exists.
In the latter case, $g>0$ and hence the system exhibits a nematic order. Both
of these two possibilities can be realized at low $T$. When the competition
between nematic and SC orders is sufficiently strong, it is in principle
possible for the nematic order to be suppressed in the low-$T$ region, leading
to a positive slope of nematic critical line in the SC dome Wang2015PRB .

We now use the RG solutions to judge whether the nematic critical line has a
positive or negative slope in the SC dome of
$\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$. In the effective field model given by
Eq. (4), the relation between $g$ and $w$ determines whether the nematic order
is present or not. As pointed out previously in Refs. Fernandes2014NPhys ;
Fernandes2012PRB ; Wang2015PRB , when $g>\max(0,-w)$, only one of the two
order parameters $M_{X}$ and $M_{Y}$ develops a finite mean value due to
tetragonal symmetry breaking, which is a clear signature for the existence of
a nematic order. On the other hand, we have $\langle M_{X}\rangle=\langle
M_{Y}\rangle$ if $g<\max(0,-w)$, which implies the absence of nematic order
Fernandes2012PRB ; Fernandes2014NPhys ; Wang2015PRB . This property will be
used to judge whether the nematic critical line bends back.

Figure 3: Two possibilities about the slopes of the nematic transition lines.

To examine how the relation between $g$ and $w$ varies with decreasing $T$, we
have solved the RG equations and obtained its dependence on the running length
scale $l$. To be specific, we have chosen the following bare values of model
parameters: $a^{0}_{s}=-0.001$, $u^{0}=0.05$, $u^{0}_{s}=0.01$, $g^{0}=0.01$,
$\lambda^{0}=0.01$, $\lambda^{0}_{\Delta A}=1.0\times 10^{-8}$. We consider
several representative values of $w^{0}$: $w^{0}=0.1u^{0}$, $0.3u^{0}$,
$0.5u^{0}$, and $1.2u^{0}$. The $l$-dependence of these parameters can be
easily converted to a $T$-dependence by utilizing the transformation
Huh2008PRB ; She2015PRB ; Wang2015PRB  $T=T_{0}e^{-l}$, where $T_{0}$ is some
reference temperature smaller than $T_{c}$. The numerical results are
presented in Fig. 2(a). We now determine whether the nematic state becomes a
non-nematic state as $T$ is lowered down to zero on the basis of these
results.

There are in principle two possibilities about the slopes of nematic
transition line $T_{m4}$, as schematically shown in Fig. 3. We consider an
arbitrary point $A$ lying slightly below the transition line $T_{m4}$. At
point $A$, the system is in the nematic state with $g>\mathrm{max}(0,-w)$. We
then lower $T$ along the route $A\rightarrow B$. If the inequality
$g>\mathrm{max}(0,-w)$ is always satisfied as $T\rightarrow 0$ along
$A\rightarrow B$, the system is always in the nematic state and the slope of
the transition line $T_{m4}$ is negative. This corresponds to the case
represented by Fig. 3(a). In contrast, if the condition $g>\mathrm{max}(0,-w)$
is violated as $T$ is reduced to certain value, the second possibility shown
in Fig. 3(b) occurs. In this case, the nematic state becomes non-nematic once
again and the slope of transition line becomes positive at lower temperatures,
exhibiting back-bending behavioe. The numerical results of Eq. (29) informs
that the ratio $g/|w|\gg 1$ for various values of $l$, wherein $w$ may be both
positive and negative (it is negative for the curve shown in Fig. 4). From the
asymptotic behaviors of $g(l)$ presented in Fig. 2(a) and $g(l)/|w(l)|$
presented in Fig. 4, we infer that the inequality $g>max(0,-w)$ remains true
as $T\rightarrow 0$ if it is satisfied at the starting point $A$. This clearly
indicates that the nematic transition line $T_{m4}$ has a negative slope
inside the SC dome and never bends backwards, which is well consistent with
the observed phase diagram Hardy2015NComm .

###  III.2 Suppression of superconductivity due to ordering competition

We now verify whether superconductivity is suppressed by ordering competition
in the vicinity of the magnetic QCP. To this end, we will compute the
superfluid density after taking into account the ordering competition among
nematic ($C_{2}$ SDW), $C_{4}$ SDW and SC orders. The $T$-dependence of
superfluid density $\rho_{s}(T)$ is computed based on the $l$-dependence of
parameter $\alpha_{A}\equiv\alpha_{A}(l)$ Huh2008PRB ; She2015PRB , which is
extracted from the coupled flow equation (29) and hence captures the ordering
competition. The superfluid density of superconductor has the generic form
$\rho_{s}(T)=\rho^{A}_{s}(T)-\rho_{n}(T)$, where $\rho^{A}_{s}(T)$ can be
evaluated by virtue of the formula $\rho^{A}_{s}(T)\propto\alpha_{A}(T)$ with
$\alpha_{A}(T)$ being the mass for vector potential $\mathbf{A}$ generated via
Anderson-Higgs mechanism Halperin1974PRL  and $\rho_{n}(T)$ is the density of
thermally excited normal (non-SC) fermionic quasiparticles.

Figure 4: (Color online) The evolutions of $g/|w|$ at some representative
initial values of parameters: $a^{0}_{s}=-0.001$, $u^{0}_{s}=0.05$,
$u^{0}=0.01$, $g^{0}=0.01$, $\lambda^{0}=0.01$, and $\lambda_{\Delta
A}^{0}=1.0\times 10^{-8}$. The basic conclusion is independent of these
initial values.

In this work, we consider only the competition between distinct order
parameters and neglect the contribution of the normal component, i.e.
$\rho_{n}(T)\ll\rho^{A}_{s}(T)$, which is possible for the $T\ll T_{c}$,
focusing on how superfluid density is modified by ordering competition. To
determine the impact of ordering competition, we suppose a specific
temperature $T_{0}$ as a reference, and then examine how superfluid density
$\rho_{s}$ varies as a function of the ratio $T/T_{0}$. We assume that $T_{0}$
is well below $T_{c}$ so that the normal fermionic quasiparticles can nearly
be neglected and hence $\rho_{s}(T)\sim\rho^{A}_{s}(T)$. From the results
displayed in Fig. 2(b), we can see that $\rho^{A}_{s}(T)$ is strongly
dependent of $T$ in the presence of ordering competition and decreases rapidly
as $T/T_{0}$ grows. It is thus clear that the superfluid density is strongly
suppressed by ordering competition and approximately goes to zero in the
vicinity of the point $C$, where $T\approx 0.8T_{0}$.

We then consider the impact of ordering competition on $T_{c}$. The value of
$T_{c}$ can be determined by solving the equation
$\rho_{s}(T_{c})=\rho^{A}_{s}(T)-\rho_{n}(T_{c})=0$. Although the contribution
$\rho_{n}(T)$ is not known, we can still infer that $T_{c}$ is suppressed by
ordering competition because $\rho_{s}$ is significantly reduced. As shown in
Fig. 2(b), $\rho_{s}(T)$ vanishes at certain point with $T<T_{0}$ ($T\approx
0.8T_{0}$). This conclusion is well consistent with recent experiment
Hardy2015NComm , in which a considerable drop of $T_{c}$ is observed near the
putative magnetic QCP. In an improved theoretic treatment, one would compute
$\rho_{n}(T)$ by incorporating the contribution of fermionic quasiparticles.
Notice that these quasiparticles are not free, but couple strongly to the SDW
order parameter at the magnetic QCP Liu2012PRB ; Wang_Chubukov2013PRL ;
Levchenko_Chubukov2013PRL ; Chowdhury2013PRL ; Nomoto2013PRL ; Kang2015PRB ;
Chubukov2016PRB . Usually, this coupling tends to excite more fermionic
quasiparticles out of the SC condensate, which further suppresses the
superfluid density and reduces $T_{c}$ Liu2012PRB ; Wang_Chubukov2013PRL ;
Levchenko_Chubukov2013PRL ; Chowdhury2013PRL ; Nomoto2013PRL ; Kang2015PRB ;
Chubukov2016PRB .

Figure 5: (Color online) $T$-dependence of parameter $w$ obtained at given
bare values of dimensionless parameters: $a^{0}_{s}=-0.001$, $u^{0}_{s}=0.05$,
$u^{0}=0.01$, $g^{0}=-0.01$, $\lambda^{0}=0.01$, and $\lambda_{\Delta
A}^{0}=1.0\times 10^{-8}$. The main conclusion is independent of these bare
values. The parameter $w$ is not very sensitive to the change of $T$, so we
provide two insets to show more clearly how $w$ varies as $T$ is decreasing.
(a) Starting point is CSDW-type $C_{4}$ state with $w^{0}<0$; (b) Starting
point is SVC-type $C_{4}$ state with $w^{0}>0$.

###  III.3 $C_{4}$ symmetric magnetic order

We finally turn to analyze the property of $C_{4}$ symmetric magnetic state.
To uncover the effects caused by ordering competition, we consider the
evolution of the system along the route $D\rightarrow E\rightarrow F$, shown
in Fig. 1, and examine how $g$ and $w$ vary along this route.

Eq. (4) clearly shows that $w$ is associated with the quadratic term of
$C_{4}$ SDW order parameter Fernandes2016PRB ; Schmalian2016PRB . In analogy
to the nematic transition, the sign of $w$ determines which sort of $C_{4}$
SDW order, either SVC or CSDW, is realized Fernandes2014NPhys ;
Fernandes2012PRB ; Wang2015PRB ; Fernandes2016PRB ; Schmalian2016PRB . In
particular, a $C_{4}$ SVC order is generated for $g<0$ and $w>0$, whereas
$g<-w$ and $w<0$ implies the occurrence of a $C_{4}$ CSDW order
Fernandes2012PRB ; Fernandes2014NPhys ; Wang2015PRB ; Fernandes2016PRB ;
Schmalian2016PRB . By paralleling the analysis made for nematic critical line,
we convert the $l$-dependence of $w$ using the transformation $T=T_{0}e^{-l}$.
If one assumes that $w$ is negative at the starting point $D$, which amounts
to supposing the system is in the CSDW-type $C_{4}$ magnetic state, it remains
negative as $T$ decreases down to the $F$ point, as can be clearly seen from
Fig. 5(a). This result implies that CSDW state is stable in the low-$T$
region. On the other hand, if one starts from a positive $w$, corresponding to
a SVC-type $C_{4}$ magnetic state, we show in Fig. 5(b) that $w$ eventually
becomes negative at some critical $T$, which can be identified as the $E$
point. It follows that the SVC state is unstable in the low-$T$ region, and
that the CSDW state is more favorable.

We conclude from the above analysis that, although in principle either SVC or
CSDW type $C_{4}$ state could be realized in the high-$T$ region, the CSDW-
type $C_{4}$ state is the only stable one in the low-$T$ region. In a recent
work, Christensen _et al._ Christensen2015PRB  have suggested the spin-orbit
coupling gives rises to the CSDW-type $C_{4}$ SDW. Additionally, Hoyer _et
al._ Schmalian2016PRB  have studied the disorder effects and demonstrated that
impurity scattering favors CSDW over SVC. Here, we provide a different
approach to determine the nature of $C_{4}$ symmetric magnetic order.
Moreover, the sudden drop of $w$ at certain critical energy scale usually
indicates the happening of a first-order transition, which is qualitatively
consistent with recent experiments Hardy2015NComm .

##  IV Comparison to $\mathrm{Ba(Fe_{1-x}Co_{x})_{2}As_{2}}$

We now compare the present RG results with a previous work Wang2015PRB , which
investigated the impact of ordering competition on the global phase diagram of
$\mathrm{Ba(Fe_{1-x}Co_{x})_{2}As_{2}}$. Both
$\mathrm{Ba(Fe_{1-x}Co_{x})_{2}As_{2}}$ and
$\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$ belong to the 122 family of FeSCs, and
display a complicated phase diagram. A common feature is that, over a large
part of their phase diagrams, superconductivity coexists and competes with a
SDW type magnetic order and a nematic order. The ordering competition and its
effects on the phase diagram can be described by deriving an effective low-
energy field theory which is supposed to contain several distinct order
parameters Fernandes2013PRL ; Fernandes2016PRB ; Schmalian2016PRB ;
Fernandes2017RPP . Such an effective theory is expected to be as general as
possible, and applicable in $\mathrm{Ba(Fe_{1-x}Co_{x})_{2}As_{2}}$,
$\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$, and other similar 122 FeSCs.

However, there are some important differences between the compounds
$\mathrm{Ba(Fe_{1-x}Co_{x})_{2}As_{2}}$ and
$\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$. In
$\mathrm{Ba(Fe_{1-x}Co_{x})_{2}As_{2}}$, there is only a $C_{2}$ symmetric
stripe-type SDW order. In contrast, there are both $C_{2}$ and $C_{4}$
symmetric magnetic states in $\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$ and a number
of other hole-doped 122 FeSCs Goldman2010PRB ; Osborn2014NatureComm ;
Wang2016PRB ; Osborn2016NaturePhys ; Hassinger2012PRB ; Hardy2015NComm ;
Allred2015PRB ; Hassinger2016PRB . Moreover, the nematic transition line
exhibits completely different doping dependence in the SC dome of these two
FeSCs: it has a positive slope inside the SC dome of
$\mathrm{Ba(Fe_{1-x}Co_{x})_{2}As_{2}}$ Nandi2010PRL , but a negative slope
inside the SC dome of $\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$ Hardy2015NComm .

To capture both the similarity and difference, the model of
$\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$ should be formally analogous but not
identical to that of $\mathrm{Ba(Fe_{1-x}Co_{x})_{2}As_{2}}$ Wang2015PRB . It
was suggested in Refs.Fernandes2016PRB ; Schmalian2016PRB  that the $C_{4}$
symmetric magnetic order that emerges in $\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$
can be described by introducing a new term
$-\left(\mathbf{M}_{X}\times\mathbf{M}_{Y}\right)^{2}$. As shown previously in
Ref.Wang2015PRB , in the absence of this term, ordering competition gives rise
to the suppression of superconductivity and in particular the positive slope
of nematic transition line in the SC dome, which are in good agreement with
experiments performed in $\mathrm{Ba(Fe_{1-x}Co_{x})_{2}As_{2}}$ Nandi2010PRL
. In the current work, we have demonstrated through RG calculations that,
adding the above new term leads to the suppression of superconductivity near
the magnetic QCP and also the negative slope of nematic transition line in the
SC dome of $\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$, which is qualitatively
consistent with recent experiments Hassinger2012PRB ; Hardy2015NComm ;
Allred2015PRB ; Hassinger2016PRB . Furthermore, our RG analysis revealed that
the CSDW-type $C_{4}$ magnetic state is more favorable than the SVC-type
$C_{4}$ magnetic state, and hence can be used to determine the nature of
$C_{4}$ magnetic state observed in $\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$
Hardy2015NComm . It is therefore clear that the effective model of
$\mathrm{Ba(Fe_{1-x}Co_{x})_{2}As_{2}}$ can be properly modified to describe
$\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$, and that the same perturbative RG scheme
used in Ref.Wang2015PRB  and here can be applied to account for both the
similarity and difference between $\mathrm{Ba(Fe_{1-x}Co_{x})_{2}As_{2}}$ and
$\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$.

##  V Summary and discussion

In summary, we have studied the impact of the competition between
superconductivity and $C_{2}$ and $C_{4}$ symmetric magnetic orders in a hole-
doped FeSC $\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$. After performing a detailed
RG analysis within an effective field theory, we have reproduced a number of
interesting features of the global phase diagram. In particular, our RG
analysis have showed that the order parameter fluctuation and ordering
competition lead to moderate suppression of superconductivity near the
magnetic QCP, maintain the negative slope of nematic critical line in the SC
dome, and also sort out the CSDW-type $C_{4}$ magnetic order as the more
stable state than a SVC-type $C_{4}$ magnetic order in the low-$T$ regime. All
these theoretic results are well consistent with the recent experiments of
Ref. Hardy2015NComm , and schematically summarized in Fig. 1.

Our RG calculations are confined to the small region surrounding the magnetic
QCP in the SC dome. To gain a better knowledge of the entire phase diagram, it
is necessary to consider the non-SC phase above $T_{c}$. A salient feature
observed in Ref. Hardy2015NComm  is the back-bending behavior of a critical
line between nematic ($C_{2}$ SDW) to pure $C_{4}$ SDW orders, namely
$T_{M4}$, which turns out to be induced by the emergence of $C_{4}$ symmetric
magnetic order. The transition line $T_{M4}$ exists well above $T_{c}$, thus
there is no SC order and the competition between SC and magnetic order
parameters is unlikely to be important. It turns out that the underlying
mechanism for the back-bending behavior of $T_{M4}$ is entirely different from
that is used to account for the slope of $T_{m4}$ in the SC dome. We believe
that an essential role is played by elementary fermionic degrees of freedom,
which are strongly suppressed below $T_{c}$ by the SC gap but should be
present above $T_{c}$. The inter-fermion interaction is expected to be
responsible for the transition between $C_{4}$ and $C_{2}$ symmetric magnetic
states. This problem is made more complicated by the uncertainty of the nature
of $C_{4}$ symmetric magnetic order. In the SC dome below $T_{c}$, ordering
competition lifts the degeneracy between CSDW and SVC states at low energies,
and chooses CSDW as the true ground state. However, order competition is much
less important above $T_{c}$. It remains unclear whether the CSDW or SVC state
is realized in the region between $T_{M4}$ and $T_{c}$. The microscopic
mechanism for the back-bending behavior of $T_{M4}$ could be properly
understood only after the nature of $C_{4}$ symmetric magnetic order is
identified, which is subject to future research.

In this paper, we have considered only one specific compound
$\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$. Recent experiments of Hardy _et al._
Hardy2015NComm  provided a clear and detailed global phase diagram of
$\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$, which gives us a good opportunity to
directly compare our RG results with experimental results. Apart from
$\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$, the $C_{4}$ symmetric magnetic order
also exists in a number of other hole-doped FeSCs, including
$\mathrm{Ba(Fe_{1-x}Mn_{x})_{2}As_{2}}$ Goldman2010PRB ,
$\mathrm{Ba_{1-x}Na_{x}Fe_{2}As_{2}}$ Osborn2014NatureComm ; Wang2016PRB , and
$\mathrm{Sr_{1-x}K_{x}Fe_{2}As_{2}}$ Osborn2016NaturePhys . It should be
possible to generalize our RG approach to study the global phase diagrams of
these three FeSCs. However, there might be important difference between
$\mathrm{Ba_{1-x}K_{x}Fe_{2}As_{2}}$ and these FeSCs. In that case, the
effective field-theoretic model given by Eq. (4) needs to be properly
modified. Once the modified effective model is specified, it is
straightforward to carry out RG calculations, just as what we have done in
this work.

##  VI Acknowledgements

J.W. and G.Z.L acknowledge the financial support from the National Natural
Science Foundation of China under Grants 11504360, 11274286, and 11574285.
J.W. is also partly supported by the China Postdoctoral Science Foundation
under Grants 2015T80655 and 2014M560510, the Fundamental Research Funds for
the Central Universities (P. R. China) under Grant WK2030040074, and the
Program of Study Abroad for Young Scholar sponsored by CSC (China Scholarship
Council). D.V.E. and J.v.d.B would like to acknowledge the financial support
provided by the German Research Foundation (Deutsche Forschungsgemeinschaft)
through priority program SPP 1458. J.v.d.B is also supported by SFB 1143 of
the Deutsche Forschungsgemeinschaft.

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