# Interplay of stripe and double-Q magnetism with superconductivity in
$\mathrm{Ba}_{1-x}\mathrm{K}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$ under the
influence of magnetic fields

K. Willa<EMAIL_ADDRESS>Institute for Quantum Materials and
Technologies, Karlsruhe Institute of Technology, Karlsruhe D-76021, Germany
R. Willa  Institute for Condensed Matter Theory, Karlsruhe Institute of
Technology, Karlsruhe D-76131, Germany  F. Hardy  Institute for Quantum
Materials and Technologies, Karlsruhe Institute of Technology, Karlsruhe
D-76021, Germany  L. Wang  Institute for Quantum Materials and Technologies,
Karlsruhe Institute of Technology, Karlsruhe D-76021, Germany  P. Schweiss
Institute for Quantum Materials and Technologies, Karlsruhe Institute of
Technology, Karlsruhe D-76021, Germany  T. Wolf  Institute for Quantum
Materials and Technologies, Karlsruhe Institute of Technology, Karlsruhe
D-76021, Germany  C. Meingast  Institute for Quantum Materials and
Technologies, Karlsruhe Institute of Technology, Karlsruhe D-76021, Germany

###### Abstract

At $x\approx 0.25$
$\mathrm{Ba}_{1-x}\mathrm{K}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$ undergoes a
novel first-order transition from a four-fold symmetric double-Q magnetic
phase to a two-fold symmetric single-Q phase, which was argued to occur
simultaneously with the onset of superconductivity (Böhmer et al., Nat. Comm.
6, 7911 (2015)). Here, by applying magnetic fields up to 10T, we investigate
in more detail the interplay of superconductivity with this magneto-structural
transition using a combination of high-resolution thermal-expansion and heat-
capacity measurements. We find that a magnetic field suppresses the reentrance
of the single-Q orthorhombic phase more strongly than the superconducting
transition, resulting in a splitting of the zero-field first-order transition.
The suppression rate of the orthorhombic reentrance transition is stronger for
out-of-plane than for in-plane fields and scales with the anisotropy of the
superconducting state. These effects are captured within a phenomenological
Ginzburg-Landau model, strongly suggesting that the suppression of the
reentrant orthorhombic single-Q phase is primarily linked to the field-induced
weakening of the superconducting order. Not captured by this model is however
a strong reduction of the orthorhombic distortion for out-of-plane fields,
which deserves further theoretical attention.

##  I Introduction

Coexistence of superconductivity and magnetism as well as their competition
for phase space is a recurring characteristic of iron-based superconductors
[1, 2]. Superconductivity emerges when the prevailing stripe-type
antiferromagnetic spin density wave (SDW)—is suppressed by either
hole/electron doping or pressure [3, 4, 5]. This SDW state is accompanied by
an orthorhombic distortion of the lattice which occurs simultaneously or
sometimes precedes it in form of a vestigial nematic transition. Moreover,
careful investigations of hole-doped systems, including
$\mathrm{Ba}_{1-x}\mathrm{Na}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$,
$\mathrm{Ba}_{1-x}\mathrm{Ka}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$,
$\mathrm{Sr}_{1-x}\mathrm{Na}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$ and
$\mathrm{Ca}_{1-x}\mathrm{Na}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$, have
revealed a plethora of competing new electronic phases near the transition
region between stripe antiferromagnetism ($C_{2}$) and superconductivity. In
particular a tetragonal ($C_{4}$) magnetic phase was discovered in a narrow
doping region [6, 7, 8, 9, 10]. Neutron studies [11, 12] showed that in this
magnetically ordered $C_{4}$ phase, the moments flip from in-plane to out-of-
plane and Mössbauer studies as well as Muon spin rotation measurements [13,
14] revealed a double-$Q$ order, in which only every other Fe atom carries a
magnetic moment. According to the classification of possible double-$Q$
magnetic orders (including, hedgehog and loop spin-vortex crystals), see Ref.
[15], this finding is consistent with the so-called spin-charge density wave.
In the case of
$\mathrm{Ba}_{1-x}\mathrm{K}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$, initial
studies [16, 17, 18, 19] did not find the small doping region where the
double-Q $C_{4}$-phase occurs, which was only later revealed upon closer
investigation [7], see Figure 1 c). In contrast to
$\mathrm{Ba}_{1-x}\mathrm{Na}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$ and
$\mathrm{Sr}_{1-x}\mathrm{Na}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$ [8, 10, 20],
the $C_{4}$ phase in
$\mathrm{Ba}_{1-x}\mathrm{K}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$ does not
extent to zero temperature; rather superconductivity surprisingly drives the
system back to a single-Q orthorhombic magnetic phase [7, 21]. Böhmer et al.
proposed that this was due to the higher electronic entropy available for
superconductivity in the single-Q compared to the double-Q phase.

Figure 1: Temperature dependence of the thermal expansion $\Delta L/L$ of
$\mathrm{Ba}_{1-x}\mathrm{K}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$ measured along
the [110] direction in different external magnetic fields where for a) H
$\parallel$ [110] and for b) H $\parallel$ [001]. The discontinuities in
$\Delta L/L$ mark the transitions between the $C_{2}$ and $C_{4}$-phases as
indicated at the bottom of the figure. c) Phase diagram of
$\mathrm{Ba}_{1-x}\mathrm{K}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$ for $0<x<0.3$
[7], showing the narrow doping region in which a C4-phase is observed. d) The
extrapolated magnetic field dependent step height $\delta L_{0K}(H)/\delta
L_{0K}(0)$ as extracted from the thermal expansion measurements.

Here, by applying magnetic fields up to 10 T, we investigate in more detail
the interplay of superconductivity with this magnetostructural transition
using a combination of high-resolution thermal-expansion and heat-capacity
measurements. We find that a magnetic field suppresses the reentrance of the
single-Q orthorhombic phase more strongly than the superconducting transition,
resulting in a splitting of the zero-field first-order transition. The
suppression rate of the orthorhombic reentrance transition, which remains
first-order, is stronger for out-of-plane than for in-plane fields and scales
with the anisotropy of the superconducting state. These effects are captured
within a phenomenological Ginzburg-Landau model, strongly suggesting that the
suppression of the reentrant orthorhombic magnetic phase is primarily linked
to the field-induced weakening of the superconducting order. Not captured by
this model is however the strong reduction of the orthorhombic distortion for
out-of-plane fields, which deserves further theoretical attention.

Figure 2: Temperature dependence of the electronic specific heat for
$\mathrm{Ba}_{1-x}\mathrm{K}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$ subject to
different applied magnetic fields up to 10T. The left (right) panel shows the
response to in-plane (out-of-plane) magnetic fields. The arrows mark the
location of the superconducting and structural transitions, respectively. The
shaded region highlights the entropy corresponding to the structural
distortion. Curves are shifted vertically by -0.03J/molK2 for every 2T for
clarity, the dashed lines mark the corresponding $C/T=0$. The inset c) shows
the entropy associated to the $C_{4}$-to-$C_{2}$ phase transition (equal to
the shaded region in $C/T$) normalized by the zero-field entropy.

##  II Experimental details

Heat-capacity and thermal expansion measurements were made on two platelet
shaped self-flux grown single crystals
$\mathrm{Ba}_{1-x}\mathrm{K}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$ ($x\approx
0.24$) with masses of 2.63mg (Sample A) and 1.93mg (Sample B), in plane
dimensions of $2.1\\!\times\\!2.6\mathrm{mm}^{2}$ (Sample A). The samples were
grown by a self-flux technique [7]. Both crystals are from the same batch as
in Ref. 7 and where more details about the crystal synthesis are provided.
Thermal-expansion measurements were performed on Sample A in a home-made high-
resolution capacitive dilatometer [22]. As demonstrated previously,
measurements of the crystal along the [110]T direction of the original
tetragonal cell ($C_{4}$-PM) produces partial ’detwinning’ due to the pressure
of the springs in the dilatometer cell, giving access to the thermal expansion
of the shorter b-axis in the low-temperature stripe phase (C2-M) [7, 8, 23].
Rotating the dilatometry cell allows to align the magnetic field along both
the [110]T and the [001] crystal directions. Specific heat measurements were
performed on sample B in a 14T Physical Property Measurement system from
Quantum Design with the heat-capacity option.

##  III Results

Figures 1a) and 1b) show the relative thermal expansion $\Delta
L_{110}/L_{110}$ of sample A for fields applied parallel to the [110]T and
[001] directions, respectively. In zero field, a pronounced reduction of
$\Delta L_{110}/L_{110}$ occurs at $T_{s,N}=65$K indicative of the transition
from the paramagnetic tetragonal state into the $C_{2}$-magnetic stripe phase.
At lower temperatures, a switch back of $\Delta L_{110}/L_{110}$ is observed
at $T_{1}=32$K, where the system enters the double-Q magnetic phase
($C_{4}$-M) that restores tetragonal symmetry, followed by another sudden
reduction at $T_{2}=19$K marking the simultaneous re-entrance of stripe
magnetism and the emergence of superconductivity. Application of an external
magnetic field of up to 10T does not affect the magnetic transitions at
$T_{s,N}$ and $T_{1}$ for either field direction, indicative of a very robust
magnetism. The locus of these first-order phase transitions and the determined
$\mathrm{K}$ concentration of sample A are in good agreement with the reported
$(x,T)$ phase diagram of Ref. 7, schematically reproduced in Fig. 1c) (dashed
line). The reduced value of $\Delta L_{110}/L_{110}$ for $T<T_{2}$ with
respect to that from extrapolation from the $C_{2}$-M stripe phase indicates
that superconductivity, which favors the single-Q stripe phase over the
double-Q phase, however still competes with the former [24]. In contrast, to
the transitions at $T_{s,N}$ and $T_{1}$, a clear shift of $T_{2}(H)$ towards
lower temperature is observed with increasing field for both field
orientations. However, the opposite evolution of $\Delta L_{110}/L_{110}$,
which is an indicator of the orthorhombic distortion, is quite different
inside this re-entrant $C_{2}$ phase for both field directions. Whereas
$\Delta L_{110}/L_{110}(H)$ is slightly enhanced for fields along [110]T, it
is drastically reduced to $\approx 10\%$ of its zero-field value for
$\mu_{0}H=10$T along the [001] direction, as illustrated in figure 1d),
suggesting an almost complete suppression of the reentrant $C_{2}$ phase.
These opposite behaviors with field orientation cannot be solely accounted for
by the re-entrance of the stripe phase, and the influence of
superconductivity, which emerges concomitantly in zero field, must be
considered. In order to gain further insight about the interplay of
superconductivity with the reentrant transition, we have also measured the
heat capacity for both field orientations.

Figures 2a) and 2b) show the respective electronic specific heat of sample B
for fields applied perpendicular and parallel to the [001] direction, obtained
after subtracting a suitable lattice contribution (see e.g. Refs. [7, 25, 26,
27] for details). In zero field, a single first-order-like peak at
$T_{c}=T_{2}=21$K confirms that the re-entrance into the $C_{2}$ stripe phase
occurs simultaneously with superconductivity, as observed in Ref. 7. This is
corroborated by the vanishing $C_{e}/T$ in the $T\rightarrow 0$ limit,
indicative of a fully-gapped Fermi surface.

With increasing field, a progressive splitting of the zero-field peak into two
distinct anomalies is unambiguously resolved for $H\perp$[001]. Here, a
broadened mean-field-like anomaly at $T_{c}(H)$ marks the transition to the
superconducting state (dark blue arrow) followed by a peak at
$T_{2}(H)<T_{c}(H)$ where reentrance of the stripe phase takes place (light
blue arrow). Thus, our data reveal that the application of a magnetic field
causes the apparition of two distinct superconducting states coexisting with
either the single-Q stripe SDW or the double-Q C-SDW, separated by the
$T_{2}(H)$ boundary. In field, the transition at $T_{2}$ broadens somewhat,
but remains first-order as illustrated by the entropy discontinuity still
observed in 10 T (see Fig. 4a)). This is also consistent with the
discontinuity observed in the thermal expansion at $T_{2}$. For
$H\parallel$[001], the splitting is also clearly observed (see Fig.2b)), but
with a more pronounced suppression of $T_{2}(H)$. In contrast to in-plane
fields, the peak at $T_{2}(H)$ (blue arrow) is rapidly suppressed with
increasing $c$-axis fields and fades away for $H>6$T. We have estimated the
entropy associated with this transition by the area between the measured curve
and an extrapolation of the low-T quasi linear behavior to $T=T_{c}$, see
shaded areas in Figure 2. Most prominently, we note that this entropy
discontinuity rapidly decreases with increasing field strength for fields
along the $c$ axis, while experiencing only moderate changes for in-plane
fields. This trend, shown in the inset of Figure 2, is in line with the field
dependence of the orthorhombic distortion, as inferred from our thermal-
expansion data, Fig. 1d). We note that we do not observe a clear signature at
$T_{c}$ in the thermal expansion even at 10 T, where there is a clear
separation between $T_{2}$ and $T_{c}$. This is surprising, since, although
small, clear thermal expansion anomalies at $T_{c}$ have been observed in both
$\mathrm{Ba}_{1-x}\mathrm{Na}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$ and
$\mathrm{Sr}_{1-x}\mathrm{Na}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$ in the
double-Q phase [8, 10].

Figure 3: a) Calculated heat capacity near the onset of superconductivity for
different magnetic field strengths. At zero field, the transition is first-
order and accompanied by a magnetic switching from the $C_{4}$ to the $C_{2}$
stripe order. For sufficiently large fields, $H\gtrsim 0.02$, this transition
is split into a second order onset of superconductivity, followed by a first-
order magnetic switching. The proxy $|m_{x}-m_{y}|(T)$ of the orthorhombic
distortion for different magnetic fields is shown below in b). Transition
temperatures are marked with arrows. c) Close-up of the phase diagram near the
tricritical point $(x^{*},T^{*})$ where the $C_{2}$ and $C_{4}$ magnetic
phases are degenerate and where superconductivity appears. For larger doping
the first-order onset of superconductivity is accompanied by a transition into
a $C_{2}$ magnetic phase. At even higher doping levels, these two transitions
split and allow for a superconducting phase with $C_{4}$ magnetic order. The
dashed line marks the doping concentration for which the specific heat and
orthorhombic distortion are shown in a) and b).  Figure 4: Panels a) and b)
show the temperature dependence of the electronic entropy in 0T and for 10T
applied in- and out-of-plane fields. Panel c) shows the calculated entropy for
the theoretical model. $T_{c}$, $T_{2}$, and $T_{1}$ are marked with arrows.
The dashed lines guide the eye along the theoretical entropy curve. Panels d)
and e) show the low temperature phase diagram with transition temperatures
$T_{c}$ and $T_{2}$ taken from specific heat (circles) and thermal expansion
(rectangles). f) Theoretically calculated phase boundary.

##  IV Theoretical model

Multiple theoretical studies [28, 29, 30, 31] have investigated the interplay
between the different possible magnetic ground states in iron-based
superconductors. For a qualitative understanding of the physical mechanisms
underlying the presently observed magnetic field dependence of the reentrant
transition in $\mathrm{Ba}_{1-x}\mathrm{K}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$,
we aim at capturing the key observations within a minimal phenomenological
model. We start from the treatment proposed by Kang _et al._ [32], and account
for a spin-density modulation
$m=m_{x}e^{i\boldsymbol{Q}_{x}\boldsymbol{r}}+m_{y}e^{i\boldsymbol{Q}_{y}\boldsymbol{r}}$
and the leading $s_{\pm}$ superconducting pairing state $\Delta$ [33, 34, 35].
The effective theory for the SDW fields
$\boldsymbol{m}_{x/y}\\!\propto\\!\frac{1}{2}\sum_{\boldsymbol{k}}c^{\dagger}_{\boldsymbol{k}}\boldsymbol{\sigma}c_{\boldsymbol{k}+\boldsymbol{Q}_{x/y}}$
($m_{x/y}\\!=\\!|\boldsymbol{m}_{x/y}|$) is obtained by applying a Hubbard-
Stratonovich transformation to decouple the fermion interaction and
subsequently integrating out the fermionic fields. The free energy functional
$\mathcal{F}[m_{x},m_{y},\Delta]=\mathcal{F}_{m}+\mathcal{F}_{s}+\mathcal{F}_{\mathrm{int}}$
consists of a magnetic part

$\displaystyle\\!\\!\\!\mathcal{F}_{m}$
$\displaystyle=\frac{\alpha}{2}(m_{x}^{2}+m_{y}^{2})+\frac{u}{4}(m_{x}^{2}+m_{y}^{2})^{2}-\frac{g}{4}(m_{x}^{2}-m_{y}^{2})^{2}$
$\displaystyle\quad+\frac{v}{6}(m_{x}^{2}+m_{y}^{2})(m_{x}^{2}-m_{y}^{2})^{2}+\frac{\gamma}{6}(m_{x}^{2}+m_{y}^{2})^{3},$
(1)

a minimal superconducting contribution

$\displaystyle\mathcal{F}_{s}$
$\displaystyle=\frac{\alpha_{s}}{2}\Delta^{2}+\frac{u_{s}}{4}\Delta^{4},$ (2)

and a mutual interaction

$\displaystyle\mathcal{F}_{\mathrm{int}}$
$\displaystyle=\frac{c_{1}}{2}\Delta^{2}(m_{x}^{2}+m_{y}^{2})+2c_{2}\Delta^{2}m_{x}^{2}m_{y}^{2}.$
(3)

The magnetic sector $\mathcal{F}_{m}$ allows either for a $C_{4}$ order with
$m_{x}=m_{y}$, or two degenerate $C_{2}$ orders with $m_{x}=m$ and $m_{y}=0$
or vice versa. The energy sheets associated with these two possibilities
evolve differently upon changing external parameters (temperature, doping) and
will trigger first-order transitions whenever they cross. Upon entering the
magnetic phase, the term $-(g/4)(m_{x}^{2}-m_{y}^{2})^{2}$ favors the $C_{2}$
magnetic order for $g>0$. For $v>0$ the sixth-order term may tip the balance
in favor of the $C_{4}$ phase. Consequently, $v$ seems a suitable tuning
parameter to capture the dominant doping dependence. The first interaction
term in $\mathcal{F}_{\mathrm{int}}$ accounts for a competition between
superconductivity and magnetism ($c_{1}>0$). The second term ($c_{2}>0$)
further enhances the competition of superconductivity with the
$C_{4}$-magnetic order while being absent within the $C_{2}$-magnetic phase.
Our model extends the one of Kang _et al._ [32] by including a magnetic field
dependence in $\alpha_{s}$. Owing to the magnetic field robustness of the
first two magnetic transitions observed in the experiment, we attribute the
leading field-dependence to the superconducting orbital pair-breaking. In
addition, the second term in $\mathcal{F}_{\mathrm{int}}$ brings a strong
energetic distinction on how the two magnetic orders couple to
superconductivity. Such an interaction has been shown to arise from the
coupling of the superconducting $s_{\pm}$ order to an underlying, yet
unfulfilled, $d$-wave gap function [32]. When the system realizes the $C_{4}$
phase with a nearly degenerate (but subordinate) $C_{2}$ state, the onset of
superconductivity can trigger a first-order transition into a superconducting
$C_{2}$ state. More details on the model and the specific parameter values are
given in Appendix A.

The phase diagram in zero magnetic field, Figure 3 c), reproduces well the
observation reported in Ref. [7]. Upon cooling, the system first enters the
$C_{2}$ stripe phase, followed, at low doping, by the onset of
superconductivity. Above a critical doping, the $C_{2}$-magnetic state
undergoes a first-order transition into a tetragonal double-$Q$ magnetic phase
which lowers the onset of superconductivity, presumably due to additional
gapping of the fermi surface in the double-Q phase [7]. This lost density of
states is recovered when, with the onset of superconductivity, the magnetism
switches back to the $C_{2}$ stripe phase. At even larger doping levels, the
last two transitions appear sequentially, with superconductivity setting in as
a second-order transition, followed by a first-order switching to the
$C_{2}$-magnetic state.

For a fixed doping level, indicated by the dashed line in figure 3c), we have
calculated the heat capacity and $|m_{x}-m_{y}|(T)$ as a proxy for the
orthorhombic lattice distortion. Both quantities associated to our model
calculation are shown in Figure 3a) and b). In zero magnetic field, we observe
the first order transition into the double-Q charge density wave at $T_{1}$
and then superconductivity drives the system back in to a single-Q SDW. This
joint transition is well captured by both a delta peak and a step in the heat
capacity as well as a step in the lattice distortion. In magnetic fields, the
transition splits apart and the system becomes first superconducting (step in
$C(T)$) and then orthorhombic (delta peak in $C(T)$). As the lattice
distortion is zero in the tetragonal phase, the onset of superconductivity is
not captured in $|m_{x}-m_{y}|(T)$. In our model calculations, we observe a
field-induced low-temperature increase of the orthorhombic distortion, like
that observed in the thermal expansion data for $H\parallel[100]$ (see Fig. 1.
In the model, this can be attributed to the field-induced weakening of the
competition between superconductivity and stripe-type magnetism.

##  V Discussion and Conclusion

The transition temperatures inferred from our thermal-expansion and heat
capacity measurements are summarized in the ($H,T$) phase diagrams of Figs 4d)
and 4e) for $H\perp$[001] and $H\parallel$[001], respectively, and are
compared to the theoretical phase diagram shown in Fig. 4(f). Here, $T_{2}(H)$
is determined experimentally as the position of the maximum in heat capacity
(light blue arrow in Fig. 2) and in the coefficient of linear thermal
expansion $\alpha_{110}(T,H)=1/L_{100}((\partial\Delta L_{110})/\partial T)$.
As the peak observed in specific heat broadens considerably with increasing
field in the [001] direction, this criterion is restricted to fields less than
6 T. On the other hand, we determine $T_{c}(H)$ as the maximum of the mean-
field discontinuity in heat capacity, rather than using an entropy-conserving
construction, because it is partially obstructed by the nearby broadened
transition to the low temperature orthorhombic single-Q state. The small
offset in $T_{2}(H)$ between the heat capacity (light blue circles) and
thermal-expansion data (light blue squares) is likely related to slight
variations in $\mathrm{K}$ content between the two samples, but this does not
affect our discussion since both lines are found to run parallel till 10 T.

We find that $T_{c}$ is weakly suppressed for both field orientations with
$((\partial T_{c})/\partial H)\approx 0.1$K/T for $H\perp$[001] with the
superconducting anisotropy being $\approx 2$, a typical value also for other
Fe-based superconductors [36, 37, 38, 39, 40]. In contrast, the suppression of
$T_{2}(H)$ is more pronounced, about a factor 4 larger than that of
$T_{c}(H)$, revealing that a minor weakening of $T_{c}(H)$ has a strong impact
on the transition into the reentrant single-Q phase. Furthermore, the
anisotropy of suppression of $T_{2}(H)$ is also close to 2 scaling with the
superconducting anisotropy which strongly suggests that the stripe-phase
reentrance is intimately coupled to superconductivity and by weakening the
superconducting order through applied magnetic fields the reentrance of the
low temperature single-$Q$ SDW is retarded. Although our model does not
account for both crystal and magnetic anisotropies, an anisotropic field
dependence of $\alpha_{S}$ in Eq. (2) matching the experimental
superconducting anisotropy immediately reproduces the observed anisotropy of
$T_{2}(H)$.

Not reproduced by our minimal model is the contrasting thermal expansion
behavior of the $b$-orthorhombic axis (see Fig.1) with respect to field
direction observed deep inside the reentrant orthorhombic single-Q phase. On
the one hand, the model reproduces quite well (in Fig.3b)) the slight increase
of orthorhombicity found for $H\parallel$[110]T. This effect can be understood
to result from the well documented competition between superconductivity and
the orthorhombicity in the stripe magnetic phase [24] i.e. the magnetic field
suppresses the superconducting order and thereby promotes the coexisting
magnetic order. On the other hand, the remarkable rapid disappearance of the
orthorhombic distortion and the vanishing entropy discontinuity for
$H\parallel$[001], which constitute one of the novel results of our study, are
quite puzzling and remain to be understood theoretically. A missing ingredient
in our model is the accounting for spin-orbit effects, which are clearly of
importance in these materials [41, 12, 28]. For example, it was found that the
$c$-axis corresponds to the direction of the magnetic moment in the similar
$C_{4}$-magnetic phase of
$\mathrm{Ba}_{1-x}\mathrm{Na}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$ [12]. Neutron
scattering experiments [41] have recently shown that spin fluctuations in
$\mathrm{Ca}\mathrm{K}(\mathrm{Fe}_{1-x}\mathrm{Ni_{x}})_{4}\mathrm{As}_{4}$
are strongly anisotropic, with the $c$ axis contributions dominating over in-
plane ones. This finding has in turn been attributed to a strong spin-orbit
effect in this material which could be shown to be responsible for the spin-
reorientation in the tetragonal double-Q state [28]. Anisotropic spin
fluctuations might be responsible for the reduced orthorhombicity in the low-
temperature $C_{2}$ magnetic phase for out-of-plane magnetic fields and would
have to be considered explicitly in any future model that wants to capture
this effect.

In conclusion, we have studied in detail the magnetic field dependence of the
transition from the tetragonal double-Q magnetic phase into the orthorhombic,
superconducting single-Q phase in
$\mathrm{Ba}_{1-x}\mathrm{K}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$ ($x\approx
0.25$). We report that a magnetic field suppresses the reentrance of the
single-Q orthorhombic phase stronger than superconductivity leading to a
splitting of the zero-field first-order transition. The transition into the
orthorhombic reentrant phase hereby remains first-order. The suppression rate
of this transition temperature is stronger for out-of-plane than for in-plane
fields showing the same anisotropy as the superconducting state. This can be
reproduced by a phenomenological Ginzburg-Landau model which suggests, that
the weakening of superconductivity with magnetic field primarily results in
the suppression of the reentrant orthorhombic single-Q phase. Furthermore, we
observe a strong reduction of the orthorhombic distortion for out-of-plane
fields, that can not be captured with our model and calls for further
theoretical investigations.

###### Acknowledgements.

We acknowledge stimulating discussions with R. Fernandes. K. W. acknowledges
support from the Swiss National Science Foundation through the
Postdoc.Mobility program.

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##  Appendix A Theoretical Model

The phenomenology in
$\mathrm{Ba}_{1-x}\mathrm{K}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$ is well
captured by the Ginzburg-Landau free energy density
$\mathcal{F}=\mathcal{F}_{m}+\mathcal{F}_{s}+\mathcal{F}_{\mathrm{int}}$, as
specified in Eqs. (IV)-(3). The magnetic texture realizes either a double-$Q$
spin-density wave order ($|m_{x}|=|m_{y}|$) that preserves the $C_{4}$
symmetry or a single-$Q$ stripe magnetic order ($m_{x}m_{y}=0$) lowering the
in-plane rotational symmetry to $C_{2}$. These two possibilities define
distinct energy manifolds, between which first order transitions can occur.
Once superconductivity appears, the leading interaction term in
$\mathcal{F}_{\mathrm{int}}$ only mildly affects the energetics between the
two magnetic sectors. The next order term $\propto c_{2}$ on the other hand
strongly differentiates between the $C_{4}$ order (where it vanishes) and the
$C_{2}$ phase.

In order to study this model system we limit ourselves to a simple
description, where the temperature-dependence is limited to the two mass terms
$\alpha$ and $\alpha_{s}$. By identifying $v=\nu x$, with $\nu\approx 32$, we
attribute the dominant doping-dependence $x$ to the term dictating the
relative energetics of the magnetic phases. Experimentally, the magnetic
field-dependence of
$\mathrm{Ba}_{1-x}\mathrm{K}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$ can to a large
extend be reduced to the superconducting sector. This motivates our choice of
coupling only the phenomenological parameter $\alpha_{s}$ to the magnetic
field $H$. The specific choice of parameters is provided in Table 1.

$\alpha(\tau)$ |  $u$ |  $g$ |  $v$ |  $\gamma$ | $\alpha_{s}$ |  $u_{s}$ | $\tau_{c}(H)$ |  $c_{1}$ |  $c_{2}$
---|---|---|---|---|---|---|---|---|---
$\tau-1$ |  4 |  0.5 |  $x$ |  4 | $\tau-\tau_{c}$ |  1 | $0.6-H$ |  1 |  0.2
Table 1: Parameters used in the phenomenological model.

Within our model, the calorimetric data can be deduced from
$C=T\partial^{2}E/\partial T^{2}$, where $E(T)$ is the system’s energy at a
given temperature $T$. Furthermore, the coupling of the stripe spin-density
state ($|m_{x}-m_{y}|\neq 0$) to the elastic degrees of freedom will result in
a length-change of the specimen. We anticipate that $\delta
L/L\propto|m_{x}-m_{y}|$ and use the latter as a proxy for the former.

Note that the orientation of the moments in the different magnetic phases is
susceptible to change due to crystal field effects. This aspect is not
accounted for within the proposed model. In particular is is not capable of
resolving the field-direction dependence at low temperature, see Fig. 1.