# Tuning superconductivity and spin-vortex fluctuations in CaKFe4As4 through
in-plane antisymmetric strains

Adrian Valadkhani1 Belén Zúñiga Céspedes2 Salony Mandloi2 Mingyu Xu3,4 Juan
Schmidt3,4 Sergey L. Bud’ko3,4 Paul C. Canfield3,4 Roser Valentí1 Elena Gati2
<EMAIL_ADDRESS>1 Institute for Theoretical Physics, Goethe University
Frankfurt, 60438 Frankfurt am Main, Germany  2 Max Planck Institute for
Chemical Physics of Solids, 01187 Dresden, Germany  3 Ames National
Laboratory, US Department of Energy, Iowa State University, Ames, Iowa 50011,
USA  4 Department of Physics and Astronomy, Iowa State University, Ames, Iowa
50011, USA

###### Abstract

Lattice strains of appropriate symmetry have served as an excellent tool to
explore the interaction of superconductivity in the iron-based superconductors
with nematic and stripe spin-density wave (SSDW) order, which are both closely
tied to an orthorhombic distortion. In this work, we contribute to a broader
understanding of the coupling of strain to superconductivity and competing
normal-state orders by studying CaKFe4As4 under large, in-plane strains of
$B_{1g}$ and $B_{2g}$ symmetry. In contrast to the majority of iron-based
superconductors, pure CaKFe4As4 exhibits superconductivity with relatively
high transition temperature of $T_{c}\,\sim\,$35 K in proximity of a non-
collinear, tetragonal, hedgehog spin-vortex crystal (SVC) order. Through
experiments, we demonstrate an anisotropic in-plane strain response of
$T_{c}$, which is reminiscent of the behavior of other pnictides with
nematicity. However, our calculations suggest that in CaKFe4As4, this
anisotropic response correlates with the one of the SVC fluctuations,
highlighting the close interrelation of magnetism and high-$T_{c}$
superconductivity. By suggesting moderate $B_{2g}$ strains as an effective
parameter to change the stability of SVC and SSDW, we outline a pathway to a
unified phase diagram of iron-based superconductivity.

###### pacs:

xxx

††preprint: APS/123-QED

The phase diagrams of high-temperature superconductors typically show various
competing ordering tendencies, the fluctuations of which might be considered
as the main pairing glue for superconductivity 1. It is often found that the
competing electronic orders are accompanied by the formation of a pronounced
in-plane anisotropy 2, 3, 4. This observation has initiated a huge surge in
using external stresses and strains of appropriate symmetry, that couple
directly to the anisotropic electronic state 5, 6, 7, 8, 9, 10, 11, as a tool
to explore the role of its fluctuations for superconductivity.

In this context, iron-based superconductors have served as prime examples to
explore and establish the intimate connection between anisotropic electronic
states and superconductivity. Many of the members of this family, such as
$Ae_{1-x}A_{x}$Fe2As2 ($Ae$ = Ba, Sr, Ca and $A$ = K, Na) or
Ba(Fe${}_{1-x}T_{x}$)2As2 ($T$ = Co, Rh, Ni, Pd) 12 with 1-2-2 stochiometry,
show superconductivity in proximity of stripe-type spin-density wave (SSDW)
magnetism13 (see Fig. 1 (a)). The SSDW order is characterized by ordering
vectors $\textbf{Q}_{\textrm{SSDW}}=(\pi,0)$ or $(0,\pi)$, which gives rise to
an inequivalence between the two in-plane directions of the high-temperature
tetragonal lattice. This type of magnetism therefore results, aside from
broken spin-rotational symmetry, in a spontaneous $B_{2g}$ lattice distortion,
which reduces the crystallographic symmetry from $C_{4}$ to $C_{2}$. The order
with broken lattice symmetry but preserved time-reversal symmetry is commonly
referred to as nematic order. It is therefore often found to be ’vestigial’ to
the SSDW order 14, 15. In other cases, like FeSe 16, the nematic phase is even
more prominent, as it is not accompanied by the formation of long-range SSDW
order at ambient pressure.

The understanding of normal-state of iron-based superconductors has been
tremendously advanced by utilizing lattice strains of different symmetry 8, 9,
10, 11. This was, for example, crucial in establishing the electronically-
driven nature of nematicity 7, 3, 17, 11. In order to provide compelling
evidence that superconductivity benefits from this unusual, anisotropic normal
state, a set of experiments recently studied the direct impact of applied
lattice strains on the superconducting critical temperature $T_{c}$ in a
series of tetragonal 122 compounds 18, 19. They found that, whereas $B_{2g}$
strains, which break the same symmetry as the nematic order, measurably
suppress $T_{c}$ both under compression and tension, the application of strain
in the $B_{1g}$ channel (i.e., a strain that is oriented 45∘ away from the
nematic axis) has resulted in a much weaker response 19. Based on a
phenomenological Landau model, the anisotropic strain-response of $T_{c}$ was
attributed to the coupling thereof to the nematic order parameter. This has
strengthened the notion that nematic fluctuations play a key role20, 11 in
boosting $T_{c}$.

However, this notion might be questioned 21, 22 by the discovery of
superconductivity with very high $T_{c}$ values in proximity to magnetic and
charge-ordered states that preserve the tetragonal $C_{4}$ symmetry. In this
context, a particular notable reference material is the quarternary compound
CaKFe4As4, which is a superconductor with high $T_{c}\,\approx\,$35 K 23, 24
in its stochiometric form, i.e., free from substitutional disorder. This
superconducting state occurs in proximity of a $C_{4}$ magnetic state 25, 26,
27, 28, 29, 30, 31, 32 (see Fig. 1 (a)). In this so-called spin-vortex type
(SVC) magnetic order, the moments rotate clock-wise/anti-clock-wise in an Fe
plaquette 33 ($\textbf{Q}_{\textrm{SVC}}=(\pi,\pi)$). Importantly, upon tuning
by doping and hydrostatic pressure 25, 28, 29, 31, so far no $C_{2}$ symmetric
order has been identified. This phenomenology has motivated proposals that
isotropic magnetic fluctuations, related to the SVC order, are sufficient to
generate high-temperature superconductivity 21, 22.

Figure 1: (a) Schematic temperature-doping phase diagrams of the two
superconductors CaKFe4As4 (’1144’, after Ref. 25) and Ba0.5K0.5Fe2As2 (’122’,
after Ref. 34). Upon adding electrons to the systems, superconductivity (SC,
light blue) is suppressed and magnetic phases emerge. For electron-doped 1144,
the magnetic phase is the so-called spin-vortex (SVC) phase that preserves the
tetragonal $C_{4}$-symmetry. In contrast, for the 122 compounds, the magnetic
phase is the stripe spin-density wave order (SSDW), which displays $C_{2}$
symmetry and is accompanied by a vestigial nematic phase. Only in proximity of
both SC and $C_{2}$-SSDW order, a small region of $C_{4}$ magnetic order can
be observed in the 122 compound; (b) Schematic representation of the symmetry
decomposition of applied strains with respect to the tetragonal unit cell.
$A_{1g}$ strains preserve the tetragonal symmetry, whereas $B_{1g}$ and
$B_{2g}$ do not (after Ref. 8). The induced $B_{1g}$ and $B_{2g}$ strains are
larger than the $A_{1g}$ strains due to the in-plane Poisson ratio, see SI.

Clearly, CaKFe4As4 presents a unique and possibly much richer platform to
explore the coupling of superconductivity and its normal-state properties to
in-plane strains, compared to the majority of iron-based superconductors with
prominent SSDW magnetism and nematicity (see Fig. 1 (a)). Yet, the evolution
of the superconductivity and magnetism in CaKFe4As4 have not been studied
under large, tunable in-plane strains, even though such studies promise key
insights for developing a unified understanding of the phase diagram of iron-
based superconductors.

In this work, we combine experiments and density-functional theory (DFT)
calculations to shed light on this issue. We first demonstrate that the
superconducting $T_{c}$ of CaKFe4As4 shows a strong anisotropic response under
antisymmetric in-plane strains of $B_{1g}$ and $B_{2g}$ symmetry, reminiscent
of the findings in 122 compounds. We then show through calculations, that in-
plane strains in CaKFe4As4 have primarily a different effect than in 122
compounds. In CaKFe4As4, moderate, experimentally achievable, in-plane strains
of $B_{2g}$ symmetry offer an excellent mean to change the preference from an
SVC to SSDW state. In contrast, $B_{1g}$ strains leave the preference for SVC
unchanged. Taken together, these results suggest that the development of in-
plane anisotropic strain response is not a unique fingerprint of coupling to
nematicity, but can also arise from the coupling of superconductivity to
fluctuations of the non-collinear SVC magnetic order. With these results, we
provide important insights to a broader understanding of strain tuning of the
multiple phases in iron-based superconductors.

For clarity, we will use the notion of irreducible strains throughout this
paper. For the tetragonal unit cell of CaKFe4As4, there are two antisymmetric
irreducible strains, denoted by $\epsilon_{B1g}$ and $\epsilon_{B2g}$, which
break $C_{4}$ symmetry. In this work, $B_{1g}$ and $B_{2g}$ refer to the
irreducible representations of the tetragonal point group associated with the
actual crystallographic unit cell rather than the one-Fe unit cell. As
schematically shown in Fig. 1 (b), $\epsilon_{B1g}$ and $\epsilon_{B2g}$
strains are primarily induced when strain is applied along the
crystallographic [1 0 0] and [1 1 0] directions, respectively, in addition to
a smaller fully symmetric strain $\epsilon_{A1g}$. The used decomposition
procedure is described in the SI 35. Tensile (compressive) strains are denoted
throughout our work by positive (negative) signs.

We first demonstrate how the superconducting critical temperature, $T_{c}$,
changes with these in-plane antisymmetric strains through experiments. To this
end, oriented CaKFe4As4 crystals36 (along [1 0 0] and [1 1 0]) were mounted on
rigid platforms37, 38. Varying strains were applied to the platform and the
sample with a piezoelectric-actuator-based uniaxial pressure cell 5. $T_{c}$
was determined through temperature-dependent measurements of the mutual
inductance of two concentric coils wound around the platform with the sample
(see SI Sec. A.1 for details).

Figure 2 shows the temperature dependence of the mutual inductance, $M$, at
different strains of type (a)-(b) $\epsilon_{B2g}$ and (c)-(d)
$\epsilon_{B1g}$. In each figure, the top (bottom) panel shows the data taken
under tensile (compressive) strains. The superconducting transition is clearly
identified in all data sets by a sharp drop of $M$ and we assign $T_{c}$ to
the temperature where $M$ has reached 50% of its full value (see grey dashed
line). The bare $M$ data reveals clearly our main experimental findings.
First, the response of $T_{c}$ to $\epsilon_{B2g}$ is larger than the one to
$\epsilon_{B1g}$ strain (Note the same scale of the temperature axes in all
plots). Second, both compressive and tensile $\epsilon_{B2g}$ strains suppress
$T_{c}$. The suppression is as large as $\Delta T_{c}\,\sim\,$-0.8 K by
$\epsilon_{B2g}\sim\,-\,0.4\%$.

Figure 2: Mutual inductance data on CaKFe4As4 as a function of temperature for
different applied, antisymmetric strains (a)-(b) $\epsilon_{B2g}$ and (c)-(d)
$\epsilon_{B1g}$. Top (bottom) panels show data under tensile (compressive)
strains. The spacing in strain between two data sets amounts to
$\sim\,0.05\%$. The grey dashed line visualizes the 50% threshold that is used
to infer $T_{c}$.

To further quantify the statements above, we compiled the phase diagram as
$\Delta T_{c}\,=\,T_{c}(\epsilon)-T_{c}(\epsilon\,=\,0)$ up to $\,\pm\,0.4\%$
antisymmetric strains (and $\,\pm\,0.2\%$ symmetric strains) in Fig. 3. The
color shaded areas around the $T_{c}(\epsilon)$ data indicate the width of the
superconducting transition at each $\epsilon$, determined from 25% and 75%
threshold values.

Figure 3: Change of superconducting critical temperature of CaKFe4As4, $\Delta
T_{c}\,=\,T_{c}(\epsilon)-T_{c}(\epsilon\,=\,0)$ with antisymmetric
$\epsilon_{B2g}$ (closed symbols) and $\epsilon_{B1g}$ (open symbols) strains
(bottom axis). In both experiments under dominating $B_{1g}$ and $B_{2g}$
strains, a finite symmetric $A_{1g}$ strain is induced as well and depicted on
the top axis. The color shading represents the width of the transition. Dashed
and dotted grey lines correspond to polynomial fits up to the second order in
strain.

By consideration of symmetry-allowed terms in a Ginzburg-Landau approach 18,
14, it is expected that, to lowest order, $\Delta T_{c}(\epsilon)\,\sim
D_{A1g}\epsilon_{A1g}+D_{i}\epsilon_{i}^{2}$ with $i=B_{1g}$ or $B_{2g}$. The
linear strain dependence can only result from the dependence of $T_{c}$ on
$\epsilon_{A1g}$, which are also induced in our experiments.

Figure 3 shows that the data of $T_{c}$ vs. $\epsilon_{B2g}$ is clearly
dominated by the quadratic strain dependence, expected for antisymmetric
strains, over almost the full strain range. A polynomial fit of order two (see
dashed line) yields the quadratic coefficient
$D_{B2g}/T_{c}(0)\,=\,-(2780\,\pm\,50)\,$. Only for high compression
($|\epsilon|\gtrsim\,0.3\%$), small deviations from the quadratic behavior are
observed, which, however, are still within the error bars of our experiment.

For the $B_{1g}$ data, a weak quadratic change of $T_{c}$ with
$\epsilon_{B1g}$ can also be identified, even though the linear contribution
to $T_{c}(\epsilon)$ due to $A_{1g}$ strains dominates. The quadratic
coefficient amounts to $D_{B1g}/T_{c}(0)\,=\,-(36\,\pm\,1)\,$, which is
unlikely to result from a small misalignment of the crystal (see SI) and is
therefore considered intrinsic to the $B_{1g}$ channel. Similar to the
$B_{2g}$ data, only small deviations from the polynomial fit within the error
bars of the experiment occur for $|\epsilon|\gtrsim\,0.3\%$.

Figure 4: (a,b) Calculated energies of CaKFe4As4 for imposed ”frozen” spin
configurations of spin-vortex crystal (SVC) order and stripe spin-density wave
(SSDW) order as a function of antisymmetric strains of (a) $B_{2g}$ and (b)
$B_{1g}$ symmetry. Whereas for $B_{1g}$ strains the SVC configuration remains
clearly energetically favorable, $B_{2g}$ strains change the preferred type of
spin fluctuations from SVC to SSDW around
$\epsilon_{B2g}\,\approx\,\pm\,0.8\,\%$, i.e., when
$E_{\textrm{SSDW}}(\epsilon)<E_{\textrm{SVC}}(\epsilon)$. The SVC (SSDW)
ordering motif is visualized by red (blue) arrows in the small cartoons; (c)
Energy of the SVC state as a function of antisymmetric strains on enlarged
scales in the experimentally studied strain range.

To explore possible correlations of $T_{c}(\epsilon)$ with the strain
dependence of the normal-state fluctuations, we discuss in the following our
results of DFT calculations under the same antisymmetric strain fields. In
CaKFe4As4 at ambient conditions, no static magnetic order 22, 39 can be found,
but strong SVC fluctuations 22, 40 exist. In previous computational studies,
it has been shown that it is crucial to take spin fluctuations into account
for accurate predictions of the ambient-pressure structure and structural
transitions at high pressures 28, 41, 42. In these works, the presence of spin
fluctuations was simulated by imposing a ”frozen” magnetic configuration
within a reduced Stoner theory, in which the size of magnetic moments is
adjusted for the values found in experiment in Ni-doped CaKFe4As4 26. Given
that this approach has proven successful in exploring the coupling of
magnetism to strain, we now calculate within DFT the energy of ”frozen” SVC
and SSDW orders in CaKFe4As4 under finite in-plane strains (see SI) and use it
as a proxy for the nature and strength of magnetic fluctuations.

Consistent with earlier DFT results 28, 41, the result at $\epsilon\,=\,0$ is
that a SVC configuration is energetically favored over the SSDW
($E_{\textrm{SVC}}(0)<E_{\textrm{SSDW}}(0)$), see Fig. 4. For finite strains,
the change in energy of a given magnetic configuration is, to lowest order,
given by $\Delta E\,\sim C_{A1g}\epsilon_{A1g}+C_{i}\epsilon_{i}^{2}$ with
$i=B_{1g},B_{2g}$. The calculated energies for both SVC and SSDW magnetic
configurations are well-described by such a linear plus quadratic strain
dependence for $\epsilon_{B2g}$ (Fig. 4 (a)) and $\epsilon_{B1g}$ (Fig. 4
(b)). The sign and strength of the quadratic term of the strain dependence,
however, strongly differ between the different orders and strains, as we
discuss below.

Specifically, $\epsilon_{B2g}$ strongly weakens the tendency towards the SVC
configuration and promotes the one towards SSDW order (Fig. 4 (a)), consistent
with a symmetry analysis within a Landau approach 43, since the SSDW couples
directly to $\epsilon_{B2g}$ (see also Fig. 1). In contrast, the energy of the
SVC configuration is only weakly increased by $\epsilon_{B1g}$. At the same
time, the SSDW configuration becomes significantly unfavorable under
increasing $\epsilon_{B1g}$ (Fig. 4 (b)).

On a quantitative level, following important conclusions can be drawn. First,
a $B_{2g}$ strain of $\epsilon_{B2g}^{crit}\approx\,\pm\,0.8\%$ changes the
preferred type of spin fluctuations from SVC and SSDW. We note that
$\epsilon_{B2g}^{crit}$ is larger than typical strains that are induced by a
spontaneous nematic/SSDW distortion in the 122 pnictides
($\epsilon_{B2g}\,\lesssim\,0.3\%$) 34. This reflects the fact that the
CaKFe4As4 is not as close to a SSDW-nematic instability at ambient conditions
as the related 122 compounds 30. Second, for $B_{1g}$ strains, there is hardly
any change in the magnetism and, the SVC configuration remains clearly
favorable. Overall, this results in a clear anisotropy of the antisymmetric
strain dependence of the magnetic energies of the SVC order of
$(C_{B2g}/C_{B1g})_{\textrm{SVC}}\sim\,52$ (Fig. 4 (c)).

The qualitatively similar anisotropic strain response of the SVC configuration
in Fig. 4 (c) and the experimentally-determined superconducting $T_{c}$ in
Fig. 3 under both types of antisymmetric strains, $\epsilon_{B1g}$ and
$\epsilon_{B2g}$, is striking and is the main result of the present work. Even
the magnitude of the measured anisotropy parameters of $T_{c}$ of
$D_{B2g}/D_{B1g}\,\sim\,$77 and of $E$ of $C_{B2g}/C_{B1g}\sim\,52$ is
similar. Even though the calculated energies can only serve as a rough proxy
for the spin fluctuations, it is natural to assume that in a scenario of
magnetically-driven superconductivity 44, 45, 46, there exists a correlation
of $E$ and $T_{c}$ (and correspondingly of $C_{i}$ and $D_{i}$). As a
consequence, in the case of CaKFe4As4 with SVC configuration, it is the fact
that $\epsilon_{B2g}$ strains primarily weaken SVC fluctuations 43, while
$\epsilon_{B1g}$ do not, that is most likely at the origin of the observed in-
plane strain anisotropy of $T_{c}$. Whereas the anisotropic strain response of
$T_{c}$ is a widely observed feature of iron-based superconductivity 18, 19,
47, it is driven in CaKFe4As4 by the coupling of superconductivity to
magnetism, rather than by nematicity, as suggested for the 122 compounds. This
conclusion strengthens the view that it is the magnetism that dominates the
properties of high-$T_{c}$ iron-based superconductivity. The observation that
bulk FeSe does not show a clear quadratic contribution 38, 48 to
$T_{c}(\epsilon_{B2g})$ further points to magnetic degrees of freedom 47 being
an important ingredient for the development of a strain-anisotropy of $T_{c}$.

At the same time, our results clearly demonstrate a route towards a unified
phase diagram of iron-based superconductivity by using antisymmetric strains
in the CaKFe4As4 family, since these strains might be used to manipulate the
relative importance of SVC and SSDW magnetism 30 for superconductivity. The
theoretical prediction of the strain tunability of magnetism (see Fig. 4)
within an experimentally-achievable strain range motivates a series of studies
in the future. For example, it would be very interesting to study
superconducting properties at larger $B_{2g}$ strains, in particular at those
strains, where SSDW fluctuations become dominant. A simple extrapolation of
the present $T_{c}(\epsilon_{B2g})$ data, using the quadratic fit function, to
$\epsilon_{B2g}^{crit}\,\pm\,\sim\,0.8\%$ would predict a quite sizable
$T_{c}$ of $\sim\,$27 K. If such a relatively high $T_{c}$ can be confirmed
and combined with microscopic insights 49, 50, 51, 52, 53, 54, 55, 56, 57, the
idea that both SVC and SSDW fluctuations promote similar superconducting
states 33 with high $T_{c}$ might be strengthened.

In summary, we have established how antisymmetric strains can be used to tune
the superconductivity and the nature of magnetic fluctuations in the
stochiometric high-$T_{c}$ superconductor CaKFe4As4. Specifically, we
demonstrated that both superconductivity and the preferred magnetic
configuration develop a highly anisotropic strain response to in-plane
antisymmetric strains of $B_{2g}$ and $B_{1g}$ type. The correlation of these
two quantities strongly suggests that the anisotropic response is driven by
the coupling of the non-collinear SVC magnetic configuration to antisymmetric
strain, rather than by nematicity. Thus, our work contributes to a broader
understanding of how antisymmetric strains impact superconductivity and its
competing states. Based on the prediction that moderate antisymmetric strains
can be used to manipulate the relative stability of non-collinear and
collinear orders, antisymmetric strain tuning is expected to be a powerful
tuning parameter for a wide range of magnetic quantum materials.

Acknowledgements - We thank William Meier and Andreas Kreyssig for insightful
discussions on the 1144 compounds, as well as Jack Barraclough for useful
discussions on the limitations of the strain platforms. Financial support by
the Max Planck Society is gratefully acknowledged. In addition, we gratefully
acknowledge funding through the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) through TRR 288—422213477 (project B05) and the SFB 1143
(project-id 247310070; project C09). Research in Dresden benefits from the
environment provided by the DFG Cluster of Excellence ct.qmat (EXC 2147,
project ID 390858940). Work at the Ames National Laboratory was supported by
the U.S. Department of Energy, Office of Science, Basic Energy Sciences,
Materials Sciences and Engineering Division. The Ames National Laboratory is
operated for the U.S. Department of Energy by Iowa State University under
Contract No. DEAC02-07CH11358.

## References

  * Fradkin et al. [2015] E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Rev. Mod. Phys. 87, 457 (2015).
  * Fradkin et al. [2010] E. Fradkin, S. A. Kivelson, M. J. Lawler, J. P. Eisenstein, and A. P. Mackenzie, Annual Review of Condensed Matter Physics 1, 153 (2010).
  * Fernandes et al. [2014] R. M. Fernandes, A. V. Chubukov, and J. Schmalian, Nat. Phys. 1, 97 (2014).
  * Hinkov et al. [2008] V. Hinkov, D. Haug, B. Fauque, P. Bourges, Y. Sidis, A. Ivanov, C. Bernhard, C. T. Lin, and B. Keimer, Science 319, 5863 (2008).
  * Hicks et al. [2014] C. W. Hicks, M. E. Barber, S. D. Edkins, D. O. Brodsky, and A. P. Mackenzie, Review of Scientific Instruments 85, 065003 (2014).
  * Kim et al. [2021] H.-H. Kim, E. Lefrançois, K. Kummer, R. Fumagalli, N. B. Brookes, D. Betto, S. Nakata, M. Tortora, J. Porras, T. Loew, et al., Phys. Rev. Lett. 126, 037002 (2021).
  * Chu et al. [2012] J.-H. Chu, H.-H. Kuo, J. G. Analytis, and I. R. Fisher, Science 337, 6095 (2012).
  * Ikeda et al. [2018] M. S. Ikeda, T. Worasaran, J. C. Palmstrom, J. A. W. Straquadine, P. Walmsley, and I. R. Fisher, Phys. Rev. B 98, 245133 (2018).
  * Willa et al. [2019] R. Willa, M. Fritz, and J. Schmalian, Phys. Rev. B 100, 085106 (2019).
  * Gati et al. [2020] E. Gati, L. Xiang, S. L. Bud’ko, and P. C. Canfield, Annalen der Physik 532, 2000248 (2020).
  * Böhmer et al. [2022] A. E. Böhmer, J.-H. Chu, S. Lederer, and M. Yi, Nature Physics 18, 1412 (2022).
  * Canfield and Bud’ko [2010] P. C. Canfield and S. L. Bud’ko, Annual Review of Condensed Matter Physics 1, 27 (2010).
  * Dai [2015] P. Dai, Rev. Mod. Phys. 87, 855 (2015).
  * Fernandes and Millis [2013] R. M. Fernandes and A. J. Millis, Phys. Rev. Lett. 111, 127001 (2013).
  * Fernandes et al. [2019] R. M. Fernandes, P. P. Orth, and J. Schmalian, Annual Review of Condensed Matter Physics 10, 133 (2019).
  * Böhmer and Kreisel [2017] A. E. Böhmer and A. Kreisel, Journal of Physics: Condensed Matter 30, 023001 (2017).
  * Worasaran et al. [2021] T. Worasaran, M. S. Ikeda, J. C. Palmstrom, J. A. W. Straquadine, S. A. Kivelson, and I. R. Fisher, Science 372, 96545 (2021).
  * Malinowski et al. [2020] P. Malinowski, Q. Jiang, J. J. Sanchez, J. Mutch, Z. Liu, P. Went, J. Liu, P. J. Ryan, J.-W. Kim, and J.-H. Chu, Nat. Phys. 16, 1189–1193 (2020).
  * Liu et al. [2019] Z. Liu, Y. Gu, W. Hong, T. Xie, D. Gong, X. Ma, J. Liu, C. Hu, L. Zhao, X. Zhou, et al., Phys. Rev. Res. 1, 033154 (2019).
  * Lederer et al. [2015] S. Lederer, Y. Schattner, E. Berg, and S. A. Kivelson, Phys. Rev. Lett. 114, 097001 (2015).
  * Zhang et al. [2018] W.-L. Zhang, W. R. Meier, T. Kong, P. C. Canfield, and G. Blumberg, Phys. Rev. B 98, 140501 (2018).
  * Ding et al. [2018] Q.-P. Ding, W. R. Meier, J. Cui, M. Xu, A. E. Böhmer, S. L. Bud’ko, P. C. Canfield, and Y. Furukawa, Phys. Rev. Lett. 121, 137204 (2018).
  * Iyo et al. [2016] A. Iyo, K. Kawashima, T. Kinjo, T. Nishio, S. Ishida, H. Fujihisa, Y. Gotoh, K. Kihou, H. Eisaki, and Y. Yoshida, Journal of the American Chemical Society 138, 3410 (2016).
  * Meier et al. [2016] W. R. Meier, T. Kong, U. S. Kaluarachchi, V. Taufour, N. H. Jo, G. Drachuck, A. E. Böhmer, S. M. Saunders, A. Sapkota, A. Kreyssig, et al., Phys. Rev. B 94, 064501 (2016).
  * Meier et al. [2018] W. R. Meier, Q.-P. Ding, A. Kreyssig, S. L. Bud’ko, A. Sapkota, K. Kothapalli, V. Borisov, C. D. B. Roser Valentí, P. P. Orth, R. M. Fernandes, et al., npj Quantum Materials 3, 5 (2018).
  * Kreyssig et al. [2018] A. Kreyssig, J. M. Wilde, A. E. Böhmer, W. Tian, W. R. Meier, B. Li, B. G. Ueland, M. Xu, S. L. Bud’ko, P. C. Canfield, et al., Phys. Rev. B 97, 224521 (2018).
  * Bud’ko et al. [2018] S. L. Bud’ko, V. G. Kogan, R. Prozorov, W. R. Meier, M. Xu, and P. C. Canfield, Phys. Rev. B 98, 144520 (2018).
  * Kaluarachchi et al. [2017] U. S. Kaluarachchi, V. Taufour, A. Sapkota, V. Borisov, T. Kong, W. R. Meier, K. Kothapalli, B. G. Ueland, A. Kreyssig, R. Valentí, et al., Phys. Rev. B 96, 140501 (2017).
  * Xiang et al. [2018] L. Xiang, W. R. Meier, M. Xu, U. S. Kaluarachchi, S. L. Bud’ko, and P. C. Canfield, Phys. Rev. B 97, 174517 (2018).
  * Böhmer et al. [2020] A. E. Böhmer, F. Chen, W. R. Meier, M. Xu, G. Drachuck, M. Merz, P. W. Wiecki, S. L. Bud’ko, V. Borisov, R. Valentí, et al., arXiv:2011.13207 (2020).
  * Xu et al. [2022] M. Xu, J. Schmidt, E. Gati, L. Xiang, W. R. Meier, V. G. Kogan, S. L. Bud’ko, and P. C. Canfield, Phys. Rev. B 105, 214526 (2022).
  * Xu et al. [2023] M. Xu, J. Schmidt, M. A. Tanatar, R. Prozorov, S. L. Bud’ko, and P. C. Canfield, Phys. Rev. B 107, 134511 (2023).
  * Fernandes et al. [2016] R. M. Fernandes, S. A. Kivelson, and E. Berg, Phys. Rev. B 93, 014511 (2016).
  * Böhmer et al. [2015] A. E. Böhmer, F. Hardy, L. Wang, T. Wolf, P. Schweiss, and C. Meingast, Nat. Commun. 6, 7911 (2015).
  * [35] _Supplemental information containing references36, 5, 37, 38, 42, 58, 59, 60, 61, 62, 63, 64, 65, 66, 41, 28, 25, 30, 25, 26, 8, 67_.
  * Meier et al. [2017] W. R. Meier, T. Kong, S. L. Bud’ko, and P. C. Canfield, Phys. Rev. Materials 1, 013401 (2017).
  * Park et al. [2020] J. Park, J. M. Bartlett, H. M. L. Noad, A. L. Stern, M. E. Barber, M. König, S. Hosoi, T. Shibauchi, A. P. Mackenzie, A. Steppke, et al., Review of Scientific Instruments 91, 083902 (2020).
  * Bartlett et al. [2021] J. M. Bartlett, A. Steppke, S. Hosoi, H. Noad, J. Park, C. Timm, T. Shibauchi, A. P. Mackenzie, and C. W. Hicks, Phys. Rev. X 11, 021038 (2021).
  * Bud’ko et al. [2017] S. L. Bud’ko, T. Kong, W. R. Meier, X. Ma, and P. C. Canfield, Philosophical Magazine 97, 2689 (2017).
  * Xie et al. [2018] T. Xie, Y. Wei, D. Gong, T. Fennell, U. Stuhr, R. Kajimoto, K. Ikeuchi, S. Li, J. Hu, and H. Luo, Phys. Rev. Lett. 120, 267003 (2018).
  * Borisov et al. [2018] V. Borisov, P. C. Canfield, and R. Valentí, Phys. Rev. B 98, 064104 (2018).
  * Song et al. [2019] G. Song, V. Borisov, W. R. Meier, M. Xu, K. J. Dusoe, J. T. Sypek, R. Valentí, P. C. Canfield, and S.-W. Lee, APL Materials 7 (2019).
  * Meier [2018] W. Meier, Ph.D. thesis, Iowa State University (2018).
  * Chubukov [2012] A. Chubukov, Annual Review of Condensed Matter Physics 3, 57 (2012).
  * Hirschfeld [2016] P. J. Hirschfeld, Comptes Rendus Physique 17, 197 (2016).
  * Liu et al. [2022] C. Liu, P. Bourges, Y. Sidis, T. Xie, G. He, F. Bourdarot, S. Danilkin, H. Ghosh, S. Ghosh, X. Ma, et al., Phys. Rev. Lett. 128, 137003 (2022).
  * Zhao et al. [2023] Z. Zhao, D. Hu, X. Fu, K. Zhou, Y. Gu, G. Tan, X. Lu, and P. Dai, arXiv:2305.04424 (2023).
  * Ghini et al. [2021] M. Ghini, M. Bristow, J. C. A. Prentice, S. Sutherland, S. Sanna, A. A. Haghighirad, and A. I. Coldea, Phys. Rev. B 103, 205139 (2021).
  * Khasanov et al. [2018] R. Khasanov, W. R. Meier, Y. Wu, D. Mou, S. L. Bud’ko, I. Eremin, H. Luetkens, A. Kaminski, P. C. Canfield, and A. Amato, Phys. Rev. B 97, 140503 (2018).
  * Jost et al. [2018] D. Jost, J.-R. Scholz, U. Zweck, W. R. Meier, A. E. Böhmer, P. C. Canfield, N. Lazarević, and R. Hackl, Phys. Rev. B 98, 020504 (2018).
  * Lochner et al. [2017] F. Lochner, F. Ahn, T. Hickel, and I. Eremin, Phys. Rev. B 96, 094521 (2017).
  * Bristow et al. [2020] M. Bristow, W. Knafo, P. Reiss, W. Meier, P. C. Canfield, S. J. Blundell, and A. I. Coldea, Phys. Rev. B 101, 134502 (2020).
  * Mou et al. [2016] D. Mou, T. Kong, W. R. Meier, F. Lochner, L.-L. Wang, Q. Lin, Y. Wu, S. L. Bud’ko, I. Eremin, D. D. Johnson, et al., Phys. Rev. Lett. 117, 277001 (2016).
  * Cho et al. [2017] K. Cho, A. Fente, S. Teknowijoyo, M. A. Tanatar, K. R. Joshi, N. M. Nusran, T. Kong, W. R. Meier, U. Kaluarachchi, I. Guillamón, et al., Phys. Rev. B 95, 100502 (2017).
  * Teknowijoyo et al. [2018] S. Teknowijoyo, K. Cho, M. Kończykowski, E. I. Timmons, M. A. Tanatar, W. R. Meier, M. Xu, S. L. Bud’ko, P. C. Canfield, and R. Prozorov, Phys. Rev. B 97, 140508 (2018).
  * Iida et al. [2017] K. Iida, M. Ishikado, Y. Nagai, H. Yoshida, A. D. Christianson, N. Murai, K. Kawashima, Y. Yoshida, H. Eisaki, and A. Iyo, Journal of the Physical Society of Japan 86, 093703 (2017).
  * Biswas et al. [2017] P. K. Biswas, A. Iyo, Y. Yoshida, H. Eisaki, K. Kawashima, and A. D. Hillier, Phys. Rev. B 95, 140505 (2017).
  * Barber et al. [2019] M. E. Barber, A. Steppke, A. P. Mackenzie, and C. W. Hicks, Review of Scientific Instruments 90, 023904 (2019).
  * Hohenberg and Kohn [1964] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
  * Kohn and Sham [1965] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
  * Blöchl [1994] P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
  * Kresse and Joubert [1999] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
  * Kresse and Hafner [1993] G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).
  * Kresse and Furthmüller [1996] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
  * Kresse and Furthmüller [1996] G. Kresse and J. Furthmüller, Computational Materials Science 6, 15 (1996).
  * Perdew et al. [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * Bradley and Cracknell [2010] C. Bradley and A. Cracknell, _The Mathematical Theory of Symmetry in Solids: Representation Theory for Point Groups and Space Groups_ , EBSCO ebook academic collection (OUP Oxford, 2010).

##  Appendix A Supplemental Information

###  A.1 Extended Methods

Experiment - Samples of CaKFe4As4 were grown following the procedure described
in Ref. 36. For measurements under varying strains, we used a piezoelectric-
actuator based uniaxial pressure cell 5. Since CaKFe4As4 is very malleable and
cleaves easily, we relied on the method of attaching thin samples to rigid
platforms that are stressed to apply large, tunable strains to mechanically
delicate samples, initially described in detail in Ref. 37, 38. This method
ensures as homogeneous strain as possible by avoiding buckling and minimizes
the risk for cleavage of the sample.

Figure 5: Experimental setup for the determination of the superconducting
transition temperature of CaKFe4As4 under tunable in-plane strains; (a) Bar-
shaped sample cut out of a larger piece of single crystal, using Plasma Focus
Ion Beam milling; (b) Sample mounted on rigid platform, and contacted for
four-probe resistance measurements; (c) Two coils wound around the section of
the platform, on which the sample is mounted, for measurements of the mutual
inductance.

To ensure good strain transmission into the bulk of the sample (see Ref. 37
for finite-element simulations), the sample dimensions were typically chosen
to be $\sim\,800\,\times\,180\,\times\,10\,\mu$m3, with longest axis either
along the [1 0 0] or the [1 1 0] direction. The in-plane dimensions and
orientation were precisely controlled using Plasma Focused Ion Beam (PFIB)
cutting (see Fig. 5 (a)). The small out-of-plane thickness was achieved by
multiple cleaving steps of the samples prior to PFIB cutting. After cleaving
and cutting, the samples were glued on the platforms using Stycast 1266 (see
Fig. 5 (b)). Using this procedure, the maximum misalignment of the sample in
the plane amounts to $\sim$ 3∘.

The maximum applied strain that was achieved in our experiments was
$\sim\,\pm\,0.6\,\%$, corresponding to $\epsilon_{B2g}\sim\,\pm\,0.4\,\%$ (see
below). This value is limited by plastic deformation of the platform material
(grade 5 Ti-V-Al in our work), but not by the elastic limit42 of CaKFe4As4,
which showed fully elastic behavior across the entire strain range studied and
reproducible $T_{c}$ behavior upon various strain sweeps. We note that the
elastic limit of the platform material here exceeds the report of Refs. 37, 38
by a factor of 2. This was achieved by work hardening the platform inside a
force-displacement cell, similar to the one described in Ref. 58, at room
temperature prior to mounting the sample. Monitoring both force and
displacement through capacitive sensors installed in the cell (see Ref. 58 for
a description of the working principle of the sensors) allows for the
detection of plastic deformation of the platform. For all measurements
reported here, the platform showed linear elastic behavior to a good
approximation, i.e., showed a linear force-displacement relation. Only very
close to the maximal strains applied, some small deviations from perfect
linearity were observed.

For the reference value $\epsilon_{0}\,=\,0$, we chose the strain at which
$T_{c}$ is maximal for [1 1 0]T strain and used the same capacitance reading
of the displacement sensor as the strain-neutral point for the data with
strain along [1 0 0]T.

The superconducting transition temperature, $T_{c}$, was determined through
temperature-dependent measurements of the mutual inductance of two concentric
coils wound around the platform with the sample (see Fig. 5 (c)). We used
about 20 turns per coil, which was sufficient to measure the superconducting
signal with high-enough signal-to-noise ratio.

In total, 10 samples were measured (8 with strain applied along the
crystallographic [1 1 0] orientation and 2 with strain applied along the [1 0
0] direction) using two different platform materials (grade 5-Titanium and
quartz). Results were found to be consistent among different samples and
platforms, so that we focus here on results from samples mounted on the Ti
platforms only.

Theory - We perform electronic structure calculations within density
functional theory (DFT)59, 60 by using the pseudo-potential augmented plane-
wave (PAW) 61, 62 Vienna Ab initio Simulation Package (VASP) 63, 64, 65. All
calculations were performed with the Perdew–Burke–Ernzerhof (PBE) generalized
gradient approximation (GGA) 66 and include spin orbit coupling. The energy
cut-off was set to $600$ eV. For the relaxations we used a $6$ $\times$ $6$
$\times$ $4$ gamma centered $k$-mesh with a force threshold of $0.001$ eV/Å
and unrestricted DFT magnetic moments. We increased the number of $k$-points
to $10$ $\times$ $10$ $\times$ $4$ and set a convergence criterion of
$10^{-6}$ eV for the self-consistent-field (SCF) total energies. For all of
the calculations we follow the protocol established in previous works 41, 28,
25, 30, 42. In order to get information about the most stable underlying
magnetic order which is responsible for the nature of the spin fluctuations at
a given strain value, we compared the total energies of the two competing
orders in this structure – antiferromagnetic stripe and vortex ”hedgehog” 41,
25 – to the zero-strain energy difference of the vortex and stripe state. The
strains were applied in our calculations on a
$\sqrt{2}\,\times\,\sqrt{2}\,\times\,1$ unit cell for a strain range from
$-0.5\,\%$ to $0.5\,\%$ in steps of $0.25\,\%$ in each direction for each
order with its unrestricted DFT magnetic moments. The DFT ground state for the
vortex order converges to a magnetic moment $\mu_{Fe,v}=1.35\mu_{B}$ per iron
site, while the stripe order converges to a value of
$\mu_{Fe,s}\,=\,1.47\,\mu_{B}$ per iron site. With these DFT magnetic moments
$\mu_{Fe,v/s}$ the stripe order is lower in energy compared to the vortex
order. Experimental observations, however, suggest the vortex fluctuations to
be the most dominant with a much lower magnetic moment
$\mu_{Fe}\,\approx\,0.4\,\mu_{B}$ per iron site 26. By carefully decreasing
the size of our DFT magnetic moments in the SCF calcuations but keeping their
direction and the structure of the unrestricted DFT calculation, each order
has been restricted to ten different magnetic moments ranging from
$0.2\,\mu_{B}$ to $1.5\,\mu_{B}$. At about $\mu_{Fe}\,\approx 0.8\,\mu_{B}$
and less the vortex order is lower in energy than the stripe order. To obtain
the energy $E$ of an order for a given magnetic moment $\mu_{i}$ we
interpolated the 10 points with cubic splines and took the respective energy
$E_{i}\,=\,E(\mu_{i})$. For the vortex order, the two strain directions are
$[100]$ and $[110]$. For the stripe order, the $[100]$ direction is treated in
the same way, however, for the $[110]$ case it is important to distinguish
between applying strain along or perpendicular to the stripes. This was tested
explicitely by initializing the stripe order once along $[110]$ and once along
$[1\bar{1}0]$. Of the two stripe orientation datasets, the datapoints lowest
in energy were used to obtain the final fit. The fits were determined using a
second order polynomial. For all of our fits we took the experimental
reference value of $\mu_{i}=0.4\mu_{B}$ 26 for Ni-doped CaKFe4As4 at zero
strain to obtain the total energy in dependence of strain. Every order has
been compared to the zero strain vortex case. Due to the restriction of the
magnetic moments to $\mu_{i}$ on a structure with very different DFT magnetic
moment $\mu_{Fe,v/s}$ (magneto-elastic) stresses were induced to the unit
cell. Within the range of our strains, the stress tensor remains unchanged
except for the component we want to change, therefore, we were able to correct
the unwanted stresses by adding a linear correction term to the fitted total
energies.

###  A.2 Decompositions of strains into irreducible representations

Here, we outline how the applied strains, $\epsilon_{[1\,0\,0]}$ and
$\epsilon_{[1\,1\,0]}$, can be decomposed into irreducible representations 8,
see Fig. 1 (b) for a schematic representation. The decomposition follows from
the character table of point group $4/mmm$ 67.

The irreducible strains are given by following decomposition rules:
$\epsilon_{A1g}\,=\,\frac{1}{2}(\epsilon_{[1\,0\,0]}+\epsilon_{[0\,1\,0]})=\,\frac{1}{2}(\epsilon_{[1\,1\,0]}+\epsilon_{[1\,\bar{1}\,0]})$,
$\epsilon_{B1g}\,=\,\frac{1}{2}(\epsilon_{[1\,0\,0]}-\epsilon_{[0\,1\,0]})$
and
$\epsilon_{B2g}\,=\,\frac{1}{2}(\epsilon_{[1\,1\,0]}-\epsilon_{[1\,\bar{1}\,0]})$.
The strains along the different crystallographic directions are related by the
Poisson ratio $\nu$.

In our experimental configuration, where thin samples are attached to a rigid
platform, $\nu$ is given by the Poisson’s ratio of the platform. Thus, for a
titanium-alloy platform,
$\nu\,=\,-\epsilon_{[0\,1\,0]}/\epsilon_{[1\,0\,0]}=-\epsilon_{[1\,\bar{1}\,0]}/\epsilon_{[1\,1\,0]}\,\sim\,0.32$
38. Consequently,
$\epsilon_{A1g}=\frac{1}{2}(1-\nu)\epsilon_{[1\,0\,0]}=\,0.34\epsilon_{[1\,0\,0]}=\,0.34\epsilon_{[1\,1\,0]}$,
$\epsilon_{B1g}=\,0.66\epsilon_{[1\,0\,0]}$ and
$\epsilon_{B2g}=\,0.66\epsilon_{[1\,1\,0]}$, which has been used to decompose
the data shown in the main text.

For the theoretical calculations, we use the DFT-calculated strain tensor and
the same decomposition rules to determine $\epsilon_{B1g}$ and
$\epsilon_{B2g}$.

###  A.3 Influence of possible small misalignments of the crystallographic
axes on analysis

In the following, we discuss how a possible, small misalignment of the crystal
axes with respect to the strain axis influences our measurement results of
$T_{c}(\epsilon)$. In the present study, this analysis is particularly
relevant for measurements under $\epsilon_{B1g}$, where both
$T_{c}(\epsilon_{B1g})$ and $T_{c}(\epsilon_{B2g})$ exhibit a quadratic
suppression of $T_{c}$ with $D_{B1g}\,\ll\,D_{B2g}$ and thus, the observed
$T_{c}(\epsilon_{B1g})$ might be significantly influenced by a small
misalignment.

As a first confirmation of the good sample aligment, we show in Fig. 6 the
elastoresistance of the normal-state, measured at $T\,=\,37\,$K, i.e., above
$T_{c}$. In general in many iron-based superconductors, the normal-state
elastoresistance is signficant and shows a strongly anisotropic behavior,
depending on whether strain is applied along the [1 0 0] or the [1 1 0]
direction. This is also the case for CaKFe4As4, as shown in previous works 25,
30, where it was shown that the longitudinal elastoresistance $m=1/R\Delta
R/\Delta\epsilon$ shows a different sign for $\epsilon_{100}$ vs.
$\epsilon_{110}$ and $m_{110}\sim-3m_{100}$. Our data of the elastoresistance,
shown in Fig. 6, is fully in line with previous results, suggesting only a
very minor, possible misalignment error.

Figure 6: Longitudinal elastoresistance of CaKFe4As4, $\Delta R/R$, measured
at $T\,=\,37\,$K $>\,T_{c}$ for strain applied along the tetragonal [110]
direction (black squares) and the [100] direction (red circles).

For an explicit error analysis, we evaluated the possibile scenario that
$D_{B1g}=0$ and the observed quadratic component in $T_{c}(\epsilon_{B1g})$
therefore solely results from a misalignment of the crystal, so that the
contribution of $B_{2g}$ strains to the experimentally applied strain is
significant. The result of this calculation showed that this scenario is only
possible, if the misalignment of the crystal is around $6-7^{\circ}$, which is
quite a bit larger than realistic error of maximally $3^{\circ}$. We therefore
conclude that a weak quadratic suppression of $T_{c}$ with $\epsilon_{B1g}$ is
intrinsic to CaKFe4As4.