The properties of an electron in a typical solid are modified by the interaction with the crystal ions, leading to the formation of a quasiparticle: the polaron. Such polarons are often described using the Fröhlich Hamiltonian, which assumes the underlying lattice phonons to be harmonic. However, this approximation is invalid in several interesting materials, including the recently discovered high-pressure hydrides which superconduct at temperatures above 200 K. In this paper, we show that Fröhlich theory can be extended to eliminate this problem. We derive four additional terms in the Fröhlich Hamiltonian to account for anharmonicity up to third order. We calculate the energy and effective mass of the new polaron, using both perturbation theory and Feynman's path-integral formalism. It is shown that the anharmonic terms lead to significant additional trapping of the electron. The derived Hamiltonian is well suited for analytical calculations, due to its simplicity and since the number of model parameters is low. Since it is a direct extension of the Fröhlich Hamiltonian, it can readily be used to investigate the effect of anharmonicity on other polaron properties, such as the optical conductivity and the formation of bipolarons.

The perturbative effects of interactions in two two-dimensional systems with a single quadratic band touching point have been studied at a transition point. The effects of interactions have been studied using the renormalization group (RG) at one-loop level and it is found that at the transition a repulsive interaction is marginally relevant and an attractive interaction is marginally irrelevant. We corroborate the results obtained from the RG calculation by studying a microscopic model the ground state and Green's functions of which can be obtained exactly. We find that away from the transition the system displays an instability towards forming an excitonic condensate.

We present a quantitative analysis of two-particle interaction effects in generalized, one-dimensional Aubry-André-Harper models with the Fermi energy placed in one of the band gaps. We investigate systems with periodic as well as open boundary conditions, for the latter focusing on the number of edge states and the boundary charge. Both these observables are important for the classification of noninteracting topological systems. In our first class of models the unit cell structure stems from periodically modulated single-particle parameters. In the second it results from the spatial modulation of the two-particle interaction. For both types of models we find that the single-particle band gaps are renormalized by the interaction in accordance with expectations employing general field theoretical arguments. While interaction-induced effective edge states can be found in the local single-particle spectral function close to a boundary, the characteristics of the boundary charge are not modified by the interaction. This indicates that our results for the Rice-Mele and Su-Schriefer-Heeger model [Phys. Rev. B 102, 085122 (2020)] are generic and can be found in lattice models with more complex unit cells as well.

We report ${\left[{\left({\mathrm{LaNiO}}_3\right)}_{\mathrm{n}}/{\left({\mathrm{LaMnO}}_3\right)}_{\mathrm{n}}\right]}_{\mathrm{M}}/{\mathrm{SrTiO}}_3\left(111\right)$ superlattices [$n=20$, 10, 5, 2, and 1 unit cells (u.c.)] as “precursor” heterostructures for a double perovskite ${\mathrm{La}}_2{\mathrm{NiMnO}}_6$. A charge transfer at the ${\mathrm{LaNiO}}_3$/${\mathrm{LaMnO}}_3$ interfaces and formation of the emergent ${\mathrm{La}}_2{\mathrm{NiMnO}}_6$ phase with ${\mathrm{Ni}}^{2+}$ and ${\mathrm{Mn}}^{4+}$ oxidation states was evidenced by in situ optical ellipsometry and supported by magnetic and Raman measurements. The 10/10 and 5/5 superlattices revealed a complex magnetic behavior with two ferromagnetic phase transitions with $T_{\mathrm{C}\text{−}\mathrm{low}}=170\mathrm{K}$ and $T_{\mathrm{C}\text{−}\mathrm{high}}\sim 250–280\mathrm{K}$, originating from the ferromagnetic ${\mathrm{LaMnO}}_3$ and the emergent interfacial ${\mathrm{La}}_2{\mathrm{NiMnO}}_6$ phase, respectively. With decreasing thickness of individual layers, the magnetism of 2/2 and 1/1 superlattices transforms into that of the ordered double perovskite ${\mathrm{La}}_2{\mathrm{NiMnO}}_6$. Furthermore, a structural reconstruction into a monoclinic $\left(P_{12}\right)$ structure of the ordered ${\mathrm{La}}_2{\mathrm{NiMnO}}_6$ at the interface, evidenced by Raman spectroscopy, goes ahead with the charge transfer and magnetic ${\mathrm{La}}_2{\mathrm{NiMnO}}_6$ behavior.

Boron-rich materials combine chemical stability with refractory properties and, consequently, are interesting for high-temperature thermoelectric applications. Therefore, the magnetic, electrical, and thermal transport properties of the ${\mathrm{Y}}_{1-x}{\mathrm{Ce}}_x{\mathrm{CrB}}_4$ series have been investigated here to employ the concept of correlation-enhanced thermoelectric properties. Combining x-ray diffraction and energy- or wavelength-dispersive spectrometry, we find a rather narrow stability range of ${\mathrm{Y}}_{1-x}{\mathrm{Ce}}_x{\mathrm{CrB}}_4$, only samples on the Y- and Ce-rich substitution limits $(x=0,0.05,0.95,\mathrm{and}1)$ were obtained. Electrical resistivity data show a change from semiconducting $(x=0)$ to metallic behavior upon Ce substitution $(x\ge 0.95)$. From magnetic susceptibility measurements and x-ray absorption spectroscopy, we find a temperature-dependent intermediate valence state of Ce of about $+3.5$. However, a fit of the magnetic susceptibility data to the Coqblin-Schrieffer model yields a surprisingly high Kondo temperature of about 1100 K. Together with the good thermal conductivity for the studied substitution series this impedes a suitable thermoelectric performance. Electronic structure calculations for ${\mathrm{YCrB}}_4$ support its narrow gap semiconducting nature in contrast to previous studies. Surprisingly, its electronic structure is characterized by pronounced van Hove singularities very close to the Fermi-level $E_{\mathrm{F}}$. They originate from nearly dispersionless Cr $3d_{z^2-r^2}$-derived bands in a large part of the Brillouin zone, suggesting the appearance of electronic instabilities upon rather small electron doping into these states.

Expanding and improving the repertoire of numerical methods for studying quantum lattice models is an ongoing focus in many-body physics. While the density matrix renormalization group (DMRG) has been established as a practically useful algorithm for finding the ground state in one-dimensional systems, a provably efficient and accurate algorithm remained elusive until the introduction of the rigorous renormalization group (RRG) by Landau et al. [Nat. Phys. 11, 566 (2015)]. In this paper, we study the accuracy and performance of a numerical implementation of RRG at first-order phase transitions and in symmetry-protected topological phases. Our study is motivated by the question of when RRG might provide a useful complement to the more established DMRG technique. In particular, despite its general utility, DMRG can give unreliable results near first-order phase transitions and in topological phases, since its local update procedure can fail to adequately explore (near-)degenerate manifolds. The rigorous theoretical underpinnings of RRG, meanwhile, suggest that it should not suffer from the same difficulties. We show this optimism is justified, and that RRG indeed determines well-ordered, accurate energies even when DMRG does not. Moreover, our performance analysis indicates that in certain circumstances seeding DMRG with states determined by coarse runs of RRG may provide an advantage over simply performing DMRG.

Breaking time-reversal symmetry in three-dimensional topological insulator thin films can lead to different topological quantum phases, such as the Chern insulator (CI) phase and the axion insulator (AI) phase. Using first-principles density functional theory methods, we investigate the onset of these two topological phases in a trilayer heterostructure consisting of a ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ $\left({\mathrm{Bi}}_2{\mathrm{Te}}_3\right)$ TI thin film sandwiched between two antiferromagnetic MnTe layers. We find that an orthogonal exchange field from the MnTe layers, stabilized by a small anisotropy barrier, opens an energy gap of the order of 10 meV at the Dirac point of the TI film. A topological analysis demonstrates that, depending on the relative orientation of the exchange field at the two interfaces, the total Chern number of the system is either $\mathcal{C}=1$ or $\mathcal{C}=0$, characteristic of the CI and AI phases, respectively. Nontopological surface states inside the energy-gap region, caused by the interface potential, complicate this identification. Remarkably though, the calculation of the anomalous Hall conductivity shows that such nontopological surface states do not affect the topology-induced transport properties. Given the size of the exchange gap, we estimate that gapless chiral edge states, leading to the quantum anomalous Hall effect, should emerge on the sidewalls of these heterostructures in the CI phase for widths $\ge 200$ nm. We also discuss the possibility of inducing transitions between the CI and the AI phases by means of the spin-orbit torque caused by the spin Hall effect in an adjacent conducting layer.

Inspired by recent advances in atomic homo and heterostructures, we consider the vertical stacking of plasmonic lattices as a new degree of freedom to create a coupled system showing a modified optical response concerning the monolayer. The precise design of the stacking and the geometrical parameters of two honeycomb plasmonic lattices tailors the interaction among their metallic nanoparticles. Based on the similarity of the lattice symmetry, analogies can be drawn with stacked atomic crystals, such as graphene. We use the multipolar spectral representation to study the plasmonic vertical stack's optical response in the near-field regime, emphasizing symmetry properties. The strong coupling of certain optical bands shows up as anticrossings in the dispersion diagram, resulting in the polarization exchange of the interacting bands. By leveraging these effects, we engineer the near-field intensity distribution. Additionally, lifting band degeneracy at specific points of the Brillouin zone is obtained with the consequent opening of minigaps. These effects are understood by quantifying the multipolar coupling among nanospheres belonging to the same and different sublattices, as well as the interlayer and intralayer nanoparticle interactions. Differences with the atomic case are also analyzed and explained in terms of the stack's interaction matrix. Finally, we predict the absorption spectrum projected on the two orthogonal linear polarizations.

We show that the entire continuous model of twisted bilayer graphene (TBG) (and not just the two active bands) with particle-hole symmetry is anomalous and hence incompatible with lattice models. Previous works [e.g., Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, Phys. Rev. Lett. 123, 036401 (2019); J. Ahn, S. Park, and B.-J. Yang, Phys. Rev. X 9, 021013 (2019); H. C. Po, L. Zou, T. Senthil, and A. Vishwanath, Phys. Rev. B 99, 195455 (2019); J. Kang and O. Vafek, Phys. Rev. X 8, 031088 (2018); M. Koshino, N. F. Q. Yuan, T. Koretsune, M. Ochi, K. Kuroki, and L. Fu, Phys. Rev. X 8, 031087 (2018); J. Liu, J. Liu, and X. Dai, Phys. Rev. B 99, 155415 (2019); L. Zou, H. C. Po, A. Vishwanath, and T. Senthil, Phys. Rev. B 98, 085435 (2018)] found that the two flatbands in TBG possess a fragile topology protected by the $C_{2z}T$ symmetry. Song et al. [Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, Phys. Rev. Lett. 123, 036401 (2019)] also pointed out an approximate particle-hole symmetry ($\mathcal{P}$) in the continuous model of TBG. In this paper, we numerically confirm that $\mathcal{P}$ is indeed a good approximation for TBG and show that the fragile topology of the two flatbands is enhanced to a $\mathcal{P}$-protected stable topology. This stable topology implies $4l+2$ ($l\in \mathbb{N}$) Dirac points between the middle two bands. The $\mathcal{P}$-protected stable topology is robust against arbitrary gap closings between the middle two bands and the other bands. We further show that, remarkably, this $\mathcal{P}$-protected stable topology, as well as the corresponding $4l+2$ Dirac points, cannot be realized in lattice models that preserve both $C_{2z}T$ and $\mathcal{P}$ symmetries. In other words, the continuous model of TBG is anomalous and cannot be realized on lattices. Two other topology related topics, with consequences for the interacting TBG problem, i.e., the choice of Chern band basis in the two flatbands and the perfect metal phase of TBG in the so-called second chiral limit, are also discussed.

We derive the explicit Hamiltonian of twisted bilayer graphene (TBG) with Coulomb interaction projected into the flat bands and study the symmetries of the Hamiltonian. First, we show that all projected TBG Hamiltonians can be written as positive semidefinite Hamiltonians, an example of which was found in work by Kang and Vafek [Phys. Rev. Lett. 122, 246401 (2019)]. We then prove that the interacting TBG Hamiltonian exhibits an exact U(4) symmetry in the exactly flat band (nonchiral-flat) limit. We further define, besides a first chiral limit where the AA stacking hopping is zero, a second chiral limit where the AB/BA stacking hopping is zero. In the first chiral-flat limit (or second chiral-flat limit) with exactly flat bands, the TBG is enhanced to have an exact U(4)$\times$U(4) symmetry, whose generators are different between the two chiral limits. While in the first chiral limit and in the nonchiral case these symmetries have been found in work by Bultinck et al. [Phys. Rev. X 10, 031034 (2020)], for the eight lowest bands, we here prove that they are valid for projection into any $8n_{\text{max}}$ particle-hole symmetric TBG bands, with $n_{\text{max}}>1$ being the practical case for small twist angles $<1^{\circ }$. Furthermore, in the first or second chiral-nonflat limit without flat bands, an exact U(4) symmetry still remains. We also elucidate the link between the U(4) symmetry presented here and the similar but different U(4) of Kang and Vafek [Phys. Rev. Lett. 122, 246401 (2019)]. Furthermore, we show that our projected Hamiltonian can be viewed as the normal-ordered Coulomb interaction plus a Hartree-Fock term from passive bands, and exhibits a many-body particle-hole symmetry which renders the physics symmetric around charge neutrality. We also provide an efficient parametrization of the interacting Hamiltonian. The existence of two chiral limits with an enlarged symmetry suggests a possible duality of the model yet undiscovered.

We derive the exact insulator ground states of the projected Hamiltonian of magic-angle twisted bilayer graphene (TBG) flat bands with Coulomb interactions in various limits, and study the perturbations away from these limits. We define the (first) chiral limit where the AA stacking hopping is zero, and a flat limit with exactly flat bands. In the chiral-flat limit, the TBG Hamiltonian has a U(4)$\times \mathrm{U}$(4) symmetry, and we find that the exact ground states at integer filling $-4\le \nu \le 4$ relative to charge neutrality are Chern insulators of Chern numbers ${\nu }_C=4-\left|\nu \right|,2-\left|\nu \right|,\cdots ,\left|\nu \right|-4$, all of which are degenerate. This confirms recent experiments where Chern insulators are found to be competitive low-energy states of TBG. When the chiral-flat limit is reduced to the nonchiral-flat limit which has a U(4) symmetry, we find $\nu =0,\pm 2$ has exact ground states of Chern number 0, while $\nu =\pm 1,\pm 3$ has perturbative ground states of Chern number ${\nu }_C=\pm 1$, which are U(4) ferromagnetic. In the chiral-nonflat limit with a different U(4) symmetry, different Chern number states are degenerate up to second-order perturbations. In the realistic nonchiral-nonflat case, we find that the perturbative insulator states with Chern number ${\nu }_C=0$ ($0<|{\nu }_C|<4-|\nu |$) at integer fillings $\nu$ are fully (partially) intervalley coherent, while the insulator states with Chern number $|{\nu }_C|=4-|\nu |$ are valley polarized. However, for $0<|{\nu }_C|\le 4-|\nu |$, the fully intervalley coherent states are highly competitive (0.005 meV/electron higher). At nonzero magnetic field $\left|B\right|>0$, a first-order phase transition for $\nu =\pm 1,\pm 2$ from Chern number ${\nu }_C=\text{sgn}\left(\nu B\right)(2-|\nu \left|\right)$ to ${\nu }_C=\text{sgn}\left(\nu B\right)(4-|\nu \left|\right)$ is expected, which agrees with recent experimental observations. Lastly, the TBG Hamiltonian reduces into an extended Hubbard model in the stabilizer code limit.

We find exact analytic expressions for the energies and wave functions of the charged and neutral excitations above the exact ground states (at rational filling per unit cell) of projected Coulomb Hamiltonians in twisted bilayer graphene. Our exact expressions are valid for any form of the Coulomb interaction and any form of $AA$ and $AB/BA$ tunneling. The single charge excitation energy is a convolution of the Coulomb potential with a quantum geometric tensor of the TBG bands. The neutral excitations are (high-symmetry group) magnons, and their dispersion is analytically calculated in terms of the form factors of the active bands in TBG. The two-charge excitation energy and wave functions are also obtained, and a sufficient condition on the graphene eigenstates for obtaining a Cooper pair from Coulomb interactions is obtained. For the actual TBG bands at the first magic angle, we can analytically show that the Cooper pair binding energy is zero in all such projected Coulomb models, implying that either phonons and/or nonzero kinetic energy are needed for superconductivity. Since Vafek and Kang [Phys. Rev. Lett. 125, 257602 (2020)] showed that the kinetic energy bounds on the superexchange energy are less ${10}^{-3}$ in Coulomb units, the phonon mechanism becomes then very likely. If nonetheless the superconductivity is due to kinetic terms which render the bands nonflat, one prediction of our theory is that the highest $T_c$ would not occur at the highest DOS.

Using exact diagonalization, we study the projected Hamiltonian with the Coulomb interaction in the eight flat bands of first magic angle twisted bilayer graphene. Employing the U(4) [$\mathrm{U}\left(4\right)\times \mathrm{U}\left(4\right)$] symmetries in the nonchiral (chiral) flat band limit, we reduced the Hilbert space to an extent that allows for study around $\nu =\pm 3,\pm 2,\pm 1$ fillings. In the first chiral limit $w_0/w_1=0$, where $w_0$ ($w_1$) is the $AA$ ($AB$) stacking hopping, we find that the ground states at these fillings are extremely well-described by Slater determinants in a so-called Chern basis, and the exactly solvable charge $\pm 1$ excitations found in Bernevig et al. [Phys. Rev. B 103, 205415 (2021)] are the lowest charge excitations up to system sizes $8\times 8$ (for restricted Hilbert space) in the chiral-flat limit. We also find that the flat metric condition (FMC) used by Bernevig et al. [Phys. Rev. B 103, 205411 (2021)], Song et al. [Phys. Rev. B 103, 205412 (2021)], Bernevig et al. [Phys. Rev. B 103, 205413 (2021)], Lian et al. [Phys. Rev. B 103, 205414 (2021)], and Bernevig et al. [Phys. Rev. B 103, 205415 (2021)] for obtaining a series of exact ground states and excitations holds in a large parameter space. For $\nu =-3$, the ground state is the spin and valley polarized Chern insulator with ${\nu }_C=\pm 1$ at $w_0/w_1\lesssim 0.9$ (0.3) with (without) FMC. At $\nu =-2$, we can only numerically access the valley polarized sector, and we find a spin ferromagnetic phase when $w_0/w_1\gtrsim 0.5t$ where $t\in [0,1]$ is the factor of rescaling of the actual TBG bandwidth, and a spin singlet phase otherwise, confirming the perturbative calculation [Lian. et al., Phys. Rev. B 103, 205414 (2021), Bultinck et al., Phys. Rev. X 10, 031034 (2020)]. The analytic FMC ground state is, however, predicted in the intervalley coherent sector which we cannot access [Lian et al., Phys. Rev. B 103, 205414 (2021), Bultinck et al., Phys. Rev. X 10, 031034 (2020)]. For $\nu =-3$ with/without FMC, when $w_0/w_1$ is large, the finite-size gap $\mathrm{\Delta }$ to the neutral excitations vanishes, leading to phase transitions. Further analysis of the ground state momentum sectors at $\nu =-3$ suggests a competition among (nematic) metal, momentum $M_M$ ($\pi$) stripe and $K_M$-CDW orders at large $w_0/w_1$.

We study the quantum metric tensor and its scalar curvature for a particular version of the Lipkin-Meshkov-Glick model. We build the classical Hamiltonian using Bloch coherent states and find its stationary points. They exhibit the presence of a ground-state quantum phase transition where a bifurcation occurs, showing a change in stability associated with an excited-state quantum phase transition. Symmetrically, for a sign change in one Hamiltonian parameter, the same phenomenon is observed in the highest-energy state. Employing the Holstein-Primakoff approximation, we derive analytic expressions for the quantum metric tensor and compute the scalar and Berry curvatures. We contrast the analytic results with their finite-size counterparts obtained through exact numerical diagonalization and find excellent agreement between them for large sizes of the system in a wide region of the parameter space except in points near the phase transition where the Holstein-Primakoff approximation ceases to be valid.

Theoretical prediction on the elastic properties of superionic material is challenging due to the fast diffusion of cation/anion in the lattice. Here, we investigated the elastic properties of ${\mathrm{Li}}_2\mathrm{O}$ at elevated temperature and pressure using ab initio molecular dynamics (AIMD). We observed the superionic transition above 1300 K, and the highly diffusive ${\mathrm{Li}}^{+}$ leads to local structure change with significant influence on the elasticity of ${\mathrm{Li}}_2\mathrm{O}$. We successfully predicted the significant $C_{11}$ softening above 1300 K, and the calculated elastic constants fit the previous experimental results very well. It suggests the anharmonic lattice vibration before superionic transition and the diffusion of ${\mathrm{Li}}^{+}$ after superionic transition are very important for the prediction on the elastic properties, and the AIMD method is able to describe the superionic behavior accurately. In addition, we calculated the bulk and shear moduli, sound velocities, as well as elastic and sound velocity anisotropies. We found that the superionic state transition also leads to the weakening of the elastic and sound velocity anisotropies in ${\mathrm{Li}}_2\mathrm{O}$. Pressure has negative effect on the mobility of ${\mathrm{Li}}^{+}$, which strengthens the elastic stiffening effect of superionic ${\mathrm{Li}}_2\mathrm{O}$ with increasing pressure.

Circular dichroism observed by resonant x-ray diffraction at the $\mathrm{Cu}{L}_3$ edge was investigated in a chiral antiferromagnet, $\mathrm{Pb}\left(\mathrm{TiO}\right){\mathrm{Cu}}_4{\left({\mathrm{PO}}_4\right)}_4$, which shows magnetic quadrupole order $(T_{\mathrm{N}}=7\mathrm{K})$. At temperatures above $T_{\mathrm{N}}$, space-group-forbidden reflection 100 is observed due to the anisotropic tensor of susceptibility (ATS) scattering. With decreasing temperature, the reflection intensity shows substantial circular dichroism below $T_{\mathrm{N}}$. We found one-to-one correspondence between the sign of the circular dichroism and that of the sample's crystallographic chirality. In addition, the reflection intensity depends on the product of poling magnetic and electric fields. The circular dichroism observed in this study is interpreted as the interference between the ATS and the magnetic scatterings. This finding shows that the interference scattering probes both the chirality and the order parameter of the magnetic quadrupole order in this chiral antiferromagnet.

We study the microscopical origin of anisotropic ferromagnetism in the van der Waals magnet ${\text{CrI}}_3$. We conclude that the nearest-neighbor exchange is well described by the Heisenberg-Kitaev-$\mathrm{\Gamma }$ ($\mathrm{HK}\mathrm{\Gamma }$) model, and we also find a nonzero Dzyaloshinskii-Moriya interaction (DMI) on next-nearest neighbors. Both Kitaev and DMI are known to generate a nontrivial topology of the magnons in the honeycomb lattice and have been used separately to describe the low-energy regime of this material. We discuss how including one or the other leads to different signs of the Chern number. Furthermore, the topological gap at the $\mathit{K}$ point seems to be mainly produced by the DMI, despite its being one order of magnitude smaller than Kitaev. Finally, we show that, by applying an external electric field perpendicular to the crystal plane, it is possible to induce DMI on nearest neighbors, and this could have consequences in noncollinear spin textures, such as domain walls and skyrmions.

Melting of a particle lattice in two dimensional systems substantially differs from the one in three dimensions. The most prominent theory of the lattice melting in 2D, formulated by Kosterlitz, Thouless, Halperin, Nelson, and Young (KTHNY), describes the solid-liquid phase transition as a two-step process, where the solid phase turns into the liquid via the hexatic phase. The microscopic nature of this process lies in the excitation and unbinding of topological defects in the 2D lattice. The KTHNY melting scenario has been observed in number of 2D physical systems. Here, we theoretically analyze the melting process of a Néel skyrmion lattice by means of numerical simulations using a model of lacunar spinel ${\mathrm{GaV}}_4{\mathrm{S}}_8$ hosting hexagonal skyrmion lattice at low temperatures. We show that topological defects are excited in such skyrmion lattice causing its melting via the hexatic phase in agreement with the KTHNY theory.

In this work we revisit the phase diagram of Co-doped LaFeAsO using single crystals and thermodynamic methods. From magnetic susceptibility studies, we track the doping evolution of the antiferromagnetic phase to reveal a continuous suppression of $T_{\mathrm{N}}$ up to 5% Co doping. To study the evolution of the so-called nematic phase, the temperature dependence of the length changes along the $a$ and $b$ orthorhombic directions, $\mathrm{\Delta }L/L_0$, was determined by high-resolution capacitance dilatometry. The results clearly show a gradual reduction of the orthorhombic distortion $\delta$ and of $T_{\mathrm{S}}$ with increasing Co content up to 4.5%, while it is completely suppressed for 7.5% Co. Bulk superconductivity was found in a small doping region around 6% Co content, while both $T_{\mathrm{c}}$ and the superconducting volume fraction rapidly drop in the neighboring doping regime. Ultimately, no microscopic coexistence between the superconducting and magnetic phases can be assessed within our resolution limit, in sharp contrast with other iron-pnictide families, e.g., electron- and hole-doped ${\mathrm{BaFe}}_2{\mathrm{As}}_2$.