At the interface between two distinct materials, desirable properties, such as superconductivity, can be greatly enhanced1, or entirely new functionalities may emerge2. Similar to in artificially engineered heterostructures, clean functional interfaces alternatively exist in electronically textured bulk materials. Electronic textures emerge spontaneously due to competing atomic-scale interactions3, the control of which would enable a top-down approach for designing tunable intrinsic heterostructures. This is particularly attractive for correlated electron materials, where spontaneous heterostructures strongly affect the interplay between charge and spin degrees of freedom4. Here we report high-resolution neutron spectroscopy on the prototypical strongly correlated metal CeRhIn5, revealing competition between magnetic frustration and easy-axis anisotropy—a well-established mechanism for generating spontaneous superstructures5. Because the observed easy-axis anisotropy is field-induced and anomalously large, it can be controlled efficiently with small magnetic fields. The resulting field-controlled magnetic superstructure is closely tied to the formation of superconducting6 and electronic nematic textures7 in CeRhIn5, suggesting that in situ tunable heterostructures can be realized in correlated electron materials.


The role of interfaces in enhancing or creating functionality is twofold; interfaces exhibit reduced dimensionality, which is known to significantly influence electronic, magnetic and optical properties8. Furthermore, crossed response functions can arise from the interplay of two distinct order parameters at the interface, and lead to entirely new properties. This is successfully utilized in bottom-up approaches to device design. For example, semiconductor heterostructures can be grown with clean, atomically flat interfaces, the basis for applications in electronics and quantum optics9. Due to the intrinsic coupling between various order parameters, heterostructures grown from strongly correlated electron materials are a promising path towards new generations of devices, as highlighted by recent discoveries1,2,8. However, despite some impressive initial success, controlling these interfaces remains a significant challenge, precisely due to the underlying complexity8. Interestingly, this complexity is also what holds the key to a top-down approach for realizing high-quality interfaces. The complex ground states of strongly correlated electron materials arise from the competition between two or more atomic-scale interactions, often leading to superstructures, which we propose to exploit as intrinsic heterostructures.

We show that heavy electron metals—that is, prototypical strongly correlated electron materials—are exceptional model systems to investigate intrinsic heterostructures. Here a frustrated Ruderman–Kittel–Kasuya–Yosida (RKKY) exchange interaction between localized f-electrons, which frequently favours spiral order, directly competes with a substantial easy-axis anisotropy enabled by the large spin–orbit interaction of lanthanide-based materials. The minimal model describing this competition is the axial next-nearest-neighbour Ising (ANNNI) Hamiltonian5, which shows that the conflict of frustration and anisotropy is universally resolved via the formation of modulated superstructures with applications in hard and soft matter.

As illustrated in Fig. 1a–c, in heavy electron metals the formation of a magnetic superstructure may also have important consequences for the electronic ground state. The presence of an additional Kondo interaction favours screening of f-electron magnetic moments by conduction electrons, leading to heavy electronic quasiparticles with an enhanced electronic density of states (DOS). Due to this strong coupling between spin and charge, the underlying magnetic superstructure is likely to induce a spatially modulated electronic texture (Fig. 1b, c). Given that the period λ of the magnetic superstructure is highly sensitive to external control parameters, our top-down approach offers the advantage that the electronic heterostructure can be tuned in situ.

Fig. 1: Interplay of magnetic superstructures and electronic textures in heavy fermion materials.
Fig. 1

a, The competition of a frustrated Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction, which typically promotes spiral order, is in direct conflict with a substantial easy-axis anisotropy enabled by the large spin–orbit interaction of lanthanide-based materials, and results in the formation of strongly modulated magnetic phases where the magnitude of the f-electron magnetic moment changes as a function of position. b, Further, the Kondo interaction tends to hybridize conduction electrons and localized f-electrons by aligning conduction electrons spins antiparallel to f-moments. In the presence of strongly modulated f-electron moments this will generate an additional modulation of the f-electron contribution to the electronic density of states (DOS). c, Illustration of extreme cases where the magnetic moment is maximum (top and bottom) and minimum (middle). The f-electron density of states at the Fermi level is represented by the blue-shaded region, where the electrons are more localized in the maximal moment case, and more itinerant in the minimal moment case. The prototypical heavy fermion material CeRhIn5 investigated here exhibits two phases with electronic textures as shown in panels d and e that arise via the mechanism illustrated in ac (see text). d, Magnetic phase diagram as a function of temperature T and pressure P (ref. 10). At ambient pressure and below the Néel temperature TN = 3.8 K, CeRhIn5 orders antiferromagnetically (AFM I). Application of pressure suppresses the AFM I order, resulting in a quantum critical point (QCP) at Pc = 2.25 GPa around which a broad dome of unconventional superconductivity (SC) emerges. TSC denotes a region of textured superconductivity6. Arrows indicate temperature regions where the magnetic ordering wavevector is k1 = (1/2 1/2 0.326) and k2 = (1/2 1/2 0.391), at P 1.48GPa16. e, Magnetic phase diagram as a function of temperature T and magnetic field H. The AFM I state can alternatively be suppressed at a second QCP by applying a critical field Hc = 50 T. Magnetic fields applied with a small in-plane component results in the formation of an electronic nematic phase above H* = 28 T (H* varies slightly as a function of θ, see Fig. 3 of ref. 11) for temperatures below T = 2.2 K.

We demonstrate that a surprisingly small magnetic field of 2 T induces a substantial uniaxial magnetic anisotropy in the magnetically frustrated heavy electron material CeRhIn5, resulting in the formation of a field-tunable magnetic heterostructure. CeRhIn5 is a tetragonal antiferromagnet (AFM), with Néel temperature TN = 3.8 K at ambient pressure and zero magnetic field. Increasing pressure enhances the Kondo interaction via a growing overlap of neighbouring Ce 4 f orbitals, eventually leading to the complete suppression of the Ce magnetic moments at a magnetic quantum critical point (QCP) at Pc = 2.25 GPa, around which a broad superconducting dome emerges (Fig. 1d)10. Remarkably, in CeRhIn5, part of the superconducting phase is textured (TSC in Fig. 1d)6. In a strikingly similar fashion, the AFM phase may also be suppressed by a magnetic field H, resulting in a QCP at Hc = 50 T, regardless of field direction11. Near this QCP, a new phase unstable towards the formation of an electronic nematic texture was recently discovered for H > H* = 28 T (Fig. 1e). An arbitrarily small in-plane field component breaks the rotational symmetry of the electronic structure, suggesting a surprisingly large nematic susceptibility7.

Interestingly, small in-plane fields also break the rotational symmetry of the AFM state, suggesting that electronic and magnetic textures are indeed related. Due to magnetic frustration arising from competing antiferromagnetic nearest- (NN) and next-nearest-neighbour (NNN) RKKY exchange along the c-axis12, the AFM order at low fields (AFM I in Fig. 2a,c) is an incommensurate spin spiral propagating along the c-axis with propagation vector kI = (1/2 1/2 0.297), which conserves in-plane rotational symmetry13. However, for Hc, a spin-flop transition occurs above the critical field \({H}_{{\rm{c}}}^{{\rm{III}}}=2.1\,{\rm{T}}\) (see Figure 2a)14, where the Ce moments align perpendicular to H, forming a commensurate collinear square-wave phase, with propagation vector kIII = (1/2 1/2 1/4), suggesting a large magnetic-field-induced in-plane easy-axis anisotropy.

Fig. 2: Signatures of highly-tunable modulated magnetic superstructures in CeRhIn5.
Fig. 2

a, Below TN = 3.8 K at ambient pressure, CeRhIn5 orders in an incommensurate antiferromagnetic spin helix (AFM I), with a propagation vector kI = (1/2 1/2 0.297), where the magnetic moments lie parallel to tetragonal basal plane13. Note that the AFM I phase conserves the four-fold rotational symmetry of the underlying crystal structure (see also c). Applying H parallel to the tetragonal basal plane of CeRhIn5 breaks the four-fold symmetry and results in the emergence of two additional magnetic phases: at high temperature, an incommensurate elliptical helix (AFM II) with strongly modulated magnetic moments and temperature-dependent propagation vector kII = (1/2 1/2 l(T)) (see also d), and at low temperature, a commensurate collinear square-wave (AFM III, 'up-up-down-down' configuration) with a propagation vector kIII = (1/2 1/2 1/4), separated from AFM I by the critical magnetic field \({H}_{{\rm{c}}}^{{\rm{III}}}\), and from AFM II by critical temperature \({T}_{{\rm{c}}}^{{\rm{III}}}\) (ref. 14). b, T versus H phase diagram for CeRhIn5 calculated on the basis of our effective spin Hamiltonian, using the exchange interaction and field-dependent uniaxial magnetic anisotropy determined via neutron scattering. Colour scale denotes the c-component of the magnetic propagation vector k = (1/2 1/2 l), derived from equation (1). c, Illustrations of the three magnetic structures. The upper sketches contain the projection of the three unit cells onto the tetragonal basal plane, clarifying the orientation of the Ce magnetic moments (red arrows) in the plane. When a magnetic field is applied in the tetragonal basal plane (here \({\boldsymbol{H}}\parallel (1\bar{1}0)\), see black arrows), all (AFM III), or most (AFM II) Ce magnetic moments align perpendicular to H. Note that for the AFM II phase, the size of the Ce magnetic moment is strongly modulated. d, The c-component of the magnetic propagation vector k = (1/2 1/2 l) at H = 3.5 T as a function of temperature from experiment and as calculated from equation (1), seen in the theoretical phase diagram in Fig. 1d. Dashed line indicates fit to the logarithmic function \(-1/{\rm{ln}}\left(T-{T}_{{\rm{c}}}^{{\rm{III}}}\right)\). The logarithmic temperature-dependence of the propagation vector is characteristic of modulated superstructures as described by the ANNNI framework5, and illustrates the highly tunable superstructure period λ = 2π/k. Note that, for CeRhIn5, λ may be tuned as a function of T or H (see b). Error bars represent standard deviations.

To elucidate the role of this field-induced easy-axis anisotropy, we investigate the magnetic interactions of CeRhIn5 using neutron spectroscopy. This reveals that the magnetic interactions of CeRhIn5 for in-plane fields are remarkably well described by the effective spin model Hamiltonian

$${\mathscr{H}}=\sum _{ij}\left[{J}_{ij}\left((1-\delta ){S}_{i}^{x}{S}_{j}^{x}+(1+\delta ){S}_{i}^{y}{S}_{j}^{y}\right)+\Delta {J}_{ij}{S}_{i}^{z}{S}_{j}^{z}\right]-{g}_{J}{\mu }_{\rm{B}}h\sum _{j}{S}_{j}^{x}$$

which is related to the ANNNI model5. S i in equation (1) is a spin-1/2 operator representing the effective magnetic moment of the \({{\rm{\Gamma }}}_{7}^{2}\) crystal field doublet. We note that the Hamiltonian in equation (1) is valid for H applied in the tetragonal basal plane, and adopt the convention that \({\boldsymbol{H}}\parallel (1\bar{1}0)\), so that the easy-axis anisotropy is along (110) (Fig. 2c). Our previous H = 0 study12 revealed that the magnetic excitations are accurately described by \({\mathscr{H}}\)(δ = 0,h = 0) with only three exchange constants J ij : a NN exchange in the tetragonal basal plane, J0, and two NN and NNN exchange interactions along c, J1 and J2, that, in combination with an easy-plane anisotropy Δ > 0 in the basal plane, generate the spiral ground state (see Figure 2c). Two additional ingredients are required to include field dependence: a conventional Zeeman term (final term in equation (1)) and a field-dependent easy-axis exchange anisotropy favouring spin alignment perpendicular to H, described by the dimensionless parameter δ. The above effective spin-1/2 Hamiltonian can be obtained by projecting the crystal field eigenstates onto the lowest-energy doublet. The exchange anisotropy arises a priori from changes in the orbital character of the Ce 4 f electronic wavefunction with H, where its strength is expected to be substantial due to the large spin–orbit coupling for Ce and vary as δ(H) = I δ H2.

In Fig. 3, we show the full spin excitation spectrum of CeRhIn5 as measured in the AFM III phase at H = 7 T (Fig. 2a,c), along the three principal directions (h00), (hh0), and (00 l), centred at the commensurate magnetic zone centre at kIII = (1/2 1/2 1/4), with additional fields presented in the Supplementary Information. Comparing data sets at various magnetic fields reveals a clear field-induced increase in the spin gap ΔS at kIII. Figure 4a presents ΔS as function of H extracted from energy cuts through the spin wave spectra shown in Fig. 3 at kIII. The dynamic susceptibility χ″(q,ω) (see Figure 3d–f and Supplementary Information), and the corresponding spin-wave dispersion is obtained from a large-S expansion:

$$\hslash {\omega }_{{\bf{q}}}=\sqrt{{\left({J}_{0{\bf{q}}}^{N}+{J}_{2{\bf{q}}}^{N}\pm \left|{J}_{1{\bf{q}}}\right|\right)}^{2}+{\left({J}_{0{\bf{q}}}^{A}+{J}_{2{\bf{q}}}^{A}\pm \left|{J}_{1{\bf{q}}}\right|\right)}^{2}}$$

Here J1q and \({J}_{0,2{\bf{q}}}^{A,N}\) are the Fourier transformation of the exchange parameters (Supplementary Eqs. S10–S14 Supplementary Information), each consisting of the exchange integrals J0, J1 and J2, easy-plane anisotropy Δ > 0, and easy-axis anisotropy δ, introduced in equation (1).

Fig. 3: Magnetic excitations of CeRhIn5 in in-plane magnetic fields.
Fig. 3

ac, Measured spin excitation spectra at H = 7 T in the AFM III phase, where kIII = (1/2 1/2 1/4), along three high-symmetry directions: (h00) (a), (hh0) (b), (00 l) (c). Dashed lines indicate spin wave dispersions (see equation (2)), resulting from fitting. df, Calculated dynamic magnetic susceptibility χ″(q,ω) using the fitted parameters. g, Three magnetic exchange integrals used for our calculations.

Fig. 4: Salient parameters of the effective spin model related to ANNNI framework5 to describe the field-tuned uniaxial anisotropy in CeRhIn5.
Fig. 4

a, Spin gap ΔS extracted from spin wave spectra measured via neutron spectroscopy (see Figure 3) as a function of magnetic field H. Squares indicate the gap was measured on LET at T = 2 K, circles on CNCS at T = 2 K, and triangles on SPINS at T = 0.3 K. Error bars reflect instrument resolution. The dark and light backgrounds denote the boundary between the incommensurate helical AFM order (AFM I, kI = (1/2 1/2 0.297)) and the commensurate sine-square wave AFM order (AFM III, kIII = (1/2 1/2 1/4)), below and above HIII = 2.1 T, respectively. The dashed line is a fit to the gap function ΔS(H) derived from the ANNNI model. The finite gap at H = 0 is not due to ANNNI physics (Supplementary Information). b, H-dependence of the magnitude of the uniaxial magnetic anisotropy δ. The dashed line denotes δ(H) = I δ H2 with I δ  = 0.0013(1) (T2). c, H-dependence of the nearest-neighbour magnetic exchange integrals J0 and J1. The next-nearest-neighbour exchange integral J2 scales as 0.809 J1 and is therefore not shown (Supplementary Information). The red and orange dashed lines denote the interpolated values of J0 and J1, respectively, that were used to calculate the values for the dashed line in a (see text for details). All error bars in b and c represent standard deviations.

The dashed lines in Fig. 3 illustrate exemplary fits of χ″(q,ω) to our data, performed for every H, showing that the Hamiltonian in equation (1) describes our data quantitatively (Supplementary Information). Due to the small size of the magnetic Brillouin zone along the c direction, Umklapp scattering occurs at the zone boundary, resulting in additional spin wave branches, \(\hslash {\omega}_{{\mathbf{{q}}}\pm {{\mathbf{{k}}}}_{{\rm{III}}}}\)12. The easy-plane anisotropy was fixed to Δ = 0.82, as determined at H = 012, and assumed to be field-independent; additional fit details are provided in the Methods section. The resulting size of δ and exchange integrals as a function of H are shown in Fig. 4b, c. Within AFM III, the parameters change smoothly with H; J0, J1 and J2 decrease, in agreement with the decreasing bandwidth of the spectrum, and δ increases in accordance with the growing spin gap. We note that the ratio of J2/J1 remains unchanged for all fields, indicating that the magnetic frustration is not affected by the applied magnetic field. Finally, as demonstrated by the red solid line in Fig. 4b, we find δ(H) = I δ H2 with I δ  = 0.0013(1) (T2). This implies that the experimental critical exchange anisotropy at \({H}_{{\rm{c}}}^{{\rm{III}}}\) is δ c  = 0.0057(5). By comparison, the critical exchange anisotropy calculated via mean-field modelling of the Hamiltonian in equation (1) (Supplementary Information), \({\delta }_{{\rm{c}}}^{{\rm{MF}}}=0.0091\), agrees well with the experiment, which is remarkable considering that our model assumes f-electron localization in CeRhIn5 (Supplementary Information), and that the mean field treatment neglects the effects of quantum fluctuations. Although the gap \({\Delta }_{\rm{S}}=\hslash {\omega }_{{{\boldsymbol{k}}}_{{\rm{III}}}}=\sqrt{2\delta \left(2{J}_{0}+{J}_{2}\right)\left[\left(2{J}_{0}+{J}_{2}\right)\left(1+\delta +\Delta \right)-{J}_{1}\Delta \right]}\) is the clearest indicator of increasing uniaxial anisotropy, it is also sensitive to the field-dependent exchange integrals J. By inserting interpolated values for the exchange integrals and δ we obtain the dashed line in Fig. 4a, demonstrating that our fits to the dynamic susceptibility quantitatively describe the observed spin gap for \(H > {H}_{{\rm{c}}}^{{\rm{III}}}\). We note that an unexpected, small spin gap ΔS ≈ 0.25 meV was observed at H = 0, but likely represents the longitudinal (or Higgs) mode that arises due to Kondo screening of the Ce magnetic moments, as explained in the Supplementary Information(this scenario assumes that there is still a gapless transverse mode). Recent neutron diffraction measurements demonstrate that the Ce magnetic form factor is significantly different from free Ce3+, with a magnetic moment that is reduced by 41% with respect to the expectation from the crystal field ground state, suggesting that the Kondo interaction in CeRhIn5 is indeed substantial, in agreement with this scenario13.

As we show now, CeRhIn5 exhibits an instability towards the formation of highly-tunable modulated magnetic superstructures. Using the exchange constants shown in Fig. 4c, and \({\delta }_{c}^{{\rm{MF}}}\), we obtain the theoretical temperature versus magnetic field phase diagram for CeRhIn5 shown in Fig. 2b, based on our spin Hamiltonian and a mean-field calculation (Supplementary Information). In addition to the remarkable agreement with the experimental phase diagram, it reveals a prominent feature of the ANNNI model, namely that the superstructure period is highly-tunable in proximity to TN5. Notably, critical magnetic fluctuations immediately below TN compete with the uniaxial anisotropy, which causes a softening of the pinning of the magnetic moments along (110), ultimately leading to a magnetic structure with moments primarily along (110), but with small components parallel to H. This high-temperature phase (AFM II) is represented by an elliptical helix in which the size of the moments is modulated (see Figure 2a,c)14. Our model predicts a change of the magnetic propagation vector kII = (1/2 1/2 l) as a function of both H and T. For the ANNNI model, the temperature dependence is given by \({\rm{\Delta }}l\left(T\right)\propto -1/{\rm{ln}}\left(T-{T}_{{\rm{c}}}^{{\rm{III}}}\right)\)(ref. 5), with l = ¼ at \(T={T}_{{\rm{c}}}^{{\rm{III}}}\) (critical temperature between AFM II and III), and slowly approaching the value dictated by NN and NNN exchange interactions along c, l = 0.297 for T TN. In Fig. 2d we show that l(T) at H = 3.5T, as determined via high-resolution neutron diffraction, indeed changes logarithmically, illustrating the ease with which the superstructure period λ = 2π/k may be tuned.

The instability towards this highly-tunable magnetic heterostructure is apparent throughout the entire temperature–field–pressure phase diagram, with significant impact on material properties. Transport measurements show that the AFM II phase continues to exist at pressures approaching the QCP15. Further, even for H = 0, the magnetic propagation vector changes from k1 = (1/2 1/2 0.326) to k2 = (1/2 1/2 0.391) near the phase boundary between textured and bulk superconducting states (indicated by the arrows in Fig. 1d)16. Here the textured superconductivity is suggested to arise due to the coexistence of k1 and k2 magnetic domains, where the superconductivity only nucleates in k26. This may be explained via the mechanism shown in Fig. 1a–c, where k1 and k2 magnetic superstructures each induce distinct electronic textures; however, with only one of them being compatible with the superconducting order parameter. This notably highlights that the tunable period of the magnetic heterostructure in CeRhIn5 enables the control of material properties.

Similarly, invoking the mechanism discussed in Fig. 1a–c for the field-induced nematic phase (Fig. 1e), an underlying modulated magnetic superstructure may generate two-dimensional (2D) electronic layers, where the direction of the local magnetic moments establishes a preferential direction that breaks rotational symmetry within the 2D layers with respect to the underlying lattice. For CeRhIn5, the large field-induced magnetic anisotropy identified here can be accessed by a slight tilting of the magnetic field away from the c axis (inset of Fig. 1e) to align the magnetic moments, providing a natural explanation for the observed large nematic susceptibility.

Quantum oscillation measurements report a crossover from a small to a large Fermi surface volume near both QCPs (Fig. 1d, e), suggesting enhanced coupling between spin and charge degrees of freedom due to the Kondo interaction in their vicinity17,18. This may explain why the magnetic superstructures that are omnipresent throughout the entire phase diagram predominantly influence material properties near the QCPs. Finally, the observed large uniaxial anisotropy arises due to changes of the orbital character of the Ce 4 f electronic wavefunction with magnetic field. Remarkably, it has been demonstrated previously in the family of materials CeMIn5 (M = Co, Rh Ir), to which CeRhIn5 belongs, that the orbital character of the 4 f wavefunctions can be also controlled via chemical substitution or pressure19. This not only affords an intrinsic mechanism for alternatively tuning the uniaxial anisotropy by pressure, but clarifies the striking similarity of the phase diagrams as functions of H and P (see Figure 1d,e).

In conclusion, via the quantitative application of an ANNNI-based effective spin model, a notable first for a heavy electron metal, we have identified a simple mechanism to create highly-tunable emergent magnetic heterostructures in CeRhIn5 via competing interactions. Through coupling of spin and charge degrees of freedom mediated via the Kondo effect, this mechanism concurrently generates electronic textures that significantly influence material properties. These textures are akin to emergent electronic heterostructures that exhibit clean interfaces and can be tuned with great ease, employing the use of external tuning parameters such as magnetic field or pressure. Our work demonstrates that strongly correlated electron materials are a promising route for top-down approaches to producing tunable and emergent heterostructures. Notably, because frustrated exchange is common to f-electron materials, and field-induced uniaxial magnetic anisotropy has been reported in various heavy electron materials20,21, the mechanism identified here may apply universally for heavy electron materials. Furthermore, other classes of strongly correlated electron materials, such as high-Tc copper oxide, iron pnictide, and ruthenate superconductors all exhibit electronic textures near magnetic QCP22,23,24, many of which exhibit instabilities towards incommensurate modulated magnetism25,26,27,28, where orbital effects29 and/or magnetic frustration30 have similarly been proposed to be their origin, suggesting intrinsic functional heterostructures may be realized more broadly.


Sample preparation

Neutron scattering measurements were all performed on a mosaic (~2.2 g) of 14 CeRhIn5 single crystals grown via the In self-flux method. To mitigate the effects of high neutron absorption by Rh and In, individual crystals were polished to a thickness of < 0.6 mm along the crystallographic c-axis and glued to a thin Al plate using a hydrogen-free adhesive (CYTOP). This sample mosaic is well-characterized and was used in our previous neutron spectroscopy study12.

Neutron spectroscopy

Time-of-flight neutron spectroscopy measurements shown in Fig. 3 and the Supplementary Information were performed on two direct geometry spectrometers: the Cold Neutron Chopper Spectrometer (CNCS)31 at the Spallation Neutron Source (SNS), for applied magnetic fields of 5 T and below, with incident neutron energy Ei = 3.315 meV, and the LET Spectrometer32 at the ISIS pulsed neutron and muon source for applied magnetic fields above 5 T, with Ei = 3.3 meV. Energy resolution in both cases was estimated to be ~0.08 meV. Inelastic slices with subtracted background were generated using Horace and fitted to the theoretical dynamic susceptibility using a least-squares method implemented in NeutronPy (http://neutronpy.github.io/). Background scans were obtained on the CeRhIn5 sample at T = 20 K. Detailed inelastic neutron scattering measurements of the gap, shown in Fig. 4a, were performed on the Spin Polarized Inelastic Neutron Spectrometer (SPINS), a cold-neutron triple-axis spectrometer at the NIST Center for Neutron Research (NCNR), using a 7 T magnet with a 3He-dipper. Constant-q scans were obtained with fixed Ef = 3.0 meV, 40′ collimation before the sample, a 60′ radial collimator after the sample, and a horizontally focused 11-blade PG(002) analyser. Higher-order neutrons were filtered using a cold Be-filter. Error bars of the gap values reflect the combined fitting error and energy resolution estimated by the quasielastic linewidth as measured on a standard vanadium sample. Error bars shown in Figs. 4b, c reflect the standard errors resulting from least-squares fitting. Diffraction data shown in Fig. 2d were also obtained on SPINS by performing scans along the (00 l) direction with Ei = Ef = 3.315 meV, 20′-S-10′ collimation in triple-axis mode, a flat monochromator, and a flat 3-blade analyser. Peak positions shown in Fig. 2d were obtained from fitting scans to a single Gaussian with constant background, and error bars represent the combination estimated error and momentum resolution calculated with the Cooper–Nathans method implemented in NeutronPy.

Calculation of phase diagram

To obtain the phase diagram, we treat spins in equation (1) as classical spins, and then numerically minimize the free energy of equation (1). We first perform numerical annealing using the Markov chain Monte Carlo method33, which minimizes the chance of trapping in a metastable state. Subsequently we use the relaxation method to determine the state with minimal free energy. Because the ordering wavevector is temperature-dependent, we continuously change the system size in the c-direction from 4 to 80, and keep the solution with the lowest free energy.

Data availability statement

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. The neutron spectroscopy raw data from the experiment performed at LET are available at https://doi.org/10.5286/ISIS.E.82355430. Data from experiments carried out at SPINS are available at ftp://ftp.ncnr.nist.gov/pub/ncnrdata/ng5/201610/Fobes/CeRhIn5_22425/ and ftp://ftp.ncnr.nist.gov/pub/ncnrdata/ng5/201509/Fobes/CeRhIn5/.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


  1. 1.

    Ge, J.-F. et al. Superconductivity above 100 K in single-layer FeSe films on doped SrTiO3. Nat. Mater. 14, 285–289 (2014).

  2. 2.

    Reyren, N. et al. Superconducting interfaces between insulating oxides. Science 317, 1196–1199 (2007).

  3. 3.

    Mohottala, H. E. et al. Phase separation in superoxygenated La2–xSr x CuO4+y. Nat. Mater. 5, 377–382 (2006).

  4. 4.

    Dagotto, E. Complexity in strongly correlated electronic systems. Science 309, 257–262 (2005).

  5. 5.

    Selke, W. The ANNNI model—Theoretical analysis and experimental application. Phys. Rep. 170, 213–364 (1988).

  6. 6.

    Park, T. et al. Textured superconducting phase in the heavy fermion CeRhIn5. Phys. Rev. Lett. 108, 077003 (2012).

  7. 7.

    Ronning, F. et al. Electronic in-plane symmetry breaking at field-tuned quantum criticality in CeRhIn5. Nature 548, 313–317 (2017).

  8. 8.

    Boschker, H. & Mannhart, J. Quantum-matter heterostructures. Annu. Rev. Condens. Matter Phys. 8, 145–164 (2017).

  9. 9.

    Alferov, Z. I. The history and future of semiconductor heterostructures. Semiconductors 32, 1–14 (1998).

  10. 10.

    Park, T. et al. Hidden magnetism and quantum criticality in the heavy fermion superconductor CeRhIn5. Nature 440, 65–68 (2006).

  11. 11.

    Moll, P. J. W. et al. Field-induced density wave in the heavy-fermion compound CeRhIn5. Nat. Commun. 6, 6663 (2015).

  12. 12.

    Das, P. et al. Magnitude of the magnetic exchange interaction in the heavy-fermion antiferromagnet CeRhIn5. Phys. Rev. Lett. 113, 246403 (2014).

  13. 13.

    Fobes, D. M. et al. Low temperature magnetic structure of CeRhIn5 by neutron diffraction using absoption optimized samples. J. Phys. Condens. Matter 29, 17LT01 (2017).

  14. 14.

    Raymond, S., Ressouche, E., Knebel, G., Aoki, D. & Flouquet, J. Magnetic structure of CeRhIn5 under magnetic field. J. Phys. Condens. Matter 19, 242204 (2007).

  15. 15.

    Knebel, G. et al. Antiferromagnetism and superconductivity in CeRhIn5. J. Phys. Soc. Jpn 80, SA001 (2011).

  16. 16.

    Aso, N. et al. Switching of magnetic ordering in CeRhIn5 under hydrostatic pressure. J. Phys. Soc. Jpn 78, 073703 (2009).

  17. 17.

    Jiao, L.et al Fermi surface reconstruction and multiple quantum phase transitions in the antiferromagnet CeRhIn5. Proc. Natl Acad. Sci. USA 112, 673–678 2015).

  18. 18.

    Shishido, H., Settai, R., Harima, H. & Onuki, Y. A drastic change of the Fermi surface at a critical pressure in CeRhIn5: dHvA study under pressure. J. Phys. Soc. Jpn 74, 1103–1106 (2005).

  19. 19.

    Willers, T.et al Correlation between ground state and orbital anisotropy in heavy fermion materials. Proc. Natl Acad. Sci. USA 112, 2384–2388 2015).

  20. 20.

    Fak, B. et al. Anomalous spin response in the non-centrosymmetric metal CePt3Si. J. Phys. Soc. Jpn 83, 063703 (2014).

  21. 21.

    Moll, P. J. W. et al. Emergent magnetic anisotropy in the cubic heavy-fermion metal CeIn3. npj Quantum Mater. 2, 46 (2017).

  22. 22.

    Borzi, R. A. et al. Formation of a nematic fluid at high fields in Sr3Ru2O7. Science 315, 214–217 (2007).

  23. 23.

    Chuang, T. M. et al. Nematic electronic structure in the 'parent' state of the iron-based superconductor Ca(Fe1-xCox)(2)As2. Science 327, 181–184 (2010).

  24. 24.

    Kivelson, S. A., Fradkin, E. & Emery, V. J. Electronic liquid-crystal phases of a doped Mott insulator. Nature 393, 550–553 (1998).

  25. 25.

    Kitagawa, K., Mezaki, Y., Matsubayashi, K., Uwatoko, Y. & Takigawa, M. Crossover from commensurate to incommensurate antiferromagnetism in stoichiometric NaFeAs revealed by single-crystal 23Na,75As-NMR experiments. J. Phys. Soc. Jpn 80, 033705 (2011).

  26. 26.

    Raymond, S. & Lapertot, G. Ising incommensurate spin resonance of CeCoIn5: A dynamical precursor of the Q phase. Phys. Rev. Lett. 115, 037001 (2015).

  27. 27.

    Tranquada, J. M. et al. Evidence for an incommensurate magnetic resonance in La2–xSr x CuO4. Phys. Rev. B 69, 174507 (2004).

  28. 28.

    Wen, J. et al. Short-range incommensurate magnetic order near the superconducting phase boundary in Fe1+dTe1–xSe x . Phys. Rev. B 80, 104506 (2009).

  29. 29.

    Baek, S. H. et al. Orbital-driven nematicity in FeSe. Nat. Mater. 14, 210–214 (2015).

  30. 30.

    Glasbrenner, J. K. et al. Effect of magnetic frustration on nematicity and superconductivity in iron chalcogenides. Nat. Phys. 11, 953–958 (2015).

  31. 31.

    Ehlers, G., Podlesnyak, A. A., Niedziela, J., Iverson, E. B. & Sokol, P. E. The new cold neutron chopper spectrometer at the Spallation Neutron Source: design and performance. Rev. Sci. Instrum. 82, 085108 (2011).

  32. 32.

    Bewley, R. I., Taylor, J. W., & Bennington, S. M. LET, a cold neutron multi-disk chopper spectrometer at ISIS. Nucl. Instrum. Methods Phys. Res. A 637, 128–134 (2011).

  33. 33.

    Suwa, H. & Todo, S. Markov chain Monte Carlo method without detailed balance. Phys. Rev. Lett. 105, 120603 (2010).

Download references


We acknowledge useful discussions with R. Baumbach, C. Pfleiderer, M. Garst, M. Votja, P. Böni and J. M. Lawrence. Work at Los Alamos National Laboratory (LANL) was performed under the auspices of the US Department of Energy. LANL is operated by Los Alamos National Security for the National Nuclear Security Administration of DOE under contract DE-AC52-06NA25396. Research supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under the project ‘Complex Electronic Materials’ (material synthesis and characterization) and the LANL Directed Research and Development program (neutron scattering, development of the spin wave model, mean-field computation and development of analysis software). Research conducted at Oak Ridge National Laboratory’s (ORNL) Spallation Neutron Source was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, US Department of Energy. Experiments at the ISIS Pulsed Neutron and Muon Source were supported by a beam time allocation from the Science and Technology Facilities Council. We acknowledge the support of the National Institute of Standards and Technology, US Department of Commerce, in providing the neutron research facilities used in this work.

Author information

Author notes

    • Pinaki Das

    Present address: Division of Materials Sciences and Engineering, Ames Laboratory, U.S. DOE, Iowa State University, Ames, IA, USA

    • N. J. Ghimire

    Present address: Argonne National Laboratory, Lemont, IL, USA


  1. MPA-CMMS, Los Alamos National Laboratory, Los Alamos, NM, USA

    • D. M. Fobes
    • , Pinaki Das
    • , N. J. Ghimire
    • , E. D. Bauer
    • , J. D. Thompson
    • , F. Ronning
    • , C. D. Batista
    •  & M. Janoschek
  2. Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN, USA

    • S. Zhang
    •  & C. D. Batista
  3. T-4, Los Alamos National Laboratory, Los Alamos, NM, USA

    • S.-Z. Lin
  4. NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD, USA

    • L. W. Harriger
  5. QCMD, Oak Ridge National Laboratory, Oak Ridge, TN, USA

    • G. Ehlers
    •  & A. Podlesnyak
  6. ISIS Facility, STFC Rutherford Appleton Laboratory, Harwell Science and Innovation Campus, Chilton, Didcot, UK

    • R. I. Bewley
  7. Institute of Crystallography, RWTH Aachen University and Jülich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), Garching, Germany

    • A. Sazonov
    •  & V. Hutanu


  1. Search for D. M. Fobes in:

  2. Search for S. Zhang in:

  3. Search for S.-Z. Lin in:

  4. Search for Pinaki Das in:

  5. Search for N. J. Ghimire in:

  6. Search for E. D. Bauer in:

  7. Search for J. D. Thompson in:

  8. Search for L. W. Harriger in:

  9. Search for G. Ehlers in:

  10. Search for A. Podlesnyak in:

  11. Search for R. I. Bewley in:

  12. Search for A. Sazonov in:

  13. Search for V. Hutanu in:

  14. Search for F. Ronning in:

  15. Search for C. D. Batista in:

  16. Search for M. Janoschek in:


N.J.G., P.D. and E.D.B. synthesized the single crystal samples; J.D.T. and F.R. carried out thermal and transport measurements; D.M.F., G.E., A.P., L.W.H., R.I.B., V.H., A.S. and M.J. performed the neutron spectroscopy measurements; D.M.F. wrote the software for analysing the neutron data; DMF and MJ analyzed the neutron data; M.J. supervised the experimental work; S.Z., S.Z.L. and C.D.B. developed the theoretical model and carried out all calculations; D.M.F., S.Z.L., C.D.B. and M.J. proposed and designed this study, and D.M.F., C.D.B. and M.J. wrote the manuscript; all authors discussed the data and commented on the manuscript.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to M. Janoschek.

Supplementary information

  1. Supplementary Information

    4 Figures, 14 References

About this article

Publication history