Introduction

Ultra-thin layers and atomic chains of transition metal atoms on high-Z substrates show a variety of non-collinear spin states1 ranging from chiral domain walls2 over spin-spirals3,4,5,6,7,8 to skyrmions9. In most of the cases, these non-collinear spin states are stabilized by the so-called Dzyaloshinskii-Moriya (DM) interaction \({\mathbf{D}}_{ij}( {{\hat{\mathbf S}}_i \times {\hat{\mathbf S}}_j})\) between neighboring spins \({\hat{\mathbf S}}_i\) and \({\hat{\mathbf S}}_j\), which is relevant due to the broken inversion asymmetry at the interface of the magnetic material and the substrate. While the usual Heisenberg interaction \(J_{ij}\,{\hat{\mathbf S}}_i \cdot {\hat{\mathbf S}}_j\) favors a parallel (ferromagnetic) or antiparallel (antiferromagnetic) alignment of neighboring spins, the DM interaction favors perpendicular orientations. The interplay of these two interactions can induce a spin-spiral state, with a rotational sense determined by the orientation of Dij, and with a period given by the ratio of Dij and Jij, and by the magnetic anisotropy10.

For many of the proposed applications of such chiral spin-spiral states, e.g., in spintronics and information technology1,2,5 or as platforms for Majorana quasiparticles11,12,13,14, a tuning of these parameters is highly desirable, but challenging15,16. A very appealing approach towards this end is to use the tip of a scanning tunneling microscope to construct chains atom by atom with tailored magnetic anisotropy and interactions. While numerous bottom-up magnetic chains have been investigated17,18,19,20,21,22, these studies have focused solely on Heisenberg interactions inducing collinear order. However, the ubiquitous Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between magnetic atoms in contact to metallic substrates additionally contains a DM component leading to non-collinear order, which is typically strong for high-Z materials23,24. Indeed, it has been shown recently that the DM component of the RKKY interaction between Fe atoms on Pt(111) is of comparable strength as the Heisenberg part25. Most notably, due to the oscillatory nature of the RKKY coupling, the two components are adjustable in their relative strengths and signs by changing the distance between the Fe atoms25. The use of similar material combinations of dilute transition metal chains on high-Z substrates therefore promises the stabilization of adjustable spin-spiral states in bottom-up fabricated chains.

Here, we have experimentally realized this idea in the form of a spin-spiral state residing in a 16 atom long Fe chain that has been assembled atom-by-atom on the (111) surface of a Pt single crystal. By comparison of the experimental results to density-matrix renormalization group (DMRG) calculations, we demonstrate that the spin-spiral’s wavelength and rotational sense can be tuned via the interatomic distances of the Fe atoms in the chain.

Results

Construction of chains of various lengths

Using tip-induced atom manipulation we built FeN chains on a Pt(111) surface with different numbers N of Fe atoms and with different nearest neighbor (NN) distances dNN (see 'Methods' for experimental details). Figure 1b and Supplementary Fig. 1 show an isolated Fe atom (Fe1) and assembled FeN chains with N = 2, 3, 4 and dNN = 4a with the shortest distance of equivalent hollow adsorption sites of a = 2.78 Å. All chains investigated in this work were built from Fe atoms sitting on the hexagonal close-packed (hcp) adsorption site, as identified by the characteristic spin-excitation at 0.19 meV in inelastic scanning tunneling spectroscopy (ISTS) taken on Fe1 (Fig. 1a). As shown in ref. 26, they exhibit a relatively weak easy-plane magnetic anisotropy, which only slightly favors an orientation of the Fe spin in the surface plane. From an investigation of pairs of an Fe-hydrogen complex and an Fe atom with different distances25, we expect that the chosen dNN results in an antiferromagnetic Heisenberg interaction of JNN ≈ −50 μeV and a comparable DM coupling D⊥,NN ≈ +30 μeV between NN Fe atoms in the chains (see Supplementary Note 1 and Supplementary Fig. 2 for the definition of the coordinates).

Fig. 1
figure 1

ISTS of bottom-up FeN chains of different numbers N of atoms. a ISTS spectra taken on top of an isolated Fe atom (Fe1, 10) and on the Fe atoms inside Fe2 (8, 9), Fe3 (5, 6, 7), and Fe4 (1, 2, 3, 4) chains with dNN = 4a. For comparison, the spectrum of the isolated Fe1 is shown in gray behind all other spectra. All spectra were normalized by dividing by a substrate spectrum. (Vstab = −6 mV, Istab = 3 nA, Vmod = 40 μV). b, c STM topographs and schematic ball models where the red and gray spheres represent the Fe and surface Pt atoms, respectively, of all chains investigated in (a) (Vs = −6 mV, Istab = 0.5 nA). The black scale bar below (b) has a length of 1 nm

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Non-collinear spin states in 4a chains

ISTS spectra taken on the Fe atoms in the FeN chains (Fig. 1a) reveal considerable shifts of the spin-excitation to higher excitation energies and a broadening of the excitation step as compared to the isolated Fe1 shown in gray as a reference25. This effect is more pronounced for the atoms with two neighbors (atoms 2, 3, 6) and we therefore tentatively assign the shifts to the mutual magnetic interactions between the atoms, which can be used to determine the corresponding coupling parameters with an accuracy of 10 μeV by comparison to model calculations25. Note, that there is a negligible residual interaction between the chains leading to a slight asymmetry within each chain (see Supplementary Fig. 1) which results in the tiny nonequivalence of the two end atoms (Fig. 1a). Due to symmetry arguments, the DM vector has its largest component in the surface plane perpendicular to the chain axis \(( {D_{||} = 0,D_z \,{\mathrm{small}}} )\)25. We therefore anticipate a tendency towards the formation of a cycloidal spin-spiral state in the (||, z)-plane. Since the chain atoms are strongly coupled to the substrate conduction electrons, these states typically do not show remanence5,19 and therefore cannot be resolved by conventional spin-polarized (SP) scanning tunneling microscopy (STM) in the absence of a magnetic field. We will therefore use two different methods, namely (i) an external homogeneous magnetic field Bz along the surface normal (z) and (ii) RKKY coupling one end of the chain to a ferromagnetic island27,28, in order to stabilize the non-collinear spin states.

First, we applied Bz to the chains and performed ISTS. For the sake of simplicity, we will focus on the data of the Fe4 chain shown in Fig. 2b, c (ISTS and calculations of all other chains are shown in Supplementary Figs. 3–6 as described in Supplementary Note 2 showing the reproducible evolution of the spectra from N = 2 to 4). The spectra reveal a small shift of the spin-excitation to lower energies for field strengths up to Bz ≈ 4, and then a nearly linear increase in the spin-excitation energy proportional to Bz, with slight differences between the spectra on the two inner and the two edge atoms of the chain. Compared to the Bz-dependent ISTS of the Fe1, where the minimum of the excitation energy appears at Bz ≈ 3 T 26 (see Supplementary Fig. 4a), this minimum is much less pronounced and shifted to larger Bz on the chain atoms. The shift again manifests the significant magnetic interactions between the Fe atoms in the chain. Note, that small differences in the spectra between the two inner and between the two outer atoms again result from the slight asymmetry in the chain due to negligible residual couplings to the neighboring chains (see Supplementary Fig. 1). For a further analysis of the spin state of the chain, we quantify the spectra utilizing a perturbation theory model29 based on an effective spin Hamiltonian that considers the Zeeman energy, the easy-plane magnetic anisotropy energy, and the Heisenberg as well as DM interactions (see 'Methods'). Astonishingly, by simply considering the atomic parameters known from single hcp Fe atoms on Pt(111)26 and using the above Heisenberg and DM interactions extracted from the pairs25 as NN interactions in the model (see the parameters in the caption of Fig. 2), we can already excellently reproduce the evolution of the excitation in the ISTS data (cf. Fig. 2c, d). The spectra are rather sensitive to changes in JNN and D⊥,NN, and neither JNN nor D⊥,NN alone can satisfactorily reproduce the data (see Supplementary Fig. 6). This result further corroborates that the pairwise interaction parameters describe the chain data fairly well. However, for this particular distance, we found that the calculated spectra for the Fe2 pair and the Fe3 and Fe4 chains fit somewhat better to the experimental data when the pairwise Heisenberg coupling25 is slightly reduced to JNN = −25 ÂµeV, which is justified as the coupling parameters in chains can slightly deviate from the ones in the according pairs19. This value was therefore used in all calculations for the dNN = 4a chains. The corresponding spin-expectation values calculated from exact diagonalization (ED, see 'Methods'), which are illustrated in Fig. 2a reveal that, as long as Bz is not too large, the underlying spin state is highly non-collinear. The spins of neighboring atoms are mutually canted within the (||, z)-plane as imposed by the strong D⊥,NN component. Note, that a pure spin-spiral state is not realized because of the homogeneous Bz driving all spins into z-direction (see Supplementary Note 4 and Supplementary Figs. 11–13). We will later see, how we can avoid this problem by using an RKKY coupling of the chain end to a ferromagnetic island.

Fig. 2
figure 2

Non-collinear spin state in the Fe4 chain with dNN = 4a. a Calculated spin-expectation values \(( {\langle {\hat S_{||}} \rangle ,\langle {\hat S_ \bot } \rangle ,\langle {\hat S_z} \rangle } )\), see arrows (\(\langle {\hat S_ \bot } \rangle = 0\) by symmetry), for each atom in the Fe4 chain at Bz = 3 T. The results are obtained by exact diagonalization (ED) of a spin model with parameters given in Table 1, JNN = −25 μeV, and D⊥,NN = +30 μeV. b, c ISTS spectra (b) and corresponding colorplots (c) measured on all four atoms in the Fe4 chain for different magnetic fields Bz up to Bz = 11 T. (Vstab = −6 mV, Istab = 3 nA, Vmod = 40 μV). d Calculated ISTS from the perturbation theory model using the same parameters as in (a)

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Non-collinear spin states in 3a chains

Next, in order to demonstrate the adjustability of the non-collinear chain state by dNN, we investigated FeN chains (N = 2, 3, 4) with smaller dNN = 3a (see the ISTS data and calculations of Fe4 in Fig. 3, all other chains are shown in Supplementary Figs. 7–10 as described in Supplementary Note 3 again showing a reproducible evolution of the spectra from N = 2 to 4). The smaller dNN induces a stronger antiferromagnetic Heisenberg component (JNN ≈ −60 ÂµeV) and a stronger DM component (D⊥,NN ≈ −50 μeV) in pairs25 as compared to the 4a-case studied above. Most notably, we know from ab initio calculations of the same pairs, that the sign of D⊥,NN is reversed25. Indeed, Bz dependent ISTS of this Fe4 chain behaves rather differently compared to the 4a-chain data (cf. Figs. 2b, c and 3c, d): (i) the zero field spin-excitation appears at a larger energy; (ii) for the two inner atoms, the shift of the spin-excitation to lower energies continues here up to at least Bz = 7 T; (iii) for the two end atoms, the excitation energy monotonously increases. These differences can be assigned to the change in JNN and D⊥,NN, as supported by the perturbation theory model calculations of the spectra shown in Fig. 3e, which nicely reproduce the data. Again, we simply used the NN interactions extracted from the according pair with d = 3a, but additionally took into account the next nearest neighbor (NNN) interactions extracted from the couplings of a pair with d = 6a25 (see the values given in the caption of Fig. 3). As corroborated by ED, the underlying spin state of the chain stabilized in Bz = 3 T (see Fig. 3b) is again strongly non-collinear.

Fig. 3
figure 3

Non-collinear spin state in the Fe4 chain with dNN = 3a. a SPSTM topograph of the Fe4 chain measured at Bz = 3 T (Vs = −6 mV, Istab = 0.5 nA). The white scale bar has a length of 1 nm. b Calculated spin-expectation values at Bz = 3 T from ED for each atom in the Fe4 chain, using the parameters given in Table 1, and JNN = −60 μeV, D⊥,NN = −50 μeV, JNNN = +15 μeV, and D⊥,NNN = −20 μeV. c, d ISTS spectra taken with a magnetic tip (SP-ISTS, c) and corresponding colorplots (d) measured on all four atoms in the Fe4 chain in a magnetic field up to Bz = 7 T. (Vstab = −6 mV, Istab = 3 nA, Vmod = 40 μV). e Calculated ISTS from the perturbation theory model using the same parameters as in (b)

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Moreover, the character of the two non-collinear spin states of the 3a- and 4a-chains is obviously different (cf. Fig. 2a–3b). While the spins of the 1st and 3rd (2nd and 4th) atom in the 4a-chain are tilted to the left (right) (cf. Fig. 2a), this tilt is opposite for the 3a-chain (cf. Fig. 3b). This is a result of the sign change of D⊥,NN between the two cases, which will lead to a different rotational sense if a pure spin-spiral state was stabilized. We finally allude to a subtle difference between the experimental (Fig. 3d) and calculated (Fig. 3e) ISTS. While the conductance levels at positive and negative bias above the excitation threshold are the same in the calculations, there is a systematic difference in the experimental data. For Bz < 2 T the positive-bias conductance is lower (higher) for the 1st and 3rd (2nd and 4th) atom, and this contrast is reversed at Bz = 3 T. Considering the alternating tilts of the spins in the ||-direction (cf. Fig. 3b), we assign this asymmetry to a non-zero ||-component of the spin-polarization of this particular tip, which is reversing at Bz = 3 T. This conclusion is further supported by the alternating heights of neighboring atoms in the SPSTM image taken on this Fe4 chain (Fig. 3a). Note, that the spin-polarization is not taken into account in the calculation of the ISTS data.

Spin-spiral state in a 16 atom 3a chain

In order to stabilize the pure spin-spiral state we have built a 3a-chain with one end RKKY-coupled to a ferromagnetic Co stripe (Fig. 4). The Co stripe has a remanent out-of-plane magnetization, which can be used to stabilize the magnetization of atoms that are adsorbed on the Pt(111) at several nanometers lateral distance27. The first atom of the chain was put at a lateral distance of dCo−Fe ≈ 1.5 nm to the stripe, where the interaction showed strongest antiferromagnetic coupling27. Starting with this atom, an Fe16 chain with dNN = 3a aligned almost perpendicular to the rim of the Co stripe has been assembled (Fig. 4a). Since the RKKY-coupling to the stripe decays rather slowly with lateral distance, we expect possible residual couplings to the 2nd and 3rd chain atoms. However, for all other chain atoms the RKKY interaction to the Co stripe is already an order of magnitude smaller than the mutual RKKY interaction between the atoms within the chain (cp. 27. and 25). We therefore anticipate that a largely undisturbed spin-spiral can form on the chain. The SPSTM image of the chain (Fig. 4b, top panel) taken with a tip with sensitivity to the out-of-plane component of the magnetization (Supplementary Note 5 and Supplementary Fig. 14) reveals an alternating height of neighboring atoms in some areas (cf. atoms 2 to 4, 7 to 9, and 11 to 14) and almost vanishing contrast in between (cf. atoms 5 to 6 and 9 to 11). To prove the magnetic origin of these height differences, the out-of-plane magnetization of the Co stripe was reversed by ramping the field Bz to −1.7 T and back to zero, while the tip was retracted. Due to the RKKY-stabilization, this procedure reverses the magnetization of atom 1 in the chain, and thereby supposedly switches the chain from a state 0 to a state 1 with a reversed magnetization of all chain atoms. Please note that the magnetic state of the tip thereby remained unchanged (Supplementary Note 5 and Supplementary Fig. 14). Indeed, in the subsequently taken SPSTM image of state 1 (Fig. 4b, bottom panel) the alternating height contrast is reversed with respect to state 0, while the vanishing contrast regions remain unaffected. For a quantitative analysis of the magnetic contrast, we measure the height on top of each atom (i) in state 1 (hi,1) and 0 (hi,0) (after correcting the slope of the image by plane subtraction) and calculate the spin-asymmetry Δi by hi,0 − hi,1 divided by the sum hi,0 + hi,1, which is directly proportional to the spin-polarization of the atoms along the magnetic orientation of the tip30. In Fig. 4c, Δi reveals a long-range beating with a wavelength of about 10 atoms superimposed on the short-range antiferromagnetic component (see the gray points in Fig. 4c for which the antiferromagnetic component was removed by reversing the sign of all even numbered atoms). This strongly indicates the stabilization of an antiferromagnetic spin-spiral in the Fe16 chain.

Fig. 4
figure 4

Spin-spiral in Fe16 chain coupled to a Co stripe. a STM topograph of an Fe16 chain with dNN = 3a RKKY-coupled to a ferromagnetic Co stripe on the left. (Is = 0.5 nA, Vs = 6 mV). The white scale bar has a length of 2 nm. b SPSTM images of the two states 0 and 1 of the Fe16 chain stabilized by the two magnetization states of the Co stripe taken with out-of-plane sensitive tip (Bz = 0 T, Is = 0.5 nA, Vs = 6 mV). The magnetic state of the Co stripe was reversed between top and bottom image. c Spin-asymmetry Δi between states 1 and 0 of atom i of the Fe16 chain taken from the images in b (see text). For the gray points, the short-range antiferromagnetic component has been removed by reversing the sign of all even numbered atoms, such that the long range spin-spiral component is visible. The error bars were determined via error propagation from the standard deviation of each measured height (measured three times)

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Comparison to DMRG calculations

This finding is further corroborated in Fig. 5b by a comparison of the Fourier transform of the measured spin-asymmetry Δi with the z-component of the spin-structure factor \(\left\langle {\hat S_z\left( q \right)} \right\rangle\) obtained from a DMRG calculation for an isolated spin S = 5/2 chain with the first spin pinned along z (see 'Methods'). We used the same NN coupling parameters as for the ED and perturbation theory model calculations, which were extracted from the experimental investigation of pairs25. The experimental and theoretical curves show a very similar trend with the formation of a clear maximum at the wavevector q ≈ 7π/(8dNN) (see the dashed vertical line). This finally substantiates the formation of a spin-spiral in the bottom-up constructed Fe16 chain with a wavelength of λ = 2π/q ≈ 2.29dNN = 19 Å, which is shown in real space in Fig. 5a. The commensuration length of this spin-spiral resulting from λ is approximately 7 atoms.

Fig. 5
figure 5

Determination of the spin-spiral wavevector from experiment and DMRG. a Illustration of the spin-spiral realized in the 3a-chain using the DMRG calculated vectors \(( {\langle {\hat S_{||,i}} \rangle ,\langle {\hat S_{z,i}} \rangle } )\) with the length set to unity. b Comparison of the Fourier transform of the experimental spin-asymmetry (orange dots) taken from Fig. 4 with the spin-structure factor \(\langle {\hat S_z( q )} \rangle\) obtained from the DMRG calculation of a dNN = 3a (N = 16, orange diamonds) and a dNN = 4a (N = 16, cyan diamonds) chain using the NN interactions experimentally determined from pairs (dNN = 3a: JNN = −60 μeV, D⊥,NN = −50 μeV; dNN = 4a: JNN = −25 μeV, D⊥,NN = +30 μeV; remaining parameters given in Table 1). The curves are normalized to the area. The colored arrows indicate the calculated wavevectors q and rotational sense (blue: positive, red: negative) of spin-spirals in chains (N = 128) with dNN given in Å by the numbers besides the arrows. They were determined from the maxima of the calculated \(\langle {\hat S_z( q )} \rangle\) using the experimental pair-interactions from ref. 25 (slightly adjusted JNN for dNN = 4a, see Supplementary Fig. 16). c \(\langle {\hat S_z( q )} \rangle\) as a function of J with a constant D⊥ = −50 μeV (N = 128). d \(\langle {\hat S_z( q )} \rangle\) as a function of D⊥ with a constant J = −60 μeV (N = 128). The experimental parameters of the 3a-chain are indicated by the dashed gray line

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Discussion

We finally discuss the tunability of the spin-spiral period via the exchange constant ratio r = |D⊥,NN|/|JNN|, which can be tailored by the distance dependent RKKY-interaction as shown above. Figure 5c, d illustrates the DMRG calculation of \(\left\langle {\hat S_z\left( q \right)} \right\rangle\) for long chains (N = 128) as a function of r while keeping D⊥,NN (c) or JNN (d) constant, revealing how q depends on r. q approaches π/(2dNN) for large r, corresponding to a spin-spiral with perpendicular orientations of neighboring spins. For r ≈ 1, which corresponds to the case experimentally realized here, the peak gets broader due to the competition of Heisenberg and DM interactions, and the magnetic anisotropy, but is still well defined, indicating that the wavevector of the antiferromagnetic (JNN < 0) spin-spiral can in principle be adjusted within the range π/(2dNN) < q < π/dNN. For the other experimentally realized 4a-chain, the DMRG calculated \(\langle {\hat S_z( q )} \rangle\) (see cyan curve in Fig. 5b) predicts a maximum at q ≈ 104π/(128dNN) corresponding to an antiferromagnetic spin-spiral with a larger wavelength of λ ≈ 27 Å and opposite rotational sense as compared to the 3a-chain.

Summarizing, we demonstrated that the rotational sense and the period of spin-spirals realized in bottom-up constructed chains of RKKY-coupled transition-metal atoms on a high-Z substrate can be tuned by the NN distance of the chain atoms. Building chains with other experimentally accessible NN distances will enable to further adjust JNN and D⊥,NN25, including ferromagnetic Heisenberg interaction. Our DMRG calculations of according chains using the experimentally determined couplings25 predict that q (see the colored arrows in Fig. 5b, Supplementary Note 7 with Supplementary Fig. 16 and Supplementary Table 1) can thereby be varied from π/(4dNN) to 114π/(128dNN), corresponding to a large variation in the wavelength from λ ≈ 17 Å to 133 Å. This tunability of non-collinear spin states could be very important for the manipulation of Majorana end modes of such chains assembled on a superconducting substrate12,13.

Methods

Experimental procedures

All experiments were performed in a home-built STM facility31 at a temperature of 0.3 K. A magnetic field of up to Bz = 12 T was applied perpendicular to the sample surface. The Pt(111) sample was cleaned in situ by subsequent sputter-and-anneal cycles as well as O2 annealing and high temperature flashes25,26. Finally the sample was flashed to 1000 °C for 1 min. In order to prepare the Co stripes, a fraction of a monolayer Co was deposited from an electron-beam heated rod during cooling down the sample to room temperature after the final flash, resulting in the decoration of the Pt step edges and terraces with one atomic layer high Co stripes and islands, respectively27. For the deposition of Fe atoms, about 1% of a monolayer Fe was evaporated onto the cold sample kept at \(T \lesssim 10\,{\mathrm{K}}\).

STM topographs were recorded in the constant-current mode of the STM at a stabilization current Istab and voltage Vs applied to the sample. For the assembly of the Fe chains, the Fe atoms were manipulated using lateral tip-induced atom manipulation32 with a current of Istab ≈ 40 nA and a voltage of Vs = 2 mV. We used flashed and Cr coated (≈50 ML) tungsten tips which were prepared as described before32. In order to achieve magnetic contrast in the SPSTM images or in SP-ISTS with a sensitivity to the out-of-plane component of the sample magnetization (along z), the tip was dipped up to 500 pm into a Co layer and several single Fe atoms were picked up, until a sufficient contrast was observed on the remanently out-of-plane magnetized Co stripes or islands (see Supplementary Fig. 14). For non-magnetic ISTS measurements, voltage pulses were applied to the tip and/or it was gently dipped into the Pt substrate until the magnetic contrast vanished and a spectroscopically flat Pt substrate spectrum within the voltage range used for ISTS was obtained32. ISTS was performed by adding a modulation voltage Vmod (rms) of frequency fmod = 4.142 kHz to Vs. After stabilizing the tip at Istab and switching the feedback off, the bias Vs was ramped from the start to the end voltage of the desired spectroscopy range. Simultaneously, the differential conductance dI/dV was recorded via a lock-in amplifier. In order to eliminate tip related features, all presented spectra of atoms were normalized by dividing by a spectrum taken with the identical tip on the bare Pt substrate.

Perturbation theory model

The experimental ISTS data was modeled by employing the perturbation theory code (v0.999) by Ternes29. The model is based on the following effective spin Hamiltonian

$$\begin{array}{*{20}{l}} {\hat {\cal H}} \hfill & = \hfill & {\hat {\cal H}_{{\mathrm{Zeeman}}} + \hat {\cal H}_{{\mathrm{anisotropy}}} + \hat {\cal H}_{{\mathrm{Heisenberg}}} + \hat {\cal H}_{{\mathrm{DM}}}} \hfill \\ {} \hfill & = \hfill & { - \mathop {\sum}\limits_i^N g\mu _{\mathrm{B}}{\mathbf{B}} \cdot {\hat{\mathbf S}}_i + \mathop {\sum}\limits_i^N K\left( {\hat S_{i,z}} \right)^{\mathrm{2}}} \hfill \\ {} \hfill & {} \hfill & { - \mathop {\sum}\limits_{i \ne j}^N J_{ij}{\kern 1pt} {\hat{\mathbf S}}_i \cdot {\hat{\mathbf S}}_j - \mathop {\sum}\limits_{i \ne j}^N {\mathbf{D}}_{ij}\left( {{\hat{\mathbf S}}_{\mathrm{i}} \times {\hat{\mathbf S}}_{\mathrm{j}}} \right),} \hfill \end{array}$$
(1)

which includes the Zeeman energy and the magnetic anisotropy of all atoms in the chain, quantified by the Landé g-factor g, Bohr magneton μB, and axial magnetic anisotropy constant K, respectively, as well as the mutual Heisenberg and the DM interactions between atoms i and j quantified by the Heisenberg exchange constant Jij and DM vector Dij, respectively. Note, that no third order effects had to be considered to nicely reproduce the data. All model parameters which are not specified in the text were kept constant and are given in Table 1 (same parameters as used in Ref.25).

Table 1 Simulation parameters
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ED and DMRG calculations

To solve the model Equation 1 for the S = 5/2 chains up to N = 4, we used ED. For the longer chains (N = 16 and N = 128) we used our DMRG code33. Note, that the Hilbert space of 16 atoms becomes prohibitively large since we are dealing with a spin S = 5/2. The DMRG method exploits the relatively small entanglement between parts of the system to compress the wavefunction into a matrix product state which is controlled by the bond dimension χ (a measure for the amount of variational parameters), and yields essentially exact results. The summations are restricted to NN.

Our algorithm adaptively chooses the bond dimension χ using the energy variance \(\varepsilon = \left| {\left\langle {\hat {\cal H}^2} \right\rangle - \left\langle {\hat {\cal H}} \right\rangle }^2 \right|/N\) as a convergence criterion for the ground state (in addition to the ground-state energy), where \(\varepsilon < 10^{ - 6}\) can be typically achieved with χ ~ 50–100 for a system of N = 128 sites. Note, that a well-defined peak in \(\left\langle {\hat S_z\left( q \right)} \right\rangle\) is already obtained for N = 16, which does not change significantly for N = 128 sites (cf. Supplementary Fig. 15 and Supplementary Note 6).

Note, that the DM term in Equation 1 breaks the SU(2) symmetry of the Heisenberg model, such that all the components \(\left\langle {\hat S_{||,i}} \right\rangle\), \(\left\langle {\hat S_{ \bot ,i}} \right\rangle\) and \(\left\langle {\hat S_{z,i}} \right\rangle\) are in general nonvanishing and the Hamiltonian is complex. To determine the wavevector q of the spiral in a chain of length N, we calculate \(\left| {\left\langle {\hat S_\alpha } \right\rangle \left( q \right)} \right| = 1/N\left| {\mathop {\sum}\nolimits_{n = 0}^{N - 1} \left\langle {\hat S_{\alpha ,n}} \right\rangle e^{{\mathrm{i}}qn}} \right|\) with \(\alpha = ||, \bot ,z\) and q taking the values \(q_n = 2\pi n/N\), where \(n = 0,1, \ldots ,N - 1\). For \(D_{||} = 0\) and \(D_z = 0\) the Hamiltonian becomes real, which speeds up the calculations considerably. Furthermore, the ground state can be chosen real as well, meaning that the perpendicular component vanishes: \(\left\langle {\hat S_{ \bot ,i}} \right\rangle = {\mathrm{Im}}\left\langle {\hat S_i^ + } \right\rangle = 0\).

Data availability

The authors declare that all relevant data are included in the paper and its Supplementary Information files. Additional data are available from the corresponding author upon reasonable request. Some references concerning data analysis and display are refs.34,35.