A Supplementary material

# Interplay between classical magnetic moments and superconductivity in quantum one-dimensional conductors: toward a self-sustained topological Majorana phase

## Abstract

We study a one-dimensional (1D) interacting electronic liquid coupled to a 1D array of classical magnetic moments and to a superconductor. We show that at low energy and temperature the magnetic moments and the electrons become strongly entangled and that a magnetic spiral structure emerges. For strong enough coupling between the electrons and magnetic moments, the 1D electronic liquid is driven into a topological superconducting phase supporting Majorana fermions without any fine-tuning of external parameters. Our analysis applies at low enough temperature to a quantum wire in proximity of a superconductor when the hyperfine interaction between electrons and nuclear spins is taken into account, or to a chain of magnetic adatoms adsorbed on a superconducting surface.

###### pacs:
75.75.-c,73.63.Nm,74.45.+c,71.10.Pm

Introduction. The interaction between localized magnetic moments and delocalized electrons contains the essential physics of many modern condensed matter systems. It is on the basis of nuclear magnets (1), heavy fermion materials of the Kondo-lattice type (2) or ferromagnetic semiconductors (3); (4); (5); (6). It often leads to new intricate physics and rich phases diagrams already when the magnetic moments behave classically. Electron systems interacting with nuclear spins through the hyperfine interaction or magnetic adatoms with large magnetic moments arranged in some array on a metallic surface enter into this class.

In 1D, the interactions between the nuclear spins and electrons lead to dramatic effects: below a cross-over temperature , a new exotic phase of matter in which the nuclear magnets are strongly tied to the electrons naturally emerges (7); (8). In this phase, the nuclear spins form a helical magnetic structure caused by the effective Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction (9) mediated by the electron system. The feedback of this nuclear Overhauser field on the electron system entirely restructures the electronic states in that it opens a gap in one half of the elementary low-energy modes. The remaining electronic degrees of freedom remain gapless and form a quasi helical Luttinger liquid with strong analogies (10) with the edge states of the 2D quantum spin Hall effect (11); (12). Due to the mutual feedback this order of strongly coupled electrons and nuclear spins is stable below a temperature , and electron-electron interactions substantially enhance the stability (8). Recent transport measurements in cleaved edge overgrowth GaAs quantum wires found a reduction of the conductance by a factor of two below independently of the density or applied magnetic field, consistent with this theory (13).

We stress that the mechanism behind this emergent helical structure is general, the essential ingredient being the RKKY interaction. Therefore, the same mechanism can apply if the nuclear spins are replaced by classical magnetic moments forming a 1D lattice (not necessarily a regular one) such as magnetic adatoms on top of a metallic surface (14).

When a finite-sized helical liquid is put in proximity of a s-wave superconductor, Majorana states can emerge at both ends (15) (see Fig. 1). This is the case for a quantum wire in presence of spin-orbit coupling and a Zeeman term (16); (17) where some possible signatures of Majorana fermion physics have been recently reported experimentally (18); (19); (20). The helical liquid, up to a gauge transformation (21), can also be obtained by coupling electrons to a spiral magnetic field (like the intrinsic nuclear Overhauser field (8)), or by manufacturing an external rotating magnetic field (22). They can appear in rare-earth compounds exhibiting coexisting helical magnetism and superconductivity (23), or emerge by arranging magnetic adatoms in 1D arrays on the surface of a superconductor (24); (25).

Since induced or intrinsic superconductivity entirely restructures the electron system, it is not a priori obvious whether the helical entangled states remain stable. In this work we provide such an investigation, carefully taking also into account electron-electron interactions and disorder. The underlying physics is the following: The RKKY interaction between the local moments has a strong component (with the Fermi wave vector), such that the local moments tend to be opposite at length , with . The RKKY energy can thus be minimized if the local moments form a spiral (7); (8). This spiral acts back on the electron as the effective spiral Zeeman field required, together with the induced superconductivity, to drive the system into the topological superconductivity phase. Yet the opening of the superconducting gap as well as renormalizations by interactions and disorder modify again the RKKY interaction. Taking this self-consistently into account, however, we demonstrate that a self-stabilizing topological phase supporting Majorana bound states naturally emerges. We emphasize that this topological phase requires no fine-tuning. It is an intrinsic effect and constitutes the thermodynamic ground state of the system.

Model Hamiltonian. We consider a 1D conduction electron liquid in proximity of a s-wave superconductor. The 1D electrons are further coupled to an array of magnetic moments. Such a generic system is of the 1D Kondo-lattice type and described by the Hamiltonian

 H=Hel+∑xiA0Sxi⋅Ixi. (1)

The first term, describes the electron system in the induced or intrinsic superconducting state and also includes electron-electron interactions. The second term describes the coupling between the electron spins and the magnetic moments , weighted by a coupling constant . The positions of the magnetic moments lie on a 1D chain. We assume that the distances between neighboring moments fulfill . Yet it is not required that the lie on a regular 1D lattice. The operators are a tight binding representation of the electron spin operator in a region of dimension centered about . The quantities are either very large spins or composites of a large number of individual magnetic moments locked to a parallel configuration in some small volume at (such as nuclear spins in the transverse section of a quantum wire (8)). The former case is treated with . We assume that is normalized with respect to such that represents the interaction constant between an electron and one of the individual moments of . It has been shown numerically in (25) and further justified in (26) that this simple model captures qualitatively the behavior of a regular array of magnetic adatoms adsorbed on a superconducting surface (see also (27) for the 2D case).

The magnetic coupling also provides an effective RKKY (9) interaction and dynamics for the magnetic moments. This interaction is carried over the response of the electron system to magnetic perturbations and consists in an almost instantaneous long-ranged Heisenberg interaction between the magnetic moments , where is the electron spin susceptibility and the lattice spacing between the moments . Note that generally can be anisotropic.

To analyze the Hamiltonian (1), we resort to a Born-Oppenheimer decoupling as in (8) which relies on the fact that the magnetic moments have a much slower dynamics than the electrons. Since the large moments allow a (quasi)classical description, the terms act like a local quasi-static Zeeman field on the electron spins . The resulting effective theory is, therefore, expressed by the pair of Hamiltonians (28); (7); (8)

 Heffel =Hel+∑xiBeff(xi)⋅Sxi, (2) Heffm =∑xi,xjJ(xi−xj)Ixi⋅Ixj. (3)

While acts only on electrons and only on the magnetic moments, we stress that both Hamiltonians are strongly coupled since depends on the state of the magnetic moment system, and on the electron state. A characterization of the physics described by Eqs. (2) and (3) must rely on a fully self-consistent approach, similar to the treatment of Refs. (28); (7); (8).

Susceptibility in the non-interacting case. Without superconductivity, the magnetic moments order in a spiral at low temperature due to the self-organization of the coupled systems (7); (8), leading to an effective spiral field of amplitude and spatial period . Since the spiral field rotates in a plane defining, for instance, the spin plane, the RKKY interaction is governed by the transverse spin susceptibility, defined as (8) , for momenta with , , and destroying an electron of spin at position ( throughout the paper).

We first neglect electron interactions. Without superconductivity, is then equivalent, by a gauge transformation (21), to a one-channel conducting wire with spin-orbit interaction in a magnetic field. The resulting band structure consists of two spin-mixing bands with dispersions , where is the single-particle dispersion and the Fermi velocity. At both bands are separated by a gap of amplitude , such that when the Fermi level lies within this gap, only band remains conducting and forms a quasi-helical conductor with roughly opposite spins at opposite Fermi points (when undoing the gauge transformation ). In the presence of superconductivity, the proximity induced gap is expanded in the eigenbasis and results in two triplet pairing terms within the ’’ and ’’ bands, , and a singlet term that mixes both bands. A similar expansion holds for the transverse electronic susceptibility, which we write as . Since both bands are separated by a gap, is dominated by the gapless ’’ band such that . The analytical form of is derived in the supplementary material. It has a deep minimum of the normal state susceptibility at (see Fig. 2). For , we find (see supplement) that it is well described by

 χ⊥−−(q)≈−12πvF[ln(2vFkFΔ)−1κarctanh(κ)], (4)

with . Since maintains the deep minimum at characteristic of the normal state, this confirms on a mean field level the consistency of the assumption of the spiral effective field, while stability under fluctuations will be considered below. We also note that the minimum can only be approximated by a Lorentzian of half-width for .

Susceptibility in the interacting case. In 1D conductors, electron-electron interactions often are detrimental. To include them, we first linearize the spectrum in the absence of the both the proximity induced gap and of the spiral magnetic field. Our treatment is therefore valid only at low energy. We then use the standard approach of bosonizing the low-energy electronic Hamiltonian (29) and incorporate the pairing term and the spiral magnetic field in the bosonized Hamiltonian. The resulting low-energy model becomes with

 H0ν=c,s=∫dx2π[vνKν(∇ϕν(r))2+vνKν(∇θν(r))2], (5)
 HB=∫dx2πaBeffcos(√2(ϕc+θs)), (6)

and

 HP=∫dx2πaΔ[sin(√2(θc+ϕs))+sin(√2(θc−ϕs))]. (7)

In Eq. (5), are the Luttinger Liquid parameters for the charge and spin density fluctuations, is the charge and spin density wave velocities, and is a short distance cutoff. The non-interacting case is described by and . Repulsive electron-electron interactions lead to . If the spin SU(2) symmetry is preserved otherwise . Strictly speaking, this symmetry is broken here by , yet only weakly such that (30); (31). The term has a scaling dimension and the terms the scaling dimension . In the non-interacting case both scaling dimensions are equal to 1 indicating the absence of any renormalization. For and , both the pairing term and effective magnetic field terms are relevant, however, dominates.

To make progress, we assume that (for strong repulsive interactions, one can relax this condition). This ensures that conduction modes unaffected by the opening of the gap are helical, and that the induced superconductivity in this helical conductor is topological (16); (17); (32). We proceed in a 2-step renormalization group (RG) analysis. The coupling reaches the strong coupling first, which opens a gap in the spectrum by pinning the field to a constant (8). This physics is best accessed by introducing the fields related to the original fields by a unitary transform (see the supplemental material), with the fields corresponding to the previous bands of the non-interacting case. In this new basis, the effective Zeeman term gives a simple dominant sine-Gordon term . Hence the gap opens only in the ’’ sector, and from the strong coupling limit we can estimate it to be , coinciding with in the non-interacting limit and increasing as a power law otherwise. After projecting out the gapped sector, the remaining Hamiltonian takes the simpler form

 H−=H−0+∫dx2πaBΔ′sin(2θ−), (8)

where is given by Eq. (5) with , . Note that the effective bandwidth in this resulting ’’ sector is now determined by such that replaces the UV cutoff . The pairing term is now replaced by the renormalized value . The Hamiltonian (8) is nothing but the bosonized version of a spinless electronic chain in proximity of a superconductor. which can can be studied by our second step RG procedure. As shown in Ref. (32), strong electron interactions further renormalize the value of , usually to . However, by noticing that also grows under the RG, we can refermionize the system at a length scale defined by (32). The refermionized Hamiltonian is nothing but a non-interacting 1D triplet superconductor with a renormalized pairing gap . The susceptibility is, therefore, given by Eq. (4) with the replacements by and by . Repulsive electron interactions therefore increase while decreasing .

Stability analysis. In the previous analysis, we have found that the RKKY interaction remains strongly peaked at in the presence of the superconductivity, such that the magnetic moments tend to form the spiral order. Through the opening of the gap in the electron system by this self-organization, this () spiral is strongly favored energetically. To investigate its stability, we perform a magnon analysis of similar to Ref. (8) analyzing the low-energy fluctuations of the moments . If the are regularly spaced and composed by individual moments of size that are ferromagnetically locked to each other, the magnetization per site (normalized to ) reads (8) , with the temperature, the Boltzmann constant, the number of sites of the chain, and the magnon dispersion with . The order is stable as long as , and we denote the crossover temperature at which the order disappears by . For an infinite chain, and order cannot be stable. For finite systems of length , this lowest mode is cut off, leading to . Contrarily to the normal state case (8), the RKKY interaction is determined by but not by (provided that ). Hence remains invariant under temperature changes, and becomes a function of the remaining scales characterizing , notably the cutoff scale , and the mean-field scale .

At very short system sizes such that only very few values fall into the dip of the RKKY interaction, we have for almost all terms in the summation, and we obtain , which leads to a vanishing at temperatures exceeding

 kBT∗=2I2|J2kF|=A20I2aI[ln(2v−kF/Δ∗)−1]/πv−. (9)

Quite remarkably, this result remains accurate up to large system lengths in which the approximation is no longer valid for most . Indeed, let us consider the limit, in which the sum to calculate is dominated by the term, . The associated temperature scale is . The length effects become influential only when , i.e., at the length . For systems with , the crossover temperature lies between and , and decays with not faster than . However, since usually , remains independent far into the regime required for obtaining isolated Majorana bound states. We notice that the unrenormalized in Eq. (9) must not to be confused with the renormalized seen by the electrons. The topological phase requires , yet within this situation both and are possible.

Disorder. We have assumed so far that the system is free of disorder. Since the RKKY magnetic interaction between the magnetic moments is mainly dominated by , this remains the case even if the magnetic adatoms do not form a regular array. More problematic is the disorder in the electronic part. Semiconducting wires made out of GaAs or InAs are not free of disorder. A chain of adatoms on the surface also naturally introduces potential scattering terms. We introduce some quenched disorder in where is the electron density and encodes Gaussian disorder characterized by with the disorder strength. Technically, disorder introduces extra terms proportional to in the bosonized Hamiltonian (29). After disorder averaging, disorder terms generally competes with and . As before, we assume again to be larger than . By comparing the scaling dimensions of the disorder and spiral magnetic field terms (33), one finds that when with (we assumed here), disorder dominates and ultimately leads to localization. However, when is the largest energy scale, we can project the disorder term into the ’’ helical state which renders the disorder term, being non-magnetic, inoperant at lowest order in (34); (35). When all scales are of the same order, this is a difficult problem which goes beyond the scope of the present analysis. Therefore, the disorder energy scale much be at least smaller than the effective Zeeman field to observe the effect.

Application. Let us first consider semiconducting GaAs or InAs wires. We take m, and , and a proximity induced gap meV (18). The hyperfine coupling for GaAs is eV and . For , we obtain meV and mK, too small to be observed. More interesting is InAs which has and eV (36). For , we obtain mK and meV (which guaranties a topological SC phase) while for , mK and mK which is within experimental reach. For Co atoms on a Nb surface (25), we take , Å, meV and meV. For , assuming a topological phase, i.e., , we find that . Therefore a local magnetic coupling on the order of meV, which is actually in the right range for magnetic exchange interactions (14), pushes the system in the topological phase.

Conclusion. We have shown that in 1D a strong entanglement between magnetic moments and the electrons leads naturally at low temperature to a magnetic spiral structure. Combined with a proximity induced superconducting gap, this structure can drive the system into a topological superconducting phase supporting Majorana fermions. A fine-tuning of external parameters is not required. This scenario applies to semiconducting wires with nuclear spins or to a chain of magnetic adatoms on top of a superconductor surface. We also demonstrated that moderate electron interactions help stabilizing the topological phase.

Acknowledgments. We thank M. Franz, D. Loss and P. Wahl for helpful discussions and correspondences. PS would like to thank the department of physics in UBC (Vancouver) for its kind hospitality during the final stage of this work. BB acknowledges the support by the EU-FP7 project SE2ND [271554].

Note added. During the final completion of this manuscript, we became aware of Ref. (37); (38) which has overlaps with the present work in the non-interacting limit.

## Appendix A Supplementary material

### a.1 Susceptibility in the non-interacting helical superconductor

In the non-interacting limit, the topological superconductor is described by an effective -wave BCS theory with the single-particle Hamiltonian

 H=∑k′(c†k↑,c−k↓)(ξkΔΔ−ξ−k)(ck↑c†−k↓), (10)

with the single electron energies measured from the Fermi surface and the electron operators for momentum and spin . We assume that the helical conductor is characterized by right moving modes with and left moving modes with . The symbol marks a restriction of the momentum sum to positive values .

A straightforward diagonalization of this model leads to the standard expressions

 H=E0+∑k′Ek(α†kαk+β†kβk), (11)

with the ground state energy, , and the quasiparticle operators related to the electron operators by

 ck↑ =ukαk+vkβ†k, (12) c†−k↓ =vkαk−ukβ†k, (13)

for the wave functions

 uk=√1+ξk/Ek/√2,  vk=√1−ξk/Ek/√2. (14)

The field operators for the left and right moving electrons with opposite spins are then expressed as

 ψ↑(x) =1√N∑k′eikx[ukαk+vkβ†k], (15) ψ↓(x) =1√N∑k′e−ikx[vkα†k−ukβk], (16)

for the number of sites. We have defined the transverse electron spin susceptibility as

 χ⊥(q) =−ia2I∫∞0dt∫dx eiqx ⟨[S+(x,t),S−(0,0)]⟩ =1NaI∑k′2ukuq−kvkuq−k−u2kv2q−k−v2ku2q−kEk+Eq−k. (17)

In the latter sum, as well as are restricted to positive values close to , which we can capture by introducing a momentum cutoff such that with . The summation in Eq. (17) is therefore restricted to . The latter sum leads to a well marked minimum of at . In the regime and for , we can let . We furthermore go over to continuous , which allows us to straightforwardly integrate Eq. (17), and we obtain with and

 χ⊥(q)=−18πvF{ln[ηkF+√Δ2+η2kF−ηkF+√Δ2+η2kF] +ln[ηkF+η¯q+√Δ2+(ηkF+η¯q)2−ηkF−η¯q+√Δ2+(ηkF+η¯q)2]+2√4Δ2+η2¯qη¯q ×ln[2Δ2−η¯qηkF+√Δ2+η2kF√4Δ2+η2¯q2Δ2+η¯qηkF+¯q+√Δ2+η2kF+¯q√4Δ2+η2¯q]}. (18)

This function has a minimum at given by

 χ⊥(2kF)= (19) −14πvF{ln[ηkF+√Δ2+η2kF−ηkF+√Δ2+η2kF]+2ηkFΔ+√Δ2+η2kF}. (20)

In the regime and for such that we can expand these results to the more speaking expressions

 χ⊥(q)≈−12πvF[ln(2vFkFΔ)−1κarctanh(κ)], (21)

where , and

 χ⊥(2kF)≈−12πvF{ln[2ηkFΔ]−1}. (22)

In the limit a further expansion of the leads to a Lorentzian shaped susceptibility

 χ⊥(q)≈¯χ−13πvF2Δ24Δ2+η2¯q, (23)

with .

### a.2 Bosonized form of Hamiltonians

The bosonization of the Hamiltonians follows the standard approach (29). The interacting electron Hamiltonian (in the absence of field and pairing amplitude) is rewritten in terms of the boson fields for charge () and spin () degrees of freedom as

 H0ν=∫dx2π[vνKν(∇ϕν)2+vνKν(∇θν)2], (24)

for velocities and Luttinger liquid parameters as described in the main text. In terms of the boson fields, the electron operators for right () and left () moving modes and spin is given by , where is the short distance cutoff, and the Klein factors taking into account the fermion statistics and satisfying .

For a spiral magnetic field , we use the bosonized form of the electron operators to rewrite the Zeeman coupling as (8)

 B(x)⋅S(x)=B4πa[ cos(√2(ϕc+θs)) + cos(√2(ϕc−θs)−4kFx)]. (25)

We have dropped eventually the Klein factors as they do not have any further influence. As long as is not commensurate with the lattice the second term in Eq. (25) oscillates quickly in space and its contribution to the Hamiltonian is irrelevant. The first term, however, is relevant and provides Eq. (6) in the main text.

In the same way, the pairing amplitude is bosonized for terms and leads to Eq. (7) in the main text.

### a.3 Change of basis

We introduce the new boson fields , , and , are defined as follows (8),

 ϕc = √KcK[√Kcϕ+−1√Ksϕ−] ϕs = −√KsK[1√Ksθ++√Kcθ−] θc = −1√KcK[√Kcθ+−1√Ksθ−] θs = 1√KsK[1√Ksϕ++√Kcϕ−], (26)

where and the fields obey the standard commutation relations with .

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