Abstract
Topological phases of matter are among the most intriguing research directions in Condensed Matter Physics. It is known that superconductivity induced on a topological insulator’s surface can lead to exotic Majorana modes, the main ingredient of many proposed quantum computation schemes. In this context, the ironbased high critical temperature superconductors are a promising platform to host such an exotic phenomenon in real condensedmatter compounds. The Coulomb interaction is commonly believed to be vital for the magnetic and superconducting properties of these systems. This work bridges these two perspectives and shows that the Coulomb interaction can also drive a canonical superconductor with orbital degrees of freedom into the topological state. Namely, we show that above a critical value of the Hubbard interaction the system simultaneously develops spiral spin order, a highly unusual triplet amplitude in superconductivity, and, remarkably, Majorana fermions at the edges of the system.
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Introduction
Topologically protected Majorana fermions—the elusive particles which are their own antiparticles—are exciting because of their potential importance in faultresistant quantum computation. From the experimental perspective, heterostructurebased setups were proposed as the main candidates to host the Majorana zeroenergy modes (MZM). For example, the topologically protected gapless surface states of topological insulators can be promoted to MZM by the proximityinduced pairing caused by an underlying superconducting (SC) substrate^{1}. However, the large spin–orbit coupling required to split the doubly degenerated bands due to the electronic spins, renders such a setup hard to engineer. Another group of proposals utilizes magnetic atoms (e.g., Gd, Cr, or Fe) arranged in a chain structure on a BCS superconductor^{2,3,4,5,6,7,8,9,10,11,12}. These important efforts have shown that creating MZM in real condensedmatter compounds is challenging and only rare examples are currently available.
Interestingly, a series of recent works have shown that doped high critical temperature ironbased superconductor Fe(Se,Te) can host MZM^{13,14,15,16,17}. Although the electron–electron interaction is believed to be relevant for the pairing, its role in the stabilization of MZM is unknown. In fact, in most theoretical proposals to realize MZM, these zeroenergy modes are a consequence of specific features in the noninteracting band structure, with the electron–electron interaction playing only a secondary role (and often even destabilizing the MZM)^{18,19}. By contrast, here we will show that a SC system with orbital degrees of freedom can be driven into a topologically nontrivial phase hosting MZM via increasing Hubbard interactions; see illustrative sketch in Fig. 1a. We will focus on a generic model with coexisting wide and narrow energy bands, relevant to lowdimensional ironbased materials^{20}. It was previously shown^{21,22,23} that the multiorbital Hubbard model can accurately capture static and dynamical properties of iron selenides, especially the blockmagnetic order^{24} of the 123 family AFe_{2}X_{3} of ironbased ladders (with A alkali metals and X chalcogenides). For example, the three and twoorbital Hubbard model on a onedimensional (1D) lattice^{23,25} successfully reproduces the inelastic neutron scattering spin spectrum, with nontrivial optical and acoustic modes. The aforementioned models exhibit^{21,26} the orbitalselective Mott phase (OSMP), with coexistent Mottlocalized electrons in one orbital and itinerant electrons in the remaining orbitals. The system is then in an exotic state with simultaneously metallic and insulating properties. Furthermore, the localized orbitals have vanishing charge fluctuations, simplifying the description^{26} into an OSMP effective model, i.e. the generalized Kondo–Heisenberg model (gKH)
The first three terms in the above Hamiltonian describe the itinerant electrons: \({c}_{\ell ,\sigma }^{\dagger }\) (\({c}_{\ell ,\sigma }\)) creates (destroys) an electron with spin projection σ = {↑, ↓} at site ℓ = {1, …, L}, t_{i} is their hopping amplitude, U is the repulsive Hubbard interaction, and μ = ϵ_{F} is the Fermi energy set by the density of itinerant electrons \(\overline{n}={\sum }_{\ell }({n}_{\ell ,\uparrow }+{n}_{\ell ,\downarrow })/L\).
The double occupancy of the localized orbital can be eliminated by the Schrieffer–Wolff transformation and the remaining degrees of freedom, the localized spins S_{ℓ} in the above model, interact with one another via a Heisenberg term with spinexchange \(K=4{t}_{{\rm{l}}}^{2}/U\) [t_{l} is the hopping amplitude within the localized band]. Finally, J_{H} stands for the onsite interorbital Hund interaction, coupling the spins of the localized and itinerant electrons, S_{ℓ} and s_{ℓ}, respectively. Figure 1b contains a sketch of the model. Here, we consider a 1D lattice and use t_{i} = 0.5 [eV] and t_{l} = 0.15 [eV], with kinetic energy bandwidth W = 2.1 [eV] as a unit of energy^{27}. Furthermore, to reduce the number of parameters in the model, we set J_{H}/U = 1/4, a value widely used when modeling iron superconductors. Systems with open boundary conditions are studied via the densitymatrix renormalization group (DMRG) method (see the “Methods” section).
The key ingredient in systems expected to host the MZM^{28} is the presence of an SC gap, modeled typically by an swave pairing field. Such a term represents the proximity effect^{29} induced on the magnetic system by an external swave superconductor. However, it should be noted that the SC proximity effect has to be considered with utmost care. For example, recent experimental investigations^{30} showed that although the interface between Nb (BCS swave SC) and Bi_{2}Se_{3} film (topological metal) leads to induced SC order, the same setup with (Bi_{1−x}Sb_{x})_{2}Se_{3} (another topological insulator) displays massive suppression of proximity pairing. On the other hand, in the class of systems studied here (lowdimensional OSMP ironbased materials), the pairing tendencies could arise from the intrinsic superconductivity of BaFe_{2}S_{3} and BaFe_{2}Se_{3} under pressure^{31,32,33} or doping^{22,34}.
In order to keep our discussion general, we will make minimal assumptions on the SC state, and consider only the simplest onsite pairing. The reader should consider it either as the intrinsic pairing tendencies of the ironbased SC material or as the pairing field induced by the proximity to an swave SC substrate, e.g., Pb or Nb. Independently of its origin, the SC in the 1D OSMP system studied here must be investigated beyond the 1D lattice since the quantum fluctuations would inevitably destroy any longrange order. Therefore, let us first consider the OSMP chain placed atop the center of a twodimensional (2D) BCS superconductor (see Fig. 2a for a sketch) and the total system described by the Hamiltonian
Here, \({\ell }^{\prime}\) represents the single site within the 2D BCS system H_{BCS} which is closest to the site ℓ in the OSMP chain, and \({a}_{i,\sigma },{a}_{i,\sigma }^{\dagger }\) stand for fermionic operators within the BCS superconductor (see the “Methods” section). The interaction between the subsystems [last term in Eq. (2)] is studied within the BCSlike decoupling scheme, where we introduce the pairing amplitudes \({\Delta }_{{\ell }^{\prime}}^{{\rm{BCS}}}=\langle {a}_{{\ell }^{\prime},\downarrow }{a}_{{\ell }^{\prime},\uparrow }\rangle\) and \({\Delta }_{\ell }^{{\rm{OSMP}}}=\langle {c}_{\ell ,\downarrow }{c}_{\ell ,\uparrow }\rangle\) for the BCS superconductor and the OSMP chain, respectively. In order to fully take into account the manybody nature of the OSMP system, we have developed a hybrid algorithm, the details of which are given in the “Methods” section. In summary: we iteratively solve the OSMP chain and the BCS system by means of the DMRG and the Bogoliubov–de Gennes (BdG) equations, respectively. This backandforth computational setup is costly but important to gain confidence in our result.
We monitor the landscapes of pairing fields in both systems and exemplary results are presented in Fig. 2b (for more results see Supplementary Note 1). Initially, only the BCS system has finite, spatially uniform, pairing amplitudes \({\Delta }_{{\ell }^{\prime}}^{{\rm{BCS}}}\) (left column in Fig. 2b), which are then used in the DMRG procedure applied to the OSMP Hamiltonian
where \({\Delta }_{\ell }=V{\Delta }_{{\ell }^{\prime}}^{{\rm{BCS}}}\). Next, the \({\Delta }_{\ell }^{{\rm{OSMP}}}\) set is calculated from DMRG and returned to the BdG equations relevant for the BCS system. The procedure is repeated until convergence is established. The results presented in Fig. 2b show that already after ~4 iterations the landscape of \({\Delta }_{{\ell }^{\prime}}^{{\rm{BCS}}}\) stabilizes to an interaction Udependent value. We found that the resulting amplitude \({\Delta }_{\ell }=V{\Delta }_{{\ell }^{\prime}}^{{\rm{BCS}}}\) is almost uniform within the OSMP chain. Furthermore, we have also confirmed that using extended swave pairing (creating pairs on nearestneighbor sites) does not influence our conclusions. Therefore, in the remainder of the paper, we use spatially uniform Δ_{ℓ} = Δ_{SC} in Eq. (3). Also, in order to emphasize the role of interaction, in the main text, we fix the pairing field to Δ_{SC}/W ≃ 0.5. The detailed Δ_{SC}dependence of our findings is discussed in Supplementary Note 1.
Results
Magnetism of OSMP
Previous work has shown that the OSMP (with Δ_{SC} = 0) has a rich magnetic phase diagram^{26}. (i) At small U the system is paramagnetic. (ii) At \(\overline{n}=1\) and \(\overline{n}=0\) standard antiferromagnetic (AFM) order develops, ↑↓↑↓, with total onsite magnetic moment 〈S^{2}〉 = S(S + 1) = 2 and 3/4, respectively. (iii) For \(0\;<\;\overline{n}\;<\;1\) and U ≫ W the system is a ferromagnet (FM) ↑↑↑↑. Interestingly, in the always challenging intermediate interaction regime \(U \sim {\mathcal{O}}(W)\) the AFM and FMtendencies (arising from superexchange and doubleexchange, respectively) compete and drive the system towards novel magnetic phases unique to multiorbital systems. (iv) For U ~ W, the system develops a socalled blockmagnetic order, consisting of FM blocks that are AFM coupled, e.g. ↑↑↓↓, as sketched in Fig. 1c. The block size appears controlled by the Fermi vector k_{F}, i.e., the propagation wavevector of the blockmagnetism is given by \({q}_{\max }=2{k}_{{\rm{F}}}\) (with \(2{k}_{{\rm{F}}}=\pi \overline{n}\) for the chain lattice geometry). In this work, we choose \(\overline{n}=0.5\) (adjusted via the chemical potential μ), as the relevant density for BaFe_{2}Se_{3 }π/2block magnetic order^{24}. Then, the latter order can be identified via the peak position of the static structure factor S(q) = 〈T_{−q} ⋅ T_{q}〉 at \({q}_{\max }=\pi /2\) or via a finite dimer order parameter D_{π/2} = ∑_{ℓ}(−1)^{ℓ}〈T_{ℓ} ⋅ T_{ℓ+1}〉/L, where we introduced the Fourier transform \({{\bf{T}}}_{q}={\sum }_{\ell }\exp (iq\ell )\ {{\bf{T}}}_{\ell }/\sqrt{L}\) of the total spin operator T_{ℓ} = S_{ℓ} + s_{ℓ}. In Fig. 1c S(q) is shown at moderate interaction: at U/W < 1.6 it displays a maximum at \({q}_{\max }=\pi /2\), consistent with ↑↑↓↓ the order.
Remarkably, it has been shown recently^{27} that there exists an additional unexpected phase in between the block and FMordering. Namely, upon increasing the interaction (1.6 < U/W < 2.4), the maximum of S(q) in Fig. 1c shifts towards incommensurate wavevectors (while for U/W > 2.4 the system is a ferromagnet). This incommensurate region reflects a novel magnetic spiral where the magnetic islands maintain their ferromagnetic character (with D_{π/2} ≠ 0) but start to rigidly rotate, forming a socalled blockspiral (see sketch Fig. 1c). The latter can be identified by a large value^{27} of the longrange chirality correlation function 〈κ_{ℓ} ⋅ κ_{m}〉 where κ_{ℓ} = T_{ℓ} × T_{ℓ+N} and N is the block size. It is important to note that the spiral magnetic order appears without any direct frustration in the Hamiltonian (1), but rather is a consequence of hidden frustration caused by competing energy scales in the OSMP regime. Finally, it should be noted that the blockspiral OSMP state is not limited to 1D chains. In Supplementary Note 2, we show similar investigations for the ladder geometry and find rigidly rotating 2 × 2 FM islands. These results are consistent with recent nuclear magnetic resonance measurements on the CsFe_{2}Se_{3} ladder compound which reported the system’s incommensurate ordering^{35}.
Interestingly, an interactioninduced spiral order is also present when SC pairing is included in the model, as evident from Fig. 1d. However, the spiral mutates from block to canonicaltype with D_{π/2} = 0 (see the sketch in Fig. 1d), indicating unusual backandforth feedback between magnetism and superconductivity. As discussed below, the pairing optimizes the spiral profile to properly create the Majoranas. The competition between many energy scales (Hubbard interaction, Hund exchange, and SC pairing) leads to novel phenomena: an interactioninduced topological phase transition into a manybody state with MZM, unconventional SC, and canonical spiral.
Majorana fermions
Figure 3 shows the effect of Δ_{SC}/W ≃ 0.5 on the singleparticle spectral function A(q, ω) (see the “Methods” section) for the two crucial phases in our study, the blockcollinear and blockspiral magnetic orders (U/W = 1 and U/W = 2, respectively). As expected, in both cases, a finite SC gap opens at the Fermi level ϵ_{F} (~0.5 [eV] for U/W = 1 and ~0.1 [eV] for U/W = 2). Remarkably, in the blockspiral phase, an additional prominent feature appears: a sharply localized mode inside the gap at ϵ_{F}, displayed in Fig. 3b. Such an ingap mode is a characteristic feature of a topological state, namely the bulk of the system is gapped, while the edge of the system contains the ingap modes. To confirm this picture, in Fig. 3c, we present a highresolution frequency data of the realspace local densityofstates (LDOS; see the “Methods” section) near the Fermi energy ϵ_{F}. As expected, for the topologically nontrivial phase, the zero energy modes are indeed confined to the system’s edges. It is important to note that this phenomenon is absent for weaker interaction U/W = 1. Furthermore, one cannot deduce this behavior from the U → ∞ or J_{H} → ∞ limits, where the system has predominantly collinear AFM or FM ordering, leading again to a trivial SC behavior. However, as shown below, at moderate U the competing energy scales present in the OSMP lead to the topological phase transition controlled by the electron–electron interaction.
Let us now identify the induced topological state. The size dependence of the LDOS presented in Fig. 3c reveals zeroenergy edge modes, namely peaks at frequency ω ≃ ϵ_{F} localized at the edges of the chain with open ends. While such modes are a characteristic property of the MZM, finding peaks in the LDOS alone is insufficient information for unambiguous identification. To demonstrate that the gKH model with superconductivity indeed hosts Majorana modes, we have numerically checked three distinct features of the MZM:

(i)
Since the Majorana particles are their own antiparticles, the spectral weight of the localized modes should be built on an equal footing from the electron and hole components. Figure 4a shows that this is indeed the case.

(ii)
The total spectral weight present in the localized modes can be rigorously derived from the assumption of the MZM’s existence (see the “Methods” section), and it should be equal to 0.5. Integrating our DMRG results in Fig. 3c over a narrow energy window and adding over the first few edge sites gives ≃ 0.47, very close to the analytical prediction. Note that the Majoranas are not strictly localized at one edge site ℓ ∈ {1, L}, as evident from Fig. 4a. Instead, the MZM is exponentially decaying over a few sites (see Fig. 5c), and we must add the spectral weight accordingly (separately for the left and right edges).

(iii)
The MZM located at the opposite edges of the system form one fermionic state, namely the edge MZM is correlated with one another over large distances. To show such behavior, consider the hole and electronlike centrosymmetric spectral functions, \({\langle \langle {c}_{\ell }\ {c}_{L\ell +1}^{\dagger }\rangle \rangle }_{\omega }^{{\rm{h}}}\) and \({\langle \langle {c}_{\ell }^{\dagger }\ {c}_{L\ell +1}\rangle \rangle }_{\omega }^{{\rm{e}}}\), respectively. These functions represent the probability amplitude of creating an electron on one end and a hole at the opposite end (or viceversa) at a given energy ω (see the “Methods” section for detailed definitions and Supplementary Note 3 for further discussion). Figure 4b shows \({\langle \langle {c}_{\ell }\ {c}_{L\ell +1}^{\dagger }\rangle \rangle }_{\omega }^{{\rm{h}}}\) and \({\langle \langle {c}_{\ell }^{\dagger }\ {c}_{L\ell +1}\rangle \rangle }_{\omega }^{{\rm{e}}}\) at the Fermi level ω = ϵ_{F}, namely in the region where the MZM should be present. As expected, the bulk of the system behaves fundamentally different from the edges. In the former, crudely when L/2 ≲ ℓ ≲ 3L/4, the aforementioned spectral functions vanish reflecting the gapped (bulk) spectrum with lack of states at the Fermi level. However, at the boundaries (ℓ ≪ L/2 and ℓ ≫ L/2) the values of the centrosymmetric spectral functions are large, with a maximum at the edges ℓ ∈ {1, L}. The longrange (across the system) correlations of the edge states strongly support their topological nature.
Finally, let us discuss the physical mechanism causing the onset of MZM. In Fig. 5a we present the Hubbard U interaction dependence of the edgeLDOS (ℓ = 1) in the vicinity of the Fermi level, ω ~ ϵ_{F}. It is evident from the presented results that the edgeLDOS acquires a finite value quite abruptly for U > U_{c} ≃ 1.5. To further clarify this matter, let us return to the magnetic states in the OSMP regime. Figure 5b shows the chirality correlation function 〈κ_{L/2} ⋅ κ_{ℓ}〉 (with κ_{ℓ} = T_{ℓ} × T_{ℓ+1}) for increasing value of the Hubbard U strength. We observe a sudden appearance of the chirality correlation exactly at U_{c}, a behavior similar to that of the edge LDOS. Interestingly, in the system without the pairing field, Δ_{SC} = 0, at a similar value of U ≃ 1.6 the system enters the blockspiral phase with rigidly rotating FM islands. However, in our setup, the tendencies of OSMP to create magnetic blocks^{26} are highly suppressed by empty and doubly occupied sites favored by the finite pairing field Δ_{SC}. As a consequence, the blockspiral order is reshaped to a canonical type of spiral without dimers D_{π/2} = 0. This behavior is similar to the MZM observed when combining swave SC with a classical magnetic moment heterostructure^{2,4,5,6}. In the latter, the Ruderman–Kittel–Kasuya–Yosida (RKKY) mechanism stabilizes a classical longrange spiral with 2k_{F} pitch (where \({k}_{{\rm{F}}}\propto \overline{n}\) is the Fermi wavevector). Within the OSMP, however, the pitch is, on the other hand, controlled by the interaction U (at fixed \(\overline{n}\)), as evident from the results presented in Fig. 1b, c.
Furthermore, analysis of the chirality correlation function 〈κ_{ℓ} ⋅ κ_{ℓ+r}〉 indicates that the spiral order decays with the distance r (see Fig. 5c), as expected in a 1D quantum system. Note, however, that the MZM decay length scale, l_{MZM}, and that of the spiral, l_{s}, differ substantially. The Majoranas are predominantly localized at the system edges, thus yielding a short localization length l_{MZM} ≃ 3. The spiral, although still decaying exponentially, has a robust correlation length l_{s} ≃ 13, of the same magnitude as the Δ_{SC} = 0 result^{27}. Then, for large but finite chains the overlap between the edge modes is negligible while the magnetic correlations on the distance L are still large enough to promote triplet pairing and the Majorana modes. In addition, we have observed that smaller values of Δ_{SC} than considered here also produce the MZM. However, since the Majoranas have an edge localization length inversely proportional to Δ_{SC}, reducing the latter leads to overlaps between the left and right Majorana states in our finite systems^{28,36}, thus distorting the physics we study. After exploration, Δ_{SC}/W ≃ 0.5 was considered an appropriate compromise to address qualitatively the effects of our focus given our practical technical constraints within DMRG (see Supplementary Note 4 for details).
Conceptually, it is important to note that the interactioninduced spiral at U/W = 2 is not merely frozen when Δ_{SC} increases. Specifically, the characteristics^{27} of the chirality correlation function 〈κ_{i} ⋅ κ_{j}〉 qualitatively differ between the trivial (Δ_{SC} = 0) and topological phases (Δ_{SC} ≠ 0): increasing Δ_{SC} suppresses the dimer order and leads to a transformation from block spiral to a standard canonical spiral with D_{π/2} = 0 in the topologically nontrivial phase. As a consequence, the proximity to a superconductor influences on the magnetic order to optimize the spin pattern needed for MZM. Surprisingly, Δ_{SC} influences on the collinear spin order as well. In fact, at U/W = 1, before spirals are induced, the proximity to superconductivity changes the block spin order into a more canonical staggered spin order to optimize the energy (see Fig. 3b). This is a remarkable, and unexpected, backandforth positive feedback between degrees of freedom that eventually causes the stabilization of the MZM.
Discussion
Our main findings are summarized in Fig. 6: upon increasing the strength of the Hubbard interaction U within the OSMP with added SC pairing field, the system undergoes a topological phase transition. The latter can be detected as the appearance of edge modes which are mutually correlated in a finite system. This in turn leads to, e.g., the sudden increase of the entanglement, as measured by the von Neumann entanglement entropy S_{vN} (see the “Methods” section). The transition is driven by the change in the magnetic properties of the system, namely by inducing a finite chirality visible in the correlation function 〈κ_{ℓ} ⋅ κ_{m}〉. The above results are consistent with the appearance of the MZM at the topological transition. It should be noted that the presence of those MZM implies unconventional pwave superconductivity^{8}. As a consequence, for our description to be consistent, the topological phase transition ought to be accompanied by the onset of triplet SC amplitudes Δ_{T}. To test this nontrivial effect, we monitored the latter, together with the singlet SC amplitude Δ_{S} (related to a nonlocal swave SC; see the “Methods” section for detailed definitions). As is evident from the results in Fig. 6, for U < U_{c} we observe only the singlet component Δ_{S} canonical for an swave SC, while for U ≥ U_{c} the triplet amplitude Δ_{T} develops a robust finite value. It is important to stress that Δ_{T} ≠ 0 is an emergent phenomenon, induced by the correlations present within the OSMP, and is not assumed at the level of the model (we use a trivial onsite swave pairing field as input).
In summary, we have shown that the many competing energy scales induced by the correlation effects present in SC multiorbital systems within OSMP lead to a topological phase transition. Differently from the other proposed MZM candidate setups, our scheme does not require frozen classical magnetic moments or, equivalently, FM ordering in the presence of the Rashba spin–orbit coupling^{3}. All ingredients necessary to host Majorana fermions appear as a consequence of the quantum effects induced by the electron–electron interaction. The pairing filed can be induced by the proximity effect with a BCS superconductor, or it could be an intrinsic property of some iron superconductors under pressure or doping. It is important to note that the coexistence of SC and nontrivial magnetic properties is mostly impossible in singleorbital systems. Here, the OSMP provides a unique platform in which this constraint is lifted by, on the one hand, spatially separating such phenomena, and, on the other hand, strongly correlating them with each other. Furthermore, our proposal allows to study the effect of quantum fluctuations on the MZM modes. There are only a few candidate materials that may exhibit the behavior found here. The blockmagnetism (a precursor of the blockspiral phase) was recently argued to be relevant for the chain compound Na_{2}FeSe_{2}^{37}, and was already experimentally found in the BaFe_{2}Se_{3} ladder^{24}. Incommensurate order was reported in CsFe_{2}Se_{3}^{35}. Also, the OSMP^{38,39,40} and superconductivity^{31,32,33} proved to be important for other compounds from the 123 family of ironbased ladders.
Our findings provide also a new perspective to the recent reports of topological superconductivity and Majorana fermions found in twodimensional compounds Fe(Se,Te)^{13,14,15,16,17}. Since orbitalselective features were observed in clean FeSe^{41,42}, it is reasonable to assume that OSMP is also relevant for doped Fe(Se,Te)^{43}. Regarding magnetism, the ordering of FeSe was mainly studied within the classical longrange Heisenberg model^{44}, where blocklike structures (e.g., double stripe or staggered dimers) dominate the phase diagram for realistic values of the system parameters. Note that the effective spin model of the blockspiral phase studied here was also argued to be longranged^{27}. The aforementioned phases of FeSe are typically neighboring (or are even degenerate with) the frustrated spirallike magnetic orders^{44}, also consistent with the OSMP magnetic phase diagram^{26}. In view of our results, the following rationale could be used to explain the behavior of the above materials: the competing energy scales present in multiorbital ironbased compounds, induced by changes in the Hubbard interaction due to chemical substitution or pressure, lead to exotic magnetic spin textures. The latter, together with the SC tendencies, lead to topologically nontrivial phases exhibiting the MZM^{45,46}. Also, similar reasoning can be applied to the heavyfermion metal UTe_{2}. It was recently shown that this material displays spintriplet superconductivity^{47} together with incommensurate magnetism^{48}.
Methods
DMRG method
The Hamiltonians and observables discussed here were studied using the density matrix renormalization group (DMRG) method^{49,50} within the singlecenter site approach^{51}, where the dynamical correlation functions are evaluated via the dynamicalDMRG^{52,53}, i.e., calculating spectral functions directly in frequency space with the correctionvector method using Krylov decomposition^{53}. We have kept up to M = 1200 states during the DMRG procedures, allowing us to accurately simulate system sizes up to L = 48 and L = 60 with truncation errors ~10^{−8} and ~10^{−6}, respectively.
We have used the DMRG++ computer program developed at Oak Ridge National Laboratory (https://g1257.github.io/dmrgPlusPlus/). The input scripts for the DMRG++ package to reproduce our results can be found at https://bitbucket.org/herbrychjacek/corrwro/ and also on the DMRG++ package webpage.
Hybrid DMRG–BdG algorithm
We consider a 2D, swave, BCS superconductor at halffilling,
Here 〈i, j〉 denotes summation over nearestneighbor sites of a square lattice and \({a}_{i,\sigma }^{\dagger }\) (\({a}_{i,\sigma }\)) creates (destroys) an electron with spin projection σ = {↑, ↓} at site i. The BCS system is coupled to the OSMP chain, as described by the last term of Hamiltonian (2) in the main text. At the BCS level, the latter term emerges as an additional (external) pairing field to the OSMP region
Here, the summation is restricted to the sites of the BCS system which are coupled to the OSMP chain. In numerical calculations, we set the hopping integral t_{BCS} = 2 [eV], fix the system size to L_{x} = 54 and L_{y} = 27 (with 1D OSMP system coupled to the \({\ell }^{\prime}=14\) row of sites), use the BCS attractive potential V_{BCS}/t_{BCS} = 2 and the coupling strength V/t_{BCS} = 2. Although we assume periodic boundary conditions for the BCS system, the translational invariance is broken by the coupling to the OSMP chain.
Our procedure consists of two alternating steps:

1.
BdG calculations: In the first step, we assume an initial set \({\Delta }_{\ell }^{{\rm{OSMP}}}\) and diagonalize the Hamiltonian H_{BCS} + H_{V}, as defined in Eqs. (4) and (5). To this end, we use the standard BdG equations at zero temperature. They yield selfconsistent results for the pairing amplitude, \({\Delta }_{i}^{{\rm{BCS}}}=\langle {a}_{i,\downarrow }{a}_{i,\uparrow }\rangle\), for all sites i within the BCS system. From among the latter results, we single out the amplitudes \({\Delta }_{{\ell }^{\prime}}^{{\rm{BCS}}}\) on the sites \(i={\ell }^{\prime}\) which are coupled to the OSMP chain.

2.
DMRG calculations: The OSMP system within Eq. (3) is evaluated using the DMRG approach. The spatially dependent amplitudes \({\Delta }_{\ell }^{{\rm{OSMP}}}\) are calculated providing a new set of external fields for the subsequent BdG calculations.
The above procedure is repeated iteratively until we obtain converged results. In the main text (see Fig. 2) we presented results of the above algorithm starting from \({\Delta }_{\ell }^{{\rm{OSMP}}}=0\). However, the procedure can also start from arbitrary pairing fields \({\Delta }_{\ell }^{{\rm{OSMP}}}\) in the first step. The right column of Fig. 2b depicts results obtained using a random initial profile \({\Delta }_{\ell }^{{\rm{OSMP}}}\in [0,1]\). It is evident from the presented results that the converged result is independent of the initial configuration (at least for the couplings studied here). See Supplementary Note 1 for further discussion and additional results.
Spectral functions
Let us define the siteresolved frequency (ω)dependent electron (e) and hole (h) correlation functions
where the signs + and − should be taken for \({\langle \langle ...\rangle \rangle }_{\omega }^{\rm{{{e}}}}\) and \({\langle \langle}\! {\ldots}\!\!{\rangle \rangle }_{\omega }^{\rm{{{h}}}}\), respectively. Here, \(\left{\rm{gs}}\right\rangle\) is the groundstate, ϵ_{0} the groundstate energy, and ω^{+} = ω + iη with η a Lorentzianlike broadening. For all results presented here, we choose η = 2δω, with δω/W = 0.001 (unless stated otherwise).
The singleparticle spectral functions A(q, ω) = A^{e}(q, ω) + A^{h}(q, ω), where A^{e} (A^{h}) represent the electron (hole) part of the spectrum, have a standard definition,
with \({c}_{\ell }={\sum }_{\sigma }{c}_{\ell ,\sigma }\). Finally, the LDOS is defined as
Spectral functions of Majorana edgestates
For simplicity, in this section, we suppress the spin index σ and assume that the lattice index j contains all local quantum numbers. The manybody Hamiltonian is originally expressed in terms of fermionic operators \({c}_{j}^{\left(\dagger \right)}\), but it may be equivalently rewritten using the Majorana fermions (not to be confused with the MZM):
where \({\gamma }_{l}^{\dagger }={\gamma }_{l}\) and {γ_{i}, γ_{j}} = 2δ_{ij}. The latter anticommutation relation is invariant under orthogonal transformations, thus we can rotate the Majorana fermions arbitrarily with
where \(\hat{V}\) are real, orthogonal matrices \({\hat{V}}^{\top }\hat{V}=\hat{V}{\hat{V}}^{\top }=1\). If the system hosts a pair of Majorana edge modes, Γ_{L} and Γ_{R}, then we can find a transformation \(\hat{V}\) such that the following Hamiltonian captures the lowenergy physics
It is important to note that \(H^{\prime}\) does not contribute to the ingap states. It contains all Majorana operators, Γ_{a}, other than the MZM (Γ_{L} and Γ_{R}). The first term in Eq. (11) arises from the overlap of the MZM in a finite system, while in the thermodynamic limit ε → 0 both Γ_{L} and Γ_{R} become strictly the zero modes. While the ground state properties obtained from the zero temperature DMRG do not allow us to formally construct the transformation \(\hat{V}\), we demonstrate below that the computed local and nonlocal spectral functions are fully consistent with the MZM. In fact, we are not aware of any other scenario that could explain the spectral functions reported in this work.
Let us investigate the retarded Green’s functions
which are related to the already introduced spectral functions
Using the transformations (9) and (10) one may express \({G}^{{\rm{e}},{\rm{h}}}\left({c}_{j},{c}_{l}^{\dagger }\right)\) as a linear combination of the Green’s functions defined in terms of the Majorana fermions \({G}^{{\rm{e}},{\rm{h}}}\left({{{\Gamma }}}_{a},{{{\Gamma }}}_{b}\right)\). However, the only contributions to the ingap spectral functions come from the zeromodes, i.e., from a, b ∈ {L, R}, and the corresponding functions can be obtained directly from the effective Hamiltonian (11),
The Green’s functions determine the ingap peak in the left part of the system
with α ∈ {e, h}, and a similar expression holds for the peak on its right side. Utilizing the orthogonality of \(\hat{V}\), one may explicitly sum up the Green’s functions over the lattice sites
where the sum over j contains few sites at the edge of the system due to the exponential decay of the \(\hat{V}\) elements. The result Eq. (16) explains why the total spectral weights originating from ∑_{j}G^{α} equal 1/4, while the total spectral weights of the peaks in LDOS equal 1/2. A similar discussion of the nonlocal centrosymmetric spectral functions \({\langle \langle {c}_{\ell }\ {c}_{L\ell +1}^{\dagger }\rangle \rangle }_{{\epsilon }_{{\rm{F}}}}^{{\rm{h}}}\) and \({\langle \langle {c}_{\ell }^{\dagger }\ {c}_{L\ell +1}\rangle \rangle }_{{\epsilon }_{{\rm{F}}}}^{{\rm{e}}}\) can be found in Supplementary Note 3.
Von Neumann entanglement entropy
S_{vN}(ℓ) measures entanglement between two subsystems containing, respectively, ℓ and L−ℓ sites, and can be easily calculated within DMRG via the reduced density matrix ρ_{ℓ}, i.e., \({S}_{{\rm{vN}}}(\ell )={\rm{Tr}}{\rho }_{\ell }{\mathrm{ln}}\,{\rho }_{\ell }\). The results presented in Fig. 6 depict the system divided into two equal halves, ℓ = L/2. The full spatial dependence of S_{vN}(ℓ) is presented in Supplementary Note 5.
SC amplitudes
The swave and pwave SC can be detected with singlet Δ_{S} and triplet Δ_{T} amplitudes, respectively, defined as
with
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
J. Herbrych and M. Środa acknowledge support by the Polish National Agency for Academic Exchange (NAWA) under contract PPN/PPO/2018/1/00035, and together with M. Mierzejewski, by the National Science Centre (NCN), Poland via project 2019/35/B/ST3/01207. The work of G. Alvarez was supported by the Scientific Discovery through Advanced Computing (SciDAC) program funded by the US DOE, Office of Science, Advanced Scientific Computer Research and Basic Energy Sciences, Division of Materials Science and Engineering. The development of the DMRG++ code by G. Alvarez was conducted at the Center for Nanophase Materials Science, sponsored by the Scientific User Facilities Division, BES, DOE, under contract with UTBattelle. E. Dagotto was supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Materials Sciences and Engineering Division. A part of the calculations was carried out using resources provided by the Wrocław Centre for Networking and Supercomputing.
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J.H., M.M., and E.D. planned the project. G.A. developed the DMRG++ computer program. J.H. and M.Ś. performed the numerical simulations. J.H., M.Ś., M.M., and E.D. wrote the manuscript. All coauthors provided comments on the paper.
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Herbrych, J., Środa, M., Alvarez, G. et al. Interactioninduced topological phase transition and Majorana edge states in lowdimensional orbitalselective Mott insulators. Nat Commun 12, 2955 (2021). https://doi.org/10.1038/s41467021232612
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DOI: https://doi.org/10.1038/s41467021232612
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