arXiv:2011.05646v1 [cond-mat.str-el] 11 Nov 2020sces.phys.utk.edu/publications/Pub2019/ArXiv.2011.05646.pdf · 2020. 11. 15. · Interaction-induced topological phase transition and

Interaction-induced topological phase transition and Majorana edge statesin low-dimensional orbital-selective Mott insulators

J. Herbrych1, M. Sroda1, G. Alvarez2, M. Mierzejewski1, and E. Dagotto3,41Department of Theoretical Physics, Faculty of Fundamental Problems of Technology,

Wroc law University of Science and Technology, 50-370 Wroc law, Poland2Computational Sciences and Engineering Division and Center for Nanophase Materials Sciences,

Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA3Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA and

4Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA(Dated: November 12, 2020)

Topological phases of matter are among the most intriguing research directions in CondensedMatter Physics. It is known that superconductivity induced on a topological insulator’s surfacecan lead to exotic Majorana modes, the main ingredient of many proposed quantum computationschemes. In this context, iron-based high critical temperature superconductors are among themain candidates to host such exotic phenomenon. Moreover, it is commonly believed that theCoulomb interaction is vital for the magnetic and superconducting properties of these systems.This work bridges these two perspectives and shows that the Coulomb interaction can also drivea trivial superconductor with orbital degrees of freedom into the topological phase. Namely, weshow that above some critical value of the Hubbard interaction, identified by the change in entropybehaviour, the system simultaneously develops spiral spin order, a highly unusual triplet amplitudein superconductivity, and, remarkably, Majorana fermions at the edges of the system.

Topologically protected Majorana fermions - the elu-sive particles which are their own antiparticles - are ex-citing because they could realize fault-resistant quan-tum computation. From the experimental perspective,heterostructure-based setups were proposed as the maincandidates to host the Majorana zero-energy modes(MZM). For example, the topologically protected gap-less surface states of topological insulators can be pro-moted to MZM by the proximity-induced pairing of anunderlying superconducting (SC) substrate [1]. However,the large spin-orbit coupling required to split the doubly-degenerated bands due to the electronic spins, renderssuch a setup hard to engineer. Another group of propos-als utilizes magnetic atoms (e.g., Gd, Cr, or Fe) arrangedin a chain structure on a BCS superconductor [2–10].These important efforts have shown that creating MZMin real condensed-matter compounds is challenging andonly rare examples are currently available.

Interestingly, a series of recent works have shown thatdoped high critical temperature iron-based superconduc-tor Fe(Se,Te) can host MZM [11–15] . Although theelectron-electron interaction is believed to be relevantfor the pairing, its role in the stabilization of MZM ismostly unknown. In fact, in most theoretical proposalsto realize MZM, these zero-energy modes are a conse-quence of specific features in the non-interacting bandstructure, with the electron-electron interaction playingonly a secondary role (and often even destabilizing theMZM) [16, 17]. By contrast, here we will show that asuperconducting system with orbital degrees of freedomcan be driven into a topologically nontrivial phase host-ing MZM via increasing Hubbard interactions; see illus-trative sketch in Fig. 1a. We will focus on a genericmodel with coexisting wide and narrow energy bands,

relevant to low-dimensional iron-based materials [18]. Itwas previously shown [19–21] that the multi-orbital Hub-bard model can accurately capture static and dynamicalproperties of iron selenides, especially the block-magneticorder [22] of the 123 family AFe2X3 of iron-based ladders(with A alkali metals and X chalcogenides). For exam-ple, the three- and two-orbital Hubbard model on a one-dimensional (1D) lattice [21, 23] successfully reproducesthe inelastic neutron scattering spin spectrum, with non-trivial optical and acoustic modes. The aforementionedmodels exhibit [19, 24] the orbital-selective Mott phase(OSMP), with coexistent Mott-localized electrons in oneorbital and itinerant electrons in the remaining orbitals.The system is then in an exotic state with simultane-ously metallic and insulating properties. Furthermore,the localized orbitals have vanishing charge fluctuations,simplifying the description [24] into an OSMP effectivemodel, i.e. the generalized Kondo-Heisenberg model(gKH)

HgKH = ti∑

`,σ

(c†`,σc`+1,σ + H.c.

)+ U

∑

`

n`,↑n`,↓

+ µ∑

`,σ

n`,σ − 2JH∑

`

S` · s` +K∑

`

S` · S`+1 .(1)

The first three terms in the above Hamiltonian de-scribe the itinerant electrons: c†`,σ (c`,σ) creates (de-stroys) an electron with spin projection σ = {↑, ↓} atsite ` = {1, . . . , L}, ti is their hopping amplitude, U isthe repulsive Hubbard interaction, and µ = εF is theFermi energy set by the density of itinerant electronsn =

∑`(n`,↑+n`,↓)/L. Furthermore, in order to keep our

discussion general, we will make minimal assumptionson the SC pairing field, and consider the simplest on-

arX

iv:2

011.

0564

6v1

[co

nd-m

at.s

tr-e

l] 1

1 N

ov 2

020

2

site term ∆SC

∑`

(c†`,↑c

†`,↓ + H.c.

)that induces s-wave

superconductivity, which the reader should consider tobe caused either by proximity or by intrinsic pairing ten-dencies (see the discussion later on).

Figure 1. Interaction-induced topological phase tran-sition and Majorana edge states. a Sketch of the edgedensity-of-states as function of the electron-electron Hubbardinteraction strength. Magnetic orders (depicted by arrows) inthe trivial and the topological superconducting (SC) phasesare also presented. b Schematic representation of the gener-alized Kondo-Heisenberg model studied here, with localizedand itinerant electrons and simultaneously active Hubbard Uand superconducting ∆SC couplings.

In the model above, the double occupancy of the lo-calized orbital is eliminated by the Schrieffer-Wolff trans-formation and the remaining degrees of freedom, the lo-calized spins S`, interact with one another via a Heisen-berg term with spin exchange K = 4t2l /U [tl is the hop-ping amplitude within the localized band]. Finally, JHstands for the on-site interorbital Hund interaction, cou-pling the spins of the localized and itinerant electrons,S` and s`, respectively. Figure 1b contains a sketchof the model. Here, we consider a 1D lattice and useti = 0.5 [eV] and tl = 0.15 [eV], with kinetic energy band-width W = 2.1 [eV] as unit of energy [25]. Furthermore,to reduce the number of parameters in the model, we setJH/U = 1/4, value widely used when modeling iron su-perconductors. Systems with open boundary conditionsare studied via the density-matrix renormalization group(DMRG) method (see Methods section).

Magnetism of orbital-selective Mott phasePrevious work has shown that the OSMP (with ∆SC = 0)has a rich magnetic phase diagram [24]. (i) At small Uthe system is paramagnetic. (ii) At n = 1 and n = 0standard antiferromagnetic (AFM) order develops, ↑↓↑↓,

with total on-site magnetic moment 〈S2〉 = S(S+ 1) = 2and 3/4, respectively. (iii) For 0 < n < 1 and U � Wthe system is a ferromagnet (FM) ↑↑↑↑. Interestingly, inthe always challenging intermediate interaction regimeU ∼ O(W ) the AFM- and FM-tendencies (arising fromsuperexchange and double-exchange, respectively) com-pete and drive the system towards novel magnetic phasesunique to multi-orbital systems. (iv) For U ∼ W , thesystem develops a so-called block-magnetic order, con-sisting of FM blocks which are AFM coupled, e.g. ↑↑↓↓,as sketched in Fig. 2a. The block size appears controlledby the Fermi vector kF, i.e., the propagation wavevectorof the block-magnetic order is given by qmax = 2kF (with2kF = πn for the chain lattice geometry). In this workwe choose n = 0.5 (adjusted via the chemical potentialµ), as the relevant density for BaFe2Se3 π/2-block mag-netic order [22]. Then, the latter order can be identifiedvia the peak position of the static structure factor S(q) =〈T−q ·Tq〉 at qmax = π/2 or via a finite dimer order pa-rameter Dπ/2 =

∑`(−1)`〈T` ·T`+1〉/L, where we intro-

duced the Fourier transform Tq =∑` exp(iq`)T`/

√L of

the total spin operator T` = S` + s`. In Fig. 2a S(q) isshown at moderate interaction: at U/W < 1.6 it displaysa maximum at qmax = π/2, consistent with ↑↑↓↓ order.

Remarkably, it has been shown recently [25] that thereexists an additional unexpected phase in between theblock- and FM-ordering. Namely, upon increasing theinteraction (1.6 < U/W < 2.4), the maximum of S(q)in Fig. 2a shifts towards incommensurate wavevectors(while for U/W > 2.4 the system is a ferromagnet). Thisincommensurate region reflects a novel magnetic spiralwhere the magnetic islands maintain their ferromagneticcharacter (with Dπ/2 6= 0) but start to rigidly rotate,forming a so-called “block-spiral” (see sketch Fig. 2a).The latter can be identified by a large value [25] of thelong-range chirality correlation function 〈κ` · κm〉 whereκ` = T`×T`+N and N is the block size. It is importantto note that the spiral magnetic order appears withoutany direct frustration in the Hamiltonian (1), but ratheris a consequence of hidden frustration caused by com-peting energy scales in the OSMP regime. Finally, itshould be noted that the block-spiral OSMP state is notlimited to one-dimensional chains. In the Supplemen-tary Information, we show similar investigations for theladder geometry and find rigidly rotating 2 × 2 FM is-lands. These results are consistent with recent nuclearmagnetic resonance measurements on the CsFe2Se3 lad-der compound which reported the system’s incommensu-rate ordering [26].

Interestingly, an interaction induced spiral order is alsopresent when SC tendencies are included in the model,see Fig. 2b. However, the spiral mutates from block-to canonical-type with Dπ/2 = 0, indicating an unusualback-and-forth feedback between magnetism and super-conductivity (see sketches in Fig. 2a and Fig. 2b). Re-markably, as shown below, pairing optimizes the spiral

3

Figure 2. Magnetism and spectral functions. a-b Interaction U dependence of the static structure factor S(q) for L = 36and n = 0.5 for a ∆SC = 0 and b ∆SC/W ' 0.5. Sketches of magnetic ordering are also provided. c-d Effect of the finitepairing field ∆SC/W ' 0.5 on the single-particle spectral function A(q, ω) for c U/W = 1 and d U/W = 2 calculated forL = 36, n = 0.5, and δω = 0.02 [eV]. Majorana zero-energy modes are indicated in d. e Local density-of-states (LDOS) in thein-gap frequency region (δω = 0.002 [eV]) vs chain site index. The sharp LDOS peaks at the edges represent Majorana edgestates, while the bulk of the system exhibits gapped behaviour.

profile to properly create the Majoranas. The competi-tion between many energy scales (Hubbard interaction,Hund exchange, and SC pairing) lead to novel phenom-ena: an interaction induced topological phase transitioninto a many-body state with MZM, unconventional SC,and canonical spiral.

Local density-of-statesThe key ingredient in systems expected to host theMZM [27] is the presence of a SC gap, modeled typicallyby an s-wave pairing field. Such a term represents theproximity effect [28] induced on the magnetic system byan external s-wave superconductor. However, it shouldbe noted that the SC proximity effect has to be consid-ered with utmost care. For example, recent experimentalinvestigations [29] showed that although the interface be-tween Nb (BCS s-wave SC) and Bi2Se3 film (topological

metal) leads to induced superconducting order, the samesetup with (Bi1−xSbx)2Se3 (another topological insula-tor) displays massive suppression of proximity pairing.On the other hand, in the class of systems studied here(low-dimensional OSMP iron-based materials), the pair-ing tendencies could arise from the intrinsic superconduc-tivity of BaFe2S3 and BaFe2Se3 under pressure [31–33]or doping [20, 30].

Figures 2c and 2d show the effect of ∆SC/W ' 0.5 onthe single-particle spectral function A(q, ω) (see Methodssection) for the two crucial phases in our study, the block-collinear and block-spiral magnetic orders (U/W = 1and U/W = 2, respectively). As expected, in bothcases, a finite superconducting gap opens at the Fermilevel εF (∼ 0.5 [eV] for U/W = 1 and ∼ 0.1 [eV] forU/W = 2). Remarkably, in the block-spiral phase an ad-

4

ditional prominent feature appears: a sharply localizedmode inside the gap at εF, displayed in Fig. 2d. Such anin-gap mode is a characteristic feature of a topologicalstate, namely the bulk of the system is gapped, while theedge of the system contains the in-gap modes. To con-firm this picture, in Fig. 2e, we present a high-resolutionfrequency data of the real-space local density-of-states(LDOS; see Methods Section) near the Fermi energy εF.As expected, for the topologically nontrivial phase, thezero energy modes are indeed confined to the system’sedges. It is important to note that this phenomenon isabsent for weaker interaction U/W = 1. Furthermore,one cannot deduce this behaviour from the U → ∞ orJH → ∞ limit, where the system has predominantlycollinear AFM or FM ordering, leading again to a trivialsuperconducting behaviour. However, as shown below, atmoderate U the competing energy scales present in theOSMP lead to the topological phase transition controlledby the electron-electron interaction.

Topological phase transitionTo formally investigate the topological aspects of theinteraction-induced phase transition consider the vonNeumann entanglement entropy SvN and the closely re-lated central charge c, i.e.,

SvN(`) =c

6ln

[2L

πsin

(π`

L

)]+ g , (2)

where g is a non-universal constant. SvN(`) measuresentanglement between two subsystems containing, re-spectively, ` and L − ` sites, and can be easily calcu-lated within DMRG via the reduced density matrix ρ`,i.e., SvN(`) = −Trρ` ln ρ`. Within conformal field the-ory [34, 35], Eq. (2) describes the system in the criticalregime and the central charge is related to the numberof critical degrees of freedom at the Fermi level [36]. Inthe case of topological superconductors, the low-energystates are the MZM. Consequently, the topological phasetransition can be identified by monitoring SvN and c [37].

Figure 3a shows the dependence of the von Neumannentropy SvN(`) on the subsystem size for 1.4 < U/W <1.6. Two characteristic behaviours emerge: for U < Uc '1.51W , SvN(`) displays an oscillatory behaviour, whilefor U > Uc the entropy increases abruptly and changesto a smooth function of `. Remarkably, the central chargec (Fig. 3b), obtained from fits to Eq. (2), also sharply in-creases at Uc, from c ' 0.5 for U < Uc to c ' 4 forU > Uc (as calculated for L = 36 sites). These sud-den changes in entropy behaviour are consistent with aninteraction-induced topological phase transition. AboveUc the central charge remains constant until U/W ∼ 3and decreases afterwards. An interesting question is theexact (thermodynamic L → ∞) value of c, especially inthe topological phase for U > Uc. However, due to theexpected logarithmic dependence on the system size L(Eq. (2)), such a subtle analysis is not practical even with

DMRG in our many-body and multi-orbital system. Nev-ertheless, using the finite lattice sizes studied here, onecan still detect a sudden change in the behaviour of theentropy, indicating that the transition is accompanied byan increased number of gapless excitations.

1

2

3

1 9 18 27 35

a

Intera

ctionU/W

topological phase

trivial phase

0

1

2

3

4

5

0 1 2 3 4

b

L = 24L=36

L = 481

2

3

1 18 35

U/W = 1.4

1

2

3

1 18 35

U/W = 1.6EntropySvN(ℓ)

Site ℓ

1.4

1.5

1.6

Cen

tralch

argec

Interaction U/W

Figure 3. Von Neumann entropy. a InteractionU ∈ {1.41, 1.42, . . . , 1.59, 1.60} dependence of the von Neu-mann entanglement entropy SvN(`) of the subsystem of size`. b Interaction dependence of central charge c. The be-haviour changes sharply at Uc/W = 1.51 indicating a topo-logical phase transition, in agreement with results in panel a.The left and right insets depict the fit of SvN(`) to Eq. (2)below (U/W = 1.4) and above (U/W = 1.6) the transition,respectively. Both panels show data for L = 36 and n = 0.5.In addition, panel b displays data for L = 24 and L = 48close to the transition Uc. In all panels, ∆SC/W = 0.5.

Majorana fermionsLet us now identify the induced topological state. Thesite-dependence of the local density-of-states presentedin Fig. 2e reveals zero-energy edge modes, namely peaksat frequency ω ' εF localized at the edges of the chainwith open ends. While such modes are a characteristicproperty of the MZM, finding peaks in the LDOS aloneis insufficient information for an unambiguous identifi-cation. To demonstrate that the gKH model with su-perconductivity indeed hosts Majorana modes, we havenumerically checked three distinct features of the MZM:(i) Since the Majorana particles are their own antipar-ticles, the spectral weight of the localized modes shouldbe built on an equal footing from the electron and holecomponents. Figure 4a shows that this is indeed the case.(ii) The total spectral weight present in the localized

5

0

4

8

12a

−4

−2

0

2

4

1 9 18 27 36

b

cℓ

c†L−ℓ+1

propagation

c†ℓ

cL−ℓ+1propagation

LDOS at ǫF

electron part 〈〈c†ℓ cℓ〉〉eǫFhole part 〈〈cℓ c†ℓ〉〉hǫF

Site ℓ

〈〈cℓ c†L−ℓ+1〉〉hǫF

〈〈c†ℓ cL−ℓ+1〉〉eǫF

Figure 4. Correlation functions of Majorana fermions.a Site ` dependence of the local density-of-states (LDOS) at

the Fermi level (ω = εF) together with its hole 〈〈c` c†`〉〉hεFand electron 〈〈c†` c`〉〉eεF contributions. b Site dependence

of the centrosymmetric spectral function 〈〈c` c†L−`+1〉〉hεF and

〈〈c†` cL−`+1〉〉eεF . Sketches represent the calculated process:the probability of creating the electron on one end of the sys-tem (site `) and a hole at the opposite end (site L − ` + 1),or vice-versa, at given energy ω. The pairs of sites where thespectral function is evaluated are represented by the samecolors. All results were calculated for L = 36, U/W = 2,∆SC/W ' 0.5, and n = 0.5.

modes can be rigorously derived from the assumptionof the MZM’s existence (see Methods section), and itshould be equal 0.5. Integrating our DMRG results inFig. 2e over a narrow energy window and adding overthe first few edge sites gives ' 0.47, very close to theanalytical prediction. Note that the Majoranas are notstrictly localized at one edge site ` ∈ {1, L}, as evidentfrom Fig. 4a. Instead, the MZM are exponentially decay-ing over a few sites (see Fig. 5c), and we must add thespectral weight accordingly (separately for the left andright edges).

(iii) The MZM located at the opposite edges of the sys-tem form one fermionic state, namely the edge MZMare correlated with one another over large distances. Toshow such a behaviour, consider the hole- and electron-like centrosymmetric spectral functions, 〈〈c` c

†L−`+1〉〉hω

and 〈〈c†` cL−`+1〉〉eω, respectively. These functions repre-sent the probability amplitude of creating an electronon one end and a hole at the opposite end (or vice-versa) at a given energy ω (see Methods section for de-tailed definitions and Supplementary Information for fur-

ther discussion). Figure 4b shows 〈〈c` c†L−`+1〉〉hω and

〈〈c†` cL−`+1〉〉eω at the Fermi level ω = εF, namely inthe region where the MZM should be present. As ex-pected, the bulk of the system behaves fundamentallydifferent from the edges. In the former, crudely whenL/2 <∼ ` <∼ 3L/4, the aforementioned spectral functionsvanish reflecting the gapped (bulk) spectrum with lackof states at the Fermi level. However, at the boundaries(` � L/2 and ` � L/2) the values of the centrosym-metric spectral functions are large, with maximum atthe edges ` ∈ {1, L}. The centrosymmetric functions,

〈〈c` c†L−`+1〉〉hω and 〈〈c†` cL−`+1〉〉eω, at the Fermi energy

shows the same spatial dependence as the local spectralfunctions, 〈〈c` c

†`〉〉hω and 〈〈c†` c`〉〉eω, up to an alternating

sign, as can be seen by comparing both panels in Fig. 4.In the Supplementary Information, we demonstrate thatthis similarity is a consequence of the presence of MZM.The long-range (across the system) correlations of theedge states strongly support their topological nature.

Finally, let us discuss the physical mechanism causingthe onset of MZM. In Fig. 5a we present the HubbardU interaction dependence of the edge-LDOS (` = 1) inthe vicinity of the Fermi level, ω ∼ εF. From our results,it is evident that the edge-LDOS acquires a finite valuequite abruptly for U > Uc. In agreement with Fig. 3b,at U = Uc the MZM emerge at the edges of the systemcontributing to the increased value of the central chargec. It is quite clear that a finite critical value of the Hub-bard interaction U > Uc is necessary to sustain the MZMwithin the studied OSMP model. To further clarify thismatter, let us return to the magnetic states in the OSMPregime. Figure 5b shows the chirality correlation function〈κL/2 ·κ`〉 (with κ` = T` ×T`+1) for increasing value ofthe Hubbard U strength. We observe a sudden appear-ance of the chirality correlation exactly at Uc, a behavioursimilar to that of the edge LDOS. Interestingly, in thesystem without the pairing field, ∆SC = 0, at a similarvalue of U ' 1.6 the system enters the block-spiral phasewith rigidly rotating FM islands. However, in our setup,the tendencies of OSMP to create magnetic blocks [24]are highly suppressed by empty and doubly occupied sitesfavored by the finite pairing field ∆SC. As a consequence,the block-spiral order is reshaped to a canonical type ofspiral without dimers Dπ/2 = 0. This behaviour is sim-ilar to the MZM observed when combining s-wave SCwith a classical magnetic moments heterostructure [2–5]. In the latter, the Ruderman–Kittel–Kasuya–Yosida(RRKY) mechanism stabilizes a classical long-range spi-ral with 2kF pitch (where kF ∝ n is the Fermi wavevec-tor). Within the OSMP, the pitch is, on the other hand,controlled by the interaction U (at fixed n), as presentedin Fig. 2b.

Furthermore, analysis of the chirality correlation func-tion 〈κ` · κ`+r〉 indicates that the spiral order decayswith the distance r (see Fig. 5c), as expected in a 1D

6

a edge-LDOS interaction U dependence

∆SC/W ≃ 0.5

Energy

ω− ǫF

[eV]

−0.05

0.00

+0.05

b chiral correlation 〈κL/2·κℓ〉 interaction U dependence

∆SC/W ≃ 0.5

Site

ℓ

1

L/2

L

10−3

10−2

10−1

100

4 8 12 16 20 24 28 32

∝exp(−

r/3)

∝ exp(−r/13)

c U/W = 2 ,∆SC/W ≃ 0.5

U/W=

3.0

U/W=

2.8

U/W=

2.6

U/W=

2.5

U/W=

2.4

U/W=

2.3

U/W=

2.2

U/W=

2.1

U/W=

2.0

U/W=

1.9

U/W=

1.8

U/W=

1.7

U/W=

1.6

U/W=

1.5

U/W=

1.4

U/W=

1.2

U/W=

1.0

U/W=

3.0

U/W=

2.8

U/W=

2.6

U/W=

2.5

U/W=

2.4

U/W=

2.3

U/W=

2.2

U/W=

2.1

U/W=

2.0

U/W=

1.9

U/W=

1.8

U/W=

1.7

U/W=

1.6

U/W=

1.5

U/W=

1.4

U/W=

1.2

U/W=

1.0

Distance r

〈κℓ·κℓ+r〉LDOS at ǫF

Figure 5. Interaction dependence of the MZM. Depen-dence on the Hubbard interaction U of a the edge-LDOSat site ` = 1 (near the Fermi level εF) and b the chiralitycorrelation function 〈κL/2 · κ`〉. All results calculated for∆SC/W ' 0.5, n = 0.5, L = 36. c Spatial decay of thelocal density-of-states at the Fermi level (LDOS at εF) andthe chirality correlation function 〈κ` · κ`+r〉 for U/W = 2.Red solid lines indicate exponential decay exp(−r/lα) withlMZM = 3 and ls = 15, for the MZM and the spiral order,respectively.

quantum system. Note, however, that the MZM decaylength scale, lMZM, and that of the spiral, ls, differ sub-stantially. The Majoranas are predominantly localizedat the system edges, thus yielding a short localizationlength lMZM ' 3. The spiral, although still decayingexponentially, has a robust correlation length ls ' 13,of the same magnitude as the ∆SC = 0 result [25]. Inaddition, we have observed that smaller values of ∆SC

than considered here also produce the MZM. However,

since the Majoranas have an edge localization length in-versely proportional to ∆SC, reducing the latter leads toan overlaps between the left and right Majorana states inour finite systems [27, 38], thus distorting the physics westudy. After exploration, ∆SC/W ' 0.5 was consideredan appropriate compromise to address the effects of ourfocus (see Supplementary Information for details).

Conceptually, it is important to note that theinteraction-induced spiral at U/W = 2 is not merelyfrozen when ∆SC increases. Specifically, the character-istics [25] of the chirality correlation function 〈κi · κj〉qualitatively differ beween the trivial (∆SC = 0) andtopological phases (∆SC 6= 0): increasing ∆SC suppressesthe dimer order and leads to a transformation from blockspiral to a standard canonical spiral with Dπ/2 = 0 in thetopologically nontrivial phase (see Supplementary Infor-mation for further discussion). As a consequence, theproximity to a superconductor influences on the mag-netic order to optimize the spin pattern needed for MZM.Surprisingly, ∆SC influences on the collinear spin orderas well. In fact, at U/W = 1, before spirals are in-duced, the proximity to superconductivity changes theblock spin order into a more canonical staggered spinorder to optimize the energy (see Fig. 2b). This is a re-markable, and unexpected, back-and-forth positive feed-back between degrees of freedom that eventually causesthe stabilization of the MZM.

DiscussionOur main findings are summarized in Fig. 6: uponincreasing the strength of the Hubbard interaction Uwithin the OSMP with added SC pairing field, the sys-tem undergoes a topological phase transition. The lattercan be detected as a sudden increase of the entangle-ment entropy or as the appearance of edge modes whichare mutually correlated in a finite system. The transitionis driven by the change in the magnetic properties of thesystem, namely by inducing a finite chirality visible inthe correlation function 〈κ` ·κm〉. The above results areconsistent with the appearance of the MZM at the topo-logical transition. It should be noted that the presenceof those MZM implies unconventional p-wave supercon-ductivity [6]. As a consequence, for our description to beconsistent, the topological phase transition ought to beaccompanied by the onset of triplet SC amplitudes ∆T.To test this nontrivial effect, we monitored the latter,together with the singlet SC amplitude ∆S (related to anonlocal s-wave SC; see the Methods section for detaileddefinitions). As is evident from the results in Fig. 6,for U < Uc we observe only the singlet component ∆S

canonical for an s-wave SC, while for U ≥ Uc the tripletamplitude ∆T develops a robust finite value. It is impor-tant to stress that ∆T 6= 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 atrivial on-site s-wave pairing field).

7

0 1 2 3 4

00000

ceL

DOSǫF

κL/2

∆S

∆T

4.2

15.8

0.05

0.23

0.12

a

0.0

0.1

0.2

b

U/W = 1 ,∆SC/W ≃ 0.5

−0.1

0.0

0.1

1 5 10 15 20 25 30 35

c

U/W = 2 ,∆SC/W ≃ 0.5

Interaction U/W

ceLDOSǫF

κL/2

∆S

∆T

∆S,ℓ

∆T0,ℓ

Site ℓ

∆S,ℓ

∆T0,ℓ

Figure 6. Phase diagram. a Hubbard interaction U de-pendence of the (i) central charge c, (ii) edge local density-of-states at the Fermi level (eLDOSεF), (iii) the value of chiralitycorrelation function at distance L/2 (i.e. 〈κL/4 · κ3L/4〉), aswell as (iv) nonlocal singlet ∆S and triplet ∆T pairing am-plitudes. See text for details. All results were calculated forL = 36, ∆SC/W ' 0.5, and n = 0.5. b-c Spatial dependenceof the singlet and triplet SC amplitudes, ∆S,` and ∆T0,`, re-spectively (see Methods section for details), with b the trivialphase (U/W = 1 ,∆SC/W ' 0.5) and c the topological phase(U/W = 2 ,∆SC/W ' 0.5). In the latter, the oscillations ofthe triplet component are related to the pitch of the underly-ing spiral magnetic order.

In summary, we have shown that the many competingenergy scales induced by the correlation effects presentin superconducting multi-orbital systems within OSMPlead to a topological phase transition. Differently fromthe other proposed MZM candidate setups, our schemedoes not require frozen classical magnetic moments orthe Rashba spin-orbit coupling. Pairing is induced bythe proximity effect with a BCS superconductor, or itcould be an intrinsic property of some iron supercon-ductors under pressure or doping. All ingredients neces-sary to host Majorana fermions appear as a consequenceof the quantum effects induced by the electron-electroninteraction. There are only a few candidate materialsthat may exhibit the behaviour found here. The block-magnetism (a precursor of the block-spiral phase) wasrecently argued to be relevant for the chain compound

Na2FeSe2 [39], and was already experimentally found inthe BaFe2Se3 ladder [22]. Incommensurate order was re-ported in CsFe2Se3 [26]. Also, the OSMP [40–42] and su-perconductivity [31–33] proved to be important for othercompounds from the 123 family of iron-based ladders.

Furthermore, our findings provide a new perspectiveto the recent reports of topological superconductivityand Majorana fermions found in two-dimensional com-pounds Fe(Se,Te) [11–15]. Since orbital-selective fea-tures were observed in clean FeSe [43, 44], it is reason-able to assume that OSMP is also relevant for dopedFe(Se,Te) [45]. Regarding magnetism, the ordering ofFeSe was mainly studied within the classical long-rangeHeisenberg model [46], where block-like structures (e.g.,double stripe or staggered dimers) dominate the phase di-agram for realistic values of the system parameters. Notethat the effective spin model of the block-spiral phasestudied here was also argued to be long-ranged [25]. Theaforementioned phases of FeSe are typically neighboring(or are even degenerate with) the frustrated spiral-likemagnetic orders [46], also consistent with the OSMPmagnetic phase diagram [24]. In view of our results,the following rationale could be used to explain the be-haviour of the above materials: the competing energyscales present in multi-orbital iron-based compounds, in-duced by changes in the Hubbard interaction due tochemical substitution or pressure, lead to exotic mag-netic spin textures. The latter, together with the su-perconducting tendencies, lead to topologically nontriv-ial phases exhibiting the MZM [47, 48]. Also, similarreasonings can be applied to the heavy-fermion metalUTe2. It was recently shown that this material displaysspin-triplet superconductivity [49] together with incom-mensurate magnetism [50].

METHODS

DMRG method. The Hamiltonians and observablesdiscussed here were studied using the density matrixrenormalization group (DMRG) method [51, 52] withinthe single-center site approach [53], where the dynamicalcorrelation functions are evaluated via the dynamical-DMRG [54, 55], i.e., calculating spectral functions di-rectly in frequency space with the correction-vectormethod using Krylov decomposition [55]. We have keptup to M = 1200 states during the DMRG procedures,allowing us to accurately simulate system sizes up toL = 48 and L = 60 with truncation errors ∼ 10−8 and∼ 10−6, respectively.

We have used the DMRG++ computer pro-gram developed at Oak Ridge National Labo-ratory (https://g1257.github.io/dmrgPlusPlus/).The input scripts for the DMRG++ pack-age to reproduce our results can be found athttps://bitbucket.org/herbrychjacek/corrwro/ andalso on the DMRG++ package webpage.

8

Spectral functions. Let us define the site-resolved fre-quency (ω) dependent electron (e) and hole (h) correla-tion functions

〈〈A`Bm〉〉e,hω = − 1

πIm〈gs|A`

1

ω+ ∓ (H − ε0)Bm|gs〉 ,

(3)where the signs “+” and “−” should be taken for 〈〈...〉〉eωand 〈〈...〉〉hω, respectively. Here, |gs〉 is the ground-state,ε0 the ground-state energy, and ω+ = ω + iη with η aLorentzian-like broadening. For all results presented herewe choose η = 2δω, with δω/W = 0.001 (unless statedotherwise).

The single-particle spectral functions A(q, ω) =Ae(q, ω)+Ah(q, ω), where Ae (Ah) represent the electron(hole) part of the spectrum, have a standard definition,

Ah(q, ω) =∑

`

e−iq(`−L/2) 〈〈c` c†L/2〉〉hω ,

Ae(q, ω) =∑

`

e+iq(`−L/2) 〈〈c†` cL/2〉〉eω , (4)

with c` =∑σ c`,σ. Finally, the local density-of-states is

defined as

LDOS(`, ω) = 〈〈c` c†`〉〉hω + 〈〈c†` c`〉〉eω . (5)

Spectral functions of Majorana edge-states. Forsimplicity, in this section we suppress the spin indexσ and assume that the lattice index j contains all lo-cal quantum numbers. The many-body Hamiltonian is

originally expressed in terms of fermionic operators c(†)j ,

but it may be equivalently rewritten using the Majoranafermions (not to be confused with the MZM):

γ2j−1 = cj + c†j , γ2j = −i(cj − c†j) , (6)

where γ†l = γl and {γi, γj} = 2δij . The latter anticom-mutation relation is invariant under orthogonal transfor-mations, thus we can rotate the Majorana fermions arbi-trarily with

Γa =∑

j

Vajγj , (7)

where V are real, orthogonal matrices V >V = V V > =1. If the system hosts a pair of Majorana edge modes,ΓL and ΓR, then we can find a transformation V suchthat the following Hamiltonian captures the low-energyphysics

H ' iε

2ΓL ΓR +H ′ . (8)

It is important to note that H ′ does not contribute tothe in-gap states. It contains all Majorana operators,Γa, other than the MZM (ΓL and ΓR). The first termin Eq. (8) arises from the overlap of the MZM in a finite

system, while in the thermodynamic limit ε → 0 bothΓL and ΓR become strictly the zero modes. While theground state properties obtained from the zero temper-ature DMRG do not allow us to formally construct thetransformation V , we demonstrate below that the com-puted local and non-local spectral functions are fully con-sistent with the MZM. In fact, we are not aware of anyother scenario that could explain the spectral functionsreported in this work.

Let us investigate the retarded Green’s functions

Gh(cj , c

†l

)= −i

∫ ∞

0

dt eiωt〈gs|cj(t)c†l |gs〉 ,

Ge(cj , c

†l

)= −i

∫ ∞

0

dt eiωt〈gs|c†l cj(t)|gs〉 , (9)

which are related to the already introduced spectral func-tions

〈〈cjc†l 〉〉hω = − 1

πImGh

(cj , c

†l

),

〈〈c†l cj〉〉eω = − 1

πImGe

(cj , c

†l

). (10)

Using the transformations (6) and (7) one may ex-

press Ge,h(cj , c

†l

)as a linear combination of the Green’s

functions defined in terms of the Majorana fermionsGe,h (Γa,Γb). However, the only contributions to the in-gap spectral functions come from the zero-modes, i.e.,from a, b ∈ {L,R}, and the corresponding functions canbe obtained directly from the effective Hamiltonian (8),

Gh (ΓL,ΓL) = Gh (ΓR,ΓR) =1

ω − |ε|+ iη,

Ge (ΓL,ΓL) = Ge (ΓR,ΓR) =1

ω + |ε|+ iη. (11)

The Green’s functions determine the in-gap peak in theleft part of the system

Gα(cj , c

†j

)=V 2L,2j + V 2

L,2j−1

4Gα (ΓL,ΓL) , (12)

with α ∈ {e,h}, and a similar expression holds for thepeak in its right side. Utilizing the orthogonality of V ,one may explicitly sum up the Green’s functions over thelattice sites

∑

j

Gα(cj , c

†j

)=

1

4Gα (ΓL,ΓL) , (13)

where the sum over j contains few sites at the edge ofthe system due to the exponential decay of the V ele-ments. The result Eq. (13) explains why the total spec-tral weights originating from

∑j G

α equal 1/4, while thetotal spectral weights of the peaks in LDOS equal 1/2.Similar discussion of the nonlocal centrosymmetric spec-tral functions 〈〈c` c

†L−`+1〉〉hεF and 〈〈c†` cL−`+1〉〉eεF can be

found in the Supplementary Information.

9

Superconducting amplitudes. The s-wave and p-wave SC can be detected with singlet ∆S and triplet ∆T

amplitudes, respectively, defined as

∆S =

3L/4∑

`=L/4

|∆S,`| ,

∆T =

3L/4∑

`=L/4

(|∆T0,`|+ |∆T↓,`|+ |∆T↓,`|) , (14)

with

∆S,` =⟨c†`,↑c

†`+1,↓ − c

†`,↓c†`+1,↑

⟩,

∆T0,` =⟨c†`,↑c

†`+1,↓ + c†`,↓c

†`+1,↑

⟩,

∆T↑,` =⟨c†`,↑c

†`+1,↑

⟩, ∆T↓,` =

⟨c†`,↓c

†`+1,↓

⟩. (15)

[1] Elliott, S. R. & Franz, M. Colloquium: Majorana fermionsin nuclear, particle, and solid-state physics, Rev. Mod.Phys. 87, 137 (2015).

[2] Nadj-Perge, S., Drozdov, I. K., Bernevig, B. A. & Yazdani,A. Proposal for realizing Majorana fermions in chains ofmagnetic atoms on a superconductor. Phys. Rev. B 88,020407(R) (2013).

[3] Braunecker, B. & Simon, P. Interplay between ClassicalMagnetic Moments and Superconductivity in QuantumOne-Dimensional Conductors: Toward a Self-SustainedTopological Majorana Phase. Phys. Rev. Lett. 111,147202 (2013).

[4] Klinovaja, J., Stano, P., Yazdani, A. & Loss, D. Topolog-ical Superconductivity and Majorana Fermions in RKKYSystems. Phys. Rev. Lett. 111, 186805 (2013).

[5] Vazifeh, M. M. & Franz, M. Self-Organized Topologi-cal State with Majorana Fermions. Phys. Rev. Lett. 111,206802 (2013).

[6] Pawlak, R., Hoffman, S., Klinovaja, J., Loss, D. & Meyer,E. Majorana fermions in magnetic chains. Prog. Part.Nucl. Phys. 107, 1 (2014).

[7] Steinbrecher, M. et al. Non-collinear spin states in bottom-up fabricated atomic chains. Nat. Commun. 9, 2853(2018).

[8] Jack, B. et al. Observation of a Majorana zero mode in atopologically protected edge channel. Science 364, 1255(2019).

[9] Kim, H. et al. Toward tailoring Majorana bound states inartificially constructed magnetic atom chains on elementalsuperconductors. Sci. Adv. 11, aar5251 (2018).

[10] Palacio-Morales, A. et al. Atomic-scale interface en-gineering of Majorana edge modes in a 2D magnet-superconductor hybrid system. Sci. Adv. 5, eaav6600(2019).

[11] Wang, D. et al. Evidence for Majorana bound states inan iron-based superconductor. Science 362, 333 (2018).

[12] Zhang, P. et al. Observation of topological superconduc-tivity on the surface of an iron-based superconductor, Sci-ence 360, 182 (2018).

[13] Machida, T. et al. Zero-energy vortex bound state inthe superconducting topological surface state of Fe(Se,Te).Nat. Mater. 18, 811 (2019).

[14] Wang, Z. et al. Evidence for dispersing 1D Majoranachannels in an iron-based superconductor. Science 367,104 (2020).

[15] Chen, C. et al. Atomic line defects and zero-energy endstates in monolayer Fe(Te,Se) high-temperature supercon-ductors. Nat. Phys. 16, 536 (2020).

[16] Thomale, R., Rachel, S. & Schmitteckert, P. Tunnelingspectra simulation of interacting Majorana wires. Phys.Rev. B 88, 161103(R) (2013).

[17] Rahmani, A., Zhu, X., Franz, M. & Affleck, I. PhaseDiagram of the Interacting Majorana Chain Model. Phys.Rev. B 92, 235123 (2015).

[18] Daghofer, M., Nicholson, A., Moreo, A. & Dagotto E.,Three orbital model for the iron-based superconductors.Phys. Rev. B 81, 014511 (2010).

[19] Rincon, J., Moreo, A., Alvarez, G. & Dagotto, E., Ex-otic Magnetic Order in the Orbital-Selective Mott Regimeof Multiorbital Systems. Phys. Rev. Lett. 112, 106405(2014).

[20] Patel, N. D. et al. Magnetic properties and pairing ten-dencies of the iron-based superconducting ladder BaFe2S3:Combined ab initio and density matrix renormalizationgroup study. Phys. Rev. B 94, 075119 (2016).

[21] Herbrych, J. et al. Spin dynamics of the block orbital-selective Mott phase. Nat. Commun. 9, 3736 (2018).

[22] Mourigal, M. et al. Block Magnetic Excitations in theOrbitally Selective Mott Insulator BaFe2Se3. Phys. Rev.Lett. 115, 047401 (2015).

[23] Herbrych, J., Alvarez, G., Moreo, A., and Dagotto, E.Block orbital-selective Mott insulators: a spin excitationanalysis. Phys. Rev. B 102, 115134 (2020).

[24] Herbrych, J. et al. Novel Magnetic Block States inLow-Dimensional Iron-Based Superconductors. Phys. Rev.Lett. 123, 027203 (2019).

[25] Herbrych, J. et al. Block-Spiral Magnetism: An ExoticType of Frustrated Order. Proc. Natl Acad. Sci. USA 117,16226 (2020).

[26] Murase, M. et al. Successive magnetic transitions andspin structure in the two-leg ladder compound CsFe2Se3observed by 133Cs and 77Se NMR. Phys. Rev. B 102,014433 (2020).

[27] Kitaev, A. Y. Unpaired Majorana fermions in quantumwires. Phys.-Usp. 44, 131 (2001).

[28] van Wees, B. J. & Takayanagi, H. The SuperconductingProximity Effect in Semiconductor-Superconductor Sys-tems: Ballistic Transport, Low Dimensionality and Sam-ple Specific Properties, Editors: Sohn, L. L., Kouwen-hoven, L. P. & Schon, G. Mesoscopic Electron Trans-port. NATO ASI Series (Series E: Applied Sciences), 345,Springer (Dordrecht, Netherlands).

[29] Hlevyack, J. et al., Massive Suppression of ProximityPairing in Topological (Bi1−xSbx)2Te3 Films on Niobium.Phys. Rev. Lett. 124, 236402 (2020).

[30] Patel, N. D., Nocera, A., Alvarez, G., Moreo, A. &Dagotto, E. Pairing tendencies in a two-orbital Hubbardmodel in one dimension. Phys. Rev. B 96, 024520 (2017).

[31] Takahashi, H. et al. Pressure-induced superconductivityin the iron-based ladder material BaFe2S3. Nat. Mat. 14,1008 (2015).

[32] Yamauchi, T., Hirata, Y., Ueda, Y. & Ohgushi, K.Pressure-Induced Mott Transition Followed by a 24-K Su-

10

perconducting Phase in BaFe2S3. Phys. Rev. Lett. 115,246402 (2015).

[33] Ying, J., Lei, H., Petrovic, C., Xiao, Y. & Struzhkin, V.V. Interplay of magnetism and superconductivity in thecompressed Fe-ladder compound BaFe2Se3. Phys. Rev. B95, 241109(R) (2017).

[34] Vidal, G., Latorre, J. I., Rico, E. & Kitaev, A. Entangle-ment in Quantum Critical Phenomena. Phys. Rev. Lett.90, 227902 (2003).

[35] Calabrese, P. & Cardy, J. Entanglement entropy andquantum field theory. J. Stat. Mech. 2004, P06002(2004).

[36] Kong, K. & Wen, X.-G. A relation between chiral centralcharge and ground state degeneracy in 2 + 1-dimensionaltopological orders. arXiv:2004.11904 (2020).

[37] Jiang, H.-C., Wang, Z. & Balents, L. Identifying topo-logical order by entanglement entropy. Nat. Phys. 8, 902(2012).

[38] Stanescu, T. D. , Lutchyn, R. M . & Das Sarma, S. Di-mensional crossover in spin-orbit-coupled semiconductornanowires with induced superconducting pairing. Phys.Rev. B 87, 094518 (2013).

[39] Pandey, B. et al. Prediction of exotic magnetic states inthe alkali-metal quasi-one-dimensional iron selenide com-pound Na2FeSe2. Phys. Rev. B 102, 035149 (2020).

[40] Caron, J. M. et al. Orbital-selective magnetism in thespin-ladder iron selenides Ba1−xKxFe2Se3. Phys. Rev. B85, 180405 (2012).

[41] Ootsuki, D. et al. Coexistence of localized and itinerantelectrons in BaFe2X3 (X=S and Se) revealed by photoe-mission spectroscopy. Phys. Rev. B 91, 014505 (2015).

[42] Craco, L. & Leoni, S. Pressure-induced orbital-selectivemetal from the Mott insulator BaFe2Se3. Phys. Rev. B101, 245133 (2020).

[43] Yu, R., Zhu, J.-X. & Si, Q. Orbital Selectivity Enhancedby Nematic Order in FeSe. Phys. Rev. Lett. 121, 227003(2018).

[44] Kostin, A. et al. Imaging orbital-selective quasiparticlesin the Hund’s metal state of FeSe. Nat. Mat. 17, 869(2018).

[45] Jiang, Q. et al. Nematic Fluctuations in an Orbital Selec-tive Superconductor Fe1+yTe1−xSex. arXiv:2006.15887(2020).

[46] Glasbrenner, J. K. et al. Effect of magnetic frustration onnematicity and superconductivity in iron chalcogenides.Nat. Phys. 11, 953 (2014).

[47] Nakosai, S., Tanaka, Y. & Nagaosa, N. Two-dimensionalp-wave superconducting states with magnetic moments ona conventional s-wave superconductor. Phys. Rev. B 88,180503(R) (2013).

[48] Steffensen, D., Andersen, B. M. & Kotetes, P.Majorana Zero Modes in Magnetic Texture Vortices.arXiv:2008.10626 (2020).

[49] Jiao, L. et al. Chiral superconductivity in heavy-fermionmetal UTe2. Nature 579, 523 (2020).

[50] Duan, C. et al. Incommensurate Spin Fluctuationsin the Spin-triplet Superconductor Candidate UTe2.arXiv:2008.12377 (2020).

[51] White, S. R. Density matrix formulation for quantumrenormalization groups. Phys. Rev. Lett. 69, 2863 (1992).

[52] Schollwock, U. The density-matrix renormalizationgroup. Rev. Mod. Phys. 77, 259 (2005).

[53] White, S. R. Density matrix renormalization group algo-rithms with a single center site. Phys. Rev. B 72, 180403

(2005).[54] Jeckelmann, E. Dynamical density-matrix

renormalization-group method. Phys. Rev. B 66,045114 (2002).

[55] Nocera, A. & Alvarez, G. Spectral functions with thedensity matrix renormalization group: Krylov-space ap-proach for correction vectors. Phys. Rev. E 94, 053308(2016).

Acknowledgments J. Herbrych and M. Sroda ac-knowledge support by the Polish National Agencyfor Academic Exchange (NAWA) under contractPPN/PPO/2018/1/00035 and, together with M. Mierze-jewski, by the National Science Centre (NCN), Polandvia project 2019/35/B/ST3/01207. The work of G. Al-varez was supported by the Scientific Discovery throughAdvanced Computing (SciDAC) program funded by theUS DOE, Office of Science, Advanced Scientific Com-puter Research and Basic Energy Sciences, Division ofMaterials Science and Engineering. The development ofthe DMRG++ code by G. Alvarez was conducted at theCenter for Nanophase Materials Science, sponsored bythe Scientific User Facilities Division, BES, DOE, undercontract with UT-Battelle. E. Dagotto was supported bythe US Department of Energy (DOE), Office of Science,Basic Energy Sciences (BES), Materials Sciences and En-gineering Division. A part of the calculations was carriedout using resources provided by the Wroc law Centre forNetworking and Supercomputing.

Author contribution J.H., M.M., and E.D. plannedthe project. G.A. developed the DMRG++ computerprogram. J.H. and M.S. performed the numerical simula-tions. J.H., M.S., M.M., and E.D. wrote the manuscript.All co-authors provided comments on the paper.

Additional information Supplementary Informationaccompanies this paper. Correspondence and requestsfor materials should be addressed to J.H.

S1

SUPPLEMENTARY INFORMATION for:

Interaction-induced topological phase transition and Majorana edge statesin low-dimensional orbital-selective Mott insulators

by J. Herbrych, M. Sroda, G. Alvarez, M. Mierzejewski, and E. Dagotto

CONTENTS

S1. Ladder geometry considerations S2

S2. Spectral functions S3

S3. Pairing-field dependence S5

S4. Entropy and dimer order S8

Supplemental References S9

S2

S1. Ladder geometry considerations

In the main text, we have shown that the generalized Kondo-Heisenberg (gKH) model on the chain geometry cansupport Majorana zero-energy modes (MZM) when in the presence of a superconducting (SC) pairing field ∆SC.Here, we will show that the key ingredients necessary to support the MZM in the gKH model are also present inthe ladder geometry, i.e. the block-spiral magnetic state (see Fig. S1a for a sketch) and single-particle spectra withparity-breaking quasi-particles. We will consider a spatially isotropic ladder with t‖ = t⊥ ≡ ti, the latter hoppingdefined in the main text, and choose filling n = 1.75, which supports block-magnetism at U ∼W [S1, S2]. Althoughaccurate calculations within the grand-canonical ensemble (needed with finite pairing field ∆SC 6= 0) are numericallytoo demanding (due to the doubling of the lattice sites on the ladder of L rungs), precise canonical calculations of thestatic structure factor can still be performed, i.e.,

S(q‖, q⊥) =∑

`,`′

∑

r,r′

e+iq⊥(r−r′)e+iq‖(`−`′)〈T`,r ·T`′,r′〉 , (S1)

where T`,r = S`,r + s`,r, and (`, r) represent the leg and rung number, respectively. Our results in Fig. S1b revealthat the S(q‖, q⊥) lies at incommensurate values of the wavevectors, the one of the block-spiral magnetic statesignatures [S3]. Another feature of the latter is the existence of two cosine-like bands in the single-particle spectral(see also the discussion in the next section) A(q‖, q⊥, ω) = Ae(q‖, q⊥, ω) + Ah(q‖, q⊥, ω) near the Fermi level ω ∼ εF,where

Ae(q‖, q⊥, , ω) =∑

`

∑

r,r′

e+iq⊥(r−r′)e−iq‖(`−L/2) 〈〈c`,r c†L/2,r′〉〉eω ,

Ah(q‖, q⊥, , ω) =∑

`

∑

r,r′

e+iq⊥(r−r′)e+iq‖(`−L/2) 〈〈c†`,r cL/2,r′〉〉hω . (S2)

The results presented in Fig. S1c (for L = 36 rungs, U/W = 2.2, JH/U = 0.25, and n = 1.75) are consistent withthis scenario and resemble the chain geometry results. Consequently, it is reasonable to assume that the influence ofa finite pairing field ∆SC 6= 0 will lead to a topological superconducting state and the emergence of the MZM also onthe ladder lattice.

0.0

5.0

0.00 0.25 0.50 0.75 1.00Staticstructure

factorS(q

‖,0)

Wavevector q‖/π

U/W=1.8=1.9=2.0=2.1=2.2=2.3=2.4=2.5=2.6=2.7=2.8

b

−1.0 −0.5 0.0 0.5 1.0

Wavevector q‖/π

−2

0

2

Energyω−

ǫ F[eV] c A(q‖, q⊥, ω) U/W = 2.2

q⊥ = 0

q⊥ = π

Figure S1. Block-spiral state on the ladder geometry. a Sketch of the block-spiral state on the ladder geometry. Color(shaded) area represents rigidly rotating 2 × 2 FM blocks. b Interaction dependence of the symmetric component q⊥ = 0 ofthe static structure factor S(q‖, q⊥ = 0). c Wavevector dependence of both components (symmetric and antisymmetric) of thesingle-particle spectral function A(q‖, q⊥ = 0, ω) +A(q‖, q⊥ = π, ω) near the Fermi level in the block-spiral phase (η = 2δω andδω = 0.02 [eV]). All results calculated for the gKH ladder of L = 36 rungs, JH/U = 0.25, n = 1.75, and ∆SC = 0.

S3

S2. Spectral functions

In Fig. 4 of the main text, we have shown the spatial dependence of the local density-of-states (LDOS) at theFermi level, together with equal contributions of the electron and hole components, as expected for MZM. Here, inFig. S2, we show that the same holds in frequency ω space. Furthermore, in the same figure we show that both spincomponents contribute equally (within our numerical precision).

0

4

8

12

−0.15 −0.10 −0.05 0.00 0.05 0.10 0.15

a

0

4

8

12

−0.15 −0.10 −0.05 0.00 0.05 0.10 0.15

b

Energy ω − ǫF [eV]

eLDOSǫF

eLDOSǫF holes

eLDOSǫF electrons

Energy ω − ǫF [eV]

eLDOSǫF

eLDOSǫF spin-↓

eLDOSǫF spin-↑

Figure S2. Components of the Majorana edge states. Frequency ω dependence of the edge-LDOS (` = 1) near the Fermilevel ω ∼ εF Panel a depicts electron and hole contributions, while panel b the ↓- and ↑-spin component. Calculated for L = 36,U/W = 2, ∆SC/W ' 0.5, and n = 0.5.

The same reasoning used for the LDOS in the main text can also be applied to the off-diagonal functions Gα(cj , c

†l

)

where sites j and l belong to the left j < L/2 and the right l > L/2 portions of the system. Then, it can be shownthat

Gα(cj , c

†l

)=VL,2j−1VR,2l−1 + VL,2jVR,2l

4Gα (ΓL,ΓR) ,

+ iVL,2jVR,2l−1 − VL,2j−1VR,2l

4Gα (ΓL,ΓR) , (S3)

with

Gh (ΓL,ΓR) = −Gh (ΓR,ΓL) =i sgn(ε)

ω − |ε|+ iη,

Ge (ΓL,ΓR) = −Ge (ΓR,ΓL) =−i sgn(ε)

ω + |ε|+ iη. (S4)

Since the considered Hamiltonian is real, the spectral functions 〈〈cjc†l 〉〉hω and 〈〈c†jcl 〉〉eω should be real as well. Giventhat the weights of 〈〈ΓLΓR〉〉αω are purely imaginary [see Eq. (S4)], the upper line in Eq. (S3) should vanish. Indeed,for real Hamiltonians, ΓL (and also ΓR) contains γj with only even or odd j. In other words, ΓL contains only γ2jand ΓR contains only γ2j−1 or vice versa. Without losing generality, we may choose the former possibility,

Gα(cj , c

†l

)= i

VL,2jVR,2l−14

Gα (ΓL,ΓR)

Gα(cl , c

†j

)= −iVR,2l−1VL,2j

4Gα (ΓR,ΓL) = Gα

(cj , c

†l

), (S5)

and obtain the spectral functions shown in Fig. S3 and Fig. 4b of the main text

〈〈cl c†L−l+1〉〉hω = −1

4

{VL,2lVR,2L−2l+1 for l < L/2

VL,2L−2l+2VR,2l−1 for l > L/2

}sgn(ε) δ(ω − |ε|) ,

〈〈c†l cL−l+1〉〉eω = +1

4

{VL,2lVR,2L−2l+1 for l < L/2

VL,2L−2`+2VR,2l−1 for l > L/2

}sgn(ε) δ(ω + |ε|) , (S6)

where the electron and hole contributions arise with opposite signs, as it is also visible in Fig. S3 and Fig. 4b of themain text. Finally, it is reasonable to assume that the spatial profiles of the MZMs at both system edges are mutually

S4

symmetric, i.e., |VR,2L−2l+1| ' |VL,2l|. Then, comparing Eq. (12) of the main text and Eq. (S6) we obtain a mirroringof the diagonal (local) and off-diagonal spectral functions

|〈〈cl c†L−l+1〉〉hω| ' |〈〈cl c

†l 〉〉hω|,

|〈〈c†L−l+1cl 〉〉eω| ' |〈〈c†l cl 〉〉eω| , (S7)

which is reasonably well reproduced by the numerical results.

−8

−4

0

4

8

−0.15 −0.10 −0.05 0.00 0.05 0.10 0.15

Energy ω − ǫF [eV]

〈〈c1 c†L〉〉hω

〈〈c†1 cL〉〉eω

+∑L/2

ℓ=1|〈〈cℓ c†L−ℓ+1〉〉hω|

−∑L/2

ℓ=1|〈〈c†ℓ cL−ℓ+1〉〉eω|

Figure S3. Off-diagonal spectral functions. Frequency dependence of centrosymmetric spectral functions, Eq. (S6), at theedge of the system ` = 1 (solid points). We also present spatially integrated, according to Eq. (13) of the main text, spectralfunctions as colored area. Results shown were calculated for L = 36, U/W = 2, ∆SC/W ' 0.5, and n = 0.5.

S5

S3. Pairing-field dependence

In this section, we will discuss the pairing field ∆SC dependence of our results. Let us first focus on the single-particle spectral function A(q, ω) [see Eq. (4) of the main text]. In Fig. S4a and Fig. S4b we show A(q, ω) for systems(at electronic filling n = 0.5) without pairing field ∆SC = 0 for two representative values of the interaction: U/W = 1and U/W = 2, i.e., in the block-collinear and block-spiral magnetic phases. Both spectra exhibit a finite density-of-states (DOS) at the Fermi level εF. In the case of the block-spiral phase at U/W = 2, Fig. S4b, one can observetwo bands of quasiparticles: left and right movers reflecting the two possible rotations of the spirals. It is obviousfrom these results that the quasiparticles break the parity symmetry; i.e., going from q → −q momentum changesthe quasiparticle character, as expected for a spiral state. It is also worth noting that for the block-magnetic order(U/W = 1 and ∆SC = 0) one can observe [S3] the V-like shape of DOS in the vicinity of εF. The latter indicates asemiconductor-like behaviour which was also experimentally found [S4] in the 2× 2 block-magnetic ladder compoundBaFe2Se3. This result shows our model’s strength and relevance for realistic investigations of the iron-based materialsfrom the 123 family.

−1 0 1

Wavevector q/π

−1

0

1

Energyω−ǫ F

[eV]

b U/W = 2 ,∆SC/W = 0

−1

0

1

Energyω−ǫ F

[eV]

a U/W = 1 ,∆SC/W = 0

−1 0 1

Wavevector q/π

d U/W = 2 ,∆SC/W ≃ 0.5

c U/W = 1 ,∆SC/W ≃ 0.5

Figure S4. Single-particle spectra for block-collinear and block-spiral magnetism. Single-particle spectra A(q, ω)of the gKH model using L = 36 sites and electronic filling n = 0.5 for a U/W = 1 ,∆SC = 0, b U/W = 2 ,∆SC = 0, c,U/W = 1 ,∆SC/W ' 0.5, and d U/W = 2 ,∆SC/W ' 0.5.

The pairing field ∆SC 6= 0 has a striking effect on these two phases, see Fig. S4c and Fig. S4d where we presentresults for ∆SC/W ' 0.5. As discussed in detail in the main text, the pairing field leads to the appearance of MZMin the spiral phase (U/W = 2, Fig. S4d), with the flat δ-mode inside the superconducting gap. On the other hand,for the collinear block-magnetic phase (U/W = 1, Fig. S4c) we observe only a trivial opening of the SC gap, withoutany in-gap states. These results indicate that the Hubbard interaction strength U , as the main driver between thetwo magnetic states, plays a crucial role in the stabilization of the MZM.

In order to investigate all these aspects in more detail, in Fig. S5 we present the pairing field ∆SC dependence ofthe quantities discussed in the main text, i.e.: (i) value of edge-LDOS at the Fermi level εF (eLDOS), (ii) chiralitycorrelation function 〈κL/4 · κ3L/4〉 at L/2 distance (κL/2), and (iii) amplitudes of extended (non-local) SC singletand triplet amplitudes, ∆S and ∆T0, respectively [see Eq. (14) of the main text]. Furthermore, in the same figure wepresent the value of the on-site pairing amplitude

∆0 =2

L

3L/4∑

`=L/4

∣∣c†`,↑c†`,↓∣∣ . (S8)

For the collinear block-magnetic phase (U/W = 1, Fig. S5a) we observe that the ∆SC does not induce any topologicalphase transitions. For all considered values of the pairing field, 0 < ∆SC/W <∼ 0.7, the eLDOS, κL/2, and ∆T arezero. Only the singlet SC amplitudes, local ∆0 and non-local ∆S, take a finite value. The behaviour of the U/W = 2

S6

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

00000

eLDOSǫF

∆0

∆S

∆T0

κL/2

19

0.36

0.36

0.09

0.13

a block phase

(U/W = 1)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

b block-spiral phase

(U/W = 2)

∆SC/W

eLDOSǫF∆0

∆S

∆T0

κL/2

∆SC/W

eLDOSǫF∆0

∆S

∆T0

κL/2

Figure S5. Phase diagram. Pairing field ∆SC dependence of: (i) the value of edge-LDOS at the Fermi level εF (eLDOS), (ii)chirality correlation function 〈κL/4 ·κ3L/4〉 at L/2 distance (κL/2), (iii) amplitudes of local and non-local SC singlet amplitudes,∆0 and ∆S, respectively, together with triplet component ∆T0. Panel a shows results for the block-collinear magnetic phase(U/W = 1), while panel b for the block-spiral phase (U/W = 2). All results were calculated for L = 36 and n = 0.5.

case is strikingly different (see Fig. S5b). The chirality correlation function κL/2 has a finite value already at ∆SC = 0,reflecting the block-spiral ordering at this interaction strength, and weakly changes till ∆SC/W ∼ 0.6, after which itdecays to zero. Simultaneously, the triplet SC amplitude ∆T0 increases smoothly with the pairing-field ∆SC (togetherwith singlet components ∆0 and ∆S). It is worth noting that this behaviour is strikingly different from the Uvariation presented in the main text, where we observed a sudden appearance of κL/2 and ∆T0 at a specific valueof Uc/W = 1.51. Moreover, we remark again that the pairing field influences on the characteristics of the spiral,optimizing its shape from block to canonical, to better host the MZM.

The behaviour of the edge-LDOS in the spiral phase needs special attention. As evident from the results presentedin Fig. S5b, the value of the latter becomes finite for ∆SC/W >∼ 0.25 and vanishes for ∆SC/W ∼ 0.7 (together withthe already discussed κL/2 and ∆T0). In order to explain the missing weight of edge LDOS for ∆SC/W <∼ 0.25 let

us investigate the frequency dependence of the hole 〈〈c1c†1〉〉hω and electron 〈〈c†1c1〉〉eω contributions to the edge-LDOS.As it is evident from the results in Fig. S6a, upon increasing ∆SC the peaks in the electron- and hole-like spectralfunctions approach each other (Fig. S6b shows in more detail the positions of both maxima). Within the accessiblefrequency resolution, both peaks are easily distinguishable for ∆SC/W ' 0.25, while they merge into a single peakfor ∆SC/W > 0.3.

a edge-LDOS pairing-field ∆SC dependence

U/W = 2

Energy

ω− ǫF

[eV]

−0.05

0.00

+0.05

−0.04

−0.02

0.00

0.02

0.04

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

bU/W = 2

∆SC/W

=0.7

5

∆SC/W

=0.7

0

∆SC/W

=0.6

5

∆SC/W

=0.6

0

∆SC/W

=0.5

5

∆SC/W

=0.5

0

∆SC/W

=0.4

5

∆SC/W

=0.4

0

∆SC/W

=0.3

5

∆SC/W

=0.3

0

∆SC/W

=0.2

5

∆SC/W

=0.2

0

∆SC/W

=0.1

5

∆SC/W

=0.1

0

∆SC/W

=0.0

5

∆SC/W

=0.0

0

Energyε[eV]

Pairing field ∆SC/W

〈〈c1 c†1〉〉hǫF

〈〈c†1 c1〉〉eǫF

Figure S6. Pairing field dependence of the edge local density-of-states. a Frequency ω dependence of edge (` = 1) localdensity-of-states (LDOS) as a function of the pairing field ∆SC/W ' {0.0 , 0.05 , . . . , 0.7}, as calculated for L = 36, n = 0.5,and U/W = 2. b Pairing field dependence of the maximum position [i.e., offset energy ε, see Eq. (11) of the main text] of the

hole 〈〈c1c†1〉〉hω and electron 〈〈c†1c1〉〉eω contributions to edge-LDOS, see Eq. (5) of the main text (based on the data presented inpanel a).

The behaviour described above is characteristic of systems hosting the MZM, i.e., despite the Majorana modesbeing located at the opposite edges of the studied chain, they overlap in any finite-L system [S5, S6]. Due to this

S7

overlap, the peaks arise at a nonzero frequency ω = ±ε, see Eq. (11) of the main text. The clear splitting in theformer case allows for a systematic finite-size study, shown in Fig. S7. In order to obtain well merged Majorana modesfor ∆SC/W ' 0.25, one needs to consider chains with at least L ∼ 60 sites (see Fig. S7c), whereas systems half thatsize are sufficient for the case that is primarily studied in the present work, i.e., for ∆SC/W ' 0.5 (see Fig. S7d).Furthermore, it is worth noting that if the chain is too short, then the remnants of the peaks become visible also inthe middle of the system, as it is visible from results for L = 24 in Fig. S7a and Fig. S7c. On the other hand, a clearsingle peak at the system’s edge always coincides with a well-developed gap in the bulk, as shown in Fig. S7b andFig. S7d. All these results consistently support the scenario that the nonzero splitting ε originates from the overlapof the edge modes. Finally, in Fig. S7e we explicitly show that ε decays exponentially with L, as expected for systemshosting the MZM [S5, S6].

10−3

10−2

10−1

12 24 36 48 60

frequency resolution δω

e

Energyε[eV]

System size L

∆SC/W ≃ 0.25

∆SC/W ≃ 0.50

Figure S7. Finite-size analysis. System lengths L = {12 , 24 . . . , 60} dependence of the local density-of-states a-b in themiddle of the chain representing the bulk (` = L/2) and c,d at the edge (` = 1) of the system, as calculated for U/W = 2, a,c∆SC/W ' 0.25 and b,Herbrych2019-2d ∆SC/W ' 0.5. e System size L dependence of the offset energy ε [see Eq. (11) ofthe main text] for ∆SC/W = {0.25, 0.5} (based on the data presented in panels c and d).

S8

S4. Entropy and dimer order

In the main text, we have shown that the interaction-induced topological phase transition at Uc may be identified viastudying the entanglement entropy of a subsystem containing ` ≤ L sites. Here, we will also argue that this transitionis accompanied by a rapid change of the dimer order Dπ/2. In Figures S8a and S8b we show the entanglement entropy,respectively for the trivial and nontrivial phases, where SvN(`) shows clear oscillations in the former case. In order toexplain the physical origin of such oscillations we have also plotted the static spin-spin correlation function 〈T` ·T`+1〉.We can observe that the maxima of |〈T` ·T`+1〉| and SvN(`) coincide. To calculate the entanglement entropy we splitthe system into two subsystems cutting the bond between sites ` and `+ 1. Whenever a bond with a large spin-spincorrelation is cut, also the entanglement entropy is large. Therefore, we expect that the oscillatory behaviour ofSvN(`) is a direct manifestation of the dimer order. Indeed, in the topological phase the spin dimerization persistsonly at the very edges of the system, as shown in Fig. S8b, so that the (bulk) dimer order vanishes presumably as1/L, see Fig. S8d. Due to the absence of bulk dimer order, the entanglement entropy smoothly changes with `, asdemonstrated also in Fig. S8b. Finally, in Fig. S8c we show the U dependence of the central charge derived fromfitting SvN(`) to the formula discussed in the main text. This quantity shows significant finite-size effects inside thetopological phase. Nevertheless, close to the transition, U/W = 1.52, the size dependence is not relevant, namely thejump of the central charge at the transition is robust and clear when changing the length of the chain.

1.0

1.2

1.4

1.6

1.8

2.0

1 9 18 27 35−1.0

−0.8

−0.6

−0.4

−0.2a

U/W = 1.4

1.0

1.5

2.0

2.5

3.0

3.5

4.0

1 9 18 27 35

−0.4

−0.2

0.0

0.2

b

U/W = 1.6

0

1

2

3

4

1.40 1.45 1.50 1.55 1.60

c

Uc

0.0

0.1

0.2

0.3

0.4

0.5

0.6

1.40 1.45 1.50 1.55 1.60

d

Uc

EntropySvN(ℓ)

〈Tℓ·T

ℓ+1〉

Site ℓ

EntropySvN(ℓ)

〈Tℓ·T

ℓ+1〉

Site ℓ

Cen

tralch

argec

Interaction U/W

L = 24

L = 36

L = 48

Dim

mer

order

Dπ/2

Interaction U/W

L = 24

L = 36

L = 48

Figure S8. Entropy SvN and dimer order Dπ/2. a-b Site ` dependence of von Neumann entanglement entropy SvN andlocal spin-spin correlation function 〈T` ·T`+1〉 (where T` = S`+s` is the total on-site spin) a below (U/W = 1.4) and b above(U/W = 1.6) the topological phase transition. Results calculated for L = 36, ∆SC/W ' 0.5, and n = 0.5. c,d System size Ldependence of topological phase transition as deduced from c the central charge c and d the dimer order parameter Dπ/2. Seetext for details.

S9

[S1] Herbrych, J. et al. Spin dynamics of the block orbital-selective Mott phase. Nat. Commun. 9, 3736 (2018).[S2] Herbrych, J. et al. Novel Magnetic Block States in Low-Dimensional Iron-Based Superconductors. Phys. Rev. Lett. 123,

027203 (2019).[S3] Herbrych, J. et al. Block-Spiral Magnetism: An Exotic Type of Frustrated Order. Proc. Natl Acad. Sci. USA 117, 16226

(2020).[S4] Lei, H., Ryu, H., Frenkel, A. I. & Petrovic, C. Anisotropy in BaFe2Se3 single crystals with double chains of FeSe tetrahedra.

Phys. Rev. B 84, 214511 (2011).[S5] Stanescu, T. D. , Lutchyn, R. M. & Das Sarma, S. Dimensional crossover in spin-orbit-coupled semiconductor nanowires

with induced superconducting pairing. Phys. Rev. B 87, 094518 (2013).[S6] Rainis, D., Trifunovic, L., Klinovaja, J. & Loss, D. Towards a realistic transport modeling in a superconducting nanowire

with Majorana fermions. Phys. Rev. B 87, 024515 (2013).