# Zero energy modes in a superconductor with ferromagnetic adatom chains and quantum phase transitions

###### Abstract

We study Majorana zero energy modes (MZEM) that occur in a s-wave superconducting surface, at the ends of a ferromagnetic (FM) chain of adatoms, in the presence of Rashba spin-orbit interaction (SOI) considering both non self-consistent and self-consistent superconducting order. We find that in the self-consistent solution the average superconducting gap function over the adatom sites has a discontinuous drop with increasing exchange interaction at the same critical value where the topological phase transition occurs. We also study the MZEM for both treatments of superconducting order and find that the decay length is a linear function of the exchange coupling strength, chemical potential and superconducting order. For wider FM chains the MZEM occur at smaller exchange couplings and the slope of the decay length as a function of exchange coupling grows with chain width. Thus we suggest experimental detection of different delocalization of MZEM in chains of varying widths. We discuss similarities and differences between the MZEM for the two treatments of the superconducting order.

###### pacs:

73.20.Hb, 74.25.Ha, 74.40.Kb, 74.45.+c## I Introduction

Non-magnetic impurities in a conventional superconductor have no pair-breaking effect and do not lead to qualitative change Anderson (1959), in contrast to the case of unconventional gapless superconductors where their effect is pair-breaking Ueda and Rice (1985). On the other hand magnetic impurities in a conventional superconductor have interesting effects Balatsky et al. (2006). A single magnetic impurity coupled to the conduction electron spin density gives rise to a local bound state, known as Yu-Shiba-Rusinov (YSR) state, that is created by breaking a Cooper pair and capturing an electron with the appropriate spin component Yu (1965); Shiba (1968); Rusinov (1969); Sakurai (1970); Salkola et al. (1997); Flatté and Byers (1997). As the exchange coupling increases, a first order quantum phase transition (QPT) occurs, at which point the gap function has a shift and the magnetization of the conduction electrons jumps from zero to . This phase transition has been detected experimentally Yazdani et al. (1997); Ji et al. (2008); Hatter et al. (2015) and is signalled in various quantities including quantum information signatures Sacramento et al. (2007a); Paunković et al. (2008). As it turns out Yu (1965); Shiba (1968); Rusinov (1969) the YSR states come in pairs that lay inside the gap (one at positive and one at negative energy) and tend to lower energies as the coupling grows. At the quantum critical point there is a level crossing, such that the bound state has a small but finite energy. Increasing the number of impurities, more states appear inside the gap (two per impurity) and as the coupling increases, a series of quantum phase transitions occurs and the magnetization of the conduction electrons changes in increasing plateaus. Interesting interference phenomena occur as the number of impurities grows and extended states appear along the chain of impurities placed on the two-dimensional superconductor Ji et al. (2008); Morr and Stavropoulos (2003, 2004); Morr and Yoon (2006); Sacramento et al. (2007b).

A quantum treatment of impurities with spin 1/2 describes the competition between the screening of the impurity spin and the superconducting correlations, which lead to a similar type of QPT between the spin-doublet and spin-singlet state for the unscreened and screened impurity spin, respectively Zittartz and Müller-Hartmann (1970); Satori et al. (1992); Sakai et al. (1993). Real magnetic impurities however, have higher spin, where magnetic anisotropy plays an important role Otte et al. (2008). The effects of magnetic anysotropy on the subgap excitations induced by quantum impurities in conventional superconductor Žitko et al. (2011) were recently measured Hatter et al. (2015) in Manganese phthalocyanine (MnPc) on top of superconductor Pb, where Mn has high spin 3/2. Different possible ways of adsorption of MnPc molecules enabled the precise spectroscopy of YSR states across the QPT using scanning tunnelling microscope (STM).

Increasing the number of magnetic impurities eventually leads to destruction of superconductivity. If the magnetic impurities have arbitrary orientations and locations on the superconductor the pair breaking effect leads to gapless superconductivity and eventually destruction of superconductivity Abrikosov and kov L. P. (1961) for small concentration of impurities of the order of a few percent. A higher robustness of superconductivity to the increasing number of magnetic impurities has been found if they are correlated, particularly if their locations are not random but organized in some patterns Sacramento et al. (2010). These regular distributions allow quite high concentrations of impurities without destruction of superconductivity.

In 2014 an experiment was reported Nadj-Perge et al. (2014) in a system of a chain of magnetic adatoms (Fe) placed on top of a two-dimensional conventional superconductor (Pb). Localized zero energy modes were detected at the edges of adatom chain using STM, which seems to provide evidence of MZEM. These zero energy modes have attracted considerable attention recently Hasan and Kane (2010); Qi and Zhang (2011); Leijnse and Flensberg (2012); Beenakker (2013, 2015) because of their non-abelian statistics, which can be used for quantum information storage and manipulation Kitaev (2001, 2003); Nayak et al. (2008); Alicea (2012); Sarma et al. (2015). A key model exhibiting such topological properties is the one-dimensional Kitaev model of spinless fermions with p-wave superconducting pairing Kitaev (2001). More feasible models based on a conventional s-wave superconductor in proximity to either topological insulator Fu and Kane (2008) or semiconducting nanowires in the presence of a Zeeman field Lutchyn et al. (2010); Oreg et al. (2010) greatly enhanced the interest in the field. The later proposal appears to be experimentaly realized in a zero-bias conductance peak measurements in a semiconducting nanowire on top of a conventional superconductor Mourik et al. (2012); Finck et al. (2013). Although all these experiments support the existence of MZEM they are for now inconclusive Finck et al. (2013); Lee et al. (2014); Peng et al. (2015); Dumitrescu et al. (2015).

In the experiment Nadj-Perge et al. (2014), the zero energy states are sharply located at the edges. Complementing the experiment was a theoretical study of the system, which in its simplest form considers 2D superconducting surface with nearest neighbour hopping, chemical potential, exchange interaction with the ferromagnetically aligned magnetic adatoms, Rashba SOI and proximity induced s-wave superconductivity. As parameters of the model change, the system evolves from trivial to topological phase, as determined by the non self-consistent solution of the problem, justifiable by a proximity induced superconducting order or by neglecting the change of the gap function and consequently of other physical quantities in the vicinity of the magnetic adatoms.

Moreover, there are many theoretical studies showing that regularly positioned chains of adatoms give rise to MZEM. These were given for the cases of random Choy et al. (2011), spiral Kjaergaard et al. (2012); Martin and Morpurgo (2012); Nadj-Perge et al. (2013); Nakosai et al. (2013); Braunecker and Simon (2013); Klinovaja et al. (2013); Vazifeh and Franz (2013); Pientka et al. (2013, 2014); Pöyhönen et al. (2014); Röntynen and Ojanen (2014); Reis et al. (2014); Kim et al. (2014); Li et al. (2016); Hoffman et al. (2016), AFM Heimes et al. (2014, 2015) and FM Nadj-Perge et al. (2014); Li et al. (2014); Heimes et al. (2015); Brydon et al. (2015); Hui et al. (2015); Peng et al. (2015); Pöyhönen et al. (2016) orderings of the magnetic adatoms. A stable spiral magnetic structure of the 1D adatom chain is established via the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction Ruderman and Kittel (1954); Kasuya (1956); Yosida (1957) mediated by the electrons in the underlaying superconductor, which has to be treated fully self-consistently Braunecker and Simon (2013); Klinovaja et al. (2013); Vazifeh and Franz (2013). A more realistic 2D model of a superconducting surface coupled to a 1D chain of magnetic moments shows Reis et al. (2014); Heimes et al. (2015), that besides spiral ordering, there are regions in the phase diagram where the magnetic moments arrange either FM or AFM. These however, can in the presence of SOI, still host MZEM which occur in multiple topological phases Heimes et al. (2015); Pöyhönen et al. (2016) even in the presence of periodic modulation of superconducting order parameter Hoffman et al. (2016). Recently magnetic chains and the competition between MZEM with the zero energy modes at the border of a triplet superconductor were also studied Sacramento (2015). The importance and effects of the self-consistent solution on the order parameter in triplet superconductor were also discussed very recently Mercaldo et al. (2016).

In this work we are interested in comparing the results obtained from the non self-consistent solution with those obtained by treating the superconducting gap function self-consistently. The case of a few impurities has shown the relevance of a full treatment Sacramento et al. (2007b, 2010) and we aim to see how the results change as we pursue a more detailed solution. As was already shown previously for spiral magnetic chains Nadj-Perge et al. (2013); Reis et al. (2014) qualitatevely both treatments give similar results and here we confirm such results for the case of FM chain in the presence of Rashba SOI, with notable quantitative shift of the regions with MZEM in chemical potential. We study also Majorana decay lengths and find that as a function of exchange interaction, chemical potential and the superconducting gap function (effective attractive interaction for the self-consistent superconductivity) the decay length is a linear function with positive, zero and negative slope, respectively. By considering FM chains of various widths, we show that the wider chains correspond to a narrow chain with increased effective exchange interaction. Due to the linear dependence of the Majorana decay length on exchange interaction different delocalizations of MZEM in the STM measurements should be detectable on chains of various widths.

An important difference between the two treatments of superconductivity is that in the self-consistent not only topological phase transition but also a series of quantum phase transitions occur. The later are due to various ingap level crossings and can be observed in changes of total magnetization of the electron spin density, local spin density, local density of states, various quantum information measures and the local order parameter. In this work we focus on the later and show that the local superconducting gap on the adatom sites experiences a discontinuous drop at the exchange coupling where the topological phase transition occurs.

The paper is organized as follows: in section II we introduce the considered model. In section III we present solutions of the model for both treatments of the superconducting order parameter. We study energy levels in subsection III.1, discuss topological invariant in subsection III.2 study the MZEM wavefunctions in subsection III.3 and consider the case of wider chains and the Majorana decay lengths in subsection III.4. In section IV we focus on the self-consistent treatment of superconductivity and study local order parameter. We present our conclusions in section V.

## Ii Model

We consider a 2D s-wave superconducting surface with adsorbed magnetic adatoms, which we describe as classical spins and assume their FM alignment. The classical spins are placed along the line of length and width and oriented along the normal to the surface. The SC surface is a 2D square lattice as shown in Fig.1 and the Hamiltonian of the system considered is

where the first term describes the electron hopping between two adjacent lattice sites, the second term is the chemical potential , the third is the Rashba SOI, forth term is the superconducting s-wave pairing, with site dependent order parameter and the last term is exchange interaction with the magnetic adatoms located at specific sites. We consider at these adatom sites. are vectors to the next neighbouring sides along x (y) direction, are Pauli spin matrices with and . Note that the indices , where is the total number of sites.

We diagonalize the Hamiltonian using the Bogoliubov transformation

(2) |

where is the complete set of states, and are related to the eigenfunctions of the Hamiltonian (II) and are the new fermionic quasiparticle creation (annihilation) operators. We calculate the coefficients and by solving Bogoliubov-de Gennes (BdG) equations de Gennes (1989), which can be written as

(3) |

with the dimensional vector

and the matrix at site is

(4) |

where , for , i.e. displacement along and and coefficients are the particle and hole coefficients . Unless stated otherwise we set the hopping to unity, i.e. and express all the energies in this unit.

We solve BdG equations by diagonalizing a matrix, with the gap either constant, which we refer to as non self-consistent solution or calculated self-consistently, imposing at each iteration that

(5) |

where is an effective attractive interaction between the electrons. The superconducting gap function is calculated via

(6) |

where is Debye frequency, the sum runs over all the positive energy eigenstates with values smaller than , is the Fermi function and is the temperature. In this work we set , and find that calculated has in all cases considered negligible imaginary part. The dimension of the systems considered are where the chain length is up to 180 sites and the lateral dimension is between 11 and 31.

## Iii Majorana zero energy modes

### iii.1 Energy levels

First let us focus on the single-particle energy levels of eq. (II) for self-consistent treatment of superconductivity. In the absence of magnetic adatoms () and SOI (), the energy states in a clean homogeneous s-wave superconductor are gapped, with the gap value of . By adding a single magnetic impurity, two YSR local bound states appear within the gap Yu (1965); Shiba (1968); Rusinov (1969), one at positive and one at a symmetric negative energy. If the exchange coupling , between the impurity spin and the conduction electron spin density is weak enough, the problem may be treated classically Satori et al. (1992) and the impurity spin acts as a local magnetic field that interacts with the superconductor through the Zeeman effect. In this work we consider such case. Note also that is actually a product of the exchange coupling and the magnitude of the impurity spin . As the exchange coupling increases, the values of the two levels drop towards zero until a critical value of is reached, where a QPT occurs. With the further increase of the levels start to repel and eventually merge into the gapped states.

In the case of impurities, there are YSR states inside the gap for small . Similar to the case of a single impurity, the energy values of these states approach zero by increasing the exchange coupling and in the case of an ordered array they form minibands Sacramento et al. (2007b). By increasing the exchange coupling several level crossings occur (all at finite energy) accompanied by a series of first order QPTs.

In Fig. 2 are shown the lowest energies as a function of exchange coupling for the chain of length and width , where the local spins are FM aligned. Increasing the exchange coupling from low value a YSR miniband approaches zero energy and at some critical exchange coupling the lowest state becomes practically zero. By increasing the exchange coupling the lowest energy state remains zero, while the energy of the first excited state increases to a value of about 1/10 of the superconducting gap. By further increasing the first excited state, together with the whole miniband returns to the zero energy state and the minigap closes. Still increasing the exchange coupling the whole miniband moves toward the superconducting gap, which has all along remained nearly constant, as will be discussed in more detail below. Also shown in Fig. 2 is similar behaviour of the lowest energies at fixed exchange coupling as a function of chemical potential. There are several minigap closings and reopenings and two regions with the lowest energy state remaining in vicinity of zero, while the minigap increases. At fixed exchange interaction the superconducting gap reduces with increasing chemical potential. We also note that the superconducting gap function is in the case of self-consistent superconductivity site-dependent and changes its average value by changing either chemical potential, SOI or effective attractive interaction. Similar effects were already observed in various magnetic and superconducting hybrid systems Yang et al. (2004); Gillijns et al. (2005); Stamopoulos et al. (2005) and studied theoretically in magnetic chains on top of a superconducting surface Sacramento et al. (2007b); Reis et al. (2014) in the absence of SOI.

The results for non self-consistent treatment of superconductivity are qualitatively similar for lowest energy states, but quite different when considering lowest energy levels. The lowest energy states are within the superconducting gap only in vicinity of zero chemical potential. Away from zero chemical potential no clear superconducting gap is observed.

Next we focus on the lowest single-particle state energy. Examples of contour plots as a function of chemical potential and exchange coupling for fixed values of superconducting gap (effective attractive interaction ) and SOI are presented in Fig. 3 for non self-consistent (self-consistent) superconductivity. Note that the energy scale used is logarithmic, with the cutoff at the energy . Blue (red) areas correspond to lower (higher) energies. The low energy area is mainly concentrated in the vicinity of the chemical potential , for non self-consistent, self-consistent superconductivity, respectively. In both cases the low energy area appears at some minimal exchange coupling, which increases with increasing superconducting gap (effective attractive interaction ) in the case of non self-consistent (self-consistent) treatment of superconductivity.

### iii.2 Topological invariant

To characterize the zero energy states we calculate the Majorana number Kitaev (2001) , which shows the presence () or absence () of MZEM at the edges of chains in the system. For translationally invariant system the Majorana number is given as

(7) |

where Pf() denotes the Pfaffian of a matrix and () is the Hamiltonian with periodic (anti-periodic) boundary conditions, rewritten in Majorana basis. Introducing Majorana fermion operators , , for which holds and the Hamiltonian (II) is

(8) | |||||

Let us first consider a 1D chain (), which is just the discrete case of a single-channel quantum wire Lutchyn et al. (2010); Oreg et al. (2010). Taking as homogeneous and using the Fourier transform and the Hamiltonian (8) can be rewritten in the basis as

(9) |

where a skew-symmetric matrix has the following nonzero elements

(10) |

with . The Majorana number can be calculated using eq. (7), where and , thus we find for

(11) |

Note that the topologically non-trivial region is for 1D case independent on the SOI. As is well known Martin and Morpurgo (2012), the 1D chain in constant magnetic field (here ) and SOI is equivalent to a 1D chain in spiral magnetic field, which is evident also by comparing the matrix to the one given in Ref. Nadj-Perge et al. (2013).

The equivalence no longer holds for the 2D system, since the SOI considered in Hamiltonian (II) is homogeneous throughout the system, whereas the spiral magnetic ordering is limited only to adatom chain. For the topologically non-trivial region becomes also SOI dependent. In Fig. 4 we show the lowest single-particle energy states as functions of the considered model parameters. Note the transitions of the lowest energy states from finite to zero values when the gaps close and reopen. When , the occurence of zero energy states coincides with the negative Majorana number, which we calculate using the algorithm developed in Ref. Wimmer (2012). Note that in some cases the Majorana number is negative, but the minigap is closed, thus in these cases the lowest energy states are not MZEM.

### iii.3 Wave functions

To study the zero energy modes in more detail we analyze their wave-functions. The wave-function is given as

(12) |

where are the particle and hole coefficients, respectively and the sum runs over spin . Majorana bound states appear at the end of the FM chain, as can be seen in Fig. 5, middle panels. In case of Majorana states the particle and hole coefficients are related due to the particle-hole symmetry Leijnse and Flensberg (2012) and the two end state peaks are separated by taking the following linear combinations,

(13) | |||||

as shown in left and right panels of Fig. 5. This way the separate end states are obtained directly from the particle and hole coefficients calculated from the BdG Hamiltonian (4).

Typical evolution of MZEM as a function of exchange coupling is presented in Fig. 6, where we show the wave-function and limit to only at the sites of the adatom chain. Note that before the closing of the gap (small exchange coupling) the wave-function is spread all along the chain, corresponding to the YSR state, whereas at the transition to topological phase, the wave-functions become strongly localized at the ends of the chain. Unlike in 1D systems Kitaev (2001); Lutchyn et al. (2010); Oreg et al. (2010) the transition here is more gradual. By further increasing the exchange coupling, the decay of the wave-function becomes larger and the energy of the lowest energy state increases (inset in Fig. 6).

### iii.4 Wider chains

We study also the wave-function decay length , which has been modeled Zyuzin et al. (2013); Nadj-Perge et al. (2014) as . Strong localization of MZEM observed in experiment Nadj-Perge et al. (2014) was explained recently by modeling system as a linear chain of Anderson impurities on top of superconductor and solving it analytically with mean-field theory Peng et al. (2015). As in the theoretical treatment supplementing experiment and experiment itself Nadj-Perge et al. (2014) we find here strongly localized MZEM. Due to the evolution of highly localized state from the YSR state, the decay length first decreases and we note that in this transition region the fitting function poorly describes the actual envelope function. After the transition, the decay length is a linear function of exchange coupling (as can be seen in Fig. 7) and roughly independent on the chemical potential. With the exception of the transition regions, the decay length shows linear behaviour with negative slope also as a function of the superconducting gap function (effective attractive interaction) for the non self-consistent (self-consistent) treatment of superconductivity.

We consider also cases with various chain widths . Note that the length dependence was studied previously Nadj-Perge et al. (2014) and also various widths were considered but here we systematically study the width effect on Majorana decay lengths and show its implications for experiment. The general observation in this case is that wider chains correspond to a narrow chain () with increased effective exchange coupling. This can be seen in the Majorana decay lengths in Fig. 7. The wider the chain, the MZEM appear at smaller exchange coupling, which is also seen in the plot of the exchange coupling where the minimum decay length is shown for different chain widths. This minimum is the same for all the widths considered and has for the chosen parameters value . Besides the shift of the decay length minimum also the slope of the decay length as the function of the exchange coupling increases with the increasing chain width.

From these results we see that at fixed exchange coupling the wider chains have longer decay lengths (in case they are in the topological regime with the MZEM). Thus we suggest that the delocalization of the edge modes for wider adatom chains could be observed in the experiments.

## Iv Quantum phase transitions

First let us review known results for few magnetic impurities. In the case of a single magnetic impurity treated classically there is a first order QPT as a function of the exchange coupling, which was first pointed out by Sakurai Sakurai (1970). The transition corresponds to a level crossing where the superconductor becomes thermodynamically unstable against spontaneous creation of localized quasiparticle excitation (with spin opposite to the local magnetic field due to impurity), which occurs for both non-self consistent and self-consistent (mean-field) treatment of superconductivity Salkola et al. (1997). In the latter at the QPT both local superconducting gap function and the local spin density have discontinous jumps as can be seen in Fig. 3 of Ref. Salkola et al. (1997). A jump in the local superconducting order is seen also in the full quantum treatment Sakai et al. (1993). In that case the QPT occurs only for antiferromagnetic coupling () Satori et al. (1992) at , where and are the Kondo temperature and superconducting gap, respectively. Recently the spectral properties of YSR states at finite temperatures were studied using the numerical renormalization group Žitko (2016).

The case of two magnetic (classicall) impurities exhibits quantum interference effects, which depend on the interimpurity distance and the relative angle between the impurities Morr and Stavropoulos (2003). In contrast to the single impurity case the two impurity system can be driven through multiple QPTs, which may or may not be accompanied by a phase shift of the onsite superconducting gap Morr and Yoon (2006). At QPTs the local superconducting gap function changes discontinuously and the spin polarization changes for . The gap function renormalization in the system of two impurities was recently revisited Meng et al. (2015). Similarly to a pair of magnetic impurities also ordered magnetic chains undergo a series of first order QPTs as the exchange interaction of magnetic moments is increased Balatsky et al. (2006); Sacramento et al. (2007b, 2010). These phase transitions are induced by numerous level crossings between the ingap states and can be observed in changes of total magnetization of the electron spin density, local spin density, local density of states, various quantum information measures and the local order parameter. In this work we focus on the latter.

In this section we consider only self-consistent treatment of superconductivity. The typical superconducting gap function profile after convergence is shown in Fig. 8. It is usually reduced to a small absolute value, which can be either positive or negative on the magnetic adatom sites and assumes a finite positive value away from them. Similar superconducting gap profile was reported also in the study of spiral magnetic chain on top of an s-wave superconducting surface Reis et al. (2014), although in that work the QPTs were not discussed.

To give a more detailed analysis we calculate two average values of the gap function, first the average over all sites in the system, , given as

(14) |

and second the average over the adatom sites, , calculated as

(15) |

The overall average of the gap function depends very little on the exchange coupling when the chemical potential is kept fixed and its maximum value at is roughly proportional to the attractive interaction strength . This value decreases with the increasing chemical potential until it vanishes and the superconductivity is destroyed at the chemical potential . This behaviour of is seen also by the -st energy state in Fig. 2 in the parameter ranges where the ingap states are away from the superconducting gap. As a function of spin-orbit interaction the gap has the largest values at small chemical potentials, which decrease towards 0 value with increasing . For larger chemical potential the value of the gap at small diminishes to 0 at and grows to some finite value after some critical which increases with increasing .

Although itself is very weakly dependent on the exchange coupling, shows strikingly different behaviour, as can be seen in Fig. 9. At small exchange coupling and it slowly decreases with increasing . At some exchange coupling the local gap function suddenly drops. This occurs at for all differrent chemical potentials and attractive electron interactions considered. The slope change is greater at smaller chemical potentials and it occurs at smaller for smaller chemical potentials and smaller attractive interactions. At higher chemical potentials ( for and for ) changes smoothly and there is no abrupt slope change. As exchange coupling is further increased passes zero value and becomes negative ( shift). This critical exchange coupling , where the supercondicting gap at the impurity sites changes sign is between for wide range of chemical potentials and attractive electron interactions. By increasing the attractive electron interaction the points and move closer. The shift is also observed by changing the SOI. The critical SOI where the transition occurs dependends on the chemical potential and is for at some finite value, which decreases to 0 at . We observe no shift as a function of attractive interaction .

Closer inspection of the region in vicinity of reveals that the superconducting gap function on the adatom sites has a finite drop as shown in Fig. 10. This drop is accompanied with the single particle gap closing and the occurence of MZEM. Consistently the Majorana number calculated using (7) becomes negative. Remarkably already the case of gives both a change in Majorana number and a drop in the superconducting gap, which happens practically at the same value of exchange coupling as for longer chains. However, the lowest single particle energy is in this case . Note that even shorter chain of length does not show a drop in the local superconducting gap function, however the Majorana number also for this case changes sign (at ). The size of the drop of superconducting gap is a bit larger for smaller system sizes and is the same for longer chains.

The coincidence of the drop in local superconducting gap function and the change of sign in Majorana number is perfect in the whole range of chemical potentials and for various attractive interactions considered where MZEM occur. As can be seen from Fig. 3 this happens around chemical potential . Thus around there are no MZEM, but as soon as they appear we find . By increasing chemical potential the abrupt decrease of adatom gap function becomes smooth and the chemical potential where this occurs coincides with the disapperance of MZEM in the phase diagram. Since , the superconducting gap function on the adatom sites changes sign already within the area of MZEM, thus the latter exist both at and .

## V Discussion and Conclusions

The self-consistent treatment of superconductivity was already addressed in systems of magnetic chains on top of conventional superconductors both in the context of QPTs Sacramento et al. (2007a, 2010) and MZEM Nadj-Perge et al. (2013); Reis et al. (2014). In the latter it was shown that qualitatively both non self-consistent and self-consistent treatment of superconductivity give very similar results, that is in both treatments MZEM are found in the phase diagrams. Here we confirm such results for a 2D s-wave superconducting surface with a FM chain of adatoms, in the presence of Rashba SOI. We have shown that quantitatively the areas where MZEM occur shift in the phase diagrams, most notable is the shift in the chemical potential (seen in Fig. 3), where in the case of non self-consistent (self-consistent) superconductivity the MZEM occur at higher (lower) chemical potential, i.e. (). In the self-consistent treatment the shift is due to the decrease of the superconducting gap with increasing chemical potential (seen in Fig. 2). Note however that whereas the bulk superconducting gap is relatively high, the local superconducting gap on the adatom sites is small (and even changes sign) in the MZEM region.

By a systematic study of Majorana decay lengths in wide chains (as presented in Fig. 7) we have shown that wider chains correspond to a narrow chain with increased effective exchange coupling. Due to the linear dependence of the Majorana decay lengths on the exchange coupling that we found, the wider chains have larger decay lengths (at fixed exchange coupling). From this analysis we suggest that STM experiments performed on wider chains should observe greater delocalization of the zero energy modes.

To study the QPTs we have focused on the local superconducting gap function (Fig. 9 and 10). As a function of exchange coupling it decreases gradually with increasing until a certain value , where there is a drop, which is already present at short adatom chain and is only weakly dependent on its length. By further increasing the exchange interaction the superconducting gap changes sign ( shift). In this work we have shown that the value coincides with the occurence of MZEM at the ends of the chain, thus there is a connection between the topological phase transition and the quantum phase transition. It is interesting to note that there is an analogy between the coincidence of a drop in superconducting gap at the site of single impurity and the QPT and the coincidence of the topological phase transition and the drop of local superconducting gap function at the adatom chain.

T. Č. would like to thank R. Žitko, P. Kotetes, I. Sega, V. Susič and R. Mondaini for fruitful discussions and acknowledge the support by Slovenian ARRS Grant No. P1-0044. Partial support from FCT through grants PEst-OE/FIS/UI0091/2014 and UID/CTM/04540/2013 is also acknowledged.

## References

- Anderson (1959) P. W. Anderson, Phys. Rev. Lett. 3, 325 (1959).
- Ueda and Rice (1985) K. Ueda and T. M. Rice, “Theory of heavy fermions and valence fluctuations,” (Springer, Berlin, 1985).
- Balatsky et al. (2006) A. V. Balatsky, I. Vekhter, and J.-X. Zhu, Rev. Mod. Phys. 78, 373 (2006).
- Yu (1965) L. Yu, Acta Phys. Sin. 21, 75 (1965).
- Shiba (1968) H. Shiba, Progress of Theoretical Physics 40, 435 (1968).
- Rusinov (1969) A. Rusinov, Sov. Phys. JETP 29, 1101 (1969).
- Sakurai (1970) A. Sakurai, Progress of Theoretical Physics 44, 1472 (1970).
- Salkola et al. (1997) M. I. Salkola, A. V. Balatsky, and J. R. Schrieffer, Phys. Rev. B 55, 12648 (1997).
- Flatté and Byers (1997) M. E. Flatté and J. M. Byers, Phys. Rev. Lett. 78, 3761 (1997).
- Yazdani et al. (1997) A. Yazdani, B. A. Jones, C. P. Lutz, M. F. Crommie, and D. M. Eigler, Science 275, 1767 (1997).
- Ji et al. (2008) S.-H. Ji, T. Zhang, Y.-S. Fu, X. Chen, X.-C. Ma, J. Li, W.-H. Duan, J.-F. Jia, and Q.-K. Xue, Phys. Rev. Lett. 100, 226801 (2008).
- Hatter et al. (2015) N. Hatter, B. W. Heinrich, M. Ruby, J. I. Pascual, and K. J. Franke, Nat Commun 6 (2015), 10.1038/ncomms9988.
- Sacramento et al. (2007a) P. D. Sacramento, P. Nogueira, V. R. Vieira, and V. K. Dugaev, Phys. Rev. B 76, 184517 (2007a).
- Paunković et al. (2008) N. Paunković, P. D. Sacramento, P. Nogueira, V. R. Vieira, and V. K. Dugaev, Phys. Rev. A 77, 052302 (2008).
- Morr and Stavropoulos (2003) D. K. Morr and N. A. Stavropoulos, Phys. Rev. B 67, 020502 (2003).
- Morr and Stavropoulos (2004) D. K. Morr and N. A. Stavropoulos, Phys. Rev. Lett. 92, 107006 (2004).
- Morr and Yoon (2006) D. K. Morr and J. Yoon, Phys. Rev. B 73, 224511 (2006).
- Sacramento et al. (2007b) P. D. Sacramento, V. K. Dugaev, and V. R. Vieira, Phys. Rev. B 76, 014512 (2007b).
- Zittartz and Müller-Hartmann (1970) J. Zittartz and E. Müller-Hartmann, Zeitschrift für Physik A Hadrons and nuclei 232, 11 (1970).
- Satori et al. (1992) K. Satori, H. Shiba, O. Sakai, and Y. Shimizu, Journal of the Physical Society of Japan 61, 3239 (1992).
- Sakai et al. (1993) O. Sakai, Y. Shimizu, H. Shiba, and K. Satori, Journal of the Physical Society of Japan 62, 3181 (1993).
- Otte et al. (2008) A. F. Otte, M. Ternes, K. von Bergmann, S. Loth, H. Brune, C. P. Lutz, C. F. Hirjibehedin, and A. J. Heinrich, Nat Phys 4, 847 (2008).
- Žitko et al. (2011) R. Žitko, O. Bodensiek, and T. Pruschke, Phys. Rev. B 83, 054512 (2011).
- Abrikosov and kov L. P. (1961) A. A. Abrikosov and G. kov L. P., Sov. Phys. JETP 12, 1243 (1961).
- Sacramento et al. (2010) P. D. Sacramento, V. K. Dugaev, V. R. Vieira, and M. A. N. AraÃºjo, Journal of Physics: Condensed Matter 22, 025701 (2010).
- Nadj-Perge et al. (2014) S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A. H. MacDonald, B. A. Bernevig, and A. Yazdani, Science 346, 602 (2014).
- Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
- Qi and Zhang (2011) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
- Leijnse and Flensberg (2012) M. Leijnse and K. Flensberg, Semiconductor Science and Technology 27, 124003 (2012).
- Beenakker (2013) C. W. J. Beenakker, Annual Review of Condensed Matter Physics 4, 113 (2013).
- Beenakker (2015) C. W. J. Beenakker, Rev. Mod. Phys. 87, 1037 (2015).
- Kitaev (2001) A. Y. Kitaev, Physics-Uspekhi 44, 131 (2001).
- Kitaev (2003) A. Kitaev, Annals of Physics 303, 2 (2003).
- Nayak et al. (2008) C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).
- Alicea (2012) J. Alicea, Rep. Prog. Phys. 75, 076501 (2012).
- Sarma et al. (2015) S. D. Sarma, M. Freedman, and C. Nayak, Npj Quantum Information 1 (2015).
- Fu and Kane (2008) L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008).
- Lutchyn et al. (2010) R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010).
- Oreg et al. (2010) Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010).
- Mourik et al. (2012) V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Science 336 (2012), 10.1126/science.1222360.
- Finck et al. (2013) A. D. K. Finck, D. J. Van Harlingen, P. K. Mohseni, K. Jung, and X. Li, Phys. Rev. Lett. 110, 126406 (2013).
- Lee et al. (2014) E. J. H. Lee, X. Jiang, M. Houzet, R. Aguado, C. M. Lieber, and S. De Franceschi, Nat. Nano. 9, 79 (2014).
- Peng et al. (2015) Y. Peng, F. Pientka, L. I. Glazman, and F. von Oppen, Phys. Rev. Lett. 114, 106801 (2015).
- Dumitrescu et al. (2015) E. Dumitrescu, B. Roberts, S. Tewari, J. D. Sau, and S. Das Sarma, Phys. Rev. B 91, 094505 (2015).
- Choy et al. (2011) T.-P. Choy, J. M. Edge, A. R. Akhmerov, and C. W. J. Beenakker, Phys. Rev. B 84, 195442 (2011).
- Kjaergaard et al. (2012) M. Kjaergaard, K. Wölms, and K. Flensberg, Phys. Rev. B 85, 020503 (2012).
- Martin and Morpurgo (2012) I. Martin and A. F. Morpurgo, Phys. Rev. B 85, 144505 (2012).
- Nadj-Perge et al. (2013) S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, and A. Yazdani, Phys. Rev. B 88, 020407 (2013).
- Nakosai et al. (2013) S. Nakosai, Y. Tanaka, and N. Nagaosa, Phys. Rev. B 88, 180503 (2013).
- Braunecker and Simon (2013) B. Braunecker and P. Simon, Phys. Rev. Lett. 111, 147202 (2013).
- Klinovaja et al. (2013) J. Klinovaja, P. Stano, A. Yazdani, and D. Loss, Phys. Rev. Lett. 111, 186805 (2013).
- Vazifeh and Franz (2013) M. M. Vazifeh and M. Franz, Phys. Rev. Lett. 111, 206802 (2013).
- Pientka et al. (2013) F. Pientka, L. I. Glazman, and F. von Oppen, Phys. Rev. B 88, 155420 (2013).
- Pientka et al. (2014) F. Pientka, L. I. Glazman, and F. von Oppen, Phys. Rev. B 89, 180505 (2014).
- Pöyhönen et al. (2014) K. Pöyhönen, A. Westström, J. Röntynen, and T. Ojanen, Phys. Rev. B 89, 115109 (2014).
- Röntynen and Ojanen (2014) J. Röntynen and T. Ojanen, Phys. Rev. B 90, 180503 (2014).
- Reis et al. (2014) I. Reis, D. J. J. Marchand, and M. Franz, Phys. Rev. B 90, 085124 (2014).
- Kim et al. (2014) Y. Kim, M. Cheng, B. Bauer, R. M. Lutchyn, and S. Das Sarma, Phys. Rev. B 90, 060401 (2014).
- Li et al. (2016) J. Li, T. Neupert, A. Bernevig, and A. Yazdani, Nat. Commun. 7 (2016).
- Hoffman et al. (2016) S. Hoffman, J. Klinovaja, and D. Loss, Phys. Rev. B 93, 165418 (2016).
- Heimes et al. (2014) A. Heimes, P. Kotetes, and G. Schön, Phys. Rev. B 90, 060507 (2014).
- Heimes et al. (2015) A. Heimes, D. Mendler, and P. Kotetes, New Journal of Physics 17, 023051 (2015).
- Li et al. (2014) J. Li, H. Chen, I. K. Drozdov, A. Yazdani, B. A. Bernevig, and A. H. MacDonald, Phys. Rev. B 90, 235433 (2014).
- Brydon et al. (2015) P. M. R. Brydon, S. Das Sarma, H.-Y. Hui, and J. D. Sau, Phys. Rev. B 91, 064505 (2015).
- Hui et al. (2015) H. Y. Hui, P. M. R. Brydon, J. D. Sau, S. Tewari, and S. Das Sarma, Sci. Rep. 5, 8880 (2015).
- Pöyhönen et al. (2016) K. Pöyhönen, A. Westström, and T. Ojanen, Phys. Rev. B 93, 014517 (2016).
- Ruderman and Kittel (1954) M. A. Ruderman and C. Kittel, Phys. Rev. 96, 99 (1954).
- Kasuya (1956) T. Kasuya, Progress of Theoretical Physics 16, 45 (1956).
- Yosida (1957) K. Yosida, Phys. Rev. 106, 893 (1957).
- Sacramento (2015) P. D. Sacramento, Journal of Physics: Condensed Matter 27, 445702 (2015).
- Mercaldo et al. (2016) M. T. Mercaldo, M. Cuoco, and P. Kotetes, arXiv:1607.03539 (2016).
- de Gennes (1989) P. G. de Gennes, “Superconductivity of metals and alloys,” (Addison-Wesley, Reading, MA, 1989).
- Yang et al. (2004) Z. Yang, M. Lange, A. Volodin, R. Szymczak, and V. V. Moshchalkov, Nature Materials 3, 793 (2004).
- Gillijns et al. (2005) W. Gillijns, A. Y. Aladyshkin, M. Lange, M. J. Van Bael, and V. V. Moshchalkov, Phys. Rev. Lett. 95, 227003 (2005).
- Stamopoulos et al. (2005) D. Stamopoulos, M. Pissas, and E. Manios, Phys. Rev. B 71, 014522 (2005).
- Wimmer (2012) M. Wimmer, ACM Trans. Math. Softw. 38, 30:1 (2012).
- Zyuzin et al. (2013) A. A. Zyuzin, D. Rainis, J. Klinovaja, and D. Loss, Phys. Rev. Lett. 111, 056802 (2013).
- Žitko (2016) R. Žitko, Phys. Rev. B 93, 195125 (2016).
- Meng et al. (2015) T. Meng, J. Klinovaja, S. Hoffman, P. Simon, and D. Loss, Phys. Rev. B 92, 064503 (2015).