Topological superconductivity at finite temperatures in proximitized magnetic nanowires
TTopological superconductivity at finite temperaturesin proximitized magnetic nanowires
Anna Gorczyca–Goraj, Tadeusz Doma´nski, and Maciej M. Ma´ska ∗ Department of Theoretical Physics, University of Silesia, Katowice, Poland Institute of Physics, M. Curie Sk(cid:32)lodowska University, Lublin, Poland (Dated: February 20, 2019)Performing Monte Carlo simulations we study the temperature dependent self–organization ofmagnetic moments coupled to itinerant electrons in a finite–size one–dimensional nanostructureproximitized to a superconducting reservoir. At low temperature an effective interaction betweenthe localized magnetic moments, that is mediated by itinerant electrons, leads to their helical or-dering. This ordering, in turn, affects the itinerant electrons, inducing the topologically nontrivialsuperconducting phase that hosts the Majorana modes. In a wide range of system parameters, thespatial periodicity of a spiral order that minimizes the ground state energy turns out to promote thetopological phase. We determine the correlation length of such spiral order and study how it is re-duced by thermal fluctuations. This reduction is accompanied by suppression of the topological gap(which separates the zero-energy mode from continuum), setting the upper (critical) temperaturefor existence of the Majorana quasiparticles. Monte Carlo simulations do not rely on any ansatzfor configurations of the localized moments, therefore they can be performed for arbitrary modelparameters, also beyond the perturbative regime.
I. INTRODUCTION
Recent progress in fabricating artificial nanostructureswith spatial constraints enabled observation of novelquantum states , where topology plays a prominentrole. Motivated by the seminal Kitaev’s paper , oneof such intensively explored fields is related to topo-logical superconductivity which occurs in semiconduct-ing nanowires proximitized to superconductors ornanochains of magnetic atoms deposited on supercon-ducting surfaces . In both cases the Majorana-typequasiparticles have been observed at boundaries of prox-imitized nanoscopic wires/chains and non–Abelian statis-tics makes them promising for realization of quantumcomputing and/or new spintronic devices .Mechanism that drives the proximitized nanowire intoa topologically non-trivial phase can originate from thespin-orbit coupling (SOC) combined with the Zeemansplitting above some critical value of magnetic field .Upon approaching this transition a pair of finite-energy(Andreev) bound states coalesces into the degenerateMajorana quasiparticles formed near the ends ofnanowire. Another scenario combines the proximity–induced superconducting state with the spiral magneticorder . The latter approach is particularly appeal-ing, because magnetic order seems to self-adjust its peri-odicity (characterized by the pitch vector q ∗ ) to supportthe topological phase. Origin of the topological phasein a system with spirally ordered magnetic moments ismathematically equivalent to the scenario based on thespin–orbit and Zeeman interactions and its topofilia has been investigated by a number of groups .Topological features of the systems with self–organizedspiral ordering have been so far studied, focusing mainlyon the zero temperature limit. Thermal effects have beenpartly addressed, taking into account magnon excitations (which suppress a magnitude of the spiral order) andinvestigating a contribution of the entropy term to thefree energy (which substantially affects the wave vectorof the spiral order, so that magnetic order might be pre-served but the electronic state could no longer be topo-logical) . Usually, however, any long–range order hardlyexists in one–dimensional systems at finite temperaturesand therefore it is important – especially for practicalapplications of such systems – to estimate the maxi-mum temperature up to which the topologically nontriv-ial states could survive. For its reliable determination weperform here the Monte Carlo (MC) simulations.Our numerical results unambiguously indicate thatthermal effects are detrimental to both the topologicalsuperconducting state and to the Majorana quasiparti-cles. This is evidenced by: (i) changeover of the topological Z number, (ii) characteristic scaling of the temperature-dependentcoherence length of the spiral magnetic order, (iii) and directly from the quasiparticle spectrum,where thermal effects suppress the topological en-ergy gap converting the zero-energy quasiparticlesinto overdamped modes.The rest of the paper is organized as follows. In Sec. IIwe introduce the microscopic model. Next, in Sec. III, webriefly revisit the topologically nontrivial superconduct-ing state at zero temperature and check if it really coin-cides with the spiral pitch q ∗ that minimizes the groundstate energy. Essential results of our study are presentedin Sec. IV, where we analyze (dis)ordering of the mag-netic moments at finite temperatures by means of the MCmethod determining the upper (critical) temperature forexistence of the topological superconducting state andthe Majorana quasiparticles. Finally, in Sec. VI, we sum-marize the main results. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b II. MODEL
We consider a chain of the localized magnetic impuri-ties whose moments are coupled to the spins of itinerantelectrons. This nanoscopic chain is deposited on a sur-face of s –wave bulk superconductor, through the prox-imity effect inducing electron pairing. Such system canbe described by the following Hamiltonian H = − t (cid:88) i,σ ˆ c † i,σ ˆ c i +1 ,σ − µ (cid:88) i,σ ˆ c † i,σ ˆ c i,σ (1)+ J (cid:88) i S i · ˆ s i + (cid:88) i (cid:16) ∆ˆ c † i ↑ ˆ c † i ↓ + H.c. (cid:17) , where ˆ c † i,σ and ˆ c i,σ are the creation and annihilation op-erators of electron at site i and ˆ s i is their spinˆ s i = 12 (cid:88) α,β ˆ c † i,α σ αβ ˆ c i,β (2)with σ being a vector of the Pauli matrices. We assumethat magnetic moments S i have much slower dynamicsthan electrons and can be treated classically. In gen-eral, they can be expressed in the spherical coordinatesin terms of the polar and azimuthal angles θ i and φ i S i = S (sin θ i cos φ i , sin θ i sin φ i , cos θ i ) . (3)In the weak coupling J limit it has been shown that the effective Ruderman-Kittel-Kasuya-Yosida inter-action induces the helical ordering between the magneticmoments of the impurities φ i = i a q ∗ (4)where a is the lattice constant and the spiral pitch q ∗ isstrongly dependent on the model parameters . SinceHamiltonian (1) has an SU(2) spin rotation symmetry,for any constant opening angle θ i without loss of gener-ality it can be assumed that θ i = π/
2. It is possible toperform the gauge transformation, upon which the local-ized magnetic moments become ferromagnetically polar-ized at expense of introducing the spin and q ∗ –dependenthopping amplitude . Here, however, we are mostly in-terested in nonzero temperatures, where the ground stateordering is affected by thermal excitations. Therefore, wewill treat φ i ’s as fluctuating degrees of freedom. This willallow us not only to describe thermal states, but also totake into account possible phase separation, where or-derings with different values of q ∗ take place in segmentsof the nanochain . We shall also check influence of θ i fluctuations on stability of the topological phase (Sec. V).In what follows we set the intersite spacing as a unit( a = 1) and impose S = 1 . For simplicity we also setthe Boltzmann constant k B ≡ t = 1) for all energies dis-cussed in our study. III. TOPOFILIA OF THE GROUND STATE
In the case of periodic boundary conditions a spin-dependent gauge transformation can convert the Hamil-tonian (1) into a translationally invariant form that canbe easily diagonalized . Here, however, we focus onthe open boundary conditions what allows us to studythe Majorana end states. Additionally, open boundaryconditions do not impose any restrictions on the spiralpitch q , what is especially important for rather shortnanochains. Most of our calculations have been per-formed for the nanowire comprising 70 sites. We havenumerically diagonalized the system, considering variousconfigurations of the local magnetic moments S i . In par-ticular, we have inspected the spiral ordering and con-sidered q ∈ [0; π ] varying the model parameters J, µ, ∆(transformation q → − q changes the chirality of the spi-ral but it neither affects the thermodynamic nor topolog-ical properties).Ground state of the Hamiltonian (1) refers to somecharacteristic pitch q = q ∗ , which is determined fromminimization of its energy. Since in 1D metals the staticspin susceptibility diverges at 2 k F , where k F is the Fermimomentum, it has been suggested that also in presence . . . . . . q/ π − − − − − E G S a) q ∗ . . . . . . q/ π − . − . − . . . . . ω b) q ∗ FIG. 1. a) The ground state energy E GS versus the spiralpitch q obtained for ∆ = 0 . , µ = 1 . J = 1. b) Evolu-tion of the quasiparticle spectrum with respect to q . The redarrows indicate q ∗ , minimizing the ground state energy. . . . . . . . . . . . . . q / π det( R ) µ = 0 . − . . . . . . . . . . . . q / π det( R ) µ = 1 . − . . . . . . . . . . . . q / π det( R ) µ = 1 . − . . . . . . . . . . . . . . q / π det( R ) µ = 2 . − . . . . . . . . . . . . . q / π det( R ) µ = 2 . −
101 0 . . . . . . . . . . . . . . q / π det( R ) µ = 3 . − FIG. 2. Zero temperature value of det( R ) [see Eq. (7)] as a function of ∆ and q obtained for 70 sites, using J = 2 and differentvalues of µ . The yellow circles show q ∗ , minimizing the ground state energy. Note, that for µ = 0 . q ∗ is equal to 0.5. of the proximity–induced pairing the system will self–organize into a helical structure with the spiral pitch q ∗ coinciding with the momentum 2 k F . However,even in absence of the induced superconductivity the spi-ral pitch that minimizes the ground state energy candeviate from 2 k F if one goes beyond the Born approx-imation in the RKKY scheme . We have investigatednumerically variation of the ground state energy with re- spect to the model parameters and found, that q ∗ ≈ k F only in some regimes, whereas generally q ∗ can vary from0 (fully polarized magnetic moments) to π/a (anitferro-magnetic ordering). Fig. 1a shows a typical example ofthe ground state energy dependence on q .To distinguish the trivial from nontrivial supercon-ducting phases we have computed the topological number Z , determining it from the scattering matrix . Here .
06 0 .
08 0 .
10 0 .
12 0 .
14 0 .
16 0 .
18 0 .
20 0 . . . . . µ det( R ) J = 0 . −
101 0 .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 .
30 0 . . . . . µ det( R ) J = 0 . − .
10 0 .
15 0 .
20 0 .
25 0 .
30 0 .
35 0 .
40 0 . . . . . . µ DCBA det( R ) J = 1 . − . . . . . . . . . . µ det( R ) J = 2 . − . . . . . . . . . . . . . . . µ det( R ) J = 5 . −
101 1 . . . . . . . . . . . . . . . µ det( R ) J = 10 . − FIG. 3. Zero temperature value of det( R ) [see Eq. (7)] as a function of ∆ and µ for different values of J , ranging from 0.5to 10. The blue regions represent the topologically nontrivial phase ( Q <
0) with the Majorana end-modes. We have chosen q to minimize the ground state energy for the chain of L = 70 lattice sites. Yellow circles labelled A, B, C, D indicate theparameters, for which results are presented in Fig. 7. we follow the procedure described in Ref. 28. We havethus computed the scattering matrix S of the chain S = (cid:18) R T (cid:48)
T R (cid:48) (cid:19) , (5)where R and T ( R (cid:48) and T (cid:48) ) are 4 × (cid:18) ψ − , L ψ + , R (cid:19) = S (cid:18) ψ + , L ψ − , R (cid:19) , (6)where ψ ± ,L/R are the right or left moving modes ( ± )at the left or right edge ( L/R ) at the Fermi level. Thetopological quantum number is given by Q = sign det( R ) = sign det( R (cid:48) ) . (7)The scattering matrix S can be obtained from multipli-cation of the individual transfer matrices of all the latticesites. Since the product of numerous transfer matrices isnumerically unstable, we converted them into a compo-sition of the unitary matrices, involving only eigenvaluesof unit absolute value.The spiral pitch q can in general be treated as an in-dependent parameter and we can study the topologicalproperties of the Hamiltonian (1) as its function. Fig. 2shows det( R ) versus q and ∆ for J = 2 and several valuesof the chemical potential µ (analogous data have beenobtained by us also for the stronger coupling J ). Ineach panel we display the spiral pitch q ∗ (yellow line),that minimizes the ground state energy. Such curves re-semble the results obtained previously in the weak cou-pling limit J (see Fig. 3 in Ref. 53). Let us remark, thatfor the wide range of model parameters the spiral pitch q ∗ (∆) indeed coincides with the topological region. Itmeans that the system has a natural tendency towardsself-adjusting the local magnetic moments in a way thatguarantees the topologically nontrivial superconductingstate . Nevertheless, closer inspection of Fig. 2 re-veals that such tendency is not universal. For instance,for µ = 0 . q ∗ . Also, for µ = 2 . . (cid:46) ∆ (cid:46) .
87, but it coincides with q ∗ only in a nar-row regime 0 . (cid:46) ∆ (cid:46) .
72. Fig. 3 shows examples ofthe topological phase diagrams with respect to ∆ and µ for the nanochain consisting of 70 sites, assuming the sta-ble spiral orderings q = q ∗ . Role of the finite–size effectsis presented in Appendix (Fig. 13). We noticed thatwith increasing length L the topological regions gradu-ally expand and their boundaries become sharper. IV. ROLE OF THERMAL EFFECTS
Influence of finite temperatures on the model (1) canbe seen in a twofold way: by thermal broadening of theFermi–Dirac distribution function of itinerant electronsand by disturbance induced among the classical localmoments S i . Since the energy resulting from rearrange-ment of the magnetic moments is much lower than costsof the thermal excitations of itinerant electrons, we fo-cus on fluctuations of the classical moments and assumethat fermions are in their ground state . Such fluctua-tions are expected to suppress ordering of the local mo-ments, indirectly affecting the topological superconduct-ing phase.To estimate the critical temperature T c up to whichthe topologically nontrivial state can persist, we haveperformed the MC simulations for the localized magneticmoments. Since the Hamiltonian (1) includes both the quantum (fermions) and classical (localized magnetic mo-ments) degrees of freedom, we apply the method used inRef. 55. At each MC step a randomly chosen localizedmagnetic moment is rotated, the Hamiltonian (1) withactual configuration of S i is diagonalized and the trialmove is accepted or rejected according to the Metropoliscriterion based on the free energy instead of the inter-nal energy. During such routine we have computed thetopological quantum number Q and various correlationfunctions. Great advantage of the MC method is that wedo not need any particular ansatz for the magnetic orderwhat is crucial for inspecting the self–organized struc-tures composed of, e.g., several coexisting phases .Most of our results refer to the magnetic moments con-fined to a plane, therefore only the azimuthal angles φ i have been varied in MC simulations. Sec. V presentssome results for the case when this constraint is relaxed.In what follows, we discuss the most interesting resultsobtained within the aforementioned algorithm. A. Correlation function
In Sec. III we have inspected the long–range spiral or-dering of the ground state. Here, we analyze how thisorder is affected by thermal fluctuations. In Fig. 4 weshow the structure factor of the magnetic order A ( q ) =1 /L (cid:80) jk e iq ( j − k ) (cid:104) S j · S k (cid:105) obtained at different tempera-tures, as indicated. At very low temperature there is anarrow peak at q = q ∗ , indicating that magnetic con-figurations are nearly identical with the perfect zero–temperature long-range order. With increasing temper- FIG. 4. The structure factor of the magnetic order obtainedfor J = 1 and the model parameters referring to the pointC in Fig. 3. Results are averaged over 10 statistically inde-pendent configurations generated during MC runs at temper-atures T = 10 − , − , − and 10 − . The arrows (whosecolors correspond to the Fourier transforms in the main) showrepresentative configurations at various temperatures. ature this peak remains at its original position, but itswidth substantially broadens and its height is reduced.This signals that thermal fluctuations are detrimentalfor the magnetic ordering. We illustrate this behaviorin the inset in Fig. 4, where spatial configurations of S j are presented for indicated temperatures.Stability of the spiral order against thermal fluctua-tions is determined by the strength and range of the effec-tive interaction between the localized magnetic moments.The interaction is mediated by itinerant electrons whichare paired through the proximity effect. Since the long–range type of the RRKY interaction in one–dimensionalsystems results from the gapless nature of excitationsnear the Fermi point, it is possible that in our case theeffective interaction can differ from the standard one typ-ical for metals. Proximity to a bulk superconductor cansubstantially affect its range, which should be importantfor any magnetic order at finite temperatures . In par-ticular, if the interaction varies as r − α the long–rangeorder could exist for α < S Heisenberg model .To get an insight into effective interactions betweenthe localized moments and role of the thermal effects we − C ( r ) a) T = 0 . − C ( r ) b) T = 0 . r − C ( r ) c) T = 0 . FIG. 5. Correlation function between the local magnetic mo-ments (8) as a function of distance r obtained at representa-tive temperatures for µ = 1 .
7. The red thick points show theMC data while the blue line is the best fit with a functiondefined in Eq. (9). − − − − − T ξ MC resultsfit ∝ T − FIG. 6. Log–log plot of the correlation length versus temper-ature for the same model parameters as in Fig. 5. The redthick dots display the MC data and the blue line is the bestfit with a function ξ ( T ) = A T − . have analyzed the correlation function defined as C ( r ) = 1 L − r − s L − r − s (cid:88) i = s (cid:104) S i · S i + r (cid:105) , (8)where L denotes the nanochain length and s is a small off-set introduced to minimize the finite size effects. Resultsof our numerical MC computations obtained for threerepresentative temperatures are presented by the thickred dots in Fig. 5. The simulations show that the expo-nential decay of the two–point correlation function hasa power law correction. The classical Ornstein–Zernikepower ( d − /
2, where d is the dimensionality of thesystem, vanishes in a one-dimensional system . Here,however, the MC results can be very well fitted by C ( r ) ∝ cos( qr ) r − α e − r/ξ ( T ) , (9)where α is small ( α (cid:28)
1) and slightly temperature de-pendent. Major influence of thermal fluctuations is seenby the correlation length ξ ( T ) (see Fig. 6). We have alsotreated q as a fitting parameter, but it turned out thateven at elevated temperatures its value was very close to q ∗ that minimizes the ground state energy.Fitting the MC results by C ( r ) defined in Eq. (9) hasenabled us to determine the temperature-dependent cor-relation length. As can be seen in Fig. 6, it diverges for T →
0, indicating that the effective interaction is tooshort–ranged to produce any long–range order at finitetemperatures. Nevertheless, at sufficiently low tempera-ture the correlation length is comparable to the nanowirelength therefore the system remains in the topologicallynontrivial state with the Majorana modes located at itsedges. For unambiguous verification of such possibilitywe have directly calculated the topological properties ofthe system at finite temperatures (Sec. IV B).
B. Topological phase at finite temperatures
Fig. 7a displays variation of the topological invariant Q during the MC runs performed at different tempera-tures (vertical axis). We have chosen the model param-eters which guarantee the system to be in topologicallynontrivial phase at zero temperature (point C in Fig. 3,corresponding to J = 1 . MC sweep . . . . . . T a) .
00 0 .
02 0 .
04 0 .
06 0 .
08 0 . T − . − . − . − . . . . . . h Q i b) µ = 0 . (point A) µ = 1 . (point B) µ = 1 . (point C) µ = 1 . (point D) µ = 1 . .
000 0 .
025 0 .
050 0 .
075 0 . T − h Q i FIG. 7. a) Variation of topological invariant Q during theMC sweeps obtained for varying temperature. Blue regionscorrespond to Q = − Q = +1, respectively. Theresults refer to J = 1 , µ = 1 . , ∆ = 0 .
27 (point C in Fig. 3).b) Temperature dependence of the invariant Q averaged over10 MC sweeps for the model parameters indicated by pointsA-D in Fig. 3. The thick black dotted line marked as “3D”shows Q calculated for point C under the assumption that themagnetic moments S i are not confined to a plane (see SectionV). The inset presents the standard deviation of Q obtainedfor point C. creasing temperature more and more frequently the sys-tem prefers the topologically trivial state. Such gradualchangeover from the topological to non-topological phasedepends on the chemical potential (Fig. 7b) and other pa-rameters as well. Roughly speaking, for the chosen set ofmodel parameters the topological phase exists up to thecritical temperature T c ∼ .
05 (in units of the hopping integral). Considering typical values t ∼
10 meV thiswould yield the critical temperature for the topologicalsuperconducting phase T c ∼
6K which is a more stringentlimitation than all previous estimations . C. Spectral functions
Another evidence for the detrimental influence of ther-mal effects on the topological superconductivity and theMajorana modes can be seen directly from the quasipar-ticle spectra of fermions. The spectral function A ( k, ω ) = − π Im G ( k, ω + i + ) (10)can be obtained using the single particle Green’s function G ( k, z ) δ ( k − k (cid:48) ) = (cid:88) m,n (cid:104)G mn ( z ) (cid:105) e i ( mk − nk (cid:48) ) . (11)Here G mn ( z ) = { [ z − H ] − } mn is defined in the real-space for a given configuration of the localized moments(recall that the lattice constant a ≡
1) and (cid:104) . . . (cid:105) denotesaveraging over configurations generated in MC runs.Let us first inspect the spectral function (10) at zerotemperature to demonstrate its characteristic featuresupon entering the topological regime. Fig. 8 presentsevolution of the low energy spectrum, showing emergenceof the zero-energy Majorana mode. For a given value of∆ we have computed the optimal pitch q ∗ of the groundstate and then determined A ( k, ω ) for the model Hamil-tonian (1) with such particular configuration of the localmoments S i . In other words, at zero temperature the av-eraging over configurations (cid:104) . . . (cid:105) defined in Eq. (11) wasnot necessary. For the chosen value µ = 2 . q ∗ (∆) is shown by the yellow line in Fig. 2. Inparticular, we can notice the qualitative change (fromtopological to nontopological phase) when ∆ varies from0 .
70 to 0 .
72, which corresponds to the abrupt jump of q ∗ (∆) displayed in Fig. 2.Influence of thermal effects of the spectral function(10) is illustrated for the representative set of model pa-rameters in Fig. 9. At zero temperature the Majoranamode (appearing near boundaries of the Brillouin zone,as shown by the inset) is protected from the finite-energyAndreev quasiparticles by the topological gap. Upon in-creasing the temperature such topological gap graduallydiminishes. This is accompanied by an ongoing disorder-ing of the local magnetic moments leading to a broad-ening of all the spectral lines. Ultimately, at temper-atures T (cid:39) .
05 the topological gap is hardly visible,and the zero-energy feature merges with a continuum.Nonetheless, even at higher temperatures we could stillresolve some remnants of the overdamped zero-energymode. This brings us to the conclusion that the topolog-ical superconductivity vanishes near such critical tem-perature in a continuous manner (like a crossover ratherthan typical phase transition). − . − . − . − . − . − . . . . ω ∆ = 0 . − . − . − . − . − . − . . . . ∆ = 0 . − . − . − . − . − . − . . . . ∆ = 0 . − . − . − . − . − . − . . . . ω ∆ = 0 . − . − . − . − . − . − . . . . ∆ = 0 . − . − . − . − . − . − . . . . ∆ = 0 . − . − . − . − . − . − . . . . ω ∆ = 0 . − . − . − . − . − . − . . . . ∆ = 0 . − . − . − . − . − . − . . . . ∆ = 0 . − . − . − . − . k − . − . . . . ω ∆ = 0 . − . − . − . − . k − . − . . . . ∆ = 0 . − . − . − . − . k − . − . . . . ∆ = 0 . FIG. 8. Evolution of the zero temperature spectral functions with respect to varying ∆ obtained for q = q ∗ which for the modelparameter µ = 2 . J = 2 is shown by the yellow line in Fig. 2. Note, that the presence of the zero-energy feature coincideswith q ∗ being in the topological region (blue area in Fig. 2). V. BEYOND COPLANAR ORDERING
Finally, we have checked whether deviation of the az-imuthal angle of the local moments (3) from its copla-nar value θ i = π/ θ i , φ i ) on equalfooting. To find the lowest energy configuration of thelocalized magnetic moments we used the simulated an-nealing method .At very low temperature the local magnetic momentsare arranged in a coplanar spiral albeit now the planeof moments rotation is arbitrarily oriented, what reflects the symmetry of the Hamiltonian (1). This situation isillustrated in Fig. 10a, where the moments have beenshifted so that their origins are in the same point. As aresult, the zero temperature phase diagrams are the sameas in Fig. 3. With increasing temperature, the momentsdeviate from their coplanar arrangement (besides intro-ducing in–plane disorder) what is illustrated in Fig. 10band c. Similarly to the previously studied case, wherethe moments were confined to a plane, it may lead todestruction of the topological state. An example of sucha behavior is illustrated by the thick dotted black linein Fig. 7b. One can notice there that the temperaturedependence of (cid:104)Q(cid:105) is almost unaffected by the presence − π − π/ π/ πk − − − − ω T = 0 . − π − π/ π/ πk − − − − ω T = 0 . − π − π/ π/ πk − − − − ω . . . . . . . T = 0 . − π − π/ π/ πk − − − − ω . . . . . . T = 0 . − π − π/ π/ πk − − − − ω . . . . . . T = 0 . − π − π/ π/ πk − − − − ω . . . . . T = ∞ FIG. 9. The spectral function [defined in Eq. (10)] averaged over 10 statistically independent configurations of the localizedmoments { S i } . MC results are obtained for J = 1 , µ = 1 . , ∆ = 0 .
27 and several temperatures, as indicated. The zoomedregion displays the zero–energy mode. of the additional degree of freedom, what may suggestthat polar angle θ i is rather irrelevant for stability of thetopologically nontrivial superconducting phase.This property, however, is not universal. Fig. 11 showsthe temperature dependence of (cid:104)Q(cid:105) for a different setof the model parameters. In this case the topologicalphase is destroyed by increasing temperature when onlyin–plane thermal fluctuations of the localized moments are allowed, but it survives to pretty high temperatureswhen they rotate freely in all three dimensions. Since athigh temperature the helical order vanishes, the modelHamiltonian (1) cannot be related to the scenario withthe spin–orbit and Zeeman interactions .However, it was shown in Ref. 28 that even withoutthe helical order this Hamiltonian can have topologi-cally nontrivial state provided the localized magnetic mo-0 FIG. 10. Orientations of the localized magnetic moments atdifferent temperatures. The model parameters correspond topoint C in Fig. 3. For the sake of clearness the origin of allthese vectors has been collected to a common point and theaverage plane of the order tilted to be horizontal. ments point in different directions. It is usually assumedthat for sufficiently large J the electron spin is paral-lel to the localized magnetic moment. We have verifiedthis assumption by calculating the correlation function L (cid:80) i (cid:104) S i · ˆ s i (cid:105) . The results show that the electron spinis almost completely polarized along the localized mag-netic moments for arbitrary value of J . In such a casethe Hamiltonian (1) can be projected onto the lowest spinband and take a form of Kitaev’s chain with additionalhopping to the next nearest neighbors . In the effec- .
00 0 .
02 0 .
04 0 .
06 0 .
08 0 . T − . − . − . − . . . . . . h Q i coplanar3D − − T − . − . . . . h Q i FIG. 11. Temperature dependence of the average topologicalinvariant Q for coplanar configurations of the localized mag-netic moments (blue solid line) and when their rotation isallowed in arbitrary direction (red dashed line marked “3D”).The inset shows the same but in a semilogarithmic scale fora wider range of temperatures. The model parameters are J = 4 , ∆ = 0 . µ = 2 . tive Kitaev Hamiltonian the pairing potential increaseswith increasing disorder of the magnetic moments andcan drive the system into the topological phase. Thiscan explain that while the spiral ordering is destroyed athigh temperature, another mechanism can still keep thesystem in the nontrivial state, as marked by the red linein Fig. 11. VI. SUMMARY
We have investigated stability of the topologically non-trivial superconducting phase of itinerant electrons cou-pled to the local magnetic moments in the finite-lengthnanowire proximitized to s -wave superconductor. Wehave performed the MC simulations, considering variousconfigurations of such local moments constrained on aplane and oriented arbitrarily in all three directions. Wehave focused on the role played by thermal fluctuations.MC simulations clearly indicate that self-organization ofthe local moments into the spiral order gradually ceasesupon increasing the temperature. We have found theuniversal scaling of the correlation function for the lo-calized magnetic moments (8) and determined the co-herence length, revealing its characteristic temperaturedependence ξ ( T ) ∝ /T .Our MC data for the topological invariant and anal-ysis of the quasiparticle spectrum both unambiguouslyshow the upper (critical) temperature T c , above whichthe topological nature of the superconducting phase nolonger exists. When approaching this critical tempera-ture from below there occurs a gradual reduction of thetopological gap, protecting the zero-energy mode fromthe finite-energy (Andreev-type) quasiparticles, so thatat T → T c the Majorana modes get overdamped. Our1quantitative estimations show that T c ∼ .
05 (in unitsof the hopping integral) what in realistic systems wouldyield T c ∼ ACKNOWLEDGMENTS
We acknowledge discussions with Jelena Klinovajaand Jens Paaske. This work is supported by theNational Science Centre (Poland) under the contractsDEC-2018/29/B/ST3/01892 (A.G.–G. and M.M.M.)and DEC-2017/27/B/ST3/01911 (T.D.).
Appendix: Finite-size scaling
In the scenario based on the Rashba nanowire proxim-itized to a bulk superconductor a sharp transition fromthe topologically trivial to nontrivial regime has been pre-dicted only for infinitely long wires and it has been em-phasized that finite–size effects would smooth it into acrossover. Due to correspondence between systems withthe spin–orbit and Zeeman interactions and systems withthe spiral ordering of localized moments, the same effectcan be expected for the present model described by theHamiltonian (1). To verify it we performed additionalcalculations for various lengths L of nanowires, compris-ing 40 to 200 sites.Pitch vector q ∗ of the ground state (Fig. 12) and dia-grams of the topological superconducting phase (Fig. 13)clearly indicate that: ( i ) q ∗ is hardly affected by nanowirelength L , ( ii ) total area of the topological phase in theparameter space increases with increasing L and ( iii )boundaries of the topological region are much sharperfor longer nanowires. 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