Self-stabilizing temperature driven crossover between topological and non-topological ordered phases in one-dimensional conductors
SSelf-stabilizing temperature driven crossover between topological and non-topologicalordered phases in one-dimensional conductors
Bernd Braunecker and Pascal Simon SUPA, School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK Laboratoire de Physique des Solides, CNRS UMR-8502,Universit´e de Paris Sud, 91405 Orsay Cedex, France (Dated: October 29, 2018)We present a self-consistent analysis of the topological superconductivity arising from the interac-tion between self-ordered localized magnetic moments and electrons in one-dimensional conductorsin contact with a superconductor. We show that due to a gain in entropy there exists a magneticallyordered yet non-topological phase at finite temperatures that is relevant for systems of magneticadatom chains on a superconductor. Spin-orbit interaction is taken into account, and we show thatit causes a modification of the magnetic order yet without affecting the topological properties.
PACS numbers: 74.20.Mn, 71.10.Pm, 75.30.Hx, 75.75.-c
Introduction.
Topological superconductors have re-ceived much attention recently, partly because they hostexotic low energy excitations such as Majorana boundstates [1–3], whose non-Abelian statistics are attractivefor topological quantum computation [4, 5]. As a re-markable feature, topological superconductivity can becreated artificially by contacting specific materials witha conventional s -wave superconductor. For instance, itarises at the interface between the surface states of athree-dimensional topological insulator and a s -wave su-perconductor [6]; in one-dimensional (1D) semiconduct-ing wires with a strong spin-orbit interaction (SOI) anda Zeeman magnetic field with proximitized superconduc-tivity [7–11]; or in arrays of magnetic nanoparticles ormagnetic adatoms on top of a superconducting surface[12–22] such as iron adatoms on lead [23–25].The systems we consider in this letter exhibit a topo-logical phase emerging from self-organization of magneticmoments embedded in 1D conductors with proximity in-duced superconductivity. This situation may apply tosemiconducting wires with extrinsic magnetic impuritiesor intrinsic moments such as nuclear spins, or a conduct-ing wire made of magnetic adatoms on a superconduct-ing surface. Due to the Ruderman-Kittel-Kasuya-Yosida(RKKY) interaction mediated through the electrons, themagnetic moments can undergo an ordering transitionbelow a temperature T ∗ and form a spiral with a spa-tial period characterized by the wave number 2 k m (see superconductor 1D conductormagnetic moments ... ... FIG. 1: Zoom on a 1D conductor with embedded magneticmoments on top of a superconductor. The magnetic momentsself-order in the form a spiral order with spatial period π/k m . Fig. 1) such that k m = k F , for k F the Fermi momen-tum. This ordering mechanism was first demonstratedfor normal conductors [26, 27], then conjectured [13] andself-consistently demonstrated [28–30] for the supercon-ducting case. These results were corroborated recentlyby showing that the spiral order persists beyond theRKKY limit, and k m stays close to k F , as long as k F isaway from commensurate band fillings and the couplingstrength A between magnetic moments and electrons re-mains smaller than the electron bandwidth [31, 32].The locking of k m to k F has important consequences.The magnetic spiral forms a periodic superstructure thatcauses a part of the electrons to undergo a spin-selectivePeierls transition [33] to a non-conducting spiral elec-tron spin density wave, whereas the remaining conduct-ing electron states become helical (spin-filtered). Theinduced superconductivity then becomes of the topolog-ical p -wave type, and Majorana bound states appear atthe two ends of the 1D wire. A system with such a spiralmagnetic order is indeed equivalent [12–18, 21, 28–30, 33–37] to the original proposals for topological superconduc-tivity in nanowires [7, 8]. Remarkably, by this mech-anism, the topological superconducting phase emergesnaturally as the ground state without any fine tuning.Although both the RKKY based and the nonperturba-tive approaches consistently predict the locking condition k m ∼ k F , it must be stressed that the former results arebased on the further condition of large magnetic and su-perconducting gap energies, whereas the latter apply onlyat zero temperature.In this letter, we provide a general analysis which in-corporates entropy and thermal fluctuations in the non-zero temperature regime and we show that there existsa previously unknown crossover to a magnetically or-dered yet non-topological phase. Furthermore, we showthat the spin-orbit interaction (SOI), which is genuinelypresent in such systems either intrinsically or throughinterface effects, causes a modification of the magneticspiral but has no effect on the topological properties. a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec (b) k m k F k m = k F J n o r m a l (cid:15) k (a) k m k F k m < k F (c) k m k F k m > k F − − k/k F (e) ∗ − − k/k F s up e r c . E k (d) ∆ ∗ = 0 − − k/k F (f) ∆ ∗ = 0 FIG. 2: Example of dispersion relations in the trans-formed basis obtained by the spin dependent shift of momenta k → k + σk m for which the spiral effective magnetic fieldbecomes ferromagnetic and the Zeeman like gap J opens at k = 0 [33, 38]. In the plots, the chemical potential µ remainsconstant, but k m varies. Top panels for the normal state( (cid:15) k ), with red and blue colors corresponding to opposite spinprojections perpendicular to the spiral field. Bottom panelsfor the induced superconducting state ( E k , with ∆ s < J ).For k m = k F , the chemical potential µ (dashed line) lies inthe middle of the J gap (b) and the superconducting sys-tem is fully gapped (e). For smaller or larger k m , the gaplies at lower or higher energies and µ eventually touches theupper (a) or lower gap edge (c). At both touching pointsthe superconducting gap closes (d,f) and the state becomesnon-topological. Heuristic considerations.
The physical origin of thenon-topological phase is illustrated in Fig. 2. At lowtemperatures T , the thermodynamic ground state is de-termined by the gain in electronic energy E obtainedby maintaining large magnetic ( J ) and superconducting(∆ s ) gaps, and the system adjusts k m to k F . As T israised, however, the ground state is dictated by the freeenergy F = E − T S , and the entropy S can play a deci-sive role. Indeed, if k m is lowered or raised to the valuesas indicated in the left and right panels of Fig. 2, suchthat the chemical potential µ touches the bottom or topof a band, the induced superconducting gap closes (for J > ∆ s ) because the touched bands are fully spin polar-ized. The effective dispersion arising from the supercon-ducting case becomes gapless, with a larger entropy thanin the gapped case. As a result, if T is large enough, typ-ically k B T < ∼ ∆ s (with k B the Boltzmann constant), theminimization of F can be dominated by the enhancementof S , and the thermodynamic ground state correspondsto situations (d) or (f) in Fig. 2, a topologically trivialyet magnetically ordered phase.Not yet taken into account in this argument is the sta-bility of this phase upon thermal fluctuations of the mag-netic moments. As shown in [26–28], for both the un-gapped and gapped cases a mean-field description of thespin-wave fluctuations captures the correct value of theordering temperature and T ∗ ∝ | χ k m | , with χ k m thetransverse spin susceptibility at momentum 2 k m . Since | χ k m | increases for a gapless dispersion, closing the su-perconducting gap by moving k m away from k F causes n o r m a l (cid:15) k (a) k m k F k m = πa − k F (b) k m k F k m = 0 . πa (c) k m k F k m = π a − . − . . . . ka/π s up e r c . E k (d) ∗ − . − . . . . ka/π (e) ∆ ∗ = 0 − . − . . . . ka/π (f) ∆ ∗ = 0 FIG. 3: Dispersion relations as in Fig. 2 for the tight bindingmodel associated with Eq. (1) and corresponding to Fig. 4.Parts (a,d) represent the low-temperature topological phase;parts (b,e) the crossover to the non-topological phase wherethe gap ∆ ∗ closes; parts (c,f) the final high-temperature phasewhere k m = π/ a and the system is magnetically ordered yetnon-topological. furthermore an enhanced stability against thermal fluc-tuations, provided that the effect occurs at T < T ∗ . Ob-viously, the latter condition depends on the consideredmaterial as specified below. For practical implementa-tions, the condition k B T ∗ ∼ ∆ s is a priori required forthe topological self-tuning phase to be accessible at highenough temperatures, and for T close to T ∗ the non-topological ordered phase may indeed be favored. Yetwe find that for semiconductor bands with an effectivemass as in Fig. 2, the value of T ∗ remains generally stilltoo low, such that S would dominate E only at temper-atures T > ∼ T ∗ where no order can persist anyway.This situation changes drastically for tight binding sys-tems such as shown in Fig. 3, which are the natural de-scription for adatom chains. Due to the cosine natureof the dispersion, magnetic gaps appear at two pointsin the Brillouin zone, and k m can self-adjust such thatthe superconducting-magnetic gaps ∆ ∗ at both pointsbecome equal and fulfill the condition ∆ ∗ < { k B T ∗ , ∆ s } [see Fig. 3 (f)]. At T ∼ ∆ ∗ the effective doubling of ther-mally accessible states provides a doubling of the valueof S . This is sufficient to push the transition to the non-topological to T < T ∗ in precisely the systems that aremost attractive for realizing a self-sustained topologicalphase. Furthermore, the equality of the two gaps ∆ ∗ leads to k m = π/ a (with a the lattice spacing), whichcorresponds to an antiferromagnetic arrangement of themagnetic moments if the latter are on the same latticesites. Quantitative analysis . For a quantitative investigationwe consider a quantum wire with induced superconduc-tivity and embedded magnetic moments, described bythe Hamiltonian H = (cid:88) k,σ ( (cid:15) k − µ ) c † k,σ c k,σ + (cid:88) k,σ,σ (cid:48) ( α · σ ) σ,σ (cid:48) kc † k,σ c k,σ (cid:48) (1)+ (cid:88) k (cid:0) ∆ s c † k, ↑ c †− k, ↓ + h.c. (cid:1) + (cid:88) k,q,σ,σ (cid:48) ( J q · σ ) σ,σ (cid:48) c † k + q,σ c k,σ (cid:48) . Here c k,σ are the operators for electron with spin σ = ↑ , ↓ and dispersion relation (cid:15) k . µ is the chemical potential,∆ s the induced superconducting gap, σ = ( σ x , σ y , σ z )the vector of Pauli matrices, and the vector α the ef-fective SOI in the system, arising from the sum of SOIcontributions due to the internal structure of the wire orto interface effects with the substrate. The vectors J q arethe Fourier transforms of the chain of magnetic scatterers J i = A I i coupling to the electron spin, where I i are themagnetic moments and A is the coupling strength. The J i are placed at positions r i that can be irregular but aresufficiently dense with respect to 2 π/k F such that theycan be considered as a continuum.For T = 0 and α = 0, the ground state energyis minimized if the vectors J i are confined to an ar-bitrary two-dimensional plane, spanned by orthogonalunit vectors (ˆ e , ˆ e ), in which they rotate as a spiral asa function of r i , J i = J [cos(2 k m r i )ˆ e + sin(2 k m r i )ˆ e ][12, 13, 18, 28–32, 34, 35]. Choosing ˆ x , ˆ y such that(ˆ x , ˆ y ) = (ˆ e , ˆ e ), the corresponding term in the Hamil-tonian becomes (cid:80) k J ( c † k + k m , ↑ c k − k m , ↓ + h . c . ). Letting c k + σk m ,σ → ˜ c k,σ , (cid:15) k → (cid:15) k ∓ σk m ,σ [33] produces a unitarytransformation diagonalizing the Hamiltonian, in which J forms a uniform ferromagnetic coupling along the spin- x direction. For a parabolic dispersion (cid:15) k = (cid:126) k / m with the band mass m we obtain Fig. 2.Since the SOI term is linear in k , a α (cid:54) = 0 produces asimilar spin-dependent momentum shift. For a quadraticdispersion (cid:15) k = (cid:126) k / m with m the band mass, andspin axes σ α such that α · σ = ασ α , the SOI can beabsorbed by letting (cid:15) k → (cid:15) k + σ α k SO , with k SO = αm/ (cid:126) [33]. If α is not perpendicular to the (ˆ e , ˆ e ) plane, thetwo shifts by k SO and k m are not compatible, and thediagonalization of H would generally mix all momenta.Such modification of the long-ranged wave functions bythe spiral order would cause an extensive energy costwhich is not favored energetically. This can be avoided,however, by an alignment of the J i spiral to the planeperpendicular to α . We thus define the spin directionssuch that α = α ˆ z , and (ˆ e , ˆ e ) = (ˆ x , ˆ y ), for which k SO and k m are parallel and can be directly added. Remark-ably, to maintain the optimal k m by minimizing the freeenergy, the spiral undergoes an adjustment of k m to a k (cid:48) m such that k m = k (cid:48) m − k SO . While the “ − ” sign arisesfrom the choice of spin axes, k m can have either sign,and so two spirals with opposite helicities and differentperiods, k (cid:48) m = ±| k m | + k SO , are possible. Therefore evena large SOI has no further influence than the pinning ofthe plane of the magnetic spiral together with the ad-justment of k (cid:48) m , provided that 2 π/k (cid:48) m does not becomesmaller than the electron lattice spacing or the averagespacing between the J i . As long as k m = ± k F (up to J dependent corrections that can be included), a mea-surement of the period and plane of the magnetic spiralcould therefore give a direct measurement of k SO ∝ α .Due to the extensive energy cost, there are furthermore no conical deformations out of the spiral plane [18, 39].In a spin-Nambu matrix representation spanned by thevectors ( c † k + k m , ↑ , c † k − k m , ↓ , c − k + k m , ↑ , c − k − k m , ↓ ) the Hamil-tonian takes then the form H = (cid:88) k> ξ k − k m J s J ξ k + k m − ∆ s − ∆ ∗ s − ξ − k − k m − J ∆ ∗ s − J − ξ − k + k m + E , (2)for ξ k = (cid:15) k − µ , k m = k (cid:48) m − k SO , J q = Jδ | q | , | k (cid:48) m | , andthe restriction of the summation to k > E = (cid:80) k> (cid:2) ξ − k + k m + ξ − k − k m + 2 J (cid:3) , and due to its k m dependence must bekept for comparison of different k m . The diagonalizationof the matrix in Eq. (2), for ξ − k = ξ k , leads to theenergies E ν,ν (cid:48) k = ν (cid:48) E k,ν , for ν, ν (cid:48) = ± , with E k, ± = J +∆ s + ξ ,k + ξ − ,k ± (cid:113) ∆ s J + ξ ,k ( J + ξ − ,k ) for ξ ± ,k =( ξ k + k m ± ξ k − k m ) /
2. This leads to the ground state energy E = E + (cid:88) k> ,ν,ν (cid:48) E ν,ν (cid:48) k f ν,ν (cid:48) k , (3)and the entropy S = − k B (cid:88) k> ,ν,ν (cid:48) (cid:104) f ν,ν (cid:48) k ln (cid:0) f ν,ν (cid:48) k (cid:1) + (cid:0) − f ν,ν (cid:48) k (cid:1) ln (cid:0) − f ν,ν (cid:48) k (cid:1)(cid:105) , (4)for f ν,ν (cid:48) k = [1 + exp( E ν,ν (cid:48) k /k B T )] − the Fermi function.Notice that the sums are restricted to k > F = E − T S determines theordering vector k m . However, analyzing only F is incom-plete to assure the stability of the ordered phase since thelong-wavelength spin-wave fluctuations smooth any mag-netization at finite T for a system of finite size [26–30].Taking this condition into account, it was demonstrated[27, 28] that for any realistic system size the mean fieldresult, k B T ∗ = 2 J | χ k m | ∼ J a (cid:48) π (cid:126) v F ln (cid:18) E F Γ (cid:19) , (5)provides the ordering temperature T ∗ for both thegapped and gapless cases, where χ k m is the static trans-verse spin susceptibility, expressed in terms of the Fermienergy E F = (cid:126) v F k F /
2, the Fermi velocity v F , and theshort distance cutoff a (cid:48) (limited by lattice spacing a or distance between the J i ). If ∆ ∗ is the gap as in-dicated in Figs. 2 and 3 the energy Γ is roughly setby max(∆ ∗ , k B T ) and its value reflects the transitionbetween the gapped (∆ ∗ > k B T ) and gapless regimes(∆ ∗ < k B T ). As a consequence, if k m departs from k F and ∆ ∗ shrinks, the value of T ∗ initially grows butthen saturates at a self-consistent value where Γ ∼ k B T ∗ .Electron interactions further modify the logarithm in Eq.(5) to a temperature dependent power-law in the gaplesscase and can lead to a considerable further increase of T ∗ [26, 27]. We note that Eq. (5) results from ana-lyzing fluctuations arising from the RKKY interaction.While the RKKY limit is insufficient to characterize theground state and F needs to be used, it correctly encodesthe fluctuations away from the ground state configurationfor J < a (cid:48) / (cid:126) v F . Application to various systems.
Since the transition tothe non-topological phase occurs at k B T < ∼ ∆ s , materialswith large k B T ∗ ∼ ∆ s shall be considered. With typical∆ s ∼ . − k B T ∗ (cid:28) ∆ s , hence a guaranteed topolog-ical phase, albeit a low transition temperature T ∗ . Formost 1D wire situations (including InAs [28]), we indeedfind that T ∗ remains too low to allow for a strong impactof the entropy, and the self-sustained topological phasesremain stable.Dense chains of magnetic adatoms on a superconduct-ing substrate have a larger coupling constant A . If neigh-boring adatom orbitals hybridize, the chains become con-ductors of the type of Eq. (1). SOI effects are gener-ally strong in such systems [23–25]. However, this causeshere just a mere rearrangement of the magnetic helix k m → k (cid:48) m . While systems such as in [23–25] likely de-pend much on the direct exchange interaction betweenneighboring moments, we focus here instead on the casewhere RKKY dominates over the latter (therefore ourresults do not a priori apply to [23–25]).As demonstrated in [28], with A ∼ . k B T ∗ > ∆ s ∼ T ∗ is, as seen in Eq. (5) a largeprefactor a (cid:48) / (cid:126) v F , which means a rather small bandwidthor a renormalization of the Fermi velocity [41]. A tightbinding model is therefore suitable, in which the factor (cid:126) v F /a (cid:48) is replaced by the hopping integral t , and thedispersion relation is (cid:15) k = − t cos( ak ). Consequently,a minimum of F at a shift k m has a particle-hole re-versed minimum corresponding to the shift ( π/a − k m ).Away from half filling, k F (cid:54) = π/ a , both minima are in-equivalent, yet for k F not too far from π/ a they canlie energetically close enough together such that furtherentropy can be gained by tuning k m through the topo-logical boundary and pin it to k m = π/ a , an antifer-romagnetic order at which the system is non-topologicaland has small gaps but with both minima contributingto S . In combination with a small bandwidth, a large∆ s , and J > ∼ ∆ s , the entropy can become large enoughto dominate F . An example is given in Fig. 4, show-ing F as a function of k m and T . The values k m ( T )minimizing F ( T ) are indicated by the red line. At lowtemperatures k m lies near, but not on ( π/a − k F ) since J and ∆ s cause a significant band deformation due tothe small band width. At k B T ≈ . s we observe a .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 .
30 0 . k B T / ∆ s . . . . . k m a / π k Fπa − k Fπ a ( a ) .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 .
30 0 . k B T / ∆ s . . . . . . k m π / a πa − k Fπ a ( b ) k m : min Fk m : min Sk m : min E − . − . − . − . − . − . − . F [ e V ] FIG. 4: (a) Free energy F for an adatom chain conductor oflength L = 1 µ m as a function of total spiral momentum k m (including SOI contributions) and temperature T [38]. Con-tour lines complement the color coding. The minima k m ( T )are marked by the red line and correspond to the groundstate configuration. The values k F , π/a − k F , and π/ a aremarked by horizontal dashed lines. Parameters for the tightbinding model are t = 10 meV, a = 3 ˚A, ∆ s = 2 meV, J = 3 meV, L = 1 µ m, µ = − . t (7/20 filling). Stability ofthe order is ensured up to k B T ∗ = 0 . s (at the right plotlimits). The gap ∆ ∗ closes at the phase boundary betweenthe topological (clear) and non-topological (hatched) regions.The system becomes non-topological at k B T ≈ . s and k m then stabilizes at π/ a , corresponding to an antiferromag-netic order. (b) Values of k m minimizing the free energy F [same as in (a)], the entropy S , and the energy E . While E alone favors a topological phase, the entropy S favors a gap-less phase, but is at higher temperatures further enhanced bytuning k m → π/a , and eventually dominates the minimum of F at k B T > . s . For k m = π/ a the two identical gapsindicated in Fig. 3 (f) are ∆ ∗ = 0 . s . crossing into the non-topological region (indicated by thehatching), well below k B T ∗ ≈ . s , and the pinningto k m = π/ a . Conclusions . We have analyzed a 1D conductor withspin-orbit interaction coupled to a 1D chain of magneticmoments. Through a self-consistent analysis taking intoaccount the full electronic free energy F and the fluctua-tions about the ordered magnetic ground states, we havedetermined the stability of the topological superconduct-ing phase at finite temperature. We showed that spin-orbit interaction causes only a pinning of the plane of themagnetic spiral and an adjustment of its spatial period.Furthermore, in some situations especially met in sys-tems of magnetic adatoms, we demonstrated that thereis a significant temperature range, in which a magneticorder persist but the electronic state is non-topological. Acknowledgments . We thank C. Carroll, J. Klinovaja,D. Loss and D. Morr for helpful discussions. PS ac-knowledges support by the French Agence Nationale dela Recherche through the contract ANR Mistral. [1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010).[2] J. Alicea, Rep. Prog. Phys. , 076501 (2012).[3] M. Leijnse and K. Flensberg, Semicond. Sci. Technol. ,124003 (2012).[4] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S.Das Sarma, Rev. Mod. Phys. , 1083 (2008).[5] J. K. Pachos, Introduction to Topological QuantumComputation (Cambridge University Press, Cambridge,2012).[6] L. Fu and C. L. Kane, Phys. Rev. Lett. , 096407(2008).[7] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev.Lett. , 077001 (2010).[8] Y. Oreg, G. Refael, and F. Oppen Phys. Rev. Lett. ,177002 (2010).[9] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A.M. Bakkers, and L. P. Kouwenhoven, Science , 1003(2012).[10] A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, andH. Shtrikman, Nano Lett. , 887 (2012).[11] M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson, P.Caroff, and H. Q. Xu, Nano Lett. , 6414 (2012).[12] T.-P. Choy, J. M. Edge, A. R. Akhmerov, and C. W. J.Beenakker, Phys. Rev. B , 195442 (2011).[13] S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, and A.Yazdani, Phys. Rev. B , 020407(R) (2013).[14] F. Pientka, L. I. Glazman, and F. von Oppen, Phys. Rev.B , 155420 (2013).[15] F. Pientka, L. I. Glazman, and F. von Oppen, Phys. Rev.B , 180505(R) (2014).[16] K. P¨oyh¨onen, A. Weststr¨om, J. R¨ontynen, and T. Oja-nen, Phys. Rev. B , 115109 (2014).[17] J. R¨ontynen and T. Ojanen, Phys. Rev. B , 180503(2014).[18] Y. Kim, M. Cheng, B. Bauer, R. M. Lutchyn, and S. DasSarma, Phys. Rev. B , 060401(R) (2014).[19] A Heimes, P Kotetes, G Sch¨on, Phys. Rev. B ,060507(R) (2014). [20] P. M. R. Brydon, S. Das Sarma, H.-Y. Hui, and J. D.Sau, Phys. Rev. B
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