Counting Majorana bound states using complex momenta

Recently, the connection between Majorana fermions bound to the defects in arbitrary dimensions, and complex momentum roots of the vanishing determinant of the corresponding bulk Bogoliubov–de Gennes (BdG) Hamiltonian, has been established (EPL, 2015, 110, 67005). Based on this understanding, a for...

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1. Verfasser:	Mandal, I.
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spelling	irk-123456789-1562212019-06-19T01:25:26Z Counting Majorana bound states using complex momenta Mandal, I. Recently, the connection between Majorana fermions bound to the defects in arbitrary dimensions, and complex momentum roots of the vanishing determinant of the corresponding bulk Bogoliubov–de Gennes (BdG) Hamiltonian, has been established (EPL, 2015, 110, 67005). Based on this understanding, a formula has been proposed to count the number (n) of the zero energy Majorana bound states, which is related to the topological phase of the system. In this paper, we provide a proof of the counting formula and we apply this formula to a variety of 1d and 2d models belonging to the classes BDI, DIII and D. We show that we can successfully chart out the topological phase diagrams. Studying these examples also enables us to explicitly observe the correspondence between these complex momentum solutions in the Fourier space, and the localized Majorana fermion wavefunctions in the position space. Finally, we corroborate the fact that for systems with a chiral symmetry, these solutions are the so-called “exceptional points”, where two or more eigenvalues of the complexified Hamiltonian coalesce. Нещодавно (EPL, 2015, 110, 67005) було встановлено зв’язок мiж фермiонами Майорани, зв’язаними з дефектами у довiльнiй вимiрностi, i комплексними iмпульсними коренями детермiнанта вiдповiдного об’ємного гамiльтонiану Боголюбова-де Жена. Базуючись на цьому розумiннi, запропоновано формулу для пiдрахунку числа (n) зв’язаних станiв Майорани з нульовою енергiєю, якi пов’язанi з топологiчною фазою системи. В цiй статтi дається вивiд формули пiдрахунку, яка застосовується до низки 1d i 2d моделей, що належать до класiв BDI, DIII i D. Показано, як можна успiшно побудувати топологiчнi фазовi дiаграми. Вивчення даних прикладiв дозволяє явно спостерiгати вiдповiднiсть мiж цими комплексними розв’язками для iмпульсу в Фур’є просторi i локалiзованими хвильовими функцiями фермiонiв Майорани в позицiйному просторi. Накiнець, пiдтверджено факт, що для систем з хiральною симетрiєю цi розв’язки є так званими “винятковими точками”, де два чи бiльше власних значень ускладненого гамiльтонiана зливаються. 2016 Article Counting Majorana bound states using complex momenta / I. Mandal // Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 33703: 1–21. — Бібліогр.: 56 назв. — англ. 1607-324X PACS: 73.20.-r, 74.78.Na, 03.65.Vf DOI:10.5488/CMP.19.33703 arXiv:1503.06804 http://dspace.nbuv.gov.ua/handle/123456789/156221 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Recently, the connection between Majorana fermions bound to the defects in arbitrary dimensions, and complex momentum roots of the vanishing determinant of the corresponding bulk Bogoliubov–de Gennes (BdG) Hamiltonian, has been established (EPL, 2015, 110, 67005). Based on this understanding, a formula has been proposed to count the number (n) of the zero energy Majorana bound states, which is related to the topological phase of the system. In this paper, we provide a proof of the counting formula and we apply this formula to a variety of 1d and 2d models belonging to the classes BDI, DIII and D. We show that we can successfully chart out the topological phase diagrams. Studying these examples also enables us to explicitly observe the correspondence between these complex momentum solutions in the Fourier space, and the localized Majorana fermion wavefunctions in the position space. Finally, we corroborate the fact that for systems with a chiral symmetry, these solutions are the so-called “exceptional points”, where two or more eigenvalues of the complexified Hamiltonian coalesce.
format	Article
author	Mandal, I.
spellingShingle	Mandal, I. Counting Majorana bound states using complex momenta Condensed Matter Physics
author_facet	Mandal, I.
author_sort	Mandal, I.
title	Counting Majorana bound states using complex momenta
title_short	Counting Majorana bound states using complex momenta
title_full	Counting Majorana bound states using complex momenta
title_fullStr	Counting Majorana bound states using complex momenta
title_full_unstemmed	Counting Majorana bound states using complex momenta
title_sort	counting majorana bound states using complex momenta
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2016
url	http://dspace.nbuv.gov.ua/handle/123456789/156221
citation_txt	Counting Majorana bound states using complex momenta / I. Mandal // Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 33703: 1–21. — Бібліогр.: 56 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT mandali countingmajoranaboundstatesusingcomplexmomenta
first_indexed	2025-07-14T08:40:43Z
last_indexed	2025-07-14T08:40:43Z
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fulltext	Condensed Matter Physics, 2016, Vol. 19, No 3, 33703: 1–21 DOI: 10.5488/CMP.19.33703 http://www.icmp.lviv.ua/journal Counting Majorana bound states using complex momenta I. Mandal Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo ON N2L 2Y5, Canada Received February 17, 2016, in final form April 14, 2016 Recently, the connection between Majorana fermions bound to the defects in arbitrary dimensions, and com- plex momentum roots of the vanishing determinant of the corresponding bulk Bogoliubov–de Gennes (BdG) Hamiltonian, has been established (EPL, 2015, 110, 67005). Based on this understanding, a formula has been proposed to count the number (n) of the zero energy Majorana bound states, which is related to the topological phase of the system. In this paper, we provide a proof of the counting formula and we apply this formula to a variety of 1d and 2d models belonging to the classes BDI, DIII and D. We show that we can successfully chart out the topological phase diagrams. Studying these examples also enables us to explicitly observe the correspon- dence between these complex momentum solutions in the Fourier space, and the localized Majorana fermion wavefunctions in the position space. Finally, we corroborate the fact that for systems with a chiral symme- try, these solutions are the so-called “exceptional points”, where two or more eigenvalues of the complexified Hamiltonian coalesce. Key words: exceptional points, Majorana fermions, BDI, DIII, D, counting PACS: 73.20.-r, 74.78.Na, 03.65.Vf 1. Introduction Topological superconductors [1] are systems which can provide the condensed matter version of Ma- jorana fermions, because they can host topologically protected zero energy states at a defect or edge, for which the creation operator (γ† E=0) is equivalent to the annihilation operator (γE=0). These localized zero-energy states obey non-Abelian braiding statistics [2, 3], which can find potential applications in designing fault-tolerant topological quantum computers [2, 4]. Although Majorana fermion bound states have not yet been conclusively found in nature, they have been theoretically shown to exist in low di- mensional spinless p-wave superconducting systems [2, 5], as well as other systems involving various heterostructures with proximity-induced superconductivity which are topologically similar to them [6– 12]. Non-interacting Hamiltonians for gapped topological insulators and topological superconductors, in arbitrary spatial dimensions, can be classified into ten topological symmetry classes [13–15], character- ized by certain topological invariants. Moreover, there exists a unified framework for classifying topo- logical defects in insulators and superconductors [16], which follows from the bulk-boundary correspon- dence and identification of the protected gapless fermion excitations with topological invariants char- acterizing the defect. Here we focus on 1d and 2d Bogoliubov-de Gennes (BdG) Hamiltonians with the particle-hole symmetry (PHS) operator squaring to +1, which can be categorized [13] into three classes: BDI, DIII and D. In our earlier work [17], we have explored the connection between the complex momentum solu- tions of the determinant of a bulk BdG Hamiltonian (HBdG) in arbitrary dimensions, and the Majorana fermion wavefunctions in the position space associated with a defect or edge. We have found that the imaginary parts of these momenta are related to the exponential decay of the wavefunctions, localized at the defects, and hence their sign-change at a topological phase transition point signals the appearance © I. Mandal, 2016 33703-1 http://dx.doi.org/10.5488/CMP.19.33703 http://www.icmp.lviv.ua/journal I. Mandal or disappearance of Majorana zero mode(s). Based on this understanding, we have proposed a formula to count the number (n) of the zero energy Majorana bound states, which is related to the topological phase of the system. This formula serves as an alternative to the familiar Z and Z2 topological invariants [13, 14, 18] and other counting schemes [19–22]. In this paper, we prove this formula and apply it to a variety of 1d and 2d models belonging to the classes BDI, DIII and D.We show that we can successfully chart out the topological phase diagrams. Study- ing these examples also enables us to explicitly observe the correspondence between these complex mo- mentum solutions in the Fourier space, and the localizedMajorana fermionwavefunctions in the position space. Finally, we also corroborate the fact that for systems with a chiral symmetry, these solutions can be identified with the so-called “exceptional points” (EP’s) [23–29], where two or more eigenvalues of the complexified Hamiltonian coalesce. EP’s are singular points at which the norm of at least one eigenvector vanishes, when certain real parameters appearing in the Hamiltonian are continued to complex values, and the complexified Hamiltonian becomes non-diagonalizable. The concept of EP’s is similar to that of a degeneracy point, but with the important difference that all the energy eigenvectors cannot be made orthogonal to each other. In previous works, EP’s have been used [30–34] to describe topological phases of matter for 1d topological superconductors/superfluids. The paper is organized as follows: in section 2, we review the results obtained earlier [17] for counting the number (n) of Majorana zero modes bound to defects, based on the bulk-edge correspondence. In section 3, we provide a proof of the counting formula. In section 4, we consider some 1d and 2d models in the class BDI and apply the EP formalism to count n. Section 5 is devoted to the study of edge states for Hamiltonians in class DIII, where we illustrate the applicability of EP solutions as the chiral symmetry exists. In section 6, we discuss some systems in the class D and conclude that EP’s cannot be related to the Majorana fermion wavefunctions for such Hamiltonians, because chiral symmetry is broken. We conclude with a summary and outlook in section 7. In appendix A, we provide a simple example to show how one should choose the correct EP solutions such that their imaginary parts are continuous functions in the parameter space in order to evaluate our counting formula. 2. Counting formula for the Majorana zero modes In this section, we review the connection [17] between the complex momentum solutions of det[HBdG(k)] = 0, and the Majorana fermion wavefunctions in the position space associated with a de- fect or edge. We consider a topological defect embedded in (or at the boundary of) a d -dimensional topologi- cal superconductor. Let m be the dimensions of the defect, parametrized by the Cartesian coordinates r⊥ = (r1, . . . ,rd−m) and r∥ = (rd−m+1, . . . ,rd ), located at r⊥ = 0. Let k⊥ = k⊥Ω̂ = (k1, . . . ,kd−m) and k ∥ = (kd−m+1, . . . ,kd ) be the corresponding conjugate momenta, where k⊥ = \|k⊥\| and Ω̂ is the unit vector when written in spherical coordinates. For a generic HBdG, let k j A and k j B ( j = 1, . . . ,Q) be the two sets of complex k⊥-solutions for det[HBdG(k)] = 0, related by {Im(k j A )} =−{Im(k j B )}, after k⊥ has been analytically continued to the complex plane. One should be careful to choose solutions such that their imaginary parts are continuous functions of the parameter(s) which tune(s) through the transition, and the solutions in one set are related to the other by changing the sign of their imaginary parts throughout. This point has been illustrated by an example in appendix A. Assuming the Majorana wavefunction to be of the form ∼ exp(−z \|r⊥\|) in the bulk, the correspondence ik⊥ ↔−z has been established [17]. At a topological phase transition point, one or more of the Im(k j A/B )’s go through zero. When Im(k j A/B ) changes sign at a topological phase transition point, the position space wavefunction of the corresponding Majorana fermion changes from exponen- tially decaying to exponentially diverging or vice versa. If the former happens, the Majorana fermion ceases to exist. A new Majorana zero mode appears in the latter case. The count (n) for the Majorana fermions for a defect is captured by the function f ({λi },k∥,Ω̂) = 1 2 ∣∣∣∣ Q∑ j=1 ( sign { Im [ k j A/B ( {λi },k∥,Ω̂ )]}− sign{ Im [ k j A/B ( {λ0 i },k0 ∥,Ω̂ 0)]})∣∣∣∣ , (2.1) 33703-2 Counting Majorana bound states using complex momenta where ({λi },k∥,Ω̂) are the parameters appearing in the expressions for k j A/B , and ({λ0 i },k0 ∥,Ω̂ 0 ) are their values at any point in the non-topological phase. If there is a chiral symmetry operator O which anticommutes with the Hamiltonian, the latter takes the form Hchiral(k) = ( 0 A (k) A †(k) 0 ) , (2.2) in the momentum space, for the corresponding bulk system with no defect. On analytically continuing themagnitude k⊥ ≡ k = \|k\| to the complex k⊥-plane, at least one of the eigenvectors of Hchiral(k) collapses to zero norm where det[A (k)] = 0 or det [ A †(k) ]= 0. (2.3) These points are associated with the solutions of EP’s for complex k⊥-values where two or more energy levels coalesce. Furthermore, these coalescing eigenvalues have zero magnitude since det[A (k)] = 0 (or det[A †(k)] = 0) also implies det[Hchiral(k)] = 0. Hchiral(k) becomes non-diagonalizable, as in the com- plex k⊥-plane, det[A (k)] = 0; det[A †(k)] = 0 (or vice versa). However, at the physical phase transition points, the imaginary parts of one or more solutions vanish, and det[A (k)] = det[A †(k)] = 0 for those so- lutions, makingHchiral(k) once again diagonalizable andmarking the disappearance of the corresponding EP’s. Since it satisfies equation (2.3), each EP solution corresponds to a Majorana fermion of a definite chirality with respect to O . IfA †(k) =A T(−k) holds, then the two sets of EP’s are related by {k j A } =−{k j B }, one set corresponding to the solutions obtained from one of the two off-diagonal blocks. In such cases, the pairs of the Majorana fermion wavefunctions are of opposite chiralities. 3. Derivation of the counting formula A simple derivation of the counting formula in equation (2.1) can be motivated as follows: 1. Let us consider one of the solutions given by j = 1. In the non-topological phase, say phase “0”, k1 A ({λ0 i },k0 ∥,Ω̂ 0 ) gives no Majorana zero mode and hence does not give rise to any decaying mode localized at a defect. On the other hand, in a topological phase, say phase “t”, with a Majorana wavefunction ∼ exp[−\| Im(k1 A )\|r⊥], k1 A ({λi },k∥,Ω̂) ∣∣ phase t localized at r⊥ = 0 and zero at r⊥ = ∞, should now give rise to an admissible decaying zero mode solution. This implies that there is a change in sign of Im(k1 A ) from −1 to +1 when we jump from phase “0” to phase “t”. 2. Majorana zero modes must occur in pairs, though they might be localized far apart. Hence, if k1 A ∣∣ phase C corresponds to a Majorana mode localized at r⊥ = 0, then k1 B ∣∣ phase t must correspond to one localized at r⊥ =∞, where k1 B = (k1 A )∗. Hence, whether or not we are in the topological phase “t” is captured by the function f1 = 1 2 ∣∣sign{ Im[k1 A/B ({λi },k∥,Ω̂)] }− sign{ Im[k1 A/B ({λ0 i },k0 ∥,Ω̂ 0 )] }∣∣ taking the value 1 or zero. 3. From the above discussion, it may seem that the counting formula should be given by 1 2 Q∑ j=1 ∣∣∣sign{ Im [ k j A/B ( {λi },k∥,Ω̂ )]}− sign{ Im [ k j A/B ( {λ0 i },k0 ∥,Ω̂ 0)]}∣∣∣ . However, this is not quite correct. To understand this, let us consider the scenario when at least two of the solutions, say k1 A and k2 A are such that sign[Im(k2 A )] ∣∣ phase0 =−sign[Im(k1 A )] ∣∣ phase0. This implies that in the trivial phase, the wavefunction given by c1 exp(ik1 A ∣∣ phase0 r⊥)+ c2 exp(ik2 A ∣∣ phase0 r⊥) is inadmissible for not being capable of satisfying the boundary conditions — the only solution is c1 = c2 = 0. In another topological phase, say “ t̃”, let sign[Im(k1 A )] ∣∣ phase t̃ =−sign[Im(k1 A )] ∣∣ phase0 and sign[Im(k2 A )] ∣∣ phase t̃ = −sign[Im(k2 A )] ∣∣ phase0. This means that both Im(k1 A ) and Im(k2 A ) change sign when we jump from phase “0” to phase “ t̃”. However, they still should not give any Majorana zero mode in the phase “ t̃”, because c̃1 exp(ik1 A ∣∣ phase t̃ r⊥)+ c̃2 exp(ik2 A ∣∣ phase t̃ r⊥) cannot satisfy the boundary conditions. So, the correct formula is given by equation (2.1). 33703-3 I. Mandal 4. EP formalism for the BDI class In this section, we consider some 1d and 2d spinless models in the BDI class, which can support multiple Majorana fermions at any end of an open chain. For systems in this class, there exists a chiral symmetry operator O , such that HBdG can be rotated to the form Hchiral in equation (2.2). After reviewing the transfer matrix scheme to find Majorana fermion solutions localized at an edge, we show how EP solutions in the complex k⊥-plane can be used to count the number of Majorana zero modes in a given topological phase. We also make emphasis on the connection of these EP solutions with the position space wavefunctions calculated in the real space lattice with open ends. 4.1. Transfer matrix approach Kitaev [2] suggested the model of a 1d p-wave superconducting chain, which can support Majorana zero modes at the two ends. For a finite and open chain with N sites, the Hamiltonian takes the form HK =− N∑ j=1 µ ( c† j c j − 1 2 ) + N−1∑ j=1 ( −w c† j c j+1 +∆c j c j+1 +h.c. ) , (4.1) where µ is the chemical potential, w and ∆ are the nearest-neighbour hopping amplitude and supercon- ducting gap, respectively. The pair of fermionic annihilation and creation operators, c j and c† j , describe the lattice site j , and obey the usual anticommutation relations {c j ,c ′j } = 0 and {c j ,c† j ′ } = δ j j ′ . The Majo- rana mode structure of the wire can be better understood by rewriting the above Hamiltonian in terms of the Majorana operators a j = c† j + c j , b j =−i ( c† j − c j ) , (4.2) satisfying a j = a† j , b j = b† j , {a j ,b j ′ } = 0, {a j , a j ′ } = {b j ,b j ′ } = 2δ j j ′ . Then, the Hamiltonian reduces to HK =− i 2 N∑ j=1 µa j b j − i 2 N−1∑ j=1 [ (w −∆) a j b j+1 − (w +∆)b j a j+1 ] . (4.3) This chain can support one Majorana bound state (MBS) at an edge for appropriate values of the parameters. More recently, a variation of the model was considered with next-nearest-neighbour hop- ping and pairing amplitudes [35]. A general version of such longer-ranged interactions with all possible hoppings and pairings was studied [36, 37] with the Hamiltonian Hl =− i 2 N∑ j=1 µa j b j − i q∑ r=1 N−q∑ j=1 [ J−r a j b j+r + Jr a j+r b j ] , (4.4) where the J±r ’s are real parameters, and 0 < q < N . These models can support multiple MBSs at an edge. If we impose periodic boundary conditions (PBC’s), the Hamiltonian can be diagonalized by a Bogoliubov transformation: Hl =−∑ k ( c† k c−k ) hl (k) ( ck c† −k ) , hl (k) =−2 q∑ r=−q ( Jr cos(kr ) −i Jr sin(kr ) i Jr sin(kr ) −Jr cos(kr ) ) , J0 =−µ 2 , (4.5) where the anticommuting fermion operators (c† k , ck ) are suitable linear combinations in the momentum space of the original (c j , c† j ) fermion operators. The energy eigenvalues are given by El (k) =±2 √[∑ r Jr cos(kr ) ]2 + [∑ r Jr sin(kr ) ]2 . (4.6) 33703-4 Counting Majorana bound states using complex momenta We now review the transfer matrix approach [35–38] to identify the number of MBSs at each end of the chain for this model. The transfer matrix can be obtained from the Heisenberg equations of motion for the Majorana operators in equation (4.4): 2i da j dt =−i q∑ r=−q J−r b j+r , 2 i db j dt = i q∑ r=−q Jr a j+r . (4.7) Assuming the time-dependence to be of the form a j = A j e−iEl t and b j = B j e−iEl t , the El = 0 (zero energy modes) are given by the recursion relation of the amplitudes: q∑ r=−q J−r b j+r = 0, q∑ r=−q Jr a j+r = 0. (4.8) Clearly, it will suffice to solve one set of the recursive equations to obtain the solutions for both. Assuming A j =λ j A and B j =λ j B , we get the polynomial equations q∑ r=−q Jr λ q+r A = 0, q∑ r=−q J−r λ q+r B = 0. (4.9) An MBS can exist if we have a normalizable solution, i.e., if \|λA\| < 1 or \|λB\| < 1, if the solution is to be localized at the left end. Similarly, for a mode to be localized at the right-hand end of the chain, we must have \|λA\| > 1 or \|λB\| > 1. Depending on the number of constraint equations (or boundary conditions on the amplitudes), one should determine the number of independent MBSs at each end of the chain. 4.2. Relation of the EP formalism with the transfer matrix approach Let us apply the EP formalism [17, 34] to the Hamiltonian in equation (4.5). First we rotate it to the off-diagonal form hl ,od(k) =U † l hl (k)Ul = ( 0 Al (k) Bl (k) 0 ) , Ul = ip 2 ( −1 −1 −1 1 ) , Al (k) =−2 q∑ r=−q [ Jr cos(kr )+ i Jr sin(kr ) ] , Bl (k) =−2 q∑ r=−q [ Jr cos(kr )− i Jr sin(kr ) ] . (4.10) The EP’s where either Al (k) or Bl (k) vanishes, are given by the solutions q∑ r=−q Jr λ̃ q+r Al = 0, where λ̃Al = exp(ikAl ) , (4.11) q∑ r=−q J−r λ̃ q+r Bl = 0, where λ̃Bl = exp(ikBl ) . (4.12) Comparing equations (4.9), (4.11) and (4.12), it is easy to see that the solutions for EP’s in the complex k-plane for the PBC’s correspond to the MBS solutions for the open boundary conditions (OBC’s). Since \|λ̃Al/Bl \| < 1 ⇒ Im(kAl/Bl ) > 0 ⇔ \|λA / B \| < 1, (4.13) \|λ̃Al /Bl \| > 1 ⇒ Im(kAl/Bl ) < 0 ⇔ \|λA / B \| > 1, (4.14) a sign change of Im(kAl/Bl ) indicates a topological phase transition, by which we move from a phase where an MBS can exist to the one where that particular zero mode gets destroyed. This is related to the fact that Im(kAl/Bl )’s are related to the exponential decay of the MBS position space wavefunctions localized at one end of the open chain. Choosing J0 = −µ 2 , J1 = J2 = 1+∆ 2 , J−1 = J−2 = 1−∆ 2 and all other Jr ’s to be zero, we can get a system supporting up to four Majorana zero modes at each end of the chain. The phase diagram obtained using equation (2.1) is shown in figure 1. 33703-5 I. Mandal Figure 1. (Color online) The topological phase diagram of the Hamiltonian described by equation (4.4), with J0 = −µ 2 , J1 = J2 = 1+∆ 2 , J−1 = J−2 = 1−∆ 2 , and all other Jr ’s set to zero. Here, n labels the number of Majorana zero modes at each end of the chain, as captured by the function f (µ,∆) defined in equa- tion (2.1). Instead, for the parameters J0 = −M 2 cos(φ2), J1 = − J 2 cos(φ1), J−1 = − J 2 sin(φ1), and all other Jr ’s set to zero, we get a system having three EP’s for either Al (k) = 0 or Bl (k) = 0. For this model, up to two Ma- jorana zero modes can appear at an edge. The phase diagrams for J/M = 0.625 and J/M = 1.3, obtained using equation (2.1), are shown in figure 2. We should note another important point: if there areQ EP solutions for either Al (k) = 0 or Bl (k) = 0, clearly there are 2Q solutions in total. However, for counting the zero modes in equation (2.1), we should consider only one set, where the two sets obey the relation λ̃Al = 1/λ̃Bl or kAl =−kBl . (4.15) (a) (b) Figure 2. (Color online) Panels (a) and (b) show the topological phase diagram of the Hamiltonian de- scribed by equation (4.4), with J0 =−M 2 cos(φ2), J1 =− J 2 cos(φ1), J−1 =− J 2 sin(φ1), and all other Jr ’s set to zero. Here, n labels the number of Majorana zero modes at each end of the chain, as captured by the function f (µ,∆) defined in equation (2.1). 33703-6 Counting Majorana bound states using complex momenta As we have already seen, these two sets correspond to the wavefunctions of the MBSs at the two opposite ends. Evidently, the MBSs exist in pairs at the two ends and the topological phase is characterized by their number at each individual end. 4.3. Single-channel ferromagnetic nanowire The 1d Hamiltonian for a ferromagnetic nanowire embedded on Pb superconductor [39] with a single spatial channel (i.e., no transverse hopping) is given by HN 1 = ∑ k Ψ† k hN 1(k)Ψk , Ψk = (ck↑,ck↓,c† −k↓,−c† −k↑)T, hN 1(k) = ξ(k)σ0τz + [ ∆s σ0 +∆p sin(k)d ·σ] τx + V ·στ0 , ξ(k) =−2t cos(k)−µ . (4.16) Here, k is the 1d crystal momentum, Ψk is the four-component Nambu spinor defined in the particle- hole (τ) and spin (σ) spaces, and V is the Zeeman field which can be induced by ferromagnetism. Also, ∆s and ∆p are proximity-induced s-wave and p-wave superconducting pairing potentials, respectively, with d determining the relative magnitudes of the components of the p-wave superconducting order parameter ∆αβ (α,β =↑,↓). In our calculations, we use d = (1,0,0) and V = (0,0,V ). This Hamiltonian belongs to the BDI class with the chiral symmetry operator given by O =σxτy . The eigenvalues of the Hamiltonian are given by: E1(k) =± √ ξ2(k)+V 2 +∆2 s +∆2 p sin2(k)− ẽ1 , E2(k) =± √ ξ2(k)+V 2 +∆2 s +∆2 p sin2(k)+ ẽ1 , ẽ1 = 2 √ V 2 [ ∆2 s +ξ2(k) ]+∆2 s ∆ 2 p sin2(k) . (4.17) A level crossing can occur if either E1(k) = 0 or E2(k) = 0. However, for a finite V and ∆s , the latter is impossible. Hence, a level crossing takes place when E1(k) = 0 for k = 0 or π for the appropriate values of the parameters, which also indicates that this corresponds to the appearance of zero energy modes. A generic complex value of k corresponding to E1(k) = 0 can be obtained by solving[ Ṽ 2 −ξ2(k)+∆2 p sin2(k) ]2 +4ξ2(k)∆2 p sin2(k) = 0, Ṽ = √ V 2 −∆2 s . (4.18) We can rotate the Hamiltonian in equation (4.16) to the chiral basis, where it takes the form Hchi N 1(k) =U † 2 hN 1(k)U2 = ( 0 hN 1 u (k) hN 1 l (k) 0 ) , U2 = 1 2  −1− i 0 1+ i 0 0 1+ i 0 −1− i 0 1− i 0 1− i 1− i 0 1− i 0  , hN 1 u (k) = ( −ξ(k)+ i∆p sin(k)−V ∆s ∆s ξ(k)− i∆p sin(k)−V ) , hN 1 l (k) = ( −ξ(k)− i∆p sin(k)−V ∆s ∆s ξ(k)+ i∆p sin(k)−V ) . (4.19) If either det[hN 1 u (k)] = 0 or det[hN 1 l (k)] = 0 for a complex k-value, this leads to the vanishing of the norm of one of the four eigenvectors of Hchi N 1(k), signalling the existence of an EP for that value of k. At an EP, Hchi N 1(k) is thus non-diagonalizable. The solutions for the EP’s are given by either det [ hN 1 u (k) ]= 0 ⇒ Ṽ 2 −ξ2(k)+∆2 p sin2(k) =−2iξ(k)∆p sin(k) ⇒ k = ku s1,s2 =−i ln {[ s1 Ṽ −µ+ s2 √( s1 Ṽ −µ)2 −4t 2 ]( 2t +∆p )−1 } , (4.20) or det [ hN 1 l (k) ]= 0 ⇒ Ṽ 2 −ξ2(k)+∆2 p sin2(k) = 2iξ(k)∆p sin(k) ⇒ k = k l s1,s2 =−i ln {[ s1 Ṽ −µ+ s2 √( s1 Ṽ −µ)2 −4t 2 ]( 2t −∆p )−1 } , (4.21) 33703-7 I. Mandal -4 -2 2 4 Μ�t -4 -2 2 4 E1,2HkL (a) (b) - 6 - 4 - 2 2 4 6 μ / t 1 2 f (μ ) (c) Figure 3. (Color online) Parameters: ∆s = ∆p = 0.1 t , Ṽ = 1.5 t corresponding to the Hamiltonian in equation (4.16). (a) Energy bands E1,2(k), given in equation (4.17), have been plotted in blue and red, respectively, as functions of µ/t . (b) Plots of Im(ku s1,s2 ) versus µ/t . (c) f (µ) giving the count of the chiral Majorana zero modes as a function of µ/t . where (s1 =±1, s2 =±1). Clearly, k = ku/l s1,s2 also solves equation (4.18), which corresponds to two coincid- ing zero energy solutions (where two levels coalesce [27] for a complex k-value). The plots of the energy bands, Im(ku s1,s2 ), and f (µ) have been shown in figure 3 , using the values ∆s =∆p = 0.1t and Ṽ = 1.5t . Now, let us try to understand the existence of the EP’s throughout a given topological phase and their disappearance right at the phase transition points, the latter being tied to the sign change of the Im(ku/l s1,s2 )’s. The Hamiltonian in equation (4.19), when written in position space, gives the following equations for the Majorana zero modes, ψ+ = (u+,0)T and ψ− = (0,u−)T (with chirality +1 and −1, re- spectively): ( ∂2 x +µ+∆p ∂x −V ∆s ∆s −∂2 x −µ−∆p ∂x −V ) u+ = 0, ( ∂2 x +µ−∆p ∂x −V ∆s ∆s −∂2 x −µ+∆p ∂x −V ) u− = 0. (4.22) Here, we have assumed a continuum for an open wire and set t = 1. For ψ− , let us assume the trial solution u− =∑ r exp(−zr x) ( u↑ r u↓ r ) . The complex zr ’s must satisfy the quartic equation det ( z2 r +µ−∆p zr −V ∆s ∆s −z2 r −µ+∆p zr −V ) = 0 ⇒ ( z2 r +µ−∆p zr )2 = Ṽ 2 , (4.23) 33703-8 Counting Majorana bound states using complex momenta whereas for small k , from equation (4.21), we get(−k2 +µ+ i∆p k )2 = Ṽ 2 , (4.24) indicating correspondence ik ↔−zr . The magnitude of zr will determine the admissible MBS solutions subject to OBC’s (as analyzed in an earlier work [40]), just as in the transfermatrix analysis for the 1d spin- less lattice case. Hence, here also we have been able to establish the relation between the existence of EP’s in the complex k-plane (for the periodic Hamiltonian) and the localized Majorana zero modes at the ends of an open chain. 4.4. Two-channel time-reversal-symmetric nanowire system MBSs in a two-channel TRS nanowire proximity-coupled to an s-wave superconductor have been re- cently studied [41]. The low-energy model for the lowest bands of the system is described by the effective 1d 4×4 BdG Hamiltonian: HN 2 = ∑ k Ψ† k hN 2(k)Ψk , Ψk = (ck↑,ck↓,c† −k↓,−c† −k↑)T, hN 2 = ξ̃(k)σ0τz + v ( kσz −pc σ0 ) τx +B ·στ0 , ξ̃(k) = k2 2m − µ̃ , (4.25) where pc is the momentumwhen the gap closes, B is a magnetic field for the Zeeman term, and (v, µ̃) are effective parameters. We have set pc = 2vm for our calculations. Since this nanowire system belongs to the BDI class when B is perpendicular to the spin-orbit-coupling direction, we will take B= (B ,0,0) in our analysis. Then, the chiral symmetry operator is given by O =σzτy . The eigenvalues of the Hamiltonian are given by: E1(k) =± √ B 2 +k2 v2 +4m2 v4 + ξ̃2(k)− ẽ2 , E2(k) =± √ B 2 +k2 v2 +4m2 v4 + ξ̃2(k)+ ẽ2 , ẽ2 = 2 √ 4m2 v4 ( B 2 +k2 v2 )+B 2 ξ̃2(k) . (4.26) We can have two levels coalescing if E1(k) = 0 for a complex k-value obtained by solving[ B 2 +k2 v2 − ξ̃2(k)−4m2 v2]2 +4 ξ̃2(k)k2 v2 = 0. (4.27) As before, we rotate the Hamiltonian in equation (4.25) to the chiral basis, where it takes the form Hchi N 2(k) =U † 3 hN 2(k)U3 = ( 0 hN 2 u (k) hN 2 l (k) 0 ) , U3 = 1 2  0 −1− i 0 1+ i 0 1− i 0 1− i 1+ i 0 −1− i 0 1− i 0 1− i 0  , hN 2 u (k) = ( −ξ̃(k)+ i ( kv +2mv2 ) B B −ξ̃(k)+ i ( kv −2mv2 ) ) , hN 2 l (k) = ( −ξ̃(k)− i ( kv +2mv2 ) B B −ξ̃(k)− i ( kv −2mv2 ) ) . (4.28) The solutions for the EP’s are then given by either det [ hN 2 u (k) ]= 0 ⇒ [ ξ̃(k)− ikv ]2 = B 2 −4m2v4 ⇒ k = ku s1,s2 =−i ln ( s1 √ m2v2 −2mµ̃+2is2m √ 4m2v4 −B 2 + imv ) , (4.29) or det [ hN 2 l (k) ]= 0 ⇒ [ ξ̃(k)+ ikv ]2 = B 2 −4m2v4 ⇒ k = k l s1,s2 =−i ln ( s1 √ m2v2 −2mµ̃+2is2m √ 4m2v4 −B 2 − imv ) , (4.30) 33703-9 I. Mandal -10 -5 5 10 Μ � -5 5 E1,2HkL (a) -6 -4 -2 2 4 6 Μ � -5 5 E1,2HkL (b) n=0 n=2 n=1 n=1 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 μ ˜ /(v pc) B /( v p c ) (c) Figure 4. (Color online) Parameters: v = 1, m = 1/(2v2), pc = 2vm corresponding to the Hamiltonian in equation (4.25). Panels (a) and (b) show the energy bands E1,2(k), given in equation (4.26), as functions of µ̃, for B = 0 and B = 3, respectively. E1,2(k) have been plotted in blue and red, respectively. Panel (c) shows the contourplot of f (µ) giving the count “n” of the MBSs in the µ̃/(v pc )−B/(v pc ) plane. where (s1 =±1, s2 =±1). Clearly, k = ku/l s1,s2 also solves equation (4.27) and hence corresponds to the coa- lescing of two energy levels at the zero value in the complex k-plane. Choosing v = 1 and m = 1/(2 v2), the energy bands for B = 0 and B = 3, and the contourplot for f (µ̃,B) [defined in equation (2.1)] have been shown in figure 4. Once again we find that f (µ̃,B) gives the correct topological phase diagram in figure 4 (c). Needless to add that here also exp(iku/l s1,s2 )’s determine the admissible solutions for the MBS wavefunctions in the position space, at the ends of an open chain. 4.5. Majorana edge modes for the Kitaev honeycomb model In this subsection, we consider the EP-formalism for a 2d lattice Hamiltonian in the class BDI. The Kitaev honeycomb model [42] can be mapped onto free spinless fermions with p-wave pairing on a hon- eycomb lattice, using the Jordan-Wigner transformation. The solutions for the edge modes for a semi- infinite lattice 1 have been studied earlier [49–52]. The momentum space Hamiltonian in terms of the 1 We would like to point out that this system is different from two-dimensional px + ipy fermionic superfluids, whose excitation spectra include gapless Majorana-Weyl fermions [43]. Volovik showed that in such chiral superfluids, the fermionic zero modes along the domain wall have the same origin as the fermion zero modes appearing in the spectrum of the Caroli-de Gennes-Matricon bound states in a vortex core [44–48]. This correspondence can be understood by picturing the chiral fermions as orbiting around the vortex axis, analogous to the motion along a closed domain boundary. 33703-10 Counting Majorana bound states using complex momenta Majorana operators is Hh =∑ k ( â† k b̂† k ) hh(k) ( âk b̂k ) , hh(k) = ( 0 A(k) B(k) 0 ) , k= (kx ,ky ) , A(k) =−2i [ J3 + J1 cos ( kx −ky 2 ) + J2 cos ( kx +ky 2 )] +2 [ J1 sin ( kx −ky 2 ) + J2 sin ( kx +ky 2 )] , B(k) = 2i [ J3 + J1 cos ( kx −ky 2 ) + J2 cos ( kx +ky 2 )] +2 [ J1 sin ( kx −ky 2 ) + J2 sin ( kx +ky 2 )] (4.31) with the eigenvalues E(k) =±2 {[ J3 + J1 cos ( kx −ky 2 ) + J2 cos ( kx +ky 2 )]2 + [ J1 sin ( kx −ky 2 ) + J2 sin ( kx +ky 2 )]2 }1/2 . (4.32) We will consider two kinds of edges [49, 50], namely, zigzag and armchair, which can support Majo- rana fermions. We will find the phase diagram using the EP’s corresponding to these edges setting either A(k) = 0 or B(k) = 0, after complexifying the momentum component perpendicular to the edge. For this 2d case, f in equation (2.1) is a function of (J1, J2, J3,k∥), where k∥ is the momentum along the 1d edge being considered. One can have a zigzag edge in the y -direction, according to the convention of Nakada et al. [49], so that we will complexify k⊥ = kx , and k∥ = ky will be one of the parameters determining the topological phase transition points. The solution for B(kx = k⊥,ky = k∥) = 0 is given by k⊥ =−2i ln [ − J1 exp(ik∥/2)+ J2 exp(−ik∥/2) J3 ] . (4.33) (a) (b) (c) (d) Figure 5. (Color online) Topological phase diagrams for edges for the 2d honeycomb lattice described by equation (4.31), as captured by the function f (J1, J2, J3,k∥) defined in equation (2.1). Panels (a) and (b) show the number of chiral Majorana zeromodes for a zigzag edge, while panels (c) and (d) show the same for an armchair edge located at the top of a semi-infinite lattice. 33703-11 I. Mandal Majorana zero modes exist for all the values of ky if J1+ J2 < J3. There is no edge state if \|J1− J2\| > J3. For J1 = J2 = J3 , edge states exist if \|ky \| > 2π/3. These results have been plotted in figures 5 (a) and 5 (b). For the armchair edge [49] in the x-direction on the top of the lattice, we will complexify k⊥ = ky , and k∥ = kx will be now one of the parameters determining the topological phase transition points. The two EP’s for A(kx = k∥,ky = k⊥) = 0 are given by k± ⊥ =−2i ln [−J3 exp(−ik∥/2)± √ J 2 3 exp(−ik∥)−4 J1 J2 2 J2 ] . (4.34) No Majorana zero mode exists for any value of kx if J1 = J2 = J3 or J1 < J2. For J1 > J2, a Majorana fermion can exist for a specific range of values for kx . Figures 5 (c) and 5 (d) show these topological phases, obtained using equation (2.1). Equations (4.33) and (4.34) are seen to coincide with the solutions of the Majorana edge states ob- tained earlier by the transfer matrix formalism [49, 52]. 5. EP formalism for the DIII class A point defect in class DIII can support a Majorana Kramers pair (MKP) corresponding to doubly de- generate Majorana zero modes, whereas a line defect can support a pair of helical Majorana edge states. Both are characterized by a Z2 topological invariant. The chiral symmetry operator O can be defined such that the Hamiltonian in class DIII can be brought to the block off-diagonal form [equation (2.2)], just like for the class BDI. 5.1. 1d model A simple 1d model of topological superconductivity in the class DIII is described by the Hamiltonian [53] Hm1 = ∑ k Ψ† k hm1(k)Ψk , Ψk = (ck↑,ck↓,c† −k↓,−c† −k↑)T, hm1(k) = [ξm1(k)σ0 +λR sin(k)σz ]τz + ∆ cos(k)σ0τx , ξm1(k) = t cos(k)−µ . (5.1) This system may be realized in a Rashba wire that is proximity-coupled to a nodeless s± wave supercon- ductor. The energy eigenvalues are given by: E1(k) =± √ [ξm1(k)−λR sin(k) ]2 +∆2 cos2(k) , E2(k) =± √ [ξm1(k)+λR sin(k) ]2 +∆2 cos2(k) , (5.2) whose plots are shown in figure 6 (a) for ∆= 0.1t and λR = 2t , as µ/t is varied along the horizontal axis. Observing that a chiral symmetry operator O = σ0τy exists in the presence of Mz , we rotate the Hamiltonian in equation (5.1) to the chiral basis, where it takes the form Hchi m1(k) =U † 4 hm1(k)U4 = ( 0 hu m1(k) hl m1(k) 0 ) , U4 = 1p 2  0 −i 0 i 0 1 0 1 −i 0 i 0 1 0 1 0  , hu m1(k) = diag ( i∆cos(k)−ξm1(k)+λR sin(k), i∆cos(k)−ξm1(k)−λR sin(k) ) , hl m1(k) = diag ( − i∆cos(k)−ξm1(k)+λR sin(k), −i∆cos(k)−ξm1(k)−λR sin(k) ) . (5.3) The solutions for the EP’s are then given by either det [ hu m1(k) ]= 0 ⇒ i∆cos(k)−ξm1(k) = s1λR sin(k) ⇒ k = km1u s1,s2 =−i ln [µ+ s2 √ µ2 −λ2 R − (t − i∆)2 t − i (∆+ s1λR) ] , (5.4) 33703-12 Counting Majorana bound states using complex momenta -6 -4 -2 2 4 6 μ /t -2 -1 1 2 E1,2(k) (a) (b) Figure 6. (Color online) Parameter: ∆ = 0.1t corresponding to the Hamiltonian in equation (5.1). (a) En- ergy bands E1,2(k), given in equation (5.2), have been plotted in blue and red, respectively as functions of µ/t , for λR = 2t . (b) f (µ/t ,λR/t ) giving the count “n” of the chiral Majorana fermions. or det [ hl m1(k) ] = 0 ⇒ i∆cos(k)+ξm1(k) = s1λR sin(k) ⇒ k = km1l s1,s2 =−i ln [µ+ s2 √ µ2 −λ2 R − (t + i∆)2 t + i (∆+ s1λR) ] , (5.5) where (s1 =±1, s2 =±1). For this model, we note that the two different sets of EP solutions, related by {Im(k)} ∣∣ set=A =−{Im(k)} ∣∣ set=B , are obtained from equations (5.4) and (5.5) whenwe set s1 = 1 and s1 =−1, respectively. This is related to the fact that [hu m1(k)]† , [hu m1(−k)]T (where hl m1(k) = [hu m1(k)]†) for real k. However, we have argued before that for equation (2.1) to work, we must take all the EP solutions from one of the sets related by a negative sign of Im(k). Using either (km1u +1,s2 ,km1l +1,s2 ) or (km1u −1,s2 ,km1l −1,s2 ) (rather than both), figure 6 (b) gives the correct topological phase diagram in the µ/t −λR/t plane, for ∆ = 0.1t . We clearly see that there exist phases with a pair of MBSs, which correspond to one Kramers doublet (MKP). 5.2. 2d model The 1d model of a Rashba semiconductor combined with a nodeless s± wave superconductor can be easily generalized to a 2d system, described by the Hamiltonian [54] Hm2 = ∑ k Ψ† khm2(k)Ψk , Ψk = (ck↑,ck↓,c† −k↓,−c† −k↑)T, hm2(k) = [ξm2(k)+ ∆m(k)]σ0τz + 2λR[sin(kx )σy − sin(ky )σx ]τz , ξm2(k) =−2t [cos(kx )+cos(ky )]−µ , ∆m(k) =∆0 +2∆1[cos(kx )+cos(ky )] , (5.6) where ∆m(k) is the s± wave singlet pairing potential that switches its sign between the centre (0,0) and the corner (π,π) of the 2d Brillouin zone, when 0 < \|∆0\| < 4∆1. The energy eigenvalues are given by: E1(k) =± √[ ξm2(k)−2λR √ sin2(kx )+ sin2(ky ) ]2 +∆2 m(k) , E2(k) =± √[ ξm2(k)+2λR √ sin2(kx )+ sin2(ky ) ]2 +∆2 m(k) , (5.7) whose plots are shown in figure 7 (a) for λR =∆0 =∆1 = 2t , as µ/t is varied along the horizontal axis. 33703-13 I. Mandal (a) (b) (c) Figure 7. (Color online) Panel (a) shows the plot of energy levels E1,2(k) of equation (5.7) in blue and red, respectively, for the Hamiltonian in equation (5.6), as functions of µ/t . We have used λR =∆0 =∆1 = 2t . Panels (b) and (c) show the contourplots of f (µ/t ,∆0/∆1, ky , λR = 2t ,∆1 = t ) in the ∆0/∆1 −µ/t plane, giving the count “n” of the Majorana edge states along the y -direction, for ky = 0 and ky =π, respectively. Rotating the Hamiltonian in equation (5.6) to the diagonal basis of the chiral symmetry operator O = σ0τy , we get Hchi m2(k) =U † 4 hm2(k)U4 = ( 0 hu m2(k) hl m2(k) 0 ) , hu m2(k) = ( i∆m(k)−ξm2(k) 2λR[sin(ky )− isin(kx )] 2λR[sin(ky )+ isin(kx )] i∆m(k)−ξm2(k) ) , hl m2(k) = ( −i∆m(k)−ξm2(k) 2λR[sin(ky )− isin(kx )] 2λR[sin(ky )+ isin(kx )] −i∆m(k)−ξm2(k) ) . (5.8) The equations for the EP’s, corresponding to edge modes along the y -direction (so that k∥ = ky and k⊥ = kx ), for ky = 0 and ky =π, are given by det [ hu m2(k) ]∣∣∣ ky=(0,π) = 0 ⇒ {i∆m(k)−ξm2(k)} ∣∣ ky=(0,π) = 2 s1λR sin(kx ) , (5.9) and det [ hl m2(k) ]∣∣∣ ky=(0,π) = 0 ⇒ {i∆m(k)+ξm2(k)} ∣∣ ky=(0,π) = 2 s1λR sin(kx ) . (5.10) 33703-14 Counting Majorana bound states using complex momenta The solutions for ky = 0 are: km2u s1,s2 =−i ln [ − 2t +µ+ i (2∆1 +∆0) 2t +2i(∆1 + s1λR) + s2 √ (iµ−∆0) (4∆1 +∆0 − iµ−4i t )−4λ2 R 2t +2i(∆1 + s1λR) ] , (5.11) and km2l s1,s2 =−i ln [−2t −µ+ i (2∆1 +∆0) 2t −2i(∆1 + s1λR) + s2 √ −(iµ+∆0) (4∆1 +∆0 + iµ+4i t )−4λ2 R 2t −2i(∆1 + s1λR) ] . (5.12) Those for ky =π are: km2u s1,s2 =−i ln [ 2t −µ+ i (2∆1 −∆0) 2t +2i(∆1 + s1λR) + s2 √ (∆0 − iµ) (4∆1 −∆0 + iµ−4i t )−4λ2 R 2t +2i(∆1 + s1λR) ] , (5.13) and km2l s1,s2 =−i ln [ 2t −µ+ i (∆0 −2∆1) 2t −2i(∆1 + s1λR) + s2 √ (∆0 + iµ) (4∆1 −∆0 − iµ+4i t )−4λ2 R 2t −2i(∆1 + s1λR) ] . (5.14) Here, km2u s1,s2 and km2l s1,s2 correspond to the vanishing of det[hu m2(k)] and det[hl m2(k)], respectively, and (s1 =±1, s2 =±1). Two distinct sets of EP solutions, related by {Im(k⊥)} ∣∣ set=A = −{Im(k⊥)} ∣∣ set=B , are ob- tained by setting s1 = 1 and s1 = −1, respectively. Using either (km2u +1,s2 ,km1l +1,s2 ) or (km2u −1,s2 ,km1l −1,s2 ) (rather than both) in equation (2.1), figures 7 (b) and 7 (c) give the desired topological phase diagrams in the µ/t −∆0/∆1 plane, for λR = 2t and ∆1 = t . The topological phases with a pair of helical Majorana edge states are clearly seen. 6. Broken time reversal symmetry: class D In this section, we consider 1d and 2d Hamiltonians in the symmetry class D, where the TRS is broken. We will see that the EP formalism in the complex k⊥-plane is not applicable for such systems. 6.1. 1d spinless model We examine the spinless model described by the Hamiltonian HD1 = N−1∑ j=1 ( −w c† j c j+1 +∆c j c j+1 −w∗ c† j+1 c j +∆∗ c j+1 c j ) − N∑ j=1 µ ( c† j c j − 1 2 ) , (6.1) which looks similar to HK in equation (4.1), but with the important difference that the TRS is broken by the fact that w and ∆ can be complex numbers [37]. Without any loss of generality, we can choose ∆ to be real and encode the entire phase-difference (φ) between ∆ and w0 by writing w = w0 eiφ , where w0 is real and positive. With PBC’s, one can write the corresponding BdG Hamiltonian in the momentum space as: HD1 =−∑ k ( c† k c−k ) hD1(k) ( ck c† −k ) , hD1(k) = [ 2 w0 sin(φ)sin(k) ] τ0 − [ 2 w0 cos(φ)cos(k)+µ] τz + 2∆sin(k)τy . (6.2) The energy eigenvalues are given by: E(k) = 2 w0 sin(φ) sin(k)± ẽt , ẽt = √[ 2 w0 cos(φ) cos(k)+µ]2 +4∆2 sin2(k) . (6.3) 33703-15 I. Mandal Hence, it follows that two levels become degenerate when ẽt = 0 ⇒ 2 w0 cos(φ) cos(k)+µ=±2i∆sin(k) . (6.4) When extended to the complex k-space, there can be EP’s where the norm of an eigenvector van- ishes. For convenience, we rotate hD1(k) to find the points (EP’s) where the Hamiltonian becomes non- diagonalizable: hD1,od(k) =U † D1 hD1(k)UD1 = ( 2 w0 sin(φ) sin(k) AD1(k) BD1(k) 2 w0 sin(φ) sin(k) ) , UD1 = ip 2 ( 1 −1 −1 −1 ) , AD1(k) = 2 w0 cos(φ) cos(k)+µ+2i∆sin(k) , BD1(k) = 2 w0 cos(φ) cos(k)+µ−2i∆sin(k) . (6.5) The EP’s are given by AD1(k) = 0 or BD1(k) = 0. At such points, two levels coalesce for a complex value of k satisfying equation (6.4). However, we immediately observe that these EP’s do not correspond to zero energy modes, which appear when det[hD1(k)] = 0 ⇒ [ 2 w0 cos(φ) cos(k)+µ]2 +4∆2 sin2(k) = 4 w2 0 sin2(φ) sin2(k) (6.6) (a) (b) (c) (d) Figure 8. (Color online) Panels (a), (b), (c) and (d) show the contourplots of f (µ,∆) corresponding to the Hamiltonian in equation (6.2), giving the count “n” of the MBSs in the µ/w0 −∆/w0 plane for φ = 0, π/10, 2π/5 and π/2, respectively. The blue regions have n = 1MBS at each end of the open chain, while the purple regions correspond to the n = 0 trivial phases. 33703-16 Counting Majorana bound states using complex momenta is satisfied. We note that since det[hD1(k)] is equal to the product of the energy eigenvalues, vanishing of det[hD1(k)] implies the condition for the existence of a zero energy solution. Although the EP description no longer applies now to the existence of MBSs, we can find the complex k-values satisfying det [ hD1,od(k) ]= 0 ⇒ 2 w0 cos(φ) cos(k)+µ=±2isin(k) √ ∆2 −w2 0 sin2(φ) . (6.7) We can solve for the k-values either with the “+” or the “−” sign on the RHS (rather than both), and plug in the roots of that equation into the formula in equation (2.1). The function f (µ,∆) will still give the number of MBSs in a given topological phase. Once again we emphasize that to count the zero modes in equation (2.1), we should include only one of the two sets of roots related by a sign change of Im(k), as these two sets correspond to the wavefunctions of the pair of MBSs at the two opposite edges. We have shown the plots of f (µ,∆) in figure 8 for four different values of φ. It indeed captures the correct Z2 topological invariant (n = 0, or 1). No zero mode exists, i.e., the system is entirely gapped in certain regions in the φ−µ plane where Im(k) vanishes, as Im(k) is related to the exponential part of the MBS wavefunction in the real space. 6.2. 2d spinless model The following Hamiltonian gives a model of a p + ip wave superconductor on a square lattice [55]: HD2 =−∑ k ( c† k c−k ) hD2(k) ( ck c† −k ) , hD2(k) = [ 2 tx cos(kx )+2 ty cos(ky )−µ] τz + dx sin(kx )τx +dy sin(ky )τy , (6.8) where (tx , ty ) are the hopping strengths and (dx ,dy ) are the pairing amplitudes along the (x, y)-direc- tions, and µ is the chemical potential. The energy eigenvalues, given by E(k) =± √[ 2 tx cos(kx )+2 ty cos(ky )−µ]2 + d 2 x sin2(kx )+d 2 y sin2(ky ) , (6.9) are plotted in figure 9 (a) for tx = ty = dx = dy = 1, as µ is varied along the horizontal axis. We consider the edges parallel to the y -axis, so that k∥ = ky and k⊥ = kx . Complexifying kx , we can compute the number of non-chiral Majorana fermions propagating along these edges with momenta ky = 0 and ky = π, using equation (2.1). The complex kx -values satisfying det[hD2(k)] = 0, for ky = 0 and ky =π, are given by 2 tx cos(kx )+2 ty −µ= i s1 dx sin(kx ) ⇒ kx = ks1,s2 =−i ln [2 s1(1− ty )+ s2 √ 4(ty −1)2 +d 2 x −4 t 2 x dx +2 s1 tx ] , (6.10) and 2 tx cos(kx )−2 ty −µ= i s1 dx sin(kx ) ⇒ kx = ks1,s2 =−i ln [2 s1(1+ ty )+ s2 √ 4(ty +1)2 +d 2 x −4 t 2 x dx +2 s1 tx ] , (6.11) respectively. Here, (s1 =±1, s2 =±1), and we need to use either s1 = 1 or s1 = −1 (rather than both) in equation (2.1), to obtain the phase diagrams shown in figures 9 (b) and 9 (c). 7. Conclusion We have established the relation of the EP solutions for complexified momenta to the Majorana fermion wavefunctions bound to a topological defect in a system with a chiral symmetry, by studying 33703-17 I. Mandal (a) (b) (c) Figure 9. (Color online) Parameters: dx = dy = 1 corresponding to the Hamiltonian in equation (6.8). Panel (a) shows the plot of energy levels E(k), given in equation (6.9), as functions of µ, for tx = ty = 1. Panels (b) and (c) show the contourplots of f (tx , ty ,ky ,µ= 2) in the tx − ty plane, giving the count “n” of the non-chiral Majorana edge states along the y -direction, for ky = 0 and ky =π, respectively. some explicit examples in 1d and 2d. These models include both spinless and spinful cases. We have shown that such EP solutions cannot exist for systems in class D, where there is no chiral symmetry. The generic formula, which was proposed earlier [17] to count the number of Majorana zero modes in arbi- trary dimensions, has been demonstrated to chart out the desired topological phase diagrams for thewide variety of systems we have considered. The detailed study of these models also helps us illustrate how one distinct set of complex k⊥-solutions for det[HBdG(k)] = 0, related by {Im(k⊥)} ∣∣ set=A =−{Im(k⊥)} ∣∣ set=B , should be used while using our formula. An explicit proof of the counting formula has also been dis- cussed. For a system with or without a chiral symmetry, the imaginary parts of these solutions in the complexified k⊥-plane are related to the exponential decay of the Majorana fermion wavefunctions in the bulk in the position space. Hence, the imaginary parts of ñ of the solutions in one set undergoes a change of sign across a topological phase transition point, if the number of Majorana zero modes at a defect changes by ñ. 8. Acknowledgements We thank Atri Bhattacharya, Fiona Burnell, Sudip Chakravarty, Sumathi Rao, Diptiman Sen, Krish- nendu Sengupta and Sumanta Tewari for stimulating discussions.We are also grateful to Chen-HsuanHsu and Arijit Saha for their valuable comments on the manuscript. This research was partially supported by the Templeton Foundation. Research at the Perimeter Institute is supported in part by the Government of Canada through Industry Canada, and by the Province of Ontario through the Ministry of Research and Information. 33703-18 Counting Majorana bound states using complex momenta (a) (b) -4 -2 2 4 Μ 1 fHΜL (c) Figure 10. (Color online) For edges of the system described by equation (A.1), corresponding to ky = 0: (a) Plots of Im(knc s1,s2 ) versus µ. (b) Plots of Im(kc s1,s2 ) versus µ. (c) f (µ) giving the count of the Majorana zero modes as a function of µ. A. Choice of EP solutions In this appendix, we provide a simple example to show how one should choose the correct EP solu- tions such that their imaginary parts are continuous functions in the parameter space for our counting formula 2 . Let us take the 2d class D Hamiltonian [56]: HD3 = ∑ k ( c† k c−k ) hD3(k) ( ck c† −k ) , hD3(k) = kx τx +ky τy + (k2 −µ)τz , (A.1) where µ is the chemical potential. The system is known to be topological for µ> 0 and non-topological for µ < 0. For non-chiral Majorana fermions along the edges parallel to the y -axis with momentum ky = 0, one should solve for det[hD3(kx ,ky = 0)] =−k2 x − (kx −µ)2 = 0. We will have four solutions which can be written as either knc s1,s2 = s1 √ −1 2 + s2 √ 1−4µ 2 +µ ; (A.2) or Fs1 ≡ ( k2 x −µ )+ i s1kx = 0 ⇒ kx = kc s1,s2 = −i s1 + s2 √ 4µ−1 2 , (A.3) where det [ hD3(kx ,ky = 0) ]≡−F+F− , (A.4) and (s1 =±1, s2 =±1). 2 We thank Victor Gurarie for suggesting to clarify this point. 33703-19 I. Mandal The plots of Im(knc s1,s2 ), Im(kc s1,s2 ) and f (µ) have been shown in figure 10. We find that Im[knc+,−(µ)] = − Im[knc−,−(µ)] and Im[knc+,+(µ)] = − Im[knc−,+(µ)]. Also, Im[kc+,+(µ)] = − Im[kc−,−(µ)] and Im[kc+,−(µ)] = − Im[kc−,+(µ)]. 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Zhang F., Kane C.L., Mele E.J., Phys. Rev. Lett., 2013, 111, No. 5, 056402; doi:10.1103/PhysRevLett.111.056402. 55. Asahi D., Nagaosa N., Phys. Rev. B, 2012, 86, 100504; doi:10.1103/PhysRevB.86.100504. 56. Read N., Green D., Phys. Rev. B, 2000, 61, 10267; doi:10.1103/PhysRevB.61.10267. Пiдрахунок зв’язаних станiв Майорани з використанням комплексних iмпульсiв I.Мандал Iнститу теоретичної фiзики “Периметр”, Ватерлоо, Онтарiо N2L 2Y5, Канада Нещодавно (EPL, 2015, 110, 67005) було встановлено зв’язок мiж фермiонами Майорани, зв’язаними з дефектами у довiльнiй вимiрностi, i комплексними iмпульсними коренями детермiнанта вiдповiдного об’ємного гамiльтонiану Боголюбова-де Жена. Базуючись на цьому розумiннi, запропоновано формулу для пiдрахунку числа (n) зв’язаних станiв Майорани з нульовою енергiєю, якi пов’язанi з топологiчною фазою системи. В цiй статтi дається вивiд формули пiдрахунку, яка застосовується до низки 1d i 2d мо- делей, що належать до класiв BDI, DIII i D. Показано, як можна успiшно побудувати топологiчнi фазовi дiаграми. Вивчення даних прикладiв дозволяє явно спостерiгати вiдповiднiсть мiж цими комплексними розв’язками для iмпульсу в Фур’є просторi i локалiзованими хвильовими функцiями фермiонiв Майорани в позицiйному просторi. Накiнець, пiдтверджено факт,що для систем з хiральною симетрiєю цi розв’язки є так званими “винятковими точками”, де два чи бiльше власних значень ускладненого гамiльтонiана зливаються. Ключовi слова: винятковi точки, фермiони Майорани, BDI, DIII, D, пiдрахунок 33703-21 http://dx.doi.org/10.1103/PhysRevLett.110.146404 http://dx.doi.org/10.1103/PhysRevB.88.165111 http://dx.doi.org/10.1088/1367-2630/13/6/065028 http://dx.doi.org/10.1103/PhysRevB.91.094505 http://dx.doi.org/10.1103/PhysRevB.86.220506 http://dx.doi.org/10.1103/PhysRevLett.112.126402 http://dx.doi.org/10.1016/j.aop.2005.10.005 http://dx.doi.org/10.1134/1.567563 http://dx.doi.org/10.1016/0550-3213(81)90044-4 http://dx.doi.org/10.1103/PhysRevB.82.094504 http://dx.doi.org/10.1103/PhysRevB.75.212509 http://dx.doi.org/10.1103/PhysRevB.69.184511 http://dx.doi.org/10.1103/PhysRevLett.99.037001 http://dx.doi.org/10.1103/PhysRevB.54.17954 http://dx.doi.org/10.1103/PhysRevB.76.205402 http://dx.doi.org/10.1103/PhysRevLett.111.146801 http://dx.doi.org/10.1103/PhysRevB.89.235434 http://dx.doi.org/10.1103/PhysRevLett.111.056403 http://dx.doi.org/10.1103/PhysRevLett.111.056402 http://dx.doi.org/10.1103/PhysRevB.86.100504 http://dx.doi.org/10.1103/PhysRevB.61.10267 Introduction Counting formula for the Majorana zero modes Derivation of the counting formula EP formalism for the BDI class Transfer matrix approach Relation of the EP formalism with the transfer matrix approach Single-channel ferromagnetic nanowire Two-channel time-reversal-symmetric nanowire system Majorana edge modes for the Kitaev honeycomb model EP formalism for the DIII class 1d model 2d model Broken time reversal symmetry: class D 1d spinless model 2d spinless model Conclusion Acknowledgements Choice of EP solutions

Counting Majorana bound states using complex momenta

Institution

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