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 (*γ**E*=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 (*H*BdG) 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

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 familiarZ^{and}Z2topological 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 localized Majorana fermion wavefunctions 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[*H*BdG(_{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**_{⊥}=(*r*1, . . . ,*r*_{d−m})^{and}**r**_{∥}=(*r*_{d−m+1}, . . . ,*r*_{d}), located at**r**_{⊥}=0^{. Let}**k**_{⊥}=*k*_{⊥}**Ω**ˆ =(*k*1, . . . ,*k*_{d}_{−m})^{and}**k**_{∥}=
(*k*_{d−m+1}, . . . ,*k*_{d})be the corresponding conjugate momenta, where*k*_{⊥}= |**k**_{⊥}|^{and} **Ω**ˆ is the unit vector
when written in spherical coordinates.

For a generic *H*BdG, let *k*_{A}^{j} ^{and} *k*_{B}^{j} ^{(}*j* = 1, . . . ,*Q*) be the two sets of complex *k*_{⊥}-solutions for
det[*H*BdG(_{k})]=0, related by{Im(*k*_{A}^{j})}= −{Im(*k*_{B}^{j})}^{, 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 correspondencei*k*_{⊥}↔ −*z*has been established [17]. At a topological phase transition point, one
or more of theIm(*k*_{A/B}^{j} )’s go through zero. WhenIm(*k*_{A/B}^{j} )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*({*λ**i*},_{k}_{∥}, ˆ**Ω**)=1
2

¯

¯

¯

¯

*Q*

X

*j*=1

µ sign

n Imh

*k*_{A/B}^{j} ¡

{*λ**i*},_{k}_{∥}, ˆ**Ω**¢io

−^{sign}n
Imh

*k*_{A/B}^{j} ¡

{*λ*^{0}_{i}},_{k}^{0}_{∥}, ˆ**Ω**^{0}¢io¶ ¯

¯

¯

¯

, ^{(2.1)}

where({*λ**i*},_{k}_{∥}, ˆ**Ω**)are the parameters appearing in the expressions for*k*_{A/B}^{j} ^{, and}({*λ*^{0}_{i}},_{k}^{0}_{∥}, ˆ**Ω**^{0})^{are their}
values at any point in the non-topological phase.

If there is a chiral symmetry operatorO which anticommutes with the Hamiltonian, the latter takes the form

*H*chiral(_{k})=

µ 0 A(**k**)
A^{†}(_{k}) 0

¶

, ^{(2.2)}

in the momentum space, for the corresponding bulk system with no defect. On analytically continuing
the magnitude*k*_{⊥}≡*k*= |**k**|to the complex*k*_{⊥}-plane, at least one of the eigenvectors of*H*chiral(_{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 sincedet[A(_{k})]=0^{(or}
det[A^{†}(_{k})]=0) also impliesdet[*H*chiral(_{k})]=0^{.}*H*chiral(_{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, anddet[A(_{k})]=det[A^{†}(_{k})]=0for those so-
lutions, making*H*chiral(_{k})once again diagonalizable and marking 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 toO^{. If}A^{†}(_{k})=A^{T}(−**k**)holds, then the two sets of EP’s are related by{*k*^{j}_{A}}= −{*k*_{B}^{j}}^{,}
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^{”,}
*k*_{A}^{1}({*λ*^{0}_{i}},_{k}^{0}_{∥}, ˆ**Ω**^{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(*k*^{1}_{A})|*r*_{⊥}]^{,}*k*^{1}_{A}({*λ**i*},_{k}_{∥}, ˆ**Ω**)¯

¯_{phase t} localized at*r*_{⊥}=0and zero at*r*_{⊥}= ∞^{,}
should now give rise to an admissible decaying zero mode solution. This implies that there is a
change in sign ofIm(*k*_{A}^{1})^{from}−1^{to}+1when 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
*k*_{A}^{1}¯

¯_{phase C}corresponds to a Majorana mode localized at*r*_{⊥}=0^{, then}*k*_{B}^{1}¯

¯_{phase t}must correspond to
one localized at*r*_{⊥}= ∞^{, where}*k*^{1}_{B}=(*k*^{1}_{A})^{∗}. Hence, whether or not we are in the topological phase “t”

is captured by the function*f*_{1}=^{1}_{2}¯

¯^{sign}

©Im[*k*^{1}_{A/B}({*λ**i*},_{k}_{∥}, ˆ**Ω**)]ª

−^{sign}©

Im[*k*_{A/B}^{1} ({*λ*^{0}_{i}},_{k}^{0}

∥, ˆ**Ω**^{0})]ª¯

¯^{taking}
the value1^{or zero.}

3. From the above discussion, it may seem that the counting formula should be given by 1

2

*Q*

X

*j*=1

¯

¯

¯^{sign}
n

Imh
*k*_{A/B}^{j} ¡

{*λ**i*},_{k}_{∥}, ˆ**Ω**¢io

−^{sign}n
Imh

*k*_{A/B}^{j} ¡

{*λ*^{0}_{i}},_{k}^{0}_{∥}, ˆ**Ω**^{0}¢io ¯

¯

¯.

However, this is not quite correct. To understand this, let us consider the scenario when at least two
of the solutions, say*k*_{A}^{1}^{and}*k*_{A}^{2}are such that sign[Im(*k*_{A}^{2})]¯

¯_{phase}_{0}= −^{sign}[Im(*k*^{1}_{A})]¯

¯_{phase}_{0}. This implies
that in the trivial phase, the wavefunction given by*c*_{1}exp(i*k*^{1}_{A}¯

¯_{phase}_{0}*r*_{⊥})+*c*_{2}exp(i*k*_{A}^{2}¯

¯_{phase}_{0}*r*_{⊥})^{is}
inadmissible for not being capable of satisfying the boundary conditions — the only solution is
*c*_{1}=*c*_{2}=0. In another topological phase, say “˜t ”, let sign[Im(*k*_{A}^{1})]¯

¯_{phase}_{˜}_{t}= −^{sign}[Im(*k*^{1}_{A})]¯

¯_{phase}_{0}^{and}
sign[Im(*k*_{A}^{2})]¯

¯_{phase}_{˜}_{t}= −^{sign}[Im(*k*^{2}_{A})]¯

¯_{phase}_{0}. This means that bothIm(*k*_{A}^{1})^{and}Im(*k*^{2}_{A})change sign
when we jump from phase “0” to phase “˜t ”. However, they still should not give any Majorana
zero mode in the phase “˜t ”, because*c*˜_{1}exp(i*k*_{A}^{1}¯

¯_{phase}_{˜}_{t}*r*_{⊥})+*c*˜_{2}exp(i*k*_{A}^{2}¯

¯_{phase}_{˜}_{t}*r*_{⊥})cannot satisfy the
boundary conditions. So, the correct formula is given by equation (2.1).

**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 operatorO, such that*H*BdGcan be rotated to the form*H*chiralin 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

*H**K*= −

*N*

X

*j*=1

*µ*
µ

*c*^{†}_{j}*c**j*−1
2

¶ +

*N*−1X

*j*=1

³

−*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*^{0}_{j}}=0^{and}{*c*_{j},*c*^{†}

*j*^{0}}=*δ**j j*^{0}. 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}= −i³
*c*^{†}_{j}−*c*_{j}´

, ^{(4.2)}

satisfying

*a*_{j}=*a*^{†}_{j}, *b*_{j}=*b*^{†}_{j}, {*a*_{j},*b*_{j}0}=0 , {*a*_{j},*a*_{j}0}={*b*_{j},*b*_{j}0}=2*δ**j j*^{0}.
Then, the Hamiltonian reduces to

*H**K*= −i
2

*N*

X

*j*=1

*µa**j**b**j*−i
2

*N*−1X

*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

*H*_{l}= −i
2

*N*

X

*j*=1

*µa*_{j}*b*_{j}−i

*q*

X

*r*=1
*N*−*q*

X

*j*=1

£*J*_{−r}*a*_{j}*b*_{j+r}+*J*_{r}*a*_{j+r}*b*_{j}¤

, ^{(4.4)}

where the*J*_{±r}’s are real parameters, and0<*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:

*H*_{l}= −X

*k*

³

*c*_{k}^{†} *c*_{−k} ´
*h*_{l}(*k*)

µ *c*_{k}
*c*_{−k}^{†}

¶ ,

*h*_{l}(*k*)= −2

*q*

X

*r*=−*q*

µ *J*_{r}cos (*kr*) −i*J*_{r} sin (*kr*)
i*J**r*sin (*kr*) −*J**r* cos (*kr*)

¶

, *J*_{0}= −*µ*

2, ^{(4.5)}

where the anticommuting fermion operators(*c*^{†}_{k},*c*_{k})are suitable linear combinations in the momentum
space of the original(*c*_{j},*c*^{†}_{j})fermion operators. The energy eigenvalues are given by

*E*_{l}(*k*)= ±2
s

hX

*r*

*J*_{r} cos (*kr*)
i2

+ hX

*r*

*J*_{r} sin (*kr*)
i2

. ^{(4.6)}

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):

2 id*a*_{j}
d*t* = −i

*q*

X

*r*=−*q*

*J*_{−}*r**b**j*+*r*, 2 id*b*_{j}
d*t* =i

*q*

X

*r*=−*q*

*J**r**a**j*+*r*. ^{(4.7)}

Assuming the time-dependence to be of the form*a*_{j}=*A*_{j}*e*^{−iE}^{l}^{t}^{and}*b*_{j}=*B*_{j}*e*^{−iE}^{l}^{t}^{, the}*E*_{l}=0(zero energy
modes) are given by the recursion relation of the amplitudes:

*q*

X

*r*=−*q*

*J*_{−r}*b*_{j}_{+r}=0 ,

*q*

X

*r*=−*q*

*J*_{r}*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*A^{j} and*B*_{j}=*λ*B^{j}, we get the polynomial equations

*q*

X

*r*=−*q*

*J*_{r}*λ*^{q+r}A =0 ,

*q*

X

*r*=−*q*

*J*_{−r}*λ*B^{q+r}=0 . ^{(4.9)}

An MBS can exist if we have a normalizable solution, i.e., if|*λ*^{A}| <1^{or}|*λ*^{B}| <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|*λ*^{A}| >1^{or}|*λ*^{B}| >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

*h*_{l}_{,}od(*k*)=*U*^{†}

*l* *h*_{l}(*k*)*U*_{l}=

µ 0 *A*_{l}(*k*)
*B*_{l}(*k*) 0

¶

, *U*_{l}= i

p2

µ −1 −1

−1 1

¶ ,

*A*_{l}(*k*)= −2

*q*

X

*r*=−*q*

£*J*_{r}cos (*kr*)+i*J*_{r}sin (*kr*)¤

, *B*_{l}(*k*)= −2

*q*

X

*r*=−*q*

£*J*_{r}cos (*kr*)−i*J*_{r} sin (*kr*)¤

. ^{(4.10)}

The EP’s where either*A*_{l}(*k*)^{or}*B*_{l}(*k*)vanishes, are given by the solutions
*q*

X

*r*=−*q*

*J**r**λ*˜A^{q}^{+}*l*^{r}=0, ^{where} *λ*˜^{A}*l*=exp (i*k*A*l*) , ^{(4.11)}

*q*

X

*r*=−*q*

*J*_{−r}*λ*˜B^{q}^{+}*l*^{r}=0, ^{where} *λ*˜^{B}*l*=exp (i*k*B*l*) . ^{(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

|*λ*˜^{A}*l*/^{B}*l*| <1 ⇒ Im (*k*A*l*/^{B}*l*)>0 ⇔ |*λ*^{A / B}| <1 , ^{(4.13)}

|*λ*˜^{A}*l*/^{B}*l*| >1 ⇒ Im (*k*A*l*/^{B}*l*)<0 ⇔ |*λ*^{A / B}| >1 , ^{(4.14)}
a sign change ofIm (*k*A*l*/^{B}*l*)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 thatIm (*k*A*l*/^{B}*l*)’s are related to the exponential decay of the MBS position space wavefunctions
localized at one end of the open chain.

Choosing *J*_{0}= −^{µ}_{2},*J*_{1}=*J*_{2}=^{1}^{+∆}_{2} ,*J*_{−1}=*J*_{−2}= ^{1}^{−∆}_{2} and all other *J*_{r}’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.

**Figure 1.** (Color online) The topological phase diagram of the Hamiltonian described by equation (4.4),
with*J*0= −^{µ}_{2},*J*1=*J*2=^{1+∆}_{2} ,*J*_{−}1=*J*_{−}2=^{1−∆}_{2} , and all other*J**r*’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 *J*0= −^{M}_{2}cos(*φ*2),*J*1= −^{J}_{2}cos(*φ*1),*J*_{−}1= −_{2}^{J}sin(*φ*1), and all other*J**r*’s set
to zero, we get a system having three EP’s for either*A*_{l}(*k*)=0^{or}*B*_{l}(*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 are*Q*EP solutions for either*A*_{l}(*k*)=0^{or}*B*_{l}(*k*)=0^{,}
clearly there are2*Q*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

*λ*˜^{A}*l*=1/ ˜*λ*^{B}*l* or *k*A*l*= −*k*B*l*. ^{(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*J*_{0}= −^{M}_{2}cos(*φ*2),*J*_{1}= −_{2}^{J}cos(*φ*1),*J*_{−1}= −_{2}^{J}sin(*φ*1), and all other*J**r*’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).

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

*H*_{N1}=X

*k*

Ψ^{†}_{k}*h*_{N1}(*k*)Ψ*k*, Ψ*k*=(*c*_{k}_{↑},*c*_{k}_{↓},*c*^{†}

−*k*↓,−*c*^{†}

−*k*↑)^{T},
*h*_{N1}(*k*)=*ξ*(*k*)*σ*0*τ**z*+£

∆*s**σ*0+∆*p*sin(*k*)**d**·* σ*¤

*τ**x*+**V**·* στ*0,

*ξ*(

*k*)= −2

*t*cos(

*k*)−

*µ*.

^{(4.16)}Here,

*k*is the 1d crystal momentum,Ψ

*k*is the four-component Nambu spinor deﬁned in the particle- hole(

*)*

**τ**^{and spin}(

*)spaces, and*

**σ****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 byO=*σ**x**τ**y*.

The eigenvalues of the Hamiltonian are given by:

*E*_{1}(*k*)= ±

q*ξ*^{2}(*k*)+*V*^{2}+∆^{2}*s*+∆^{2}*p*sin^{2}(*k*)−*e*˜_{1}, *E*_{2}(*k*)= ±

q*ξ*^{2}(*k*)+*V*^{2}+∆^{2}*s*+∆^{2}*p*sin^{2}(*k*)+*e*˜_{1},

*e*˜1=2
q

*V*^{2}£

∆^{2}*s*+*ξ*^{2}(*k*)¤

+∆^{2}*s*∆^{2}*p*sin^{2}(*k*) . ^{(4.17)}

A level crossing can occur if either*E*_{1}(*k*)=0^{or}*E*_{2}(*k*)=0. However, for a ﬁnite*V* ^{and}∆*s*, the latter is
impossible. Hence, a level crossing takes place when*E*_{1}(*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*E*_{1}(*k*)=0can be obtained by solving

h*V*˜^{2}−*ξ*^{2}(*k*)+∆^{2}*p*sin^{2}(*k*)i2

+4*ξ*^{2}(*k*)∆^{2}*p*sin^{2}(*k*)=0 , *V*˜=
q

*V*^{2}−∆^{2}*s*. ^{(4.18)}
We can rotate the Hamiltonian in equation (4.16) to the chiral basis, where it takes the form

*H*_{N1}^{chi}(*k*)=*U*_{2}^{†}*h*_{N}_{1}(*k*)*U*_{2}=

µ 0 *h*^{N1}_{u} (*k*)
*h*^{N1}_{l} (*k*) 0

¶

, *U*_{2}=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

,

*h*^{N1}_{u} (*k*)=

µ −*ξ*(*k*)+i∆*p*sin(*k*)−*V* ∆*s*

∆*s* *ξ*(*k*)−i∆*p*sin(*k*)−*V*

¶ ,

*h*_{l}^{N1}(*k*)=

µ −*ξ*(*k*)−i∆*p*sin(*k*)−*V* ∆*s*

∆*s* *ξ*(*k*)+i∆*p*sin(*k*)−*V*

¶

. ^{(4.19)}

If eitherdet[*h*^{N}_{u}^{1}(*k*)]=0^{or}det[*h*_{l}^{N1}(*k*)]=0for a complex*k*-value, this leads to the vanishing of the norm
of one of the four eigenvectors of*H*_{N1}^{chi}(*k*), signalling the existence of an EP for that value of*k*. At an EP,
*H*_{N1}^{chi}(*k*)is thus non-diagonalizable.

The solutions for the EP’s are given by either det£

*h*_{u}^{N1}(*k*)¤

=0 ⇒ *V*˜^{2}−*ξ*^{2}(*k*)+∆^{2}*p*sin^{2}(*k*)= −2 i*ξ*(*k*)∆*p*sin(*k*)

⇒ *k*=*k*^{u}_{s}_{1}_{,s}_{2}= −i ln

½·

*s*_{1}*V*˜−*µ*+*s*_{2}
q

¡*s*_{1}*V*˜−*µ*¢2

−4*t*^{2}

¸

¡2*t*+∆*p*¢_{−1}

¾

, ^{(4.20)}

or

det£
*h*^{N}_{l} ^{1}(*k*)¤

=0 ⇒ *V*˜^{2}−*ξ*^{2}(*k*)+∆^{2}*p*sin^{2}(*k*)=2 i*ξ*(*k*)∆*p*sin(*k*)

⇒ *k*=*k*_{s}^{l}_{1}_{,s}_{2}= −i ln

½·

*s*1*V*˜−*µ*+*s*2

q

¡*s*1*V*˜−*µ*¢2

−4*t*^{2}

¸

¡2*t*−∆*p*¢_{−}1¾

, ^{(4.21)}

**-4** **-2** **2** **4**

**Μ** **** **t**

**-4**
**-2**
**2**

**E**

**E**

**1,24**

**H** **k** **L**

(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*^{,}*V*˜ =1.5*t* corresponding to the Hamiltonian in
equation (4.16). (a) Energy bands*E*1,2(*k*), given in equation (4.17), have been plotted in blue and red,
respectively, as functions of*µ*/*t*. (b) Plots ofIm (*k*^{u}_{s}_{1}_{,s}_{2})^{versus}*µ*/*t*^{. (c)}*f*(*µ*)giving the count of the chiral
Majorana zero modes as a function of*µ*/*t*^{.}

where(*s*_{1}= ±1^{,}*s*_{2}= ±1)^{. Clearly,}*k*=*k*^{u/l}_{s}

1,*s*2also 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 (*k*^{u}_{s}_{1}_{,s}_{2})^{, and} *f*(*µ*)have been shown in ﬁgure 3 , using the values

∆*s*=∆*p*=0.1*t*^{and}*V*˜=1.5*t*^{.}

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 (*k*^{u/l}_{s}

1,*s*_{2})’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):

µ *∂*^{2}*x*+*µ*+∆*p**∂**x*−*V* ∆*s*

∆*s* −*∂*^{2}*x*−*µ*−∆*p**∂**x*−*V*

¶
*u*_{+}=0 ,

µ *∂*^{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*_{−}=X

*r*

exp (−*z*_{r}*x*)
Ã *u*^{↑}_{r}

*u*^{↓}_{r}

!

. The complex*z*_{r}’s must satisfy the quartic equation

det

µ *z*^{2}_{r}+*µ*−∆*p**z*_{r}−*V* ∆*s*

∆*s* −*z*_{r}^{2}−*µ*+∆*p**z**r*−*V*

¶

=0 ⇒ ¡

*z*_{r}^{2}+*µ*−∆*p**z*_{r}¢2

=*V*˜^{2}, ^{(4.23)}

whereas for small*k*, from equation (4.21), we get

¡−*k*^{2}+*µ*+i∆*p**k*¢2

=*V*˜^{2}, ^{(4.24)}

indicating correspondencei*k*↔ −*z**r*. The magnitude of*z**r* will determine the admissible MBS solutions
subject to OBC’s (as analyzed in an earlier work [40]), just as in the transfer matrix 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
1d4×4BdG Hamiltonian:

*H**N*2=X

*k*

Ψ^{†}_{k}*h**N*2(*k*)Ψ*k*, Ψ*k*=(*c*_{k↑},*c*_{k↓},*c*_{−k↓}^{†} ,−*c*_{−k↑}^{†} )^{T},

*h**N*2=*ξ*˜(*k*)*σ*0*τ**z*+*v*¡

*kσ**z*−*p**c**σ*0¢

*τ**x*+**B**·* στ*0,

*ξ*˜(

*k*)=

*k*

^{2}

2*m*−*µ*˜, ^{(4.25)}

where*p*_{c}is the momentum when the gap closes,**B**is a magnetic ﬁeld for the Zeeman term, and(*v*, ˜*µ*)^{are}
effective parameters. We have set*p**c*=2*vm*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 byO=*σ**z**τ**y*.

The eigenvalues of the Hamiltonian are given by:

*E*_{1}(*k*)= ±
q

*B*^{2}+*k*^{2}*v*^{2}+4*m*^{2}*v*^{4}+*ξ*˜^{2}(*k*)−*e*˜_{2}, *E*_{2}(*k*)= ±
q

*B*^{2}+*k*^{2}*v*^{2}+4*m*^{2}*v*^{4}+*ξ*˜^{2}(*k*)+*e*˜_{2},

˜
*e*2=2

q

4*m*^{2}*v*^{4}¡

*B*^{2}+*k*^{2}*v*^{2}¢

+*B*^{2}*ξ*˜^{2}(*k*) . ^{(4.26)}

We can have two levels coalescing if*E*_{1}(*k*)=0for a complex*k*-value obtained by solving

£*B*^{2}+*k*^{2}*v*^{2}−*ξ*˜^{2}(*k*)−4*m*^{2}*v*^{2}¤2

+4 ˜*ξ*^{2}(*k*)*k*^{2}*v*^{2}=0 . ^{(4.27)}
As before, we rotate the Hamiltonian in equation (4.25) to the chiral basis, where it takes the form

*H*_{N2}^{chi}(*k*)=*U*_{3}^{†}*h*_{N}_{2}(*k*)*U*_{3}=

µ 0 *h*^{N2}_{u} (*k*)
*h*^{N2}_{l} (*k*) 0

¶

, *U*_{3}=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

,

*h*^{N2}_{u} (*k*)=

µ −*ξ*˜(*k*)+i¡

*kv*+2*mv*^{2}¢

*B*
*B* −*ξ*˜(*k*)+i¡

*kv*−2*mv*^{2}¢

¶ ,

*h*_{l}^{N2}(*k*)=

µ −*ξ*˜(*k*)−i¡

*kv*+2*mv*^{2}¢

*B*
*B* −*ξ*˜(*k*)−i¡

*kv*−2*mv*^{2}¢

¶

. ^{(4.28)}

The solutions for the EP’s are then given by either det£

*h*^{N2}_{u} (*k*)¤

=0 ⇒ £*ξ*˜(*k*)−i*kv*¤2

=*B*^{2}−4*m*^{2}*v*^{4}

⇒ *k*=*k*^{u}_{s}_{1}_{,s}_{2}= −i ln
µ

*s*_{1}
q

*m*^{2}*v*^{2}−2*mµ*˜+2i*s*_{2}*m*p

4*m*^{2}*v*^{4}−*B*^{2}+i*mv*

¶

, ^{(4.29)}

or

det£
*h*^{N2}_{l} (*k*)¤

=0 ⇒ £*ξ*˜(*k*)+i*kv*¤2

=*B*^{2}−4*m*^{2}*v*^{4}

⇒ *k*=*k*^{l}_{s}_{1}_{,s}_{2}= −i ln
µ

*s*1

q

*m*^{2}*v*^{2}−2*mµ*+˜ 2i*s*2*m*p

4*m*^{2}*v*^{4}−*B*^{2}−i*mv*

¶

, ^{(4.30)}

**-10** **-5** **5** **10****Μ**

**-5**
**5**

**E**_{1,2}**HkL**

(a)

**-6** **-4** **-2** **2** **4** **6** **Μ**

**-5**
**5**

**E**_{1,2}**HkL**

(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****p****c****)**
**B****/(****v****p****c****)**

(c)

**Figure 4.** (Color online) Parameters:*v*=1^{,}*m*=1/(2*v*^{2})^{,}*p*_{c}=2*vm*corresponding to the Hamiltonian in
equation (4.25). Panels (a) and (b) show the energy bands*E*_{1,2}(*k*), given in equation (4.26), as functions
of*µ*˜^{, for}*B*=0^{and}*B*=3, respectively.*E*_{1,2}(*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*µ*/(*v p*˜ *c*)−*B*/(*v p**c*)^{plane.}

where(*s*_{1}= ±1,*s*_{2}= ±1)^{. Clearly,}*k*=*k*^{u/l}_{s}_{1}_{,s}_{2} 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*v*^{2})^{,}
the energy bands for*B*=0^{and}*B*=3, and the contourplot for*f*( ˜*µ*,*B*)[deﬁned in equation (2.1)] have
been shown in ﬁgure 4. Once again we ﬁnd that *f*( ˜*µ*,*B*)gives the correct topological phase diagram in
ﬁgure 4 (c). Needless to add that here alsoexp(i*k*^{u/l}_{s}

1,*s*2)’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

1We would like to point out that this system is different from two-dimensional*p**x*+i*p**y*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.

Majorana operators is
*H*_{h}=X

**k**

³
ˆ
*a*^{†}

**k** *b*ˆ^{†}

**k**

´
*h*_{h}(_{k})

µ *a*ˆ_{k}
*b*ˆ_{k}

¶

, *h*_{h}(_{k})=

µ 0 *A*(_{k})
*B*(_{k}) 0

¶

, _{k}=(*k**x*,*k**y*) ,

*A*(_{k})= −2i

·

*J*_{3}+*J*_{1}cos

µ*k**x*−*k**y*

2

¶
+*J*_{2}cos

µ*k**x*+*k**y*

2

¶¸

+2

·
*J*_{1}sin

µ*k**x*−*k**y*

2

¶
+*J*_{2}sin

µ*k**x*+*k**y*

2

¶¸

,

*B*(**k**)=2i

·

*J*3+*J*1cos

µ*k*_{x}−*k*_{y}
2

¶
+*J*2cos

µ*k*_{x}+*k*_{y}
2

¶¸

+2

·
*J*1sin

µ*k*_{x}−*k*_{y}
2

¶
+*J*2sin

µ*k*_{x}+*k*_{y}
2

¶¸

(4.31) with the eigenvalues

*E*(_{k})= ±2
(·

*J*_{3}+*J*_{1}cos

µ*k**x*−*k**y*

2

¶
+*J*_{2}cos

µ*k**x*+*k**y*

2

¶¸2

+

·
*J*_{1}sin

µ*k**x*−*k**y*

2

¶
+*J*_{2}sin

µ*k**x*+*k**y*

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(*J*1,*J*2,*J*3,*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*_{⊥}=*k*_{x}^{, and}*k*_{∥}=*k*_{y} will be one of the parameters determining the topological
phase transition points. The solution for*B*(*k*_{x}=*k*_{⊥},*k*_{y}=*k*_{∥})=0is given by

*k*_{⊥}= −2 i ln

·

−*J*1exp(i*k*_{∥}/2)+*J*2exp(−i*k*_{∥}/2)
*J*3

¸

. ^{(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*(*J*_{1},*J*_{2},*J*_{3},*k*_{∥})deﬁned in equation (2.1). Panels (a) and (b)
show the number of chiral Majorana zero modes 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.

Majorana zero modes exist for all the values of*k*_{y}^{if}*J*_{1}+*J*_{2}<*J*_{3}. There is no edge state if|*J*_{1}−*J*_{2}| >*J*_{3}^{. For}
*J*_{1}=*J*_{2}=*J*_{3}, edge states exist if|*k*_{y}| >2*π*/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*_{⊥}=*k*_{y}^{, and}
*k*_{∥}=*k*_{x} will be now one of the parameters determining the topological phase transition points. The two
EP’s for*A*(*k*_{x}=*k*_{∥},*k*_{y}=*k*_{⊥})=0are given by

*k*_{⊥}^{±}= −2 i ln

·−*J*_{3}exp(−i*k*_{∥}/2)±
q

*J*_{3}^{2}exp(−i*k*_{∥})−4*J*_{1}*J*_{2}
2*J*_{2}

¸

. ^{(4.34)}

No Majorana zero mode exists for any value of*k*_{x} ^{if} *J*_{1}=*J*_{2}=*J*_{3}^{or} *J*_{1}<*J*_{2}^{. For} *J*_{1}>*J*_{2}, a Majorana
fermion can exist for a speciﬁc range of values for*k*_{x}. 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 aZ2topological invariant. The chiral symmetry operatorO 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]

*H*_{m1}=X

*k*

Ψ^{†}_{k}*h*_{m1}(*k*)Ψ*k*, Ψ*k*=(*c*_{k↑},*c*_{k↓},*c*^{†}

−*k*↓,−*c*^{†}

−*k*↑)^{T},

*h*_{m1}(*k*)=[*ξ**m*1(*k*)*σ*0+*λ*^{R}sin(*k*)*σ**z*]*τ**z*+∆cos(*k*)*σ*0*τ**x*, *ξ**m*1(*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:

*E*_{1}(*k*)= ±
q

[*ξ**m*1(*k*)−*λ*^{R}sin(*k*) ]^{2}+∆^{2}cos^{2}(*k*) , *E*_{2}(*k*)= ±
q

[*ξ**m*1(*k*)+*λ*^{R}sin(*k*) ]^{2}+∆^{2}cos^{2}(*k*) , ^{(5.2)}
whose plots are shown in ﬁgure 6 (a) for∆=0.1*t*^{and}*λ*^{R}=2*t*^{, as}*µ*/*t*is varied along the horizontal axis.

Observing that a chiral symmetry operatorO =*σ*0*τ**y* exists in the presence ofM*z*, we rotate the
Hamiltonian in equation (5.1) to the chiral basis, where it takes the form

*H*_{m1}^{chi}(*k*)=*U*_{4}^{†}*h*_{m1}(*k*)*U*_{4}=

µ 0 *h*^{u}_{m1}(*k*)
*h*_{m1}^{l} (*k*) 0

¶

, *U*_{4}= 1

p2

0 −i 0 i

0 1 0 1

−i 0 i 0

1 0 1 0

,

*h*^{u}_{m1}(*k*)=^{diag}

³

i∆cos(*k*)−*ξ**m*1(*k*)+*λ*^{R}sin(*k*), i∆cos(*k*)−*ξ**m*1(*k*)−*λ*^{R}sin(*k*)

´
,
*h*_{m1}^{l} (*k*)=^{diag}

³

−i∆cos(*k*)−*ξ**m*1(*k*)+*λ*^{R}sin(*k*), −i∆cos(*k*)−*ξ**m*1(*k*)−*λ*^{R}sin(*k*)´

. ^{(5.3)}

The solutions for the EP’s are then given by either det£

*h*^{u}_{m1}(*k*)¤

=0 ⇒ i∆cos(*k*)−*ξ**m*1(*k*)=*s*_{1}*λ*^{R}sin(*k*)

⇒ *k*=*k*^{m1u}_{s}_{1}_{,s}_{2}= −i ln

·*µ*+*s*_{2}q

*µ*^{2}−*λ*^{2}R−(*t*−i∆)^{2}
*t*−i (∆+*s*_{1}*λ*^{R})

¸

, ^{(5.4)}