Fermi arc criterion for surface Majorana modes in superconducting time-reversal symmetric Weyl semimetals
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Fermi arc criterion for surface Majorana modes in superconducting time-reversal symmetric
Weyl semimetals
Rauf Giwa, Pavan Hosur1
1
Department of Physics, University of Houston, Houston 77204, USA
In recent years, many clever realizations of Majorana fermions in condensed matter have been predicted
– and some largely verified – by exploiting the interplay between superconductivity and band topology in
metals and insulators. However, realizations in semimetals remain less explored. We ask, “under what
conditions do superconductor vortices in time-reversal symmetric Weyl semimetals – three-dimensional
band semimetals with only time-reversal symmetry – trap Majorana fermions on the surface?” If each
constant-kz plane, where z is the vortex axis, contains equal numbers of Weyl nodes of each chirality,
arXiv:2006.03613v2 [cond-mat.str-el] 10 May 2021
we predict a generically gapped vortex and derive a topological invariant ν = ±1 in terms of the Fermi
arc structure that signals the presence or absence of surface Majorana fermions. In contrast, if certain
constant-kz planes contain a net chirality of Weyl nodes, the vortex is gapless. We analytically calculate
ν within a perturbative scheme and provide numerical support with an orthorhombic lattice model. Using
our criteria, we predict phase transitions between trivial, critical and topological vortices by simply tilting
the vortex, and propose Li(Fe0.91 Co0.09 )As and Fe1+y Se0.45 Te0.55 with broken inversion symmetry as
candidates for realizing our proposals.
Over the last decade, the interplay of band topology, most of which stem from the chiral anomaly or the non-
spin-orbit coupling and superconductivity has paved a new conservation of chiral charge under parallel electric and
route to Majorana fermions (MFs) – as zero energy bound magnetic fields [57]. On the surface, the bulk band topol-
states trapped in topological defects such as domain walls ogy manifests as Fermi arcs that resemble disjoint seg-
and superconductor vortices [1–20]. Following strong evi- ments of a 2D Fermi surface and connect projections of
dence of MFs in several types of experiments in semicon- bulk WNs of opposite chirality onto the surface Brillouin
ductor nanowire-superconductor heterojunctions [11, 14, zone.
21], recent experiments have seen signatures of surface Motivated by the quest for MFs, in this work, we ask and
MFs at the ends of vortices in the bulk superconductor answer the question, “what is the fate of a superconductor
FeSe0.45 Te0.55 , making it the first three-dimensional (3D) vortex in a T-WSM?” We show that the vortex assumes
system with experimentally detected MFs [22, 23, 25, 55]. one of three possible phases – (i) gapped, with end-MFs;
A fundamental theoretical question that drives the search (ii) gapped, without end-MFs; (iii) gapless, with a series
for such MFs is, “if a 3D material develops conventional of topologically protected chiral Majorana modes (CMMs)
superconductivity, what properties of its normal state band dispersing along the vortex axis ẑ. Crucially, we prove
structure ensure that vortices in the superconductor trap that the phase of the vortex relies solely on the Fermi arc
MFs at their ends?” Restricting to band structures with configuration on the surface normal to ẑ, and the locations
time-reversal symmetry (T ), since T enables a conven- of the bulk WNs, thus allowing the phase to be predicted
tional superconducting instability in the first place, suffi- simply from experimentally accessible band structure data.
cient conditions are known in two generic cases. First, if Remarkably, simply tilting the vortex can drive transitions
the material is a band insulator, MFs at vortex ends exist if between the three phases.
the insulator is topological [5]. Second, a metal will host The criteria for the phases, depicted in Fig. 1, are as
MFs if it can be obtained by doping a topological insulator follows. Within each constant-kz plane, identify the pair
upto a threshold [6]. FeSe0.45 Te0.55 belongs to the latter (or pairs) of WNs of opposite chirality that are closest to
class, which is how MFs in it were predicted [26] before each other in periodic k-space. Connect the partners with
they were seen in experiments. a geodesic and project it onto the surface. From the re-
maining WNs, identify the next closest pair and project
A third type of generic 3D band structure that preserves their geodesic onto the surface, and so on for all WNs and
T is that of a time-reversal symmetric Weyl semimetal (T- constant-kz planes. If all the WNs find partners in the pro-
WSM) [27–30]. Here, point intersections between non- cess, the surface Brillouin zone will contain a set of lines
degenerate bands create Weyl nodes (WNs) with well- that, along with the Fermi arcs, will form M closed loops
defined chirality of ±1 based on whether they emit or or Fermi-geodesic surfaces (FGSs). We predict that the
absorb unit Berry flux, and the interplay of T and Bril- vortex in this case will be gapped and, when viewed as a
louin zone periodicity ensures a total of 4N WNs, where 1D superconductor, is characterized by a topological in-
N ∈ Z ≥ 1. Weyl semimetals are a topological state variant:
of matter in the sense that they are immune to any pertur-
bation that does not hybridize WNs of opposite chirality. ν = (−1)M (1)
The WNs spawn numerous topological responses [31–56], In other words, an odd integer M will result in a topo-
2
logically protected MF trapped in the vortex core on each
surface whereas even M will not trap surface MFs. For
all WNs to find partners, each constant-kz plane must
contain equals numbers of left- and right-handed WNs.
A preserved mirror or glide plane parallel to ẑ can en-
sure this since chirality is odd under reflection; in fact,
most known T-WSMs respect such a reflection symmetry
[45, 58–60]. In addition, for a minimal T-WSM with WNs
at (±K 1 , ±K 2 ), one can always orient the vortex such
that |K1z | = |K2z |, making (1) applicable.
On the other hand, if some WNs do not find partners,
they will yield open curves or Fermi-geodesic arcs (FGAs)
in the surface Brillouin zone. In this case, the vortex will Figure 1. Schematic of the main result. Orange (blue) cir-
be gapless and protected simply by kz -conservation. Each cles denote right(left)-handed WNs in the bulk, which pro-
unpartnered WN will project onto an end-point of an FGA duce right(left)-moving CMMs inside the vortex, colored
on the surface and contribute one 1D CMM to the bulk red (green). Dotted sheets are guides highlighting whether
vortex spectrum with a chirality equal to its own chirality WNs of opposite chiralities are in the same or different
times the vortex winding number of ±1. constant-kz layers, resulting in a gapped or gapless vortex,
These criteria survive doping as long as the resulting respectively. To determine the topological state of the vor-
Fermi surfaces are well-separated, mixing between sin- tex, identify pairs of WNs of opposite chirality and same
glet and triplet pairing provided the 3D superconductor is kz , draw a geodesic (black dashed lines) connecting each
gapped, and the presence of additional trivial Fermi sur- pair, and project the geodesic onto the surface. If the sur-
faces with rare exceptions mentioned later. Thus, they en- face projections of the geodesics (black solid lines) along
compass generic T-WSMs with arbitrary WN locations and with the FAs (red curves) form M closed loops, as shown
Fermi arc structures. The criteria are also insensitive to any in (a) for two different Fermi arc configurations with the
surface effects as long as the surface is exposed to vacuum, same bulk WN positions, the vortex is gapped and has a
because any reconstruction of the Fermi arcs under these topological invariant ν = (−1)M , whereas open arcs pro-
circumstances cannot change the parity of M or intercon- duce a gapless vortex, as shown in (b).
vert FGAs and FGSs. The parity of M can change only if negligible. If equi-chiral CMMs hybridize more strongly
the surface is exposed to a topological insulator; then, the than anti-chiral CMMs, the vortex will be gapless with
odd number of surface Fermi surfaces of the topological some modes crossing zero energy, but these modes can be
insulator effectively change M by an odd number. smoothly deformed to produce a gapped vortex that satis-
Eq. (1) is our main non-trivial result. We arrive at it fies (1).
by viewing the vortex as a 1D superconductor with only To understand (1) intuitively, imagine moving the WNs
particle-hole symmetry [61, 62] and computing the Z2 in k-space at fixed kz along the geodesics and annihilating
topological invariant derived by Kitaev for such a system them in pairs. If all WNs get annihilated in the process, the
[7]. In deriving it, we require two mild assumptions in the resulting insulator will be topological (trivial) if the sur-
clean limit: (i) for a given WN, if the two nearest nodes of face Fermi arcs evolve into an odd (even) number of sur-
opposite chiralities in the same kz plane are at distances face Fermi surfaces, while the superconducting vortex will
2
∆K1 and ∆K2 , respectively, then e−~vξ(∆K1 ) /∆0 be topological (trivial). However, the vortex spectrum re-
2
e−~vξ(∆K2 ) /∆0 or ∆K1 & ∆K2 , where ξ is the super- mains gapped in the process, so its topological state before
conducting coherence length, ∆0 is the pairing amplitude and after WN-annihilation must be the same. Heuristically,
far from the vortex and v is the typical quasiparticle ve- (1) says that vortex-end MFs are present (absent) if the T-
locity along the line connecting the WNs. This condition WSM normal state is “closer” to a topological (trivial) in-
ensures that the dominant hybridization is between CMMs sulator, where the “closeness” is defined by the distances
contributed by neighboring WNs of opposite chirality, as- WNs need to move in k-space at fixed kz to annihilate in
suming hybridization is driven by band curvature terms. If pairs and yield the insulator.
the hybridization is driven by non-magnetic disorder that Recent works have addressed similar questions. Ref.
is smooth over a length scale `D , the requirement acquires [63, 64] studied superconducting vortices in Dirac semi-
a relatively more restrictive form, ∆K1 `−1 D ∆K2 , metals, and showed that the vortex traps gapless, helical
while magnetic disorder renders the criterion (1) invalid. Majorana modes protected by crystal symmetries. This can
Fortunately, disorder can always be suppressed in princi- be viewed as a special case of our gapless vortex, in which
ple by creating cleaner samples whereas band curvature is CMMs of opposite chirality intersect at zero energy but do
unavoidable, ensuring a physical regime of validity of (1); not hybridize because of crystal symmetries. Ref. [65]
(ii) hybridization between CMMs of the same chirality is focused on T-WSMs and showed numerically on a lattice
3
model with N = 1 quadruplet of WNs that MFs appear on The CMMs remain robust in the limit |K n ξ| → ∞, but
the surface if the chemical potential µ is below a critical hybridize for finite |K n ξ|. However, the hybridization
value µc away from the WNs. At µ ∼ µc , the fact that of equichiral CMMs does not open a gap in the vortex
WNs are connected at higher energies becomes relevant, spectrum and can be adiabatically turned off while de-
the normal state itself begins to lose its essential Weyl- termining the topological nature of the vortex. Retain-
semimetal character, and the vortex is pushed into a trivial ing only hybridization between CMMs of opposite chiral-
state. Our work, which can capture arbitrary locations and ities, results in a gapped vortex, and the anti-commutation
numbers of WN quadruplets, contains the µ < µc results of Majorana operators then ensures that a generic per-
T
of Ref. [65] as a special case where N = 1, all the nodes turbation H1 in the basis (ψ+1 , ψ−1 , ψ+2 , ψ−2
) has the
have kz = 0 and neighboring WNs actually coincide, thus
0 0 iQ q12 q12̄
reducing the T-WSM to a Dirac semimetal. form H = where Q =
−iQ† 0 q1̄2 q1̄2̄
Continuum analytical result:- First, consider a sin- and qmn = hψm |H1 | ψn i. Moreover, qmn = qm̄n̄ if
gle WN described
P by the canonical Weyl Hamiltonian H1 preserves T , in which case the topological invari-
HW (P ) = h j=X,Y,Z vj Σj Pj − µ, where Σj are Pauli ant of the vortex, viewed as a 1D superconductor,
is
matrices in the pseudospin basis that labels the low en- 0
ν = sgn (Pf[H ]) = sgn det Q = sgn |q12 | − |q12̄ |
2 2
ergy bands, h = ±1 denotes the handedness of HW and
[7]. For a spatially smooth perturbation, qmn decays with
the momentum P is relative to the position of the WN.
|K m − K n |; for instance, band curvature terms in the
At PZ = 0, HW defines a single 2D Dirac node and
Bloch Hamiltonian are translationally invariant and yield
hence, resembles the surface Hamiltonian of a 3D topo- 1 2
logical insulator [3, 5, 6, 66]. In the presence of uni- qmn ∼ e− 2 |Km −Kn | ξ/∆0 for a linear vortex profile with
form s-wave pairing ∆, it formally yields a fully gapped slope ∆0 /ξ (see App. B 3 ). Then, |K 1 − K 2 | .
p + ip superconductor, while a vortex in this supercon- |K 1 + K 2 | produces a trivial vortex while |K 1 − K 2 | &
ductor, ∆(R) = ∆(R)eiΘ , traps a single MF in its core. |K 1 + K 2 | corresponds to a topological vortex with end
This was shown explicitly in Ref. [5] for vX = vY MFs. In terms of the surface states, geodesics connect-
and µ = 0, which further found that the ing K 1 to K 2 and −K 1 to −K 2 , along with the Fermi
MF ´
is given
arcs, form M = 2 FGSs. In contrast, geodesics connect-
(h) 1 † − 0R ∆(R0 )dR0
by ϕ̂ (R) = √
2
ic↑ (R) − hc↑ (R) e ing K 1 to −K 2 and −K 1 to K 2 form M = 1 FGS with
upto normalization and thus, is fully pseudospin-polarized: the Fermi arcs. Thus, there is a one-to-one correspondence
hϕ̂ |Σz | ϕ̂i = 1. The MF is topologically protected between ν and the number of FGSs, M that is captured
and hence, is expected to survive – albeit with partial by (1). Note that due to the Gaussian form of qmn , O(1)
pseudospin-polarization, 0 < hϕ̂ |Σz | ϕ̂i < +1 – even pre-factors will only produce logarithmic corrections to the
when µ 6= 0, vX 6= vY and the pairing is an arbitrary com- above inequalities.
bination of singlet and triplet but real and non-vanishing Next, consider moving the nodes away from kz = 0 in
everywhere on the Fermi surface. Indeed, Ref. [6] showed pairs while preserving T in the normal state. If K1z =
that the MF only requires a Berry phase of π on the 2D K2z , the CMM ψ+1 (r) can hybridize only with ψ+2 (r)
Fermi surface in the limit of weak pairing and a smooth but not with ψ−2 (r) so that the resulting vortex is adia-
vortex, which is guaranteed when the Fermi surface en- batically connected to one where all WNs are at kz = 0
closes a 2D Dirac node and the dispersion is linear. When and q12 6= 0 but q12̄ = 0. This vortex is trivial, since
PZ 6= 0, this MF disperses as Eh = hvZ PZ hϕ̂ |Σz | ϕ̂i, ν = sgn det |q12 |2 = 1. In contrast, if K1z = −K2z , the
thus realizing a CMM with chirality h that crosses E = 0 adiabatic equivalent with all WNs at kz = 0 has q12̄ 6= 0
at PZ = 0, or equivalently, at the kz of the parent WN. but q12 = 0, so that ν = sgn (−|q12̄ |2 ) = −1, indicating
In an actual T-WSM with multiple WNs and hence mul- a topological vortex. These conclusions extend straightfor-
tiple CMMs, kz -conservation ensures that right- and left- wardly to more quadruplets of WNs as well.
moving CMMs will not hybridize if their parent WNs are
in different constant-kz planes. The upshot is a gapless The conclusions also almost always survive the presence
vortex as depicted in Fig. 1(b). of trivial Fermi surfaces, i.e., Fermi surfaces that do not
enclose WNs or other band intersections, as long as T is
Next, consider a minimal T-WSM with one quadruplet preserved and the superconductor is fully gapped in the ab-
of WNs in the kz = 0 plane, at (±K 1 , 0), (±K 2 , 0), sence of vortices. For such superconductors, a vortex con-
so that the WNs at ±K n are related by T and have chi-
rality (−1)n . Moreover, suppose the Fermi arcs on the tains a set of bands εn (kz ) ∼ ξlF (kz ) n + 12 + φF2π
∆0 (kz )
z = 0 surface connect K 1 to K 2 and −K 1 to −K 2 , for each trivial Fermi surface, where n ∈ Z and lF (kz )
and the Fermi surfaces around the WNs are well-separated. (φF (kz )) is the perimeter (Berry phase) of the Fermi sur-
In the presence of a superconductor vortex along ẑ, each face cross-section at fixed kz [6]. Importantly, trivial Fermi
WN produces a CMM dispersing along (−1)n ẑ with a surfaces at each kz must appear in pairs with equal and op-
wavefunction ψ±n = eiK n ·r ϕn (r), where ϕn (r) is the posite Berry phases ±φF (kz ) to ensure that the constant-
zero mode of the vortex Hamiltonian near the nth WN. kz plane is a well-defined 2D metal in the absence of WNs
4
in that plane. Clearly, εn (kz ) 6= 0 as long as φF (kz ) 6= π , by
in which case these bands will not affect the physics at en- 2
ergy scales below the minigap min[∆0 /ξlF (kz )], thereby (E(k) + µ) =vz2 sin kz2 + m2 (k) + (3)
leaving the topological state of the vortex untouched. On q 2
the other hand, if φF (kz ) = π for some kz , the pair of vx2 sin kx2 + vy2 sin ky2 ± `
trivial Fermi surfaces at that kz will contribute a pair of
Varying βx,y,z and ` allows us to tune the model into trivial
counter-propagating CMMs that will generically hybridize
and topological insulating phases, as well as T-WSMs with
and gap out without affecting the vortex topological state.
N = 1, 2, 3, 4 quadruplets of WNs. The nodes all occur
The only scenario where our predictions can fail is if one or
in the kz = 0 or kz = π planes with up to two quadru-
more pairs of WNs of opposite chirality accidentally exist
plets in each plane, and are at the Fermi level when µ = 0.
at precisely this kz in such a way that at least one WN is
App. A contains further details of the model and simple,
closer to a trivial Fermi surface than to another WN. This
graphical methods for determining its normal state phases.
scenario is highly unlikely in real materials.
For simplicity, we restrict to µ = 0 and choose parameters
such that the kz = π plane is gapped, and tune the normal
state across trivial and topological insulators as well as T-
WSMs with N = 1, 2 quadruplets in the kz = 0 plane.
In the superconducting state, we restrict the pairing to s-
waveh via the Bogoliubov-deGennes
HamiltonianiH BdG =
P † P † †
k ck H(k)ck + τ =± ∆ckτ ↑ c−kτ ↓ + h.c. and as-
sume a unit vortex ∆(r) = |∆(r)| eiθ .
Fig. 2 shows the vortex phase diagram as a function of I -
breaking parameter, `, and the effective mass in the kz = 0
plane, m0 − βz . Considering a straight vortex along ẑ,
we compute the vortex topological invariant using Kitaev’s
criterion [7] for a 1D superconductor in Altland-Zirnbauer
class D [61, 62] and find excellent agreement with predic-
tions based on (1). The mismatch decreases with increasing
system size or decreasing pairing strength, suggesting that
Figure 2. Predicted (yellow mask) and calculated phase it is due to finite size of the lattice and departure from the
(black dots) diagram of the topological state of the vor- weak-pairing limit.
tex as a function of the normal state band structure defined To further establish our results, we show in Fig. 3 the
by (2). Black lines separate normal state phases which in- FGSs in the normal state and the probability density of
clude T-WSMs with N = 1 and N = 2 quadruplets of the lowest few vortex modes for selected points in Fig.
WNs, trivial insulator (N = 0+ ) and topological insulator 2. The Fermi arcs are obtained by plotting the lowest en-
(N = 0− ). We fix band parameters vx = 1.18, vy = .856, ergy at each surface momentum in the normal phase and
βx = .856, βy = 1.178, βz = 3.0, and choose a supercon- the geodesics are simply straight lines connecting proxi-
ductor vortex profile ∆(r) = 0.42 tanh (0.3r) and system mate WNs of opposite chirality in the kz = 0 plane. In
size Lx = Ly = 31 sites. Points marked t, m and b are each case, we find that the number of MFs localized to the
investigated in Fig. 3 vortex ends equals M , the number of FGSs, of which M
mod 2 are topologically protected.
Lattice numerics:- We support our general claims with
Tilting-driven phase transitions:- Next, we show that
numerics on an orthorhombic lattice model defined by
simply tilting the vortex can drive transitions between triv-
H = k c†k H(k)ck with
P
ial, topological and gapless vortex phases. We begin with
H(k) = τx σ · d (k) + τz m (k) − τy σz ` − µ (2) the trivial vortex with M = 2 corresponding to Fig. 3
(bottom) and rotate it about two separate axes as shown in
where di = vi sin ki , i = x, y, z , m(k) = m0 − Fig. 4. Rotating about the x-axis ensures each constant-
kz0 plane, where z 0 is the vortex axis, has the same number
P
i βi cos ki , τi and σi are Pauli matrices acting on or-
bital and spin space, respectively, and ck,τ,σ annihilates of WNs. However, the geodesic structure changes, result-
an electron with momentum k, spin σ and orbital index ing in M = 3, thus predicting a topological vortex. On
τ . The symmetries preserved by H(k) are time-reversal the other hand, rotating about a non-crystalline axis such as
(T = iσy K), reflection about the xz and yz planes the x = −y line results in all WNs having different kz0 and
(Mi→−i = τz σi , i = x, y, z ) and twofold rotation about hence, a critical vortex. In the weak-pairing, smooth-vortex
the z -axis (Ri = σi ), whereas inversion (I = τz ), reflec- limit, the trivial-to-topological or trivial-to-critical vortex
tion about the xy plane, and twofold rotation about the x- transition is expected at infinitesimal tilting. In the numer-
and the y -axes are broken. The spectrum of H(k) is given ics, we find transitions at θc ≈ 0.06π and θc ≈ 0.1π ,
5
Figure 4. Topological phase transition upon tilting the trivial
vortex in Fig. 3 (bottom). (a) Energy vs tilt angle about the
x-axis for the lowest few levels, obtained by diagonalizing
the vortex Hamiltonian in real space. One out of two zero
modes moves away for E = 0 at θc ≈ 0.06π , indicating a
trivial-to-topological phase transition. Inset shows that tilt-
ing about the x-axis results in M = 3 since only CMMs
coming from WNs with the same kz0 , z 0 being the vortex
Figure 3. Left column: Color plots of the lowest band for a axis, hybridize. This predicts a topological vortex, consis-
Lz = 45 layer slab in the normal state. Red filled (empty) tent with the observation. (b) Dispersion at θp = 0.1π ob-
circles denote projections of right-(left-)handed WNs onto tained by diagonalizing the vortex Hamiltonian in k-space,
the surface. Red lines mark Fermi arcs while black lines are showing that the vortex is gapped in the bulk and hence, the
projections of geodesics connecting nearest WNs of oppo- remaining zero mode in (a) at θ > θc is protected. (c and d)
site chiralities with the same bulk kz , which together form Analogous figures for tilting about the x = −y line. In (c),
M FGSs. Right column: Probability densities of six low- a small gap opens for one of the zero modes at θc ≈ 0.1π .
est energy states along a z -oriented superconductor vor- Inset shows that the surface has open FGAs since the bulk
tex calculated for a 31×31×45-site system. Bold (dot- CMMs cannot hybridize due to their parent WNs being in
ted) lines denote states with energies E < 5.0 × 10−3 different planes. (d) The vortex is gapless in the bulk, sug-
(> 1.0 × 10−2 ). The number of “zero” (E < 5.0 × 10−3 ) gesting that the small gap in (c) is a finite size gap for the
energy vortex modes localized at the vortex ends equals M , bulk critical mode.
of which M mod 2 are topologically protected MFs. All realizes a doped topological insulator that turns into a type-
figures use the same parameters as the ones in 2. Varying II superconductor below Tc ≈ 14.5K [25, 55], but the
band parameters are (top) l = 0.942, m0 = 6.28 (middle) normal state also has a pair of Dirac nodes along the c-
l = 0.972, m0 = 5.48 (bottom) l = 0.552, m0 = 6.18. axis ∼ 15meV above the Fermi level that may be ac-
cessed with naturally occuring Fe-dopants in this system
respectively. [67]. Perturbatively breaking I while preserving T will
Candidate material:- We propose Li(Fe0.91 Co0.09 )As transition either Dirac semimetal into a T-WSM with four
and Fe1+y Se0.45 Te0.55 with broken I as candidate mate- WNs at ±K 1 , ±K 2 with K1c ≈ K2c |K 1 − K 2 |.
rials for realizing our proposal. Li(Fe0.91 Co0.09 )As is a If superconductivity survives I -breaking, a vortex along ẑ
Dirac semimetal with two Dirac nodes on the c-axis of the will be topological (trivial) according to (1) if (ẑ × K 1 ) ·
crystal [67], has Fermi arcs on any surface not normal to (ẑ × K 2 ) > 0 (< 0) and |K1z | = |K2z |, whereas a vor-
the c-axis connecting projections of the Dirac nodes onto tex in any other direction will be critical. Assuming typi-
the surface, and shows strongly type-II superconductivity cal values v ∼ 105 m/s for the Dirac velocity, chemical
below Tc ≈ 9K at ambient pressure [68]. FeSe0.45 Te0.55 potential µ ∼ 100K relative to the WNs, ∆0 ∼ 5K,
6
ξ ∼ 5nm the penetration depth d ∼ 102 nm ob- logical superconductivity and majorana fermions. Semi-
served in LiFeAs [64] which guarantees negligible inter- conductor Science and Technology, 27(12), 2012. ISSN
vortex tunneling (∝ e−d/ξ ), and |K 1 − K 2 |/K1c ≈ 0.1 02681242. doi:10.1088/0268-1242/27/12/124003.
due to I -breaking, we crudely estimate a vortex gap of [9] Qin Liu, Chen Chen, Tong Zhang, Rui Peng, Ya Jun
∼ 0.1K . In other words, one must cool below ∼ 0.1K to Yan, Chen Hao Ping Wen, Xia Lou, Yu Long Huang,
Jin Peng Tian, Xiao Li Dong, Guang Wei Wang,
observe the MF. On the other hand, the estimate depends
Wei Cheng Bao, Qiang Hua Wang, Zhi Ping Yin,
exponentially on ξ , |K 1 − K 2 | and ∆0 and can change Zhong Xian Zhao, and Dong Lai Feng. Robust and
substantially with slight changes in these parameters (see clean majorana zero mode in the vortex core of high-
App. B 4 ). temperature superconductor (li0.84fe0.16)ohfese. Physi-
Conclusion:- We have derived a simple Fermi arc-based cal Review X, 8(4):41056, 2018. ISSN 21603308. doi:
criterion for the topological state of a superconductor 10.1103/PhysRevX.8.041056. URL https://doi.org/
vortex when the parent normal state is a T-WSM. By 10.1103/PhysRevX.8.041056.
merely tilting the magnetic field creating the vortex, we [10] Xiao Ping Liu, Yuan Zhou, Yi Fei Wang, and Chang De
Gong. Characterizations of topological superconductors:
propose transitions between trivial, topological and criti-
Chern numbers, edge states and majorana zero modes. New
cal vortices. Finally, we predict Li(Fe0.91 Co0.09 )As and Journal of Physics, 19(9), 2017. ISSN 13672630. doi:
Fe1+y Se0.45 Te0.55 with broken I as candidate materials 10.1088/1367-2630/aa8022.
that can realize our proposals. [11] Roman M. Lutchyn, Tudor D. Stanescu, and S. Das Sarma.
We acknowledge financial support from the Department Search for majorana fermions in multiband semiconducting
of Physics and the College of Natural Sciences and Mathe- nanowires. Physical Review Letters, 106(12):1–4, 2011.
matics, University of Houston and NSF-DMR-2047193. ISSN 00319007. doi:10.1103/PhysRevLett.106.127001.
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[12] Ning Ma. Majorana fermions in condensed matter: An out-
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10
Appendix A: Orthorhombic lattice model of a T-WSM
In this section, we analyze the orthorhombic lattice model studied in the main text and describe how to determine its
topological nature in the normal state. To recapitulate, the Bloch Hamiltonian is
H(k) = τx σ · d (k) + τz m (k) − τy σz ` − µ (A1)
P
where di = vi sin ki , i = x, y, z , m(k) = m0 − i βi cos ki and τi and σi are Pauli matrices acting on orbital and spin
space, respectively. H(k) preserves time-reversal (T = iσy K), reflection about the xz and yz planes (Mi→−i = τz σi ,
i = x, y, z ) and twofold rotation about the z -axis (Ri = σi ), but breaks inversion (I = τz ), reflection about the xy plane,
and twofold rotation about the x and the y axes are broken. Its spectrum is given by
q 2
2
(E(k) + µ) = vz2 sin kz2 + m2 (k) + vx2 sin kx2 + vy2 sin ky2 ± ` (A2)
Defining X = cos kx , Y = cos ky , a quadruplet of Weyl nodes (WNs) appears in the kz = 0 or π plane at (Kx , Ky ) =
(± cos−1 X, ± cos−1 Y ) for each intersection between the following ellipse and lines within the unit square X ∈ [−1, 1],
Y ∈ [−1, 1]
vx2 X 2 + vy2 Y 2 = vx2 + vy2 − `2 (A3)
βx X + βy Y = Mkz = m0 − βz cos kz (A4)
When the ellipse and line do not intersect within the unit square, the system is an T -symmetric insulator. These behaviors
are depicted in the top panel of Fig. 5
At ` = 0, I is restored, the system is necessarily insulating since the ellipse circumscribes the unit square and the
topological nature of the insulator can be deduced from the parity criterion which only depends on sgn[m(k)] at the
eight time-reversal invariant momenta (0/π, 0/π, 0/π). For larger `, the strong topological index of an insulating state
can be obtained easily by observing the connectivity of the Fermi arcs on an xy -surface, as shown in the bottom panel
of Fig. 5. Imagine tuning a parameter that creates and subsequently annihilates a quadruplet of WNs. Now, nodes are
always created as well as annihilated in pairs of opposite chirality. Moreover, creating a pair of nodes and moving them
apart leaves behind a surface Fermi arc that connects the surface projections of the nodes. If the nodes switch partners
between creation and annihilation – in other words, if a given right-handed WN is created along with a left-handed WN
but annihilates a different left-handed WN – a non-degenerate, T -invariant Fermi surface is left behind on the surface.
Such a Fermi surface can be viewed as the surface state of a topological insulator doped away from charge neutrality.
Therefore, each time WNs switch partners between creation and annihilation, the strong topological index of the bulk
insulator toggles.
In the main paper, we choose parameters such that the line defined by m(kz = π) = 0 never intersects the ellipse.
Then, all the normal state phase transitions occur via crossings in the kz = 0 plane, which gives access to trivial and
topological insulators as well as T-WSMs with N = 1, 2.
Appendix B: Vortex topological invariant in a minimal lattice model
In this section, we use a perturbative scheme to explicitly determine the topological state of the vortex in the lattice
model (A1) in the range of parameters which gives N = 1 quadruplet of WNs.
1. Reduction to a canonical Weyl Hamiltonian
We begin with the Bloch Hamiltonian (A1) and assume the parameters are chosen so that there is a single quadruplet of
WNs, at (±Kx , ±Ky , 0). The Bloch Hamiltonian at these points has a higher symmetry, namely, [H(K), τy σz ] = 0, so
it is convenient to work in the eigenbasis of τy σz . For convenience, let us perform a rotation
H 0 (K) = eiτx π/4 H(K)e−iτx π/4 = τx σ · d(k) + τz σz ` − µ (B1)
2
which explicitly diagonalizes the term proportional to `. Since |d(K)| = `2 at the nodes according to Eq. (4) of the
main paper, the four states at each WN have energies 2`, 0, 0, −2`. The two zero energy states explicitly are
1 T 1 T
|A0 i = √ 1, 0, 0, −eiθd , |B 0 i = √ 0, eiθd , 1, 0 (B2)
2 211
(a) (b)
(c) (d)
Figure 5. Prescription to determine the number of WN quadruplets (N ), the Fermi arc structure on the surface and the
Z2 invariant in the insulating phase in the lattice model (A1). Top: X = cos kz , Y = cos ky and the ellipses and lines
are given by (A3) and (A4), respectively, with smaller |`| defining larger ellipses. Each ellipse-line intersection within
the defines a quadruplet of WNs in the plane defining the line. Green arrows indicate the path of the intersections as the
ellipse is enlarged. Solid (dashed) ellipses denote T-WSMs with N quadruplets (insulators with N = 0). The ` = 0
ellipse circumscribes the square and defines an I -preserving insulator with Z2 indices given by the parity criterion [1].
It has the opposite (same) strong index as the innermost ellipse if exactly one line (no or both lines) intersects a vertical
and a horizontal edge of the unit square, as shown on the left (right). Bottom: Brillouin zone of the (001) surface and the
effect of moving WNs along the paths indicated in the top panel on the Fermi arcs. For simplicity, only the effect of WNs
in the kz = 0 plane is shown; the effects of kz = π WNs are identical. Circles with ± denote the surface projections
of right/left-handed WNs, and their trajectories as the ellipse in the top panel is enlarged are indicated by green arrows.
These trajectories trace out the Fermi arcs. If a quadruplet is created at a kx = −kx plane and annihilated on a ky = −ky
plane or vice-versa, the Fermi arcs close into a single Fermi surface, implying a change of the bulk strong Z2 topological
index. If a quadruplet is created and destroyed on a kx = −kx (or ky = −ky ) plane, the Z2 invariants corresponding to
the ellipse shrunk to a point and the ellipse circumscribing the unit square are the same.12
where θd = arg(dx + idy ) and the primes serve as reminders that we have performed a eiτx π/4 rotation. The low energy
Hamiltonian near the WN in the (|A0 i, |B 0 i)T basis is given by
0 0 vz px
0
HW (p) = (Σx , Σy , Σz ) βx sin Kx βy sin Ky 0 py − µ (B3)
2 −1 2 −1
−vx ` sin Kx cos Kx −vy ` sin Ky cos Ky 0 pz
where Σi are Pauli operators in the |A0 i, |B 0 i basis. Note that reversing KJ to get to a different WN is equivalent to
0 0
reversing pJ in HW . At pz = 0, HW contains only Σz and Σy . For convenience, we rotate Σz → −Σx , Σx → Σz to
00 iΣy π/4 0 −iΣy π/4 00
define HW = e HW e . HW in the pz = 0 plane is
00 −1 px
HW (pz = 0) = ` (Σx , Σy ) M̂ −µ (B4)
py
2 −1
vx ` sin Kx cos Kx vy2 `−1 sin Ky cos Ky
where M̂ = . To bring this into a canonical form, we perform a singular
βx sin Kx βy sin Ky
value decomposition of M̂
vX 0
M̂ = R(φΣ ) RT (φp ) (B5)
0 vY
cos φ − sin φ
where R(φ) = and vX,Y > 0. We have assumed that the WNs at ±(Kx , Ky ) can be brought into
sin φ cos φ
a canonical form by proper rotations. This automatically means that the nodes at ±(Kx , −Ky ) need improper rotations.
The necessity of singular value decomposition indicates that the principal axes for p and Σ are different, and both differ
from the Cartesian axes of the original problem. Moreover, vX 6= vY , implying that the WN is anisotropic. Nonetheless,
00
this can be brought into a canonical form HW = vX ΣX PX + vY ΣY PY − µ through the rotations
PX T px
= R (φp ) (B6)
PY py
X x
= RT (φp ) (B7)
Y y
ΣX Σx Σx
= RT (φΣ ) = e−iΣz φΣ /2 eiΣz φΣ /2 (B8)
ΣY Σy Σy
2. Vortex modes of anisotropic vortex
In the presence of s-wave superconductivity, the Bogoliubov-deGennes Hamiltonian is given by
00
00 HW (P ) ∆(R)
HBdG (P ) = (B9)
∆∗ (R) −HW 00
(P )
T
in the basis √12 cA0 + cB 0 , −cA0 + cB 0 , −c†A0 + c†B 0 , −c†A0 − c†B 0 . Furthermore, if the superconductivity develops a
00
vortex ∆(r) = ∆0 (r)eiθ , where θ = arg(x + iy), the pairing term in HBdG becomes ∆(R) = ei(φΣ +Θ) ∆0 (R), where
Θ = arg(X + iY ). If vX = vY , the problem has a rotational symmetry which can be used to obtain the eigenmodes of
00
HBdG analytically. This result is well-known [2–4]. When vX 6= vY , we can still obtain the eigenmodes analytically in
the linear approximation ∆0 (R) = ∆0 R/ξ , where ξ is the superconducting coherence length.
We explicitly write
00 00 ∆0 R
HBdG (P ) = Πz HW (P ) + (Πx cos (Θ + φΣ ) − Πy sin (Θ + φΣ )) (B10)
ξ
The φΣ -dependence can be eliminated by a Πz -rotation:
000
HBdG (P ) = e−iΠz φΣ /2 H 00 (P )eiΠz φΣ /2 (B11)
00 ∆0
= Πz HW (P ) + (Πx X − Πy Y ) (B12)
ξ13
At µ = 0, we can separate the X and Y parts of the problem via another rotation. Specifically, define
0000 000
HBdG (P ) = eiΠy ΣY π/4 HBdG (P ) e−iΠy ΣY π/4 (B13)
∆0 ∆0
= Πz vX ΣX PX + ΣY X − Πx vY PY + Πy Y (B14)
ξ ξ
√ √
−i vX aX i vY aY
s √ † √
2∆0 i √vX aX† i vY aY
= √ (B15)
ξ −i v a
Y Y i vX aX
√ √
−i vY a†Y −i vX a†X
q
ξ ∆0
where aJ = 2∆0 vJ ξ
J + ivJ PJ , J = X, Y is the usual annihilation operator for a quantum harmonic oscillator.
0000 T
The eigenstates of HBdG are of the form (|nX − 1, nY − 1i, |nX , nY − 1i, |nX − 1, nY i, |nX , nY i) . In this basis,
√ √
−i vX nX i vY nY
√ √
s
0000 2∆0 i vX nX i√ vY nY
HBdG (nX , nY ) = −i√v n i vX nX
(B16)
ξ Y Y √ √
−i vY nY −i vX nX
s
2∆0 √ √
= (Πz ΣY vX nX + Πx vY nY ) (B17)
ξ
Thus, it has the spectrum
s
2∆0
E (nX , nY ) = ± (vX nX + vY nY ) (B18)
ξ
In particular, the zero mode is given by nX = nY = 0 and has the wavefunction
ϕ0000 (R) = (0, 0, 0, |0, 0i)T ≡ (0, 0, 0, 1)T f00 (X, Y ) (B19)
s
∆0 X 2 Y 2
∆0
where f00 (X, Y ) = √ exp − + is the wavefunction for the (nX = 0, nY = 0) mode of
πξ vx vy 2ξ vX vY
the 2D harmonic oscillator. Undoing the rotations generated by Πy ΣY , Πz , Σy and the singular value decomposition gives
1 T
ϕ0 (R) = e−iΣy π/4 eiΠz Σz φΣ /2 ϕ00 = √ e−iΣy π/4 ieiφΣ , 0, 0, 1 f00 (X, Y ) (B20)
2
1 T
ϕ0 (x, y) = ieiφΣ , ieiφΣ , −1, 1 f˜(x, y) (B21)
2
q h 2
n oi
in the basis (cA , cB , c†B , −c†A ), where f˜(x, y) = πξ√∆v0X vY exp − ∆4ξ0r 1
vX
+ 1
vY
+ vX
1
− vY
1
cos[2(θ + φ p )] ≡
q h 2
n oi
∆
√ 0
πξ vX vY
exp − ∆4ξ 0r 1
v+
+ v1− cos[2(θ + φp )] . ϕ0 is an eigenstate of charge conjugation: Cϕ0 ≡ Πy Σy ϕ0∗ =
T
ie−iφΣ ϕ0 and hence, represents a Majorana mode. In the original basis cs↑ , cs↓ , cp↑ , cp↓ , c†s↓ , −c†s↑ , c†p↓ , −c†p↑ ,
e−iπ/4 iφΣ i(θd +φΣ ) T
ϕ(x, y) = √ −e , ie , −eiφΣ , −iei(θd +φΣ ) , −ie−iθd , 1, ie−iθd , 1 f˜(x, y) (B22)
2 2
≡ χf˜(x, y)
Finally, χ† Πz τx σz χ = 1, so non-zero kz induces a dispersion E(kz ) = vz sin kz and thus produces a CMM.You can also read
