Supplemental Material for Survival of the fractional Josephson effect in time-reversal-invariant
topological superconductors
Christina Knapp,
1, 2,
∗
Aaron Chew,
1,
∗
and Jason Alicea
1, 2
1
Department of Physics and Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, California 91125 USA
2
Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125 USA
(Dated: October 5, 2020)
LOCAL MIXING REVIEW
Reference 1 derived that when the operators describing a
Majorana Kramers pair depend on some parameters
η
, the lo-
cal mixing angle is given by
θ
=
1
2
∮
d
η
{
γ
(
η
)
,
∇
η
̃
γ
(
η
)
}
.
(1)
We review a simple example of how a non-zero mixing angle
can arise microscopically.
Consider a TRITOPS wire modeled by two Kitaev chains.
In the dimerized limit, the Hamiltonian is given by
H
0
=
ε
2
∑
j,σ
iγ
jbσ
γ
j
+1
aσ
(2)
where
σ
∈ {↑
,
↓}
labels the two time-reversed copies,
a
and
b
the two Majorana flavors that form the spinful fermion, and
j
the site. The two Majorana operators corresponding to the
same
j
and
σ
transform oppositely under
T
: here we take the
signs in Eq. (
??
) to be
s
a
=
−
1
, s
b
= 1
. This model has four
MZMs,
γ
1
aσ
, γ
Nbσ
. Let us assume that local perturbations on
the left end of the wire take the form of a chemical potential
H
μ
and
s
-wave pairing
H
∆
:
H
μ
(
t
) =
β
cos
α
(
t
)
2
∑
σ
(
iγ
1
aσ
γ
1
bσ
+ 1)
(3)
H
∆
(
t
) =
β
sin
α
(
t
)
2
(
iγ
1
a
↑
γ
1
b
↓
−
iγ
1
a
↓
γ
1
b
↑
)
,
(4)
where
α
parametrizes the ratio of the two terms and is time-
dependent. Both Eq. (3) and Eq. (4) commute with
T
. In the
presence of these perturbations, the new zero mode operators
become time-dependent as well:
γ
1
(
t
) = cos
ζγ
1
a
↑
−
sin
ζ
(cos
α
(
t
)
γ
2
a
↑
+ sin
α
(
t
)
γ
2
a
↓
)
(5)
̃
γ
1
(
t
) = cos
ζγ
1
a
↓
−
sin
ζ
(cos
α
(
t
)
γ
2
a
↓
−
sin
α
(
t
)
γ
2
a
↑
)
,
(6)
where
tan
ζ
=
β/ε
. Solving Eq. (1), we have
θ
1
=
−
sin
2
ζ
∮
dα
=
−
sin
2
ζ
∫
T
0
dt
̇
α
(
t
)
,
(7)
where
α
(
T
) =
α
(0) + 2
πn
, with
n
∈
Z
. Therefore, provided
α
has non-trivial winding,
θ
1
6
= 0
and
γ
1
,
̃
γ
1
undergo local
mixing.
a
a
...
...
...
...
FIG. 1. TRITOPS Josephson junction with each wire modeled by
two Kitaev chains in the dimerized limit. Dotted lines indicate
hybridization of Majorana Kramers pairs (dark purple) leading to
Eq. (
??
). Local mixing arises when Majorana Kramers pairs undergo
time-dependent coupling to gapped Majorana modes.
TRITOPS JOSEPHSON JUNCTION
A TRITOPS wire can be thought of as two topological su-
perconductors related by time reversal symmetry. Labeling
the two copies with a spin degree of freedom
σ
∈ {↑
,
↓}
,
time reversal acts on the fermionic operators of the
J
th wire
as
c
J
= (
c
J,
↑
,c
J,
↓
)
T
as [2]
T
c
J
T
−
1
=
is
J
(
φ
)
e
iφ
J
σ
y
c
J
.
(8)
The sign
s
J
(
φ
) =
±
1
represents a
Z
2
gauge-freedom
when defining symmetry transformations of superconductors.
When multiple TRITOPS are present but disconnected, each
satisfies its own time reversal symmetry according to Eq. (8).
When two TRITOPS are connected, e.g. by a Josephson junc-
tion, the global symmetry transformation must be consistent
between the two. Therefore, a TRITOPS Josephson junction
is only symmetric under
T
when the phase difference between
the left and right superconductors is a multiple of
π
. We label
these discrete values the “time-reversal-invariant points” and
fix
s
L
(
φ
L
) = +1
and
s
R
(
φ
R
=
nπ
+
φ
L
) =
−
(
−
1)
n
below.
A simple model of a TRITOPS Josephson junction is
H
JJ
=
̃
λc
†
L
↑
c
R
↑
−
̃
λ
∗
c
†
L
↓
c
R
↓
+
λc
†
L
↑
c
R
↓
+
λ
∗
c
†
L
↓
c
R
↑
+
h.c.
(9)
where
L/R
denote whether the fermion belongs to the wire
on the left/right end of the junction and we can generically
allow for different tunneling amplitudes between wires with
the same and different
σ
labels.
Each fermionic operator can be written as
c
Jσ
=
e
−
iφ
J
2
2
(
γ
Jaσ
+
iγ
Jbσ
)
,
(10)
2
where
φ
J
is the superconducting phase of the wire on the
J
th
side of the junction (
J
∈ {
L,R
}
), and the operators
γ
cσ
,
c
∈{
a,b
}
satisfy the Majorana anticommutation relation
{
γ
cσ
,γ
c
′
σ
′
}
= 2
δ
cc
′
δ
σσ
′
.
(11)
Equations (8) and Eq. (10) imply Eq. (
??
).
Each copy of a topological superconductor has a single
MZM at its end point. Projecting to the low-energy subspace
takes
c
Lσ
→
e
−
iφ
L
2
2
γ
Laσ
, c
Rσ
→
ie
−
iφ
R
2
2
γ
Rbσ
.
(12)
Fixing
φ
L
= 0
and
φ
R
=
φ
and dropping the
a/b
label of the
Majorana operators, Eq. (
??
) becomes
H
JJ
=
1
2
∑
σ
=
↑
/
↓
[
cos(
φ/
2)
(
σ
Re
[
̃
λ
]
iγ
Lσ
γ
Rσ
+
Re
[
λ
]
iγ
Lσ
γ
R
̄
σ
)
+ sin(
φ/
2)
(
Im
[
̃
λ
]
iγ
Lσ
γ
Rσ
+
σ
Im
[
λ
]
iγ
Lσ
γ
R
̄
σ
)]
.
(13)
In the above, we have written
̄
σ
to indicate the oppo-
site choice of
σ
for the subscripts, and
σ
as a coefficient
to correspond to
±
for
↑
/
↓
.
We recover Eq. (
??
)
by setting
̃
λ
= 0
for simplicity, denoting the real and
imaginary parts of
λ
with subscripts
e/o
, and identifying
(
γ
L,
↑
,γ
L,
↓
,γ
R,
↑
,γ
R,
↓
)
→
(
γ
L
,
̃
γ
L
,
̃
γ
R
,γ
R
)
.
Deriving
H
eff
We now derive Eq. (
??
) in the main text. The Majorana op-
erators in Eqs. (
??
) and (
??
) can be written in terms of com-
plex fermionic operators as
f
†
=
1
2
(
γ
′
ε
−
iγ
ε
)
(14)
̃
f
†
=
1
2
( ̃
γ
′
ε
−
i
̃
γ
ε
)
(15)
c
†
=
1
2
(
γ
L
+
iγ
R
)
(16)
̃
c
†
=
1
2
( ̃
γ
L
+
i
̃
γ
R
)
(17)
Then, defining even-parity basis states so that
|
0
〉
corresponds
to the vacuum state annihilated by
c,
̃
c,f,
̃
f
and
|
1
〉
=
c
†
̃
c
†
|
0
〉
(18)
|
2
〉
=
f
†
c
†
|
0
〉
(19)
|
3
〉
=
f
†
̃
c
†
|
0
〉
(20)
|
4
〉
=
̃
f
†
c
†
|
0
〉
(21)
|
5
〉
=
̃
f
†
̃
c
†
|
0
〉
(22)
|
6
〉
=
f
†
̃
f
†
|
0
〉
(23)
|
7
〉
=
f
†
̃
f
†
c
†
̃
c
†
|
0
〉
(24)
so that
f
†
f
=
1
2
(1
−
iγ
ε
γ
′
ε
)
and
c
†
c
=
1
2
(1
−
iγ
L
γ
R
)
and
similarly for the time-reversed partners. In this basis, the full
Hamiltonian can be written in first-quantized form as
H
=
λ
e
0
β
2
cos
α
−
β
2
sin
α
β
2
sin
α
β
2
cos
α
0
0
0
−
λ
e
β
2
sin
α
β
2
cos
α
−
β
2
cos
α
β
2
sin
α
0
0
β
2
cos
α
β
2
sin
α
ε
−
λ
o
0
0
0
β
2
sin
α
−
β
2
cos
α
−
β
2
sin
α
β
2
cos
α
0
ε
+
λ
o
0
0
β
2
cos
α
β
2
sin
α
β
2
sin
α
−
β
2
cos
α
0
0
ε
−
λ
o
0
−
β
2
cos
α
−
β
2
sin
α
β
2
cos
α
β
2
sin
α
0
0
0
ε
+
λ
o
β
2
sin
α
−
β
2
cos
α
0
0
β
2
sin
α
β
2
cos
α
−
β
2
cos
α
β
2
sin
α
2
ε
+
λ
e
0
0
0
−
β
2
cos
α
β
2
sin
α
−
β
2
sin
α
−
β
2
cos
α
0
2
ε
−
λ
e
(25)
where we have adopted the shorthand
λ
e
= 2
λ
e
cos
(
φ
2
)
and
λ
o
= 2
λ
o
sin
(
φ
2
)
and suppressed the time-dependence of
φ
and
α
. Working to order
ε
−
2
, the two lowest energies are
1
/
2
(
t
) =
±
2
λ
e
(
1
−
β
2
2
ε
2
)
cos
(
φ
(
t
)
2
)
(26)
3
with corresponding instantaneous eigenstates
|
ψ
1
(
t
)
〉
= +
|
1
〉−
β
2
ε
(sin
α
(
t
) [
ν
−
+
(
t
)
|
2
〉
+
ν
−−
(
t
)
|
5
〉
]
−
cos
α
(
t
) [
ν
−−
(
t
)
|
3
〉−
ν
−
+
(
t
)
|
4
〉
])
−
(
β
2
ε
)
2
(
|
1
〉−|
6
〉
)
.
(27)
|
ψ
2
(
t
)
〉
=
−|
0
〉
+
β
2
ε
(cos
α
(
t
) [
ν
++
(
t
)
|
2
〉
+
ν
+
−
(
t
)
|
5
〉
]
−
sin
α
(
t
) [
ν
+
−
(
t
)
|
3
〉−
ν
++
(
t
)
|
4
〉
]) +
(
β
2
ε
)
2
(
|
0
〉
+
|
7
〉
)
(28)
We have defined
ν
p,p
′
= 1 + (
−
p
λ
e
(
t
) +
p
′
λ
o
(
t
))
/ε
with
p,p
′
=
±
1
.
As described in the main text, when
β
̇
α
(
t
)
ε
2
, transi-
tions between the low and high-energy states are negligible.
Solutions to the Schr
̈
odinger equation for a state initialized in
the low-energy subspace take the form
|
Φ(
t
)
〉
=
v
1
(
t
)
|
ψ
1
(
t
)
〉
+
v
2
(
t
)
|
ψ
2
(
t
)
〉
.
(29)
The coefficients satisfy the equation of motion
i∂
t
v
= [
H
inst
(
t
) +
H
B
(
t
)]
v
=
H
eff
(
t
)
v
(30)
where
H
inst
(
t
) = 2
λ
e
(
1
−
β
2
2
ε
2
)
cos
(
φ
(
t
)
2
)
σ
z
(31)
H
B
(
t
) =
〈
ψ
1
(
t
)
|
∂
t
|
ψ
2
(
t
)
〉
σ
y
=
̇
α
2
β
2
ε
2
σ
y
,
(32)
recovering Eq. (
??
):
H
eff
(
t
) = 2
λ
′
e
cos [
φ
(
t
)
/
2]
σ
z
+ ̇
α
(
t
)
β
2
2
ε
2
σ
y
.
(33)
Note that all
λ
o
dependence drops out at order
ε
−
2
.
Time evolution according to
H
eff
(
t
)
Equation (
??
) has the general form
H
(
t
) =
a
(
t
)
σ
z
+
b
(
t
)
σ
y
with instantaneous eigenvalues and eigenstates
±
(
t
) =
±
Ω(
t
) =
±
√
a
(
t
)
2
+
b
(
t
)
2
(34)
|±
(
t
)
〉
=
∓
iβ
±
|
0
〉
+
β
∓
|
1
〉
(35)
where
|
0
〉
,
|
1
〉
are the eigenstates of
σ
z
corresponding
to eigenvalues
±
1
, respectively, and we have defined
β
±
(
t
) =
√
Ω(
t
)
±
a
(
t
)
2Ω(
t
)
. Consider the parameter
A
=
max
t
|〈
+(
t
)
|
̇
H
eff
(
t
)
|−
(
t
)
〉|
4
min
t
′
Ω(
t
′
)
2
(36)
=
max
t
[
|−
̇
a
(
t
)
b
(
t
) +
̇
b
(
t
)
a
(
t
)
|
/
Ω(
t
)
]
4
min
t
′
Ω(
t
′
)
2
.
(37)
The adiabatic theorem asserts that when
A
1
, the system
initialized in an energy eigenstate remains in that eigenstate
throughout the time evolution [3, 4].
For
H
eff
(
t
)
,
A
and
Ω(
t
)
evaluate to
A
=
λ
′
e
|−
cos[
φ
(
̃
t
)
/
2]
̈
θ
(
̃
t
)
−
̇
φ
(
̃
t
) sin[
φ
(
̃
t
)
/
2]
̇
θ
(
̃
t
)
/
2
|
/
Ω(
̃
t
)
4Ω(
̄
t
)
2
(38)
Ω(
t
) =
√
4
λ
′
2
e
cos
2
[
φ
(
t
)
/
2] +
̇
θ
(
t
)
2
/
4
,
(39)
where
̃
t
is the time that maximizes the numerator and
̄
t
the
time that minimizes the denominator. When
̈
θ
(
t
) = 0
,
A
re-
duces to the inverse of the Landau-Zener parameter
x
:
A
LZ
=
λ
′
e
̇
φ
(
t
∗
)
̇
θ
2
=
1
x
.
(40)
Alternatively, for the quench considered in Fig.
??
,
̇
φ
→
0
and
A
becomes
A
quench
=
λ
′
e
cos[
φ
(
t
1
)
/
2]
̈
θ
(
t
1
)
/
Ω(
t
1
)
4Ω(
̄
t
)
2
,
(41)
where
t
1
is
the
location
of
the
quench.
If
̇
θ
max
2
λ
′
e
cos[
φ
(
t
1
)
/
2]
, then the denominator reduces
to
̇
θ
2
max
and
A
quench
≈
̈
θ
(
t
1
)
2
̇
θ
2
max
≈
1
2
τ
̇
θ
max
.
(42)
Thus the transition probability approaches zero for
τ
̇
θ
max
1
/
2
.
If instead
̇
θ
max
/
2
and
2
λ
′
e
cos[
φ
(
t
1
)
/
2]
are comparable (as is the case in Fig.
??
), the adiabatic
criterion becomes
8
τ
Ω(
t
1
)
1
.
To analyze the time evolution according to Eq. (
??
) more
generally, we can consider the Schr
̈
odinger equation for a state
|
ψ
(
t
)
〉
=
∑
σ
=
±
c
σ
(
t
)
|
σ
(
t
)
〉
. The coefficients
c
±
(
t
)
satisfy
i
(
̇
c
+
(
t
)
̇
c
−
(
t
)
)
= (Ω(
t
)
σ
z
+
v
(
t
)
σ
y
)
(
c
+
(
t
)
c
−
(
t
)
)
(43)
where
v
(
t
)
≡〈
+(
t
)
|
∂
t
|−
(
t
)
〉
=
̇
a
(
t
)
b
(
t
)
−
̇
b
(
t
)
a
(
t
)
2Ω(
t
)
2
(44)
=
λ
′
e
(
̇
φ
sin[
φ/
2]
̇
θ/
2 + cos[
φ/
2]
̈
θ
)
2Ω
2
.
(45)
4
When
|
v
(
t
)
|
Ω(
t
)
, the coefficients evolve according to a
diagonal Hamiltonian and the system initialized in the instan-
taneous ground state will remain in the instantaneous ground
state at later times, resulting in the conventional
2
π
-periodic
Josephson effect. (Note that max
t
|
v
(
t
)
|
/
Ω(
t
)
corresponds to
A
when
̃
t
=
̄
t
.) When
|
v
(
t
)
|
Ω(
t
)
, the instantaneous en-
ergy states undergo Rabi oscillations, and the current-phase
relation will generally be aperiodic.
APERIODICITY FROM LOCAL MIXING
Consider a junction described by Eq. (13). In the even par-
ity sector
iγ
L
γ
R
=
i
̃
γ
L
̃
γ
R
, we can define Pauli matrices
σ
x
=
iγ
L
γ
R
=
i
̃
γ
L
̃
γ
R
(46)
σ
y
=
iγ
L
̃
γ
R
=
−
i
̃
γ
L
γ
R
(47)
σ
z
=
iγ
L
̃
γ
L
=
iγ
R
̃
γ
R
(48)
so that
H
(
e
)
JJ
(
t
) = 2
√
λ
2
e
+
̃
λ
2
e
cos
(
φ
(
t
)
2
)(
0
a
−
ib
a
+
ib
0
)
(49)
for
a
=
λ
e
/
√
λ
2
e
+
̃
λ
2
e
, b
=
̃
λ
e
/
√
λ
2
e
+
̃
λ
2
e
. When
φ
is not
equal to an odd multiple of
π
, the junction eigenstates are
|
I
±
〉
=
1
√
2
(
a
−
ib
±
1
)
.
(50)
Note that
|
I
−
〉
is the instantaneous ground state of the junc-
tion for
φ < π
, while
|
I
+
〉
is the instantaneous ground state
for
φ > π
. Consider a thought experiment of a phase-biased
TRITOPS Josephson junction, undergoing the following pro-
tocol. Initialize the system at
φ
= 0
in the state
|
I
+
〉
, then
evolve the phase
φ
such that at the
k
th time invariant point
one of the Majorana Kramers pairs accrues a local mixing an-
gle
θ
k
. In the absence of any other noise sources, between the
k
−
1
th and
k
th time-reversal invariant points, the system is in
a superposition of junction eigenstates
|
ψ
12
〉
= cos
(
∑
k
θ
k
2
)
|
I
+
〉
+
i
sin
(
∑
k
θ
k
2
)
|
I
−
〉
(51)
with current expectation value
〈
I
(
φ
)
〉
=
−
e
~
√
λ
2
e
+
̃
λ
2
e
cos
k
∑
j
=1
θ
j
sin
(
φ
2
)
.
(52)
When
θ
k
6
= 2
π
,
cos
k
−
1
∑
j
=1
θ
j
6
= cos
k
+1
∑
j
=1
θ
j
(53)
thus the current expectation value is not
4
π
periodic. More
generally,
〈
I
(
φ
)
〉
is aperiodic except for fine-tuned choices of
the
θ
j
.
The phase-biased system is not necessarily the most ex-
perimentally accessible, as usually phase would be tuned by
a magnetic field, whose presence would break the time re-
versal symmetry of the junction. A more physically rele-
vant setup is for the junction to be voltage-biased, so that the
DC Josephson equation implies a constant phase sweep speed
̇
φ
= 2
eV/
~
=
ω
J
. When the system undergoes a
4
π
periodic
fractional Josephson effect, the power spectrum of the current
P
(
ω
) = lim
C
→∞
∫
C
0
dt
∫
C
0
dt
′
〈
I
(
t
′
)
I
(
t
)
〉
e
iω
(
t
′
−
t
)
(54)
exhibits a peak at
ω
=
±
ω
J
/
2
. An aperiodic current-phase
relation manifests as no peak in the power spectrum.
If the only source of noise is local mixing, then the proba-
bility
q
±
of occupying junction eigenstates
|
I
±
〉
only changes
after passing through a time reversal invariant point.
If
s
k
= 1
−
p
k
is the probability of transitioning between junc-
tion eigenstates (i.e.
p
k
is the probability of transitioning be-
tween instantaneous energy eigenstates) at the
k
th such point,
and
q
±
(
t
k
)
is the occupation probability of
|
I
±
〉
preceding
that point, then
(
q
+
(
t
k
+1
)
q
−
(
t
k
+1
)
)
=
(
1
−
s
k
s
k
s
k
1
−
s
k
)(
q
+
(
t
k
)
q
−
(
t
k
)
)
.
(55)
Approximating
s
k
= sin
2
(
θ
k
/
2)
by its average value,
̄
s
(
q
+
(
t
k
+1
)
q
−
(
t
k
+1
)
)
=
1
2
([
1 + (1
−
2 ̄
s
)
k
]
1
1
+
[
1
−
(1
−
2 ̄
s
)
k
]
σ
x
)
(
q
+
(
t
0
)
q
−
(
t
0
)
)
.
(56)
The matrix in Eq. (56) defines the propagator from
t
j
to
t
j
+
k
:
U
(
t
=
2
πk
ω
J
)
=
1
2
([
1 + (1
−
2 ̄
s
)
k
]
1
1
+
[
1
−
(1
−
2 ̄
s
)
k
]
σ
x
)
.
(57)
Note that in the large
k
limit the system approaches the maximally mixed state at a rate
ω
J
ln[1
−
2 ̄
s
]
/
2
π
.
5
The current is
I
±
(
t
) =
±
e
~
√
λ
2
e
+
̃
λ
2
e
sin
(
ω
J
2
t
)
,
(58)
corresponding to correlator for
t
′
> t
[5]
〈
I
(
t
′
)
I
(
t
)
〉
=
∑
ij
=
±
I
i
(
t
′
)
I
j
(
t
)
U
ij
(
t
′
−
t
) = 2
I
2
0
sin
(
ω
J
t
2
)
sin
(
ω
J
t
′
2
)
(1
−
2 ̄
s
)
ω
J
(
t
′
−
t
)
2
π
(59)
= 2
I
2
0
sin
(
ω
J
t
2
)
sin
(
ω
J
t
′
2
)
(
e
ω
J
2
π
ln[1
−
2 ̄
s
](
t
′
−
t
)
Θ (1
−
2 ̄
s
) +
e
ω
J
2
π
(
iπ
+ln[2 ̄
s
−
1])(
t
′
−
t
)
Θ (2 ̄
s
−
1)
)
(60)
for
I
0
=
e
~
√
λ
2
e
+
̃
λ
2
e
. Therefore, the power spectrum is
P
(
ω
) = lim
C
→∞
2
I
2
0
C
∫
C
0
dt
∫
C
0
dt
′
e
iω
(
t
′
−
t
)
sin
(
ω
J
t
′
2
)
sin
(
ω
J
t
2
)
×
(
e
ω
J
2
π
ln[1
−
2 ̄
s
]
|
t
−
t
′
|
Θ(1
−
2 ̄
s
) +
e
i
ω
J
2
(
t
−
t
′
)
e
ω
J
2
π
ln[2 ̄
s
−
1]
|
t
−
t
′
|
Θ(2 ̄
s
−
1)
)
(61)
=
I
2
0
2
πω
J
∑
a
=
±
1
ln[1
−
2 ̄
s
]
(
ω
ω
J
+
a
2
)
2
+
(
ln[1
−
2 ̄
s
]
2
)
2
Θ(1
−
2 ̄
s
) +
ln[2 ̄
s
−
1]
(
ω
ω
J
+
a
−
1
2
)
2
+
(
ln[2 ̄
s
−
1]
2
)
2
Θ(2 ̄
s
−
1)
.
(62)
As
̄
s
→
0
(
̄
p
→
1
,
r
1
), the power spectrum has two
peaks at
ω
=
±
ω
J
/
2
, corresponding to a fractional Joseph-
son effect. When
̄
s
→
1
(
̄
p
→
0
,
r
1
), the power spec-
trum peaks at
ω
= 0
, ω
J
, corresponding to the standard
2
π
-
periodic Josephson effect. As
̄
s
→
1
/
2
from either side,
P
(
ω
)
flattens- signaling an aperiodic current-phase relation.
T
2
= +1
JOSEPHSON JUNCTIONS
Consider the model for a topological superconductor sug-
gested by Refs. 6 and 7
H
=
∫
x
ψ
†
(
−
∂
2
x
2
m
−
μ
−
hσ
x
−
iασ
y
∂
x
)
ψ
+ ∆
ψ
↑
ψ
↓
+
H.c.
(63)
where spin indices have been suppressed,
h
is a Zeeman term,
α
is the spin-orbit coupling, and
∆
is the superconduting
gap. This Hamiltonian is symmetric under
T
=
K
time-
reversal-symmetry [8, 9], which in this model is simply com-
plex conjugation. This symmetry is an artifact of the low-
energy Hamiltonian and can be broken by adding higher-order
hopping terms or interactions. Nonetheless, such terms are
expected to be weak and for low energies the wire satisfies
T
2
= +1
.
We now derive the Josephson junction Hamiltonian for the
setup shown in Fig.
??
when each Majorana nanowire indi-
vidually satisfies
T
. Label the fermionic operators by
c
Jj
,
J
∈ {
L,R
}
labeling the left/right side of the junction, and
j
∈ {
1
,
2
}
labeling the top or bottom wire. The
c
J,j
trans-
form trivially under
T
; thus the most general non-interacting
Hamiltonian describing the Josephson junction that is even
under
T
is
H
(+1)
JJ
=
∑
J,j
6
=
k
Λ
Jjk
c
†
Jj
c
Jk
+
∑
j,k
(
2
λ
jk
c
†
Lj
c
Rk
+
h.c
)
.
(64)
where all tunneling amplitudes are real:
Λ
Jjk
,
λ
jk
∈
R
.
Time reversal symmetry acts on the complex fermionic
operators
c
Jj
=
e
−
iφ
J
/
2
2
(
γ
Jaj
+
iγ
Jbj
)
as
c
Jj
→
s
J
(
φ
J
)
e
iφ
J
c
Jj
. Thus, we once again see that
φ
=
φ
R
−
φ
L
=
nπ
are the time-reversal invariant points. Fixing
φ
L
= 0
and
φ
R
=
φ
, the transformation on the Majorana operators is
γ
Jaj
→
s
J
γ
Jaj
, γ
Jbj
→−
s
J
γ
Jbj
(65)
with signs
s
L
= 1
,
s
R
= (
−
1)
n
for
φ
=
nπ
.
Projection to the low-energy subspace takes the same form
as Eq. (12)
c
Lj
→
e
−
iφ
L
2
2
γ
Laj
c
Rj
→
ie
−
iφ
R
2
2
γ
Rbj
.
(66)
From here on, we drop the
a/b
labels and write the zero mode
operators as
γ
Jj
. Under
T
,
iγ
J
1
γ
J
2
→−
iγ
J
1
γ
J
2
(67)
iγ
Lj
γ
Rk
→
s
L
s
R
iγ
Lj
γ
Rk
= (
−
1)
n
iγ
Lj
γ
Rk
.
(68)
Equation (67) implies
Λ
J
= 0
(and is precisely why in the
presence of
T
the quantum dot-based MZM parity measure-
ment proposed in Ref. 10 does not work). Therefore, we re-
6
cover Eq. (
??
)
H
(+1)
JJ
=
∑
j
iλ
jk
cos
(
φ
2
)
γ
Lj
γ
Rk
.
(69)
The model given in Eq. (25) is purely real and thus also sat-
isfies
T
2
= +1
symmetry. As such, the derivation of Eq. (
??
)
similarly holds for this system as well.
Multiwire topological Josephson junctions
We investigate the effect of local mixing on a Josephson
junction between two sets of
m
Majorana wires. Above we
argued that
m
= 2
reproduces the aperiodic behavior of a
TRITOPS junction. We now demonstrate that interactions re-
store
4
π
periodicity for
m
= 3
, and
2
π
periodicity for
m
= 4
.
Such Josephson junctions offer a testbed for probing the
Z
8
classification of Majorana nanowires theorized by Ref. 8.
We consider the low-energy Hamiltonian
H
=
∑
j
E
j
cos
(
φ
2
)
iγ
Lj
γ
Rj
,
(70)
where
j
runs over each of the
m
wires and
L
and
R
signify the
wires to the left and right of the junction. After one evolution
γ
Rj
→−
γ
Rj
. We can combine Majorana fermions into Dirac
fermions as
c
j
=
γ
Lj
+
iγ
Rj
. After one evolution the occupa-
tion of this bound state switches. Notice that for
m
wires we
track
2
m
bound states, which for free fermions all intersect at
φ
=
π
(where all energies are
0
).
•
m
= 1
. The standard fractional Josephson is immune
to local mixing, as the two bound states differ by local
fermion parity. No local mixing terms are allowed that
mix the states at
φ
=
π
.
•
m
= 2
. The model posited in previous Appendices still
respects
T
2
= +1
symmetry. The four states in ques-
tion split into even and odd parity states. Unlike the
m
= 1
wire, however, fermion parity in the junction
remains the same after a
2
π
evolution (as both bound
states switch occupation) and so we can restrict our-
selves to the even parity sector. The crossing at
φ
=
π
is protected by our symmetry, but that does not prevent
local mixing.
Interactions do not play an important role for
m
= 2
.
The only acceptable interaction at
φ
=
π
reads
H
int
=
w
1
(
iγ
L
1
γ
R
1
)(
iγ
L
2
γ
R
2
)
,
(71)
which only splits the even and odd parity sectors and
does not affect the Josephson periodicity. We recover
local mixing, implying (for certain parameter regimes)
the loss of
4
π
periodicity.
•
m
= 3
. While it may seem that
m
= 3
wires will
suffer from local mixing as well, interactions conspire
to restore
4
π
periodicity (in much the same way that
interactions stabilize an
8
π
-periodic fractional Joseph-
son effect in the absence of local mixing for a junction
of proximitized quantum spin Hall edges [11]). Notice
that after a
2
π
evolution, the local fermion parity in the
junction changes. We track
8
states,
4
with even parity
and
4
with odd parity, and these states all intersect at
φ
=
π
.
However, adding interactions
H
int
=
w
1
(
iγ
L
1
γ
R
1
)(
iγ
L
2
γ
R
2
)
+
w
2
(
iγ
L
1
γ
R
1
)(
iγ
L
3
γ
R
3
)
(72)
will shift the different bands up or down. Instead of
crossing at
π
, many crossings are now shifted away,
and so symmetry-breaking perturbations may be added
that open up avoided crossings. Not all crossings are
avoided; recall that even parity states get mapped to odd
parity states and vice versa. Crossings between these
states are protected by fermion parity; we recover the
4
π
periodic Josephson effect.
•
m
= 4
. As predicted by Ref. 8, adding interactions to a
system with
8
Majoranas makes the system trivial; the
term
H
int
=
w
1
(
iγ
L
1
γ
R
1
)(
iγ
L
2
γ
R
2
)
+
w
2
(
iγ
L
1
γ
R
1
)(
iγ
L
3
γ
R
3
)
+
w
3
(
iγ
L
1
γ
R
1
)(
iγ
L
4
γ
R
4
)
+
w
4
(
iγ
L
1
γ
L
2
)(
iγ
L
3
γ
L
4
)
completely removes any degeneracy at the crossing
while respecting time reversal. The Josephson effect
is
2
π
periodic.
∗
These authors contributed equally to this work.
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