# One-dimensional $\mathbb{Z}_{4}$ topological superconductor

Max Tymczyszyn  Edward McCann<EMAIL_ADDRESS>Department of
Physics, Lancaster University, Lancaster, LA1 4YB, UK

###### Abstract

We describe the mean-field model of a one-dimensional topological
superconductor with two orbitals per unit cell. Time-reversal symmetry is
absent, but a nonsymmorphic symmetry, involving a translation by a fraction of
the unit cell, mimics the role of time-reversal symmetry. As a result, the
topological superconductor has $\mathbb{Z}_{4}$ topological phases, two which
support Majorana bound states and two which do not, in agreement with a
prediction based on K-theory classification [K. Shiozaki et al., Phys. Rev. B
93, 195413 (2016)]. As with the Kitaev chain, the presence of Majorana bound
states gives rise to the $4\pi$-periodic Josephson effect. The K-theory
classification describes two $\mathbb{Z}_{4}$ subclasses, but, by identifying
models in both subclasses, we show that they are related by a unitary
transformation and are equivalent.

Figure 1: (a) The $\mathbb{Z}_{4}$ model shown in the Bogoliubov de Gennes
(BdG) representation with two orbitals $\alpha$ and $\beta$ per cell, and
particle-hole chains with onsite energies $\pm\mu$, nearest-neighbor hoppings
$\pm t$, and superconducting couplings of magnitude $\Delta$ and phase $\phi$.
(b) The noninteracting CDW model where $\Delta$ acts as nearest-neighbor
hopping. (c) The Kitaev chain in the BdG representation. Phase diagram of (d)
the $\mathbb{Z}_{4}$ topological superconductor with phases $\theta=1,2,3,4$
and (e) the Kitaev chain with a real order parameter. In (d), each phase is
labeled using a phase of a two-band model. The CDW model has two phases
denoted $\nu=0$ and $\nu=1$, and the Kitaev chain with nearest-neighbor
hopping has three possible winding numbers $W=0$, $1$, and $-1$. For (d) and
(e), in the designation of the Kitaev chain phases, we assume $t>0$.

The ten-fold way classification of topological insulators and superconductors
[1, 2, 3, 4, 5] is based on the presence or absence of three nonspatial
symmetries which act upon the degrees of freedom within a unit cell: Time-
reversal symmetry, particle-hole (or charge-conjugation) symmetry, and chiral
symmetry. Subsequently, this classification was generalized to describe
crystalline symmetries including nonsymmorphic symmetries which incorporate a
translation by a fraction of a unit cell [6, 7, 8, 9, 10, 11, 12, 13].
Recently, there has been interest in systems where some nonspatial symmetries
are broken, but a nonsymmorphic (NS) symmetry mimics the effect of the absent
nonspatial symmetry [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27].
We refer to these as NS time-reversal, NS particle-hole or NS chiral symmetry.
Examples include three-dimensional antiferromagnetic topological insulators
with NS time-reversal symmetry [14, 20, 21, 22, 23] and the charge-density
wave (CDW) model which is a one-dimensional two-band tight-binding model with
constant nearest-neighbor hopping strengths and alternating onsite energies
[28, 16, 25, 29, 30, 26, 31] possessing NS chiral symmetry.

In this paper we describe the mean-field model of a one-dimensional
topological superconductor with two orbitals per unit cell, yielding a four
component Hamiltonian in the Bogoliubov de Gennes (BdG) representation. It
possesses particle-hole symmetry, NS time-reversal symmetry and NS chiral
symmetry. We find that it has a $\mathbb{Z}_{4}$ topological index, in
agreement with the prediction of K-theory classification [10]. This is unique
for one dimension, where topological classes in the ten-fold way are described
by either $\mathbb{Z}$ or $\mathbb{Z}_{2}$ indices [2, 3, 4, 5], although
nontrivial topological phases have been discussed in higher spatial dimensions
with surface states referred to as an hourglass or Möbius strip [16, 9, 10,
32, 18, 33, 34, 35].

Figure 1(a) shows the model in the BdG representation where the magnitude
$\Delta$ and phase $\phi$ of the superconducting order parameter are shown as
couplings between particle-hole chains with onsite energies $\pm\mu$ and
nearest-neighbor hoppings $\pm t$, where $\mu$ is the chemical potential. We
assume that generally $\mu$, $t$, $\Delta$ are nonzero and that the order
parameter is complex (i.e., $\phi$ is not equal to an integer multiple of
$\pi/2$). The two orbitals per unit cell are denoted $\sigma=\alpha$ or
$\sigma=\beta$ which could be spin labels $\uparrow,\downarrow$ or, for
spinless fermions, sublattice labels $A,B$. Related two band models are the
noninteracting charge-density wave (CDW) model [28, 16, 25, 29, 30, 26, 31],
Fig. 1(b), and the Kitaev chain [36], shown in the BdG representation in Fig.
1(c). For the $\mathbb{Z}_{4}$ model, Figure 1(a), the intercell
superconducting order parameter is the complex conjugate of the intracell
order parameter. This is the only difference from the Kitaev model where the
order parameter is the same between every pair of sites, Fig. 1(c).

The phase diagram of the $\mathbb{Z}_{4}$ model is shown in Fig. 1(d) with
four phases distinguished by topological index $\theta=1,2,3,4$. We label the
four phases by relating them to phases of the two band models. For
$\mu^{2}+4\Delta^{2}\sin^{2}\phi<4t^{2}$, the phases ($\theta=1$ and
$\theta=3$) may be reduced to phases of the Kitaev chain with a real order
parameter [37] by setting $\phi=0$ or $\phi=\pi$, respectively, corresponding
to winding number $W=1$ or $W=-1$. For comparison, the phase diagram of the
Kitaev chain with nearest-neighbor coupling [38] is shown in Fig. 1(e). For
$\mu^{2}+4\Delta^{2}\sin^{2}\phi>4t^{2}$, the phases ($\theta=4$ and
$\theta=2$) may be reduced to phases of the CDW model by setting $t=0$ with
$\mu>0$ or $\mu<0$, respectively, corresponding to $\mathbb{Z}_{2}$
topological index $\nu=1$ or $\nu=0$ [16]. This correspondence occurs because,
for $t=0$, the phase $\phi$ may be gauged away [37] and the superconducting
chain in the BdG represention may be interpreted as a noninteracting model
(the CDW model).

The mean-field Hamiltonian of the $\mathbb{Z}_{4}$ topological superconductor
is given by

$\displaystyle\\!\\!\\!H$ $\displaystyle=$
$\displaystyle-\mu\sum_{n,\sigma}c_{n,\sigma}^{\dagger}c_{n,\sigma}+t\sum_{n}\left(c_{n,\alpha}^{\dagger}c_{n,\beta}+c_{n+1,\alpha}^{\dagger}c_{n,\beta}+\mathrm{H.c.}\right)$
(1)
$\displaystyle+\Delta\sum_{n}\Big{(}e^{i\phi}c_{n,\alpha}^{\dagger}c_{n,\beta}^{\dagger}+e^{-i\phi}c_{n,\beta}^{\dagger}c_{n+1,\alpha}^{\dagger}+\mathrm{H.c.}\Big{)},$

where $c_{n,\sigma}^{\dagger}$, $c_{n,\sigma}$ are creation and annihilation
operators for orbital $\sigma$ in unit cell $n$. The Hamiltonian (1) may be
written in the BdG representation [39, 40, 41, 42, 43] as
$H=\frac{1}{2}\sum_{k}\Psi_{k}^{\dagger}{\cal
H}(k)\Psi_{k}+\frac{1}{2}\sum_{k}\mathrm{tr}(h(k))$ where
$\Psi_{k}=\begin{pmatrix}c_{k\alpha}&c_{k\beta}&c_{-k\alpha}^{\dagger}&c_{-k\beta}^{\dagger}\end{pmatrix}^{T}$
for wave vector $k$ and the $4\times 4$ Bloch Hamiltonian ${\cal H}(k)$ [37]
is

$\displaystyle{\cal H}(k)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\hat{h}(k)&\hat{\Delta}(k)\\\
\hat{\Delta}^{\dagger}(k)&-\hat{h}^{\ast}(-k)\end{pmatrix},$ (2)
$\displaystyle\hat{h}(k)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}-\mu&2t\cos(ka/2)\\\
2t\cos(ka/2)&-\mu\end{pmatrix},$ (3) $\displaystyle\hat{\Delta}(k)$
$\displaystyle=$ $\displaystyle\begin{pmatrix}0&2i\Delta\sin(ka/2+\phi)\\\
2i\Delta\sin(ka/2-\phi)&0\end{pmatrix}.$ (4)

The Bloch Hamiltonian incorporates particle-hole symmetry as

$\displaystyle\text{particle-hole:}\qquad C^{\dagger}{\cal
H}^{\ast}(k)C=-{\cal H}(-k),$ (5)

where $C=\tau_{x}\sigma_{0}$, and we use $\tau_{0}$ and $\tau_{i}$ for the
identity and Pauli matrices in the particle-hole space, and $\sigma_{0}$ and
$\sigma_{i}$ for the identity and Pauli matrices in the spin space, where
$i=x,y,z$.

The Hamiltonian breaks time-reversal symmetry [44], but the NS time-reversal
symmetry acts as

$\displaystyle\text{NS\,\,time:}\qquad T^{\dagger}{\cal H}^{\ast}(k)T={\cal
H}(-k),$ (6)

where $T=\tau_{0}\sigma_{x}$. The combination of particle-hole symmetry and NS
time-reversal symmetry yields a NS chiral symmetry,

$\displaystyle\text{NS\,\,chiral:}\qquad S^{\dagger}{\cal H}(k)S=-{\cal
H}(k),$ (7)

where $S=\tau_{x}\sigma_{x}$. Written in this basis in $k$ space [45], the NS
symmetries appear to have a straightforward representation. In position space,
however, they involve a translation by half of a unit cell [37] in which
orbital $\alpha$ moves to the position of one of the $\beta$ orbitals, Fig.
1(a).

With the symmetry constraints, the gap function may be written generically in
spin space as

$\displaystyle\hat{\Delta}(k)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}id_{y}(k)-d_{x}(k)&id_{z}(k)+i\Delta_{s}(k)\\\
id_{z}(k)-i\Delta_{s}(k)&id_{y}(k)+d_{x}(k)\end{pmatrix}.$ (8)

where the $\Delta_{s}$, $d_{x}$, $d_{y}$, $d_{z}$ components are real,
$\Delta_{s}(k)=\Delta_{s}(-k)$ has even parity and
$\mathbf{d}(k)=(d_{x}(k),d_{y}(k),d_{z}(k))$ has odd parity
$\mathbf{d}(k)=-\mathbf{d}(-k)$. This is consistent with the constraint of the
Pauli exclusion principle $\hat{\Delta}(-k)=-\hat{\Delta}^{T}(k)$. In terms of
the gap function, NS time-reversal symmetry is
$\sigma_{x}\hat{\Delta}^{\ast}(k)\sigma_{x}=\hat{\Delta}(-k)$. In comparision,
real time-reversal symmetry would be
$\sigma_{y}\hat{\Delta}^{\ast}(k)\sigma_{y}=\hat{\Delta}(-k)$ and the allowed
form of the gap function (8) would be the same except that the off-diagonal
elements would be real $\hat{\Delta}_{12}(k)=d_{z}(k)+\Delta_{s}(k)$.

Figure 2: The energy spectrum $E(k)$ of the $\mathbb{Z}_{4}$ model in $k$
space showing (a) phase $\theta=1$ with $\phi=0$ and (b) phase $\theta=4$ with
$\phi=\pi/2$. Plots are obtained by diagonalizing Eq. (2) with parameters
$\mu=t=\Delta=1.0$ [46].

For the $\mathbb{Z}_{4}$ topological superconductor, we could include diagonal
spin terms $d_{x}$, $d_{y}$ in the gap function (8), but they are not needed
to describe the $\mathbb{Z}_{4}$ topology. Hence we restrict the immediate
discussion to the off-diagonal gap function (4). The energy spectrum $E(k)$ of
the BdG Hamiltonian (2) has four bands given by

$\displaystyle E^{2}(k)$ $\displaystyle=$
$\displaystyle\mu^{2}+f^{2}+d_{z}^{2}+\Delta_{s}^{2}\pm
2\sqrt{\mu^{2}f^{2}+\Delta_{s}^{2}(f^{2}+d_{z}^{2})},$ $\displaystyle f(k)$
$\displaystyle=$ $\displaystyle 2t\cos(ka/2),$ $\displaystyle\Delta_{s}(k)$
$\displaystyle=$ $\displaystyle 2\Delta\sin\phi\cos(ka/2),$ $\displaystyle
d_{z}(k)$ $\displaystyle=$ $\displaystyle 2\Delta\cos\phi\sin(ka/2).$

Typical spectra are plotted in Fig. 2 for phases $\theta=1$ and $\theta=4$.
Despite the mixed parity of the gap function (4), the presence of particle-
hole symmetry and NS time-reversal symmetry ensure particle-hole symmetry of
the energy spectrum and its evenness in $k$. Furthermore, NS time-reversal
symmetry imposes Kramer’s degeneracy [17, 31] with the two particle bands
being degenerate at the Brillouin zone edge, likewise the two hole bands. The
degeneracy occurs at only one time-reversal invariant $k$ value (either $k=0$
or $k=\pi/a$) as opposed to both, which happens for symmorphic time-reversal
symmetry with $T^{2}=-1$ [17, 31]. Despite the Kramer’s degeneracy, there is
generally a band gap between the particle and hole bands; parameter values
leading to a vanishing band gap are indicated in the phase diagram, Fig. 1(d).

Figure 3: Energy levels of the $\mathbb{Z}_{4}$ model in position space. (a)
and (b) are as a function of the chemical potential $\mu$ with (a) $\Delta=2$,
$\phi=\pi/4$, and (b) $\Delta=1$, $\phi=\pi/4$. (c) and (d) are as a function
of the superconducting phase $\phi$ with (c) $\mu=1.0$, $\Delta=0.5$, and (d)
$\mu=\Delta=1.0$. Numbers in each plot show the phases $\theta$ of the
$\mathbb{Z}_{4}$ model. All plots are obtained by diagonalizing the BdG
Hamiltonian in position space with open boundary conditions, $48$ unit cells
and $t=1.0$ [46].

The $\mathbb{Z}_{4}$ topological index $\theta$ has been defined [10] as

$\displaystyle\theta$ $\displaystyle=$
$\displaystyle-\frac{2}{\pi}\arg\big{\\{}\mathrm{Pf}[\sigma_{x}Z(\pi/a)]\big{\\}}$
(9)
$\displaystyle\qquad+\frac{1}{\pi}\int_{0}^{\pi/a}dk\frac{\partial}{\partial
k}\arg\\{\det[\sigma_{x}Z(k)]\\},$

where $\theta$ is defined modulo $4$ and $\arg(z)=\arg(re^{i\varphi})=\varphi$
for $-\pi<\varphi\leq\pi$ [47]. For a $2\times 2$ antisymmetric matrix, the
Pfaffian is $\mathrm{Pf}(i\alpha\sigma_{y})=\alpha$. The $2\times 2$ matrix
$Z(k)=X(k)+iY(k)$ consists of a $2\pi$ periodic Hermitian part $X(k)$ and an
aperiodic Hermitian part $Y(k)$ with $X(\pi/a)=X(-\pi/a)$ and
$Y(\pi/a)=-Y(-\pi/a)$ such that the BdG Hamiltonian ${\cal H}(k)$ may be
rotated by a unitary transformation to $\tilde{{\cal
H}}(k)=\tilde{U}^{\dagger}{\cal H}(k)\tilde{U}$ where

$\displaystyle\tilde{{\cal H}}(k)=\begin{pmatrix}X(k)&iY(k)\\\
-iY(k)&-X(k)\end{pmatrix}.$ (10)

For the BdG Hamiltonian (2), we find that

$\displaystyle\\!\\!\\!Z(k)=\begin{pmatrix}-\mu+2i\Delta\sin(ka/2+\phi)&2t\cos(ka/2)\\\
-2t\cos(ka/2)&\mu-2i\Delta\sin(ka/2-\phi)\end{pmatrix},$

and, substituting into Eq. (9), this gives the values of $\theta$ shown in the
phase diagram, Fig. 1(d). Note the presence of a discontinuity in the integral
of Eq. (9) in the case $\mu^{2}+4\Delta^{2}\sin^{2}\phi<4t^{2}$ and
$\mu\cos\phi>0$.

To explore the phase diagram, Fig. 1(d), we plot the energy levels in position
space in Fig. 3. We find that the Kitaev-like phases, $\theta=1,3$, support
Majorana bound states whereas the CDW-like phases, $\theta=2,4$, do not. Fig.
3(a) shows the energy levels as a function of $\mu$ for
$\Delta^{2}\sin^{2}\phi>t^{2}$. There is a phase transition at $\mu=0$ between
the CDW-like phases $\theta=2$ and $\theta=4$ that has no analog in the Kitaev
model. For $\Delta^{2}\sin^{2}\phi<t^{2}$, Fig. 3(b), there are two phase
transitions, from $\theta=2$ to $\theta=1$ at $\mu<0$ and from $\theta=1$ to
$\theta=4$ at $\mu>0$. The Kitaev-like phase, $\theta=1$, supports edge states
at zero energy. Fig. 3(c) shows the energy levels as a function of the
superconducting phase $\phi$ with $\mu^{2}+4\Delta^{2}\sin^{2}\phi<4t^{2}$ for
all $\phi$. Thus, there are transitions between the Kitaev-like phases
$\theta=1,3$. In Fig. 3(d), there are also transitions across the boundary
$\mu^{2}+4\Delta^{2}\sin^{2}\phi=4t^{2}$ to the CDW-like phase $\theta=4$.
Note that the same plots for the Kitaev model would be independent of $\phi$
(being equal to the plots shown at $\phi=0$ for all $\phi$) because $\phi$ can
be gauged away in the Kitaev model.

Figure 4: Properties of a Josephson junction in a ring with phase $\theta=4$
(left column with $\mu=2.0$) and $\theta=1$ (right column with $\mu=1.0$). All
plots are as a function of the magnetic flux $\Phi_{B}$ where
$\Phi_{\mathrm{s}}=h/2e$ is the superconducting flux quantum. (a) and (b) show
energy levels $E$ in position space. (c), (d) show the energy $E_{\mathrm{J}}$
of the system (arbitrary units) and (e),(f) show the current $I_{\mathrm{J}}$
(arbitrary units). All plots are obtained by diagonalizing the BdG Hamiltonian
in position space with $48$ unit cells in the ring, $t=\Delta=1.0$,
$\phi=\pi/4$, and $t^{\prime}=0.1$ [46].

As in the Kitaev chain [36, 48, 39, 49, 41], Majorana bound states in the
$\mathbb{Z}_{4}$ model give rise to a $4\pi$-periodic Josephson junction. To
illustrate this, we generalize a model of a weak link in a closed ring [49]
applied previously to the Kitaev chain. For a ring with $N$ unit cells, the
Hamiltonian (1) is supplemented by a term describing hopping of magnitude
$t^{\prime}$ across the weak link,

$\displaystyle\delta
H=t^{\prime}c_{1,\alpha}^{\dagger}c_{N,\beta}e^{-i\pi\Phi_{B}/\Phi_{\mathrm{s}}}+\mathrm{H.c.},$
(11)

where $\Phi_{B}$ is the magnetic flux through the ring and
$\Phi_{\mathrm{s}}=h/2e$ is the superconducting flux quantum. We then
diagonalize the BdG Hamiltonian in position space [37] to obtain the energy
levels $E$ and the energy of the junction $E_{\mathrm{J}}$ determined by
summing the energy levels below zero energy. We calculate the derivative
$dE_{\mathrm{J}}/d\Phi_{B}$ which is proportional to the Josephson current
$I_{\mathrm{J}}=(2\pi/\Phi_{\mathrm{s}})dE_{\mathrm{J}}/d\Phi_{B}$. Fig. 4(a)
shows the energy levels for the CDW-like phase $\theta=4$ where Majorana bound
states are absent. Although the levels appear to have little dependence on
$\Phi_{B}$, Fig. 4(c) and (e) show that the energy $E_{\mathrm{J}}$ and
current $I_{\mathrm{J}}$ are periodic in $\Phi_{B}$ with period
$\Phi_{\mathrm{s}}$ as expected for a phase with tunneling of Cooper pairs and
no Majorana bound states.

Fig. 4(b) shows the energy levels for the Kitaev-like phase $\theta=1$ which
shows two subgap states. We assume that the level crossings of these states
(at $\Phi_{B}=\Phi_{\mathrm{s}}/2$ and $\Phi_{B}=3\Phi_{\mathrm{s}}/2$) are
protected by conservation of fermion parity [36, 48, 39, 49, 41] so that these
two levels have a period of $2\Phi_{\mathrm{s}}$, i.e., the single particle
flux quantum. Hence, we do not determine the thermodynamic ground state
energy, but include the level which is below zero energy at $\Phi_{B}=0$ for
all $\Phi_{B}$ in the calculation of $E_{\mathrm{J}}$, and omit the energy of
the level above zero energy at $\Phi_{B}=0$ for all $\Phi_{B}$. Fig. 4(d) and
(f) show that the resulting energy $E_{\mathrm{J}}$ and current
$I_{\mathrm{J}}$ are periodic in $\Phi_{B}$ with period $2\Phi_{\mathrm{s}}$
as expected for single-electron tunneling mediated by Majorana bound states
[36, 48, 39, 49, 41].

Figure 5: A model in a $\mathbb{Z}_{4}$ subclass in which the time-reversal
operator anticommutes with the particle-hole operator. (a) The model shown in
the Bogoliubov de Gennes (BdG) representation with two orbitals $\alpha$ and
$\beta$ per cell, and particle-hole chains with onsite energies $\pm\mu$,
nearest-neighbor hoppings $\pm t$, and superconducting couplings of magnitude
$\Delta$ and phase $\varphi$. (b) Phase diagram with phases $\theta=1,2,3,4$.
In the designation of the Kitaev chain phases, we assume $t>0$.

The K-theory classification [10] describes two $\mathbb{Z}_{4}$ subclasses
distinguished by whether the NS time-reversal symmetry operator $T$ commutes
with the particle-hole symmetry operator $C$ or anticommutes with it. The
model discussed thus far, Fig. 1(a), belongs to the commuting subclass. A
model in the anticommuting subclass is shown in Fig. 5(a) in the BdG
representation, and it is related to the commuting model by a shift of
intercell superconducting phase by $\pi$. Its mean-field Hamiltonian is

$\displaystyle\\!\\!\\!H_{\mathrm{a}}$ $\displaystyle=$
$\displaystyle-\mu\sum_{n,\sigma}c_{n,\sigma}^{\dagger}c_{n,\sigma}+t\sum_{n}\left(c_{n,\alpha}^{\dagger}c_{n,\beta}+c_{n+1,\alpha}^{\dagger}c_{n,\beta}+\mathrm{H.c.}\right)$
(12)
$\displaystyle+\Delta\sum_{n}\Big{(}e^{i\varphi}c_{n,\alpha}^{\dagger}c_{n,\beta}^{\dagger}-e^{-i\varphi}c_{n,\beta}^{\dagger}c_{n+1,\alpha}^{\dagger}+\mathrm{H.c.}\Big{)},$

where $\varphi$ is the superconducting phase. In $k$ space, the Hamiltonian in
the BdG representation, ${\cal H}_{\mathrm{a}}(k)$, takes the same form as Eq.
(2) with the noninteracting part $h(k)$ as in Eq. (3), and the superconducting
part given by

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\hat{\Delta}_{\mathrm{a}}(k)=\begin{pmatrix}0&2\Delta\cos(ka/2+\varphi)\\\
-2\Delta\cos(ka/2-\varphi)&0\end{pmatrix}\\!.$ (13)

This has particle-hole symmetry (5) with $C=\tau_{x}\sigma_{0}$ and NS time-
reversal symmetry (6) with $T_{\mathrm{a}}=\tau_{z}\sigma_{x}$. The phase
diagram for the anticommuting model is shown in Fig. 5(b). It is the same as
for the commuting model, Fig. 1(d), with the replacement $\varphi=\phi-\pi/2$
such that $\sin\varphi=-\cos\phi$ and $\cos\varphi=\sin\phi$ [37].

In fact, we find that the anticommuting Hamiltonian ${\cal H}_{\mathrm{a}}(k)$
is the same as the commuting one ${\cal H}(k)$ under the variable shift
$\varphi=\phi-\pi/2$ and a unitary transformation,

$\displaystyle U^{\dagger}{\cal H}_{\mathrm{a}}(k)U={\cal H}(k).$ (14)

The commuting and anticommuting algebras are related by

$\displaystyle TU^{\dagger}$ $\displaystyle=$ $\displaystyle U^{\dagger}T,$
(15) $\displaystyle CU^{\ast}U^{\dagger}$ $\displaystyle=$
$\displaystyle-U^{\ast}U^{\dagger}C.$ (16)

Applying the transformation (14) to the time-reversal operators gives
$T_{\mathrm{a}}=U^{\ast}TU^{\dagger}$ and, substituting this into the
anticommutation relation $CT_{\mathrm{a}}=-T_{\mathrm{a}}C$, yields
$CU^{\ast}TU^{\dagger}=-U^{\ast}TU^{\dagger}C$. Then, using Eq. (15), writing
$CU^{\ast}U^{\dagger}T=-U^{\ast}U^{\dagger}TC$ followed by Eq. (16) gives the
commutation relation $CT=TC$. For the $4\times 4$ BdG representation in $k$
space considered here,
$U=\text{diag}(1,1,i,i)=(1/2)[(1+i)\tau_{0}+(1-i)\tau_{z}]\sigma_{0}$, but the
same algebra holds in position space for Hamiltonians with arbitrary matrix
elements that preserve the relevant symmetries [37]. Thus, we conclude that
Fig. 1(a) shows the only unique $4\times 4$ model in the $\mathbb{Z}_{4}$
class in one dimension, although unitary transformations allow the model to be
expressed in alternative forms, e.g., Fig. 5(a).

Different methods of creating the Kitaev chain, or similar systems, have been
investigated, usually with a conventional $s$-wave superconductor in proximity
to a system with strong spin-orbit coupling and/or magnetic field. Examples
include hybrid superconductor-semiconductor nanowires [50, 51, 40, 52], a
superconductor near to an array of magnetic atoms or nanoparticles [53, 54] or
in a spiraling magnetic field [55, 56], a chain of quantum dots with spin-
orbit coupling and connected to superconductors [57, 58, 59, 60], and planar
semiconductor-superconductor heterostructures [61, 62, 63]. The latter two
platforms provide the option of tuning the phase $\phi$ of the order parameter
on each site separately and, with them, it may be possible to engineer a
system described by one of the $\mathbb{Z}_{4}$ Hamiltonians (1,12).

To summarize, we have identified the mean-field model of a topological
superconductor with two orbitals per cell in one dimension belonging to a
$\mathbb{Z}_{4}$ class. In the phase diagram, Figs. 1(d), two phases have no
Majorana bound states and they may be related to the phases of a
$\mathbb{Z}_{2}$ noninteracting model, the CDW model. Two phases, which
support Majorana bound states, may be related to topological phases of the
Kitaev chain and, like the Kitaev chain, the presence of Majorana bound states
gives rise to the $4\pi$-periodic Josephson effect.

###### Acknowledgements.

The authors thank C. Y. Leung, A. Romito, K. Shiozaki, D. Varjas, and J. H.
Winter for helpful discussions.

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Supplementary material: One-dimensional $\mathbb{Z}_{4}$ topological
superconductor

##  I Bogoliubov de Gennes representation in position space

###  I.1 Hamiltonian

The mean-field Hamiltonian of the $\mathbb{Z}_{4}$ topological superconductor
is given by

$\displaystyle
H=-\mu\sum_{n,\sigma}c_{n,\sigma}^{\dagger}c_{n,\sigma}+t\sum_{n}\left(c_{n,\alpha}^{\dagger}c_{n,\beta}+c_{n+1,\alpha}^{\dagger}c_{n,\beta}+\mathrm{H.c.}\right)+\Delta\sum_{n}\Big{(}e^{-i\phi}c_{n,\beta}c_{n,\alpha}+e^{i\phi}c_{n+1,\alpha}c_{n,\beta}+\mathrm{H.c.}\Big{)},$
(S1)

where $c_{n,\sigma}^{\dagger}$, $c_{n,\sigma}$ are creation and annihilation
operators for orbital $\sigma$ in unit cell $n$. Parameter $\mu$ is the
chemical potential, $t$ is nearest-neighbor hopping, and $\Delta$ and $\phi$
are the magnitude and phase of the superconducting order parameter. The two
orbitals per unit cell are denoted $\sigma=\alpha$ or $\sigma=\beta$ which
could be spin labels $\uparrow,\downarrow$ or, for spinless fermions,
sublattice labels $A,B$.

To give an example of the Bogoliubov de Gennes (BdG) representation of Eq.
(S1) in position space, we consider $N=3$ unit cells with open boundary
conditions. The expressions may be easily generalized to other $N$ values.
$N=3$ has six orbitals, or twelve orbitals in the BdG representation giving a
$12\times 12$ matrix in position space. In particular, we write
$H=(1/2)\Psi^{\dagger}{\cal H}\Psi-N\mu$, where

$\displaystyle\Psi^{\dagger}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}c_{1,\alpha}^{\dagger}&c_{1,\alpha}&c_{1,\beta}^{\dagger}&c_{1,\beta}&c_{2,\alpha}^{\dagger}&c_{2,\alpha}&c_{2,\beta}^{\dagger}&c_{2,\beta}&c_{3,\alpha}^{\dagger}&c_{3,\alpha}&c_{3,\beta}^{\dagger}&c_{3,\beta}\end{pmatrix};\qquad\qquad\Psi=\begin{pmatrix}c_{1,\alpha}\\\
c_{1,\alpha}^{\dagger}\\\ c_{1,\beta}\\\ c_{1,\beta}^{\dagger}\\\
c_{2,\alpha}\\\ c_{2,\alpha}^{\dagger}\\\ c_{2,\beta}\\\
c_{2,\beta}^{\dagger}\\\ c_{3,\alpha}\\\ c_{3,\alpha}^{\dagger}\\\
c_{3,\beta}\\\ c_{3,\beta}^{\dagger}\end{pmatrix},$ (S2)
$\displaystyle\\!\\!\\!{\cal
H}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}-\mu&0&t&\Delta
e^{i\phi}&0&0&0&0&0&0&0&0\\\ 0&\mu&-\Delta e^{-i\phi}&-t&0&0&0&0&0&0&0&0\\\
t&-\Delta e^{i\phi}&-\mu&0&t&\Delta e^{-i\phi}&0&0&0&0&0&0\\\ \Delta
e^{-i\phi}&-t&0&\mu&-\Delta e^{i\phi}&-t&0&0&0&0&0&0\\\ 0&0&t&-\Delta
e^{-i\phi}&-\mu&0&t&\Delta e^{i\phi}&0&0&0&0\\\ 0&0&\Delta
e^{i\phi}&-t&0&\mu&-\Delta e^{-i\phi}&-t&0&0&0&0\\\ 0&0&0&0&t&-\Delta
e^{i\phi}&-\mu&0&t&\Delta e^{-i\phi}&0&0\\\ 0&0&0&0&\Delta
e^{-i\phi}&-t&0&\mu&-\Delta e^{i\phi}&-t&0&0\\\ 0&0&0&0&0&0&t&-\Delta
e^{-i\phi}&-\mu&0&t&\Delta e^{i\phi}\\\ 0&0&0&0&0&0&\Delta
e^{i\phi}&-t&0&\mu&-\Delta e^{-i\phi}&-t\\\ 0&0&0&0&0&0&0&0&t&-\Delta
e^{i\phi}&-\mu&0\\\ 0&0&0&0&0&0&0&0&\Delta e^{-i\phi}&-t&0&\mu\end{pmatrix}.$
(S3)

###  I.2 Symmetry operators

Particle-hole symmetry (charge conjugation) is
${\mathcal{C}}^{\dagger}{\mathcal{H}}^{\ast}{\mathcal{C}}=-{\mathcal{H}}$. For
the position space Hamiltonian (S3),

$\displaystyle{\mathcal{C}}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}0&1&0&0&0&0&0&0&0&0&0&0\\\
1&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0&0&0&0\\\
0&0&1&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0&0&0&0&0\\\
0&0&0&0&1&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&1&0&0&0&0\\\
0&0&0&0&0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&1&0&0\\\
0&0&0&0&0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&1\\\
0&0&0&0&0&0&0&0&0&0&1&0\end{pmatrix}.$ (S4)

Time reversal symmetry is
${\mathcal{T}}^{\dagger}{\mathcal{H}}^{\ast}{\mathcal{T}}={\mathcal{H}}$. For
the position space Hamiltonian (S3), the operator ${\mathcal{T}}$ is
nonsymmorphic (NS), involving a translation,

$\displaystyle{\mathcal{T}}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}0&0&1&0&0&0&0&0&0&0&0&0\\\
0&0&0&1&0&0&0&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0&0&0&0&0\\\
0&0&0&0&0&1&0&0&0&0&0&0\\\ 0&0&0&0&0&0&1&0&0&0&0&0\\\
0&0&0&0&0&0&0&1&0&0&0&0\\\ 0&0&0&0&0&0&0&0&1&0&0&0\\\
0&0&0&0&0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&0&0&0&0&0&1&0\\\
0&0&0&0&0&0&0&0&0&0&0&1\\\ 1&0&0&0&0&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0&0&0&0&0\end{pmatrix}.$ (S5)

Chiral symmetry is
${\mathcal{S}}^{\dagger}{\mathcal{H}}{\mathcal{S}}=-{\mathcal{H}}$. For the
position space Hamiltonian (S3), the operator ${\mathcal{S}}$ is NS, involving
a translation, where ${\mathcal{S}}={\mathcal{T}}^{\ast}{\mathcal{C}}$,

$\displaystyle{\mathcal{S}}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}0&0&0&1&0&0&0&0&0&0&0&0\\\
0&0&1&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0&0&0&0&0\\\
0&0&0&0&1&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&1&0&0&0&0\\\
0&0&0&0&0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&1&0&0\\\
0&0&0&0&0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&1\\\
0&0&0&0&0&0&0&0&0&0&1&0\\\ 0&1&0&0&0&0&0&0&0&0&0&0\\\
1&0&0&0&0&0&0&0&0&0&0&0\end{pmatrix}.$ (S6)

When applying NS time-reversal and NS chiral symmetry operators to the
position space Hamiltonian (S3) with open boundary conditions, there are
errors at the boundary because the boundary breaks the NS symmetry.

###  I.3 Relation to the Kitaev chain

The Kitaev chain [1] is given by

$\displaystyle\\!\\!\\!\\!\\!\\!{\cal
H}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}-\mu&0&t&\Delta
e^{i\phi}&0&0&0&0&0&0&0&0\\\ 0&\mu&-\Delta e^{-i\phi}&-t&0&0&0&0&0&0&0&0\\\
t&-\Delta e^{i\phi}&-\mu&0&t&\Delta e^{i\phi}&0&0&0&0&0&0\\\ \Delta
e^{-i\phi}&-t&0&\mu&-\Delta e^{-i\phi}&-t&0&0&0&0&0&0\\\ 0&0&t&-\Delta
e^{i\phi}&-\mu&0&t&\Delta e^{i\phi}&0&0&0&0\\\ 0&0&\Delta
e^{-i\phi}&-t&0&\mu&-\Delta e^{-i\phi}&-t&0&0&0&0\\\ 0&0&0&0&t&-\Delta
e^{i\phi}&-\mu&0&t&\Delta e^{i\phi}&0&0\\\ 0&0&0&0&\Delta
e^{-i\phi}&-t&0&\mu&-\Delta e^{-i\phi}&-t&0&0\\\ 0&0&0&0&0&0&t&-\Delta
e^{i\phi}&-\mu&0&t&\Delta e^{i\phi}\\\ 0&0&0&0&0&0&\Delta
e^{-i\phi}&-t&0&\mu&-\Delta e^{-i\phi}&-t\\\ 0&0&0&0&0&0&0&0&t&-\Delta
e^{i\phi}&-\mu&0\\\ 0&0&0&0&0&0&0&0&\Delta e^{-i\phi}&-t&0&\mu\end{pmatrix}.$
(S7)

In this Hamiltonian, the phase $\phi$ can be gauged away. The Kitaev model
(with the phase $\phi$ gauged away) is in symmetry class BDI with a
$\mathbb{Z}$ topological index, the winding number W. With only nearest-
neighbor hopping, the winding number can take three values [2],

$\displaystyle
W=\begin{cases}1&\text{for}\,\,|\mu|<2|t|\,\,\text{and}\,\,t\Delta>0\\\
0&\text{for}\,\,|\mu|>2|t|\\\
-1&\text{for}\,\,|\mu|<2|t|\,\,\text{and}\,\,t\Delta<0\end{cases}.$ (S8)

For the $\mathbb{Z}_{4}$ model in phases $\theta=1$ and $\theta=3$, then
$\mu^{2}+4\Delta^{2}\sin^{2}\phi<4t^{2}$. Then we can set $\sin\phi=0$ (i.e.
$\phi=\pi$ or $\phi=0$) without causing a phase transition, i.e. $\phi$ is
real. For example, if we set $\phi=0$ in the $\mathbb{Z}_{4}$ Hamiltonian
(S3), it will be the same as the Kitaev model (S7) with the phase $\phi$
gauged away.

###  I.4 Relation to the CDW model

For phases $\theta=2$ and $\theta=4$, then
$\mu^{2}+4\Delta^{2}\sin^{2}\phi>4t^{2}$. Without causing a phase transition,
we can then set $t=0$. Then, as long as $\Delta\neq 0$ and $\sin\phi\neq 0$,
we could have a phase transition between $\theta=2$ and $\theta=4$ by flipping
the sign of $\mu$ (via $\mu=0$). With $t=0$, the $\mathbb{Z}_{4}$ Hamiltonian
(S3) is

$\displaystyle\\!\\!\\!\\!\\!\\!{\cal
H}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}-\mu&0&0&\Delta
e^{i\phi}&0&0&0&0&0&0&0&0\\\ 0&\mu&-\Delta e^{-i\phi}&0&0&0&0&0&0&0&0&0\\\
0&-\Delta e^{i\phi}&-\mu&0&0&\Delta e^{-i\phi}&0&0&0&0&0&0\\\ \Delta
e^{-i\phi}&0&0&\mu&-\Delta e^{i\phi}&0&0&0&0&0&0&0\\\ 0&0&0&-\Delta
e^{-i\phi}&-\mu&0&0&\Delta e^{i\phi}&0&0&0&0\\\ 0&0&\Delta
e^{i\phi}&0&0&\mu&-\Delta e^{-i\phi}&0&0&0&0&0\\\ 0&0&0&0&0&-\Delta
e^{i\phi}&-\mu&0&0&\Delta e^{-i\phi}&0&0\\\ 0&0&0&0&\Delta
e^{-i\phi}&0&0&\mu&-\Delta e^{i\phi}&0&0&0\\\ 0&0&0&0&0&0&0&-\Delta
e^{-i\phi}&-\mu&0&0&\Delta e^{i\phi}\\\ 0&0&0&0&0&0&\Delta
e^{i\phi}&0&0&\mu&-\Delta e^{-i\phi}&0\\\ 0&0&0&0&0&0&0&0&0&-\Delta
e^{i\phi}&-\mu&0\\\ 0&0&0&0&0&0&0&0&\Delta e^{-i\phi}&0&0&\mu\end{pmatrix}.$
(S9)

We change basis with ${\cal H}^{\prime}=\hat{U}{\cal H}\hat{U}^{\dagger}$
where

$\displaystyle\hat{U}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}1&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&1&0&0&0&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&1&0&0&0&0\\\ 0&0&0&0&0&0&0&0&1&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&1\\\ 0&1&0&0&0&0&0&0&0&0&0&0\\\
0&0&1&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0&0&0&0&0\\\
0&0&0&0&0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&1&0&0\\\
0&0&0&0&0&0&0&0&0&0&1&0\end{pmatrix},$ (S10)

giving

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\cal
H}^{\prime}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}-\mu&\Delta
e^{i\phi}&0&0&0&0&0&0&0&0&0&0\\\ \Delta e^{-i\phi}&\mu&-\Delta
e^{i\phi}&0&0&0&0&0&0&0&0&0\\\ 0&-\Delta e^{-i\phi}&-\mu&\Delta
e^{i\phi}&0&0&0&0&0&0&0&0\\\ 0&0&\Delta e^{-i\phi}&\mu&-\Delta
e^{i\phi}&0&0&0&0&0&0&0\\\ 0&0&0&-\Delta e^{-i\phi}&-\mu&\Delta
e^{i\phi}&0&0&0&0&0&0\\\ 0&0&0&0&\Delta e^{-i\phi}&\mu&0&0&0&0&0&0\\\
0&0&0&0&0&0&\mu&-\Delta e^{-i\phi}&0&0&0&0\\\ 0&0&0&0&0&0&-\Delta
e^{i\phi}&-\mu&\Delta e^{-i\phi}&0&0&0\\\ 0&0&0&0&0&0&0&\Delta
e^{i\phi}&\mu&-\Delta e^{-i\phi}&0&0\\\ 0&0&0&0&0&0&0&0&-\Delta
e^{i\phi}&-\mu&\Delta e^{-i\phi}&0\\\ 0&0&0&0&0&0&0&0&0&\Delta
e^{i\phi}&\mu&-\Delta e^{-i\phi}\\\ 0&0&0&0&0&0&0&0&0&0&-\Delta
e^{i\phi}&-\mu\end{pmatrix},$ (S11)

It is possible to gauge away the phases of $\Delta$ with ${\cal
H}^{\prime\prime}=R{\cal H}^{\prime}R^{\dagger}$ where

$\displaystyle
R=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}1&0&0&0&0&0&0&0&0&0&0&0\\\
0&e^{i\phi}&0&0&0&0&0&0&0&0&0&0\\\ 0&0&-e^{2i\phi}&0&0&0&0&0&0&0&0&0\\\
0&0&0&-e^{3i\phi}&0&0&0&0&0&0&0&0\\\ 0&0&0&0&e^{4i\phi}&0&0&0&0&0&0&0\\\
0&0&0&0&0&e^{5i\phi}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&1&0&0&0&0&0\\\
0&0&0&0&0&0&0&-e^{-i\phi}&0&0&0&0\\\ 0&0&0&0&0&0&0&0&-e^{-2i\phi}&0&0&0\\\
0&0&0&0&0&0&0&0&0&e^{-3i\phi}&0&0\\\ 0&0&0&0&0&0&0&0&0&0&e^{-4i\phi}&0\\\
0&0&0&0&0&0&0&0&0&0&0&-e^{-5i\phi}\end{pmatrix},$ (S12)

so that

$\displaystyle{\cal
H}^{\prime\prime}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}-\mu&\Delta&0&0&0&0&0&0&0&0&0&0\\\
\Delta&\mu&\Delta&0&0&0&0&0&0&0&0&0\\\ 0&\Delta&-\mu&\Delta&0&0&0&0&0&0&0&0\\\
0&0&\Delta&\mu&\Delta&0&0&0&0&0&0&0\\\ 0&0&0&\Delta&-\mu&\Delta&0&0&0&0&0&0\\\
0&0&0&0&\Delta&\mu&0&0&0&0&0&0\\\ 0&0&0&0&0&0&\mu&\Delta&0&0&0&0\\\
0&0&0&0&0&0&\Delta&-\mu&\Delta&0&0&0\\\ 0&0&0&0&0&0&0&\Delta&\mu&\Delta&0&0\\\
0&0&0&0&0&0&0&0&\Delta&-\mu&\Delta&0\\\ 0&0&0&0&0&0&0&0&0&\Delta&\mu&\Delta\\\
0&0&0&0&0&0&0&0&0&0&-\Delta&-\mu\end{pmatrix}.$ (S13)

This is block diagonal, effectively describing two separate CDW models with
nearest-neighbor hopping $\Delta$ and alternating onsite energies $\pm\mu$.
Each of them is in a $\mathbb{Z}_{2}$ phase [3, 4, 5] depending on the sign of
$\mu$. We define the topological index $\nu$ for the CDW chain with the first
site of onsite energy $-\mu$ as

$\displaystyle\nu=\begin{cases}0&\text{for}\,\,\mu<0\\\
1&\text{for}\,\,\mu>0\end{cases}$

Hence phase $\theta=4$ of the $\mathbb{Z}_{4}$ model corresponds to $\nu=1$,
and $\theta=2$ corresponds to $\nu=0$.

##  II Bogoliubov de Gennes representation in $k$ space: Type II tight-binding
model

There are two different possible choices of Fourier transform [6], either one
can use lattice vectors (Type I) or orbital positions (Type II). For
describing NS symmetries, we find it more convenient to use the Type II
representation, as in the main text, because the NS symmetry operators are
generally dependent on $k$ in the Type I representation, but not in the Type
II representation. The two representations are related by a unitary
transformation. Here we express the Hamiltonian and the symmetry operators in
both representations.

###  II.1 Hamiltonian

We Fourier transform creation and annihilation operators using

$\displaystyle c_{n,\alpha}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{k}c_{k\alpha}e^{ikna},\qquad\qquad
c_{n,\alpha}^{\dagger}=\frac{1}{\sqrt{N}}\sum_{k}c_{k\alpha}^{\dagger}e^{-ikna},$
(S14) $\displaystyle c_{n,\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{k}c_{k\beta}e^{ikna+ika/2},\qquad\qquad
c_{n,\beta}^{\dagger}=\frac{1}{\sqrt{N}}\sum_{k}c_{k\beta}^{\dagger}e^{-ikna-
ika/2},$ (S15)

where $N$ is the number of unit cells and $a$ is the lattice constant. The
Hamiltonian (S1) may be written in the BdG representation as
$H=\frac{1}{2}\sum_{k}\Psi_{k}^{\dagger}{\cal
H}(k)\Psi_{k}+\frac{1}{2}\sum_{k}\mathrm{tr}(h(k))$ where
$\Psi_{k}=\begin{pmatrix}c_{k\alpha}&c_{k\beta}&c_{-k\alpha}^{\dagger}&c_{-k\beta}^{\dagger}\end{pmatrix}^{T}$
for wave vector $k$ and the $4\times 4$ Bloch Hamiltonian ${\cal H}(k)$ is

$\displaystyle{\cal
H}(k)=\begin{pmatrix}-\mu&2t\cos(ka/2)&0&2i\Delta\sin(ka/2+\phi)\\\
2t\cos(ka/2)&-\mu&2i\Delta\sin(ka/2-\phi)&0\\\
0&-2i\Delta\sin(ka/2-\phi)&\mu&-2t\cos(ka/2)\\\
-2i\Delta\sin(ka/2+\phi)&0&-2t\cos(ka/2)&\mu\end{pmatrix}.$ (S16)

###  II.2 Symmetry operators

The Bloch Hamiltonian (S16) incorporates particle-hole symmetry as

$\displaystyle\text{particle-hole:}\qquad C^{\dagger}{\cal
H}^{\ast}(k)C=-{\cal H}(-k),$ (S17)

where $C=\tau_{x}\sigma_{0}$ is a $4\times 4$ matrix,

$\displaystyle C=\begin{pmatrix}0&0&1&0\\\ 0&0&0&1\\\ 1&0&0&0\\\
0&1&0&0\end{pmatrix}.$ (S18)

The Hamiltonian (S16) breaks time-reversal symmetry, but the NS time-reversal
symmetry acts as

$\displaystyle\text{NS\,\,time:}\qquad T^{\dagger}{\cal H}^{\ast}(k)T={\cal
H}(-k),$ (S19)

where $T=\tau_{0}\sigma_{x}$ is a $4\times 4$ matrix,

$\displaystyle T=\begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{pmatrix}.$ (S20)

The combination of particle-hole symmetry and NS time-reversal symmetry yields
NS chiral symmetry,

$\displaystyle\text{NS\,\,chiral:}\qquad S^{\dagger}{\cal H}(k)S=-{\cal
H}(k),$ (S21)

where $S=\tau_{x}\sigma_{x}$ is a $4\times 4$ matrix,

$\displaystyle S=\begin{pmatrix}0&0&0&1\\\ 0&0&1&0\\\ 0&1&0&0\\\
1&0&0&0\end{pmatrix}.$ (S22)

##  III Bogoliubov de Gennes representation in $k$ space: Type I tight-binding
model

###  III.1 Hamiltonian

Instead of using Eqs. (S14,S15), we Fourier transform creation and
annihilation operators using

$\displaystyle c_{n,\alpha}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{k}c_{k\alpha}e^{ikna},\qquad\qquad
c_{n,\alpha}^{\dagger}=\frac{1}{\sqrt{N}}\sum_{k}c_{k\alpha}^{\dagger}e^{-ikna},$
(S23) $\displaystyle c_{n,\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{k}c_{k\beta}e^{ikna},\qquad\qquad
c_{n,\beta}^{\dagger}=\frac{1}{\sqrt{N}}\sum_{k}c_{k\beta}^{\dagger}e^{-ikna},$
(S24)

where $N$ is the number of unit cells and $a$ is the lattice constant. Then
the $4\times 4$ Bloch Hamiltonian ${\cal\tilde{H}}(k)$ is

$\displaystyle{\cal\tilde{H}}(k)=\begin{pmatrix}-\mu&t(1+e^{-ika})&0&\Delta\big{(}e^{i\phi}-e^{-i\phi-
ika}\big{)}\\\
t(1+e^{ika})&-\mu&-\Delta\big{(}e^{i\phi}-e^{-i\phi+ika}\big{)}&0\\\
0&-\Delta\big{(}e^{-i\phi}-e^{i\phi-ika}\big{)}&\mu&-t(1+e^{-ika})\\\
\Delta\big{(}e^{-i\phi}-e^{i\phi+ika}\big{)}&0&-t(1+e^{ika})&\mu\end{pmatrix}.$
(S25)

This is related to ${\cal H}(k)$ (S16) by a unitary transformation
${\cal\tilde{H}}(k)=\tilde{U}_{k}{\cal H}(k)\tilde{U}_{k}^{\dagger}$ where

$\displaystyle\tilde{U}_{k}=\begin{pmatrix}1&0&0&0\\\ 0&e^{ika/2}&0&0\\\
0&0&1&0\\\ 0&0&0&e^{ika/2}\end{pmatrix}.$ (S26)

###  III.2 Symmetry operators

The Bloch Hamiltonian (S25) incorporates particle-hole symmetry as

$\displaystyle\text{particle-
hole:}\qquad\tilde{C}^{\dagger}{\cal\tilde{H}}^{\ast}(k)\tilde{C}=-{\cal\tilde{H}}(-k),$
(S27)

where $\tilde{C}=\tau_{x}\sigma_{0}$ is a $4\times 4$ matrix,

$\displaystyle\tilde{C}=\begin{pmatrix}0&0&1&0\\\ 0&0&0&1\\\ 1&0&0&0\\\
0&1&0&0\end{pmatrix}.$ (S28)

The Hamiltonian (S25) breaks time-reversal symmetry, but the NS time-reversal
symmetry acts as

$\displaystyle\text{NS\,\,time:}\qquad\tilde{T}^{\dagger}(k){\cal\tilde{H}}^{\ast}(k)\tilde{T}(k)={\cal\tilde{H}}(-k),$
(S29)

where $\tilde{T}(k)$ is a $4\times 4$ matrix,

$\displaystyle\tilde{T}(k)=\begin{pmatrix}0&e^{ika/2}&0&0\\\
e^{-ika/2}&0&0&0\\\ 0&0&0&e^{ika/2}\\\ 0&0&e^{-ika/2}&0\end{pmatrix}.$ (S30)

The combination of particle-hole symmetry and NS time-reversal symmetry yields
NS chiral symmetry,

$\displaystyle\text{NS\,\,chiral:}\qquad\tilde{S}^{\dagger}(k){\cal\tilde{H}}(k)\tilde{S}(k)=-{\cal\tilde{H}}(k),$
(S31)

where $\tilde{U}_{S}(k)$ is a $4\times 4$ matrix,

$\displaystyle\tilde{S}(k)=\begin{pmatrix}0&0&0&e^{-ika/2}\\\
0&0&e^{ika/2}&0\\\ 0&e^{-ika/2}&0&0\\\ e^{ika/2}&0&0&0\end{pmatrix}.$ (S32)

##  IV Hamiltonian for a Josephson junction in a ring

We follow [7] to describe a Josephson junction as a weak link in a ring
enclosing magnetic flux $\Phi_{B}$, writing the $\mathbb{Z}_{4}$ Hamiltonian
for $N$ unit cells as

$\displaystyle H$ $\displaystyle=$
$\displaystyle-\mu\sum_{n=1}^{N}\sum_{\sigma=\alpha,\beta}c_{n,\sigma}^{\dagger}c_{n,\sigma}+t\sum_{n=1}^{N}\left(c_{n,\alpha}^{\dagger}c_{n,\beta}+\mathrm{H.c.}\right)+t\sum_{n=1}^{N-1}\left(c_{n+1,\alpha}^{\dagger}c_{n,\beta}+\mathrm{H.c.}\right)$
(S33) $\displaystyle+\sum_{n=1}^{N}\Big{(}\Delta
e^{i\phi}c_{n,\alpha}^{\dagger}c_{n,\beta}^{\dagger}+\mathrm{H.c.}\Big{)}+\sum_{n=1}^{N-1}\Big{(}\Delta
e^{-i\phi}c_{n,\beta}^{\dagger}c_{n+1,\alpha}^{\dagger}+\mathrm{H.c.}\Big{)}+t^{\prime}\left(c_{1,\alpha}^{\dagger}c_{N,\beta}e^{-i\varphi_{B}}+\mathrm{H.c.}\right),$

where $t^{\prime}$ is the strength of hopping across the weak link and the
phase $\varphi_{B}$ is given by

$\displaystyle\varphi_{B}=\pi\frac{\Phi_{B}}{\Phi_{\mathrm{s}}}=\frac{e}{\hbar}\Phi_{B},$
(S34)

where $\Phi_{\mathrm{s}}=h/(2e)$ is the superconducting flux quantum. For
$N=3$ unit cells, the weak link modifies Eq. (S3) as

$\displaystyle{\cal
H}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}-\mu&0&t&\Delta
e^{i\phi}&0&0&0&0&0&0&t^{\prime}e^{-i\varphi_{B}}&0\\\ 0&\mu&-\Delta
e^{-i\phi}&-t&0&0&0&0&0&0&0&-t^{\prime}e^{i\varphi_{B}}\\\ t&-\Delta
e^{i\phi}&-\mu&0&t&\Delta e^{-i\phi}&0&0&0&0&0&0\\\ \Delta
e^{-i\phi}&-t&0&\mu&-\Delta e^{i\phi}&-t&0&0&0&0&0&0\\\ 0&0&t&-\Delta
e^{-i\phi}&-\mu&0&t&\Delta e^{i\phi}&0&0&0&0\\\ 0&0&\Delta
e^{i\phi}&-t&0&\mu&-\Delta e^{-i\phi}&-t&0&0&0&0\\\ 0&0&0&0&t&-\Delta
e^{i\phi}&-\mu&0&t&\Delta e^{-i\phi}&0&0\\\ 0&0&0&0&\Delta
e^{-i\phi}&-t&0&\mu&-\Delta e^{i\phi}&-t&0&0\\\ 0&0&0&0&0&0&t&-\Delta
e^{-i\phi}&-\mu&0&t&\Delta e^{i\phi}\\\ 0&0&0&0&0&0&\Delta
e^{i\phi}&-t&0&\mu&-\Delta e^{-i\phi}&-t\\\
t^{\prime}e^{i\varphi_{B}}&0&0&0&0&0&0&0&t&-\Delta e^{i\phi}&-\mu&0\\\
0&-t^{\prime}e^{-i\varphi_{B}}&0&0&0&0&0&0&\Delta
e^{-i\phi}&-t&0&\mu\end{pmatrix}.$

##  V Hamiltonian for the anticommuting $\mathbb{Z}_{4}$ subclass

###  V.1 Bogoliubov de Gennes representation in position space

The mean-field Hamiltonian of the model in the anticommuting subclass is

$\displaystyle
H_{\mathrm{a}}=-\mu\sum_{n,\sigma}c_{n,\sigma}^{\dagger}c_{n,\sigma}+t\sum_{n}\left(c_{n,\alpha}^{\dagger}c_{n,\beta}+c_{n+1,\alpha}^{\dagger}c_{n,\beta}+\mathrm{H.c.}\right)+\Delta\sum_{n}\Big{(}e^{i\varphi}c_{n,\alpha}^{\dagger}c_{n,\beta}^{\dagger}-e^{-i\varphi}c_{n,\beta}^{\dagger}c_{n+1,\alpha}^{\dagger}+\mathrm{H.c.}\Big{)}.$
(S35)

Using the basis (S2), the BdG representation of this model with open boundary
conditions and $N=3$ unit cells is

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\cal
H}_{\mathrm{a}}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}-\mu&0&t&\Delta
e^{i\varphi}&0&0&0&0&0&0&0&0\\\ 0&\mu&-\Delta
e^{-i\varphi}&-t&0&0&0&0&0&0&0&0\\\ t&-\Delta e^{i\varphi}&-\mu&0&t&-\Delta
e^{-i\varphi}&0&0&0&0&0&0\\\ \Delta e^{-i\varphi}&-t&0&\mu&\Delta
e^{i\varphi}&-t&0&0&0&0&0&0\\\ 0&0&t&\Delta e^{-i\varphi}&-\mu&0&t&\Delta
e^{i\varphi}&0&0&0&0\\\ 0&0&-\Delta e^{i\varphi}&-t&0&\mu&-\Delta
e^{-i\varphi}&-t&0&0&0&0\\\ 0&0&0&0&t&-\Delta e^{i\varphi}&-\mu&0&t&-\Delta
e^{-i\varphi}&0&0\\\ 0&0&0&0&\Delta e^{-i\varphi}&-t&0&\mu&\Delta
e^{i\varphi}&-t&0&0\\\ 0&0&0&0&0&0&t&\Delta e^{-i\varphi}&-\mu&0&t&\Delta
e^{i\varphi}\\\ 0&0&0&0&0&0&-\Delta e^{i\varphi}&-t&0&\mu&-\Delta
e^{-i\varphi}&-t\\\ 0&0&0&0&0&0&0&0&t&-\Delta e^{i\varphi}&-\mu&0\\\
0&0&0&0&0&0&0&0&\Delta e^{-i\varphi}&-t&0&\mu\end{pmatrix}\\!.$ (S36)

For this model, the particle-hole symmetry operator ${\mathcal{C}}$ is given
by Eq. (S4). The time-reversal symmetry operator is given by
${\mathcal{T}}_{\mathrm{a}}={\mathcal{T}}_{\mathrm{c}}{\mathcal{S}}_{z}$,
where ${\mathcal{S}}_{z}=\text{diag}(1,-1,1,-1,1,-1,1,-1,1,-1,1,-1)$ and

$\displaystyle{\mathcal{T}}_{\mathrm{a}}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}0&0&1&0&0&0&0&0&0&0&0&0\\\
0&0&0&-1&0&0&0&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0&0&0&0&0\\\
0&0&0&0&0&-1&0&0&0&0&0&0\\\ 0&0&0&0&0&0&1&0&0&0&0&0\\\
0&0&0&0&0&0&0&-1&0&0&0&0\\\ 0&0&0&0&0&0&0&0&1&0&0&0\\\
0&0&0&0&0&0&0&0&0&-1&0&0\\\ 0&0&0&0&0&0&0&0&0&0&1&0\\\
0&0&0&0&0&0&0&0&0&0&0&-1\\\ 1&0&0&0&0&0&0&0&0&0&0&0\\\
0&-1&0&0&0&0&0&0&0&0&0&0\end{pmatrix}.$ (S37)

Chiral symmetry is
${\mathcal{S}}_{\mathrm{a}}^{\dagger}{\mathcal{H}}_{\mathrm{a}}{\mathcal{S}}_{\mathrm{a}}=-{\mathcal{H}}_{\mathrm{a}}$.
The operator ${\mathcal{S}}_{\mathrm{a}}$ is NS, involving a translation,
where
${\mathcal{S}}_{\mathrm{a}}={\mathcal{T}}_{\mathrm{a}}^{\ast}{\mathcal{C}}$,

$\displaystyle{\mathcal{S}}_{\mathrm{a}}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}0&0&0&1&0&0&0&0&0&0&0&0\\\
0&0&-1&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0&0&0&0&0\\\
0&0&0&0&-1&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&1&0&0&0&0\\\
0&0&0&0&0&0&-1&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&1&0&0\\\
0&0&0&0&0&0&0&0&-1&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&1\\\
0&0&0&0&0&0&0&0&0&0&-1&0\\\ 0&1&0&0&0&0&0&0&0&0&0&0\\\
-1&0&0&0&0&0&0&0&0&0&0&0\end{pmatrix}.$ (S38)

###  V.2 Bogoliubov de Gennes representation in $k$ space: Type II tight-
binding model

Using the Fourier transforms (S14,S15), the Bloch Hamiltonian is

$\displaystyle{\cal
H}_{\mathrm{a}}(k)=\begin{pmatrix}-\mu&2t\cos(ka/2)&0&2\Delta\cos(ka/2+\varphi)\\\
2t\cos(ka/2)&-\mu&-2\Delta\cos(ka/2-\varphi)&0\\\
0&-2\Delta\cos(ka/2-\varphi)&\mu&-2t\cos(ka/2)\\\
2\Delta\cos(ka/2+\varphi)&0&-2t\cos(ka/2)&\mu\end{pmatrix}.$ (S39)

The Bloch Hamiltonian (S39) incorporates particle-hole symmetry as

$\displaystyle\text{particle-hole:}\qquad C_{\mathrm{a}}^{\dagger}{\cal
H}_{\mathrm{a}}^{\ast}(k)C_{\mathrm{a}}=-{\cal H}_{\mathrm{a}}(-k),$ (S40)

where $C_{\mathrm{a}}=\tau_{x}\sigma_{0}$ is a $4\times 4$ matrix,

$\displaystyle C_{\mathrm{a}}=\begin{pmatrix}0&0&1&0\\\ 0&0&0&1\\\ 1&0&0&0\\\
0&1&0&0\end{pmatrix}.$ (S41)

The Hamiltonian (S39) breaks time-reversal symmetry, but the NS time-reversal
symmetry acts as

$\displaystyle\text{NS\,\,time:}\qquad T_{\mathrm{a}}^{\dagger}{\cal
H}_{\mathrm{a}}^{\ast}(k)T_{\mathrm{a}}={\cal H}_{\mathrm{a}}(-k),$ (S42)

where $T_{\mathrm{a}}=\tau_{z}\sigma_{x}$ is a $4\times 4$ matrix,

$\displaystyle T_{\mathrm{a}}=\begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&-1\\\
0&0&-1&0\end{pmatrix}.$ (S43)

The combination of particle-hole symmetry and NS time-reversal symmetry yields
NS chiral symmetry,

$\displaystyle\text{NS\,\,chiral:}\qquad S_{\mathrm{a}}^{\dagger}{\cal
H}_{\mathrm{a}}(k)S_{\mathrm{a}}=-{\cal H}_{\mathrm{a}}(k),$ (S44)

where $S_{\mathrm{a}}=\tau_{y}\sigma_{x}$ is a $4\times 4$ matrix,

$\displaystyle S_{\mathrm{a}}=\begin{pmatrix}0&0&0&-i\\\ 0&0&-i&0\\\
0&i&0&0\\\ i&0&0&0\end{pmatrix}.$ (S45)

###  V.3 Bogoliubov de Gennes representation in $k$ space: Type I tight-
binding model

Using the Fourier transforms in Eqs. (S23,S24), the Bloch Hamiltonian
${\cal\tilde{H}}_{\mathrm{a}}(k)$ is

$\displaystyle{\cal\tilde{H}}_{\mathrm{a}}(k)=\begin{pmatrix}-\mu&t(1+e^{-ika})&0&\Delta\big{(}e^{i\varphi}+e^{-i\varphi-
ika}\big{)}\\\
t(1+e^{ika})&-\mu&-\Delta\big{(}e^{i\varphi}+e^{-i\varphi+ika}\big{)}&0\\\
0&-\Delta\big{(}e^{-i\varphi}+e^{i\varphi-ika}\big{)}&\mu&-t(1+e^{-ika})\\\
\Delta\big{(}e^{-i\varphi}+e^{i\varphi+ika}\big{)}&0&-t(1+e^{ika})&\mu\end{pmatrix}.$
(S46)

The Bloch Hamiltonian (S46) incorporates particle-hole symmetry as

$\displaystyle\text{particle-
hole:}\qquad\tilde{C}_{\mathrm{a}}^{\dagger}{\cal\tilde{H}}_{\mathrm{a}}^{\ast}(k)\tilde{C}_{\mathrm{a}}=-{\cal\tilde{H}}_{\mathrm{a}}(-k),$
(S47)

where $\tilde{C}_{\mathrm{a}}=\tau_{x}\sigma_{0}$ is a $4\times 4$ matrix,

$\displaystyle\tilde{C}_{\mathrm{a}}=\begin{pmatrix}0&0&1&0\\\ 0&0&0&1\\\
1&0&0&0\\\ 0&1&0&0\end{pmatrix}.$ (S48)

The Hamiltonian (S25) breaks time-reversal symmetry, but the NS time-reversal
symmetry acts as

$\displaystyle\text{NS\,\,time:}\qquad\tilde{T}_{\mathrm{a}}^{\dagger}(k){\cal\tilde{H}}_{\mathrm{a}}^{\ast}(k)\tilde{T}_{\mathrm{a}}(k)={\cal\tilde{H}}_{\mathrm{a}}(-k),$
(S49)

where $\tilde{T}_{\mathrm{a}}(k)$ is a $4\times 4$ matrix,

$\displaystyle\tilde{T}_{\mathrm{a}}(k)=\begin{pmatrix}0&e^{ika/2}&0&0\\\
e^{-ika/2}&0&0&0\\\ 0&0&0&-e^{ika/2}\\\ 0&0&-e^{-ika/2}&0\end{pmatrix}.$ (S50)

The combination of particle-hole symmetry and NS time-reversal symmetry yields
NS chiral symmetry,

$\displaystyle\text{NS\,\,chiral:}\qquad\tilde{S}_{\mathrm{a}}^{\dagger}(k){\cal\tilde{H}}_{\mathrm{a}}(k)\tilde{S}_{\mathrm{a}}(k)=-{\cal\tilde{H}}_{\mathrm{a}}(k),$
(S51)

where $\tilde{S}_{\mathrm{a}}(k)$ is a $4\times 4$ matrix,

$\displaystyle\tilde{S}_{\mathrm{a}}(k)=\begin{pmatrix}0&0&0&-ie^{-ika/2}\\\
0&0&-ie^{ika/2}&0\\\ 0&ie^{-ika/2}&0&0\\\ ie^{ika/2}&0&0&0\end{pmatrix}.$
(S52)

###  V.4 Equivalence of the commuting and anticommuting models

In position space, we use operator ${\cal U}$ where

$\displaystyle{\cal
U}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}1&0&0&0&0&0&0&0&0&0&0&0\\\
0&i&0&0&0&0&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0&0&0&0&0\\\
0&0&0&i&0&0&0&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0&0&0&0&0\\\
0&0&0&0&0&i&0&0&0&0&0&0\\\ 0&0&0&0&0&0&1&0&0&0&0&0\\\
0&0&0&0&0&0&0&i&0&0&0&0\\\ 0&0&0&0&0&0&0&0&1&0&0&0\\\
0&0&0&0&0&0&0&0&0&i&0&0\\\ 0&0&0&0&0&0&0&0&0&0&1&0\\\
0&0&0&0&0&0&0&0&0&0&0&i\end{pmatrix}.$ (S53)

With the commuting Hamiltonian ${\cal H}$ (S3) and the anticommuting
Hamiltonian ${\cal H}_{\mathrm{a}}$ (S36), we have

$\displaystyle{\cal U}^{\dagger}{\cal H}_{\mathrm{a}}(\varphi=\phi-\pi/2){\cal
U}={\cal H}.$ (S54)

In $k$ space, with the type II representation, we use operator $U$ where

$\displaystyle U=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&i&0\\\
0&0&0&i\end{pmatrix}.$ (S55)

With the commuting Hamiltonian ${\cal H}$ (S16) and the anticommuting
Hamiltonian ${\cal H}_{\mathrm{a}}$ (S39), we have

$\displaystyle U^{\dagger}{\cal H}_{\mathrm{a}}(\varphi=\phi-\pi/2)U={\cal
H}.$ (S56)

###  V.5 Equivalence of the commuting and anticommuting algebras

We consider the type II representations in $k$ space, and consider particle-
hole symmetry and NS time-reversal symmetry. For the commuting algebra:

$\displaystyle\text{particle-hole:}\qquad C^{\dagger}{\cal H}^{\ast}(k)C$
$\displaystyle=$ $\displaystyle-{\cal H}(-k),$ (S57)
$\displaystyle\text{NS\,\,time:}\qquad T^{\dagger}{\cal H}^{\ast}(k)T$
$\displaystyle=$ $\displaystyle{\cal H}(-k),$ (S58)
$\displaystyle\text{commutation:}\qquad\qquad CT$ $\displaystyle=$
$\displaystyle TC,$ (S59)

and, for the anticommuting algebra,

$\displaystyle\text{particle-hole:}\qquad C^{\dagger}{\cal
H}_{\mathrm{a}}^{\ast}(k)C$ $\displaystyle=$ $\displaystyle-{\cal
H}_{\mathrm{a}}(-k),$ (S60) $\displaystyle\text{NS\,\,time:}\qquad
T_{\mathrm{a}}^{\dagger}{\cal H}_{\mathrm{a}}^{\ast}(k)T_{\mathrm{a}}$
$\displaystyle=$ $\displaystyle{\cal H}_{\mathrm{a}}(-k),$ (S61)
$\displaystyle\text{anticommutation:}\qquad\qquad CT_{\mathrm{a}}$
$\displaystyle=$ $\displaystyle-T_{\mathrm{a}}C.$ (S62)

The unitary transformation between them is (S56)

$\displaystyle U^{\dagger}{\cal H}_{\mathrm{a}}(k)U={\cal H}(k),$ (S63)

where complex conjugation occurs because time-reversal is antinunitary. The
transformation (S63) gives

$\displaystyle T_{\mathrm{a}}=U^{\ast}TU^{\dagger}.$ (S64)

We substitute this into the anticommuting equation (S62),

$\displaystyle CU^{\ast}TU^{\dagger}=-U^{\ast}TU^{\dagger}C.$ (S65)

We find that the algebras are related by

$\displaystyle TU^{\dagger}$ $\displaystyle=$ $\displaystyle U^{\dagger}T,$
(S66) $\displaystyle CU^{\ast}U^{\dagger}$ $\displaystyle=$
$\displaystyle-U^{\ast}U^{\dagger}C.$ (S67)

Hence, from Eq. (S65), using Eq. (S66),

$\displaystyle CU^{\ast}U^{\dagger}T=-U^{\ast}U^{\dagger}TC.$ (S68)

Then use Eq. (S67),

$\displaystyle-U^{\ast}U^{\dagger}CT=-U^{\ast}U^{\dagger}TC,$ (S69)

which is equivalent to the commutating relation (S59).

In the type II representation, for the specific models considered here, we
have,

$\displaystyle C$ $\displaystyle=$ $\displaystyle\tau_{x}\sigma_{0},$ (S70)
$\displaystyle T$ $\displaystyle=$ $\displaystyle\tau_{0}\sigma_{x},$ (S71)
$\displaystyle T_{\mathrm{a}}$ $\displaystyle=$
$\displaystyle\tau_{z}\sigma_{x},$ (S72) $\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{[}(1+i)\tau_{0}+(1-i)\tau_{z}\big{]}\sigma_{0},$
(S73)

###  V.6 Equivalence of arbitrary Hamiltonians in position space

Since the nonsymmorphic symmetry involves a translation, tight-binding
parameters must be constant throughout the whole system and, hence, the
Hamiltonian is not robust in the presence of disorder [8]. We consider the
form of an arbitrary Hermitian Hamiltonian in position space which preserves
particle-hole symmetry and nonsymmorphic time-reversal symmetry. Including all
possible terms (e.g., long-range hopping parameters), a $J\times J$
Hamiltonian is parameterized by only $J$ real numbers [8]. We write out the
form of the Hamiltonians explicitly for $N=3$ unit cells ($J=12$).

A generic Hermitian Hamiltonian in position space satisying particle-hole
symmetry (S4) and commuting NS time-reversal symmetry (S5) may be written as

$\displaystyle{\mathcal{H}}_{\mathrm{c}}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}h_{1}&0&h_{3}&-h_{2}^{\ast}&h_{5}&-h_{4}^{\ast}&h_{7}&ih_{6}&h_{5}^{\ast}&h_{4}^{\ast}&h_{3}&h_{2}\\\
0&-h_{1}&h_{2}&-h_{3}^{\ast}&h_{4}&-h_{5}^{\ast}&ih_{6}&-h_{7}^{\ast}&-h_{4}&-h_{5}&-h_{2}^{\ast}&-h_{3}^{\ast}\\\
h_{3}^{\ast}&h_{2}^{\ast}&h_{1}&0&h_{3}^{\ast}&-h_{2}&h_{5}^{\ast}&-h_{4}&h_{7}^{\ast}&-ih_{6}&h_{5}&h_{4}\\\
-h_{2}&-h_{3}&0&-h_{1}&h_{2}^{\ast}&-h_{3}&h_{4}^{\ast}&-h_{5}&-ih_{6}&-h_{7}&-h_{4}^{\ast}&-h_{5}^{\ast}\\\
h_{5}^{\ast}&h_{4}^{\ast}&h_{3}&h_{2}&h_{1}&0&h_{3}&-h_{2}^{\ast}&h_{5}&-h_{4}^{\ast}&h_{7}&ih_{6}\\\
-h_{4}&-h_{5}&-h_{2}^{\ast}&-h_{3}^{\ast}&0&-h_{1}&h_{2}&-h_{3}^{\ast}&h_{4}&-h_{5}^{\ast}&ih_{6}&-h_{7}^{\ast}\\\
h_{7}^{\ast}&-ih_{6}&h_{5}&h_{4}&h_{3}^{\ast}&h_{2}^{\ast}&h_{1}&0&h_{3}^{\ast}&-h_{2}&h_{5}^{\ast}&-h_{4}\\\
-ih_{6}&-h_{7}&-h_{4}^{\ast}&-h_{5}^{\ast}&-h_{2}&-h_{3}&0&-h_{1}&h_{2}^{\ast}&-h_{3}&h_{4}^{\ast}&-h_{5}\\\
h_{5}&-h_{4}^{\ast}&h_{7}&ih_{6}&h_{5}^{\ast}&h_{4}^{\ast}&h_{3}&h_{2}&h_{1}&0&h_{3}&-h_{2}^{\ast}\\\
h_{4}&-h_{5}^{\ast}&ih_{6}&-h_{7}^{\ast}&-h_{4}&-h_{5}&-h_{2}^{\ast}&-h_{3}^{\ast}&0&-h_{1}&h_{2}&-h_{3}^{\ast}\\\
h_{3}^{\ast}&-h_{2}&h_{5}^{\ast}&-h_{4}&h_{7}^{\ast}&-ih_{6}&h_{5}&h_{4}&h_{3}^{\ast}&h_{2}^{\ast}&h_{1}&0\\\
h_{2}^{\ast}&-h_{3}&h_{4}^{\ast}&-h_{5}&-ih_{6}&-h_{7}&-h_{4}^{\ast}&-h_{5}^{\ast}&-h_{2}&-h_{3}&0&-h_{1}\end{pmatrix}.$
(S74)

where $h_{1}$ and $h_{6}$ are real, $h_{2}$, $h_{3}$, $h_{4}$, $h_{5}$,
$h_{7}$ are complex. Hamiltonian (S3) is an example of this form (without
periodic boundary conditions) with $h_{1}=-\mu$, $h_{2}=-\Delta e^{-i\phi}$,
and $h_{3}=t$. A Hamiltonian satisying particle-hole symmetry (S4) and
anticommuting NS time-reversal symmetry (S37) is given by

$\displaystyle{\mathcal{H}}_{\mathrm{a}}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}g_{1}&0&g_{3}&-g_{2}^{\ast}&g_{5}&-g_{4}^{\ast}&g_{7}&-g_{6}&g_{5}^{\ast}&g_{4}^{\ast}&g_{3}&-g_{2}\\\
0&-g_{1}&g_{2}&-g_{3}^{\ast}&g_{4}&-g_{5}^{\ast}&g_{6}&-g_{7}^{\ast}&-g_{4}&-g_{5}&g_{2}^{\ast}&-g_{3}^{\ast}\\\
g_{3}^{\ast}&g_{2}^{\ast}&g_{1}&0&g_{3}^{\ast}&g_{2}&g_{5}^{\ast}&g_{4}&g_{7}^{\ast}&g_{6}&g_{5}&-g_{4}\\\
-g_{2}&-g_{3}&0&-g_{1}&-g_{2}^{\ast}&-g_{3}&-g_{4}^{\ast}&-g_{5}&-g_{6}&-g_{7}&g_{4}^{\ast}&-g_{5}^{\ast}\\\
g_{5}^{\ast}&g_{4}^{\ast}&g_{3}&-g_{2}&g_{1}&0&g_{3}&-g_{2}^{\ast}&g_{5}&-g_{4}^{\ast}&g_{7}&-g_{6}\\\
-g_{4}&-g_{5}&g_{2}^{\ast}&-g_{3}^{\ast}&0&-g_{1}&g_{2}&-g_{3}^{\ast}&g_{4}&-g_{5}^{\ast}&g_{6}&-g_{7}^{\ast}\\\
g_{7}^{\ast}&g_{6}&g_{5}&-g_{4}&g_{3}^{\ast}&g_{2}^{\ast}&g_{1}&0&g_{3}^{\ast}&g_{2}&g_{5}^{\ast}&g_{4}\\\
-g_{6}&-g_{7}&g_{4}^{\ast}&-g_{5}^{\ast}&-g_{2}&-g_{3}&0&-g_{1}&-g_{2}^{\ast}&-g_{3}&-g_{4}^{\ast}&-g_{5}\\\
g_{5}&-g_{4}^{\ast}&g_{7}&-g_{6}&g_{5}^{\ast}&g_{4}^{\ast}&g_{3}&-g_{2}&g_{1}&0&g_{3}&-g_{2}^{\ast}\\\
g_{4}&-g_{5}^{\ast}&g_{6}&-g_{7}^{\ast}&-g_{4}&-g_{5}&g_{2}^{\ast}&-g_{3}^{\ast}&0&-g_{1}&g_{2}&-g_{3}^{\ast}\\\
g_{3}^{\ast}&g_{2}&g_{5}^{\ast}&g_{4}&g_{7}^{\ast}&g_{6}&g_{5}&-g_{4}&g_{3}^{\ast}&g_{2}^{\ast}&g_{1}&0\\\
-g_{2}^{\ast}&-g_{3}&-g_{4}^{\ast}&-g_{5}&-g_{6}&-g_{7}&g_{4}^{\ast}&-g_{5}^{\ast}&-g_{2}&-g_{3}&0&-g_{1}\end{pmatrix}.$
(S75)

where $g_{1}$ and $g_{6}$ are real, $g_{2}$, $g_{3}$, $g_{4}$, $g_{5}$,
$g_{7}$ are complex. Hamiltonian (S36) is an example of this form (without
periodic boundary conditions) with $g_{1}=-\mu$, $g_{2}=-\Delta
e^{-i\varphi}$, and $g_{3}=t$. For these Hamiltonians, we find that ${\cal
U}^{\dagger}{\mathcal{H}}_{\mathrm{a}}{\cal U}={\mathcal{H}}_{\mathrm{c}}$,
where ${\cal U}$ is given in Eq. (S53), with the replacements

$\displaystyle g_{1}=h_{1},\quad g_{2}=ih_{2},\quad g_{3}=h_{3},\quad
g_{4}=ih_{4},\quad g_{5}=h_{5},\quad g_{6}=-h_{6},\quad g_{7}=h_{7}.$

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