# Disorder and non-Hermiticity in Kitaev spin liquids with a Majorana Fermi
surface

Lukas Rødland  Department of Physics, Stockholm University, AlbaNova
University Center, SE-106 91 Stockholm, Sweden  Carlos Ortega-Taberner  School
of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland  Trinity
Quantum Alliance, Unit 16, Trinity Technology and Enterprise Centre, Pearse
Street, Dublin 2, Ireland  Megha Agarwal  Department of Physics, Stockholm
University, AlbaNova University Center, SE-106 91 Stockholm, Sweden  Maria
Hermanns  Department of Physics, Stockholm University, AlbaNova University
Center, SE-106 91 Stockholm, Sweden

###### Abstract

We study the effect of disorder on Z2 quantum spin liquids with a Majorana
Fermi line (respectively surface in three dimensions) and show that depending
on the symmetries that are preserved _on average_ qualitatively different
scenarios will occur. In particular, we identify the relevant non-Hermitian
symmetries for which disorder will effectively split the Fermi line into two
exceptional lines, with $\real(E)=0$ states filling the area in between. We
demonstrate the different scenarios using both toy models as well as large-
scale numerical simulations.

##  I Introduction

Ever since the original proposal by Anderson [1], quantum spin liquids have
continued to fascinate theorists and experimentalists alike [2]. They show
several unexpected properties, such as the lack of magnetic ordering even at
zero temperatures, the presence of long-range entanglement [3, 4] or the
fractionalization of spins. Since they necessarily are strongly correlated
phases, much of our current theoretical understanding of quantum spin liquids
has been driven by simple model systems, such as the toric code [5] or
Kitaev’s honeycomb model [6]. The latter has gained an increasing amount of
attention lately as it may be realized in certain materials [7], see also [8]
and references therein.

The Kitaev honeycomb model is exactly solvable in terms of Majorana fermions
hopping in a static emergent Z2 gauge field [6]. It allows for a variety of
different quantum spin liquid phases — in the following referred to as Kitaev
spin liquids (KSL) — depending on the underlying lattice [9, 10]. At the same
time it is sufficiently simple to allow for the computation of many,
experimentally accessible properties, such as the dynamical structure factor
[11] or neutron scattering signatures [12]. Here, we study the effect of
hopping disorder on KSLs with a Majorana Fermi surface, both in a two-
dimensional lattice [9] and a three-dimensional one [13]. We also comment on
the interesting properties of the effective Hamiltonian, obtained after
disorder average, which links to the field of non-Hermitian physics.

The effects of disorder are analytically studied using Green’s function
formalism, a powerful tool for examining quantum states in disordered systems.
Disorder typically leads to a non-Hermitian self-energy of the system, and
hence a broadening of the spectral function. This self-energy is then
integrated into an effective non-Hermitian Hamiltonian [14, 15, 16] at low
energies, achieved through the Feshbach projection method [17, 14, 18]. The
imaginary part of this Hamiltonian’s eigenvalues is directly related to the
inverse lifetime of quasiparticles and thus to the broadening of the spectral
function [19], reflecting the dynamic effects of disorder in the spectrum.

The existence of exceptional points where the Hamiltonian is non-
diagonalizable is an inherent feature of non-Hermitian systems [15]. A Dirac
point in two dimensions can either be split into a pair of exceptional points
or an exceptional ring under a non-Hermitian perturbation [20, 21, 22, 23,
24]. Between two paired exceptional points there is a Fermi arc [19, 25, 26,
27, 28, 29], where the real part of the energy vanishes. Another intriguing
aspect of non-Hermitian systems is the splitting of discrete symmetries into
two different types, due to the difference between transposition and
conjugation for non-Hermitian operators [30, 22], giving rise to unique
‘transpose-type’ symmetries that are inherently non-Hermitian. The two
variations of time-reversal and particle-hole symmetries create different
pathways to achieving chiral symmetry.

In Refs. [31, 32] the authors studied the effect of bond disorder on a Kitaev
quantum spin liquid in the gapless phase on a honeycomb lattice, where they
found that each gapless point splits into two exceptional points for generic
(time-reversal preserving) disorder and opened into an exceptional ring for
disorder that preserved time-reversal only on average. Inside the exceptional
ring, the real part of the energy eigenvalue vanishes. This area of $Re[E]=0$
states is often called Fermi surface. However, since we need the (original)
concept of Fermi surface for the three-dimensional model, we will instead call
it a _Fermi area_ for the remainder of this manuscript. The Fermi area of Ref.
[32] was very small and hard to observe. In the following, we show a
systematic way to obtain this effect in a much more robust way.

In this paper, our focus is on the gapless phase of the Kitaev spin liquid on
the square octagon model in two dimensions and on the hyperoctagon model in
three dimensions [13]. The non-trivial implementation of time-reversal
symmetry in the square-octagon (hyperoctagon) model leads [10] to a Majorana
Fermi line (surface), which is inherently stable. Any non-Hermitian
perturbation can only deform the Majorana Fermi line, not destroy it. This is
the generic, though slightly boring, scenario that prevails if all the
symmetries are broken. Here, we show that there are two additional, symmetry-
protected scenarios that occur. For instance, introducing a time-reversal
invariant hopping disorder (which partially breaks translation symmetry on
average) leads to the Majorana Fermi line splitting into two exceptional
lines. This splitting creates a large Fermi area located between the two
rings, which is protected by chiral symmetry. This Fermi area can be
interpreted as the broadening of the spectral function, where the amount of
broadening, e.g., the size of the Fermi area, is related to the lifetime of
the single-particle states. When chiral symmetry is broken, the exceptional
lines gap out to pairs of exceptional points that are connected by Fermi arcs
at $Re[E]=0$.

_Outline of this manuscript_ In Sec. II we introduce a non-Hermitian toy model
on the extended square octagon lattice that shows the effect in a transparent
way, followed by the symmetry analysis in II.2. We also discuss another toy
model in Section II.3 as an aid for the disorder analysis in Section III. We
end our manuscript with a short section on the generalization to three-
dimensional models and some concluding remarks. Most of the technical details
as well as a short introduction to the relevant KSLs can be found in the
appendices.

##  II 2D Model Hamiltonian and Symmetries

###  II.1 Simple model

Figure 1: Unit cell and translation vectors of the square-octagon model
obtained after doubling the unit cell. Green/red/blue bonds correspond to
nearest-neighbor Ising interactions with the Ising axis along
$\hat{x}$/$\hat{y}$/$\hat{z}$.

We will consider a non-Hermitian version of the Kitaev spin liquid (KSL) on a
square-octagon lattice. Readers unfamiliar with KSLs are referred to Appendix
C, which contains a short summary of KSLs in general as well as details for
the two particular examples of KSLs relevant to this manuscript. The general
solution of Kitaevs model [6] relies on the fact that we can phrase the
nearest neighbor Ising interaction of spin-$1/2$ local moments in terms of a
nearest neighbor hopping model of Majorana fermions as $\mathcal{H}=i\sum
J_{\gamma}u_{jk}c_{j}c_{k}$. This model is Hermitian as long as
$J_{\gamma}\in\mathbb{R}$. The square-octagon lattice as well as the choice of
Ising interactions are depicted in Fig. 1. The original square-octagon lattice
has a 4-site unit cell, given by sites 1 to 4. In the following, we will
partially break translation symmetry which doubles the unit cell in the way
shown in Fig. 1.

Using the 4-site unit cell, the Hermitian version of this model has two Fermi
lines in the spectrum, shown in Fig. 2(a). A symmetry analysis along the lines
of Ref. [10] shows that there is perfect nesting between the two surfaces, as
long as time-reversal symmetry or inversion symmetry remain intact. In
particular, they can be mapped onto each other by a translation of
$\mathbf{k}_{0}$, where $\mathbf{k}_{0}$ is half a reciprocal lattice vector;
see Appendix C. Any translationally invariant two-Majorana term, even a
generic non-Hermitian one, can only deform the Fermi lines. In the absence of
any symmetry, the Kitaev model on the square-octagon lattice will exhibit
Majorana Fermi lines independent of the non-Hermitian perturbation we add.
However, when retaining some symmetries, the scenario can become widely
different. To see this, we need to enlarge the unit cell to contain 8 sites,
as shown in Fig. 1. By doubling the unit cell, the Majorana Fermi lines are
mapped onto each other, as depicted in Fig. 2(b). Furthermore, they are
protected by the full translation symmetry, i.e., under translations
$(\mathbf{a}_{1}\pm\mathbf{a}_{2})/2$.

Figure 2:  Real part of the energy gap for the square octagon model with
$\phi=\theta=0$ using the (a) 4 site unit cell and (b) the extended 8 site
unit cell, where the two Majorana Fermi lines are mapped into each other.

We now consider perturbations that mimic a dimerization in the system, i.e.,
that break the original translation symmetry, but keep the translation
symmetry of the 8-site model. These may lead to two different scenarios.
Hermitian perturbations that preserve time-reversal but break inversion reduce
the doubly degenerate Fermi line to an even number of Dirac points. When
adding a non-Hermitian perturbation, the Dirac points will split into a pair
of exceptional points — a behavior already seen in Ref. [31]. However, we can
also imagine a much more interesting scenario where the Fermi line splits into
two exceptional lines that delimit a Fermi area with $Re[E]=0$. Before
discussing the symmetry that protects the exceptional rings, we first
introduce a simple explicit model that showcases both scenarios.

We introduce non-Hermiticity in the square-octagon model by allowing the
couplings to acquire an additional phase, similar to what was considered in
Ref. [31], except that we allow the phases to be different in the two squares
of the extended unit cell. In particular, let us consider the Hamiltonian
given by

$\displaystyle
H=\left(\begin{array}[]{cccccccc}0&iJ_{x}e^{i\phi}&0&iJ_{y}&0&0&iJ_{z}&0\\\
-iJ_{x}e^{i\phi}&0&iJ_{y}&0&0&0&0&ie^{-ik_{2}}J_{z}\\\
0&-iJ_{y}&0&-iJ_{x}&-ie^{i\left(-k_{1}-k_{2}\right)}J_{z}&0&0&0\\\
-iJ_{y}&0&iJ_{x}&0&0&-ie^{-ik_{1}}J_{z}&0&0\\\
0&0&ie^{i\left(k_{1}+k_{2}\right)}J_{z}&0&0&iJ_{x}e^{i\theta}&0&iJ_{y}\\\
0&0&0&ie^{ik_{1}}J_{z}&-iJ_{x}e^{i\theta}&0&iJ_{y}&0\\\
-iJ_{z}&0&0&0&0&-iJ_{y}&0&-iJ_{x}\\\
0&-ie^{ik_{2}}J_{z}&0&0&-iJ_{y}&0&iJ_{x}&0\\\ \end{array}\right),$ (9)

where a phase $\phi$ was added in the hopping between sites $1$ and $2$, and a
phase $\theta$ in the hopping between sites $5$ and $6$. Note that the phase
acquired is the same for the hopping in both directions, leading to non-
Hermiticity.

If the two phases are exactly opposite, i.e., $\phi=-\theta$, the resulting
non-Hermitian perturbation indeed splits the Majorana Fermi lines into two
exceptional rings, enclosing a Fermi area of zero real energy states, as can
be seen in Fig. 3(a). Moving away from $\theta=-\phi$, as shown in Fig. 3(b),
causes the exceptional rings to gap into isolated exceptional points that
delimit Fermi arcs. In the following section, we will show that the existence
of exceptional rings is not fine-tuned, as one might suspect from our model
system, but symmetry-protected. Even the second scenario, where we find
isolated exceptional points at zero energy, is protected by a symmetry, albeit
a weaker one. While the exceptional rings require time-reversal symmetry
(alternatively inversion), the latter relies on sublattice symmetry (with
broken time-reversal and inversion), see Appendix D. Note that time-reversal
and sublattice symmetry cannot be treated as two independent symmetries for
Hermitian Kitaev spin liquids; for non-Hermitian models, as well as for the
effective Hamiltonian of the disordered systems, they are distinct.

Figure 3:  (a) Real part of the energy gap for the non-Hermitian model using
$\phi=1/2=-\theta$, showing Fermi areas. Exceptional lines are indicated by
the gray boundaries. (b) The logarithm of the real part of the gap for the
model with $\phi=1/2,\theta=0$, where the exponential rings split into points
at the ends of Fermi arcs with $Re[E]=0$.

###  II.2 Symmetry analysis

We will now proceed to show that a version of chiral symmetry is responsible
for the protection of the exceptional lines. Its action on the Hamiltonian is
given by

$UH^{\dagger}(\mathbf{k})U^{\dagger}=-H(\mathbf{k}),$ (10)

where $U$ is unitary. This means that the Hamiltonian is anti-pseudo-
hermitian, that is, equal to $iH_{PsH}$, for some pseudo-Hermitian $H_{PsH}$.
Pseudo-Hermitian Hamiltonians possess eigenvalues that are either completely
real or form complex conjugate pairs [33]. Thus, the eigenvalues of an anti-
pseudo-Hermitian Hamiltonian can be either purely imaginary or occur in pairs
with opposite real parts. This difference causes the system to divide into two
regions. The first region has zero real energy parts, while the second
displays opposite real energy parts. Notably, these regions are separated by
exceptional lines [33]. The anti-pseudo-Hermitian symmetry, therefore,
protects the Fermi areas where the real part of the energy vanishes.

Chiral symmetry is typically achieved through a combination of particle-hole
symmetry and time-reversal symmetry [34]. In our system, particle-hole
symmetry is inherently present due to the Majorana representation, as
described by the equation:

$\displaystyle\operatorname{\mathcal{C}}_{-}H^{T}(\mathbf{k})\operatorname{\mathcal{C}^{-1}}_{-}=-H(-\mathbf{k}),$
(11)

with $\operatorname{\mathcal{C}}_{-}=\text{diag}\left(1,1,1,1,1,1,1,1\right)$.

Conversely, the implementation of time-reversal symmetry, though standard, is
less straightforward compared to particle-hole symmetry. In the many-body
framework, time-reversal symmetry is defined by:

$\displaystyle\tilde{\operatorname{\mathcal{T}}}_{+}\mathcal{H}\tilde{\operatorname{\mathcal{T}}}_{+}^{-1}=\mathcal{H},$
(12)

where $\mathcal{H}$ is the second quantized Hamiltonian and
$\tilde{\operatorname{\mathcal{T}}}_{+}$ is an anti-unitary operator. This
operator acts on the creation operators $c_{j}$ as follows:

$\displaystyle\tilde{\operatorname{\mathcal{T}}}_{+}c_{j}\tilde{\operatorname{\mathcal{T}}}_{+}^{-1}=\begin{cases}c_{j},\quad
j\in A\\\ -c_{j},\quad j\in B.\end{cases}$

Applying this to the many-body Hamiltonian, $\mathcal{H}=i\sum
J_{\gamma}u_{jk}c_{j}c_{k}$ yields:

$\displaystyle\tilde{\operatorname{\mathcal{T}}}_{+}\mathcal{H}\tilde{\operatorname{\mathcal{T}}}_{+}^{-1}=i\sum
J_{\gamma}^{*}u_{jk}c_{j}c_{k}\neq\mathcal{H}\text{ for }J_{\gamma}\neq
J_{\gamma}^{*},$ (13)

and therefore any non-Hermiticity we consider will break this symmetry.
However, in our model, the symmetry can be restored by setting $\theta=-\phi$
and supplementing time reversal with a half-unit cell translation:

$\displaystyle(\operatorname{\mathcal{T}_{+}}U_{1/2})H^{*}(-\mathbf{k})(\operatorname{\mathcal{T}_{+}}U_{1/2})^{-1}=H(\mathbf{k}),$
(14)

where

$\displaystyle\operatorname{\mathcal{T}_{+}}=\text{diag}\left(1,-1,1,-1,-1,1,-1,1\right)$
(15)

and

$\displaystyle U_{1/2}=\left(\begin{array}[]{cc}0&\mathds{1}_{4\times 4}\\\
e^{i(k_{1}+k_{2})}\mathds{1}_{4\times 4}&0\\\ \end{array}\right).$ (18)

Combining Eq. (14) with particle-hole symmetry from Eq. (11) we obtain the
symmetry (10) with
$U=\operatorname{\mathcal{C}}_{-}\operatorname{\mathcal{T}_{+}}U_{1/2}$ which
protects the Fermi area. The particle-hole symmetry is inherent in the
Majorana formulation, so the protection of the Fermi area hinges on the
combined half-translation and time-reversal symmetry.

There are various variations of Hamiltonians (9) that keep above symmetry and,
thus, exhibit exceptional lines. Obviously, we could have added phases with
opposite signs for other bonds, e.g., between sites 2 and 3, and between 6 and
7. Another alternative is to make the hopping strength non-reciprocal, e.g.,
by changing the hopping from sites 1 and 7 to $J_{z}(1+\delta_{1})$, the one
from sites 3 and 5 to $J_{z}(1+\delta_{1}^{*})$, while keeping the hopping
from 7 to 1, as well as from 5 to 3, unchanged. Not all of the possibilities
to keep the symmetry (10) are physical in the system at hand, but they may be
relevant for other realizations. In the next subsection, we discuss another
way to make the Hamiltonian non-Hermitian while retaining (10) that at first
glance seems unphysical, but is closely related to what happens when
considering disorder in the model.

We can illustrate Fermi areas by observing how chiral symmetry maps states
within these areas to themselves, while states outside are mapped to
orthogonal states. This results in an overlap quantized to either 1 or 0. The
formula for calculating this overlap is:

$\displaystyle\bra{\psi^{R}}U^{-1}\ket{\psi^{R}}\frac{\norm{\psi^{L}}}{\norm{U^{-1}\ket{\psi^{R}}}},$
(19)

as outlined in Appendix B. This technique effectively differentiates between
regions inside and outside the Fermi area, offering a clear visualization of
the $Re[E]=0$ surfaces. This is illustrated in Fig. 4 (b), for adding the
onsite imaginary terms discussed in the following section. Additionally, Fig.9
in Appendix B showcases the self-symmetric states for the model Hamiltonian
(9) when $\phi=-\theta=1/2$.

###  II.3 On-site imaginary terms

In the following section, we study how non-Hermiticity appears in an effective
description when disorder is introduced. However, non-Hermitian perturbations
like the ones in Eq. (9) are difficult to obtain using this procedure.
Typically, non-Hermiticity in this context appears as imaginary on-site
potentials. Before addressing the full disorder calculation in the next
section, we will model this effect by setting $\phi=\theta=0$ in Eq. (9) and
adding a diagonal term

$\Sigma=i\textrm{diag}(s_{1},s_{2},...,s_{8}),$ (20)

where $s_{j}\in\mathbb{R}$. Note that diagonal terms such as this do not make
sense from the Majorana perspective, as $isc_{j}c_{j}=0$ [35]. However, as we
will see later, they appear in the Green’s function description and have
relevant consequences.

Both particle-hole symmetry and time-reversal symmetry split into two distinct
symmetry classes for non-Hermitian systems [34], due to a distinction between
transposition and complex conjugation for non-Hermitian operators. Adding
imaginary terms to the diagonal breaks both time-reversal and particle-hole
symmetry as defined in Sec. II.2, but it does not break the transpose type
symmetries TRS†, given by

$\displaystyle\operatorname{\mathcal{C}}_{+}H^{T}(\mathbf{k})\operatorname{\mathcal{C}^{-1}}_{+}=H(-\mathbf{k})$
$\displaystyle\operatorname{\mathcal{C}}_{+}=\text{diag}\left(1,-1,1,-1,-1,1,-1,1\right),$
(21)

or PHS†, given by

$\displaystyle\operatorname{\mathcal{T}}_{-}H^{*}(\mathbf{k})\operatorname{\mathcal{T}^{-1}}_{-}=-H(-\mathbf{k})$
$\displaystyle\operatorname{\mathcal{T}}_{-}=\text{diag}\left(1,1,1,1,1,1,1,1\right).$
(22)

The combination of both symmetries results again in a chiral symmetry required
to protect Fermi areas,

$\displaystyle UH^{\dagger}(\mathbf{k})U^{-1}=-H(\mathbf{k}),$ $\displaystyle
U=\operatorname{\mathcal{C}}_{+}\operatorname{\mathcal{T}}_{-}.$ (23)

To split the Fermi line into exceptional lines, we need to, as discussed in
Sec. II.1, either break time-reversal symmetry or the translation of the
original 4-site unit cell model. Since adding imaginary terms to the diagonal
preserves a form of time-reversal symmetry, it becomes necessary to break the
half-translation symmetry $U_{1/2}$. The half-translation symmetry is broken
when the onsite term is of different strength on different squares, that is,
$s_{j}\neq s_{j+4}$. The extent of the Fermi area depends on the degree to
which the half-translation symmetry is broken. By choosing
$\Sigma=is\times\text{diag}\left(1,1,1,1,0,0,0,0\right)$, the effect is
maximized, but the results remain qualitatively the same regardless of the
specific choice of $\Sigma$, as long as the original translation symmetry
$U_{1/2}$ is broken. The spectrum resulting from this non-Hermitian term is
shown in Fig. 4(a), using $s=1/2$. The system hosts Fermi areas with $Re[E]=0$
that are protected by the symmetry in Eq. (23), as illustrated in Fig. 4(b).

Figure 4:  (a) Real part of the energy gap for the non-Hermitian model with
on-site imaginary terms on one square with $s=1/2$, showing Fermi areas. (b)
The corresponding self-symmetric states from Eq. (19).

##  III Disorder

In this section, we investigate the impact of introducing disorder to the
nearest-neighbor bond hoppings in the Hermitian KSL on a square-octagon
lattice. Our analysis is confined to disordering the hoppings along bonds that
lie within the 8-site unit cell. While this assumption is not essential for
the effect, it makes the analysis simpler. Including only nearest neighbor
hopping disorder ensures that time-reversal symmetry, in the form of
Eq.(II.3), remains intact, which is crucial for protecting the Fermi area.

The Hamiltonian for this system can be expressed as:

$\displaystyle H=$ $\displaystyle H_{0}+V$ $\displaystyle=$
$\displaystyle\sum_{\bm{r}\bm{r}^{\prime},\alpha\beta.}t_{\alpha\beta}(\bm{r}-\bm{r}^{\prime})\gamma_{\alpha}(\bm{r})\gamma_{\beta}(\bm{r}^{\prime})$
$\displaystyle+$
$\displaystyle\sum_{\bm{r},\alpha\beta}V_{\alpha\beta}(\bm{r})\gamma_{\alpha}(\bm{r})\gamma_{\beta}(\bm{r}),$
(24)

where $\bm{r}$ denotes the unit cell, and $\alpha,\beta,\dots$ represents the
position within the unit cell. The term
$t_{\alpha\beta}(\bm{r}-\bm{r}^{\prime})$ denotes the hopping strength between
sites $\alpha$ and $\beta$ in the Hamiltonian. The disorder term
$V_{\alpha\beta}(\bm{r})$ characterizes the randomness in the hopping along
the bond between sites $\alpha$ and $\beta$. The hopping, and thus also the
disorder $V$, is only nonzero for nearest neighbors in our model. The average
value of this disorder term is assumed to be zero, i.e., $\langle
V_{\alpha\beta}\rangle=0$. This assumption is based on the premise that any
non-zero average disorder could be effectively incorporated into the hopping
parameters $t_{\alpha\beta}$, thus altering the baseline hopping strengths in
the Hamiltonian. Note that equal hopping strength $J_{x}=J_{y}=J_{z}$ is not
at all important for our analysis.

The KSL with disorder can be effectively described by a non-Hermitian
Hamiltonian using the Feshbach projection [17, 32]. The Green’s function in
the momentum space of a disordered system is given by

$\displaystyle
G(\mathbf{k},\omega)=\frac{1}{\omega-(H_{0}(\mathbf{k})+\Sigma(\mathbf{k},\omega))}.$
(25)

Here $\Sigma(\mathbf{k},\omega)$ is the self-energy of the system, which
captures the information from the disorder. The effective Hamiltonian is given
by $H_{eff}(\mathbf{k},\omega)=H_{0}(\mathbf{k})+\Sigma(\mathbf{k},\omega)$,
which at low energies can be approximated as
$H_{eff}(\mathbf{k}):=H_{eff}(\mathbf{k},0)$, which in general is a non-
Hermitian operator.

The self-energy is formally given by

$\displaystyle\Sigma(\mathbf{k},\omega)=g^{-1}(\mathbf{k},\omega)-G^{-1}(\mathbf{k},\omega),$
(26)

where $g$ is the retarded Green’s function for the system without disorder.
Employing the self-consistent Born approximation [32] reveals the self-energy
resulting from this disorder. Importantly, this approximation gives an
iterative method to calculate the self-energy. In Appendix A we show that the
self-energy for disorder within one unit cell can be expressed as:

$\displaystyle\Sigma_{\alpha\beta}(\mathbb{k},\omega)=\sum_{\lambda\delta}F_{\alpha\lambda\delta\beta}G_{\lambda\delta}(\mathbb{k},\omega),$
(27)

where
$F_{\alpha\lambda\delta\beta}=\left<V_{\alpha\lambda}V_{\delta\beta}\right>$
captures the characteristics of the disorder.

Figure 5: Spectral function at zero frequency, showing the annulus Fermi area.
The red line shows the cut in momentum space used in Fig. 6. Figure 6:
Spectral function of the disordered system at $k_{1}=\frac{\pi}{2}$ and the
real part of energy bands of the effective Hamiltonian in red. The Fermi area
can be seen as the flat regions at zero frequency.

We now describe one way to obtain Fermi areas in the effective Hamiltonian in
a similar way as in the toy model with on-site imaginary terms. By adding
random disorder on the x- and y-bonds, one gets a non-Hermitian contribution
to the effective Hamiltonian only on the diagonal. One can break the half-
translation symmetry by having a different disorder strength on the two
squares of the unit cell, which breaks the Fermi line into exceptional lines.
The Fermi area is, therefore, protected by a symmetry of the form of equation
(23), and the area of the surface is related to the difference in disorder
strength on the two squares, e.g., how strong the breaking of translation
symmetry is. The Fermi area forms an annulus enclosed by exceptional lines.

The spectral function, defined as
$A(\mathbb{k},\omega)=-2\imaginary\Tr(G(\mathbb{k},\omega))$, shows the
density of states [19]. In Fig. 5 we show the spectral function at $\omega=0$,
which shows the Fermi area as an annulus in momentum space with a high density
of states. The red line in Fig. 5 indicates the cut in momentum space at
$k_{1}=\frac{\pi}{2}$ that is used in Fig. 6 to show the spectral function at
different frequencies and the real part of the eigenvalues of $H_{eff}$ (red
line). Here one can see the Fermi area as a flat region at zero frequency.

##  IV 3D Model Hamiltonian

Up to this point, we solely discussed two-dimensional systems. But the same
physics plays out in a similar fashion for three-dimensional ones. Here, the
starting point is a system with a doubly-degenerate Majorana Fermi surface.
These can be realized in a variety of Kitaev spin liquids [10], but also in
metals [36] or superconductors [37] in the presence of inversion symmetry.
Making the system non-Hermitian while retaining the symmetry (10) splits the
Fermi surfaces into two exceptional surfaces that now delimit a volume of
$\real[E]=0$ states. A suitable model system to illustrate this is the Kitaev
spin liquid on the hyperoctagon lattice [13]. The following discussion is very
similar to the two-dimensional case. Its main objective is to showcase an
explicit, three-dimensional example.

The KSL on the hyperoctagon lattice harbors two Fermi surfaces that are again
perfectly nested. The Fermi surfaces are again stable against any
translationally invariant two-Majorana term (with respect to the standard
4-site unit cell). The perfect nesting is protected by time-reversal symmetry.
We now proceed as before, we double the unit cell and introduce a non-
Hermitian perturbation that breaks the original translation symmetry in such a
way that the symmetry (10) remains intact.

For the sake of an explicit example, we consider imaginary onsite terms, as in
Sec. II.3, for which the corresponding Bloch Hamiltonian is given by

$\left(\begin{array}[]{cccccccc}is&-iJ_{y}&0&-iJ_{z}e^{-ik_{1}}&0&0&-iJ_{x}e^{-i(k_{1}+k_{2})}&0\\\
iJ_{y}&0&iJ_{z}&0&0&0&0&-iJ_{x}e^{-i(k_{1}+k_{2}+k_{3})}\\\
0&-iJ_{z}&is&iJ_{y}&iJ_{x}e^{-ik_{3}}&0&0&0\\\
iJ_{z}e^{ik_{1}}&0&-iJ_{y}&0&0&iJ_{x}&0&0\\\
0&0&-iJ_{x}e^{ik_{3}}&0&0&-iJ_{y}&0&-iJ_{z}e^{-ik_{1}}\\\
0&0&0&-iJ_{x}&iJ_{y}&is&iJ_{z}&0\\\
iJ_{x}e^{i(k_{1}+k_{2})}&0&0&0&0&-iJ_{z}&0&iJ_{y}\\\
0&iJ_{x}e^{i(k_{1}+k_{2}+k_{3})}&0&0&iJ_{z}e^{ik_{1}}&0&-iJ_{y}&is,\\\
\end{array}\right)$ (28)

where the diagonal terms correspond to the non-Hermitian perturbation, with
strength $s$. Such a scenario is again, as seen in Sec. III, related to the
effective Hamiltonian of a disordered system. However, since disorder
calculations are prohibitively expensive for three-dimensional systems, we
restrict the discussion to these model systems.

Figure 7: Volume of states at $\real[E]=0$ for the model in Eq. (28) with
$s=1/4$.

This Hamiltonian has the same non-spatial symmetries as the two-dimensional
model with on-site imaginary terms studied above, TRS†

$\displaystyle\operatorname{\mathcal{C}}_{+}H^{T}(\mathbf{k})\operatorname{\mathcal{C}^{-1}}_{+}=H(-\mathbf{k})$
$\displaystyle\operatorname{\mathcal{C}}_{+}=\text{diag}\left(1,-1,1,-1,-1,1,-1,1\right),$
$\displaystyle\\{\epsilon(\mathbf{k})\\}=\\{\epsilon(-\mathbf{k})\\},$ (29)

and PHS†,

$\displaystyle\operatorname{\mathcal{T}}_{-}H^{*}(\mathbf{k})\operatorname{\mathcal{T}^{-1}}_{-}=-H(-\mathbf{k})$
$\displaystyle\operatorname{\mathcal{T}}_{-}=\text{diag}\left(1,1,1,1,1,1,1,1\right),$
$\displaystyle\\{\epsilon^{*}(\mathbf{k})\\}=\\{-\epsilon(-\mathbf{k})\\}.$
(30)

Combining both of these symmetries results again in a chiral symmetry that can
protect volumes with $Re(E)=0$,

$\displaystyle UH^{\dagger}(\mathbf{k})U^{-1}=-H(\mathbf{k}),$ $\displaystyle
U=\operatorname{\mathcal{C}}_{+}\operatorname{\mathcal{T}}_{-}.$ (31)

In Fig. 7 we show that this is the case, for the model in Eq. (28) with
$s=1/4$. A cut has been made in the plot to better visualize that the region
is indeed a volume in $\mathbf{k}$ space.

Note that, as in the square-octagon case, we have various ways to build non-
Hermitian systems with such exceptional surfaces. We could instead have
introduced phases on the hoppings or introduced nonreciprocal hoppings, as in
Sec. II.1. In that case, the chiral symmetry requires an explicit half-
translation $U_{1/2}$, which is obtained from Eq. (18) by replacing
$e^{i(k_{1}+k_{2})}\rightarrow e^{i(k_{1}+k_{2}+k_{3})}$. The symmetry in (10)
is fulfilled with
$U=\operatorname{\mathcal{C}}_{-}\operatorname{\mathcal{T}_{+}}U_{1/2}$, with
$\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{T}_{+}}$ as given
in (11) and (15), respectively.

##  V Conclusions

We examined the effects of (weak) disorder on gapless Kitaev spin liquids in
the square-octagon lattice in two dimensions and the hyperoctagon lattice in
three dimensions, employing an effective non-Hermitian Hamiltonian derived
from Green’s function formalism. Unlike the work done in [32], where they
considered a lattice with a Dirac point, we studied lattices where the
codimension of gapless points is 1, that is, lines in two dimensions and
surfaces in three dimensions.

The effect of disorder crucially depends on the kinds of symmetries that are
either retained or broken on average. Disorder that does not break translation
symmetry on average will, of course, result in a Majorana metal, reflecting
the stability of the Majorana Fermi surface. Interesting effects appear when
we consider disorder that induces ‘dimerization’ in the system and treats the
two squares in the 8-site unit cell differently. Chiral symmetry protects
pairs of exceptional lines that are connected by a Fermi area at $Re[E]=0$. In
this manuscript, we discuss the case where chiral symmetry is achieved by
combining time-reversal and particle-hole symmetry. However, for the square-
octagon model, we could have equally well replaced time-reversal with
inversion. Broken chiral symmetry results in isolated exceptional points if
sublattice symmetry is retained, or in Fermi lines if it is not. The latter
scenario can be most easily understood by noting that sublattice symmetry pins
the exceptional points to $Re[E]=0$. When it is also broken, the exceptional
points move up or down in $Re[E]$, and the Fermi arcs of Fig. 3 evolve into
Fermi lines.

There has been an extensive amount of work has been done on studying the
topological properties of non-Hermitian systems in the last couple of years.
One such topological phenomenon is the non-Hermitian skin effect [38, 39, 40,
41, 42], where the bulk states are localized at the boundary of the system.
The existence of isolated exceptional points directly leads to the skin effect
[16]. Disorder that breaks time-reversal symmetry will therefore lead to an
accumulation of modes on the boundary. Disorder that preserves time-reversal
symmetry could therefore be clearly distinguished from symmetry-breaking
disorder by studying the boundary of the system. Future work into effective
non-Hermitian Hamiltonians could therefore give new insights into the
topological phenomena of quantum spin liquids.

_Acknowledgements_ — The research in this grant was supported by the Swedish
Research Council under grant no. 2017-05162 and the Knut and Alice Wallenberg
Foundation under grant no. 2017.0157.

##  Appendix A Self-consistent Born approximation

Here we derive the expression in equation (27), which is the self-energy for a
system with disorder. For the sake of simplicity, we assume that the disorder
only acts on bonds within the unit cell, not on those coupling neighboring
unit cells. The retarded Green’s function is defined as

$\displaystyle
G^{r}_{\alpha\beta}(t,t^{\prime};\bm{r},\bm{r}^{\prime})=-i\expectationvalue{\\{\gamma_{\alpha}(\bm{r},t),\gamma_{\beta}(\bm{r}^{\prime},t^{\prime})\\}}\theta(t-t^{\prime}).$
(32)

Before proceeding, it is convenient to compute the time evolution of the
Majorana operators

$\displaystyle i\partial_{t}\gamma_{\alpha}(\bm{r},t)=$
$\displaystyle-[H,\gamma_{\alpha}(\bm{r})](t)$ $\displaystyle=$ $\displaystyle
t_{\beta\lambda}(\bm{r}^{\prime}-\bm{r}^{\prime\prime})[\gamma_{\alpha}(\bm{r}),\gamma_{\beta}(\bm{r}^{\prime})\gamma_{\lambda}(\bm{r}^{\prime\prime})](t)$
$\displaystyle+V_{\beta\lambda}(\bm{r}^{\prime})[\gamma_{\alpha}(\bm{r}),\gamma_{\beta}(\bm{r}^{\prime})\gamma_{\lambda}(\bm{r}^{\prime})](t)$
$\displaystyle=$ $\displaystyle
2\gamma_{\beta}(\bm{r}^{\prime})\left(t_{\alpha\beta}(\bm{r}-\bm{r}^{\prime})-t_{\beta\alpha}(\bm{r}^{\prime}-\bm{r})\right)$
$\displaystyle+$ $\displaystyle
2\gamma_{\beta}(\bm{r^{\prime}})\left(V_{\alpha\beta}-V_{\beta\alpha}\right)\delta(\bm{r}-\bm{r^{\prime}})$
$\displaystyle=$
$\displaystyle\gamma_{\beta}(\bm{r}^{\prime})\left(h^{t}_{\alpha\beta}(\bm{r},\bm{r}^{\prime})+h^{V}_{\alpha\beta}(\bm{r},\bm{r}^{\prime})\right)$
$\displaystyle=$
$\displaystyle\gamma_{\beta}(\bm{r}^{\prime})h_{\alpha\beta}(\bm{r},\bm{r}^{\prime}).$
(33)

Using the definition of the Green’s function and the evolution of Majorana
operators, we can calculate the time evolution of the retarded Green’s
function

$\displaystyle
i\partial_{t}G^{r}_{\alpha\beta}(t,t^{\prime};\bm{r},\bm{r^{\prime}})$
$\displaystyle=\delta(t-t^{\prime})\delta(\bm{r}-\bm{r^{\prime}})\delta_{\alpha\beta}$
$\displaystyle+h_{\alpha\nu}(\bm{r},\bm{r^{\prime\prime}})G^{r}_{\nu\beta}(t,t^{\prime};\bm{r^{\prime\prime}},\bm{r^{\prime}}),$
(34)

which can be rewritten as the Dyson equation

$\displaystyle
G^{r}_{\nu\beta}(t^{\prime\prime},t^{\prime};\bm{r^{\prime\prime\prime}},\bm{r^{\prime}})=g^{r}_{\nu\beta}(t^{\prime\prime},t^{\prime};\bm{r^{\prime\prime\prime}},\bm{r^{\prime}})$
$\displaystyle+\int
g^{r}_{\nu\alpha}(t^{\prime\prime},t;\bm{r^{\prime\prime\prime}},\bm{r})h^{V}_{\alpha\lambda}(\bm{r,r^{\prime\prime\prime}})G^{r}_{\lambda\beta}(t,t^{\prime};\bm{r^{\prime\prime}},\bm{r^{\prime}})dt\,.$
(35)

By Fourier transforming into the frequency domain we obtain

$\displaystyle G^{r}_{\alpha\beta}$
$\displaystyle(\omega;\bm{r},\bm{r}^{\prime})=g^{r}_{\alpha\beta}(\omega;\bm{r},\bm{r}^{\prime})$
$\displaystyle+\sum_{\gamma\lambda;\bm{r^{\prime\prime}}}\,g^{r}_{\alpha\gamma}(\omega;\bm{r},\bm{r}^{\prime\prime})h^{V}_{\gamma\lambda}(\bm{r^{\prime\prime}})G^{r}_{\lambda\beta}(\omega;\bm{r^{\prime\prime}},\bm{r}^{\prime}).$
(36)

By iteratively expanding the Dyson equation we can write the average of the
retarded Green’s function as

$\displaystyle\expectationvalue{G}=\expectationvalue{g}+\expectationvalue{gh^{V}gh^{V}G},$
(37)

where we have set the term linear in $h^{V}$ to zero since it can be absorbed
into the hopping strength. Multiplying with $g^{-1}$ from the left and
$G^{-1}$ from the right gives us the self energy

$\displaystyle\begin{aligned}
\Sigma_{\alpha\beta}(\bm{r-r^{\prime}})&=\expectationvalue{h^{V}gh^{V}}\\\
&=\expectationvalue{h^{V}_{\alpha\lambda}(\bm{r,r^{\prime\prime}})h^{V}_{\delta\beta}(\bm{r^{\prime\prime\prime},r^{\prime}})}g_{\lambda\delta}(\omega,\bm{r^{\prime\prime}-r^{\prime\prime\prime}})\\\
&=F_{\alpha\lambda\delta\beta}g_{\lambda\delta}(\omega,\bm{r-r^{\prime}}),\end{aligned}$
(38)

where we used that $h^{V}$ depends on the position as a delta function. This
can be used to calculate the self-energy iteratively by using a self-
consistent Born approximation:

$\displaystyle\Sigma_{\alpha\beta}(\bm{r-r^{\prime}})=F_{\alpha\lambda\delta\beta}G^{r}_{\lambda\delta}(\omega,\bm{r-r^{\prime}}).$
(39)

The constants $F_{\alpha\beta\alpha\beta}$ are defined as the correlation
between the disorder between different bonds

$\displaystyle
F_{\alpha\lambda\delta\beta}=16\expectationvalue{V_{\alpha\lambda}(\bm{r})V_{\delta\beta}(\bm{r^{\prime}})}.$
(40)

We also assume that the correlations in Eq. (40) vanish for bonds in different
unit cells, i.e., $\mathbf{r}\neq\mathbf{r^{\prime}}$. From this definition,
one can readily see that $F$ acquires a minus sign when flipping one of the
bond directions $F_{\alpha\beta\gamma\delta}=-F_{\beta\alpha\gamma\delta}$,
and that swapping bonds is equivalent to complex conjugation
$F^{*}_{\alpha\beta\gamma\delta}=F_{\gamma\delta\alpha\beta}$. For all bonds
except adjacent ones, the disorder operators commute, and one finds
$F_{\alpha\beta\gamma\delta}=F_{\gamma\delta\alpha\beta}$. For adjacent bonds,
we get an extra minus sign from the swapping due to the Majoranas, such that
$F_{\alpha\beta\beta\delta}$ must be imaginary.

Going to momentum space we write the self-energy as

$\displaystyle\Sigma_{\alpha\beta}(\bm{k})=F_{\alpha\lambda\delta\beta}G^{r}_{\lambda\delta}(\omega,\bm{k}),$
(41)

which can be used in a self-consistent Born approximation to iteratively
calculate the Green’s function of the system. The spectral function, which
carries information about the density of states, is related to the Green’s
function as

$\displaystyle A(\mathbb{k},\omega)=-2\imaginary\Tr(G(\mathbb{k},\omega)).$
(42)

The imaginary part of the eigenvalues of the effective Hamiltonian
$H_{eff}(k)=H_{0}(\mathbb{k})+\Sigma(\mathbb{k},0)$ broadens the single
particle states of the system without disorder.

To illustrate the effects of disorder In Figs. 6, 5 and 8 we used the
following values for the disorder strength

$\displaystyle F_{1212}=-0.12,\quad F_{2323}=-0.0533,$ $\displaystyle
F_{3434}=-0.03,\quad F_{1414}=-0.0833,$ $\displaystyle F_{1234}=0.016133,\quad
F_{1423}=-0.0133,$ $\displaystyle F_{1214}=-0.05i,\quad F_{2334}=0.04i$
$\displaystyle F_{1423}=-0.06i,\quad F_{1434}=0.03i,$ (43)

where all the terms on the square consisting of sites 5, 6, 7, and 8 are set
to zero, to break translation symmetry. Where the values are chosen randomly
to avoid any accidental symmetries.

Figure 8: The real part of the energy difference between the lowest two bands.

##  Appendix B Anti-pseudo-Hermitian symmetry

Here, we derive Eq. (19) that shows that chiral symmetry maps eigenstates to
themselves for momenta within the Fermi area and to orthogonal states for
momenta outside of the Fermi area. Eq. (19) is used to visualize the Fermi
area in Fig. 9 and Fig. 4 b)

Figure 9:  Self-symmetric states obtained from Eq. (19) shows the Fermi area
of the model with $\phi=1/2=-\theta$.

Starting from the symmetry (10) and acting on a right eigenstate of the non-
Hermitian Hamiltonian, one finds

$\displaystyle
UH^{\dagger}(\mathbf{k})U^{\dagger}\ket{\psi^{R}_{\mu}}=-E_{\mu}\ket{\psi^{R}_{\mu}}..$
(44)

Acting with $U^{\dagger}$ on the left we obtain

$\displaystyle
H^{\dagger}(\mathbf{k})U^{\dagger}\ket{\psi^{R}_{\mu}}=-E_{\mu}U^{\dagger}\ket{\psi^{R}_{\mu}},$
(45)

or equivalently

$\displaystyle\left(\bra{\psi^{R}_{\mu}}U\right)H^{\dagger}(\mathbf{k})=-E_{\mu}^{*}\left(\bra{\psi^{R}_{\mu}}U\right).$
(46)

The eigenvalue equation above relates, up to a constant, each right eigenstate
to a left eigenstate, which we assume biorthogonalized,

$\displaystyle U^{\dagger}\ket{\psi^{R}_{\mu}}=z\ket{\psi^{L}_{\nu}},$ (47)

with the opposite real part of the energy, $E_{\nu}=-E_{\mu}^{*}$. The complex
number $z$ appears because the eigenstates themselves are not normalized, and
in general, $\mu\neq\nu$. However, if the eigenstate is itself symmetric,
$\mu=\nu$, its energy has to fulfill $E_{\mu}=-E_{\mu}^{*}$ or in other words,
$\real[E_{\mu}]=0$.

To confirm that the Fermi area states are protected by the symmetry, we
compute the overlap

$\displaystyle\bra{\psi^{R}}U^{-1}\ket{\psi^{R}}\frac{\norm{\psi^{L}}}{\norm{U^{-1}\ket{\psi^{R}}}}$
(48)

which is quantized to $0$ for general states and $1$ for the self-symmetric
states in the Fermi area. The normalization factor is needed to get rid of the
constant $z$.

##  Appendix C Short introduction to the Kitaev model

The Kitaev model introduced in Ref. [6] is one of the rare instances of an
exactly solvable spin model that harbors a variety of interesting quantum spin
liquid phases. It features spin-$\frac{1}{2}$ local moments on a tri-
coordinated lattice (originally the honeycomb lattice), interacting with their
three nearest neighbors via a bond-directional Ising interaction:

$\displaystyle\hat{H}$ $\displaystyle=\sum_{\langle
i,j\rangle\in\gamma-\mbox{\tiny
bonds}}J_{\gamma}\sigma_{i}^{\gamma}\sigma_{j}^{\gamma},$ (49)

i.e., depending on the bond direction, the spins aim to align their
$\hat{x}$,$\hat{y}$ or $\hat{z}$ components, see Fig. 1 for an illustration of
the square-octagon lattice. In the following, we will refer to a nearest-
neighbor interaction Ising interaction with Ising axis $\hat{x}$ as an
$x$-bond, and similarly for $\hat{y}$ and $\hat{z}$. A very thorough and
pedagogical treatment of how to solve the Kitaev model in general can be found
in Kitaev’s original paper [6]. Here, we will review the most important steps
and the relevant physics emerging on the square-octagon lattice.

The model is exactly solvable on any tri-coordinated lattice as long as every
spin interacts with its neighbors via exactly one $x$, one $y$, and one $z$
bond. This ensures that there are a macroscopic number of conserved
quantities, one for each plaquette of the lattice. We can define a plaquette
operator

$\displaystyle\hat{W}_{p}$ $\displaystyle=\prod_{\langle
i,j\rangle_{\gamma}\in p}\sigma_{i}^{\gamma}\sigma_{j}^{\gamma},$ (50)

where $\langle i,j\rangle_{\gamma}$ are the bonds in the plaquette $p$. For
instance, for the square consisting of atoms 1 to 4 in Fig. 1, we find the
plaquette operator

$\displaystyle\hat{W}$
$\displaystyle=(\sigma_{1}^{x}\sigma_{2}^{x})(\sigma_{2}^{y}\sigma_{3}^{y})(\sigma_{3}^{x}\sigma_{4}^{x})(\sigma_{4}^{y}\sigma_{1}^{y}).$
(51)

Note that the plaquette operators commute with each other, as well as with the
Hamiltonian. Thus, for each plaquette in the lattice, we find a conserved
quantity. Since $\hat{W}_{p}^{2}=1$ for plaquettes of even length, we can
identify the corresponding eigenvalue $w_{p}$ with a $Z_{2}$ flux through the
plaquette. Determining which flux configuration allows the lowest energy is,
in general, a non-trivial problem that requires extensive numerics [43]. For
certain lattices, however, including the square-octagon lattice, a theorem by
Lieb [44] can be used to determine that the ground state sector is obtained by
choosing $w_{p}=1$ (i.e., 0 flux) for plaquettes of length 2 mod 4, whereas
plaquettes of length 0 mod 4 require $w_{p}=-1$ (i.e., $\pi$ flux). For the
square-octagon lattice, Lieb’s theorem can be used and the lowest flux sector
has $\pi$ flux through each square and octagon [9]. The hyperoctagon lattice,
whose plaquettes all have length 10, does not fulfill the requirements for
Lieb’s theorem. Nevertheless, the ground state resides in the zero flux
sector, see, e.g., [45].

A standard way to solve the Kitaev model is to rewrite the spins in terms of
Majorana fermions

$\displaystyle\sigma_{j}^{\gamma}$ $\displaystyle=ia_{j}^{\gamma}c_{j}$ (52)

where we introduced 4 Majorana fermions: $a_{j}^{x}$, $a_{j}^{y}$,
$a_{j}^{z}$, and $c_{j}$ per site $j$. This enlarges the Hilbert space, and
one needs to project to the physical subspace by enforcing
$a_{j}^{x}a_{j}^{y}a_{j}^{z}c_{j}|\xi\rangle=|\xi\rangle$ for all physical
states $|\xi\rangle$ and all sites $j$. For a thorough discussion of that
issue, we refer the reader to [46].

Introducing a $Z_{2}$ variable $\hat{u}_{j,k}=ia_{j}^{\gamma}a_{k}^{\gamma}$
for each nearest neighbor bond $\langle j,k\rangle_{\gamma}$ of type $\gamma$,
we can then rewrite the Hamiltonian (49) as

$\displaystyle\hat{H}$ $\displaystyle=\sum_{\langle
j,k\rangle\in\gamma-\mbox{\tiny bonds}}ic_{j}c_{k}\hat{u}_{j,k}.$ (53)

Note that the $\hat{u}_{j,k}$’s commute with each other and with the
Hamiltonian. In fact, they are directly related to the conserved quantities
(50) introduced earlier. We can thus interpret the effective Hamiltonian as
Majorana fermions hopping in a _static_ emergent $Z_{2}$ gauge field (which
moreover turns out to be gapped!). The latter is a non-interacting system and
can easily be solved.

For the square-octagon, the ground state sector has $\pi$ flux per plaquette,
and the corresponding Majorana system is gapped, see Ref. [9]. However, by
adding additional terms to the Hamiltonian that penalize $\pi$-flux, one can
instead force the $0$-flux sector to be the ground state, see Ref. [47]. The
latter harbors a Majorana metal, with two Majorana Fermi lines. Time-reversal
symmetry and/or inversion symmetry ensures that the two surfaces are perfectly
nested and can be translated onto each other by half a reciprocal lattice
vector (see [10] for a general discussion of such ‘projective symmetries’).
Enlarging the unit cell from containing 4 to 8 sites (see Fig. 1), we obtain a
system with a single, doubly-degenerate Fermi line — the starting point in the
discussion of Sec. II.

The Kitaev model on the three-dimensional hyperoctagon lattice was discussed
in detail in Ref. [13]. For further information about the lattice itself, we
refer the reader to the original publication. For this manuscript, it is
sufficient to know that the ground state resides in the zero flux sector and
harbors two Majorana Fermi surfaces that are perfectly nested. The perfect
nesting condition can only be destroyed by breaking time-reversal symmetry
[10]. Similarly to the square-octagon model above, we can map the Fermi
surfaces onto each other by enlarging the unit cell to contain 8 sites instead
of the original 4. This doubly-degenerate Majorana Fermi surface is the
starting point of the discussion in Sec. IV.

##  Appendix D Symmetries of the square-octagon model

As the square-octagon model has quite a number of symmetries, which moreover
are implemented differently for the 4 vs the 8-site unit cell, we will give a
short overview of the relevant symmetries and their physical consequences.

###  D.1 4-site unit cell

The square-octagon lattice is a bi-partite lattice. However, the bi-
partiteness is not compatible with the 4-site unit cell, in the sense that if
we choose e.g. sites 1 and 3 to be sites in the A-sublattice for one unit
cell, in all the four neighboring unit cells sites 1 and 3 need to be in the
B-sublattice. The Kitaev model on a bi-partite lattice is invariant under
sublattice symmetry, which is implemented by

$\displaystyle c_{j}(\mathbf{r})$
$\displaystyle\rightarrow\left\\{\begin{array}[]{cc}c_{j}(\mathbf{r})&\mbox{
for $\mathbf{r}\in$ sublattice A}\\\ -c_{j}(\mathbf{r})&\mbox{ for
$\mathbf{r}\in$ sublattice B}\end{array}\right..$ (56)

Since the bi-partiteness is not compatible with the 4-site unit cell of the
square-octagon lattice, the corresponding transformation becomes non-local in
momentum space:

$\displaystyle\hat{h}(\mathbf{k})$
$\displaystyle=-U_{SLS}\hat{h}(\mathbf{k}+\mathbf{k}_{0})U_{SLS}^{-1}$ (57)

where $\mathbf{k}_{0}$ is half a reciprocal lattice vector and $U_{SLS}$ is a
unitary matrix. This feature has profound consequences for the physical
properties of the Kitaev spin liquid as explained in Ref. [10]. In particular,
implementing time-reversal symmetry for Majorana systems requires a sublattice
transformation in the gauge sector. Thus, time-reversal inherits the same
transformation symmetries from sublattice symmetry:

$\displaystyle\hat{h}(\mathbf{k})$
$\displaystyle=U_{TR}\hat{h}^{*}(-\mathbf{k}+\mathbf{k}_{0})U_{TR}^{-1},$ (58)

where $U_{TR}$ is a unitary matrices. The non-trivial transformation under
time-reversal immediately implies that the stable zero modes for this model
form lines, i.e. the Majorana Fermi lines seen in Fig. 2.

Apart from several mirror symmetries, and a four-fold rotation symmetry — both
of which are not relevant for our discussion — the square-octagon model also
possesses inversion symmetry. The inversion center can sit in the middle of
the squares, the middle of the octagons, or the middle of the z-bonds
connecting two squares. For the 4-site unit cell, all these choices lead to
the same constraint on the Hamiltonian, namely

$\displaystyle\hat{h}(\mathbf{k})$
$\displaystyle=U_{I}\hat{h}(-\mathbf{k}+\mathbf{k}_{0})U_{I}^{-1},$ (59)

where $U_{I}$ is a unitary matrix. Both time-reversal and inversion symmetry
imply that if there is a zero-energy state at $\mathbf{k}$, there must be
another zero-energy state at $-\mathbf{k}+\mathbf{k}_{0}$, i.e. they map one
of the Majorana Fermi surfaces to the other. Only by breaking both these
symmetries can we destroy the perfect nesting condition.

###  D.2 8-site unit cell

We now assume that translation symmetry is partially broken, such that the
8-site unit cell is appropriate. This unit cell is compatible with the bi-
partiteness of the lattice, so both time-reversal and inversion will only map
$\mathbf{k}\rightarrow-\mathbf{k}$, without an additional translation in
$\mathbf{k}$-space. For the 8-site unit cell, the inversion symmetry with its
center on the z-bond is automatically broken, so we only consider those with
centers in the square/octagon.

In particular, time-reversal is now given by

$\displaystyle\hat{h}(\mathbf{k})$
$\displaystyle=U_{TR}\hat{h}^{*}(-\mathbf{k})U_{TR}^{-1}$ (60)

and inversion symmetry by

$\displaystyle\hat{h}(\mathbf{k})$
$\displaystyle=U_{I}\hat{h}(-\mathbf{k})U_{I}^{-1},$ (61)

where $U_{TR}$ and $U_{I}$ are now 8x8 unitary matrices. The general symmetry
analysis in Ref. [10] implies that this implementation of time-reversal has
stable zero modes at discrete points in $\mathbf{k}$-space, while lines of
zero modes are generically unstable. To understand this, notice that (60)
implies that the time-reversal invariant Hamiltonian can be brought into the
form

$\displaystyle\hat{h}(\mathbf{k})$
$\displaystyle=\left(\begin{array}[]{cc}\mathbf{0}&q(\mathbf{k})\\\
q^{\dagger}(\mathbf{k})&\mathbf{0}\end{array}\right).$ (64)

The zero modes of $\hat{h}$ are obtained by requiring
$det(\mathbf{q}(\mathbf{k}))=0$. The latter are in fact two constraints,
$Re[det(\mathbf{q})]=0$ and $Im[det(\mathbf{q})]=0$, both of which form closed
lines in $\mathbf{k}$-space. The zeroes of $\hat{h}$ are located at the
intersections of these lines. The existence, though not the position, of these
intersections are stable against all perturbations that do not break time-
reversal.

This is the general behavior, but the existence of inversion symmetry adds a
twist to the story. Inversion symmetry forces $det(\mathbf{q})$ to be real.
Thus, one of the constraints mentioned above is trivially satisfied, implying
that the stable zero modes instead form lines. One should note here, that the
zero mode lines for the 4-site unit cell arise from Majorana Fermi lines
crossing zero energy. Here, in contrast, the lines arise from two bands
touching at zero energy. The former situation is stable to any symmetry
breaking, while the latter needs inversion symmetry.

We also want to note another twist, this time arising from the non-hermitian
perturbation of the model. In the Hermitian system, breaking time-reversal
necessarily implies broken sublattice symmetry. This is no longer the case for
the non-Hermitian model. For instance, non-zero phases in Eq. (9) break time-
reversal, but not sublattice symmetry. Thus we can have the novel situation
where time-reversal is broken, but sublattice symmetry is intact. The off-
diagonal structure of the Hamiltonian in (64) relies in fact on sublattice
symmetry. For the non-Hermitian model of Eq. (9), above analysis about the
stable zero modes remains valid even in the absence of time-reversal. The
effective Hamiltonian of the disordered system, on the other hand, does break
sublattice symmetry explicitly, though not time-reversal.

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