# Topological phase diagrams of exactly solvable
non-Hermitian interacting Kitaev chains

Sharareh Sayyad<EMAIL_ADDRESS>Max Planck Institute for the
Science of Light, Staudtstraße 2, 91058 Erlangen, Germany  Jose L. Lado
Department of Applied Physics, Aalto University, FI-00076 Aalto, Espoo,
Finland

(August 28, 2024)

###### Abstract

Many-body interactions give rise to the appearance of exotic phases in
Hermitian physics. Despite their importance, many-body effects remain an open
problem in non-Hermitian physics due to the complexity of treating many-body
interactions. Here, we present a family of exact and numerical phase diagrams
for non-Hermitian interacting Kitaev chains. In particular, we establish the
exact phase boundaries for the dimerized Kitaev-Hubbard chain with complex-
valued Hubbard interactions. Our results reveal that some of the Hermitian
phases disappear as non-Hermiticty is enhanced. Based on our analytical
findings, we explore the regime of the model that goes beyond the solvable
regime, revealing regimes where non-Hermitian topological degeneracy remains.
The combination of our exact and numerical phase diagrams provides an
extensive description of a family of non-Hermitian interacting models. Our
results provide a stepping stone toward characterizing non-Hermitian topology
in realistic interacting quantum many-body systems.

## Introduction.

Many-body interactions play a crucial role in Hermitian quantum systems. The
emergent correlation effects in these systems give rise to a variety of
collective phenomena, such as spontaneous symmetry breaking Koma and Tasaki
(1994); Belyaev (2006); Brauner (2010); Dong  _et al._ (2017), phase
transitions Santos  _et al._ (2016); Heyl (2018); Carollo  _et al._ (2020);
Serwatka  _et al._ (2023), and the emergence of fractionalized quasiparticles
Zhang  _et al._ (2020); Hashisaka  _et al._ (2021); Kaskela and Lado (2021);
Wouters  _et al._ (2022). Understanding these rich phenomena often requires
the combination of analytical and numerical techniques due to the scarcity of
exact solutions for many-body models. Nonetheless, especially in one-
dimensional systems, analytical solutions in specific regimes are attainable
Gangadharaiah  _et al._ (2011); Katsura  _et al._ (2015); Ezawa (2017); Wang
_et al._ (2017); Miao  _et al._ (2017); Zvyagin (2022). Away from these
parameter regimes, employing various numerical methods Lin (1990); Vidal
(2004); Schollwöck (2005); Kotliar  _et al._ (2006); Gangadharaiah  _et al._
(2011); Stoudenmire  _et al._ (2011); Silvi  _et al._ (2019); Tuovinen (2021)
allows unveiling the underlying physics of many-body systems in generic
scenarios.

The presence of losses and dissipation in real systems provide natural
platforms realizing non-Hermitian models Esaki  _et al._ (2011); Malzard  _et
al._ (2015); Gong  _et al._ (2018); Yao and Wang (2018); Shen  _et al._
(2018); Kawabata  _et al._ (2019); Yokomizo and Murakami (2019); Zhou and Lee
(2019); Borgnia  _et al._ (2020); Ashida  _et al._ (2020). Non-Hermitian
quantum models have risen as a new paradigm to manipulate and interpret
various emergent phenomena Sayyad  _et al._ (2022a); Okuma and Sato (2023).
Here, Non-Hermiticity emerges as the effective description Prosen (2008); Lieu
_et al._ (2020); Sayyad  _et al._ (2021); Jr  _et al._ (2022); Yang  _et al._
(2022); Talkington and Claassen (2022); Starchl and Sieberer (2022); Gneiting
_et al._ (2022) of out-of-equilibrium and open quantum systems, e.g., in
superconducting qubits Chen  _et al._ (2021); Abbasi  _et al._ (2022); Chen
_et al._ (2022), giving rise to various phenomena absent in the Hermitian
counterpart. Paradigmatic examples are occurrence of various non-Hermitian
degeneracies Leykam  _et al._ (2017); Bergholtz  _et al._ (2021); Sayyad and
Kunst (2022); Sayyad  _et al._ (2022b); Sayyad (2022) and non-Hermitian (bulk)
skin states Song  _et al._ (2019); Zhang  _et al._ (2022). Both of these
phenomenologies are mainly explored in effectively single-particle non-
Hermitian Hamiltonians while non-Hermitian many-body effects have remained
relatively unexplored, partly due to limitations of numerical methodsFreund
(1992); Freund  _et al._ (1993); Guo  _et al._ (2022); Carden (2011); Zhang
and Dai (2016); Chen  _et al._ (2022). In particular, recent efforts have
addressed one-dimensional non-Hermitian fermionic Fukui and Kawakami (1998);
Buča  _et al._ (2020); Zhang and Song (2021); Nakagawa  _et al._ (2021);
Yoshida and Katsura (2022); Hyart and Lado (2022) and bosonic Yamamoto  _et
al._ (2022); Wang  _et al._ (2022) Hubbard models. Here, the non-Hermiticity
is incorporated by having nonreciprocal hopping Fukui and Kawakami (1998),
complex Hopping Zhang and Song (2021), or complex Hubbard interaction Buča
_et al._ (2020); Nakagawa  _et al._ (2021); Yoshida and Katsura (2022). The
latter form of non-Hermiticity provides effective descriptions for experiments
on open quantum systems with two-body loss Lewenstein  _et al._ (2007); Zhang
_et al._ (2016); Gross and Bloch (2017); Rosso  _et al._ (2022).

In this Letter, combining exact analytical results and numerical calculations,
we establish the phase diagram of a family of non-Hermitian Kitaev chains
Rainis and Loss (2012); Lieu (2019); Sayyad  _et al._ (2021); Sakaguchi  _et
al._ (2022). Our interacting model consists of a complex-valued many-body
interaction that may host Majorana modes, in particular, realizable in an
array of Josephson junctions Hassler and Schuricht (2012). Our results reveal
that, depending on the relative couplings, non-Hermitian interacting phases
with topological degeneracies emerge in the system. We show how increasing the
non-Hermiticity parameters affect some of the many-body topological phases. We
furthermore present that the topological degeneracies of the model remain in
the non-analytically solvable regime by numerically solving the interacting
problem.

Figure 1:  Schematic illustration of a 1D dimerized Kitaev-Hubbard chain with
dimerized hopping $t$, pairing $\Delta$, and complex-valued Hubbard
interaction $U-\mathrm{i}\delta$. Here, $\eta$ denotes the dimerization
parameter and ${\cal O}\in\\{t,\Delta,U,\delta\\}$.  Figure 2:  Phase diagrams
of the non-Hermitian Hamiltonian and associated orders of degeneracies on the
$(U/t-\eta)$ plane at $\mu=0$ and $\Delta=t$. Non-Hermiticity parameter is set
to $\delta/t=$ $0$ (a), $0.5$ (b), $1.0$ (c), $1.5$ (d). The yellow dashed
lines display the exact phase boundaries given by Eq. (6). The heat map shows
the degeneracies obtained numerically for a chain with length $L=16$, shown in
red, green, and blue, respectively, for fourfold, twofold, and no
degeneracies.

## Model.

The Hamiltonian for the non-Hermitian dimerized Kitaev-Hubbard chain,
schematically shown in Fig. 1, reads

$\displaystyle{\cal
H}=-\sum_{j=1}^{L-1}\left[t_{j}\left(c^{\dagger}_{j}c_{j+1}+c^{\dagger}_{j+1}c_{j}\right)+\Delta_{j}\left(c^{\dagger}_{j}c^{\dagger}_{j+1}+c_{j+1}c_{j}\right)\right]$
$\displaystyle+\sum_{j=1}^{L-1}(U_{j}-\mathrm{i}\delta_{j})\left(2n_{j}-1\right)\left(2n_{j+1}-1\right)-\mu\sum_{j=1}^{L}(n_{j}-\frac{1}{2}),$
(1)

where $L$ denotes the length of the chain,
$c^{\dagger}_{j}\leavevmode\nobreak\ (c_{j})$ creates (annihilates) an
spinless fermion at site $j$ associated with the fermion density
$n_{j}=c^{\dagger}_{j}c_{j}$. Here $\mu$ adjusts the onsite energy, and
$t_{j}$, $\Delta_{j}$ and $U_{j}$ are, respectively, real-valued site-
dependent hopping amplitude, superconducting pairing amplitude, and Hubbard
interaction. The dimerized parameter ${\cal
O}_{j}\in\\{t_{j},\Delta_{j},U_{j},\delta_{j}\\}$ for $1\leq j\leq L$ reads

$\displaystyle{\cal O}_{j}$ $\displaystyle=\begin{cases}{\cal
O}(1-\eta),\quad\,j\,{\rm mod}\,2=0,\\\ {\cal O}(1+\eta),\quad\,j\,{\rm
mod}\,2=1,\end{cases}$ (2)

where $\eta$ is the real-valued dimerization parameter and ${\cal
O}\in\\{t,\Delta,U,\delta\\}$ stands for site-independent parameters; see also
Fig. 1.

The Hamiltonian in Eq. (1) at $\mu=0$ remains invariant under
$c_{j}\rightarrow(-1)^{j}c^{\dagger}_{j}$ which enforces the charge
conjugation symmetry. We note that respecting this symmetry ensures that
eigenvalues of the Hamiltonian come in complex-conjugate pairs making this
model not directly realizable in open quantum systems Lieu  _et al._ (2020),
which can be resolved by a negative imaginary shift of all eigenvalues
Joglekar and Harter (2018). Nevertheless, preserving the charge conjugation
symmetry in Hermitian models ($\delta=0$) at the symmetric point $\Delta=t$
allows calculating exact phase diagrams for arbitrary $\eta$ Gangadharaiah
_et al._ (2011); Katsura  _et al._ (2015); Ezawa (2017); Wang  _et al._
(2017); Miao  _et al._ (2017). In the following, we present that there exists
an analytical solution for the non-Hermitian model when the charge conjugation
symmetry is respected, i.e., at $\mu=0$ and $\Delta=t$. When $\Delta\neq t$ or
$\mu\neq 0$, we compute the topological phase diagram numerically.

Figure 3:  Phase diagrams of the non-Hermitian Hamiltonian and associated
orders of degeneracies on the $(U/t-\eta)$ plane at $\mu=0$ and $\Delta\neq
t$. The heat map corresponds to the degeneracies obtained numerically for a
system with $L=16$. The superconducting pairing amplitude and non-Hermiticity
are set to $(\Delta/t,\delta)=$ $(0.5,0.0)$ (a), $(1.5,0.0)$ (b), $(0.5,1.0)$
(c), $(1.5,1.0)$ (d). The yellow dashed lines are phase boundaries at
$\Delta=t$ and are shown as a guide for eye.

## Exact solution at $\mu=0$ and $\Delta=t$.

To obtain the exact phase diagram of the model Hamiltonian at $\mu=0$, we
employ two Jordan-Wigner transformations and one spin rotation; see the
Supplemental materials (SM) for details SuppMat . This procedure map our
initial non-Hermitian interacting Hamiltonian into a non-Hermitian quadratic
fermionic model given by

$\displaystyle{\cal H}$
$\displaystyle=\sum_{j}-t_{j}\left[f^{\dagger}_{j+1}f_{j}+f^{\dagger}_{j}f_{j+1}+f^{\dagger}_{j}f^{\dagger}_{j+1}+f_{j+1}f_{j}\right]$
$\displaystyle+\sum_{j}\tilde{U}_{j}\left[f^{\dagger}_{j+1}f_{j}+f^{\dagger}_{j}f_{j+1}-f^{\dagger}_{j}f^{\dagger}_{j+1}-f_{j+1}f_{j}\right],$
(3)

which in the momentum-space casts

$\displaystyle{\cal H}$
$\displaystyle=\sum_{k}(-t+\tilde{U})\left[z_{k}f^{\dagger}_{kB}f_{kA}+z_{k}^{*}f^{\dagger}_{kA}f_{kB}\right]$
$\displaystyle-\sum_{k}(t+\tilde{U})\left[w_{k}f^{\dagger}_{-kA}f^{\dagger}_{kB}+w_{k}^{*}f_{kB}f_{-kA}\right],$
(4)

where $\tilde{U}=U-\mathrm{i}\delta$. Diagonalizing this Hamiltonian, we
obtain the energy spectrum of this four-band system given by

$\displaystyle\frac{\Lambda_{k}^{2}}{4}$
$\displaystyle=\begin{cases}\tilde{U}^{2}(1+\eta)^{2}+t^{2}(1-\eta)^{2}-2t\tilde{U}(1-\eta^{2})\cos(k),\\\
\tilde{U}^{2}(1-\eta)^{2}+t^{2}(1+\eta)^{2}-2t\tilde{U}(1-\eta^{2})\cos(k),\end{cases}$
(5)

where $\tilde{U}^{2}=U^{2}-\delta^{2}-2\mathrm{i}\delta U$. As eigenvalues of
our system appear in complex-conjugate pairs, due to the charge-conjugation,
zero modes in $\Lambda_{k}$ emerge when $\mathop{\mathrm{Re}}[\Lambda_{k}]=0$
at

$\displaystyle\frac{U}{t}=\sqrt{\frac{\delta^{2}}{t^{2}}-\frac{(1\pm\eta)^{2}}{(1\mp\eta)^{2}}},$
(6)

which is obtained at $k=\pm\pi/2$. In the Hermitian limit ($\delta=0$), Eq.
(6) reproduces the Hermitian results Gangadharaiah  _et al._ (2011); Katsura
_et al._ (2015); Ezawa (2017); Wang  _et al._ (2017); Miao  _et al._ (2017).
Eq. (6) shows the boundaries between various phases in our system, presented
in yellow dashed lines in Fig. 3. These boundaries delineate phases with
fourfold, twofold, and no degeneracies, respectively, shown in red, green, and
blue in Fig. 2. The order of degeneracy is determined by
$\chi=\sum_{i=0}^{L}\exp[-\lambda|\varepsilon_{i}-\varepsilon_{0}|]$. This
quantity measures the degeneracy between first $\varepsilon_{0}$ and the
$i$the smallest eigenvalue $\varepsilon_{i}$ within the energy resolution of
$1/\lambda(=0.05)$. Here the eigenvalues are calculated using the numerical
exact diagonalization method. The difference between the numerical boundaries
and the exact solution should be attributed to the finite size effect. In the
thermodynamic limit ($L=\infty$), one can recover the exact phase boundaries;
see the SM SuppMat . The topological superconducting phase residing in the
boundaries surrounding $U=0$ Kitaev (2001) shrinks as non-Hermiticty
increases. At the critical value $\delta=t$, this topological phase fades
away, resulting in the mixing of other two-fold degenerate phases. It is also
worth noting that the exact phase boundaries in Eq. (6) are associated with
transitions between non-Hermitian spectra with different types of the gap in
the non-Hermitian effective model in Eq. (4); see also the SM SuppMat .

Figure 4:  Phase diagrams of the non-Hermitian Hamiltonian and associated
orders of degeneracies on the $(U/t-\eta)$ plane with $\mu\neq 0$. The
superconducting pairing amplitude is set to $\Delta=t$. The chain size is set
to $L=16$ in (a,b), and extrapolated data in the thermodynamic limit are shown
in (c,d). The yellow dashed lines are phase boundaries at $\mu=0$ and are
shown as a guide for the eye. We took $\delta,\mu=0,0.25t$ in (a,c) and
$\delta,\mu=0.5,0.25t$ in (b,d)

## Many-body Majorana edge modes.

To identify Majorana modes in our model, in the next step, we rewrite the
Hamiltonian in Eq. (3) in terms of Majorana fermions
${\Upsilon}_{j}^{A}=f^{\dagger}_{j}+f_{j}$ and
${\Upsilon}_{j}^{B}=\mathrm{i}(f^{\dagger}_{j}-f_{j})$.The Hamiltonian then
reads

$\displaystyle{\cal
H}=\mathrm{i}\sum_{j}\left[t_{j}{\Upsilon}_{j}^{B}\Upsilon_{j+1}^{A}+\tilde{U}_{j}\Upsilon_{j}^{A}\Upsilon_{j+1}^{B}\right].$
(7)

We note that interactions and hopping amplitudes between the same sublattices
or within each unit cell vanish. Hence, Eq. (7) can be decoupled into two
independent non-interacting Kitaev chains with length $L/2$ such that ${\cal
H}={\cal H}_{\rm I}+{\cal H}_{\rm II}$ where

$\displaystyle{\cal H}_{\rm I}$
$\displaystyle=\sum_{j=1}^{L/2}\left[-\mathrm{i}t_{2j}\Phi_{{\rm
I},j+1}^{A}\Phi_{{\rm I},j}^{B}+\mathrm{i}\tilde{U}_{2j-1}\Phi_{{\rm
I},j}^{A}\Phi_{{\rm I},j}^{B}\right],$ (8) $\displaystyle{\cal H}_{\rm II}$
$\displaystyle=\sum_{j=1}^{L/2}\left[\mathrm{i}\tilde{U}_{2j}\Phi_{{\rm
II},j+1}^{B}\Phi_{{\rm II},j}^{A}-\mathrm{i}t_{2j-1}\Phi_{{\rm
II},j}^{B}\Phi_{{\rm II},j}^{A}\right],$ (9)

with $\Phi^{A}_{{\rm I},j}=\Upsilon_{2j-1}^{A}$, $\Phi^{B}_{{\rm
I},j}=\Upsilon^{B}_{2j}$, $\Phi^{B}_{{\rm II},j}=\Upsilon^{B}_{2j-1}$, and
$\Phi^{B}_{{\rm II},j}=\Upsilon^{A}_{2j}$. Introducing Majorana particles from
the electron operators as $\gamma_{j}^{A}=c^{\dagger}_{j}+c_{j}$ and
$\gamma_{j}^{B}=\mathrm{i}(c^{\dagger}_{j}-c_{j})$, one can show that
${\Upsilon}$ operators are products of $\gamma$ operators such that Goldstein
and Chamon (2012); Yang and Feldman (2014); Kells (2015); Miao  _et al._
(2017); McGinley  _et al._ (2017); Ezawa (2017)

$\displaystyle\Upsilon_{j}^{A}$
$\displaystyle=\begin{cases}\prod^{j-1}_{k=\text{odd}}\left[\mathrm{i}\gamma_{k}^{B}\gamma_{k+1}^{A}\right]\gamma_{j}^{A},\quad&j=\text{odd},\\\
&\\\
\prod_{k=\text{odd}}^{j-3}\left[\mathrm{i}\gamma_{k}^{A}\gamma_{k+1}^{B}\right](\mathrm{i}\gamma^{A}_{j-1}\gamma_{j}^{A}),\quad&j=\text{even},\\\
\end{cases}$ (10) $\displaystyle\Upsilon_{j}^{B}$
$\displaystyle=\begin{cases}\prod^{j-2}_{k=\text{odd}}\left[\mathrm{i}\gamma_{k}^{A}\gamma_{k+1}^{B}\right]\mathrm{i}\gamma_{j}^{A}\gamma^{B}_{j},\quad&j=\text{odd},\\\
&\\\
\prod_{k=\text{odd}}^{j-1}\left[\mathrm{i}\gamma_{k}^{B}\gamma_{k+1}^{A}\right]\gamma^{B}_{j},\quad&j=\text{even}.\end{cases}$
(11)

These relations keep the Majorana anti-commutation relations unchanged, i.e.,
$\\{\Upsilon^{\alpha}_{i},\Upsilon^{\beta}_{j}\\}=\\{\gamma^{\alpha}_{i},\gamma^{\beta}_{j}\\}=2\delta_{i,j}\delta^{\alpha,\beta}$.
We note that $\Upsilon$ operators, which are comprised of odd (even) number of
Majorana fermions ($\gamma$s), belong to subsystem ${\rm
I}\leavevmode\nobreak\ ({\rm II})$ described by ${\cal H}_{\rm
I}\leavevmode\nobreak\ ({\cal H}_{\rm II})$.

Figure 5:  Spatial distribution of the zero modes for different system lengths
for the non-Hermitian model computed with the local response function given by
Eq. (13). It is observed that both in the presence of repulsive and attractive
interactions, an edge response appears, accounting for the topological
degeneracy of the model. The edge excitations emerge both in the case of two-
fold (a,d) and four-fold (b,c) degeneracy. We took $\mu=0$, $\Delta=t$,
$\eta=-0.6$ and $\delta=0.5t$.

The quadratic Hamiltonian in Eq. (7) may host two types of boundary modes
($\cal Q$). These boundary modes are constructed from linear combinations of
$\Phi^{A/B}_{\rm I/II}$ operators, i.e., ${\cal
Q}^{\alpha}_{\beta}=\sum_{j\geq 0}{a}_{\beta,j}\Phi^{\alpha}_{\beta,j}$ with
$\alpha=A,B$ and $\beta={\rm I,II}$. As ${\cal Q}$ consists of higher-order
multiple Majorana fermions, it is dubbed ”many-body Majorana operator”
McGinley  _et al._ (2017). In the Hermitian limit, ${\cal Q}$ operators are
conserved, $[{\cal H},{\cal Q}]=0$, and using the iteration procedure, one can
determine the coefficients ($a$) as Fendley (2012); Ezawa (2017); McGinley
_et al._ (2017)

$\displaystyle{a}_{{\rm
I},j}=-\left(\frac{{U}(1+\eta)}{t(1-\eta)}\right)^{j-1},\quad{a}_{{\rm
II},j}=-\left(\frac{t(1+\eta)}{{U}(1-\eta)}\right)^{j-1}.$ (12)

In non-Hermitian systems, the operator $\cal O$ is conserved if it satisfies
$[{\cal H}_{R},{\cal O}]=\\{{\cal H}_{I},{\cal O}\\}=0$ where ${\cal H}={\cal
H}_{R}+\mathrm{i}{\cal H}_{I}$ Sayyad (2022). We note that based on the
structure of ${\cal H}_{\rm I/II}$ in Eqs. (8) and (9), $\\{{\cal H}_{I},{\cal
O}\\}=0$ is by construction satisfied and fulfilling $[{\cal H}_{R},{\cal
O}]=0$ results in obtaining Eq. (12). Hence, the boundary modes in our non-
Hermitian system are continuously ($\delta\rightarrow 0$) connected to the
zero-energy boundary modes.

The Majorana boundary mode ${\cal Q}_{\rm I}$ consists of odd numbers of
higher-order Majorana operators ($\gamma$) is fermionic and satisfies
$\\{{\cal Z}_{2},{\cal Q}_{\rm I}\\}=0$, in the infinite chain limit Fendley
(2012). However, the ${\cal Q}_{\rm II}$ mode comprises even numbers of
higher-order Majorana operators ($\gamma$) is bosonic as $[{\cal Z}_{2},{\cal
Q}_{\rm II}]=0$ Katsura  _et al._ (2015); McGinley  _et al._ (2017); Ezawa
(2017). Regions with fourfold degeneracies in Fig. 2 host both $({\cal Q}_{\rm
I},{\cal Q}_{\rm II})$. The topological superconducting phase enclosing $U=0$
merely hosts ${\cal Q}_{\rm I}$, in agreement with Hermitian noninteracting
intuition Kitaev (2001). The other two twofold degenerate phases accommodate
${\cal Q}_{\rm II}$.

## Beyond exact solutions.

Let us now present the phase diagram of our system away from the integrable
regime $\Delta=t$ and $\mu=0$. First, we consider $\Delta/t\in\\{0.5,1.5\\}$
and plot the associated phase diagrams with $\delta\in\\{0.0,1.0\\}$ in Fig. 3
for a finite size system with $L=16$. Similar to the phase diagram of the
system at $\Delta=t$, we witness phases with fourfold, twofold, and no
degeneracies. Comparing the exact phase boundaries at $\Delta=t$, in yellow
dashed lines, with boundaries of regions with $2n$fold degeneracies, in red or
green, we identify deformation of the phase boundaries toward $U<0$ ($>0$) for
$\Delta<t\leavevmode\nobreak\ (>t)$.

At finite chemical potential and $\delta=0$, all fourfold degeneracies are
lifted, and merely phases with twofold degeneracies remain in the phase
diagram; see Fig. 4. The Hermitian phase boundary $U=-0.5t$ at $\eta=0$ is
consistent with previous calculations on the Kitaev-Hubbard chain Mahyaeh and
Ardonne (2020); see panels (a) at $L=16$, (c) in the thermodynamics limit and
the SM SuppMat . Witnessing merely twofold degeneracies in phases with $U>1$
in the Hermitian limit also persists as non-Hermiticty is increased; see Fig.
4(b) at $L=16$. While no portion of the topological superconducting phase,
region encircling $U=0$, is present in (b) obtained at $\mu=0.25$ and $L=16$,
extrapolating the phase diagram using different system sizes, shown in (d),
reveals the survival of this phase at $L=\infty$.

## Experimental measurement.

We now address the signatures of the zero modes from the experimental point of
view. In a tunneling experiment with a local probe Drost  _et al._ (2017);
Kempkes  _et al._ (2019, 2018); Huda  _et al._ (2020); Dvir  _et al._ (2023),
the conductance at zero bias $G(\omega=0,n)$ depends on the probability of
extracting (injecting) an electron in site $n$ at energy $\omega$ as
$\text{dI/dV}(n,\omega)\sim\sum_{\alpha}|\langle
GS|c^{(\dagger)}_{n}|\Psi_{\alpha}\rangle|^{2}\delta(\omega-
E_{\alpha}+E_{GS})$, where $|GS\rangle$ is the ground state and
$H|\Psi_{\alpha}\rangle=E_{\alpha}|\Psi_{\alpha}\rangle$ are the many-body
excited states of the system. In the presence of topological degeneracy, the
ground state presents zero mode excitations that distinguish different ground
states. In that scenario, the zero bias conductance at site $n$ can be written
as $\text{dI/dV}(n,\omega=0)\sim\Xi(n)$ where

$\Xi(n)=\sum_{\alpha}|\langle\Psi_{\alpha}|c_{n}|GS\rangle|^{2}+|\langle\Psi_{\alpha}|c^{\dagger}_{n}|GS\rangle|^{2}$
(13)

which directly images the probability of a local excitation between the ground
state and its degenerate manifold. Here, $\alpha$ runs over the ground state
manifold. In particular, $\Xi(n)$ allows one to directly observe the emergence
of topological zero modes associated with the topological degeneracy of the
ground state. With the previous quantity, the emergence of topological zero
modes associated to the topological degeneracy of the non-Hermitian model can
be directly imaged. We show in Fig. 5 the local correlator computed for the
interacting non-Hermitian model for different system sizes. As the system
becomes larger, an edge excitation emerges in the model, which in the
thermodynamic limit leads to decoupled modes between the two edges; see also
the SM SuppMat . The previous quantity has been directly imaged in the
realization of the current model in the topological phase with $\delta=U=0$
and $\eta\neq 0$ Drost  _et al._ (2017), and minimal a chain with $\Delta\neq
0$Dvir  _et al._ (2023).

## Conclusion.

To summarize, we have presented a family of non-Hermitian interacting models
featuring different classes of topological degeneracies. While non-interacting
non-Hermitian models can be studied with conventional methodologies, the
inclusion of many-body interactions renders exploring non-Hermitian systems
greatly challenging. Our manuscript establishes a family of solvable
interacting non-Hermitian models, providing ideal systems for benchmarking
methodologies to treat interacting non-Hermitian models. Besides showing the
emergence of different topological phases in the solvable limit, we provided
the many-body operators accounting for the topological degeneracy of the
model. We showed how non-Hermiticity modifies topological many-body models,
substantially impacting the topological phases of Hermitian systems. We
benchmarked our analytical construction with exact numerical calculations of
the full many-body system, demonstrating that even for finite systems, the
emergence of topological modes and topological degeneracies can be observed.
Our results establish a versatile family of models featuring interacting
topology, providing a starting point for higher dimensional solvable
interacting models.

## Acknowledgement.

S.S. thanks Vittorio Paeno and Abolhassan Vaezi for helpful discussions.
J.L.L. acknowledges the computational resources provided by the Aalto Science-
IT project, and the financial support from the Academy of Finland Projects No.
331342 and No. 336243 and the Jane and Aatos Erkko Foundation.

##  Appendix A Mapping the non-Hermitian interacting Kitaev chain into a
noninteracting Hamiltonian

Following Refs. Ezawa (2017); Wang  _et al._ (2017), we now present the
generalization of mapping our non-Hermitian interacting model into a non-
interacting Hamiltonian when $t=\Delta$ and $\mu=0$.

In the first step, we represent the fermion operators in terms of the spin
operators, described by Pauli matrices, using the Jordan-Wigner transformation
Sela  _et al._ (2011); Katsura  _et al._ (2015); Miao  _et al._ (2017);
Mahyaeh and Ardonne (2020); Liu  _et al._ (2021); Bi  _et al._ (2021); Zvyagin
(2022) given by $c_{i}=K_{i}\sigma^{-}_{i}$,
$c_{i}^{\dagger}=\sigma^{+}_{i}K^{+}_{i}$, and $2n_{i}-1=\sigma_{i}^{z}$ with
$\sigma^{\pm}=\sigma^{x}\pm\mathrm{i}\sigma^{y}$ and
$K_{i}=\prod_{j<i}(-\sigma^{z}_{j})$. As a result, the spin representation of
the Hamiltonian in Eq. (1) casts

${\cal
H}=\sum_{j}\left[-t_{j}\sigma^{x}_{j}\sigma^{x}_{j+1}+\tilde{U}_{j}\sigma^{z}_{j}\sigma^{z}_{j+1}\right],$
(S1)

where $\tilde{U}_{j}=U_{j}-\mathrm{i}\delta_{j}$. As the XY spin model is
exactly solvable Schultz  _et al._ (1964), one can transform the above XZ spin
model, which may not be exactly solvable, in Eq. (S1) into an XY model. For
this purpose, we perform a $\pi/2$ spin rotation on all spin operators around
the $x$ axis using $R=\exp[-\mathrm{i}\pi\sum_{j}\sigma^{x}_{j}/2]$ Miao  _et
al._ (2017). The subsequent Hamiltonian yields

$\displaystyle{\cal
H}=\sum_{j}\left[-t_{j}\sigma^{x}_{j}\sigma^{x}_{j+1}+\tilde{U}_{j}\sigma^{y}_{j}\sigma^{y}_{j+1}\right].$
(S2)

To diagonalize this spin Hamiltonian, we introduce a second Jordan-Wigner
transformation Wang  _et al._ (2017); Ezawa (2017); Ding and Zhong (2021);
Wada  _et al._ (2021) given by

$\displaystyle\sigma^{x}_{i}$
$\displaystyle=(f^{\dagger}_{i}+f_{i})\exp\left[\mathrm{i}\pi\sum_{i<j}f^{\dagger}_{i}f_{i}\right],$
(S3) $\displaystyle\sigma^{y}_{i}$
$\displaystyle=-\mathrm{i}(f^{\dagger}_{i}-f_{i})\exp\left[\mathrm{i}\pi\sum_{i<j}f^{\dagger}_{i}f_{i}\right],$
(S4) $\displaystyle\sigma^{z}_{i}$ $\displaystyle=2f^{\dagger}_{i}f_{i}-1.$
(S5)

Using the above relations, we rewrite the XY spin model as a fermionic
quadratic model, which reads

$\displaystyle{\cal H}$
$\displaystyle=\sum_{j}-t_{j}\left[f^{\dagger}_{j+1}f_{j}+f^{\dagger}_{j}f_{j+1}+f^{\dagger}_{j}f^{\dagger}_{j+1}+f_{j+1}f_{j}\right]$
$\displaystyle+\sum_{j}\tilde{U}_{j}\left[f^{\dagger}_{j+1}f_{j}+f^{\dagger}_{j}f_{j+1}-f^{\dagger}_{j}f^{\dagger}_{j+1}-f_{j+1}f_{j}\right].$
(S6)

As the bulk-boundary correspondence is not violated in our non-Hermitian
system, searching for the presence of zero modes can be done in the momentum
space ($k$) using the Fourier transformation
$f_{k}=\sum_{j}f_{j}\exp[\mathrm{i}kja]$ with $a(=1)$ be the lattice constant.

Figure S1:  Phase boundaries of the effective Hamiltonian in Eq. (S12).

The quadratic Hamiltonian in the momentum space on a bipartite unit cell with
sublattices $(A,B)$ casts

$\displaystyle{\cal H}$
$\displaystyle=\sum_{k}(-t+\tilde{U})\left[z_{k}f^{\dagger}_{kB}f_{kA}+z_{k}^{*}f^{\dagger}_{kA}f_{kB}\right]$
$\displaystyle-\sum_{k}(t+\tilde{U})\left[w_{k}f^{\dagger}_{-kA}f^{\dagger}_{kB}+w_{k}^{*}f_{kB}f_{-kA}\right].$
(S7)

Here $\tilde{U}=U-\mathrm{i}\delta$ and $z_{k}$ and $w_{k}$ read

$\displaystyle z_{k}$ $\displaystyle=(1+\eta)+e^{-\mathrm{i}ka}(1-\eta),$ (S8)
$\displaystyle w_{k}$ $\displaystyle=(1+\eta)-e^{-\mathrm{i}ka}(1-\eta).$ (S9)

The non-Hermitian Hamiltonian in Eq. (S7) can be written in a matrix form
using the vector operator ${\cal
C}_{k}^{\dagger}=(f^{\dagger}_{kA},f^{\dagger}_{kB},f_{-kA},f_{-kB})$ as

$\displaystyle{\cal H}$ $\displaystyle=\frac{1}{2}\sum_{k}{\cal
C}^{\dagger}_{k}\begin{pmatrix}0&a_{1}z_{k}^{*}&0&a_{2}w^{*}_{k}\\\
a_{1}z_{k}&0&-a_{2}w_{k}&0\\\ 0&-a_{2}w_{k}^{*}&0&-a_{1}z_{k}^{*}\\\
a_{2}w_{k}&0&-a_{1}z_{k}&0\end{pmatrix}{\cal C}_{k},$ (S10)

where $a_{1}=-t+\tilde{U}$ and $a_{2}=-(t+\tilde{U})$. Expressing ${\cal
H}=\frac{1}{2}\sum_{k}\Lambda_{k}{\cal C}^{\dagger}_{k}{\cal C}_{k}$, four
eigenvalues $\Lambda_{k}$ then read

$\displaystyle\frac{\Lambda_{k}^{2}}{4}$
$\displaystyle=\begin{cases}\tilde{U}^{2}(1+\eta)^{2}+t^{2}(1-\eta)^{2}-2t\tilde{U}(1-\eta^{2})\cos(ka),\\\
\tilde{U}^{2}(1-\eta)^{2}+t^{2}(1+\eta)^{2}-2t\tilde{U}(1-\eta^{2})\cos(ka),\end{cases}$
(S11)

where $\tilde{U}^{2}=U^{2}-\delta^{2}-2\mathrm{i}\delta U$. The gap closure in
$\Lambda_{k}$ occurs when we impose $\mathop{\mathrm{Re}}[\Lambda_{k}]=0$
resulting in

$\displaystyle\frac{U}{t}=\sqrt{\frac{\delta^{2}}{t^{2}}-\frac{(1\pm\eta)^{2}}{(1\mp\eta)^{2}}},$
(S12)

which is obtained at $k=\pm\pi/2$. We emphasize that due to the charge
conjugation symmetry enforcing $\mathop{\mathrm{Re}}[\Lambda_{k}]=0$ ensures
$\mathop{\mathrm{Im}}[\Lambda_{k}]=0$.

Figure S2:  Phase diagrams of the non-Hermitian Hamiltonian in Eq. (S12) at
$\mu=0$ on the $(U/t-\eta)$ plane for $\delta/t=$0.0 (a), 0.5 (b), 1.0 (c) and
1.5 (d). At $\delta=0$, the phase diagram consists of seven phases, including
the topological superconducting (TSC), (topological) superconducting dimer
((top-)SCD), (topological) single-electron dimer ((top-)SED), charge density
wave (CDW), and Schrödinger cat (CAT) phases. Dashed dotted lines in (a) and
(c) are at $\eta=\pm 0.2$. The circle points in (b) are located at
$(U/t,\eta)\in\\{(0,0),(0,-0.5)(0,0.5),(2.0,0.2)\\}$.

While Eq. (S12) determines phase boundaries, we note that characterizing the
properties of each phase cannot be addressed from the spectra of the
Hamiltonian in Eq. (S10). This is because the topological character of various
phases may not be preserved after performing nonlocal (Wigner) transformations
McGinley  _et al._ (2017); Ezawa (2017). Nevertheless, as non-Hermiticity does
not violate the bulk-boundary correspondence, we can deduce some pieces of
information regarding zero modes from effective noninteracting Hamiltonians.

For this purpose, we start with presenting the phase boundaries in Eq. (S12)
as a function of $(\eta,U,\delta)$ in Fig. S1. The two-dimensional
intersection of this phase boundaries at $\delta=$ $0.0$ (a), $0.5$ (b), $1.0$
(c), and $1.5$ (d) is shown in Fig. S2. The Hermitian phase diagram in Fig.
S2(a) comprises seven phases including the topological superconducting (TSC),
(topological) superconducting dimer ((top-)SCD), (topological) single-electron
dimer ((top-)SED), charge density wave (CDW), and Schrödinger cat (CAT) phases
Ezawa (2017). The CAT phase is a superposition of two superconducting states
Miao  _et al._ (2017); Ezawa (2017).

Figure S3:  Spectra of the Hermitian Hamiltonian as a function of $U/t$ at
$\mu=\delta=0$ and $\eta=-0.2$ (a) and $0.2$ (b). Two pairs of the smallest
absolute values of eigenvalues are shown in red and blue.

To explore the zero modes in the phase diagram, we now look at the spectra of
the system with an open boundary condition at $\mu=0$. We first consider the
Hermitian limit with $\delta=0.0$ and at $\eta\pm 0.2$ along the cuts shown by
dashed-dotted lines in Fig. S2. The associated spectra for $\eta=-0.2(0.2)$
are shown in Fig. S3. Comparing two pairs of smallest absolute values of
eigenvalues in (a, b), shown in red and blue, we realize that the CDW, CAT,
and TSC phases are twofold degenerate. The spectra also exhibit fourfold
degeneracies in top-SCD and top-SED; see panel (a). However, these fourfold
degeneracies are lifted for $\eta>0$ in the SCD and SED phases; see panel (b).

Figure S4:  Real(a,b) and imaginary (c,d) parts of the spectra as a function
of $U/t$ at $\mu=0$, $\delta=1.0$ and $\eta=-0.2$(a,c) and $0.2$(b,d). Two
pairs of the smallest absolute values of eigenvalues are shown in red and
blue.

Further looking at Fig. S2 reveals that the boundaries of the TSC phase shrink
toward diminishing this phase as $\delta$ increases. Reducing the regions with
the TSC phase results in lifting the separation between pairs of phases,
namely (top-SCD, top-SED), (SCD, SED), and (CAT, CDW). As the order of
degeneracies in both components of these pairs are identical, we still detect
fourfold, twofold, and no degeneracies, respectively, in the green, white, and
blue regions in all panels of Fig. S2. This can be seen in the spectra of the
system at $\delta=1.0$ along cuts at $\eta=\pm 0.2$ in Fig. S4. Note that the
real and imaginary parts of eigenvalues are zero at degenerate points due to
the particle-hole symmetry in the system.

Figure S5:  Spectra at $\mu=0$, $\delta=0.5$ and $(U/t,\eta)=(0,0)$(a),
$(2,0.2)$(b), $(0,-0.5)$(c), and $(0,0.5)$(d). The real (imaginary) line gaps
are shown in red (yellow). The point gap is shown by a circle with a unit
radius.

Aside from $2n$-fold degeneracies with $n\in\\{0,1,2\\}$, each non-Hermitian
phase in Fig. S2 exhibits a particular non-Hermitian gap in their spectrum. We
demonstrate this point by plotting the complex spectra of systems belonging to
the brown, blue, lime, and white regions, shown by circle points in Fig.
S2(b). In the TSC phases, the spectrum displays a real line gap shown by a red
rectangle in Fig. S5 (a). The spectra exhibit the imaginary line gap within
the CDW and CAT phases as exemplified in Fig S5(b). The gap becomes the point
gap both in phases with fourfold or no degeneracies, e.g., in the top-SED
shown in Fig S5(c) and in the SED phase presented in Fig. S5(d).

##  Appendix B Topological modes in the thermodynamic limit

In the previous sections, we have focused on relatively small-length systems
that can be solved with exact diagonalization. In this section, we show that
the edge modes associated with topological degeneracies evolve into fully
decoupled modes in the thermodynamic limit. In order to surpass the length
limitations of exact diagonalization, in this section, we solve the
interacting quantum many-body model using a non-Hermitian tensor network
formalism dmr ; ITe ; Fishman  _et al._ (2022), targeting the ground state and
lowest excited states Hyart and Lado (2022); Chen  _et al._ (2022). We show in
Fig. S6 the spatially resolved correlator computed with the non-Hermitian
tensor network formalism. It is observed that as the system becomes larger,
the local edge modes become more decoupled (Fig. S6(a)), giving rise to fully
decoupled modes for large systems; see panels (b) and (c) in Fig. S6.

Figure S6:  Spatially resolved local correlators, showing the emergence of
edge modes. Panel (a) shows the scaling with the length of the system,
highlighting the emergence of decoupled modes for large lengths. Panels (b,c)
show the spatially resolved correlators for two specific lengths, showing the
localization of the excitations on the edges. We set $U=0.5t$, $\Delta=t$,
$\mu=0.25t$, $\delta=0.5t$ and $\eta=0$. Figure S7: Degeneracy of the non-
Hermitian model for different system sizes, $L=12$ (a), $L=14$ (b) and $L=16$
(c), and extrapolation in the thermodynamic limit (d). It is observed that
while for finite systems, finite size effects affect the inferred boundaries
between different phases, the extrapolation scheme allows to the recovery of
the correct phase diagram. The yellow dashed lines display the exact phase
boundaries. We took $\Delta=t$, $\delta=\mu=0$, and $\lambda=0.01t$.

##  Appendix C Finite size effects and extrapolation to the thermodynamic
limit

In the main text, we have focused on studying the case of finite systems
numerically. Due to finite size effects, the degeneracy of the ground state is
lifted due to the coupling of the topological excitations. In order to extract
the topological degeneracy in the thermodynamic limit, a finite size scaling
of the energy splittings can be performed. In this section, we show that with
a size scaling, we can recover the exact analytic results in the interacting
limit from exact numerical calculations in finite systems. The finite size
scaling relies on taking a scaling for the excited state energies as
$\epsilon_{\alpha}(L)=a_{\alpha}+b_{\alpha}/L$, so that in the thermodynamic
limit $\epsilon_{\alpha}(L=\infty)=a_{\alpha}$. The excited state energies are
taken as $\epsilon_{\alpha}=|E_{\alpha}-E_{0}|$, where $E_{\alpha}$ is the
complex eigenenergy of the many-body Hamiltonian and $E_{0}$ the ground state
energy. By computing the energies for a set of finite-size systems, the
coefficients $a,b$ can be extracted, and the energies in the thermodynamic
limit are obtained. With the extracted energy differences, the degeneracy of
the ground state can be computed as
$\chi=\sum_{\alpha}e^{-\lambda\epsilon_{\alpha}}$ with $\lambda$ being the
energy smearing.

We present in Fig. S7 how this methodology allows obtaining the degeneracies
in the thermodynamic limit. It is observed that for finite-size systems, the
phase boundaries are substantially shifted and are size dependent; see panels
(a), (b), and (c) in Fig. S7). Using the extrapolation scheme noted above, the
correct phase boundaries known from the analytic solution are recovered, as
shown in Fig. S7(d).

Figure S8: Degeneracy of the non-Hermitian model for different system sizes,
$L=12$ (a), $L=14$ (b) and $L=16$ (c), and extrapolation in the thermodynamic
limit (d) with $\Delta=t$, $\delta=0$, $\mu=0.25t$, and $\lambda=0.01t$. The
vertical white line in (d) is at $U=-0.5t$.

We further perform a similar extrapolation scheme to the Hermitian phase
diagram at $\mu=0.25t$. Here we employ the exact diagonalization methods in
Fig. S8 and the tensor-network formalism in Fig. S9. As it is evident,
increasing the length of the chain in panels (a)-(c) results in improving the
boundaries of twofold degenerate topological phases. In the thermodynamic
limit, the extrapolated phase diagrams, presented in Fig. S8(d) and Fig.
S9(d), recover the exact phase boundaries at $\mu=0$ and $U\geq-0.5t$. We note
that the phase boundary at $\eta=0$ is located at $U=-0.5t$ (vertical white
line), in agreement with the previous Hermitian calculations Mahyaeh and
Ardonne (2020).

Figure S9: Degeneracy of the non-Hermitian model obtained from the DMRG
calculations for different system sizes, $L=20$ (a), $L=30$ (b) and $L=40$
(c), and extrapolation in the thermodynamic limit (d) with $\Delta=t$,
$\delta=0$, $\mu=0.25t$ and $\lambda=0.01t$. The vertical white line in (d) is
at $U=-0.5t$.

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