mycases $\displaystyle{##}$$\displaystyle{##}$ {. ††thanks: These two authors
contributed equally

††thanks: These two authors contributed equally

# Fate of the non-Hermitian skin effect in many-body fermionic systems

Faisal Alsallom  Department of Physics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA  Institute of Physics, Ecole Polytechnique
Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland  Loïc Herviou
Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015
Lausanne, Switzerland  Oleg V. Yazyev  Institute of Physics, Ecole
Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Marta Brzezińska<EMAIL_ADDRESS>Institute of Physics, Ecole
Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

###### Abstract

We revisit the fate of the skin modes in many-body non-Hermitian fermionic
systems. Contrary to the single-particle case, the many-body ground state
cannot exhibit an exponential localization of all eigenstates due to the Pauli
exclusion principle. However, asymmetry can still exist in the density
profile, which can be quantified using the imbalance between the two halves of
the system. Using the non-Hermitian Su-Schrieffer-Heeger (SSH) chain as an
illustration, we show the existence of two distinct scaling regimes for the
imbalance. In the first one, the imbalance grows linearly with the system
size, as generically expected. In the second one, the imbalance saturates to a
finite value. By combining high-precision exact diagonalization calculations
and analytical arguments, we observe that the imbalance does not scale when
the occupied bands can be deformed to their Hermitian limit. This suggests a
direct connection between the corresponding bulk topological invariants and
the skin effect in many-body systems. Importantly, this relation also holds
for interacting systems.

Over the years, the interplay between non-Hermiticity and topology has
gathered an immense interest, both from experimental and theoretical
perspectives. Non-Hermitian (nH) Hamiltonians can serve as an effective
description of open systems, accurately modeling various non-conservative
classical and quantum platforms, including metamaterials Barton  _et al._
(2018); Brandenbourger  _et al._ (2019); Ghatak  _et al._ (2020); Scheibner
_et al._ (2020); Zhou and Zhang (2020); Rao  _et al._ (2021); Chen  _et al._
(2021), optics and photonics Regensburger  _et al._ (2012); Peng  _et al._
(2014); Zeuner  _et al._ (2015); Feng  _et al._ (2017); Zhou  _et al._ (2018);
Harari  _et al._ (2018); Bandres  _et al._ (2018); Pan  _et al._ (2018); El-
Ganainy  _et al._ (2018); Miri and Alù (2019), or electric circuits Ezawa
(2019a, b, c); Hofmann  _et al._ (2020); Helbig  _et al._ (2020). Within the
framework of topological band theory, non-Hermiticity gives rise to exotic
phenomena without Hermitian counterparts. This includes exceptional points
Heiss (2004); Leykam  _et al._ (2017); Martinez Alvarez  _et al._ (2018) –
degeneracies where eigenstates coalesce – and novel topological phases
characterized by a winding of the eigenspectrum in the complex plane Rudner
and Levitov (2009); Esaki  _et al._ (2011); Lee (2016); Shen  _et al._ (2018);
Liu  _et al._ (2019); Zhou and Lee (2019); Kawabata  _et al._ (2019); Zhang
_et al._ (2020a).

Another unique feature of nH systems is the anomalous localization of all
eigenstates dubbed the non-Hermitian skin effect Lee (2016); Yao and Wang
(2018); Lee  _et al._ (2019); Lee and Thomale (2019); Borgnia  _et al._
(2020). All single-particle eigenstates become exponentially localized at the
boundaries of the system. Generally, skin modes arise due to the breakdown of
reciprocity, i.e., introducing asymmetric hoppings. The conventional bulk-
boundary correspondence is then broken, obscuring the prediction of universal
boundary phenomena from the periodic bulk properties Xiong (2018); Kawabata
_et al._ (2018); Gong  _et al._ (2018); Yao  _et al._ (2018); Kunst  _et al._
(2018); Herviou  _et al._ (2019a); Jin and Song (2019); Song  _et al._ (2019);
Yokomizo and Murakami (2019); Ozcakmakli Turker and Yuce (2019); Edvardsson
_et al._ (2019); Zirnstein  _et al._ (2021). Indeed, the phase diagram of the
single particle Hamiltonian with periodic boundary conditions (PBC) can
significantly differ from the one with open boundary conditions (OBC). The
skin effect is directly linked to the structure of the single-particle
spectrum with PBC Okuma  _et al._ (2020); Zhang  _et al._ (2020a) . A non-zero
winding of the PBC energy bands around any point in the complex plane
guarantees that the related OBC modes are exponentially localized. The
corresponding OBC eigenenergies are (nearly all) located within these non-zero
winding regions. The existence of the skin effect was confirmed experimentally
Brandenbourger  _et al._ (2019); Song  _et al._ (2020); Scheibner  _et al._
(2020); Helbig  _et al._ (2020); Xiao  _et al._ (2020); Zhang  _et al._
(2021a); Liu  _et al._ (2021). Recently, the notion of skin effect has been
expanded by introducing different types of skin states Yi and Yang (2020) such
as symmetry-enforced variants Rui  _et al._ (2019); Okuma  _et al._ (2020);
Yoshida  _et al._ (2020), reciprocal skin effect Hofmann  _et al._ (2020) in
two or higher dimensions, higher-order skin effect Kawabata  _et al._ (2020a);
Okugawa  _et al._ (2020); Fu  _et al._ (2021); Palacios  _et al._ (2021);
Zhang  _et al._ (2021b); Zou  _et al._ (2021), or the critical skin effect Li
_et al._ (2020); Yokomizo and Murakami (2021).

So far, studies of the skin effect have largely focused on the single-particle
Hamiltonian. Most of the experiments described by these effective nH
Hamiltonians are either weakly interacting or non-interacting bosonic systems
(e.g. quantum optics) or classical systems (e.g. the various types of circuits
and active media). Moreover, in Hermitian non-interacting systems, the
properties of the many-body states can be straightforwardly deduced from those
of the single-particle Hamiltonian; this is not the case for nH models. Due to
the non-orthogonality of the single-particle eigenmodes, the Pauli principle
no longer acts trivially. The fermionic repulsion reshapes the occupied
orbitals so that no more than one fermion (equivalently one hard-core boson)
populates each physical site. Hence, the exponential localization of all
fermions at a boundary becomes impossible Lee  _et al._ (2020); Liu  _et al._
(2020). Recently, several authors have investigated the fate of the skin modes
in the many-body context Mu  _et al._ (2020); Lee  _et al._ (2020); Shen and
Lee (2021); Zhang  _et al._ (2021c); Yoshida (2021); Xi  _et al._ (2021). For
instance, for interacting nH systems such as topological Mott insulators, the
skin effect was observed in fermionic systems Liu  _et al._ (2020); Cao  _et
al._ (2021), but not in the bosonic case Zhang  _et al._ (2020b); Xu and Chen
(2020). Nonetheless, no general criterion for the survival of the skin effect
has been derived so far.

In this Letter, we investigate the connection between topological properties
of non-interacting nH systems and the skin effect in a many-body wave-
function. The exponential localization of the skin modes naturally translates
to an asymmetry of the many-body density profile that grows linearly with the
system size. Through numerical simulations, using high-precision exact
diagonalization (ED) and density matrix renormalization group (DMRG), we show
that the many-body eigenstates of OBC systems exhibit a transition from a
regime with an infinite asymmetry in the thermodynamic limit to one where it
saturates. The critical anisotropy corresponds to a change in the topology of
the _periodic_ single-particle spectrum, even though the open system remains
gapped at all times. In addition, we show that both scaling regimes survive
disorder and interactions.

Conventions and model:  To exemplify our findings, we consider the nH chiral
SSH model Lieu (2018); Kunst  _et al._ (2018):

$\mathcal{H}_{\mathrm{SSH}}\\{g\\}=-t_{1}\sum\limits_{j}\left(e^{g}c^{\dagger}_{j,B}c_{j,A}+e^{-g}c^{\dagger}_{j,A}c_{j,B}\right)\\\
-t_{2}\sum\limits_{j}\left(e^{g}c^{\dagger}_{j+1,A}c_{j,B}+e^{-g}c^{\dagger}_{j,B}c_{j+1,A}\right),$
(1)

where $c^{(\dagger)}$ are fermionic annihilation (creation) operators, $j$
denotes the lattice position, $A/B$ corresponds to the sublattice, and
$t_{1/2}$ is the hopping amplitude. $g\geq 0$ is a hopping anisotropy that
breaks Hermiticity, see Fig. 1(a). We consider three boundary conditions: PBC,
OBC (with $2L$ sites) or ABA-BC (OBC with $2L+1$ sites). ABA-BC has an
explicit analytical solution (see Appendix A and Ref. Kouachi, 2006). For PBC,
the single-particle gap closes for $|t_{1}e^{\pm 2g}|=|t_{2}|$, with two
phases adiabatically connected to the trivial and topological Hermitian
phases, and an intermediate phase where the system has no line gap. In all
cases, the single-particle spectrum has a non-trivial winding, and therefore
the nH skin effect is always present. Indeed, for OBC and ABA-BC, all single-
particle eigenstates are exponentially localized to the right side of the
system. The model defined in Eq. (1) is related to the Hermitian SSH chain via
the similarity transformation

$\mathcal{U}_{g}=\exp\left(g\sum\limits_{j}(2j-1)n_{j,A}+g\sum\limits_{j}2jn_{j,B}\right)$
(2)

such that

$\mathcal{H}_{\mathrm{SSH}}\\{g\\}=\mathcal{U}_{g}\,\mathcal{H}_{\mathrm{SSH}}\\{0\\}\,\mathcal{U}_{g}^{-1}.$
(3)

Note that the OBC and ABA-BC phase diagrams are therefore independent of $g$
with a gap closing at $t_{1}=t_{2}$, and their spectrum is always real.

In the following, we focus on the ground state properties of the OBC and ABA-
BC systems. We define the ground state as the right eigenstate that minimizes
the real part of the energy:

$\mathcal{H}_{\mathrm{SSH}}\\{g\\}\ket{\Psi\\{g\\}}=E_{\mathrm{GS}}\ket{\Psi\\{g\\}}\propto
E_{\mathrm{GS}}\,\mathcal{U}_{g}\ket{\Psi\\{0\\}}.$ (4)

Our results generalize straightforwardly to other right eigenstates and to the
left eigenvectors. The observables are taken to be:

$\braket{O}_{g}=\braket{\Psi\\{g\\}}{O}{\Psi\\{g\\}}.$ (5)

This expectation value is relevant when considering nH Hamiltonians obtained
from post-selection Dalibard  _et al._ (1992); Mølmer  _et al._ (1993). The
quantum state described by such an approach remains a proper Hermitian density
matrix. As $\mathcal{H}_{\mathrm{SSH}}\\{g\\}$ is non-interacting, observables
can be obtained efficiently from the diagonalization of the single-particle
Hamiltonian Herviou  _et al._ (2019b) (see Appendix B). We numerically
diagonalize the Hermitian system with high precision before applying the
similarity transformation to obtain the eigenstates of the nH models.

Imbalance as a marker of the skin effect:  As can be seen from the
transformation $\mathcal{U}_{g}$, all the single-particle orbitals are
exponentially localized at the right boundary of the chain. We expect the
asymmetry of the orbitals to transfer to the many-body density distribution
$\braket{n_{j}}_{g}=\braket{n_{j,A}+n_{j,B}}_{g}$. Due to the Pauli principle,
only one fermion can be located at a given site and therefore the many-body
version of the skin effect is not a sum of the exponential orbitals. To
quantify the degree of asymmetry of the density distribution, we use the
imbalance

$\mathcal{I}=\sum_{j\in\text{right
half}}\braket{n_{j}}_{g}-\sum_{j\in\text{left half}}\braket{n_{j}}_{g}.$ (6)

Working at fixed charge density
$N_{F}=\sum\limits_{j}\braket{n_{j}}_{g}/2L\leq 0.5$111$N_{F}>0.5$ can be
deduced from $N_{F}<0.5$, the imbalance can at most scale linearly with system
size. We consider this linear scaling to be the manifestation of the skin
effect in many-body systems. In Fig. 2, we show the scaling of the imbalance
in the limit $t_{1}=t_{2}$ where the SSH model maps to the Hatano-Nelson model
Hatano and Nelson (1996); Mu  _et al._ (2020). We observe a linear scaling of
the imbalance at all fillings. At small $g$, the slope increases with the
anisotropy $g$, but saturates at a value equal to $N_{F}$. The large $g$ limit
corresponds to a situation where all particles are located in the right half
of the system. We note that the imbalance can be generalized to the higher
moments of the density with similar results.

Figure 1: (a) Pictorial representation of the non-Hermitian SSH model. (b)
Imbalance $\mathcal{I}$ as a function of the system size for $t_{1}/t_{2}=2$
and three representative values of $g$. At $g<g_{c}$, $\mathcal{I}$ saturates
to a finite value. At $g>g_{c}$, the scaling becomes linear. It also exhibits
a series of damped oscillations whose frequency diverges when $g\rightarrow
g_{c}^{+}$. The inset shows the corresponding PBC spectra in the complex
plane. Figure 2: (a) Scaling of the imbalance in the Hatano-Nelson model with
$t_{1}=t_{2}=1$ and $g=0.25$ at three different fillings $N_{F}$. Due to the
structure of the PBC spectrum (see inset), the band is always partially
occupied, which leads to a linear scaling with system size. (b) Slopes of the
imbalance as a function of $g$. For small values of $g$, the slope does not
depend on $N_{F}$. As $g$ is increased, all particles localize on the right
boundary.

Skin effect and band topology:  We now turn to the general case where
$t_{1}\neq t_{2}$ and focus on half-filled systems ($L/(2L+1)$ filling for
ABA-BC). Fig. 1 summarizes our results for ABA-BC. Below and at the critical
value $g_{c}=0.5\,|\log t_{2}/t_{1}|$, the imbalance saturates with system
size. For $g<g_{c}$, the saturating value is reached exponentially fast. At
the critical point, the convergence becomes algebraic, with an exponent which
depends on the $t_{1}/t_{2}$ ratio and appears to peak at $t_{1}=t_{2}$ (see
Appendix C for more details). Above $g_{c}$, the imbalance scales linearly
with $L$, marking a fundamental anisotropy in the thermodynamic limit. We also
observe slowly decaying oscillations around the linear regime. The relative
amplitude of the oscillations, as well as their period, increases with
decreasing $g$. The limit $g\rightarrow g_{c}^{+}$ appears to correspond to a
divergence of the period of the oscillation, as well as a vanishing of the
linear term. Due to the instability of the non-normal Hamiltonians, or
alternatively the high amplitude of the coefficients in the similarity
transformation $\mathcal{U}_{g}$, we stress the importance of using high-
precision libraries to distinguish between finite-size Chen  _et al._ (2019)
and finite-precision effects. The system sizes shown here require a precision
of up to several hundred of digits, hence the use of the ABA-BC where we can
bypass diagonalization of the single-particle Hamiltonian. The same results
were obtained for OBC.

Remarkably, the critical value $g_{c}$ for the _open_ system corresponds to
the gap closing of the _periodic_ system. The single-particle Hamiltonian with
OBC or ABA-BC remains gapped when crossing $g_{c}$222In fact the energy
spectrum is independent of $g$, and its eigenvectors vary smoothly. The abrupt
change in the imbalance (discontinuous in the thermodynamic limit) directly
arises from the Pauli principle.

To understand the source of the discontinuity, we summarize here how the many-
body state is obtained from the single-particle orbitals. Let us denote
$\\{\ket{R_{j}}\\}$ the right eigenvectors of the single-particle Hamiltonian,
and $c^{\dagger}_{R,j}=\vec{c}\,^{\dagger}\ket{R_{j}}$, the corresponding
orbitals. The eigenvectors of the many-body Hamiltonian are given by

$\ket{\Psi}\propto c^{\dagger}_{R,i_{1}}c^{\dagger}_{R,i_{2}}...\ket{0}.$ (7)

Due to the non-orthonormality of the $\\{\ket{R_{j}}\\}$’s, the orbitals do
not respect the conventional fermionic anticommutation relation and the right
side of Eq. (7) is not normalized. It is convenient to instead work with
$\\{\ket{Q_{i_{j}}}\\}$ — an orthonormal family generating the occupied
$\\{\ket{R_{i_{j}}}\\}$ — and the corresponding orbitals
$\\{q^{\dagger}_{i_{j}}\\}$. They can be obtained by Gram-Schmidt
orthonormalization. We then have

$\ket{\Psi}=q^{\dagger}_{i_{1}}q^{\dagger}_{i_{2}}...\ket{0}.$ (8)

The $q_{i_{j}}$ now obey standard anticommutation relations. Due to the
orthonormalization, the smooth deformation of the original eigenvectors when
$g$ crosses $g_{c}$ is amplified as the norm of the intermediate states can be
arbitrarily close to zero.

We now focus on the saturating regime, $g<g_{c}$. Note that the saturation of
the imbalance depends on the fine structure of the orbitals. Taking the
corresponding PBC Hermitian state instead leads to a linear scaling of the
imbalance after applying $\mathcal{U}_{g}$. Generalized Brillouin zone (GBZ)
approaches Yao and Wang (2018); Yokomizo and Murakami (2019); Yang  _et al._
(2020); Kawabata  _et al._ (2020b) give the same results. It is informative to
briefly study a simple example of saturation. Consider $L$ independent
orbitals

$\ket{R_{n}}=\sum\limits_{j=1}^{L}a_{j,n}\ket{j}\otimes\ket{A}+b_{j,n}\ket{j}\otimes\ket{B}$
(9)

such that $a_{j,n}=\alpha b_{j,n}$ for $1\leq n\leq L$. By dimensional
arguments, we obtain

$\displaystyle\text{Span}(\ket{R_{1}},...,\ket{R_{L}})=\text{Span}(\ket{1}\otimes(\ket{A}$
(10)
$\displaystyle+\alpha\ket{B}),...,\ket{L}\otimes(\ket{A}+\alpha\ket{B})).$

The many-body state with $L$ occupied orbitals can therefore be rewritten as

$\ket{\Psi}=\frac{1}{(\sqrt{1+\alpha^{2}})^{L}}\prod_{j}(c^{\dagger}_{j,A}+\alpha
c^{\dagger}_{j,B})\Ket{0}$ (11)

independently of the exact form of $a_{j,n}$333We only require them to form an
independent family.. This also holds when the orbitals are all exponentially
localized at a boundary; the imbalance is constant and equal to zero. As all
orbitals have the same local structures, the Pauli principle forces electrons
to spread out through the system.

The suppression of the scaling is observed only at half-filling in two-band
models. If the low-energy band is only partially occupied (resp. the high-
energy band is partially occupied), the imbalance grows linearly with system
size. This can be immediately deduced from the toy model in Eq. (9): if we do
not occupy the full translation-invariant subspace in Eq. (10), the imbalance
survives. In the nH-SSH model, for $g<g_{c}$, the periodic single-particle
spectrum has a line gap, while for $g>g_{c}$, it forms a single band with no
line gap and therefore its many-body spectrum is gapless. Let
$\\{\mathcal{E}_{j}^{\mathrm{PBC/OBC}}\\}$ be the set of energy bands with PBC
or OBC, respectively, and the bands are separated by line gaps. Following Ref.
Okuma  _et al._ , 2020, the OBC and PBC bands can be deformed into each other
(merging several OBC bands to form a single one if required), up to some small
set of states. We propose the following conjecture:

###### Conjecture 1

For non-interacting systems exhibiting single-particle skin effect, if the
sets of orbitals occupied in a many-body state $\ket{\Psi}$ can be mapped to
fully occupied PBC bands (up to a number of orbitals of measure 0 in the
thermodynamic limit), the many-body skin effect is suppressed. Otherwise, the
imbalance in $\ket{\Psi}$ will scale linearly with system size.

Beyond the SSH model: disorder, interactions and symmetries:  Our conjecture
implies that the many-body skin effect has a topological origin that can be
predicted from considering the periodic system. It is generally not the case
for topological properties of nH systems which can be obtained from a GBZ
approach or from the bulk of the OBC system.

To further show the resilience of the skin effect, we investigate the effect
of disorder Hatano and Nelson (1996); Goldsheid and Khoruzhenko (1998);
Yusipov  _et al._ (2017); Tzortzakakis  _et al._ (2020); Huang and Shklovskii
(2020); Weidemann  _et al._ (2021). For single-particle states, skin effect
and disorder compete to localize states either at the boundaries or within the
bulk of the system. When the disorder-induced localization length is larger
than the one induced by the breaking of reciprocity ($g^{-1}$ in our example),
the single-particle states are Anderson-localized. They then occupy arbitrary
positions in the lattice which therefore limits the scaling of the imbalance.
On the other hand, the saturation is highly sensitive to the structure of the
orbitals and therefore its survival is unclear. In Fig. 3, we show the
imbalance in a disordered nH-SSH model with a random chemical potential

$-\sum\limits_{j}\mu_{j}^{A}c^{\dagger}_{j,A}c_{j,A}+\mu_{j}^{B}c^{\dagger}_{j,B}c_{j,B},\quad\mu_{j}^{\alpha}\in[-W,W]$
(12)

for several values of the disorder strength $W$. We consider two different
disorder configurations: (a) $\mu_{j}^{B}=-\mu_{j}^{A}$, acting as a local
chemical potential, or (b) $\mu_{j}^{B}$ and $\mu_{j}^{A}$ independent.
Saturated and linear regimes are resilient to the presence of small disorder
in both setups. For configuration (a), the disorder increases the critical
value of $g$ by opening a line gap between the bands of the corresponding PBC
models (see Fig. 3(c, d)). At small $W$, similar results are obtained for
configuration (b). However, at large $W$, the disorder closes the line gap at
(and below) $g_{c}$ and the imbalance scales linearly accordingly.

Figure 3: (a, b) Scaling of the imbalance for $t_{1}/t_{2}=2$, $g=g_{c}$ and
$g=g_{c}\pm 0.01$ for the two disorder configurations defined in Eq. (12) with
W = $0.1,0.5,2$, averaged over $10\,000$ disorder realizations. (c, d) Typical
PBC energy spectrum for the two disorder configurations at $g=g_{c}$ and (c)
$W=0.5$ or (d) $W=2$. The imbalance remains robust at small $W$ as the
disorder opens a line gap. For (b), at $W=2$, the line gap closes and the
imbalance grows linearly.

In addition, we investigate the effect of interactions in the two scaling
regimes. We use DMRG Hauschild and Pollmann (2018) to access larger system
sizes with OBC before applying the similarity transformation which is a matrix
product operator of bond dimension $1$. We are mainly limited by the floating
point precision in the DMRG. We add to the Hamiltonian in Eq. (1) the
following interaction:

$\displaystyle
H_{\mathrm{int}}=U\sum\limits_{j}\left(n_{j,A}-\frac{1}{2}\right)\left(n_{j,B}-\frac{1}{2}\right)$
(13)
$\displaystyle+\left(n_{j,B}-\frac{1}{2}\right)\left(n_{j+1,A}-\frac{1}{2}\right).$

For $U$ real, the system remains gaugeable to a Hermitian model by the
similarity transformation given in Eq. (2). In the limit $t_{1}=t_{2}$ (and
$g=0$), the model is equivalent to the XXZ chain without transverse field. In
Fig. 4, we show the scaling of the imbalance for (a) attractive and (b)
repulsive interactions. Both linear and saturated regimes survive the presence
of small interactions. For attractive interactions ($U<0$), fermions prefer to
occupy neighbouring orbitals, and therefore the critical $g_{c}$ decreases.
Conversely, for repulsive interactions ($U>0$), the interactions favor the
electrons spreading throughout the system, so $g_{c}$ increases. The notion of
bands is no longer well-defined in the presence of interactions, but systems
without a single-particle line gap have a gapless many-body spectrum. We
verify that the variation of the critical $g_{c}$ matches a shift of the gap
closing point. We use the response of the periodic system to flux insertion to
detect the presence of a gap given the limited system sizes we have access to.

Figure 4: Imbalance scaling in the presence of (a) attractive and (b)
repulsive interactions with $t_{1}/t_{2}=\sqrt{2}$. A smaller $t_{1}/t_{2}$
ratio was used to reach larger sizes. The lines correspond to the DMRG data,
while the superimposed points are the ED data up to 24 sites. Due to
oscillations with a period four, we show the results only for every forth
length. When $U<0$, the critical $g_{c}$ decreases and the slope of
$\mathcal{I}$ increases. It is in contrast to positive $U$.

In the Appendices, we show that our results are also valid when the system is
no longer gaugeable to a Hermitian limit or when we break particle-hole
symmetry by introducing a third band.

Conclusions:  In this Letter, we have investigated the skin effect in many-
body wavefunctions. Generically, for fermionic systems, the single-particle
skin effect translates into a linear scaling of the imbalance. However, we
have numerically shown that if the orbitals occupied in an OBC many-body state
correspond to fully occupied PBC bands, the skin effect is suppressed in the
thermodynamic limit. Remarkably, this sharp transition occurs while the OBC
spectrum remains completely gapped. We have demonstrated that the suppression
of the skin effect, despite being sensitive to the details of the orbitals,
survives both the presence of disorder and interactions. This strongly implies
a topological origin of this phenomenon, similar to the topological nature of
the nH skin effect in single-particle Hamiltonians. An analytical proof of
this connection to topology is left for future work. Our results pave the way
to a better understanding of the skin effect in higher dimensions, with the
interplay between the dimensionality of the Fermi surface and other forms of
skin effect.

###### Acknowledgements.

The authors thank Jens Bardarson, Jérémy Bensadon, Tomáš Bzdušek, Frédéric
Mila, Titus Neupert, Nicolas Regnault, and Songbo Zhang for the useful
discussions. Exact diagonalization studies were performed with QuSpin Weinberg
and Bukov (2017, 2019) and DMRG calculations were done using TeNPy Library
(version 0.8.4) Hauschild and Pollmann (2018). To achieve an arbitrary
precision, we employed mpmath library Johansson  _et al._ (2021). This work
was supported by a grant from the Swiss National Supercomputing Centre (CSCS)
under project s1008. F.A. acknowledges support from the KAUST Gifted Student
Program and the EPFL Research Internship Program.

## References

  * Barton  _et al._ (2018) David R. Barton, Hadiseh Alaeian, Mark Lawrence, and Jennifer Dionne, “Broadband and wide-angle nonreciprocity with a non-hermitian metamaterial,” Phys. Rev. B 97, 045432 (2018).
  * Brandenbourger  _et al._ (2019) Martin Brandenbourger, Xander Locsin, Edan Lerner, and Corentin Coulais, “Non-reciprocal robotic metamaterials,” Nature Communications 10, 4608 (2019).
  * Ghatak  _et al._ (2020) Ananya Ghatak, Martin Brandenbourger, Jasper van Wezel, and Corentin Coulais, “Observation of non-hermitian topology and its bulk–edge correspondence in an active mechanical metamaterial,” Proceedings of the National Academy of Sciences 117, 29561–29568 (2020), https://www.pnas.org/content/117/47/29561.full.pdf .
  * Scheibner  _et al._ (2020) Colin Scheibner, William T. M. Irvine, and Vincenzo Vitelli, “Non-hermitian band topology and skin modes in active elastic media,” Phys. Rev. Lett. 125, 118001 (2020).
  * Zhou and Zhang (2020) Di Zhou and Junyi Zhang, “Non-hermitian topological metamaterials with odd elasticity,” Phys. Rev. Research 2, 023173 (2020).
  * Rao  _et al._ (2021) J.W. Rao, Y.T. Zhao, Y.S. Gui, X.L. Fan, D.S. Xue, and C.-M. Hu, “Controlling microwaves in non-hermitian metamaterials,” Phys. Rev. Applied 15, L021003 (2021).
  * Chen  _et al._ (2021) Yangyang Chen, Xiaopeng Li, Colin Scheibner, Vincenzo Vitelli, and Guoliang Huang, “Realization of active metamaterials with odd micropolar elasticity,” Nature Communications 12, 5935 (2021).
  * Regensburger  _et al._ (2012) Alois Regensburger, Christoph Bersch, Mohammad-Ali Miri, Georgy Onishchukov, Demetrios N. Christodoulides, and Ulf Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167–171 (2012).
  * Peng  _et al._ (2014) Bo Peng, Şahin Kaya Özdemir, Fuchuan Lei, Faraz Monifi, Mariagiovanna Gianfreda, Gui Lu Long, Shanhui Fan, Franco Nori, Carl M. Bender, and Lan Yang, “Parity–time-symmetric whispering-gallery microcavities,” Nature Physics 10, 394–398 (2014).
  * Zeuner  _et al._ (2015) Julia M. Zeuner, Mikael C. Rechtsman, Yonatan Plotnik, Yaakov Lumer, Stefan Nolte, Mark S. Rudner, Mordechai Segev, and Alexander Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
  * Feng  _et al._ (2017) Liang Feng, Ramy El-Ganainy, and Li Ge, “Non-hermitian photonics based on parity–time symmetry,” Nature Photonics 11, 752–762 (2017).
  * Zhou  _et al._ (2018) Hengyun Zhou, Chao Peng, Yoseob Yoon, Chia Wei Hsu, Keith A. Nelson, Liang Fu, John D. Joannopoulos, Marin Soljačić, and Bo Zhen, “Observation of bulk fermi arc and polarization half charge from paired exceptional points,” Science 359, 1009–1012 (2018), https://www.science.org/doi/pdf/10.1126/science.aap9859 .
  * Harari  _et al._ (2018) Gal Harari, Miguel A. Bandres, Yaakov Lumer, Mikael C. Rechtsman, Y. D. Chong, Mercedeh Khajavikhan, Demetrios N. Christodoulides, and Mordechai Segev, “Topological insulator laser: Theory,” Science 359, eaar4003 (2018), https://www.science.org/doi/pdf/10.1126/science.aar4003 .
  * Bandres  _et al._ (2018) Miguel A. Bandres, Steffen Wittek, Gal Harari, Midya Parto, Jinhan Ren, Mordechai Segev, Demetrios N. Christodoulides, and Mercedeh Khajavikhan, “Topological insulator laser: Experiments,” Science 359, eaar4005 (2018), https://www.science.org/doi/pdf/10.1126/science.aar4005 .
  * Pan  _et al._ (2018) Mingsen Pan, Han Zhao, Pei Miao, Stefano Longhi, and Liang Feng, “Photonic zero mode in a non-hermitian photonic lattice,” Nature Communications 9, 1308 (2018).
  * El-Ganainy  _et al._ (2018) Ramy El-Ganainy, Konstantinos G. Makris, Mercedeh Khajavikhan, Ziad H. Musslimani, Stefan Rotter, and Demetrios N. Christodoulides, “Non-hermitian physics and pt symmetry,” Nature Physics 14, 11–19 (2018).
  * Miri and Alù (2019) Mohammad-Ali Miri and Andrea Alù, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019), https://www.science.org/doi/pdf/10.1126/science.aar7709 .
  * Ezawa (2019a) Motohiko Ezawa, “Electric circuits for non-hermitian chern insulators,” Phys. Rev. B 100, 081401 (2019a).
  * Ezawa (2019b) Motohiko Ezawa, “Non-hermitian boundary and interface states in nonreciprocal higher-order topological metals and electrical circuits,” Phys. Rev. B 99, 121411 (2019b).
  * Ezawa (2019c) Motohiko Ezawa, “Non-hermitian higher-order topological states in nonreciprocal and reciprocal systems with their electric-circuit realization,” Phys. Rev. B 99, 201411 (2019c).
  * Hofmann  _et al._ (2020) Tobias Hofmann, Tobias Helbig, Frank Schindler, Nora Salgo, Marta Brzezińska, Martin Greiter, Tobias Kiessling, David Wolf, Achim Vollhardt, Anton Kabaši, Ching Hua Lee, Ante Bilušić, Ronny Thomale, and Titus Neupert, “Reciprocal skin effect and its realization in a topolectrical circuit,” Phys. Rev. Research 2, 023265 (2020).
  * Helbig  _et al._ (2020) T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, “Generalized bulk–boundary correspondence in non-hermitian topolectrical circuits,” Nature Physics 16, 747–750 (2020).
  * Heiss (2004) WD Heiss, “Exceptional points of non-hermitian operators,” Journal of Physics A: Mathematical and General 37, 2455 (2004).
  * Leykam  _et al._ (2017) Daniel Leykam, Konstantin Y. Bliokh, Chunli Huang, Y. D. Chong, and Franco Nori, “Edge modes, degeneracies, and topological numbers in non-hermitian systems,” Phys. Rev. Lett. 118, 040401 (2017).
  * Martinez Alvarez  _et al._ (2018) V. M. Martinez Alvarez, J. E. Barrios Vargas, and L. E. F. Foa Torres, “Non-hermitian robust edge states in one dimension: Anomalous localization and eigenspace condensation at exceptional points,” Phys. Rev. B 97, 121401 (2018).
  * Rudner and Levitov (2009) M. S. Rudner and L. S. Levitov, “Topological transition in a non-hermitian quantum walk,” Phys. Rev. Lett. 102, 065703 (2009).
  * Esaki  _et al._ (2011) Kenta Esaki, Masatoshi Sato, Kazuki Hasebe, and Mahito Kohmoto, “Edge states and topological phases in non-hermitian systems,” Phys. Rev. B 84, 205128 (2011).
  * Lee (2016) Tony E. Lee, “Anomalous edge state in a non-hermitian lattice,” Phys. Rev. Lett. 116, 133903 (2016).
  * Shen  _et al._ (2018) Huitao Shen, Bo Zhen, and Liang Fu, “Topological band theory for non-hermitian hamiltonians,” Phys. Rev. Lett. 120, 146402 (2018).
  * Liu  _et al._ (2019) Tao Liu, Yu-Ran Zhang, Qing Ai, Zongping Gong, Kohei Kawabata, Masahito Ueda, and Franco Nori, “Second-order topological phases in non-hermitian systems,” Phys. Rev. Lett. 122, 076801 (2019).
  * Zhou and Lee (2019) Hengyun Zhou and Jong Yeon Lee, “Periodic table for topological bands with non-hermitian symmetries,” Phys. Rev. B 99, 235112 (2019).
  * Kawabata  _et al._ (2019) Kohei Kawabata, Ken Shiozaki, Masahito Ueda, and Masatoshi Sato, “Symmetry and topology in non-hermitian physics,” Phys. Rev. X 9, 041015 (2019).
  * Zhang  _et al._ (2020a) Kai Zhang, Zhesen Yang, and Chen Fang, “Correspondence between winding numbers and skin modes in non-hermitian systems,” Phys. Rev. Lett. 125, 126402 (2020a).
  * Yao and Wang (2018) Shunyu Yao and Zhong Wang, “Edge states and topological invariants of non-hermitian systems,” Phys. Rev. Lett. 121, 086803 (2018).
  * Lee  _et al._ (2019) Ching Hua Lee, Linhu Li, and Jiangbin Gong, “Hybrid higher-order skin-topological modes in nonreciprocal systems,” Phys. Rev. Lett. 123, 016805 (2019).
  * Lee and Thomale (2019) Ching Hua Lee and Ronny Thomale, “Anatomy of skin modes and topology in non-hermitian systems,” Phys. Rev. B 99, 201103 (2019).
  * Borgnia  _et al._ (2020) Dan S. Borgnia, Alex Jura Kruchkov, and Robert-Jan Slager, “Non-hermitian boundary modes and topology,” Phys. Rev. Lett. 124, 056802 (2020).
  * Xiong (2018) Ye Xiong, “Why does bulk boundary correspondence fail in some non-hermitian topological models,”  2, 035043 (2018).
  * Kawabata  _et al._ (2018) Kohei Kawabata, Ken Shiozaki, and Masahito Ueda, “Anomalous helical edge states in a non-hermitian chern insulator,” Phys. Rev. B 98, 165148 (2018).
  * Gong  _et al._ (2018) Zongping Gong, Yuto Ashida, Kohei Kawabata, Kazuaki Takasan, Sho Higashikawa, and Masahito Ueda, “Topological phases of non-hermitian systems,” Phys. Rev. X 8, 031079 (2018).
  * Yao  _et al._ (2018) Shunyu Yao, Fei Song, and Zhong Wang, “Non-hermitian chern bands,” Phys. Rev. Lett. 121, 136802 (2018).
  * Kunst  _et al._ (2018) Flore K. Kunst, Elisabet Edvardsson, Jan Carl Budich, and Emil J. Bergholtz, “Biorthogonal bulk-boundary correspondence in non-hermitian systems,” Phys. Rev. Lett. 121, 026808 (2018).
  * Herviou  _et al._ (2019a) Loïc Herviou, Jens H. Bardarson, and Nicolas Regnault, “Defining a bulk-edge correspondence for non-hermitian hamiltonians via singular-value decomposition,” Phys. Rev. A 99, 052118 (2019a).
  * Jin and Song (2019) L. Jin and Z. Song, “Bulk-boundary correspondence in a non-hermitian system in one dimension with chiral inversion symmetry,” Phys. Rev. B 99, 081103 (2019).
  * Song  _et al._ (2019) Fei Song, Shunyu Yao, and Zhong Wang, “Non-hermitian topological invariants in real space,” Phys. Rev. Lett. 123, 246801 (2019).
  * Yokomizo and Murakami (2019) Kazuki Yokomizo and Shuichi Murakami, “Non-bloch band theory of non-hermitian systems,” Phys. Rev. Lett. 123, 066404 (2019).
  * Ozcakmakli Turker and Yuce (2019) Z. Ozcakmakli Turker and C. Yuce, “Open and closed boundaries in non-hermitian topological systems,” Phys. Rev. A 99, 022127 (2019).
  * Edvardsson  _et al._ (2019) Elisabet Edvardsson, Flore K. Kunst, and Emil J. Bergholtz, “Non-hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence,” Phys. Rev. B 99, 081302 (2019).
  * Zirnstein  _et al._ (2021) Heinrich-Gregor Zirnstein, Gil Refael, and Bernd Rosenow, “Bulk-boundary correspondence for non-hermitian hamiltonians via green functions,” Phys. Rev. Lett. 126, 216407 (2021).
  * Okuma  _et al._ (2020) Nobuyuki Okuma, Kohei Kawabata, Ken Shiozaki, and Masatoshi Sato, “Topological origin of non-hermitian skin effects,” Phys. Rev. Lett. 124, 086801 (2020).
  * Song  _et al._ (2020) Yiling Song, Weiwei Liu, Lingzhi Zheng, Yicong Zhang, Bing Wang, and Peixiang Lu, “Two-dimensional non-hermitian skin effect in a synthetic photonic lattice,” Phys. Rev. Applied 14, 064076 (2020).
  * Xiao  _et al._ (2020) Lei Xiao, Tianshu Deng, Kunkun Wang, Gaoyan Zhu, Zhong Wang, Wei Yi, and Peng Xue, “Non-hermitian bulk–boundary correspondence in quantum dynamics,” Nature Physics 16, 761–766 (2020).
  * Zhang  _et al._ (2021a) Li Zhang, Yihao Yang, Yong Ge, Yi jun Guan, Qiaolu Chen, Qinghui Yan, Fujia Chen, Rui Xi, Yuanzhen Li, Ding Jia, Shou qi Yuan, Hong xiang Sun, Hongsheng Chen, and Baile Zhang, “Acoustic non-hermitian skin effect from twisted winding topology,” (2021a), arXiv:2104.08844 [physics.app-ph] .
  * Liu  _et al._ (2021) Shuo Liu, Ruiwen Shao, Shaojie Ma, Lei Zhang, Oubo You, Haotian Wu, Yuan Jiang Xiang, Tie Jun Cui, and Shuang Zhang, “Non-hermitian skin effect in a non-hermitian electrical circuit,” Research 2021, 5608038 (2021).
  * Yi and Yang (2020) Yifei Yi and Zhesen Yang, “Non-hermitian skin modes induced by on-site dissipations and chiral tunneling effect,” Phys. Rev. Lett. 125, 186802 (2020).
  * Rui  _et al._ (2019) W. B. Rui, Moritz M. Hirschmann, and Andreas P. Schnyder, “$\mathcal{PT}$-symmetric non-hermitian dirac semimetals,” Phys. Rev. B 100, 245116 (2019).
  * Yoshida  _et al._ (2020) Tsuneya Yoshida, Tomonari Mizoguchi, and Yasuhiro Hatsugai, “Mirror skin effect and its electric circuit simulation,” Phys. Rev. Research 2, 022062 (2020).
  * Kawabata  _et al._ (2020a) Kohei Kawabata, Masatoshi Sato, and Ken Shiozaki, “Higher-order non-hermitian skin effect,” Phys. Rev. B 102, 205118 (2020a).
  * Okugawa  _et al._ (2020) Ryo Okugawa, Ryo Takahashi, and Kazuki Yokomizo, “Second-order topological non-hermitian skin effects,” Phys. Rev. B 102, 241202 (2020).
  * Fu  _et al._ (2021) Yongxu Fu, Jihan Hu, and Shaolong Wan, “Non-hermitian second-order skin and topological modes,” Phys. Rev. B 103, 045420 (2021).
  * Palacios  _et al._ (2021) Lucas S. Palacios, Serguei Tchoumakov, Maria Guix, Ignacio Pagonabarraga, Samuel Sánchez, and Adolfo G. Grushin, “Guided accumulation of active particles by topological design of a second-order skin effect,” Nature Communications 12, 4691 (2021).
  * Zhang  _et al._ (2021b) Xiujuan Zhang, Yuan Tian, Jian-Hua Jiang, Ming-Hui Lu, and Yan-Feng Chen, “Observation of higher-order non-hermitian skin effect,” Nature Communications 12, 5377 (2021b).
  * Zou  _et al._ (2021) Deyuan Zou, Tian Chen, Wenjing He, Jiacheng Bao, Ching Hua Lee, Houjun Sun, and Xiangdong Zhang, “Observation of hybrid higher-order skin-topological effect in non-hermitian topolectrical circuits,” (2021), arXiv:2104.11260 [cond-mat.mes-hall] .
  * Li  _et al._ (2020) Linhu Li, Ching Hua Lee, Sen Mu, and Jiangbin Gong, “Critical non-hermitian skin effect,” Nature Communications 11, 5491 (2020).
  * Yokomizo and Murakami (2021) Kazuki Yokomizo and Shuichi Murakami, “Scaling rule for the critical non-hermitian skin effect,” Phys. Rev. B 104, 165117 (2021).
  * Lee  _et al._ (2020) Eunwoo Lee, Hyunjik Lee, and Bohm-Jung Yang, “Many-body approach to non-hermitian physics in fermionic systems,” Phys. Rev. B 101, 121109 (2020).
  * Liu  _et al._ (2020) Tao Liu, James Jun He, Tsuneya Yoshida, Ze-Liang Xiang, and Franco Nori, “Non-hermitian topological mott insulators in one-dimensional fermionic superlattices,” Phys. Rev. B 102, 235151 (2020).
  * Mu  _et al._ (2020) Sen Mu, Ching Hua Lee, Linhu Li, and Jiangbin Gong, “Emergent fermi surface in a many-body non-hermitian fermionic chain,” Phys. Rev. B 102, 081115 (2020).
  * Shen and Lee (2021) Ruizhe Shen and Ching Hua Lee, “Non-hermitian skin clusters from strong interactions,” (2021), arXiv:2107.03414 [cond-mat.str-el] .
  * Zhang  _et al._ (2021c) Weixuan Zhang, Fengxiao Di, Hao Yuan, Haiteng Wang, Xingen Zheng, Lu He1, Houjun Sun, and Xiangdong Zhang, “Observation of non-hermitian many-body skin effects in hilbert space,” (2021c), arXiv:2109.08334 [cond-mat.mes-hall] .
  * Yoshida (2021) Tsuneya Yoshida, “Real-space dynamical mean field theory study of non-hermitian skin effect for correlated systems: Analysis based on pseudospectrum,” Phys. Rev. B 103, 125145 (2021).
  * Xi  _et al._ (2021) Wenjie Xi, Zhi-Hao Zhang, Zheng-Cheng Gu, and Wei-Qiang Chen, “Classification of topological phases in one dimensional interacting non-hermitian systems and emergent unitarity,” Science Bulletin 66, 1731–1739 (2021).
  * Cao  _et al._ (2021) Kui Cao, Qian Du, Xiao-Ran Wang, and Su-Peng Kou, “Physics of many-body nonreciprocal model: From non-hermitian skin effect to quantum maxwell’s pressure-demon effect,” (2021), arXiv:2109.03690 [cond-mat.other] .
  * Zhang  _et al._ (2020b) Dan-Wei Zhang, Yu-Lian Chen, Guo-Qing Zhang, Li-Jun Lang, Zhi Li, and Shi-Liang Zhu, “Skin superfluid, topological mott insulators, and asymmetric dynamics in an interacting non-hermitian aubry-andré-harper model,” Phys. Rev. B 101, 235150 (2020b).
  * Xu and Chen (2020) Zhihao Xu and Shu Chen, “Topological bose-mott insulators in one-dimensional non-hermitian superlattices,” Phys. Rev. B 102, 035153 (2020).
  * Lieu (2018) Simon Lieu, “Topological phases in the non-hermitian su-schrieffer-heeger model,” Phys. Rev. B 97, 045106 (2018).
  * Kouachi (2006) Said Kouachi, “Eigenvalues and eigenvectors of tridiagonal matrices,” The Electronic Journal of Linear Algebra 15, 115–133 (2006).
  * Dalibard  _et al._ (1992) Jean Dalibard, Yvan Castin, and Klaus Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
  * Mølmer  _et al._ (1993) Klaus Mølmer, Yvan Castin, and Jean Dalibard, “Monte carlo wave-function method in quantum optics,” J. Opt. Soc. Am. B 10, 524–538 (1993).
  * Herviou  _et al._ (2019b) Loïc Herviou, Nicolas Regnault, and Jens H. Bardarson, “Entanglement spectrum and symmetries in non-Hermitian fermionic non-interacting models,” SciPost Phys. 7, 69 (2019b).
  * Note (1) $N_{F}>0.5$ can be deduced from $N_{F}<0.5$.
  * Hatano and Nelson (1996) Naomichi Hatano and David R. Nelson, “Localization transitions in non-hermitian quantum mechanics,” Phys. Rev. Lett. 77, 570–573 (1996).
  * Chen  _et al._ (2019) Rui Chen, Chui-Zhen Chen, Bin Zhou, and Dong-Hui Xu, “Finite-size effects in non-hermitian topological systems,” Phys. Rev. B 99, 155431 (2019).
  * Note (2) In fact the energy spectrum is independent of $g$.
  * Yang  _et al._ (2020) Zhesen Yang, Kai Zhang, Chen Fang, and Jiangping Hu, “Non-hermitian bulk-boundary correspondence and auxiliary generalized brillouin zone theory,” Phys. Rev. Lett. 125, 226402 (2020).
  * Kawabata  _et al._ (2020b) Kohei Kawabata, Nobuyuki Okuma, and Masatoshi Sato, “Non-bloch band theory of non-hermitian hamiltonians in the symplectic class,” Phys. Rev. B 101, 195147 (2020b).
  * Note (3) We only require them to form an independent family.
  * Goldsheid and Khoruzhenko (1998) Ilya Ya. Goldsheid and Boris A. Khoruzhenko, “Distribution of eigenvalues in non-hermitian anderson models,” Phys. Rev. Lett. 80, 2897–2900 (1998).
  * Yusipov  _et al._ (2017) I. Yusipov, T. Laptyeva, S. Denisov, and M. Ivanchenko, “Localization in open quantum systems,” Phys. Rev. Lett. 118, 070402 (2017).
  * Tzortzakakis  _et al._ (2020) A. F. Tzortzakakis, K. G. Makris, and E. N. Economou, “Non-hermitian disorder in two-dimensional optical lattices,” Phys. Rev. B 101, 014202 (2020).
  * Huang and Shklovskii (2020) Yi Huang and B. I. Shklovskii, “Anderson transition in three-dimensional systems with non-hermitian disorder,” Phys. Rev. B 101, 014204 (2020).
  * Weidemann  _et al._ (2021) Sebastian Weidemann, Mark Kremer, Stefano Longhi, and Alexander Szameit, “Coexistence of dynamical delocalization and spectral localization through stochastic dissipation,” Nature Photonics 15, 576–581 (2021).
  * Hauschild and Pollmann (2018) Johannes Hauschild and Frank Pollmann, “Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy),” SciPost Phys. Lect. Notes , 5 (2018), code available from https://github.com/tenpy/tenpy, arXiv:1805.00055 .
  * Weinberg and Bukov (2017) Phillip Weinberg and Marin Bukov, “QuSpin: a Python Package for Dynamics and Exact Diagonalisation of Quantum Many Body Systems part I: spin chains,” SciPost Phys. 2, 003 (2017).
  * Weinberg and Bukov (2019) Phillip Weinberg and Marin Bukov, “QuSpin: a Python Package for Dynamics and Exact Diagonalisation of Quantum Many Body Systems. Part II: bosons, fermions and higher spins,” SciPost Phys. 7, 20 (2019).
  * Johansson  _et al._ (2021) Fredrik Johansson _et al._ ,  _mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 1.2.0)_ (2021), https://github.com/fredrik-johansson/mpmath.
  * Peschel (2003) Ingo Peschel, “Calculation of reduced density matrices from correlation functions,” Journal of Physics A: Mathematical and General 36, L205–L208 (2003).
  * Krause (1994) Gerd M. Krause, “Bounds for the variation of matrix eigenvalues and polynomial roots,” Linear Algebra and its Applications 208-209, 73–82 (1994).

##  Appendix A Analytical solution for ABA-BC nH-SSH model

The nH-SSH model with ABA-BC takes the form:

$H=\begin{pmatrix}0&t_{1}e^{-g}&0&0&\cdots&0\\\
t_{1}e^{g}&0&t_{2}e^{-g}&0&\cdots&0\\\ 0&t_{2}e^{g}&0&\ddots&\ddots&\vdots\\\
0&0&\ddots&\ddots&\ddots&0\\\
\vdots&\vdots&\ddots&\ddots&\ddots&t_{2}e^{-g}\\\
0&\cdots&\cdots&0&t_{2}e^{g}&0\end{pmatrix}.$ (14)

Following Ref. Kouachi, 2006, we can write explicitly its eigenvalues and
eigenstates. Note that due to the choice of boundary conditions, there is
always a zero-energy edge mode. The eigenvalues are given by:

$E_{n}=\begin{cases}\sqrt{t_{1}^{2}+t_{2}^{2}+2t_{1}t_{2}\cos\theta_{n}},\,&n=1,\ldots,L\\\
-\sqrt{t_{1}^{2}+t_{2}^{2}+2t_{1}t_{2}\cos\theta_{n}},\,&n=L+1,\ldots,2L\\\
0,\,&n=2L+1\end{cases}$ (15)

and the corresponding eigenstates $u_{j}^{(n)}$ are

$u_{j}^{(n)}=\rho_{j}\begin{cases}t_{1}t_{2}\sin\left(\left[\frac{2L+1-j}{2}+1\right]\theta_{n}\right)+t_{1}^{2}\sin\left(\frac{2L+1-j}{2}\theta_{n}\right),\\\
\qquad\qquad\qquad\qquad\qquad\qquad\text{if }j\text{ is odd}\\\
-\sqrt{t_{1}t_{2}}E_{n}\sin\left(\left[\frac{2L+1-j}{2}+\frac{1}{2}\right]\theta_{n}\right),\\\
\qquad\qquad\qquad\qquad\qquad\qquad\text{if }j\text{ is even}\end{cases}$
(16)

with

$\rho_{j}=\mycases(\sqrt{t_{1}t_{2}})^{2L}\,e^{g(j-1)},\,&j\text{ is odd}\\\
-(\sqrt{t_{1}t_{2}})^{2L}\sqrt{\frac{t_{1}}{t_{2}}}\,e^{g(j-1)},\,j\text{ is
even}$ (17)

and

$\theta_{n}=\begin{cases}\frac{n\pi}{L+1},\,&1\leq n\leq L\\\
\frac{(n-L)\pi}{L+1},\,&L+1\leq n\leq 2L.\end{cases}$ (18)

The zero energy eigenstate is then given by

$u_{j}^{(2L+1)}=\mycases\left(t_{1}t_{2}e^{2g}\right)^{(j-1)/2}\left(-t_{2}^{2}\right)^{(2L+1-j)/2},\,&j\text{
is odd}\\\ 0,\,j\text{ is even.}$ (19)

##  Appendix B QR decomposition for right eigenvectors

We summarize here the main arguments of Ref. Herviou  _et al._ , 2019b. We
focus on the properties of the right eigenstates of the many-body Hamiltonian.
All observables are taken as in Eq. (4). For non-interacting Hamiltonians, we
can determine the properties of the ground states (and all eigenstates) by
computing the correlation matrix Peschel (2003). The correlation matrix can be
obtained from the eigenvectors of the single-particle Hamiltonian. Let us
consider a Hamiltonian of the form:

$\mathcal{H}=\vec{c}\,^{\dagger}H\vec{c},$ (20)

with $\vec{c}=(c_{1},c_{2},...)$ the vector of annihilation operators whose
index run over all degrees of freedom. If $H$ is diagonalizable, with

$H=\sum\limits_{n}E_{n}\ket{\psi_{n}^{R}}\bra{\psi_{n}^{L}},$ (21)

then we define the operators:

$\vec{r}_{n}^{\dagger}=\vec{c}\,^{\dagger}\ket{\psi^{R}_{n}},\qquad\vec{l}_{n}^{\dagger}=\vec{c}\,^{\dagger}\ket{\psi^{L}_{n}}.$
(22)

They verify the biorthogonal fermionic anticommutation relations:

$\\{r_{m}^{\dagger},l_{n}\\}=\delta_{m,n},$ (23)
$\\{r_{m},l_{n}\\}=\\{r_{m},r_{n}\\}=\\{l_{m},l_{n}\\}=0.$ (24)

The many-body Hamiltonian $\mathcal{H}$ can be rewritten as

$\mathcal{H}=\sum\limits_{n}E_{n}r_{n}^{\dagger}l_{n}.$ (25)

With this expression, we see directly that its right eigenstates are
proportional to

$\ket{\Psi}\propto\prod\limits_{j}r^{\dagger}_{i_{j}}\ket{0}.$ (26)

Due to the non-orthogonality of the diagonalizing basis of nH operators, these
states are not normalized. Moreover, contrary to the Hermitian case,
observables cannot be directly computed from the expression above. Instead,
one needs to orthonormalize the family of occupied orbitals. If
$\\{\ket{\phi_{j}}\\}$ is an orthonormal basis of
$\mathrm{Span}(\\{\ket{\psi^{R}_{i_{j}}}\\}$, and we define

$\vec{q}_{n}^{\dagger}=\vec{c}\,^{\dagger}\ket{\phi_{n}},$ (27)

we see that the $q_{i_{j}}$’s are conventional fermionic operators. The
eigenstate in the form

$\ket{\Psi}=\prod\limits_{j}q^{\dagger}_{j}\ket{0}\propto\prod\limits_{j}r^{\dagger}_{i_{j}}\ket{0}$
(28)

is indeed normalized. $\ket{\Psi}$ is a conventional fermionic Gaussian state,
on which we can apply all the standard tricks.

##  Appendix C Scaling analysis

As discussed in the main text, the imbalance as a function of the system size
exhibits intriguing features depending on the value of $g$. To underline the
universality of our results, we perform a scaling analysis for three $g$
regimes. Below $g_{c}$, the imbalance saturates to a finite and constant
value. Away from the dimerized limits ($t_{1}=0$ or $t_{2}=0$) and from the
gapless point $t_{1}=t_{2}$, the imbalance $\mathcal{I}$ can be well
approximated by an exponential $\mathcal{I}(L)=\mathcal{I}_{\infty}+\lambda
e^{-L/\xi}$, as shown in Fig. 5(a). At $g=g_{c}$, the correlations become
algebraic with an exponent that depends on $t_{1}/t_{2}$, as presented in Fig.
5(b).

Figure 5: Scaling of the imbalance as a function of the system size $L$. (a)
Below $g_{c}$, the imbalance saturates exponentially fast with a correlation
length that varies with both $g$ and $t_{1}/t_{2}$. (b) At $g_{c}$, the
convergence becomes algebraic with an exponent which depends on $t_{1}/t_{2}$.

Finally, in the $g>g_{c}$ regime, we observe oscillations of the imbalance
around the linear scaling (see Fig. 6(a)). To analyze these oscillations, we
study the derivative $d\mathcal{I}/dL$, which clearly shows distinct peaks as
demonstrated in Fig. 6(b). The amplitude of the oscillations slowly decays
with $L$, while their period appears to diverge as $(g-g_{c})^{-1}$: the
saturation can be seen as an infinitely long plateau (see Fig. 6(c)).

Figure 6: (a) Imbalance scaling for several $g>g_{c}$. Each curve exhibits a
sequence of plateaus, with a period depending on the exact value of $g$. To
quantify this dependence, we compute the numerical derivative
$d\mathcal{I}/dL$ and perform a spline interpolation to estimate the positions
of the peaks, shown in (b). The distance between peaks $T$ diverges as
$\sim(g-g_{c})^{-1}$.

##  Appendix D Requirements for numerical precision

Apart from the errors originating from too small system sizes, i.e., finite
size effects, numerical instabilities in nH matrices pose additional
difficulties. 64-bits floating point precision becomes insufficient due to a
combination of large system sizes and large values of the reciprocity-breaking
terms. It is therefore necessary to systematically study how numerical
accuracy affects the results, as it is known that nH matrices are
exponentially unstable Krause (1994); Herviou  _et al._ (2019a). In Fig. 7, we
compare the many-body ground state densities obtained from Gram-Schmidt
orthonormalization at different numerical precisions. At low precision (up to
$150$ digits), the local density close to the edge exceeds $1$, which is
mathematically incorrect for normalized fermionic states. The minimal number
of required decimal digits scales roughly as $gL$.

Figure 7: Many-body densities $|\Psi_{RR}|^{2}$ obtained from the analytical
solutions for ABA-BC with $g=g_{c}+0.1$ and $L=400$ ($801$ sites) at different
numerical accuracies in the Gram-Schmidt procedure. The numerical inaccuracies
lead to incorrect density profiles at the edges of the system. For the given
system size, at least 200 digits precision is required to faithfully represent
the density. A standard double type variable uses 64 bits to store the value,
which translates to around 16 decimal digits.

##  Appendix E Non-gaugeable models

So far, we showed that the relation between occupied orbitals and the scaling
imbalance works for gaugeable models, i.e., models with real OBC spectrum on
which we can apply a local similarity transformation. The single-particle
Hamiltonian (represented in momentum space)

$H(k)=H_{\mathrm{SSH}}(k)+\begin{pmatrix}0&t_{p}e^{g_{p}}e^{2\mathrm{i}k}\\\
t_{p}e^{g_{p}}e^{-2\mathrm{i}k}&0\\\ \end{pmatrix}$ (29)

is not gaugeable when $g_{p}\neq 3g$. The bulk gap closes under the condition
$t_{p}=(t_{1}e^{\pm g}-t_{2}e^{\mp g})e^{\mp g_{p}}$. We verified that the
imbalance indeed changes scaling at the critical value above.

##  Appendix F Three-band models

The existence of two distinct scaling regimes of $\mathcal{I}$ is not
restricted to two-band models. In this Appendix, we build a three-band model
that can have either three bands separated by line gaps, or two asymmetric
bands with a different number of orbitals. These regimes non-trivially break
particle-hole symmetry. To do so, we consider two elementary three-band
models. $H_{1}$ corresponds to a generalized nH-SSH model with three sites per
unit cell:

$H_{1}(k)=\begin{pmatrix}0&t_{1}e^{g}&t_{3}e^{-g}e^{-\mathrm{i}k}\\\
t_{1}e^{-g}&0&t_{2}e^{g}\\\
t_{3}e^{g}e^{\mathrm{i}k}&t_{2}e^{-g}&0\end{pmatrix}.$ (30)

$H_{2}$ is a conventional nH-SSH model to which we attach an extra decoupled
site within a unit cell:

$H_{2}(k)=\begin{pmatrix}\mu_{1}&\tilde{t}_{1}e^{g}+\tilde{t}_{2}e^{-2g}e^{-2\mathrm{i}k}&0\\\
\tilde{t}_{2}e^{2g}e^{2\mathrm{i}k}+\tilde{t}_{1}e^{-g}&\mu_{2}&0\\\
0&0&\mu_{3}\end{pmatrix}.$ (31)

We then study the interpolation $H=(1-\alpha)H_{1}+\alpha H_{2}$. The models
and their eigenspectra are represented in Fig. 8. Whether we consider $1/3$,
$1/2$ or $2/3$-filled ground states, the imbalance follows the conjecture in
the main text for all parameter values. In particular, we verified that the
transition from saturated to linear regime occurs when the system goes from
three separate bands (with PBC, $L$ orbitals per band) to two bands (one with
$2L$ orbitals and another one with $L$ orbitals) in the $1/3$-filled case.

Figure 8: (a) A sketch of real-space hoppings for the two three-band models,
$H_{1}$ and $H_{2}$, together with (b) their spectra. We set $t_{1}=t_{2}=1$,
$t_{3}=0.5$ for $H_{1}$ and $\tilde{t}_{1}=\tilde{t}_{2}=0.5$,
$\mu_{1}=\mu_{2}=-1$, $\mu_{3}=0.5$ for $H_{2}$; $g=0.2$. By linear
interpolation between $H_{1}$ and $H_{2}$, we obtain a model for which the
eigenvalues go from three circles to two.

##  Appendix G Flux insertion in the nH periodic Hamiltonians

To check whether our interacting periodic systems have a many-body gap, we use
flux insertion. Due to the limited system sizes and the lack of similarity
transformation to map the systems back to Hermitian equivalents, scalings of
the gap are unreliable. Instead, we introduce a flux $\phi$ at the boundary of
the system, and study the evolution of the lowest energies when we tune $\phi$
from $0$ to $2\pi$. If the system is gapped, the lowest energy remains
detached from the upper one (and its trajectory is a closed contour in the
complex plane). In the gapless case, the trajectory merges with those
corresponding to higher energy and is no longer closed. In Fig. 9, we show the
trajectories of the two lowest energies in the model given in Eqs. (1) and
(13) with PBC for several values of $U$. The change of behavior (from gapped
to gapless, and vice-versa) matches the change in the scaling of the imbalance
(from saturated to linear regime).

Figure 9: Evolution under flux insertion of the ground energy (dots) and first
excited energy (crosses) in the model given in Eqs. (1) and (13) with PBC for
(a) $g<g_{c}$, (b) $g=g_{c}$, and (c) $g>g_{c}$. The computations were
performed for $U=-1$, $0$ and $1$ and at fixed system size $L=12$ ($24$
orbitals).

##  Appendix H Many-body computation of the imbalance

In this Appendix, we briefly introduce some convenient formulas for computing
the imbalance and its variants in many-body systems. The imbalance
$\mathcal{I}$ can be expressed as:

$\mathcal{I}=\frac{\braket{\Psi(0)}{I\mathcal{U}_{g}^{2}}{\Psi(0)}}{\braket{\Psi(0)}{\mathcal{U}_{g}^{2}}{\Psi(0)}},$
(32)

where $\ket{\Psi(0)}$ is the Hermitian ground state. From this structure, it
is mathematically more convenient to work with

$\tilde{I}_{1}=\frac{1}{L}\sum\limits_{j}\left(2j-1n_{j,A}+2jn_{j,B}\right)-(2L-1).$
(33)

This is a minor generalization of the imbalance introduced in the main text,
with essentially the same properties. It allows us to write:

$\tilde{\mathcal{I}}_{1}=\braket{\tilde{I}_{1}}=\frac{1}{2gL}\partial_{g}\log\braket{\Psi(0)}{\exp(2gL\tilde{I}_{1})}{\Psi(0)}.$
(34)

$\tilde{\mathcal{I}}_{1}$ is therefore proportional to the derivative of the
cumulant generating function of $L\tilde{I}_{1}$. We numerically check that in
the non-interacting case, the probability distribution is

$p(\tilde{I}_{1}=i_{1})\propto\exp\left(-\beta|Li_{1}|\right)\exp\left(-\alpha
i_{1}^{2}\right)$ (35)

with

$\beta=|\log t_{1}/t_{2}|,$ (36)

which explains our results.