#  Non-Hermitian Mott Skin Effect

Tsuneya Yoshida  Department of Physics, Kyoto University, Kyoto 606-8502,
Japan  Song-Bo Zhang  International Center for Quantum Design of Functional
Materials (ICQD), Hefei National Research Center for Physical Sciences at the
Microscale, University of Science and Technology of China, Hefei, Anhui
230026, China  Titus Neupert  Department of Physics, University of Zürich,
Winterthurerstrasse 190, 8057, Zürich, Switzerland  Norio Kawakami  Department
of Physics, Kyoto University, Kyoto 606-8502, Japan  Department of Materials
Engineering Science, Osaka University, Toyonaka 560-8531, Japan  Department of
Physics, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan

###### Abstract

We propose a novel type of skin effects in non-Hermitian quantum many-body
systems which we dub a non-Hermitian Mott skin effect. This phenomenon is
induced by the interplay between strong correlations and the non-Hermitian
point-gap topology. The Mott skin effect induces extreme sensitivity to the
boundary conditions only in the spin degree of freedom (i.e., the charge
distribution is not sensitive to boundary conditions), which is in sharp
contrast to the ordinary non-Hermitian skin effect in non-interacting systems.
Concretely, we elucidate that a bosonic non-Hermitian chain exhibits the Mott
skin effect in the strongly correlated regime by closely examining an
effective Hamiltonian. The emergence of the Mott skin effect is also supported
by numerical diagonalization of the bosonic chain. The difference between the
ordinary non-Hermitian skin effect and the Mott skin effect is also reflected
in the time-evolution of physical quantities; under the time-evolution spin
accumulation is observed while the charge distribution remains spatially
uniform.

Introduction–. Since the discovery of topological insulators, extensive
efforts have been devoted to understanding topological aspects of condensed
matter systems Hasan and Kane (2010); Qi and Zhang (2011); Thouless  _et al._
(1982); Halperin (1982); Hatsugai (1993); Kitaev (2001); Kane and Mele (2005a,
b); Qi  _et al._ (2008). While topological insulators are originally reported
for free fermions, it has turned out that the interplay between strong
correlations and non-trivial topology triggers further exotic phenomena Tsui
_et al._ (1982); Laughlin (1983); Jain (1989); Haldane (1983, 1985); Wen
(1995); Kit (2003); Wen (2004); Levin and Wen (2005); Kitaev (2003, 2006);
Tang  _et al._ (2011); Sun  _et al._ (2011); Neupert  _et al._ (2011); Sheng
_et al._ (2011); Regnault and Bernevig (2011); Bergholtz and Liu (2013);
Fidkowski and Kitaev (2010); Turner  _et al._ (2011); Fidkowski and Kitaev
(2011); Gu and Wen (2014); Yao and Ryu (2013); Ryu and Zhang (2012); Qi
(2013). For instance, strong correlations can induce fractional quantum Hall
states Tsui  _et al._ (1982); Laughlin (1983); Wen (1995); Tang  _et al._
(2011); Sun  _et al._ (2011); Neupert  _et al._ (2011); Sheng  _et al._
(2011); Regnault and Bernevig (2011); Bergholtz and Liu (2013). Furthermore,
topological Mott insulators exhibit the unique bulk-edge correspondence due to
the interplay between correlations and the non-trivial topology Pesin and
Balents (2010); Manmana  _et al._ (2012); Yoshida  _et al._ (2014). Namely,
corresponding to the non-trivial topology in the bulk, gapless edge modes
emerge only in the spin excitation spectrum (i.e., the charge excitation
spectrum is gapped even around edges).

Along with the above progress, the topological band theory of non-Hermitian
systems has been developed Bergholtz  _et al._ (2021); Ashida  _et al._
(2020); Gong  _et al._ (2018); Kawabata  _et al._ (2019a, b); Zhou and Lee
(2019); Lieu  _et al._ (2020); Hatano and Nelson (1996, 1997); Bender and
Boettcher (1998); Fukui and Kawakami (1998); Ashida  _et al._ (2016); Hu and
Hughes (2011); Esaki  _et al._ (2011); Sato  _et al._ (2012); Diehl  _et al._
(2011); Bardyn  _et al._ (2013); Rivas  _et al._ (2013); Zhu  _et al._ (2014);
Budich  _et al._ (2015); Lieu (2018); Rui  _et al._ (2019); Denner  _et al._
(2021); Nakamura  _et al._ (2022); Yokomizo and Murakami (2019); Kawabata  _et
al._ (2020a); Tonielli  _et al._ (2020); Kawabata  _et al._ (2021); Sayyad
_et al._ (2022); Chang  _et al._ (2020); Hsieh and Chang (2023) and revealed
unique phenomena due to the non-Hermitian point-gap topology which do not have
Hermitian counterpartsShen  _et al._ (2018); Kozii and Fu (2017); Yoshida  _et
al._ (2018); Zhou  _et al._ (2018); Yang  _et al._ (2021a); Budich  _et al._
(2019); Okugawa and Yokoyama (2019); Zhou  _et al._ (2019); Yoshida  _et al._
(2019a, 2020a); Kawabata  _et al._ (2019c); Carlström  _et al._ (2019); Xu
_et al._ (2017); Delplace  _et al._ (2021); Mandal and Bergholtz (2021). A
representative example is the non-Hermitian skin effect Lee (2016); Martinez
Alvarez  _et al._ (2018); Yao and Wang (2018); Yao  _et al._ (2018); Kunst
_et al._ (2018); Edvardsson  _et al._ (2019); Borgnia  _et al._ (2020); Lee
and Thomale (2019); Zhang  _et al._ (2020a); Okuma  _et al._ (2020); Okuma and
Sato (2023); Xiao  _et al._ (2020); Helbig  _et al._ (2020); Liang  _et al._
(2022); Yoshida  _et al._ (2020b); Okugawa  _et al._ (2020); Kawabata  _et
al._ (2020b); Fu and Wan (2020); Okuma and Sato (2019); Song  _et al._ (2019);
Haga  _et al._ (2021); Yang  _et al._ (2022); Hwang and Obuse (2023) which
results in the novel bulk-edge correspondence unique to non-Hermitian systems;
because of the non-trivial point-gap topology in the bulk, the eigenstates and
eigenvalues exhibit extreme sensitivity to the presence or absence of
boundaries Zhang  _et al._ (2020a); Okuma  _et al._ (2020). Especially, in
one-dimensional systems, most of eigenstates are localized only around one of
the edges under open boundary conditions (OBC), while eigenstates extend to
the bulk under periodic boundary conditions (PBC). The above localized
eigenstates are known as skin modes.

The above progress on correlated systems and the non-interacting non-Hermitian
topology naturally leads us to the following crucial question: how do strong
correlations affect the non-Hermitian skin effects? The significance of this
issue is further enhanced by resent advances in experiments with cold atoms
Tomita  _et al._ (2017, 2019); Takasu  _et al._ (2020) and quantum circuits Ma
_et al._ (2019) where both dissipation and correlations can be introduced.
Correlation effects on the non-Hermitian topological properties have been
studied extensively Yoshida  _et al._ (2019b, 2020c); Guo  _et al._ (2020a);
Matsumoto  _et al._ (2020); Zhang  _et al._ (2020b); Guo  _et al._ (2020b);
Shackleton and Scheurer (2020); Yang  _et al._ (2021b); Wang  _et al._ (2023);
Zhang  _et al._ (2020c); Liu  _et al._ (2020); Xu and Chen (2020); Pan  _et
al._ (2020); Xi  _et al._ (2021); Yoshida and Hatsugai (2021, 2022, 2023);
Orito and Imura (2022); Shen and Lee (2022); Mu  _et al._ (2020); Lee (2021);
Zhang  _et al._ (2022); Kawabata  _et al._ (2022); Alsallom  _et al._ (2022);
Faugno and Ozawa (2022); Hamanaka  _et al._ (2023); Tsubota  _et al._ (2022);
Sayyad and Lado (2023a); Chen  _et al._ (2023); Sayyad (2023); Sayyad and Lado
(2023b), but not on non-Hermitian skin effects, which is particularly the case
for bosonic systems.

In this paper, we address this question for a one-dimensional bosonic system
and discover a novel type of skin effects, a non-Hermitian Mott skin effect,
which induces striking skin modes. Namely, the non-trivial point-gap topology
results in the skin modes in which only the spin degree of freedom is involved
(i.e., charges are distributed uniformly even under OBC). This behavior is in
sharp contrast to non-Hermitian skin effects in non-interacting systems where
bosons are localized around the edge. We elucidate the emergence of the Mott
skin effect by examining an effective spin model in the strong correlation
regime, which hosts skin modes and possesses the non-trivial point-gap
topology characterized by the spin winding number. We also support the
emergence of the Mott skin effect by employing numerical diagonalization. Our
numerical analysis also elucidates unique real-time dynamics induced by the
Mott skin effect; dynamical spin accumulation is observed while the charge
distribution remains spatially uniform.

Model–. Let us consider a one-dimensional chain of interacting bosons. The
Hamiltonian reads

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\hat{H}_{0}+\hat{H}_{\mathrm{int}},$ (1)
$\displaystyle\hat{H}_{0}$ $\displaystyle=$
$\displaystyle\sum_{j\sigma}(-t_{\mathrm{R}\sigma}\hat{b}^{\dagger}_{j+1\sigma}\hat{b}_{j\sigma}-t_{\mathrm{L}\sigma}\hat{b}^{\dagger}_{j\sigma}\hat{b}_{j+1\sigma}),$
(2) $\displaystyle\hat{H}_{\mathrm{int}}$ $\displaystyle=$ $\displaystyle-
iV\sum_{j\sigma}\hat{n}_{j\sigma}(\hat{n}_{j\sigma}-1)-iU\sum_{j}\hat{n}_{j\uparrow}\hat{n}_{j\downarrow},$
(3)

where $\hat{b}^{\dagger}_{j\sigma}$ ($\hat{b}_{j\sigma}$) creates
(annihilates) a boson at site $j=0,1,2\ldots,L-1$ and in the spin state
$\sigma=\uparrow,\downarrow$ pse . The non-reciprocal hopping integrals are
denoted by $t_{\mathrm{R}\uparrow}=t_{\mathrm{L}\downarrow}=t_{+}$ and
$t_{\mathrm{L}\uparrow}=t_{\mathrm{R}\downarrow}=t_{-}$ with real numbers
$t_{+}$ and $t_{-}$. The operator $\hat{n}_{j\sigma}$ denotes the number
operator of bosons of spin $\sigma$ at site $j$;
$\hat{n}_{j\sigma}:=\hat{b}^{\dagger}_{j\sigma}\hat{b}_{j\sigma}$. The first
(second) term of $\hat{H}_{\mathrm{int}}$ describes the on-site interaction
between bosons with the same (opposite) spin. The parameters $V$ and $U$ are
non-negative numbers. Throughout this paper, we suppose the relation
$t_{+}>t_{-}$.

Symmetry and a topological invariant–. The above model preserves charge
$\mathrm{U}(1)$ symmetry and spin $\mathrm{U}(1)$ symmetry

$\displaystyle{}[\hat{H},\hat{N}]$ $\displaystyle=$ $\displaystyle 0,$ (4)
$\displaystyle{}[\hat{H},\hat{S}^{z}]$ $\displaystyle=$ $\displaystyle 0,$ (5)

with $\hat{N}=\hat{N}_{\uparrow}+\hat{N}_{\downarrow}$ and
$2\hat{S}^{z}=\hat{N}_{\uparrow}-\hat{N}_{\downarrow}$. Here,
$\hat{N}_{\sigma}$ denotes the number operator of bosons in the spin state
$\sigma$; $\hat{N}_{\sigma}=\sum_{j}\hat{n}_{j\sigma}$
($\sigma=\uparrow,\downarrow$). Therefore, the Hamiltonian can be block-
diagonalized into $N_{\uparrow}$ and $N_{\downarrow}$ sectors. We denote
eigenvalues of $\hat{N}_{\uparrow}$ and $\hat{N}_{\downarrow}$ as
$N_{\uparrow}$ and $N_{\downarrow}$, respectively.

In order to characterize the point-gap topology Def , we introduce the many-
body spin winding number

$\displaystyle W_{\mathrm{s}}$ $\displaystyle=$
$\displaystyle\int^{2\pi}_{0}\frac{d\theta_{\mathrm{s}}}{2\pi
i}\frac{\partial}{\partial\theta_{\mathrm{s}}}\log\left[\mathrm{det}\left(\hat{H}_{[N_{\uparrow},N_{\downarrow}]}(\theta_{\mathrm{s}})-E_{\mathrm{ref}}\right)\right],$
(6)

which is a variant of the previously introduced many-body winding number Zhang
_et al._ (2022); Kawabata  _et al._ (2022); Yoshida and Hatsugai (2022). Here,
in order to compute $W_{\mathrm{s}}$, we have imposed twisted boundary
conditions where the twist angle of the down-spin state $\theta_{\downarrow}$
is opposite to that of the up-spin state $\theta_{\uparrow}$ stw ;
$\hat{b}^{\dagger}_{0\sigma}\hat{b}_{L-1\sigma}\to
e^{i\theta_{\sigma}}\hat{b}^{\dagger}_{0\sigma}\hat{b}_{L-1\sigma}$ with
$(\theta_{\uparrow},\theta_{\downarrow})=(\theta_{\mathrm{s}},-\theta_{\mathrm{s}})$
and the twist angle $\theta_{\mathrm{s}}$. For the Fock space specified by
$[N_{\uparrow},N_{\downarrow}]$, the block-diagonalized Hamiltonian is denoted
by $\hat{H}_{[N_{\uparrow},N_{\downarrow}]}(\theta_{\mathrm{s}})$.

Analysis in the strong coupling region–. Before presenting numerical results,
we derive an effective model which provides an intuition for the Mott skin
effect. In the strong coupling region (i.e., $V,U\gg t_{+},t_{-}$), longest
lived states of the system are described by an effective spin model, i.e.,
eigenstates shift in the negative direction of the imaginary axis if bosons
occupy the same site. Applying the second order perturbation theory yields the
following effective spin model for the states where each site is occupied by
one boson

$\displaystyle\hat{H}_{\mathrm{spin}}(\theta_{\mathrm{s}})$ $\displaystyle=$
$\displaystyle\sum^{L-2}_{j=0}\left[J_{z}\hat{S}^{z}_{j+1}\hat{S}^{z}_{j}+J_{+}\hat{S}^{+}_{j+1}\hat{S}^{-}_{j}+J_{-}\hat{S}^{-}_{j+1}\hat{S}^{+}_{j}\right]$
$\displaystyle+J_{+}e^{2i\theta_{\mathrm{s}}}\hat{S}^{+}_{0}\hat{S}^{-}_{L-1}+J_{-}e^{-2i\theta_{\mathrm{s}}}\hat{S}^{-}_{0}\hat{S}^{+}_{L-1}+E_{0},$

with $J_{z}=-4i(t_{+}t_{-})(\frac{1}{V}-\frac{1}{U})$,
$J_{\pm}=-2i\frac{t^{2}_{\pm}}{U}$, and
$E_{0}=-iL(t_{+}t_{-})(\frac{1}{V}+\frac{1}{U})$. Here, $\hat{S}^{\mu}_{j}$
($\mu=x,y,z$) denotes spin operators at site $j$. The spin raising and
lowering operators are defined as $S^{\pm}_{j}=S^{x}_{j}\pm iS^{y}_{j}$. In
terms of creation and annihilation operators of bosons, the spin operators are
written as $\hat{S}^{z}_{j}=(\hat{n}_{j\uparrow}-\hat{n}_{j\downarrow})/2$,
$\hat{S}^{+}_{j}=\hat{b}^{\dagger}_{j\uparrow}\hat{b}_{j\downarrow}$ and
$\hat{S}^{-}_{j}=\hat{b}^{\dagger}_{j\downarrow}\hat{b}_{j\uparrow}$ for the
subspace where each site is occupied by one boson. Details of the derivation
are provided in Sec. S1 of Supplemental Material sup . The above Hamiltonian (
Non-Hermitian Mott Skin Effect) preserves spin $\mathrm{U(1)}$ symmetry and
can be block-diagonalized with $S^{z}$, the eigenvalue of
$\hat{S}^{z}=\sum_{j}\hat{S}^{z}_{j}$.

Applying the Jordan-Wigner transformation elucidates that the effective model
( Non-Hermitian Mott Skin Effect) gives rise to the Mott skin effect (for the
detailed derivation, see Sec. S1 of Supplemental Material sup ); the spin
model [Eq. ( Non-Hermitian Mott Skin Effect)] can be mapped to the following
spinless fermion model

$\displaystyle\hat{H}_{\mathrm{spin}}(\theta)$ $\displaystyle=$
$\displaystyle\sum^{L-2}_{j=0}\left(J_{+}\hat{f}^{\dagger}_{j+1}\hat{f}_{j}+J_{-}\hat{f}^{\dagger}_{j}\hat{f}_{j+1}\right)$
(8)
$\displaystyle-(-1)^{\hat{N}^{\mathrm{f}}}\left(J_{+}e^{2i\theta_{\mathrm{s}}}\hat{f}^{\dagger}_{0}\hat{f}_{L-1}+J_{-}e^{-2i\theta_{\mathrm{s}}}\hat{f}^{\dagger}_{L-1}\hat{f}_{0}\right)$
$\displaystyle+J_{z}\sum^{L-1}_{j=0}\hat{n}^{\mathrm{f}}_{j+1}\hat{n}^{\mathrm{f}}_{j}-J_{z}\left(\hat{N}^{\mathrm{f}}-\frac{L}{4}\right)+E_{0},$

with $\hat{N}^{\mathrm{f}}=\sum_{j}\hat{n}^{\mathrm{f}}_{j}$,
$\hat{S}^{+}_{j}=e^{i\pi\hat{N}^{<}_{j}}\hat{f}^{\dagger}_{j}$, and
$\hat{S}^{z}_{j}=(\hat{n}^{\mathrm{f}}_{j}-\frac{1}{2})$. Here,
$\hat{N}^{<}_{j}$ and $\hat{n}^{\mathrm{f}}_{j}$ are defined as
$\hat{N}^{<}_{j}=\sum^{L-1}_{j=0}\hat{n}^{\mathrm{f}}_{j}$ and
$\hat{n}^{\mathrm{f}}_{j}=\hat{f}^{\dagger}_{j}\hat{f}_{j}$. Operators
$\hat{f}^{\dagger}_{j}$ ($\hat{f}_{j}$) create (annihilate) a spinless fermion
at site $j$.

In particular, for the subspace with $S^{z}=1-\frac{L}{2}$ (i.e.,
$[N_{\uparrow},N_{\downarrow}]=[1,L-1]$), there exists only one fermion
created by $\hat{f}^{\dagger}_{j}$. Therefore, the above Hamiltonian is
simplified as

$\displaystyle\hat{H}_{\mathrm{spin}}(\theta)$ $\displaystyle=$
$\displaystyle\sum^{L-2}_{j=0}\left(J_{+}\hat{f}^{\dagger}_{j+1}\hat{f}_{j}+J_{-}\hat{f}^{\dagger}_{j}\hat{f}_{j+1}\right)$
(9)
$\displaystyle+\left(J_{+}e^{2i\theta_{\mathrm{s}}}\hat{f}^{\dagger}_{0}\hat{f}_{L-1}+J_{-}e^{-2i\theta_{\mathrm{s}}}\hat{f}^{\dagger}_{L-1}\hat{f}_{0}\right)$
$\displaystyle+J_{z}(\frac{L}{4}-1)+E_{0}.$

This model is nothing but the Hatano-Nelson chain Hatano and Nelson (1996,
1997) which exhibits the skin effect. Specifically, substituting the
Hamiltonian (9) to $\hat{H}_{[1,L-1]}(\theta_{\mathrm{s}})$, we obtain the
winding number $W_{\mathrm{s}}=2$ for the subspace with
$[N_{\uparrow},N_{\downarrow}]=[1,L-1]$. Here, we have set the reference
energy $E_{\mathrm{ref}}$ located inside of the loop formed by the
eigenvalues. We recall that $iJ_{+}>iJ_{-}$ holds for $t_{+}>t_{-}$ [see below
Eq. ( Non-Hermitian Mott Skin Effect)]. Corresponding to this non-trivial
point-gap topology, skin modes emerge only around the right edge which are
described by fermions created by $\hat{f}^{\dagger}_{j}$. We note that the
eigenvalues are aligned on the imaginary axis because $J_{+}$ and $J_{-}$ are
purely imaginary com .

Recalling the relation between spin operators and operators
$\hat{f}^{\dagger}_{j}$, we can conclude that the system exhibits the Mott
skin effect. Namely, only the spin degree of freedom is involved in the skin
modes induced by the non-trivial point-gap topology.

Numerical results: non-interacting case–. In order to numerically analyze this
model, we employ exact diagonalization. Unless otherwise noted, we set
$(t_{+},t_{-})=(1,0.1)$.

In the non-interacting case, the system is decomposed into two bosonic Hatano-
Nelson models where the hopping in the right (left) direction is dominant for
$\sigma=\uparrow$ ($\sigma=\downarrow$). Such non-reciprocal hoppings result
in the ordinary non-Hermitian skin effect as discussed below. In the
following, we analyze the many-body Hamiltonian (1) for the Fock space with
$(N_{\uparrow},N_{\downarrow})=(1,L-1)$ with $L=6$ (for analysis of the one-
body Hamiltonian, see Sec. S2 of Supplemental Material sup ).

Figure 1(a) displays the spectral flow of $\hat{H}(\theta_{\mathrm{s}})$ for
$0\leq\theta_{\mathrm{s}}\leq 2\pi$ where $E_{m}(\theta_{\mathrm{s}})$
$(m=0,1,2,\ldots)$ are eigenvalues of $\hat{H}(\theta_{\mathrm{s}})$. In this
figure, we can see that the spectral flow forms the loop structure, which
indicates the point-gap topology characterized by the many-body spin winding
number taking a non-zero value for $E_{\mathrm{ref}}=-0.01i$ (for more
details, see Sec. S3 of Supplemental Material sup ).

Figure 1:  Energy eigenvalues and expectation values for $V=U=0$. (a): Energy
eigenvalues under twisted boundary conditions and OBC (red), respectively.
(b): A magnified version of the range $-1.2\leq\mathrm{Im}E\leq 1.2$. The data
obtained under OBC are represented by red dots. With increasing
$\theta_{\mathrm{s}}$ from $0$ to $2\pi$, the eigenvalues wind the origin of
the complex plane, which indicates $W_{\mathrm{s}}$ takes a non-zero value for
$E_{\mathrm{ref}}=-0.01i$. (c) [(d)]: Expectation values
$\langle\hat{n}_{j}\rangle$ [$\langle\hat{S}^{z}_{j}\rangle$] at each site
under OBC. Here, the expectation values are computed from right eigenvectors
of the many-body Hamiltonian labeled by $m$ [see Eq. (10)]. These data are
obtained for subspace with $(N_{\uparrow},N_{\downarrow})=(1,5)$ and for
$(t_{+},t_{-})=(1,0.1)$ and $L=6$.

Corresponding to this non-trivial point-gap topology, the spectrum shows
extreme sensitivity to the boundary conditions. Imposing OBC significantly
changes the spectrum of $\hat{H}$; as shown in Fig. 1(a) the eigenvalues under
OBC are aligned on the real axis in contrast to the eigenvalues under PBC.
Eigenstates also show such sensitivity to the boundary conditions Garbe  _et
al._ (2023). In order to show this, we compute the expectation values of
$\hat{n}_{j}=\sum_{\sigma}\hat{n}_{j\sigma}$

$\displaystyle\langle\hat{n}_{j}\rangle$ $\displaystyle=$
$\displaystyle\frac{{}_{\mathrm{R}}\langle\Psi_{m}|\hat{n}_{j}|\Psi_{m}\rangle_{\mathrm{R}}}{{}_{\mathrm{R}}\langle\Psi_{m}|\Psi_{m}\rangle_{\mathrm{R}}},$
(10)

with $|\Psi_{m}\rangle_{\mathrm{R}}$ ($m=0,1,2,\ldots$) being right
eigenvectors of the many-body Hamiltonian,
$\hat{H}|\Psi_{m}\rangle_{\mathrm{R}}=E_{m}|\Psi_{m}\rangle_{\mathrm{R}}$. As
displayed in Fig. 1(c), bosons are localized around edges under OBC in
contrast to the case of PBC (the data for PBC is provided in Sec. S4 of
Supplemental Material sup ).

Due to the localization of bosons, spin polarization is observed only in the
presence of boundaries. Figure 1(c) displays the expectation values of
$\hat{S}^{z}_{j}=(\hat{n}_{j\uparrow}-\hat{n}_{j\downarrow})/2$. Spin
polarization is observed under OBC in contrast to the case of PBC.

As discussed above, in the non-interacting cases, the system shows the non-
Hermitian skin effect which results in extreme sensitivity of the eigenvalues
and eigenstates to the presence or absence of the boundaries. Accordingly, the
localization of bosons is observed only under OBC.

Figure 2:  Energy eigenvalues and expectation values for $(V,U)=(100,20)$.
(a): Energy eigenvalues under twisted boundary conditions and OBC,
respectively. (b): A magnified version of the range $-0.14\leq\mathrm{Im}E\leq
0.1$. In these figures the data obtained under OBC are represented by red
dots. With increasing $\theta_{\mathrm{s}}$ from $0$ to $2\pi$, the
eigenvalues wind the origin of the complex plane, which indicates
$W_{\mathrm{s}}=2$ for $E_{\mathrm{ref}}=-0.01i$. (c) [(d)]: Expectation
values $\langle\hat{n}_{j}\rangle$ [$\langle\hat{S}^{z}_{j}\rangle$] at each
site under OBC. As is the case of $V=U=0$, the expectation values are computed
from the right eigenvectors (see Fig. 1). These data are obtained for
$(t_{+},t_{-})=(1,0.1)$ and $L=6$.

Numerical results: interacting case–. Now, we demonstrate that the interplay
between strong correlations and non-reciprocal hoppings induces the Mott skin
effect. Namely, the non-trivial point-gap topology results in extreme
sensitivity to the boundary conditions only in the spin degree of freedom The
spin-charge separation plays an essential role in the emergence of the Mott
skin effect. In the following, we focus on the Fock space with
$(N_{\uparrow},N_{\downarrow})=(1,L-1)$ Nup .

Figures 2(a) and 2(b) display the spectral flow of
$\hat{H}(\theta_{\mathrm{s}})$ for $(V,U)=(100,20)$ with increasing
$\theta_{\mathrm{s}}$ from $0$ to $2\pi$. The interactions $V$ and $U$ shift
most of the loops in the negative direction of the imaginary axis [see Fig.
2(a)]. However, one of the loops remains around the origin of the complex
plane [see Fig. 2(b)]. This is because bosons do not feel the on-site
interactions unless multiple bosons occupy the same site.

The states remaining around the origin of the complex plane exhibit the Mott
skin effect. Because of the loop structure, the many-body spin winding number
takes $W_{\mathrm{s}}=2$ for $E_{\mathrm{ref}}=-0.01i$ (for more details, see
Sec. S3 of Supplemental Material sup ). Corresponding to the non-trivial
point-gap topology, eigenstates exhibit extreme sensitivity to the presence or
absence of the boundaries. As shown in Fig. 2(b), the eigenvalues are aligned
on the imaginary axis under OBC in contrast to the case of PBC where
eigenvalues are distributed on the complex plane Ptb .

Such sensitivity to the boundary conditions is also observed for eigenstates.
However, only the spin degree of freedom is involved in the sensitivity of the
boundary conditions, which is in sharp contrast to the non-interacting case.
Figure 2(c) displays the expectation values $\langle\hat{n}_{j}\rangle$. As
shown in this figure, strong correlations prevent bosons from localizing
around the edges, which suppresses the sensitivity to the boundary conditions
for the charge degree of freedom. Instead, the spin degree of freedom exhibits
extreme sensitivity to the boundary conditions. Figure 2(d) displays the
expectation values $\langle\hat{S}^{z}_{j}\rangle$. As shown in this figure,
spin polarization is observed under OBC in contrast to the case of PBC (the
data for PBC are provided in Sec. S4 of Supplemental Material sup ).

The above results reveal that the system exhibits the Mott skin effect
resulting in extreme sensitivity to the boundary conditions only in the spin
degree of freedom (i.e., such sensitivity is not observed in the charge degree
of freedom). The emergence of the Mott skin effect is due to the interplay
between strong correlations and non-Hermitian point-gap topology. We note that
the Mott skin effect is observed also for the subspace with
$(N_{\uparrow},N_{\downarrow})=(2,4)$ and $L=6$ (for more details see Sec. S5
of Supplemental Material sup ).

Real-time dynamics–. As seen above, the Mott skin effect involves only the
spin degree of freedom in contrast to the skin effect in the non-interacting
case. This difference is also reflected in the time-evolution of physical
quantities.

Figure 3:  (a) and (c) [(b) and (d)]: The time-evolution of the expectation
values $\langle\hat{n}_{j}(t)\rangle$ [$\langle\hat{S}^{z}_{j}(t)\rangle$]
under OBC [see Eq. (11)]. Panels (a) and (b) [(c) and (d)] display data for
$(V,U)=(0,0)$ [$(V,U)=(100,20)$]. These data are obtained for
$(t_{+},t_{-})=(1,0.1)$ and $L=6$.

Let us start with the non-interacting case. Figure 3(a) displays the
expectation values

$\displaystyle\langle\hat{n}_{j}(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{\langle\Phi(t)|\hat{n}_{j}|\Phi(t)\rangle}{\langle\Phi(t)|\Phi(t)\rangle},$
(11)

with $|\Phi(t)\rangle=e^{-i\hat{H}t}|\Phi(0)\rangle$ and $t$ being time nHS .
The initial state is chosen so that all of the bosons occupy site $j=0$;
$\sqrt{(L-1)!}|\Phi(0)\rangle=\hat{b}^{\dagger}_{0\uparrow}(\hat{b}^{\dagger}_{0\downarrow})^{L-1}|0\rangle$.
Figure 3(a) indicates that the bosons remain localized around the left edge
due to the skin effect osc . Figure 3(b) indicates that the above localization
is also observed in the expectation values $\langle
S^{z}_{j}(t)\rangle=\langle\Phi(t)|\hat{S}^{z}_{j}|\Phi(t)\rangle/\langle\Phi(t)|\Phi(t)\rangle$.
This is because bosons with spin up (spin down) are localized around the right
(left) edge (see Sec. S2 of Supplemental Material sup ). The above results
demonstrate that in the non-interacting case, the dynamical properties of the
spin degree of freedom are tied to those of charge degree of freedom.

Now, we turn to the real-time dynamics in the correlated case. In contrast to
the non-interacting case, dynamics of the spin degree of freedom exhibit an
essential difference from those of the charge degree of freedom. Figure 3(c)
displays the time-evolution of $\langle\hat{n}_{j}(t)\rangle$ for
$(V,U)=(100,20)$. As seen in Fig. 3(c), bosons immediately extend to the bulk.
In particular, for $0.125\lesssim t$, each site is occupied by one boson. This
is because the states where each site is occupied by one boson have a longer
lifetime $\tau\sim 1/\mathrm{Im}E_{m}$ than others. Contrary to the sudden
change in the charge degree of freedom, the dynamics of the spin degree of
freedom show gradual changes. The time-evlution of
$\langle\hat{S}^{z}_{j}\rangle$ is plotted in Fig. 3(d). In this figure,
dynamical spin accumulation is observed for $20\lesssim t$. This behavior is
observed only under OBC (for data under PBC, see Sec. S6 of Supplemental
Material sup ).

As seen above, the Mott skin effect induces the dynamical spin accumulation
while the charge degree of freedom remains spatially uniform. These dynamical
properties in the strongly correlated case are in sharp contrast to those in
the non-interacting case. The above dynamical properties are considered to be
observed for open quantum systems even in the presence of quantum jumps.

Conclusion–. We have proposed a non-Hermitian Mott skin effect induced by the
interplay between strong correlations and the non-Hermitian point-gap
topology. In contrast to the ordinary non-Hermirtian skin effect, the Mott
skin effect results in extreme sensitivity to the boundary conditions only in
the spin degree of freedom. We have demonstrated the emergence of the Mott
skin effect by analyzing the bosonic non-Hermitian Hamiltonian with non-
reciprocal hoppings and on-site interactions. The difference from the ordinary
skin effect is also reflected in the dynamical properties; spin accumulation
is observed while charge discribution remains spatially uniform.

We finish this paper with a remark on the relevance to open quantum systems
described by the Lindblad equation. In this paper, we have analyzed the non-
Hermitian Hamiltonian as a toy model. However, we argue that our Hamiltonian
is relevant to open quantum systems, and that the above dynamical spin
accumulation can be observed even under the presence of the jump term.

Note added–. While finising this paper, we noticed Ref. Kim  _et al._ (2023)
posted on arXiv which has some overlap with our results.

Acknowledgments–. T.Y. thanks Manfred Sigrist for fruitful discussion. T.Y.
particulary thanks Shunsuke Furukawa and Yoshihito Kuno for fruitful
discussions on the technical details of exact diagonalization. T.Y. is
grateful to the long term workshop YITP-T-23-01 held at YITP, Kyoto
University, where a part of this work was done. This work is supported by JSPS
KAKENHI Grants No. JP21K13850 and No. JP22H05247.

## References

  * Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
  * Qi and Zhang (2011) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
  * Thouless  _et al._ (1982) D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
  * Halperin (1982) B. I. Halperin, Phys. Rev. B 25, 2185 (1982).
  * Hatsugai (1993) Y. Hatsugai, Phys. Rev. Lett. 71, 3697 (1993).
  * Kitaev (2001) A. Y. Kitaev, Physics-Uspekhi 44, 131 (2001).
  * Kane and Mele (2005a) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005a).
  * Kane and Mele (2005b) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005b).
  * Qi  _et al._ (2008) X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, 195424 (2008).
  * Tsui  _et al._ (1982) D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982).
  * Laughlin (1983) R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).
  * Jain (1989) J. K. Jain, Phys. Rev. Lett. 63, 199 (1989).
  * Haldane (1983) F. D. M. Haldane, Phys. Rev. Lett. 51, 605 (1983).
  * Haldane (1985) F. D. M. Haldane, Phys. Rev. Lett. 55, 2095 (1985).
  * Wen (1995) X.-G. Wen, Advances in Physics 44, 405 (1995).
  * Kit (2003) Annals of Physics 303, 2 (2003).
  * Wen (2004) X.-G. Wen,  _Quantum field theory of many-body systems: from the origin of sound to an origin of light and electrons_ (OUP Oxford, 2004).
  * Levin and Wen (2005) M. A. Levin and X.-G. Wen, Phys. Rev. B 71, 045110 (2005).
  * Kitaev (2003) A. Kitaev, Annals of Physics 303, 2 (2003).
  * Kitaev (2006) A. Kitaev, Annals of Physics 321, 2 (2006), january Special Issue.
  * Tang  _et al._ (2011) E. Tang, J.-W. Mei, and X.-G. Wen, Phys. Rev. Lett. 106, 236802 (2011).
  * Sun  _et al._ (2011) K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, Phys. Rev. Lett. 106, 236803 (2011).
  * Neupert  _et al._ (2011) T. Neupert, L. Santos, C. Chamon, and C. Mudry, Phys. Rev. Lett. 106, 236804 (2011).
  * Sheng  _et al._ (2011) D. N. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Nature Communications 2, 389 EP (2011), article.
  * Regnault and Bernevig (2011) N. Regnault and B. A. Bernevig, Phys. Rev. X 1, 021014 (2011).
  * Bergholtz and Liu (2013) E. J. Bergholtz and Z. Liu, International Journal of Modern Physics B 27, 1330017 (2013).
  * Fidkowski and Kitaev (2010) L. Fidkowski and A. Kitaev, Phys. Rev. B 81, 134509 (2010).
  * Turner  _et al._ (2011) A. M. Turner, F. Pollmann, and E. Berg, Phys. Rev. B 83, 075102 (2011).
  * Fidkowski and Kitaev (2011) L. Fidkowski and A. Kitaev, Phys. Rev. B 83, 075103 (2011).
  * Gu and Wen (2014) Z.-C. Gu and X.-G. Wen, Phys. Rev. B 90, 115141 (2014).
  * Yao and Ryu (2013) H. Yao and S. Ryu, Phys. Rev. B 88, 064507 (2013).
  * Ryu and Zhang (2012) S. Ryu and S.-C. Zhang, Phys. Rev. B 85, 245132 (2012).
  * Qi (2013) X.-L. Qi, New J. Phys. 15, 065002 (2013).
  * Pesin and Balents (2010) D. Pesin and L. Balents, Nature Physics 6, 376 (2010).
  * Manmana  _et al._ (2012) S. R. Manmana, A. M. Essin, R. M. Noack, and V. Gurarie, Phys. Rev. B 86, 205119 (2012).
  * Yoshida  _et al._ (2014) T. Yoshida, R. Peters, S. Fujimoto, and N. Kawakami, Phys. Rev. Lett. 112, 196404 (2014).
  * Bergholtz  _et al._ (2021) E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Rev. Mod. Phys. 93, 015005 (2021).
  * Ashida  _et al._ (2020) Y. Ashida, Z. Gong, and M. Ueda, Advances in Physics 69, 249 (2020).
  * Gong  _et al._ (2018) Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Phys. Rev. X 8, 031079 (2018).
  * Kawabata  _et al._ (2019a) K. Kawabata, S. Higashikawa, Z. Gong, Y. Ashida, and M. Ueda, Nature Communications 10, 297 (2019a).
  * Kawabata  _et al._ (2019b) K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Phys. Rev. X 9, 041015 (2019b).
  * Zhou and Lee (2019) H. Zhou and J. Y. Lee, Phys. Rev. B 99, 235112 (2019).
  * Lieu  _et al._ (2020) S. Lieu, M. McGinley, and N. R. Cooper, Phys. Rev. Lett. 124, 040401 (2020).
  * Hatano and Nelson (1996) N. Hatano and D. R. Nelson, Phys. Rev. Lett. 77, 570 (1996).
  * Hatano and Nelson (1997) N. Hatano and D. R. Nelson, Phys. Rev. B 56, 8651 (1997).
  * Bender and Boettcher (1998) C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
  * Fukui and Kawakami (1998) T. Fukui and N. Kawakami, Physical Review B 58, 16051 (1998).
  * Ashida  _et al._ (2016) Y. Ashida, S. Furukawa, and M. Ueda, Phys. Rev. A 94, 053615 (2016).
  * Hu and Hughes (2011) Y. C. Hu and T. L. Hughes, Phys. Rev. B 84, 153101 (2011).
  * Esaki  _et al._ (2011) K. Esaki, M. Sato, K. Hasebe, and M. Kohmoto, Phys. Rev. B 84, 205128 (2011).
  * Sato  _et al._ (2012) M. Sato, K. Hasebe, K. Esaki, and M. Kohmoto, Progress of Theoretical Physics 127, 937 (2012).
  * Diehl  _et al._ (2011) S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, Nature Physics 7, 971 (2011).
  * Bardyn  _et al._ (2013) C.-E. Bardyn, M. A. Baranov, C. V. Kraus, E. Rico, A. İmamoğlu, P. Zoller, and S. Diehl, New Journal of Physics 15, 085001 (2013).
  * Rivas  _et al._ (2013) A. Rivas, O. Viyuela, and M. A. Martin-Delgado, Phys. Rev. B 88, 155141 (2013).
  * Zhu  _et al._ (2014) B. Zhu, R. Lü, and S. Chen, Phys. Rev. A 89, 062102 (2014).
  * Budich  _et al._ (2015) J. C. Budich, P. Zoller, and S. Diehl, Phys. Rev. A 91, 042117 (2015).
  * Lieu (2018) S. Lieu, Phys. Rev. B 97, 045106 (2018).
  * Rui  _et al._ (2019) W. B. Rui, Y. X. Zhao, and A. P. Schnyder, Phys. Rev. B 99, 241110 (2019).
  * Denner  _et al._ (2021) M. M. Denner, A. Skurativska, F. Schindler, M. H. Fischer, R. Thomale, T. Bzdusek, and T. Neupert, Nature Communications 12 (2021).
  * Nakamura  _et al._ (2022) D. Nakamura, T. Bessho, and M. Sato, arXiv preprint arXiv:2205.15635 (2022).
  * Yokomizo and Murakami (2019) K. Yokomizo and S. Murakami, Phys. Rev. Lett. 123, 066404 (2019).
  * Kawabata  _et al._ (2020a) K. Kawabata, N. Okuma, and M. Sato, Phys. Rev. B 101, 195147 (2020a).
  * Tonielli  _et al._ (2020) F. Tonielli, J. C. Budich, A. Altland, and S. Diehl, Phys. Rev. Lett. 124, 240404 (2020).
  * Kawabata  _et al._ (2021) K. Kawabata, K. Shiozaki, and S. Ryu, Phys. Rev. Lett. 126, 216405 (2021).
  * Sayyad  _et al._ (2022) S. Sayyad, J. D. Hannukainen, and A. G. Grushin, Phys. Rev. Res. 4, L042004 (2022).
  * Chang  _et al._ (2020) P.-Y. Chang, J.-S. You, X. Wen, and S. Ryu, Phys. Rev. Res. 2, 033069 (2020).
  * Hsieh and Chang (2023) C.-T. Hsieh and P.-Y. Chang, SciPost Phys. Core 6, 062 (2023).
  * Shen  _et al._ (2018) H. Shen, B. Zhen, and L. Fu, Phys. Rev. Lett. 120, 146402 (2018).
  * Kozii and Fu (2017) V. Kozii and L. Fu, arXiv preprint arXiv:1708.05841 (2017).
  * Yoshida  _et al._ (2018) T. Yoshida, R. Peters, and N. Kawakami, Phys. Rev. B 98, 035141 (2018).
  * Zhou  _et al._ (2018) H. Zhou, C. Peng, Y. Yoon, C. W. Hsu, K. A. Nelson, L. Fu, J. D. Joannopoulos, M. Soljačić, and B. Zhen, Science 359, 1009 (2018).
  * Yang  _et al._ (2021a) Z. Yang, A. P. Schnyder, J. Hu, and C.-K. Chiu, Phys. Rev. Lett. 126, 086401 (2021a).
  * Budich  _et al._ (2019) J. C. Budich, J. Carlström, F. K. Kunst, and E. J. Bergholtz, Phys. Rev. B 99, 041406 (2019).
  * Okugawa and Yokoyama (2019) R. Okugawa and T. Yokoyama, Phys. Rev. B 99, 041202 (2019).
  * Zhou  _et al._ (2019) H. Zhou, J. Y. Lee, S. Liu, and B. Zhen, Optica 6, 190 (2019).
  * Yoshida  _et al._ (2019a) T. Yoshida, R. Peters, N. Kawakami, and Y. Hatsugai, Phys. Rev. B 99, 121101 (2019a).
  * Yoshida  _et al._ (2020a) T. Yoshida, R. Peters, N. Kawakami, and Y. Hatsugai, Progress of Theoretical and Experimental Physics 2020, 12A109 (2020a).
  * Kawabata  _et al._ (2019c) K. Kawabata, T. Bessho, and M. Sato, Phys. Rev. Lett. 123, 066405 (2019c).
  * Carlström  _et al._ (2019) J. Carlström, M. Stålhammar, J. C. Budich, and E. J. Bergholtz, Phys. Rev. B 99, 161115 (2019).
  * Xu  _et al._ (2017) Y. Xu, S.-T. Wang, and L.-M. Duan, Phys. Rev. Lett. 118, 045701 (2017).
  * Delplace  _et al._ (2021) P. Delplace, T. Yoshida, and Y. Hatsugai, Phys. Rev. Lett. 127, 186602 (2021).
  * Mandal and Bergholtz (2021) I. Mandal and E. J. Bergholtz, Phys. Rev. Lett. 127, 186601 (2021).
  * Lee (2016) T. E. Lee, Phys. Rev. Lett. 116, 133903 (2016).
  * Martinez Alvarez  _et al._ (2018) V. M. Martinez Alvarez, J. E. Barrios Vargas, and L. E. F. Foa Torres, Phys. Rev. B 97, 121401 (2018).
  * Yao and Wang (2018) S. Yao and Z. Wang, Phys. Rev. Lett. 121, 086803 (2018).
  * Yao  _et al._ (2018) S. Yao, F. Song, and Z. Wang, Phys. Rev. Lett. 121, 136802 (2018).
  * Kunst  _et al._ (2018) F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Phys. Rev. Lett. 121, 026808 (2018).
  * Edvardsson  _et al._ (2019) E. Edvardsson, F. K. Kunst, and E. J. Bergholtz, Phys. Rev. B 99, 081302 (2019).
  * Borgnia  _et al._ (2020) D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, Phys. Rev. Lett. 124, 056802 (2020).
  * Lee and Thomale (2019) C. H. Lee and R. Thomale, Phys. Rev. B 99, 201103 (2019).
  * Zhang  _et al._ (2020a) K. Zhang, Z. Yang, and C. Fang, Phys. Rev. Lett. 125, 126402 (2020a).
  * Okuma  _et al._ (2020) N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Phys. Rev. Lett. 124, 086801 (2020).
  * Okuma and Sato (2023) N. Okuma and M. Sato, Annual Review of Condensed Matter Physics 14, 83 (2023).
  * Xiao  _et al._ (2020) L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, Nature Physics 16, 761 (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, Nature Physics 16, 747 (2020).
  * Liang  _et al._ (2022) Q. Liang, D. Xie, Z. Dong, H. Li, H. Li, B. Gadway, W. Yi, and B. Yan, Phys. Rev. Lett. 129, 070401 (2022).
  * Yoshida  _et al._ (2020b) T. Yoshida, T. Mizoguchi, and Y. Hatsugai, Phys. Rev. Research 2, 022062 (2020b).
  * Okugawa  _et al._ (2020) R. Okugawa, R. Takahashi, and K. Yokomizo, Phys. Rev. B 102, 241202 (2020).
  * Kawabata  _et al._ (2020b) K. Kawabata, M. Sato, and K. Shiozaki, Phys. Rev. B 102, 205118 (2020b).
  * Fu and Wan (2020) Y. Fu and S. Wan, arXiv preprint arXiv:2008.09033 (2020).
  * Okuma and Sato (2019) N. Okuma and M. Sato, Phys. Rev. Lett. 123, 097701 (2019).
  * Song  _et al._ (2019) F. Song, S. Yao, and Z. Wang, Phys. Rev. Lett. 123, 170401 (2019).
  * Haga  _et al._ (2021) T. Haga, M. Nakagawa, R. Hamazaki, and M. Ueda, Phys. Rev. Lett. 127, 070402 (2021).
  * Yang  _et al._ (2022) F. Yang, Q.-D. Jiang, and E. J. Bergholtz, Phys. Rev. Res. 4, 023160 (2022).
  * Hwang and Obuse (2023) G. Hwang and H. Obuse, arXiv preprint arXiv:2305.08548 (2023).
  * Tomita  _et al._ (2017) T. Tomita, S. Nakajima, I. Danshita, Y. Takasu, and Y. Takahashi, Science Advances 3, e1701513 (2017).
  * Tomita  _et al._ (2019) T. Tomita, S. Nakajima, Y. Takasu, and Y. Takahashi, Phys. Rev. A 99, 031601 (2019).
  * Takasu  _et al._ (2020) Y. Takasu, T. Yagami, Y. Ashida, R. Hamazaki, Y. Kuno, and Y. Takahashi, Progress of Theoretical and Experimental Physics 2020, 12A110 (2020).
  * Ma  _et al._ (2019) R. Ma, B. Saxberg, C. Owens, N. Leung, Y. Lu, J. Simon, and D. I. Schuster, Nature 566, 51 (2019).
  * Yoshida  _et al._ (2019b) T. Yoshida, K. Kudo, and Y. Hatsugai, Scientific Reports 9, 16895 (2019b).
  * Yoshida  _et al._ (2020c) T. Yoshida, K. Kudo, H. Katsura, and Y. Hatsugai, Phys. Rev. Research 2, 033428 (2020c).
  * Guo  _et al._ (2020a) C.-X. Guo, X.-R. Wang, C. Wang, and S.-P. Kou, Phys. Rev. B 101, 144439 (2020a).
  * Matsumoto  _et al._ (2020) N. Matsumoto, K. Kawabata, Y. Ashida, S. Furukawa, and M. Ueda, Phys. Rev. Lett. 125, 260601 (2020).
  * Zhang  _et al._ (2020b) Q. Zhang, W.-T. Xu, Z.-Q. Wang, and G.-M. Zhang, Communications Physics 3, 209 (2020b).
  * Guo  _et al._ (2020b) C.-X. Guo, X.-R. Wang, and S.-P. Kou, EPL (Europhysics Letters) 131, 27002 (2020b).
  * Shackleton and Scheurer (2020) H. Shackleton and M. S. Scheurer, Phys. Rev. Research 2, 033022 (2020).
  * Yang  _et al._ (2021b) K. Yang, S. C. Morampudi, and E. J. Bergholtz, Phys. Rev. Lett. 126, 077201 (2021b).
  * Wang  _et al._ (2023) Z. Wang, X.-D. Dai, H.-R. Wang, and Z. Wang, arXiv preprint arXiv:2306.12482 (2023).
  * Zhang  _et al._ (2020c) D.-W. Zhang, Y.-L. Chen, G.-Q. Zhang, L.-J. Lang, Z. Li, and S.-L. Zhu, Phys. Rev. B 101, 235150 (2020c).
  * Liu  _et al._ (2020) T. Liu, J. J. He, T. Yoshida, Z.-L. Xiang, and F. Nori, Phys. Rev. B 102, 235151 (2020).
  * Xu and Chen (2020) Z. Xu and S. Chen, Phys. Rev. B 102, 035153 (2020).
  * Pan  _et al._ (2020) L. Pan, X. Wang, X. Cui, and S. Chen, Phys. Rev. A 102, 023306 (2020).
  * Xi  _et al._ (2021) W. Xi, Z.-H. Zhang, Z.-C. Gu, and W.-Q. Chen, Science Bulletin 66, 1731 (2021).
  * Yoshida and Hatsugai (2021) T. Yoshida and Y. Hatsugai, Phys. Rev. B 104, 075106 (2021).
  * Yoshida and Hatsugai (2022) T. Yoshida and Y. Hatsugai, Phys. Rev. B 106, 205147 (2022).
  * Yoshida and Hatsugai (2023) T. Yoshida and Y. Hatsugai, Phys. Rev. B 107, 075118 (2023).
  * Orito and Imura (2022) T. Orito and K.-I. Imura, Phys. Rev. B 105, 024303 (2022).
  * Shen and Lee (2022) R. Shen and C. H. Lee, Communications Physics 5, 238 (2022).
  * Mu  _et al._ (2020) S. Mu, C. H. Lee, L. Li, and J. Gong, Phys. Rev. B 102, 081115 (2020).
  * Lee (2021) C. H. Lee, Phys. Rev. B 104, 195102 (2021).
  * Zhang  _et al._ (2022) S.-B. Zhang, M. M. Denner, T. c. v. Bzdušek, M. A. Sentef, and T. Neupert, Phys. Rev. B 106, L121102 (2022).
  * Kawabata  _et al._ (2022) K. Kawabata, K. Shiozaki, and S. Ryu, Phys. Rev. B 105, 165137 (2022).
  * Alsallom  _et al._ (2022) F. Alsallom, L. Herviou, O. V. Yazyev, and M. Brzezińska, Phys. Rev. Res. 4, 033122 (2022).
  * Faugno and Ozawa (2022) W. N. Faugno and T. Ozawa, Phys. Rev. Lett. 129, 180401 (2022).
  * Hamanaka  _et al._ (2023) S. Hamanaka, K. Yamamoto, and T. Yoshida, arXiv preprint arXiv:2305.19697 (2023).
  * Tsubota  _et al._ (2022) S. Tsubota, H. Yang, Y. Akagi, and H. Katsura, Phys. Rev. B 105, L201113 (2022).
  * Sayyad and Lado (2023a) S. Sayyad and J. L. Lado, Phys. Rev. Res. 5, L022046 (2023a).
  * Chen  _et al._ (2023) G. Chen, F. Song, and J. L. Lado, Phys. Rev. Lett. 130, 100401 (2023).
  * Sayyad (2023) S. Sayyad, arXiv preprint arXiv:2306.14766 (2023).
  * Sayyad and Lado (2023b) S. Sayyad and J. L. Lado, arXiv preprint arXiv:2309.06303 (2023b).
  * (141) To be precise, “spin” denotes pseudo-spin because the system is bosonic. However, for simplicity, we denote it by “spin” .
  * (142) The point-gap of the many-body Hamiltonian opens for $\mathrm{det}\left(\hat{H}_{[N_{\uparrow},N_{\downarrow}]}(\theta_{\mathrm{s}})-E_{\mathrm{ref}}\right)$ .
  * (143) This is because the Hamiltonian (1) is decomposed into two bosonic Hatano-Nelson model where the hopping to the right (left) is larger than the other for $\sigma=\uparrow$ ($\sigma=\downarrow$).
  * (144) Supplemental Material for details of the perturbation theory, detailed results of the bosonic Hubbard model.
  * (145) It is known that when hopping integrals of the Hatano-Nelson model are real, all of the eigenvalues take real values, which can be seen by applying an imaginary gauge transformation (see Sec. S2 of Supplemental Material sup ). Combining this fact and realness of $iJ_{+}$ and $iJ_{-}$, we can see that all of the eigenvalues of the effective spin model (9) are purely imaginary.
  * Garbe  _et al._ (2023) L. Garbe, Y. Minoguchi, J. Huber, and P. Rabl, arXiv preprint arXiv:2301.11339 (2023).
  * (147) Here, we note that the Mott skin effect is also observed for other subspaces of half-filling $N_{\uparrow}+N_{\downarrow}=L$. For instance, the Mott skin effect is observed for the subspace with $[N_{\uparrow},N_{\downarrow}]=[2,4]$ and $L=6$ (see Sec. S5 of Supplemental Material sup ) .
  * (148) The results of eigenvalues are consistent with the results of perturbation theory, which can be seen by noting that the parameters of $\hat{H}_{\mathrm{spin}}$ take the following values: $(J_{+},J_{-})=(-0.1i,-0.001i)$ and $J_{z}(\frac{L}{4}-1)+E_{0}=-0.028i$ for $(t_{+},t_{-},V,U)=(-1,0.1,100,20)$ and $L=6$ .
  * (149) We note that post-selected real-time dynamics of open quantum systems are described by $i\partial_{t}|\Phi(t)\rangle=\hat{H}|\Phi(t)\rangle$ Ashida  _et al._ (2016).
  * (150) We note that Bloch oscillation is observed around the left edge ($j=0$) as illustrated by the arrow in Fig. 3(a). This oscillation is due to the interference between skin modes whose time-evolution is given by $e^{-i\hat{H}t}|\Phi_{m}\rangle_{\mathrm{R}}=e^{-iE_{m}t}|\Phi_{m}\rangle_{\mathrm{R}}$.
  * Kim  _et al._ (2023) B. H. Kim, J.-H. Han, and M. J. Park, arXiv preprint arXiv:2309.07894 (2023).

Supplemental Materials:
Non-Hermitian Mott Skin Effect

##  S1  Details of the effective spin model

###  S1.1  Derivation of Equation ( Non-Hermitian Mott Skin Effect)

Here, based on the second order perturbation theory, we derive the effective
spin model [see Eq. ( Non-Hermitian Mott Skin Effect)] for $V,U\gg
t_{+},t_{-}$. In the following, we impose PBC for the sake of simplicity.

Let us consider a Fock space where each site is occupied by a boson. For such
a Fock space, the effective Hamiltonian can be written as

$\displaystyle\hat{H}_{\mathrm{ptb}}$ $\displaystyle=$
$\displaystyle\hat{H}_{\mathrm{ptb},\mathrm{RL}}+\hat{H}_{\mathrm{ptb},\mathrm{LR}},$
(S1) $\displaystyle\hat{H}_{\mathrm{ptb},\mathrm{RL}}$ $\displaystyle=$
$\displaystyle\sum_{j\sigma
s}\hat{P}\left[t_{\mathrm{R}\sigma}t_{\mathrm{L}s}\hat{b}^{\dagger}_{j+1\sigma}\hat{b}_{j\sigma}\frac{1}{E_{\mathrm{g}}-\hat{H}_{\mathrm{int}}}\hat{b}^{\dagger}_{js}\hat{b}_{j+1s}\right]\hat{P},$
$\displaystyle\hat{H}_{\mathrm{ptb},\mathrm{LR}}$ $\displaystyle=$
$\displaystyle\sum_{j\sigma
s}\hat{P}\left[t_{\mathrm{L}\sigma}t_{\mathrm{R}s}\hat{b}^{\dagger}_{j\sigma}\hat{b}_{j+1\sigma}\frac{1}{E_{\mathrm{g}}-\hat{H}_{\mathrm{int}}}\hat{b}^{\dagger}_{j+1s}\hat{b}_{js}\right]\hat{P},$

where $\hat{P}$ is the projection operator of the Fock space where a boson
occupies each site. The ground state energy $E_{\mathrm{g}}$ takes zero
$E_{\mathrm{g}}=\hat{P}\hat{H}_{\mathrm{int}}\hat{P}=0$.

By a straightforward calculation, we obtain

$\displaystyle\hat{H}_{\mathrm{ptb},\mathrm{RL}}$ $\displaystyle=$
$\displaystyle-i\sum_{j\sigma}\frac{t_{\mathrm{R}\sigma}t_{\mathrm{L}\sigma}}{V}\hat{P}\left(\hat{n}_{j+1\sigma}\hat{n}_{j\sigma}\right)\hat{P}$
$\displaystyle-i\sum_{j\sigma}\frac{t_{\mathrm{R}\sigma}t_{\mathrm{L}\sigma}}{U}\hat{P}\left(\hat{n}_{j+1\sigma}\hat{n}_{j\bar{\sigma}}\right)\hat{P}$
$\displaystyle-i\sum_{j\sigma}\frac{t_{\mathrm{R}\sigma}t_{\mathrm{L}\bar{\sigma}}}{U}\hat{P}\left(\hat{b}^{\dagger}_{j+1\sigma}\hat{b}_{j+1\bar{\sigma}}\hat{b}^{\dagger}_{j\bar{\sigma}}\hat{b}_{j\sigma}\right)\hat{P},$

where $\bar{\sigma}$ takes $\downarrow$ ($\uparrow$) for $\sigma=\uparrow$
($\sigma=\downarrow$).

By using the following relations

$\displaystyle 4\hat{S}^{z}_{j+1}\hat{S}^{z}_{j}+\hat{n}_{j+1}\hat{n}_{j}$
$\displaystyle=$ $\displaystyle
2(\hat{n}_{j+1\uparrow}\hat{n}_{j\uparrow}+\hat{n}_{j+1\downarrow}\hat{n}_{j\downarrow}),$
(S3)
$\displaystyle-4\hat{S}^{z}_{j+1}\hat{S}^{z}_{j}+\hat{n}_{j+1}\hat{n}_{j}$
$\displaystyle=$ $\displaystyle
2(\hat{n}_{j+1\uparrow}\hat{n}_{j\downarrow}+\hat{n}_{j+1\downarrow}\hat{n}_{j\uparrow}),$
(S4)

we obtain

$\displaystyle\hat{H}_{\mathrm{ptb},\mathrm{RL}}$ $\displaystyle=$
$\displaystyle\sum_{j}\left[-i\frac{t_{+}t_{-}}{2V}\hat{P}\left(4\hat{S}^{z}_{j+1}\hat{S}^{z}_{j}+\hat{n}_{j+1}\hat{n}_{j}\right)\hat{P}\right.$
$\displaystyle-i\frac{t_{+}t_{-}}{2U}\hat{P}\left(-4\hat{S}^{z}_{j+1}\hat{S}^{z}_{j}+\hat{n}_{j+1}\hat{n}_{j}\right)\hat{P}$
$\displaystyle\left.-i\frac{t^{2}_{+}}{U}\hat{P}\left(\hat{S}^{+}_{j+1}\hat{S}^{-}_{j}\right)\hat{P}-i\frac{t^{2}_{-}}{U}\hat{P}\left(\hat{S}^{-}_{j+1}\hat{S}^{+}_{j}\right)\hat{P}\right],$

where we have also chosen the parameters as
$(t_{\mathrm{R}\uparrow},t_{\mathrm{R}\downarrow})=(t_{+},t_{-})$ and
$(t_{\mathrm{L}\uparrow},t_{\mathrm{L}\downarrow})=(t_{-},t_{+})$.

In a similar way, we obtain

$\displaystyle\hat{H}_{\mathrm{ptb},\mathrm{LR}}$ $\displaystyle=$
$\displaystyle\sum_{j}\left[-i\frac{t_{+}t_{-}}{2V}\hat{P}\left(4\hat{S}^{z}_{j+1}\hat{S}^{z}_{j}+\hat{n}_{j+1}\hat{n}_{j}\right)\hat{P}\right.$
(S6)
$\displaystyle-i\frac{t_{+}t_{-}}{2U}\hat{P}\left(-4\hat{S}^{z}_{j+1}\hat{S}^{z}_{j}+\hat{n}_{j+1}\hat{n}_{j}\right)\hat{P}$
$\displaystyle\left.-i\hat{P}\left(\frac{t^{2}_{+}}{U}\hat{S}^{+}_{j+1}\hat{S}^{-}_{j}+\frac{t^{2}_{-}}{U}\hat{S}^{-}_{j+1}\hat{S}^{+}_{j}\right)\hat{P}\right].$

Therefore, the effective Hamiltonian is obtained as

$\displaystyle\hat{H}_{\mathrm{ptb}}$ $\displaystyle=$
$\displaystyle\sum_{j}\left[J_{+}\hat{S}^{+}_{j+1}\hat{S}^{-}_{j}+J_{-}\hat{S}^{-}_{j+1}\hat{S}^{+}_{j}+J_{z}\hat{S}^{z}_{j+1}\hat{S}^{z}_{j}\right]+E_{0},$

with

$\displaystyle J_{+}$ $\displaystyle=$ $\displaystyle-2i\frac{t^{2}_{+}}{U},$
(S8) $\displaystyle J_{-}$ $\displaystyle=$
$\displaystyle-2i\frac{t^{2}_{-}}{U},$ (S9) $\displaystyle J_{z}$
$\displaystyle=$ $\displaystyle-4it_{+}t_{-}(\frac{1}{V}-\frac{1}{U}),$ (S10)
$\displaystyle E_{0}$ $\displaystyle=$
$\displaystyle-4it_{+}t_{-}L(\frac{1}{V}+\frac{1}{U}).$ (S11)

Equation (S1.1) corresponds to the spin model [Eq. ( Non-Hermitian Mott Skin
Effect)].

Under twisted boundary conditions, the following replacement results in the
effective Hamiltonian $\hat{H}_{\mathrm{spin}}(\theta_{\mathrm{s}})$:

$\displaystyle\hat{S}^{+}_{0}\hat{S}^{-}_{L-1}$ $\displaystyle\to$
$\displaystyle e^{2i\theta}\hat{S}^{+}_{0}\hat{S}^{-}_{L-1},$ (S12)
$\displaystyle\hat{S}^{-}_{0}\hat{S}^{+}_{L-1}$ $\displaystyle\to$
$\displaystyle e^{-2i\theta}\hat{S}^{-}_{0}\hat{S}^{+}_{L-1}.$ (S13)

Therefore, we obtain the spin model shown in Eq. ( Non-Hermitian Mott Skin
Effect).

###  S1.2  Derivation of Eq. (9)

Applying the Jordan-Wigner transformation to the effective spin model [see Eq.
( Non-Hermitian Mott Skin Effect)] yields the spinless fermion model [see Eq.
(8)].

With the Jordan-Wigner transformation, $\hat{S}^{+}_{j}$ and $\hat{S}^{z}_{j}$
are rewritten as

$\displaystyle\hat{S}^{+}_{j}$ $\displaystyle=$ $\displaystyle
e^{i\pi\hat{N}^{<}_{j}}\hat{f}^{\dagger}_{j},$ (S14)
$\displaystyle\hat{S}^{z}_{j}$ $\displaystyle=$
$\displaystyle(\hat{n}^{\mathrm{f}}_{j}-\frac{1}{2}),$ (S15)

with creation (annihilation) operators of spinless fermions
$\hat{f}^{\dagger}_{j}$ ($\hat{f}_{j}$). Here, the operators are defined as
$\hat{N}^{<}_{j}=\sum^{j-1}_{i=0}\hat{n}^{\mathrm{f}}_{j}$,
$\hat{N}^{\mathrm{f}}=\sum_{j}\hat{n}^{\mathrm{f}}_{j}$, and
$\hat{n}^{\mathrm{f}}_{j}=\hat{f}^{\dagger}_{j}\hat{f}_{j}$.

Thus, each term of the spin model is transformed as follows: for $0\leq
j<L-1$,

$\displaystyle\hat{S}^{+}_{j+1}\hat{S}^{-}_{j}$ $\displaystyle=$
$\displaystyle\hat{f}^{\dagger}_{j+1}\hat{f}_{j},$ (S16)
$\displaystyle\hat{S}^{-}_{j+1}\hat{S}^{+}_{j}$ $\displaystyle=$
$\displaystyle\hat{f}^{\dagger}_{j}\hat{f}_{j+1},$ (S17)

and for $j=L-1$,

$\displaystyle\hat{S}^{+}_{j+1}\hat{S}^{-}_{j}$ $\displaystyle=$
$\displaystyle\hat{f}^{\dagger}_{0}e^{i\pi\hat{N}^{<}_{L-1}}\hat{f}_{L-1}$
(S18) $\displaystyle=$
$\displaystyle\hat{f}^{\dagger}_{0}e^{i\pi(\hat{N}^{\mathrm{f}}+\hat{n}^{\hat{f}}_{L-1})}\hat{f}_{L-1}$
$\displaystyle=$
$\displaystyle-e^{i\pi\hat{N}^{\mathrm{f}}}\hat{f}^{\dagger}_{0}e^{i\pi\hat{n}^{\hat{f}}_{L-1}}\hat{f}_{L-1}$
$\displaystyle=$
$\displaystyle-e^{i\pi\hat{N}^{\mathrm{f}}}\hat{f}^{\dagger}_{0}\hat{f}_{L-1},$
$\displaystyle\hat{S}^{-}_{j+1}\hat{S}^{+}_{j}$ $\displaystyle=$
$\displaystyle\hat{f}_{0}e^{i\pi\hat{N}^{<}_{L-1}}\hat{f}^{\dagger}_{L-1}$
(S19) $\displaystyle=$
$\displaystyle\hat{f}_{0}e^{i\pi(\hat{N}^{\mathrm{f}}+\hat{n}^{\hat{f}}_{L-1})}\hat{f}^{\dagger}_{L-1}$
$\displaystyle=$
$\displaystyle-e^{i\pi\hat{N}^{\mathrm{f}}}\hat{f}^{\dagger}_{L-1}\hat{f}_{0}.$

In a similar way, we obtain

$\displaystyle\hat{S}^{z}_{j+1}\hat{S}^{z}_{j}$ $\displaystyle=$
$\displaystyle(\hat{n}^{\mathrm{f}}_{j+1}-\frac{1}{2})(\hat{n}^{\mathrm{f}}_{j}-\frac{1}{2}),$
(S20)

for $0\leq j\leq L-1$.

Therefore, the effective spin model is rewritten as

$\displaystyle\hat{H}_{\mathrm{spin}}(\theta)$ $\displaystyle=$
$\displaystyle\sum^{L-2}_{j=0}\left(J_{+}\hat{f}^{\dagger}_{j+1}\hat{f}_{j}+J_{-}\hat{f}^{\dagger}_{j}\hat{f}_{j+1}\right)$
(S21)
$\displaystyle-(-1)^{\hat{N}^{\mathrm{f}}}\left(J_{+}e^{2i\theta_{\mathrm{s}}}\hat{f}^{\dagger}_{0}\hat{f}_{L-1}+J_{-}e^{-2i\theta_{\mathrm{s}}}\hat{f}^{\dagger}_{L-1}\hat{f}_{0}\right)$
$\displaystyle+J_{z}\sum^{L-1}_{j=0}\hat{n}^{\mathrm{f}}_{j+1}\hat{n}^{\mathrm{f}}_{j}-J_{z}(\hat{N}^{\mathrm{f}}-\frac{L}{4})+E_{0}.$

In particular, for the Fock space with $S^{z}=1-\frac{L}{2}$ (i.e.,
$[N_{\uparrow},N_{\downarrow}]=[1,L-1]$), there exists only one fermion
created by $\hat{f}^{\dagger}_{j}$. In this case, the following terms are
simplified as

$\displaystyle\sum^{L-1}_{j=0}\hat{n}^{\mathrm{f}}_{j+1}\hat{n}^{\mathrm{f}}_{j}$
$\displaystyle\to$ $\displaystyle 0$ (S22)
$\displaystyle\hat{N}^{\mathrm{f}}-\frac{L}{4}$ $\displaystyle\to$
$\displaystyle 1--\frac{L}{4}.$ (S23)

Therefore, we obtain Eq. (9) for the subspace with $S^{z}=1-\frac{L}{2}$.

##  S2  Properties of the one-body Hamiltonian

By analyzing the first quantized Hamiltonian, we see that the system shows the
non-Hermitian skin effect at the non-interacting case. The Hamiltonian (2) is
written as

$\displaystyle\hat{H}_{0}$ $\displaystyle=$
$\displaystyle\sum_{k}h_{\sigma}(k)\hat{b}^{\dagger}_{k\sigma}\hat{b}_{k\sigma}$
(S24)

with
$\hat{b}^{\dagger}_{k\sigma}=\frac{1}{\sqrt{L}}\sum_{j}e^{-ikj}\hat{b}^{\dagger}_{j\sigma}$
and $0\leq k<2\pi$. Here, we have imposed PBC. The first quantized Hamiltonian
$h_{\uparrow}(k)$ and $h_{\downarrow}(k)$ are written as
$h_{\uparrow}(k)=t_{+}e^{ik}+t_{-}e^{-ik}$ and
$h_{\downarrow}(k)=t_{-}e^{ik}+t_{+}e^{-ik}$, respectively.

Now, we introduce the spin winding number $w_{\mathrm{s}}$ characterizing the
point-gap topology of the first quantized Hamiltonian
$h(\bm{k})=\left(\begin{array}[]{cc}h_{\uparrow}(\bm{k})&0\\\
0&h_{\downarrow}(\bm{k})\end{array}\right).$ The spin winding number
$w_{\mathrm{s}}$ is defined as

$\displaystyle w_{\mathrm{s}}(\epsilon_{\mathrm{ref}})$ $\displaystyle=$
$\displaystyle\int^{2\pi}_{0}\frac{dk}{2\pi
i}\partial_{k}\mathrm{tr}[s_{z}\mathrm{log}(h(k)-\epsilon_{\mathrm{ref}}\mbox{1}\mbox{l}_{2\times
2})],$ (S25)

with $\epsilon_{\mathrm{ref}}\in\mathbb{C}$, $\mbox{1}\mbox{l}_{2\times 2}$
being the $2\times 2$ identity matrix, and
$s^{z}=\frac{1}{2}\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right)$. The
derivative with respect to $k$ is denoted by $\partial_{k}$.

Figure S1:  (a) [(b)]: Eigenvalues of the one-body Hamiltonian for
$\sigma=\uparrow$ [$\sigma=\downarrow$]. Open circles (closed triangle) denote
the data under PBC (OBC) for $L=6$. Closed circles denote the data under PBC
for $L=100$. The color of the closed circes represent the momentum $k$. (c)
[(d)]: Amplitude of the right eigenvectors for $\sigma=\uparrow$
[$\sigma=\downarrow$]. The gray (red) lines denote the data under PBC (OBC)
for $L=6$.

Figures S1(a) and S1(b) display the eigenvalues of the one-body Hamiltonian
$\epsilon_{k\sigma}=h_{\sigma}(k)$ under PBC. This figure indicates that the
spin winding number takes $1$ for $\epsilon_{\mathrm{ref}}=0$.

This non-trivial point-gap topology induces extreme sensitivity of eigenvalues
and eigenstates to the presence or absence of the boundaries. Figures S1(a)
and S1(b) indicate that the eigenvalues are aligned on the real axis under OBC
in contrast to the case of PBC. Correspondingly, the system under OBC exhibits
the skin modes localized around the boundaries [see Figs. S1(c) and S1(d)].

The above results indicate that the first quantized Hamiltonian shows the non-
Hermitian skin effect. We finish this part with a remark on the realness of
eigenvalues under OBC which can be understood by applying an imaginary gauge
transformation. Introducing operators

$\displaystyle\bar{\hat{a}}_{j}$ $\displaystyle=$ $\displaystyle e^{-\phi
j}\hat{c}^{\dagger}_{j},$ (S26) $\displaystyle\hat{a}_{j}$ $\displaystyle=$
$\displaystyle e^{\phi j}\hat{c}_{j},$ (S27)

we can rewrite the non-interacting Hamiltonian $\hat{H}_{0}$ under OBC

$\displaystyle\hat{H}_{0}$ $\displaystyle=$ $\displaystyle-
t_{\mathrm{a}}\sum_{j=0,\cdots,L-2}[\bar{\hat{a}}_{j+1\sigma}\hat{a}_{j\sigma}+\bar{\hat{a}}_{j\sigma}\hat{a}_{j+1\sigma}].$
(S28)

with $t_{\mathrm{a}}=\sqrt{t_{\mathrm{R}}t_{\mathrm{L}}}$ and
$\phi=\frac{1}{2}\ln\left(\frac{t_{L}}{t_{R}}\right)$. Here
$\bar{\hat{a}}_{j}$ and $\hat{a}_{j}$ satisfy the anti-commutation relation
$\\{\hat{a}_{i},\bar{\hat{a}}_{j}\\}=\delta_{ij}$. Equation (S28) can be
regarded as the Hermitian Hamiltonian. Thus, all of the eigenvalues are real.

##  S3  Details of computation of $W_{\mathrm{s}}$

Let us discuss the computation of $W_{\mathrm{s}}$ in details. Under twisted
boundary conditions, the non-interacting Hamiltonian is written as

$\displaystyle\hat{H}_{0}(\theta_{\mathrm{s}})$ $\displaystyle=$
$\displaystyle\sum_{\sigma=\uparrow,\downarrow}\left[\sum^{L-2}_{j=0}\left(-t_{\mathrm{R}\sigma}\hat{b}^{\dagger}_{j+1\sigma}\hat{b}_{j\sigma}-t_{\mathrm{L}\sigma}\hat{b}^{\dagger}_{j\sigma}\hat{b}_{j+1\sigma}\right)\right.$
(S29)
$\displaystyle\left.-t_{\mathrm{R}\sigma}e^{i\theta_{\sigma}}\hat{b}^{\dagger}_{0\sigma}\hat{b}_{L-1\sigma}-t_{\mathrm{L}\sigma}e^{i\theta_{\sigma}}\hat{b}^{\dagger}_{L-1\sigma}\hat{b}_{0\sigma}\right]$

with $\theta_{\uparrow}=\theta_{\mathrm{s}}$ and
$\theta_{\downarrow}=-\theta_{\mathrm{s}}$. As mentioned below Eq. (2),
$t_{\mathrm{R}\sigma}$ and $t_{\mathrm{L}\sigma}$ are chosen as
$t_{\mathrm{R}\uparrow}=t_{\mathrm{L}\downarrow}=t_{+}$ and
$t_{\mathrm{L}\uparrow}=t_{\mathrm{R}\downarrow}=t_{-}$.

Figure S2:  Argument
$\mathrm{Arg}[\mathrm{det}(\hat{H}_{{[1,L-1]}}-E_{\mathrm{ref}})]$ as
functions of $\theta_{\mathrm{s}}$ with $E_{\mathrm{ref}}=-0.01i$. Panel (a)
[(b)] displays the data for $(V,U)=(0,0)$ [(100,20)].

Equation (6) indicates that the many-body winding number $W_{\mathrm{s}}$ can
be extracted from $\theta_{\mathrm{s}}$ dependence of the argument,
$\mathrm{Arg}[\mathrm{det}(\hat{H}_{{[1,L-1]}}-E_{\mathrm{ref}})]$. Figure
S2(a) displays the argument as a function of $\theta$ for $(V,U)=(0,0)$ and
$E_{\mathrm{ref}}=-0.01i$. This figure indicates that the winding number takes
$W_{\mathrm{s}}=245$ for $E_{\mathrm{ref}}=-0.01i$ in the non-interacting
case. Figure S2(b) displays the argument as a function of $\theta$ for
$(V,U)=(100,20)$ and $E_{\mathrm{ref}}=-0.01i$. This figure indicates that the
winding number takes $W_{\mathrm{s}}=2$ for $E_{\mathrm{ref}}=-0.01i$ for the
correlated case.

##  S4  Expectation values for PBC

We provide the expectation values under PBC. Under PBC, the expectation values
$\langle\hat{n}_{j}\rangle$ and $\langle\hat{S}^{z}_{j}\rangle$ are
homogeneous for the non-interacting case (see Fig. S3),

Figure S3:  The expectation values for $(V,U)=(0,0)$ under PBC. Panel (a)
[(b)] displays $\langle\hat{n}_{j}\rangle$ [$\langle\hat{S}^{z}_{j}\rangle$].
These data are obtained for $(t_{+},t_{-})=(1,0.1)$ and $L=6$. These figures
are plotted in a similar way as Figs. 1(c) and 1(d).

These properties do not change even in the presence of the interactions.
Figure S4 displays the expectation values $\langle\hat{n}_{j}\rangle$ and
$\langle\hat{S}^{z}_{j}\rangle$ under PBC for $(V,U)=(100,20)$.

Figure S4:  The expectation values for $(V,U)=(100,20)$ under PBC. Panel (a)
[(b)] displays $\langle\hat{n}_{j}\rangle$ [$\langle\hat{S}^{z}_{j}\rangle$].
These data are obtained for $(t_{+},t_{-})=(1,0.1)$ and $L=6$. These figures
are plotted in a similar way as Figs. 2(c) and 2(d).

##  S5  Results for $(N_{\uparrow},N_{\downarrow})=(2,4)$

Here, we provide the results for subspace with
$(N_{\uparrow},N_{\downarrow})=(2,4)$ and $L=6$ which indicates that the Mott
skin effect is still observed.

Let us start with the non-interacting case where the ordinary non-Hermitian
skin effect is observed. Specifically, under PBC, the expectation values
$\langle\hat{n}_{j}\rangle$ and $\langle\hat{S}^{z}_{j}\rangle$ are
homogeneous [see Figs. S5(a) and S5(b)]. Imposing OBC results in the
localization of eigenstates which is reflected in both expectation values
$\langle\hat{n}_{j}\rangle$ and $\langle\hat{S}^{z}_{j}\rangle$ [see Figs.
S5(c) and S5(d)].

Figure S5:  The expectation values for the subspace with
$(N_{\uparrow},N_{\downarrow})=(2,4)$ and for $(V,U)=(0,0)$. Panels (a) and
(b) [(c) and (d)] are data under PBC. Panels (a) and (c) [(b) and (d)] display
$\langle\hat{n}_{j}\rangle$ [$\langle\hat{S}^{z}_{j}\rangle$]. These figures
are plotted in a similar way as Figs. 1(c) and 1(d). The data are obtained for
$(t_{+},t_{-})=(1,0.1)$ and $L=6$.

In contrast to the non-interacting case, for $(V,U)=(100,20)$, the emergence
of the skin modes is observed only in the spin degree of freedom. Firstly, we
note that under PBC, the expectation values $\langle\hat{n}_{j}\rangle$ and
$\langle\hat{S}^{z}_{j}\rangle$ are homogeneous [see Figs. S6(a) and S6(b)].
Imposing OBC results in localization in the spin degree of freedom although
such localization is not observed in the charge degree of freedom [see Figs.
S6(c) and S6(d)].

Figure S6:  The expectation values for the subspace with
$(N_{\uparrow},N_{\downarrow})=(2,4)$ and for $(V,U)=(100,20)$. Panels (a) and
(b) [(c) and (d)] are data under PBC. Panels (a) and (c) [(b) and (d)] display
$\langle\hat{n}_{j}\rangle$ [$\langle\hat{S}^{z}_{j}\rangle$]. These figures
are plotted in a similar way as Figs. 1(c) and 1(d). The data are obtained for
$(t_{+},t_{-})=(1,0.1)$ and $L=6$.

The above results indicate that the Mott skin effect is observed even for the
subspace with $(N_{\uparrow},N_{\downarrow})=(2,4)$ and $L=6$.

##  S6  Time-evolution under PBC

We discuss the time-evolution of the expectation values
$\langle\hat{n}_{j}(t)\rangle$ and $\langle\hat{S}^{z}_{j}(t)\rangle$ under
PBC. In a similar way to the case of OBC, we compute the expectation values
$\langle\hat{n}_{j}(t)\rangle$ and $\langle\hat{S}^{z}_{j}(t)\rangle$ [see Eq.
(11)]. The initial state is chosen so that all the bosons occupy site $j=0$.

Figures S7(a) and S7(b) display the data for the non-interacting case. In
contrast to the case of OBC, bosons do not localize around the left edge $j=0$
due to the absence of the skin modes. In addition, the bosons propagate to
left, which is attributed to the fact that the eigenstates with the longest
lifetime $\tau$ ($\tau\sim 1/\mathrm{Im}E$) are composed of bosons propagating
to left.

Figure S7:  The time-evolution of the expectation values
$\langle\hat{n}_{j}(t)\rangle$ and $\langle\hat{S}^{z}_{j}(t)\rangle$ under
PBC [see Eq. (11)] These figures are plotted in a similar way to Fig. 3 These
data are obtained for $(t_{+},t_{-})=(1,0.1)$ and $L=6$.

Figures S7(c) and S7(d) display the data for $(V,U)=(100,20)$. In this case,
increasing time $t$, the expectation values $\langle\hat{n}_{j}(t)\rangle$ and
$\langle\hat{S}^{z}_{j}(t)\rangle$ become homogeneous. This is because the
eigenstate with the longest lifetime $\tau$ ($\tau\sim 1/\mathrm{Im}E$) is the
state described by a fermion with zero momentum in terms of the effective
fermion model (9).