# Engineering skin effect across a junction of Hermitian and non-Hermitian
lattice

Ranjan Modak<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Tirupati, Tirupati, India 517619

###### Abstract

We study a system where the two edges of a non-Hermitian lattice with
asymmetric nearest-neighbor hopping are connected with two Hermitian lattices
with symmetric nearest-neighbor hopping. In the absence of those Hermitian
lattices, the majority of the eigenstates of the system will be localized at
the edges, the phenomena known as the non-Hermitian skin effect. We show that
once we connect it with the Hermitian lattices, for open boundary conditions
(OBC), the localized states exist at the junction of the non-Hermitian and
Hermitian lattice; moreover, the spectrum shows mobility edges that separate
delocalized and localized states. On the contrary, mobility edges vanish for
periodic boundary conditions (PBC), and the delocalized phase turns into a
scale-invariant localized phase, where the localized states are still peaked
at the junctions. We also find that if the connected Hermitian lattices are
thermodynamically large, in OBC, most of the states become delocalized, while
in PBC, the system still shows the scale-invariant localized phase.

##  I Introduction

Non-Hermitian systems have been of great interest in physics in recent years,
uncovering a wide range of phenomena Makris  _et al._ (2008); Klaiman  _et
al._ (2008); Lin  _et al._ (2011); Wiersig (2014); Pal  _et al._ (2022); Modak
and Aravinda (2023). This surge started with the inception of PT-symmetric
non-Hermitian quantum mechanics more than two decades ago. While in general
non-Hermitian Hamiltonians have complex eigenvalues, by replacing the
Hermiticity condition on the Hamiltonian with a much less constraining
condition of space-time reflection symmetry known as PT-symmetry Bender  _et
al._ (2002); Bender (2005); Khare and Mandal (2000), it was shown that the
spectrum can be completely real. Such systems described by PT-invariant non-
Hermitian Hamiltonians can typically be divided into two categories, one in
which the eigenvectors respect PT symmetry and the entire spectrum is real,
and the other in which the whole spectrum or a part of it is complex and the
eigenvectors do not respect the PT symmetry. These are known as PT-unbroken
and broken phases, respectively. The phase transition point is known as the
exceptional point (EP). Recently, it has been demonstrated that there exists a
Hermitian counterpart for the non-Hermitian systems in the PT-symmetric phase
by defining a suitable inner-product Ju  _et al._ (2019); Shukla  _et al._
(2023), while the PT-broken phase can’t be mapped into any Hermitian system.
This field of PT-symmetric non-Hermitian physics received a huge boost when
the consequence of PT transition was observed experimentally in various
systems Chitsazi  _et al._ (2017); Biesenthal  _et al._ (2019); Kremer  _et
al._ (2019); Konotop  _et al._ (2016); Ding  _et al._ (2021); Yu  _et al._
(2020).

Moreover, in the last few years, significant importance has been given to the
understanding of the topological phases of non-Hermitian systems Bergholtz
_et al._ (2021); Gong  _et al._ (2018); Kawabata  _et al._ (2019), which shows
a breakdown of the correspondence between bulk topology and the existence of
boundary modes Lee (2016). Subsequently, theoretical studies were successful
in formulating a generalized bulk-boundary correspondence which is also
consistent with experiments Kunst  _et al._ (2018); Yao and Wang (2018); Kunst
and Dwivedi (2019); Edvardsson  _et al._ (2020); Borgnia  _et al._ (2020);
Xiao  _et al._ (2020); Helbig  _et al._ (2020). In this context, the non-
Hermitian skin effect (NHSE) has also brought lots of attention to the
community. It has been shown that for certain non-Hermitian systems, the
majority of eigenstates of a non-Hermitian Hamiltonian are localized at
boundaries Borgnia  _et al._ (2020); Lee and Thomale (2019); Jiang  _et al._
(2019); Zhang  _et al._ (2020). The interplay of the NHSE with topology is
something that makes this effect particularly interesting Lee  _et al._
(2019), and has been observed in experiments as well Zou  _et al._ (2021).
These localized edge states in non-Hermitian systems have a topological origin
that makes them different from Anderson localized states in disordered systems
Abrahams  _et al._ (1979); Anderson (1958). Moreover, understanding the
interplay between NHSE and disordered potential also has been an active area
of research Molignini  _et al._ (2023); Zeng  _et al._ (2017). All these
notions are not only theoretically enriching; there are many potential
technological applications of the non-Hermitian skin effect. Recently, a
topological switch driven by the non-Hermitian skin effect has been proposed
in cold-atom systems Li  _et al._ (2020), which is controlled by atom loss.
Due to the extreme sensitivity of the boundary condition, NHSE can also lead
to applications in quantum sensors Budich and Bergholtz (2020); Koch and
Budich (2022).

Figure 1: Schematic diagram of our system (Hamiltonian Eqn. (II)). $-t$, $-t-\delta$ and $-t-\gamma$ indicate nearest-neighbour hopping strength. For $\delta\neq\gamma$, the system becomes non-Hermitian. | OBC | PBC
---|---|---
$|Re[E]|<E_{c}$ | Delocalized phase | Scale invariant localized phase; $\langle n_{j}\rangle\sim e^{-|j-(L/2-k)|/\xi(L)}$; $\xi(L)\propto L$
$|Re[E]|>E_{c}$ | Localized phase; $\langle n_{j}\rangle\sim e^{-|j-(L/2-k)|/\xi}$ | Localized phase; $\langle n_{j}\rangle\sim e^{-|j-(L/2-k)|/\xi}$
Table 1: Table shows the energy-dependent phase diagram of the eigenstates of
the Hamiltonian (II) for both open boundary condition (OBC) and periodic
boundary condition (PBC).

However, for all practical application purposes, one needs to find out a fine-
tuned version of the skin effect. Usually, the skin effect manifests localized
states at the edges. Our main goal here is to engineer a system that gives
control to the position of the localized state. Hence, we study a system where
a non-Hermitian lattice is connected with the Hermitian lattices on both sides
(see Fig. 1 for a schematic diagram); we find that at the junction of the
Hermitian and non-Hermitian lattice, one observes the localized states. Hence,
by changing the location of the non-Hermitian lattice, one can control the
position of such localized states. The systems also seem extremely sensitive
to boundary conditions, as most non-hermitian systems are. We outline our main
results in Table. 1. We find for open boundary conditions (OBC), the systems
support mobility edges Sarma  _et al._ (1988); Izrailev and Krokhin (1999);
Wang  _et al._ (2020); Chatterjee and Modak (2022) (coexistence of localized
and delocalized states in the spectrum), and for periodic boundary conditions
(PBC), the delocalized phase vanishes, and it turns into scale-invariant
localized phase. However, remarkably, all the localized states peak at the
junction.

The manuscript is organized as follows. We define our system in Sec. II. We
elaborate on our main results in Sec. III. Finally, we discuss the conclusion
and future prospect of our study in Sec. IV.

Figure 2: Imaginary part of energy eigenvalues vs. real part of the
eigenvalues for different values of $\gamma$ and $L=200$. Symbols correspond
to PBC, and lines correspond to OBC.

##  II Model

We study a system where edges of the non-Hermitian lattice of size $2k$ with
asymmetric nearest-neighbor hopping is connected with two Hermitian lattices
of sizes $L/2-k$ each with symmetric nearest-neighbor hopping (see the
schematic in Fig. 1). The Hamiltonian (which is also non-Hermitian as a whole)
is given by,

$\displaystyle{H}$ $\displaystyle=$ $\displaystyle
H_{1}+H_{nh}+H_{2},~{}~{}\text{where}$ $\displaystyle H_{1}$ $\displaystyle=$
$\displaystyle-\sum_{j=1}^{L/2-k-1}t(\hat{c}^{{\dagger}}_{j}\hat{c}_{j+1}+\text{H.c.})$
$\displaystyle H_{nh}$ $\displaystyle=$
$\displaystyle-\sum_{j=L/2-k}^{L/2+k}(t+\gamma)\hat{c}^{{\dagger}}_{j}\hat{c}_{j+1}+(t+\delta)\hat{c}^{{\dagger}}_{j+1}\hat{c}_{j}$
(1) $\displaystyle H_{2}$ $\displaystyle=$
$\displaystyle-\sum_{j=L/2+k}^{L-1}t(\hat{c}^{{\dagger}}_{j}\hat{c}_{j+1}+\text{H.c.}),$

where, $\hat{c}^{{\dagger}}_{j}$ ($\hat{c}_{j}$) is the fermionic creation
(annihilation) operator at site $j$,
$\hat{n}_{j}=\hat{c}^{{\dagger}}_{j}\hat{c}_{j}$ is the number operator. We
introduce asymmetric hopping by choosing $\gamma\neq\delta$. In the results
section, we report the results for both open boundary conditions (OBC) and
periodic boundary conditions (PBC). Note that we add a Hermitian boundary term
$-t(\hat{c}^{{\dagger}}_{L}\hat{c}_{1}+\hat{c}^{{\dagger}}_{1}\hat{c}_{L})$
with $H$ to implement PBC. The Hamiltonian (II) under PBC is exactly solvable
in the Hermitian limit, i.e., $\gamma=0$ and $\delta=0$. The Hamiltonian reads
as,

$\displaystyle
H(\gamma=0,\delta=0)=-\sum_{i}t(\hat{c}_{i}^{{\dagger}}\hat{c}_{i+1}+\text{H.c}).$
(2)

The Hamiltonian (2) can be diagonalized in the momentum basis and can be
written as,

$\displaystyle
H(\gamma=0,\delta=0)=\sum_{k}\epsilon_{k}\hat{c}_{k}^{{\dagger}}\hat{c}_{k}$
(3)

where, $\epsilon_{k}=-2t\cos k$. $\hat{c}_{k}^{{\dagger}}$ and $\hat{c}_{k}$
are fermionic creation and annihilation operators in momentum basis. Also, in
the limit $t=0$, the Hamiltonian shows a non-Hermitian skin effect; as a
result, all states become localized at the edge Zhang  _et al._ (2022); Lin
_et al._ (2023). We choose $L$ as an even number, $\gamma>0$, $\delta=0$, and
$t=1$ for all the calculations in the main text. Next, we discuss the
properties of the eigenstates of the Hamiltonian (II) in the following
section.

Figure 3: (Left panel) Shows the variation of participation ratio (PR) with
the real part of the energy for different values of $L$ and fixed $\gamma=4$.
Symbols are for OBC, and lines are for PBC. (Right panel) Shows the variation
of PR for OBC for different values of $\gamma$ and for fixed $L=200$. We fix
$k=25$. Figure 4: (Left panel) shows the variation of $\langle
n_{L/2-k}\rangle$ with the real part of energy eigenvalues for different
values of $k$. (Right panel) shows the variation of $\langle n_{j}\rangle$
with $n_{j}$ for $4^{th}$ eigenstates for different values of $k$. We fix
$L=200$, $\gamma=4$, and use OBC. Figure 5: Variation of $\langle
n_{j}\rangle$ with site $j$ for $L=150$, $k=25$, $\gamma=4$, and $m^{th}$
right-eigenstates. (Note eigenstates are arranged in ascending order of real
part of energy eigenvalues) Inset shows the variation of $\langle
n_{j=50}\rangle$ with the real part energy eigenvalues. These results are for
PBC.  Figure 6: Variation of $\langle n_{j}\rangle$ with site $j$ for
different values of $L$, $k=25$, $\gamma=4$. Results are for states
$|Re[E]|<E_{c}$ (left panel) and $|Re[E]|>E_{c}$ (right panel). Dashed lines
are exponential fitting curves i.e. $\langle
n_{j}\rangle\sim\exp(-|x-(L/2-k)|/\xi)$. Inset shows linear scaling of
localization length $\xi$ with $L$ for states $|Re[E]|<E_{c}$. These results
are for PBC.

##  III Results

First, we focus on the energy spectrum of the Hamiltonian (II). Note that our
Hamiltonian is non-Hermitian; hence eigenvalues $E_{m}$ in principle can be
complex numbers as well. We show in Fig. 2 the variation of the imaginary
parts of the eigenvalues $E_{m}$ with its own real parts for both OBC and PBC.
We find the eigenvalues are almost real for OBC; however, as soon as we
introduce PBC, they become complex and form a loop in the complex plane of
$E_{m}$. Usually, non-Hermitian systems are extremely sensitive to boundary
conditions, and a similar type of eigenvalue spectrum has also been observed
for non-Hermitian models that manifest skin effect Gong  _et al._ (2018).

Next, we focus on the properties of eigenstates. Note that our system as a
whole is also non-Hermitian. Hence, its right eigenstates and left eigenstates
are not the same. We define an energy-dependent localization-delocalization
measure using the right eigenvectors, commonly known as participation ratio
(PR) Tang  _et al._ (2020); Johri and Bhatt (2014),

$\displaystyle\text{PR}(E_{m})=\frac{(\sum_{j}|c^{m}_{j}|^{2})^{2}}{\sum_{j}|c^{m}_{j}|^{4}},$
(4)

where, $|\psi_{m}^{R}\rangle=\sum_{j}c^{m}_{j}|j\rangle$ the right eigenvector
satisfies $H|\psi_{m}^{R}\rangle=E_{m}|\psi_{m}^{R}\rangle$ (where $|j\rangle$
represents the site basis). PR is expected to be small $O(1)$ (does not scale
with $L$) in the localized phase and scales linearly with $L$ in the
delocalized phase. Figure. 3 (left panel) shows that for OBC and
$|Re[E]|<E_{c}=2$ (which is the half bandwidth of the Hermitian lattice), all
states are delocalized, and the rest of the states are localized. Hence, the
spectrum shows mobility edges at $E=\pm E_{c}$. Remarkably, as we introduce
PBC, mobility edges disappear (PR for PBC is shown using lines in Fig. 3 left
panel). In the right panel of Fig. 3, we plot the data for different values of
$\gamma$, proving the robustness of our results.

While the value of PR and its scaling with $L$ tell us whether a state is
localized or delocalized, it does not tell us the structure of the states. The
question, like where the localized states are peaked, can’t be addressed for
PR. Hence, next, we investigate structures of localized states in the case of
OBC. In the limit, $t=0$, one observes the non-Hermitian skin effect, and the
eigenstates are localized at site $j=L/2-k$ (for $\gamma>\delta$) and
$j=L/2+k$ (for $\delta>\gamma$, which we discuss in the Appendix). Here, we
find similar evidence for the localized states (see Fig. 4). The right panel
of Fig. 4 clearly shows that for localized states under OBC $\langle
n_{j}\rangle\sim\exp(-|j-(L/2-k)|/\xi)$. On the other hand, in case of PBC,
for eigenstates $|Re[E]|>E_{c}$. once again $\langle
n_{j}\rangle\sim\exp(-|j-(L/2-k)|/\xi)$ (see Fig. 5 the results for $m=4$ and
$140$). However, for eigenstates $|Re[E]|<E_{c}$,

$\displaystyle\langle n_{j}\rangle$ $\displaystyle\simeq$
$\displaystyle\exp(-(L/2-k-j)/\xi_{<}),~{}~{}\text{for}~{}~{}j<L/2-k$
$\displaystyle\simeq$
$\displaystyle\exp(-(j-L/2+k)/\xi_{>}),~{}~{}\text{for}~{}~{}j>L/2-k.$

Figure. 6 confirms that $\xi$, $\xi_{>}$ don’t depend on $L$, while $\xi_{<}$
scales linearly with $L$ (see inset of Fig. 6). Hence, we identify the
$|Re[E]|<E_{c}$ phase (for PBC) as the scale-invariant localized phase. To
strengthen our claim that all the localized states for both PBC and OBC are
peak at the junction, i.e., at site $L/2-k$, we compute $\langle
n_{L/2-k}\rangle=\langle\psi^{R}|\hat{n}_{L/2-k}|\psi^{R}\rangle/\langle\psi^{R}|\psi^{R}\rangle$
for all the states. Figure.4 (left panel) clearly shows that for the OBC, the
states which are localized ( when $|Re[E]|>E_{c}$), $\langle n_{L/2-k}\rangle$
very close to $1$, while for delocalized states the value is much smaller
(which is expected given the states are not peaked at the junction). The inset
of Fig. 5 shows similar results for the PBC. Due to the absence of a true
delocalized state (mobility edge is also absent) and all the states are
localized and peaked at the junction, under PBC, $\langle n_{L/2-k}\rangle$
value is $O(1)$ for all the states. It confirms that the localized states peak
at the junction of the non-Hermitian and Hermitian lattice.

Thermodynamic limit: Next, we discuss what happens in the thermodynamic limit.
Note that one can approach the thermodynamic limit for our system in the
following two ways, 1) taking $L,k\to\infty$ limit keeping $k/L$ finite, 2)
taking only $L\to\infty$ limit but $k$ is finite, hence $k/L\to 0$. For the
first scenario, the conclusions obtained from our finite-size results will
remain true, i.e., for OBC, mobility edges exist (coexistence of both
localized and delocalized states). For PBC, the delocalized phase turns into a
scale-invariant localized phase. On the other hand, for the 2nd scenario, for
OBC, the faction of localized states $f\to 0$. It implies that the number of
localized states will be ”measure zero” in the limit $L\to\infty$, effectively
suppressing the effect of localized state in the system. The same thing will
also happen for Hermitian systems (see Appendix for details). On the other
hand, under PBC, the majority of the states will behave as scale-invariant
localized states, and all the states will peak at the junction.

##  IV Conclusions

In this work, we study the fate of the non-Hermitian skin effect if a non-
Hermitian lattice with asymmetric hopping is connected with a Hermitian
lattice on both sides. It is well established that without the Hermitian
lattice connection, a non-Hermitian lattice of asymmetric hopping usually
shows localized edge modes at the boundary, known as the non-Hermitian skin
effect. We find here that under OBC, the localized states still exist at the
junction of the non-Hermitian and Hermitian lattice. There also exist
delocalized states separated by the so-called mobility edges in the spectrum.
On the contrary, under PBC, mobility edges vanish, and the delocalized phase
turns into a scale-invariant localized phase, i.e., localization length of the
right wave-function scales with system size. Hence, we show by connecting the
non-Hermitian lattice with the Hermitian one; it’s possible to engineer the
localized states at the junction of two such lattices, which gives us
additional freedom to control the position of the localized states, which
typically is only observed at the boundary for the non-Hermitian skin effect.
In the thermodynamic limit (keeping non-Hermitian lattice size fixed), for
OBC, the system will be completely delocalized; however, strikingly, for PBC,
the system will remain in a scale-invariant localized phase, where states will
be peaked at the Hermitian and non-hermitian lattice junction. We also like to
point out that such a scale-invariant localized phase has also recently been
observed in disordered non-Hermitian lattices, and it was referred to as the
”Anomalous skin effect”.

Also, theoretically, it raises a broader question of the stability of the non-
Hermitian skin effect of a finite-sized non-Hermitian lattice once connected
with a delocalized Hermitian infinite bath. Unlike the Hermitian system, here,
the fate of the non-Hermitian skin effect will depend on the boundary
condition. For open boundary conditions, for a true infinite delocalized bath,
the majority of states of the whole non-Hermitian system will be delocalized,
but in a ring geometry (PBC), the states will display scale-invariant
localizations, and all the states will be peaked at the junction between
Hermitian and non-Hermitian lattice. In contrast, for the Hermitian system
(assuming an Anderson-localized system is placed between two Hermitian
delocalized lattices), independent of the boundary conditions, in the
thermodynamic limit, most of the states will always be delocalized (see
Appendix for the comparison with Hermitian systems). Further, It will be
interesting to investigate the role of interactions and quasiperiodic
potentials in such cases in future studies Hamazaki  _et al._ (2019); Zhai
_et al._ (2020); Wang  _et al._ (2022); Suthar  _et al._ (2022).

##  V Acknowledgements

RM acknowledges the DST-Inspire fellowship by the Department of Science and
Technology, Government of India, SERB start-up grant (SRG/2021/002152).

##  Appendix A Results for $\delta>0$ and $\gamma=0$

Figure 7: Variation of PR with real part of the energy $E_{m}$ for different
values of $L$ and for fixed $k=25$, $\delta=4$ and $\gamma=0$. Inset shows the
variation of $\langle n_{L/2+k}\rangle$ for different eigenstates. These
results are for OBC.  Figure 8: Variation of PR with real part of the energy
$E_{m}$ for different values of $L$ and for fixed $k=25$, $V=5$ for the
Hermitian Hamiltonian (LABEL:hamiltonian_h) for PBC. Inset shows the variation
of the fraction of localized states $f$ vs $L$ for the Hamiltonian $\tilde{H}$
(PBC) and the non-Hermitian Hamiltonian ${H}$ (OBC) using circle and diamond
symbols, respectively.

While in the main text, we choose $\delta=0$ and $\gamma>0$, in this section,
we demonstrate the results for $\delta>0$ but $\gamma=0$. Figure. 7 clearly
shows that similar to the main text results, the spectrum shows mobility edges
for OBC. PR increases with system size for $|Re[E]|<E_{c}$ indicating
delocalized states, while PR of the rest of the states do not scale with
system size. In the inset, we plot the variation of $\langle n_{L/2+k}\rangle$
for different states. Results suggest that the localized states peaked at site
$j=L/2+k$. While for the $\gamma>\delta=0$ case, the peaks of the localized
states were at another junction i.e. at site $j=L/2-k$ as shown in the main
text.

##  Appendix B Comparison with Hermitian systems

We study a system that is described by the following Hermitian Hamiltonian,

$\displaystyle\tilde{H}$ $\displaystyle=$
$\displaystyle\tilde{H}_{1}+H_{h}+\tilde{H}_{2},~{}~{}\text{where}$
$\displaystyle\tilde{H}_{1}$ $\displaystyle=$
$\displaystyle-\sum_{j=1}^{L/2-k-1}t(\hat{c}^{{\dagger}}_{j}\hat{c}_{j+1}+\text{H.c.})$
$\displaystyle H_{h}$ $\displaystyle=$
$\displaystyle-\sum_{j=L/2-k}^{L/2+k}t(\hat{c}^{{\dagger}}_{j}\hat{c}_{j+1}+\hat{c}^{{\dagger}}_{j+1}\hat{c}_{j})+V\sum_{j=L/2-k}^{L/2+k}\cos(2\pi\beta
j)$ $\displaystyle\tilde{H}_{2}$ $\displaystyle=$
$\displaystyle-\sum_{j=L/2+k}^{L-1}t(\hat{c}^{{\dagger}}_{j}\hat{c}_{j+1}+\text{H.c.}),$

where, $\hat{c}^{{\dagger}}_{j}$ ($\hat{c}_{j}$) is the fermionic creation
(annihilation) operator at site $j$,
$\hat{n}_{j}=\hat{c}^{{\dagger}}_{j}\hat{c}_{j}$ is the number operator,
$\beta=(\sqrt{5}-1)/2$. For $V>2$ ($V<2$), all the states of $H_{h}$ are
localized (delocalized) Aubry and André (1980). This model is famously known
as the Aubry-Andre model, which in contrast to the true disorder, shows a
delocalization-localization transition even in one dimension Modak and
Aravinda (2022); Kumar and Modak (2022). Figure. 8 shows the variation of PR
for fixed $k=25$, $V=5$ but different values $L$ for PBC. Unlike our results
in the main text, for Hermitian Hamiltonian, even in PBC, eigenstates in the
middle of the spectrum remain delocalized. In the inset, we plot the fraction
of localized state $f$ vs $L$. In the same inset, we also show the variation
of $f$ for the non-Hermitian Hamiltonian (II) for OBC, find that $f\to 0$ as
$L\to\infty$ as $1/L$. Note that to compute $f$, those states which don’t
scale with $L$ have been identified as localized states. Hence, our main
results in the main text, i.e., the existence of scale-invariant localized
phase under PBC while the absence of localization in OBC in the thermodynamic
limit (keeping the size of the non-Hermitian lattice fixed) is purely a non-
Hermitian quantum phenomena; it does not have any Hermitian counterpart.

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