Non-Hermitian skin eﬀects on many-body localized and thermal phases Yi-Cheng Wang1 2Kuldeep Suthar1H. H. Jen1 3Yi-Ting Hsu4yand Jhih-Shih You5z 1Institute of Atomic and Molecular Sciences Academia Sinica Taipei 10617 Taiwan

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Non-Hermitian skin eﬀects on many-body localized and thermal phases

Yi-Cheng Wang,1, 2, ∗Kuldeep Suthar,1, ∗H. H. Jen,1, 3 Yi-Ting Hsu,4, †and Jhih-Shih You5, ‡

1Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617, Taiwan

2Department of Physics, National Taiwan University, Taipei 10617, Taiwan

3Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan

4Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA

5Department of Physics, National Taiwan Normal University, Taipei 11677, Taiwan

(Dated: January 12, 2023)

Localization in one-dimensional interacting systems can be caused by disorder potentials or non-

Hermiticity. The former phenomenon is the many-body localization (MBL), and the latter is the

many-body non-Hermitian skin eﬀect (NHSE). In this work, we numerically investigate the interplay

between these two kinds of localization, where the energy-resolved MBL arises from a deterministic

quasiperiodic potential in a fermionic chain. We propose a set of eigenstate properties and long-

time dynamics that can collectively distinguish the two localization mechanisms in the presence

of non-Hermiticity. By computing the proposed diagnostics, we show that the thermal states are

vulnerable to the many-body NHSE while the MBL states remain resilient up to a strong non-

Hermiticity. Finally, we discuss experimental observables that can probe the diﬀerence between

the two localizations in a non-Hermitian quasiperiodic fermionic chain. Our results pave the way

toward experimental observations on the interplay of interaction, quasiperiodic potential, and non-

Hermiticity.

Introduction.—Many-body localization (MBL) [1–3]

can exist in one-dimensional (1D) isolated quantum

systems in the presence of interaction and disorders,

where thermalization fails to occur [4–7] and the

information encoded in the initial state is preserved [8–

10]. Besides the well-known cases with random

disorders [5,6,11–15], where the thermal to MBL

transition occurs as the disorder strength increases,

numerical [16–27] and experimental [8,28,29] evidences

have suggested that MBL can also occur in the

presence of deterministic but quasiperiodic potentials. In

particular, in quasiperiodic systems with a single-particle

mobility edge, MBL and thermal phases have been found

to coexist at a given intermediate potential strength

in low- and mid-spectrum regimes, respectively [17–

20]. Such an energy-resolved localization-delocalization

transition is originated from the non-trivial interplay

between the interaction and quasiperiodic potential,

where the latter provides a localization mechanism in

many-body Hermitian systems.

In non-Hermitian systems, a distinct localization

mechanism dubbed non-Hermitian skin eﬀect (NHSE)

has recently attracted rapidly growing theoretical [30–44]

and experimental [45–54] attention, where an extensive

number of eigenstates are localized at open boundaries.

In the non-interacting limit, single-particle NHSE has

been shown to occur under open boundary condition

when the eigenspectrum under periodic boundary

condition exhibits nontrivial winding [37,38]. This

can be viewed as the non-Hermitian analogue of the

“bulk-boundary correspondence” in topological systems.

In terms of speciﬁc models, it has been shown that

such winding and localization can arise when the non-

Hermiticity is introduced by nonreciprocal hoppings [55].

In the presence of interactions, although the relation

between the winding in eigenspectrum and many-body

NHSE remains an interesting but elusive topic, recent

theoretical works have investigated the existence [56,57]

and entanglement dynamics [58] of many-body NHSE

in fermionic systems, how MBL is aﬀected by non-

Hermiticity [59,60], as well as NHSE in random

disordered systems [61]. Although the many-body NHSE

does not exhibit strictly exponential localization in the

real space as the single-particle NHSE does due to Pauli

exclusion principle [57], particles in all the many-body

eigenstates are still expected to accumulate on one end

of open boundaries in the strong non-Hermiticity limit.

Therefore, in stark contrast to MBL, where particles

in a given initial states stay localized at their initial

positions, many-body NHSE tends to push all particles

towards one of the open boundaries such that the initial

information is lost. Yet, the competition between these

two localizations in interacting 1D systems with both

non-Hermiticity and quasiperiodic potentials remains

elusive.

In this Letter, we investigate how many-body NHSE

aﬀects the MBL and thermal phases in 1D quasiperiodic

systems, focusing on a case study of the generalized

Aubry-André (GAA) model [17,18,62,63] in the

presence of nonreciprocal hoppings. To distinguish the

two localization mechanisms, namely the quasiperiodic

potential and the non-Hermiticity, we propose a set

of eigenstate properties and dynamical responses that

can collectively diagnose the many-body NHSE and

the non-Hermitian localized phase connected to the

Hermitian MBL, dubbed non-Hermitian MBL. Our key

ﬁnding is that the thermal phase is vulnerable to

non-Hermiticity such that the volume-law entanglement

entropy vanishes and the many-body NHSE appears

already at a small asymmetry between the left and

arXiv:2210.12998v2 [cond-mat.dis-nn] 11 Jan 2023



JL = Jeg

JR = Je-g

1.0

0.5

0.5

1.0

1.0

0.5

0.5

1.0

Non-Hermi�city g

Energy

Many-body localized Thermal

Many-body NHSE

Quasiperiodic poten�al

Non-

Hermi�an

MBL

FIG. 1. (a) Schematic of an open fermionic chain subject

to generalized Aubry-André potential (brown, φ= 0). The

nonreciprocal hoppings JL(R)=Je±g(blue and red) and

nearest-neighbor interaction V(black) are shown according

to the conﬁguration of occupied (•) and unoccupied (◦) sites.

(b) The phase diagram as a function of energy and non-

Hermiticity parameter gexplored in this Letter. The red and

blue dashed lines roughly separate the parameter regimes with

the initial state memory and localization of particles at the

open boundary, respectively.

right hopping. In contrast to the thermal phase,

MBL dominates over the many-body NHSE up to an

extremely large hopping asymmetry, where the many-

body eigenstates across the full spectrum exhibit strong

NHSE. Importantly, we emphasize that although both

quasiperiodic potentials and non-Hermiticity can drive

localization, the many-body NHSE does not preserve

the information of the initial state as does the MBL

phase. Finally, we discuss possible experimental

detection scheme that distinguishes non-Hermitian MBL

and many-body NHSE using the long-time behaviors of

generalized imbalance [64–66] and local particle numbers.

We expect that our observation of how the many-body

NHSE aﬀects the thermal and MBL phases is generic for

other non-Hermitian systems with many-body NHSE in

the presence of quasiperiodic as well as random disorder

potentials.

Model.—In the absence of interactions, models

with non-reciprocal hoppings are well-known to

exhibit winding in the eigenspectrum and thus single-

particle NHSE, and could realized experimentally

in various systems [67–69]. Now in the presence of

interactions, we therefore investigate many-body NHSE

by introducing non-Hermiticity with non-reciprocal

hoppings. Speciﬁcally, we consider a non-Hermitian

interacting GAA model that has a non-reciprocal factor

e±gin the nearest-neighbor hopping terms [Fig. 1(a)]

L−1

j=1

h−Jegc†

jcj+1 +e−gc†

j+1cj+V njnj+1i

j=1

2λcos(2πqj +φ)

1−αcos(2πqj +φ)nj,(1)

where c†

jcreates a fermion on site jin an open chain

with Lsites, nj=c†

jcjis the density operator, Vis the

nearest-neighbor density-density interaction, and g6= 0

controls the strength of non-Hermiticity. The model has

a quasiperiodic potential with strength 2λ, an irrational

wave number q= (√5−1)/2, a randomly chosen global

phase φ, and a dimensionless parameter α=−0.8that

controls the potential shape. In the noninteracting limit,

this model exhibits an exact single-particle mobility edge

at energy E= 2sgn(λ)(|J| − |λ|)/α [62]. At nonzero

interaction, previous numerical studies [17,20] on a

Hermitian GAA model (g= 0) with a moderate potential

strength found MBL and thermal states in the low-

and mid-spectrum regimes, respectively, along with an

intermediate phase between them [Fig. 1(b)]. In the

following, we choose V/J = 1 and λ/J = 0.45 such

that both MBL and thermal states exist in the low-

and mid-spectrum regimes in the Hermitian limit g= 0,

respectively.

Non-Hermiticity induced localization: NHSE.—

To investigate the role of non-Hermiticity in an

interacting quasiperiodic system, we ﬁrst note that the

eigenspectrum of Eq. (1) is independent of gregardless

of total particle number. This is because it can be

mapped from a Hermitian GAA model by the imaginary

gauge transformation [55], cj→egj cjand c†

j→e−gj c†

As a result, the level spacing statistics, a commonly

used diagnostic that indicates the MBL and thermal

phases in Hermitian system [5,16], cannot reﬂect the

occurrence of NHSE or the fate of MBL and thermal

states at g6= 0 [70]. Nevertheless, we expect that the

inﬂuence of NHSE can be identiﬁed from eigenstate-

related quantities since the probability of many-body

product states |niwith particles accumulated near one

of the open ends is enhanced as |ni → e−gPL

j=1 jnj|ni

followed by normalization. We therefore investigate two

eigenstate properties in the following to capture the

impact of NHSE.

The two eigenstate-based quantities we study are

the energy-resolved real-space local density hnjiand

Rényi entropies S2(l, L) = −log[Trρ2

A], where ρA=

TrB|ψihψ|is the reduced density matrix of eigenstate

|ψifor the left subsystem Athat consists of sites

i= 1,2, ..., l ≤Lin an L-site open chain,

obtained by tracing out the right subsystem B. Both

energy-resolved quantities are computed by eigenstates

uniformly sampled throughout the spectrum using exact

diagonalization (ED) for an L= 30 chain with N= 5

particles. Since the eigen-energies remain unchanged

under the non-Hermiticity magnitude g, we can study

how hnjiand S2(l, L)at diﬀerent energies change from

the Hermitian g= 0 to the non-Hermitian g > 0cases.

We ﬁrst show the drastic energy dependence in how

the real-space local density hnjievolves with the non-

Hermiticity g(see Fig. 2(a)). At g= 0 in the

left panel, the low- and mid-spectrum eigenstates show

symmetric hnjiwith respect to the center of 1D chain,

such that any asymmetric local density at nonzero g

reﬂects the inﬂuence of NHSE. As we turn on the non-

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Non-Hermitianskineectsonmany-bodylocalizedandthermalphasesYi-ChengWang,1,2,KuldeepSuthar,1,H.H.Jen,1,3Yi-TingHsu,4,yandJhih-ShihYou5,z1InstituteofAtomicandMolecularSciences,AcademiaSinica,Taipei10617,Taiwan2DepartmentofPhysics,NationalTaiwanUniversity,Taipei10617,Taiwan3PhysicsDivision,NationalCe...

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Non-Hermitian skin eﬀects on many-body localized and thermal phases Yi-Cheng Wang1 2Kuldeep Suthar1H. H. Jen1 3Yi-Ting Hsu4yand Jhih-Shih You5z 1Institute of Atomic and Molecular Sciences Academia Sinica Taipei 10617 Taiwan.pdf

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Non-Hermitian skin eﬀects on many-body localized and thermal phases Yi-Cheng Wang1 2Kuldeep Suthar1H. H. Jen1 3Yi-Ting Hsu4yand Jhih-Shih You5z 1Institute of Atomic and Molecular Sciences Academia Sinica Taipei 10617 Taiwan

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