Numerical study on non-Hermitian quantum dynamics
非エルミート量子系の動力学の数値的研究

byTakahiro Orito
折戸 隆寛

A Doctor Thesis
博士論文

Submitted to
the Graduate School of the Advanced Science and Engineering,
Hiroshima University
on January 26, 2023
in Partial Fulﬁllment of the Requirements
for the Degree of Engineering
Thesis Supervisor: Yositake Takene 高根 美武Professor of the Graduate School of the Advanced Science andEngineering, Hiroshima University

Acknowledgements
I would like to thank my supervisor, Professor Yositake Takane, for his advice
on my studies. He gave opportunities to enable me to deal with the problem andtopic of my interest throughout my three years of study at Hiroshima University.
I am also grateful to Dr. Ken-Ichiro Imura for his patience and sincere support. He has always been available for consulting on personal and scientiﬁc
problems. The discussion with him not only advanced my studies but also comprised the valuable, enjoyable time spent. Moreover, I am grateful to him forreading my thesis and providing constructive comments.
I would like to thank all members of the thesis committee, Prof. Yutaka
Kadoya, Prof. Holger Friedrich Hofmann, Prof. Seiji Higashitani, and Prof.
Munehiro Nishida, for the acceptance of the review of my thesis.
Outside of Hiroshima University, I am grateful to Prof. Ikuo Ichinose of
Nagoya Institute of Technology and Prof. Yoshihito Kuno of Akita University
for various discussions and advice.
Furthermore, I am thankful to my family for their understanding and support.
Finally, I would like to acknowledge the ﬁnancial support received from theJapan Science and Technology Agency (JST) through the Program for Developing and Supporting the Next-Generation of Innovative Researchers at Hiroshima
University (JST SPRING, Grant Number JPMJSP2132).

Contents

1 Introduction

12 (De)localization transition of (non-)Hermitian systems
5
2.1 Quantiﬁcation of the degree of localization and delocalization inquantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Absence of delocalization in one space dimension and case of the
Aubry–André model . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Case of the Hatano–Nelson model: a non-Hermitian extension ofquantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Localization/delocalization phenomena from the perspective ofwave packet dynamics . . . . . . . . . . . . . . . . . . . . . . . . 13
3 ETH, MBL, and entanglement entropy
14
3.1 Eigenstate thermalization hypothesis (ETH) . . . . . . . . . . . . 14
3.2 Many-body localization (MBL) . . . . . . . . . . . . . . . . . . . 17
3.3 Quantum entanglement and entanglement entropy . . . . . . . . . 19
3.3.1 General deﬁnition of the entanglement . . . . . . . . . . . 19
3.3.2 Two types of contributions to the entanglement entropy:
number and conﬁguration entropies . . . . . . . . . . . . . 20
3.4 Static and dynamical properties of the ETH and MBL phases . . 22
3.4.1 Static property of Stot in the ETH and MBL phases . . . . 22
3.4.2 Dynamical property of the ETH and MBL phases: Imbalance and Stot . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.5 Slow logarithmic growth of entanglement entropy in the MBL phase 25
3.6 Non-Hermitain many-body localization . . . . . . . . . . . . . . . 29
4 Non-Hermitian wave packet dynamics
31
4.1 Single-particle case: Hatano-Nelson model and Aubry-André model 31
4.2 Case of an interacting system . . . . . . . . . . . . . . . . . . . . 38
5 Non-Hermitian entanglement dynamics
5.1 Entanglement dynamics . . . . . . . . . . . . . . . . . . . . . . .
5.2 Entanglement entropy in the non-interacting limit: wave packet
spreading versus entanglement entropy . . . . . . . . . . . . . . .
5.3 Case of V ̸= 0 and g ̸= 0: non-monotonic behavior . . . . . . . . .
5.4 Eﬀect of ﬁnite size and nonequilibrium steady state . . . . . . . .
5.5 Proﬁle of the reduced density matrix . . . . . . . . . . . . . . . .

42
42

6 Concluding remarks

53

iii

43
45
47
50

A Notes on the choice of the boundary condition

56

B Free particle wave-packet dynamics in a Hermitian system

58

C Free particle wave-packet dynamics in a non-Hermitian systemwith Im ϵk ̸= 0
60
D Eﬀects of anticommutation relation on entanglement dynamics 62
E The entanglement entropy in the ETH and MBL phases (Hermitian case)64
References

65iv

Chapter 1
Introduction
Does a quantum mechanical system thermalize? The answer is certainly yes;
research is underway to establish that an isolated many-body quantum system is thermalized by its dynamics. Here, thermalization refers to the process
via which, through time evolution, the quantum expectation value of an observable tends to approach and eventually becomes practically identical to thecorresponding microcanonical expectation value. The eigenstate thermalization
hypothesis (ETH) [1, 2, 3] proposes suﬃcient conditions for such thermalization. Subsequent theoretical, numerical, and experimental studies have shown
the validity of the hypothesis under various conditions.
On contrary, many-body localization (MBL) [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]phase is rather an exceptional case wherein this ETH breaks down. In thepresence of strong disorder, the wave function is known to localize at the level ofsingle-particle quantum mechanics (Anderson localization) [14]. In the presenceof inter-particle interaction, localization still occurs; however, it occurs in theHilbert/Fock space; as in case of MBL. In time evolution the initial density (orspin) pattern remains essentially unchanged, while the expectation value of anobservable does not reach its microcanonical ensemble averaged value. This lackof thermalization stems from the emergence of an extensive number of locallyconserved quantities, each of which is called a local integral of motion (LIOM).
Since a conserved quantity commutes with the Hamiltonian, LIOM acts as aconstraint to dynamics and prohibits the quantum state from thermalization.
Hereafter, we state that a system is in the ETH phase if ETH holds therein.
The entanglement entropy S is a measure of non-local correlation (entanglement) between subsystems; here, it is presumed that the system is divided into
two parts and only one subsystem is focused on (for a more precise deﬁnitionof the entanglement, see Chapter 3). If the entire system is in a pure state, thesubsystems attain a mixed state if S ̸= 0, implying the emergence of a certain ensemble. The entanglement entropy S measures the entropy of such an ensemble,
and the expectation value of an observable in the subsystem is equal to its statistical average over this ensemble. The behaviors of S is qualitatively diﬀerent in
the MBL and ETH phases in terms of both their static [15, 16, 17, 18, 19] and dynamical [20, 21, 22, 23, 24, 25] properties. Therefore, the entanglement entropy
S can be used for characterizing the MBL and ETH phases. For an eigenstateobeying ETH, the ensemble describing a subsystem is akin to a thermal (statistical) ensemble, and thus the expectation value of an observable is regarded
as a thermal expectation value. This implies that S of the eigenstate obeys a1

volume-law scaling since thermodynamic entropy is extensive. In contrast, inthe MBL phase, an eigenstate only possesses a correlation within a localization
length, and thus S of the eigenstate obeys an area-law scaling [15, 16, 17, 18, 19].
Therefore, the ETH-MBL transition involves the transition of entanglement fromvolume-law to area-law scaling.
Particle transport generally involves the growth of entanglement entropy withtime evolution of an initial state expressed in the form of a wave packet. Such
particle transport is hindered by the random potential, and thus, a power-law
growth of S [25] in the ETH phase gradually decreases as the disorder strength
increases. However, even in the MBL phase, a logarithmic growth of S with time
still exists. This relaxation dynamics, i.e., the logarithmic growth of S [20, 21,22], is unique to the MBL phase. For the Anderson localization (AL) phase in anoninteracting system, in an early time scale, S exhibits similar behavior to the
MBL phase, although S ceases to increase after its initial growth. An extensive
set of LIOMs provides an intuitive explanation for this behavior: even if theinteraction is exponentially small, the time evolution of a wave packet becomes
more complicated than that in the AL phase owing to the mutual interaction of
LIOMs [see Eq. (3.17)].
Another motivation of this work lies in recent renewed interests [26, 27, 28, 29,
30, 31, 32] in the previously proposed (late 1990’s) non-Hermitian extension ofquantum mechanics [33, 34, 35]. Remarkable advances in experiments in the last
decades have changed and is still changing the shape of theoretical research. 1
One of the advantages of such experiments is their high controllability. This
realizes a bonaﬁde example of the so-called open quantum system, that is essentially a quantum system coupled to the measuring equipment or surrounding
environment. Such a system is eﬀectively described by a non-Hermitian Hamiltonian, whose characteristics are essentially diﬀerent from those of the original
closed systems, excluding the environment. In the eﬀective tight-binding modelin Refs. [33, 34], lies in the non-reciprocity of hopping amplitudes; 2 the amplitude with which an electron hops to the right neighboring site is diﬀerent fromthe one with which it hops to the left neighboring site.
In this thesis, we consider the case of the many-body Hatano–Nelson model,
wherein nearest-neighbor short-range interactions are incorporated into the original tight-binding model; see Eq. (2.12). In this non-Hermitian many-body system, we have examined how/whether existing scenarios on Hermitian ETH-MBL
systems function, particularly focusing on the behavior of entanglement entropy.
Few existing studies regarding this system include Refs. [36, 37], typical entanglement dynamics are stated to be qualitatively similar to the Hermitian case (atleast in an early time scale). In this thesis, I show that this statement is validated
based on observations only in the extreme cases of exceedingly weak and strong
disorder, without considering the typical behavior of the entanglement entropyS(t) observed in the moderate disorder case. In the moderate disorder case, S(t)typically turns to decrease after an initial growth, exhibiting a remarkable non1

The theoretical research on thermalization (ETH) and MBL have also been much pushedforward by such remarkable advances in the experiments, which has enabled us (experimetalists) to investigate thermalization and MBL in an isolated many-body quantum system in realexperimental setups.
2
Another typical example of the non-Hermitian eﬀective Hamiltonian is the case of complex
potential, which has been much studied from the viewpoint of PT symmetry [35].

2monotonic time dependence. An interesting feature of the Hatano–Nelson model,i.e., the localization length ξ in the Hermitian limit is deﬁnitely related to thecritical value gc of a non-reciprocal parameter at the localization–delocalizationtransition point in the non-Hermitian regime (see Chapter 2 for details). This
relation has not been examined in the presence of many-body eﬀects.
Speciﬁcally, the following unsolved issues have been clariﬁed in this thesis:i the extent to which non-reciprocal hopping qualitatively changes the entanglement behavior, andii whether the relation between ξ and gc still holds in a many-body system.
Note the result reported in Ref. [38]: the saturation value of S(t) in a manybody system with non-reciprocity increases as the disorder strength increases
under a speciﬁc parameter setting. This indicates that non-reciprocity aﬀects
the entanglement behavior and is related to issue (i). However, the authorsof Ref. [38] did not investigate this behavior; instead, they focused on anotherfeature: non-monotonic time evolution of S(t). In this study, the increase in
the saturation value of S(t) is found to disappear if the system approaches the
localization transition point, which is determined via a separate analysis [38].
This suggests that one can detect the transition between the MBL and ETHphases by observing the time evolution of S(t), paving a way for tackling issue(ii) from the perspective of dynamics.
In the Hermitian case, the spreading of a wave packet (quasiparticle) is closely
related to the growth of S(t) over time [39, 40, 41]. Assume that a pair of entangled quasiparticles is given as an initial state in the system comprising of two
subsystems. S(t) grows if the two quasiparticles propagate in opposite directions
in diﬀerent subsystems. Consequently, in the clean limit, the entanglement entropy increases as S(t) ∝ 2vf t, where vf is the velocity of the quasiparticles.
For a disordered system, a wave packet ceases to spread, and thus the growth ofentanglement entropy is suppressed. The author of Ref. [42] has reported thata wave-packet spreading in non-Hermitian systems with nonreciprocal hopping
is substantially diﬀerent from that in Hermitian systems. In the presence of
non-reciprocity, a wave packet does not spread; rather it slides in the direction
determined by the non-reciprocity. This behavior is immune to disorder, andthe wave packet spreads through the entire system only in the vicinity of thetransition point. Therefore, the non-reciprocal hopping should strongly aﬀect
the entanglement entropy.
Here, we systematically study the entanglement dynamics in non-Hermitiansystems with non-reciprocal hopping. Notably, non-reciprocal hopping qualitatively changes the entanglement dynamics compared with that in the Hermitiansystem. Interestingly, disorder generates entanglement entropy S in the delocalized phase, suppressing S in the localized phase, enabling estimation of thelocalization–delocalization transition from S(t). Additionally, we validate that a
non-reciprocal parameter g relates to the localization length in a non-interacting
many-body system. This relation holds even in an interacting many-body system. We also provide a more detailed reason why entanglement dynamics leads
to non-monotonic behavior over time. We employ number and conﬁguration entropies (Snum and Sconf ) that play diﬀerent roles in entanglement dynamics andconstitute the entanglement entropy Stot as Stot = Snum + Sconf .
3

In addition to the work described in this thesis, the following relevant topics
have also been studied:1. Fock-space localization in many-body systems [43, 44]
2. Many-body localization in a Flat-band system [45, 46]
3. Many-body localization with topological order [47, 48]
4. Information spreading and scrambling in many-body systems [49, 50].
This thesis is structured as follows. In Chapter 2, we review the localization phenomenon in Hermitian and non-Hermitian (Hatano–Nelson model) systems. In Chapter 3, we review the characteristic features of the ETH and MBL
phases in the Hermitian and non-Hermitian systems. We particularly focus
on the properties of entanglement entropy in these phases. In Chapter 4, we
demonstrate how a wave packet spreads in a non-Hermitian disordered systemwith non-reciprocal hopping. Employing single and few particle (three-particle)
cases, we show that the spreading of wave-packet is considerably diﬀerent fromits behavior in the Hermitian system. In Chapter 5, we examine entanglement
dynamics, highlighting its non-monotonic time evolution and a characteristic
disorder-dependence. We also study the size dependence of the results, predicting an unusual area-volume-area law crossover of the maximal entanglement
entropy, which is not observed in the Hermitian case and allows us to estimate
the localization transition even in dynamics. Finally, we examine and visualize the characteristic stages in the evolution of the reduced density matrix,
which closely relates to the entanglement entropy. We emphasize that the diﬀerence in its behavior between Hermitian and non-Hermitian cases arises from theimaginary part of eigenenergy. Chapter 6 incorporates the concluding remarks.
Details are provided in Appendices. In this thesis, Quspin [51, 52] is used tocompute various physical quantities of a quantum many-body system.

4

Chapter 2(De)localization transition of(non-)Hermitian systems
In quantum mechanics, disorder in the form of a random potential tends tolocalize the wave function of a quantum particle [14]. Figure 2.1 shows such a
localized wave function (plotted in orange) and a delocalized wave function (plotted in blue)for comparison, both in real space [panel (a)] and the corresponding
reciprocal space [panel (b)]. Here, W represents the strength of disorder, 1 i.e.,the magnitude of the random potential. Naturally, the case of W = 0 falls within
the clean limit, while the relatively strong disorder case of W = 4 corresponds
to the localized regime. In the case of W = 0, the eigen wave function ψ(j) isa superposition of plane waves eikj and e−ikj , extending uniformly to the entire
system, so that |ψ(j)|2 is almost constant in real space (or in the local basis)[panel (a)], where j speciﬁes a site in real space. Contrastingly, it shows only the
δ-function peaks in the reciprocal space [panel (b)]. This is precisely prescribedby the uncertainty principle of quantum mechanics. Similarity, in case of W = 4,
the eigen wave function ψ(j) becomes a localized wave function, expressed in theform of a localized function in the local basis [orange plots in panel (a)], while
it is almost uniformly distributed (up to a weak modulation associated with thequasi periodicity) in the reciprocal space.
(a)

(b)

Figure 2.1: A (de)localized eigenstate in the (a) local basis (|j⟩) and (b) reciprocal space (|k⟩) with a speciﬁc disorder conﬁguration.
In this chapter, we ﬁrst distinguish such localized and delocalized wave functions in a quantitative manner, i.e., the approach to quantify the degree of localization in quantum mechanics (Section 2.1). We introduce a common knowledge
1Here, the strength W of disorder is in the sense of Eq. (2.7).

5

of the ﬁeld: in the model case of one spatial dimension (speciﬁcally focused on inthis study), as soon as moderate randomness or disorder (in the form of a very
weak random potential) is introduced into a system, the wave function is immediately localized, so that delocalization never occurs in one spatial dimension,
except in the clean limit 2 of W = 0. We argue the existence of certain (known)
exceptions to this common knowledge, even in one spatial dimension. Furthermore, we also argue that the indices describing the localization–delocalizationtransition introduced in Section 2.1 are indeed useful in the description of thesesystems. The case of correlated disorder [Aubry–André (AA) model] [53] (Section 2.2) and that of non-reciprocal hopping [Hatano–Nelson model] [33, 34, 54](Section. 2.3) include such exceptional cases. Finally, we introduce a perspectiveon the localization/delocalization phenomena from quantum dynamics, thus far
described as properties of the eigen wave function.

2.1

Quantiﬁcation of the degree of localization and delocalization in quantum mechanics

The probability amplitude of a localized eigenstate features an imbalanced distribution, whereas a delocalized eigenstate exhibits a uniform distribution. Let
us introduce a normalized wave function ψj , where j speciﬁes a site in the system. One of the eﬀective measures to quantify the degree of localization anddelocalization is the inverse participation ratio (IPR):
X
(2.1)
IPR =
|ψj |4jor multifractal dimension (MFD):
MFD = −ln(IPR)
.
ln(L)

(2.2)

In the delocalized phase, the probability amplitude |ψj |2 is uniformly distributed
over the entire system, and thus, the probability amplitude is |ψj |2 ∼ L1 . Substituting L12 for |ψj |4 in Eq. (2.1), we obtain
IPR =

X

|ψj |4 ∼

j

X 1
j

L2

∼

1
,
L

(2.3)

which results in
MFD = −ln(IPR)ln(1/L)
∼−
= 1.
ln(L)
ln(L)

(2.4)

While in the localized phase, the probability amplitude exponentially localizes
in a domain of width ξ centered at a point called the localization center, where ξis referred to as the localization length. If the probability amplitude is uniformly

2

In this sense, the clean limit is an exceptional and unstable limit, at least in the Hermitiancase with uncorrelated disorder [cf. Eq. (2.7)].

6

distributed in the domain, the typical probability amplitude becomes |ψj |2 ∼ 1ξ .
Substituting ξ12 for |ψj |4 in Eq. (2.1), we obtain
IPR =

X

|ψj |4 ∼X

11|ψj |4 = ξ 2 = ,ξξj|j−j0 |∈ξ/2

MFD = −ln(IPR)
ln(ξ)
∼
.
ln(L)
ln(L)

(2.5)

which results in

(a)

(2.6)

(b)

Figure 2.2: IPR and MFD in the AA model as a function of W with ϕ = 0. In
calculating (a) IPR and (b) MFD, a speciﬁc disorder conﬁguration is used foreach value of L.
Figure 2.2 shows the IPR (panel (a)) and MFD (panel (b)) as a function of W
with various values of L. At weak disorder (W ∼ 0), IPR shows size-dependent
behavior, which is consistent with Eq. (2.3). As W increases, IPR gradually
increases in a weakly disordered regime, suddenly increases near W = 2J (localization transition point), and it exhibits size-independent behavior (∼ 1/ξ),
which is consistent with Eq. (2.5). In contrast to IPR, under weak disorder(W ∼ 0), MFD exhibits size-independent behavior (cf. Eq. (2.4)). As W increases, MFD gradually decreases in a weakly disordered regime, suddenly decreases near W = 2J, and it wxhibits size-dependent behavior (∼ ln(ξ)/ ln(L))(cf. Eq. (2.6)). Remarkably, MFD’s with diﬀerent system sizes cross at W = 2J,
represented by the black dashed line. This implies that MFD is an eﬀective good
indicator, enabling the estimation of the localization–delocalization transition.

2.2

Absence of delocalization in one space dimension andcase of the Aubry–André model

We have discussed in the previous section that disorder tends to localize thewave function in quantum mechanics. This is known as Anderson localization,and is quantiﬁed [14, 55] through analyses of the so-called Anderson model:
XH=−J(|j⟩⟨j + 1| + |j + 1⟩⟨j|) + Wj |j⟩⟨j| ,(2.7)
jwhere J represents the strength of hopping and Wj denotes the random on-site
potential disorder at site j. As mentioned at the beginning of this Chapter,7

the localization–delocalization transition dis commonly known to not occur in
one spatial dimension since the wave function tends to be localized if negligible
amount of disorder is introduced into the system. In higher dimensions, even in
the presence of such random potential, a particle (usually an electron) can still
avoid it and move almost freely, since there is suﬃcient room/space or allowed
paths to evade the randomness. The corresponding wave function in this caseremains delocalized/extended unless the random potential is exceedingly strong.
This common knowledge is well-established through analyses of the Andersonmodel (2.7) in the case of the random potential Wj distributed randomly but
uniformly in the range Wj ∈ [−W, W ], where the (half) width W of this uniform
distribution represents the strength of disorder.
The uniform distribution, which is the most standard form of random potential in the study of Anderson localization, is referred to as uncorrelated disorder,
in contrast to the case of correlated disorder, which we will discuss next. The
case of quasi-periodic disorder or the Aubry–André (AA) model falls within thissecond category:
Wj = W cos(2παj + ϕ),(2.8)

where α is an irrational number, considered typically to be the following inverse
golden ratio:
√
5−1
α=
.
(2.9)
2
An unimportant constant ϕ in Eq. (2.8) is set equal to 0 in this chapter. Sincean irrational number α may be suitably approximated by a fraction M/N withboth M and N being a large integer: M, N ≫ 1, so that αj mod 1 assumes analmost random number between 0 and 1 for integer j’s. Correspondingly, 2παj
represents an almost random phase, and the periodic potential (2.8) exhibits an
extremely large period. We can regard it as a quasi-periodic potential representing correlated disorder. The Anderson Hamiltonian (2.7) with the quasi-periodic
potential (2.8) is referred to as the AA model. Previous studies [56, 57, 58] showthat in this model, the delocalized phase continues to exist even in one spacedimension and at a ﬁnite disorder strength, indicating that the eﬀect of correlated disorder is weak compared with that of uncorrelated disorder. This implies
that by adopting the quasi-periodic potential (2.8), we can compare diﬀerences
between the delocalized and localized phases in one spatial dimension under various conditions, such as with and without interaction and non-reciprocity. For
this reason, we employ the AA model in the remainder of this thesis.
In the AA model, the existence of a delocalized phase and that of a localized
phase are related to duality of the model. In the following, we see that thisduality of the AA model allows us to identify the critical disorder strength Wc
at the delocalization–localization transition. We introduce the following unitarytransformation with integer k̃:
1 X
exp(2πiαk̃j/L)|k̃⟩,|j⟩ = √
L

(2.10)k̃

which is diﬀerent from the Fourier transformation (|j⟩ = √1LPk̃ exp(2πik̃j/L)|k̃⟩).

Substituting Eq. (2.10) into Eq. (2.7), we obtain the Hamiltonian in k̃-space (|k̃⟩)8

as follows:


X W
2J
HAA =
− J(|k̃⟩⟨k̃ + 1| + |k̃ + 1⟩⟨k̃|) +W cos(2παk̃)|k̃⟩⟨k̃| . (2.11)
2JWk̃Comparing Eq. (2.7) to (2.11), both Hamiltonians exhibit the same structure,Wand the unitary transformation changes the hopping amplitude from J to 2J
J
2J
and the disorder strength from W to W W . Since both Hamiltonians include thesame delocalization-localization transition point, the hopping amplitude and disorder strength in Eq. (2.7) must become identical to those in Eq. (2.11) at thetransition point, i.e., J = (W/2J)J and W = (2J/W )W . Let Wc be the critical
disorder strength of the Hamiltonian in Eq. (2.7) for a given J. Accordingly
Wc = 2J. The transition occurs in the local basis (|j⟩) and k̃-space (|k̃⟩) simultaneously.
In the previous section we have seen that IPR and MFD are suitable measures
for quantifying the extent to which a wave function is localized or delocalized.
In Fig. 2.2 we have also visualized localization–delocalization transition in adisordered system. The results illustrated in Fig. 2.1 and Fig. 2.2 are based on
the simulation performed for the AA model. As noted in Chapter 2, we cannot
observe such localization–delocalization transition in one spatial dimension if theAnderson model with uncorrelated disorder is employed.

2.3

(a)

Case of the Hatano–Nelson model: a non-Hermitian
extension of quantum mechanics(b)

Figure 2.3: Role of the similarity transformation U given in Eq. (2.13), whichtransforms an eigen wave function Ψ(j) to Ψg (j) = U Ψ(j). Panel (a) schematically shows how |Ψg (j)| varies with increasing g. Panel (b) shows numericalresults of |Ψg (j)| for several values of g with a speciﬁc disorder conﬁguration for
W = 10.
Non-Hermitian quantum mechanics has attracted considerable attention invarious ﬁelds. The Hatano–Nelson (HN) model [33, 34, 54] is one example.
Although the HN model was originally introduced to describe the phenomenonof ﬂux-line depinning of the type-II superconductors, this model also entails aninteresting implication for a disordered quantum system. In this section, wesummarize the basic properties of the HN model and highlight the emergence of9delocalization transition in a disordered quantum system as an exception to theAnderson localization. The Hatano–Nelson model is expressed as follows:
Hns (g) =

L−1
X


−g



−J(e |j⟩⟨j + 1| + e |j + 1⟩⟨j|) +
g

j=1

L
X

Wj |j⟩⟨j|,(2.12)

j=1

where the ﬁrst two terms are hopping terms with g being a parameter quantifying
the degree of non-reciprocity, and the third term is the onsite potential. If
g ̸= 0, the Hamiltonian is non-Hermitian H ̸= H † , and thus, the eigenenergy
can become a complex number.
To understand the role of g, the following similarity transformation may beintroduced:
U |j⟩ = e−gj |j⟩,⟨j|U −1 = egj ⟨j|.

(2.13)

Equation (2.13) represents the gauge transformation if g is iA (A ∈ R). Thus, thesimilarity transformation is referred to as the imaginary gauge transformation.
Using this transformation, we can reduce the non-Hermitian Hamiltonian Hns (g)to the Hermitian Hamiltonian Hns (0) = U −1 Hns (g)U . This means that the
eigenenergies of Hns (g) are real numbers [59]. Additionally, the nth left andright eigenvectors of Hns (g) are expressed as|ΨRn ⟩ = U |Ψn ⟩,(2.14)⟨ΨLn | = ⟨Ψn |U −1 ,(2.15)

andrespectively, where |Ψn ⟩ is the nth eigenvector of Hns (0). These results do not
apply under the periodic boundary condition (PBC).
Under PBC, Eq. (2.12) transforms Hns (g) asU−1

Hns (g)U =

L−1
X




−J(|j⟩⟨j + 1| + |j + 1⟩⟨j|) +

j=1

L
X

Wj |j⟩⟨j|j−J(eLg |L⟩⟨1| + e−Lg |1⟩⟨L|).

(2.16)The non-Hermiticity of the HN model is described by only the last two hopping
terms of Eq. (2.16). In the delocalized phase, these terms dominate the Hamiltonian and generate a complex eigenenergy. However, these terms become unimportant in the localized phase. To verify this, we consider
of nonP a 1perturbation|j−L/2|0√reciprocal hopping to the localized eigenstate |Ψ ⟩ ∼ j ξ exp(− ξ )|j⟩, 3
where ξ is a localization length of Hns (0). The correction ∆E to an eigenenrgy
due to the perturbation is expressed as
∆E =
∼
=
=
3

⟨Ψ0 | − J((eLg − 1)|L⟩⟨1| + (e−Lg − 1)|1⟩⟨L|)|Ψ0 ⟩−JeLg ⟨Ψ0 |L⟩⟨1|Ψ0 ⟩
J
L/2
L/2
− eLg exp(−) exp(−
)
ξ
ξ
ξ
J
−1
− eL(g−ξ ) .
ξWe assume that the localization center of |Ψ0 ⟩ is located at L/2.

10

(2.17)

When g < ξ −1 , ∆E is exponentially small as per Eq. (2.17) and hence, theeigenenergy remains a real number. Contrastingly, when g > ξ −1 , Eq. (2.17)
shows that ∆E is exponentially large. Note that Eq. (2.17) captures only themagnitude of ∆E. Since the perturbation is non-Hermitian, ∆E should containan imaginary part. Consequently, the eigenenergy becomes a complex number.
These results indicate that the critical value gc is expressed as follows:
gc = ξ −1 .

(2.18)

Equation (2.18) suggests that the non-Hermiticity enables us to evaluate the localization length. Interestingly, the normalization condition of a localized eigenstate also presents the same condition of the critical value gc . Under the similarity transformation of Eq. (2.13), a localized eigen wave function 4 is transformedtoΨg (j) ≡ U Ψ(x) ∼ exp(−|x|− gx),
ξ(2.19)

Figure 2.3(a) schematically shows the variation of Ψg (j) with increasing g. Untilg exceeds gc , Ψg (j) maintains a tendency to localize. The critical value gc canbe determined by the normalization condition of the eigenstate. If ξ −1 > g, thenormalization constant is expressed as follows:
Z ∞Z ∞−|x| − ξgx
2
2
2
)
1=C
|Ψg (x)| dx ∼ C
dx exp(2
ξ
−∞
−∞
Z ∞

Z 0
−|x| − ξgx
2
= C
dx +
dx exp(2
)
ξ
0
−∞
ξ
ξ
= C̃ 2 (+)
2(1 + ξg) 2(1 − ξg)ξ,= C̃ 2
1 − ξ 2g2
leading topC̃ =1 − ξ 2g2√.
ξ

(2.20)

If ξ −1 < g, the integration diverges, i.e., Ψg (j) is not square integrable, indicatingthat the eigenstate is delocalized. Therefore, critical gc is expressed as
gc = ξ −1 .

(2.21)

Using Eq. (2.12), we numerically examine the above results. Figure 2.3(b)shows |Ψg (j)| as a function of j for various values of g with a speciﬁc disorder
conﬁguration (W = 10). As g increases, the eigenstate changes according to
Eq. (2.19), and the eigenstate extends the entire system beyond gc . In theactual computation, we obtain eigenstates at g = 0 and select one eigenstate|Ψg=0 ⟩ at random. Next, we calculate eigenstate |Ψg ⟩ for several values of g.
The resulting |Ψg ⟩’s are plotted in Fig. 2.3(b).
4

We assume that the localization center of Ψ(x) is located at 0.

11

IPR

Figure 2.4: Typical behavior of IPR and eigenenergies for several values of gwith L = 100.
We characterize the delocalization transition due to g by employing IPR
and complex spectrum. Figure 2.4 shows complex eigenenergies and IPRs in
a complex plane as a function of g. Here, the color map represents values of
IPR. When g is small, eigenenergies do not exhibit an imaginary part, andIPR ∼ 1. As g increases, IPRs gradually decrease reﬂecting the fact that aneigenstate becomes more extended than that of the g = 0 case. Eigenenergies
remain real numbers as long as g < gc . Beyond gc , eigenenergies become complex
numbers, and IPRs take small values as IPR ∼ L1 . These results suggest that the
localization–delocalization transition accompanies the real–complex transition.

12

Figure 2.5: Wave packet spreading in the HN model with several values of disorder strength V /J with JR = 0 and J = JL = 1. The behavior of wave packet
spreading is qualitatively similar to that reported in Ref. [42].

2.4Localization/delocalization phenomena from the perspective of wave packet dynamics

Dynamical behavior of the HN model has been studied in Ref. [42]: the author
employs a modiﬁed version of the HN model expresssed asX(2.22)H=[−JR |j + 1⟩⟨j| − JL |j⟩⟨j + 1| + V cos(2παj)|j⟩⟨j|] .
jEigenstates of H are delocalized as long as V /J < 1. Time evolution of a wave
packet is shown in Fig. 2.5 for several values of V /J. This ﬁgure shows thatthe wave packet moves in the positive direction (unidirectional motion) when
V /J < 1 and stops moving when 1 < V /J. These features indicate that thediﬀusion constant β, deﬁned as
P 2
2
j j |ψj |2
P∝ tβ ,(2.23)⟨x ⟩ =
2
j |ψj |
takes a positive value when V /J < 1 and vanishes when 1 < V /J. The authorof Ref. [42] has mainly focused on the relationship between β and the complex
spectrum of the system. Cascade-like spreading of the wave packet near thetransition point (V /J = 1) is also reported in Ref. [42].
At this stage, an intuitive explanation of unidirectional motion has not been
provided. Why does the wave packet show a unidirectional motion against disorder? We will provide an intuitive explanation of the above features in Chapter 4.

13

Chapter 3
ETH, MBL, and entanglement entropy
In this chapter, we explain ETH, MBL, and its extension to the non-Hermitiancase. First, we discuss the thermalization of an isolated many body system andits relationship to ETH. We subsequently introduce a phenomenological viewof MBL. We show diﬀerences in static and dynamical features of entanglement
entropy between ETH and MBL phases. We also discuss diﬀerences between AL
and MBL phases and discuss why the MBL phase exhibits relaxation behavior inits dynamics. We comment on open problems of the ETH–MBL transition andelucidate why estimating its transition point is diﬃcult. Finally, we summarize
the previous study of the many-body HN model and highlight potential problem
related to its dynamics.

3.1

Eigenstate thermalization hypothesis (ETH)

In an isolated quantum many-body system, its energy is conserved in the courseof time evolution. At equilibrium, the system with certai energy E is describedby a microcanonical ensemble:
ρ̂mic (E) =

X

1
ZE,δE

|EN ⟩⟨EN |,(3.1)|EN −E|<δE

where |EN ⟩ is an eigenstate in the microcanonical energy shell |EN − E| < δE,
and ZE,δE is the number of such eigenstates. Using ρ̂mic (E), the expectation
value of an observable Ô in the microcanonical ensemble is expressed as⟨Ô⟩mic = Tr(ρ̂mic (E)Ô).

(3.2)

The expectation value of Ô evolves as⟨Ô(t)⟩ = ⟨Ψ(t)|Ô|Ψ(t)⟩X=eiEN t ϕ∗N ⟨EN |Ô|EN ′ ⟩e−iEN ′ t ϕN ′
N,N ′

=

X

ei(EN −EN ′ )t ϕ∗N ϕN ′ ÔN,N ′ ,

(3.3)

N,N ′

where ϕN = ⟨EN |Ψ(0)⟩ and ÔN,N ′ = ⟨EN |Ô|EN ′ ⟩. We expect that ⟨Ô(t)⟩ approaches the microcanonical ensemble averaged value expressed in Eq. (3.2).
As per Eq. (3.3), ⟨Ô(t)⟩ oscillates reﬂecting the phase factor arising from the14

Figure 3.1: Time evolution of ⟨Ô(t)⟩ (a schematic picture); the broken redline
represents the time average of ⟨Ô(t)⟩, and ∆O2 represents the variance of ⟨Ô(t)⟩.
time evolution operator. Figure 3.1 shows the behavior of ⟨Ô(t)⟩ in a schematicmanner.
Considering the thermalization of a given state, we are interested in thestatistical behavior of ⟨Ô⟩. Thus, we focus on the time average of ⟨Ô(t)⟩ deﬁned
asZ
1 T⟨Ψ(t)|Ô|Ψ(t)⟩dt.
⟨Ô⟩ = lim(3.4)
T →∞ T 0
Generally, one deﬁnes the thermalization in such a manner that ⟨Ô⟩ becomesequal to ⟨Ô⟩mic . This results in ⟨Ô(t)⟩ ∼ ⟨Ô⟩mic because a quantum state shouldreach the equilibrium state if it is thermalized. To satisfy the above condition,
the variance of ⟨Ô(t)⟩ deﬁned as∆Ô2 = [⟨Ô(t)⟩ − ⟨Ô⟩]2(3.5)

must be small.
In considering the average over t, we assume that EN and EN ′ are not degenerate
EN ̸= EN ′(3.6)

and that arbitrary two pairs of eigenstates (N and N ′ , and N ′′ and N ′′′ ) satisfy
EN − EN ′ ̸= EN ′′ − EN ′′′ .

15

(3.7)

Accordingly we can rewrite Eq. (3.4) as follows:Z
1 T⟨Ô⟩ = lim⟨Ψ(t)|Ô|Ψ(t)⟩dt
T →∞ T 0Z
1 T X i(EN −EN ′ )t ∗= lime
ϕN ϕN ′ ÔN,N ′T →∞ T 0
N,N ′X=|ϕN |2 ÔN,N

(3.8)

N

and Eq. (3.5) as∆Ô2 = [⟨Ô(t)⟩ − ⟨Ô(t)⟩]2= ⟨Ô⟩2 − [⟨Ô⟩]2
Z
X
1 T
= lim
[ei[(EN −EN ′ )+(EN ′′ −EN ′′′ )]t
T →∞ T 0
N,N ′ ,N ′′ ,N ′′′×(ϕ∗N ϕN ′ ϕ∗N ′′ ϕN ′′′ ÔN,N ′ ÔN ′′ ,N ′′′ )] − [⟨Ô⟩]2
X X=
ϕ∗N ϕN ′ ϕ∗N ′′ ϕN ′′′N ̸=N ′ N ′′ ̸=N ′′′∗×ei[(EN −EN ′ )−(EN ′′′ −EN ′′ )]t ÔN,N ′ ÔN′′′ ,N ′′X X=
ϕ∗N ϕN ′ ϕ∗N ′′ ϕN ′′′N ̸=N ′ N ′′ ̸=N ′′′∗×δN,N ′′′ δN ′ ,N ′′ ÔN,N ′ ÔN′′′ ,N ′′X|ϕN |2 |ϕN ′ |2 |ÔN,N ′ |2 .
=

(3.9)

N ̸=N ′

One of the promising hypotheses for realization of thermalization is the eigenstate thermalization hypothesis (ETH) that comprises two parts: oﬀ-diagonal
ETH and diagonal ETH. The oﬀ-diagonal ETH assumes that ON,N ′ = ⟨N |Ô|N ′ ⟩with N ̸= N ′ in the microcanonical energy shell satisfying
lim ON,N ′ → 0.

(3.10)

L→∞

Under the oﬀ-diagonal ETH, ∆Ô2 vanishes in the limit of L → ∞. This can be
expressed as follows:
X∆Ô2 =|ϕN |2 |ϕN ′ |2 |ÔN,N ′ |2
N ̸=N ′≤X

|ϕN |2 |ϕN ′ |2 maxN ̸=N ′ {|ÔN,N ′ |}

2

N,N ′

≤

maxN ̸=N ′ {|ÔN,N ′ |}

2

→ 0.

(3.11)The diagonal ETH assumes that ON,N = ⟨N |Ô|N ⟩ in the microcanonical energy
shell satisfying
lim ON,N → ⟨Ô⟩mic (E).

L→∞

16

(3.12)

Under the diagonal ETH, ⟨Ô⟩ ∼ ⟨Ô⟩mic is expressed as
X⟨Ô⟩ =|ϕN |2 ÔN,NN≃

X|ϕN |2 ⟨Ô⟩mic (E)N= ⟨Ô⟩mic (E).

(3.13)

Next, we ﬁnally comment on the physical representation of thermalization ina isolated quantum system without a heat bath. A possible interpretation is theoccurrence of thermalization as a consequence of entanglement. Let us assume
that the system can be divided into two subsystems A and B, and we consider
the expectation value of an operator ÔA that belongs to A. The expectation
value is expressed as ⟨ÔA × IB ⟩, where IB is the identity operator in B. The
expectation value is expressed as follows:⟨Ψ(t)|ÔA × IB |Ψ(t)⟩ = Tr(ÔA × IB ρ)= TrA TrB (ÔA × IB ρ)= TrA (ÔA ρA ),(3.14)

where ρA in the last line is the reduced density matrix of the subsystem A
deﬁned as ρA = TrB {ρ}. If |Ψ(t)⟩ is entangled, the quantum state describedby ρA becomes a mixed state. One can interpret ⟨ÔA × IB ⟩ as the expectation
value of ÔA with a statistical distribution ρA . This implies that the subsystem
B plays the role of a bath for subsystem A [6], leading to the thermalization of
A. Conversely, the subsystem A leads to the thermalization of subsystem B.

3.2

Many-body localization (MBL)

Under the ETH, the expectation value of an observable O in an eigenstate of
H depends on only its eigenenergy E without large ﬂuctuation. The ﬂuctuation
vanishes in the thermodynamic limit. However, in a strongly disordered system,
such a quantity ﬂuctuates and does not depend on E [60]. We examine such
behavior, employing the Hamiltonian expressed as
H=

X Jxy
−
+
z
(Sj+ Sj+1+ Sj+1Sj− ) + Jzz Sjz Sj+1[
+ Wj Sjz ],2
j(3.15)

where the ﬁrst two terms represent exchange interaction, the third term represents Ising interaction, and the last term represents an onsite random magnetic
ﬁeld term with uniform random distribution Wj ∈ [−W, W ]. In considering aquantum system, attention is typically focused on the properties of the groundstate. Notably, we focus on highly excited states in the studies on ETH–MBL
transition. We introduce normalized eigenenergy:ϵ=E − Min(E),
Max(E) − Min(E)(3.16)

where Max(E) and Min(E) are the maximum and minimum eigenenergies, rezsoectively. Figure 3.2 shows ⟨SL/2⟩ as a function of ϵ at weak (W = 0.4) and17

zstrong (W = 8.0) disorder strength. In the weak disorder case, ⟨SL/2⟩ depends
on ϵ, and its ﬂuctuation is suﬃciently small in the middle of the spectrum. Thesetendencies become more apparent as the system size increases [60]. In the strongz
disorder case, ⟨SL/2⟩ is independent of ϵ, and its ﬂuctuation is considerably large.
That is, ETH does not hold. This failure of ETH stems from the many-bodyz
Figure 3.2: Expectation value ⟨SL/2⟩ as a function of ϵ, where L = 14, Jxy =1, and Jzz = 1 withPa speciﬁc disorder conﬁguration. We consider the zeromagnetization sector j Sjz = 0.

localization (MBL), which is intrinsic to a quantum many-body system. An
eigenstate of the MBL phase localizes in the Fock space [61, 62, 63, 43] andtypically involves localization in real space [64] as well. Here, the localization inreal space means that the system is separated into several domains: ⟨Sjz ⟩ ∼ 12 incertain domains and ⟨Sjz ⟩ ∼ − 21 in others. This localization behavior is shownin Fig. 3.2 in the strong disorder case of W = 8.0. The authors of Refs. [22, 65]
have suggested that an ideal MBL phase is described by a phenomenological
Hamiltonian that is expressed as
XXX
HM BL = E0 +hj τjz +(3.17)
Jj,k τjz τkz +Jj,k,l τjz τkz τlz + · · · ,
j

j,k

j,k,l

where E0 is a constant value, and τiz is a locally conserved quantity called a local
integral of motion (LIOM). Equation (3.17) has been justiﬁed for a certain classof spin chains [66]. LIOM’s are operator which exponentially localize in Fockspace and are expressed as
X X
X X α,βpaαi1 σxα / ξ +τiz =bi1 ,i2 σiα1 σiβ2 + · · · ,(3.18)
i1 α=x,y,z

ii ,i2 α,β=x,y,z

18

where ξ is the localization length, ik denotes the kth localization center, and−|i−i1 |/ξ
aαi1 and bα,βand e−(|i−i1 |+|i−i2 |)/ξ , respectively [67,i1 ,i2 are proportional to e
68, 69]. Hamiltonian (3.17) incorporates only the terms with τjz ’s and therefore
commutes with τjz :[HM BL , τjz ] = 0.

(3.19)

This means that each τjz is a conserved quantity [22, 65], and the ideal MBL phasecan be regarded as an integrable system [8, 65]. LIOMs act as constraint in thequench dynamics without particle transport under the breaking of ETH [68, 69].
The existence of such operators explains the relaxation behavior of observables
and entanglement entropy [22, 70], which we discuss in Section 3.5.

3.3

Quantum entanglement and entanglement entropy

3.3.1

General deﬁnition of the entanglement

Let us introduce Fock basis |{n}⟩ ≡ |n1 n2 · · · nL−1 nL ⟩, where nj (j = 1, 2, · · · , L)denotes the number of electrons in the jth orbital 1 . One can describe a generalmany-body quantum state |Ψ⟩ as a superposition of basis states corresponding
to diﬀerent particle conﬁgurations with many-body coeﬃcients ψ{n} :X(3.20)|Ψ⟩ =ψ{n} |{n}⟩.
{n}Here, we divide the system of length L into two subsystems A and B of length
L/2, where A consists of the sites with j ∈ jA = {1, 2, · · · , L/2}, while B consistsof the sites with j ∈ jB = {L/2 + 1, L/2 + 2, · · · , L}. By tracing out subsystemB, we calculate the entanglement entropy Stot for subsystem A. To concretize
this procedure, we employ the density matrix Ω,Ω = |Ψ⟩⟨Ψ|X∗′
ψ{n} ψ{n=′ } |{n}⟩⟨{n }|{n},{n′ }=X{nA },{nB },{n′A },{n′B }∗′′ψ{nA },{nB } ψ{n′ },{n′ } |{nA }⟩|{nB }⟩⟨{nA }|⟨{nB }|,A
B(3.21)
where {nA } = n1 , n2 , · · · , nL/2 and {nB } = nL/2+1 , · · · , nL 2 . We perform partial
trace over {nB }:ΩA = TrB Ω
X=⟨{n′′B }|Ω|{n′′B }⟩
{n′′
B}

=

X

{nA },{n′A },{nB }∗′ψ{nA },{nB } ψ{n′ },{n } |{nA }⟩⟨{nA }|.
B
A(3.22)For a spin 1/2 system, ni = 0 and ni = 1 is interpreted as ↓i and ↑i , respectively.
Additionally, {n} corresponds to the conﬁguration of spins in real space.
2Here, ψ{n} ≡ ψ{nA },{nB } and |{n}⟩ ≡ |{nA }⟩|{nB }⟩.
1

19

The entanglement entropy is deﬁned as
Stot = −Tr ΩA log ΩA ,(3.23)

which is written in terms of the eigenvalues λα of the reduced density matrix ΩAasX
Stot = −λα log λα .
(3.24)
α3.3.2

Two types of contributions to the entanglement entropy: number and conﬁguration entropies

Figure 3.3: Schematic derivation of the reduced density matrix from a given
quantum state in the system with four sites at half ﬁlling.
We consider that a quantum state |Ψ⟩ is described by a superposition of theFock bases |{n}⟩ = |{nA }⟩|{nB }⟩. We assume that the total number of particles
is L2 that is half of the total number of sites (i.e., half ﬁlling). The number ofparticles in |{nA }⟩ deﬁned as
X
NA ≡
nj
(3.25)
j∈jA

20

relates to that in |{nB }⟩ deﬁned as

X

NB ≡

nj ,

(3.26)

j∈jB

since they satisfy the half ﬁlling condition: NA + NB = L2 . The vector space
HA spanned by |{nA }⟩’s can be decomposed into subspaces speciﬁed by NA ∈{0, 1, · · · , L2 }. The vector space HB spanned by |{nB }⟩’s can also be decomposed
into subspaces speciﬁed by NB ∈ {0, 1, · · · , L2 }. Under the constraint of NA +NB = L2 , the subspace of HA with NA = j is paired with the subspace of HB
with NB = L2 − j. Indeed, the density matrix includes non-zero components onlyin such paired subspaces, which are mutually isolated in the full vector space.
Reﬂecting this isolated structure, the reduced density matrix is expressed in a
block diagonal form, as can be observed in Fig. 3.3. Here, each block corresponds
to a subspace of HA and the total number of the blocks is L/2 + 1. The reduced
density matrix is expressed as follows:ΩA = ΩN0 ⊕ ΩN1 ⊕ · · · ⊕ ΩN L −1 ⊕ ΩN L ,2(3.27)

2

where ΩNj describes the subspace speciﬁed by NA = j (cf. Fig. 3.3). Using theblock diagonal representation of ΩA , we can rewrite Eq. (3.23) as
Stot = −Tr [ΩA log ΩA ]
= −

L/2
X




Tr ΩNj log ΩNj

j=0
dim(Ω

= −

L/2
X
XNj
j=0

)
j)
j)
λ(N
log λ(N,αα(3.28)

α=1(N )where λα j is the αth eigenvalue of ΩNj , and dim(ΩNj ) is the dimension of ΩNj .
We deﬁne pNj as
X
j)
λ(N
,
p Nj =(3.29)αα(N )and λ̃α j as
1 (Nj )λ .
p Nj α

j)
λ̃(N
=
α

(3.30)

The contributions to the entanglement entropy Stot can be divided into twoparts: the number and conﬁguration entropies (Snum and Sconf ) [71]:dim(ΩStot = −

L/2
X
XNj
j=0

)
j)
j)
λ(N
log λ(Nααα=1L/2 dim(ΩNj )

= −

X X
j=0

= −

L/2
X

j)
j)
pNj λ̃(N(log pNj + log λ̃(N
)
α
α

α=1
dim(Ω

pNj log pNj −

j=0

L/2
X
XNj
j=0

= Snum + Sconf ,)j)j)log λ̃(NpNj λ̃(Nααα=1(3.31)
21

whereSnum = −L/2
X

pNj log pNj ,(3.32)j=0
dim(ΩSconf = −

L/2
X
XNj
j=0

)
j)
j)
.
pNj λ̃(Nlog λ̃(Nαα(3.33)α=1

Let us consider the properties of Snum and Sconf . The number entropy Snum isessentially the Shannon entropy of pNj , which is a probability that one observes
Nj particles in the subsystem A; therefore, Snum quantiﬁes the ﬂuctuation of
the number of particles in the subsystem A. Particle transport generates thistype of entropy; i.e., Snum is controlled by a local process. The conﬁguration
entropy Sconf arises from non-local correlations in the many-body state that are(N )intrinsic to a many-body state. In contrast to pNj , λ̃α j can change without
particle transport, indicating that non-particle transport (i.e., evolution of nonlocal correlations) can generate Sconf in a many-body localized state.

3.4

Static and dynamical properties of the ETH and MBL
phases

3.4.1

Static property of Stot in the ETH and MBL phases

The authors of Ref. [16] have reported on the characteristic scaling transition of
Stot . They have studied Stot in the random ﬁeld spin 1/2 XXZ model, which isthe same as Eq. (3.15), employing a sophisticated exact diagonalization method.
The deﬁnition of Stot is the same as Eq. (3.24) (cf. Sec. 3.3). The behavior of
Stot shown in Fig. 3.4 is qualitatively similar to that reported in Ref. [16]. The
top panel of Fig. 3.4 show Stot /L as a function of L for various values of disorderstrength h. In the weakly disordered regime of h ∼ 1, Stot /L takes a constantvalue for large L, which implies the volume-law scaling of entanglement entropy
Stot ∝ L. This behavior is consistent with the view that the system is in the
ETH phase, where an eigenstate is thermalized and yields the volume-law scalingof entanglement entropy. In the strongly disordered regime of h ∼ 4, Stot /Ldecreases as L increases, which implies the area-law scaling of entanglement
entropy Stot ∼ const. This behavior is consistent with the view that the system
is in the MBL phase, where a localized eigenstate only generates an entanglement
when the localization length exceeds the subsystem size.
They have estimated this volume-to-area law scaling transition point and theexponent of entanglement entropy, employing ﬁnite size scaling. Additionally,
they have attempted to conduct ﬁnite size scaling of the variance of Stot . The
variance of Stot , shown in the bottom panel of Fig. 3.4, can be interpreted as
speciﬁc heat if Stot acts as a thermodynamic entropy. Notably, the actual phasetransition point and exponent remain unaddressed problems; indeed, such atransition has recently been regarded as a Kosterlitz–Thouless-like transition.
One of the possible directions for a more detailed analysis of this transition is
to calculate Stot in a quantum many-body system whose system size is larger
than that of the previous study. Unfavorably, the Hilbert space of a manybody system exponentially increases as a function of system size L, and thus,
22Figure 3.4: Scaling of Stot in the case of ϵ = 0.5: top panel; Stot /L as a functionof L, bottom panel; varianceP z of Stot as a function of h. We focus on the zeromagnetization sector j Sj = 0 as in Ref. [16].
calculating Stot in a large system size is a challenging problem. The tensor
network method is one of the methods that has addressed such an inevitableproblem and thus has attracted considerable attention recently.
3.4.2

Dynamical property of the ETH and MBL phases: Imbalanceand Stot

The authors of Ref. [72] have reported on the characteristic dynamical behavior
in a disordered quantum many-body system. They have studied the random
ﬁeld spin 1/2 XXZ model [Eq. (3.15)], employing a sophisticated tensor network
method. In the study of the time evolution of a many-body quantum system, thetensor-network method enables treating the system size up to L = 100, which isconsiderably larger than L ≤ 28 that can be treated by an exact diagonalization
method.
Imbalance I is often employed to characterize the ETH-MBL transition point.

23

Figure 3.5: Imbalance as a function of time: left, middle, and right panels show
the W = 2, 4, and 8 P
cases with chain length L = 18. We focus on the zeromagnetization sector j Sjz = 0 as in Ref. [72].
I is deﬁned byI=

4X⟨Ψ(0)|Sjz |Ψ(0)⟩⟨Ψ(t)|Sjz |Ψ(t)⟩,L j(3.34)

where |Ψ(0)⟩ is an initial state, and |Ψ(t)⟩ = e−iHt |Ψ(0)⟩. This quantity characterizes the time period for which the quantum state memorizes an initial state
after time evolution. In the delocalized phase, an initially provided spin patternmelts with increasing time, and I becomes 0. While in the localized phase, an
initially provided spin pattern persists, and I becomes ﬁnite. Figure 3.5 shows
the Imbalance I as a function of time obtained by an exact diagonalization
method. The behavior of I is qualitatively similar to that reported in Ref. [72].
They employed the Néel state |Ψ(t = 0)⟩ = | ↑↓↑↓ · · · ⟩ as an initial state. In
the weakly disordered regime (left panel of Fig. 3.5), I reaches 0 as time evolves,
which implies thermalization of the quantum many-body system. As the disorder increases (middle panel of Fig. 3.5), I becomes ﬁnite, and in the strongly
disordered regime (right panel of Fig. 3.5), I does not decrease, implying thatthe system localizes.

Figure 3.6: Stot (t) as a function of time: left, middle, and right panels show
the W = 2, 4, and 8 P
cases with chain length L = 18. We focus on the zeromagnetization sector j Sjz = 0 as in Ref. [72].
Entanglement entropy also characterizes the dynamical behavior of both theETH and MBL phases. Figure 3.6 shows Stot (t) as a function of time obtained24

by using an exact diagonalization method. The deﬁnition of Stot is the sameas Eq. (3.24) (cf. Section 3.3). The behavior of Stot (t) is qualitatively similar to that reported in Ref. [72]. In a weakly disordered regime (left panel ofFig. 3.6), Stot exhibits a power-law growth Stot ∝ tβ . As the disorder strength
W increases (middle panel of Fig. 3.6), an exponent β gradually decreases (i.e.,
Stot decreases). This decrease in β is consistent with the behavior of I, whichcharacterizes a relaxation time of a quantum many-body system. As disorderstrength W increases, the relaxation time duration of such a system increase,leading to potential divergence in the MBL phase. However, in a strongly disordered regime (right panel of Fig. 3.6), Stot still shows a logarithmic growth intime evolution, although I does not decrease, i.e., without particle transport.
This slow logarithmic growth of entanglement entropy stems from dephasing,(discussed in Section 3.5) and characterizes the MBL phase.
We have seen that the entanglement entropy characterizes the dynamical behavior of both ETH and MBL phases. However, this quantity is unsuitable for
estimating the ETH–MBL transition because one cannot distinguish between alogarithmic growth and an extremely slow power-law growth of Stot . Additionally, the saturation value of Stot obeys the volume law in both ETH and MBL
phases, i.e., the non-scaling transition of entanglement entropy. Unfavorably, I
is also unsuitable for estimating the ETH–MBL transition because its saturation
value cannot be calculated. Using the tensor network method, we can study aquantum many-body system for large system sizes, but cannot evaluate a sufﬁcient long-time scale. Therefore, estimating the ETH–MBL transition point
requires a quantity or method which enables estimation of such a transition
point from behavior in a ﬁnite time.

3.5

Slow logarithmic growth of entanglement entropy in
the MBL phase

The static properties of the MBL phase, characterized by IPR, gap ratio, andStot , show qualitatively similar behavior to the AL phase. Here an obvious
question arises: does interaction cause any diﬀerence between the AL and MBL
phases? The answer is Yes. The dynamical properties of the MBL phase isqualitatively diﬀerent from those in the AL phase.
The authors in Refs. [20, 21] have examined the dynamical behavior of Stotin the AL and MBL phases. The top panel of Fig. 3.7 shows Stot as a function oftime in a strongly disordered regime. In contrast to the non-interacting case (ALcase), Stot exhibits a slow logarithmic growth in the case including interaction.
Additionally, the saturation value of Stot shown in the bottom panel of Fig. 3.7
exhibits a volume law in the interacting cases, which is diﬀerent from the AL
case showing area law.
Based on Ref. [22], we consider a two-particle system as a simple example
to examine the eﬀect of interaction on Stot .
We consider the conventional
Hamiltonian with open boundary condition expressed asX(3.35)H=[−J(c†j cj+1 + c†j+1 cj ) + Wj nj + V nj nj+1 ],jwhere c†j (cj ) creates (annihilates) a fermion at site j, while n̂j = c†j cj counts the25

Figure 3.7: Quench dynamics of Stot in the strongly disordered regime. Top panel
represents Stot as a function of a normalized time, where J⊥ is the strength of
exchange interaction and Jz is the strength of Ising interaction. Bottom panel
represents the saturation value of Stot as a function of system size L.

L1

R1

L2

R2

Figure 3.8: Schematic of the initial state [Eq. (3.36)].

26

Figure 3.9: Time evolution of Stot with various disorder strengths with L = 10
and J = 1, and Wj ’s are uniformly distributed in [−6, 6].
number nj of fermions at site j. Even if the interaction is suﬃciently weak, alogarithmic growth of Stot emerges, and thus we consider the weak interaction
V ≪ J and strong disorder W ≫ J limit. In the non-interacting limit of V = 0,
we introduce two eigenstates localized near the left end of the system. Theireigenenergies are expressed as ϵLi (i = 1, 2), where i is used to specify the twostates. We also introduce two eigenstates localized near the right end with ϵRi(i = 1, 2). Let c†Li and c†Ri (i = 1, 2) be the creation operators of the eigenstates
localized near the left and right ends, respectively. Let us consider the time
evolution of an initial state deﬁned as1|Ψ(t = 0)⟩ = (c†L1 + c†L2 )(c†R1 + c†R2 )|0⟩.
(3.36)
2
A schematic of the initial state is shown in Fig. 3.8. The energy of this state isexpressed as
Eα,β = ϵα + ϵβ + δEα,β ,(3.37)

where α ∈ {L1 , L2 }, β ∈ {R1 , R2 }, and δEα,β represents the correction arisingfrom the interaction term with V . If only the ﬁrst order correction with respect
to the interaction is considered,−|xα − xβ |δEα,β ∼ Cα,β V exp,(3.38)
2ξ
where xα and xβ represent the localization centers of the states speciﬁed by α
and β, respectively, ξ is a localization length, and Cα,β is a constant that is
not signiﬁcant in the following argument. Since V ≪ J is assumed, the time
evolution of the state is expressed asX1
exp(−iEα,β t)|α, β⟩,|Ψ(t)⟩ =(3.39)
2 α∈L ,L and β∈R ,R
1

2

1

2

where |α, β⟩ = |α⟩|β⟩ with |α⟩ = c†α |0⟩ and |β⟩ = c†β |0⟩. To deﬁne Stot in thiscase, we introduce the reduced density matrixXρL =⟨β|Ψ(t)⟩⟨Ψ(t)|β⟩,(3.40)
β∈R1 ,R2

27

which is expressed as
1
ρL =
2



1
F (t)/2∗
F (t)/2
1

,(3.41)

where F (t) = e−iωt (1 + e−iδωt ), δω = δEL1 ,R2 − δEL2 ,R2 + δEL1 ,R1 + δEL2 ,R1 , andω = ϵL1 + ϵL2 + δEL1 ,R1 − δEL2 ,R1 . Let λ1 (t) and λ2 (t) be the eigenvalues of ρL ,in terms of which we deﬁne Stot as
X
Stot = −λi (t) ln λi (t).
(3.42)
i=1,2
π
When t ∼ 0, Stot ∼ 0, since the eigenvalues are 0 and 1. When t ∼ T = δω
∼
1 x/ξ
1
e
,
S
∼
ln
2,
since
both
eigenvalues
are
.
tot
2V
2
Using Eq. (3.35), we numerically examine the above results in an initialstate that is compatible with Eq. (3.36). Figure 3.9 shows Stot as a function oftime/V with various strengths of V . Except for the case of V = 0, Stot gradually
increases with time, and takes the maximal value ln 2, which is consistent withthe perturbation analysis discussed above.

Figure 3.10: Time evolution of the density ⟨nj ⟩ as a function of time/V. We
employ the same data used to draw Fig. 3.9.
Note that ⟨nj (t)⟩ exhibits localization tendency in dynamics as shown in
Fig. 3.10, which suggests that a non-particle transport generates Stot . Employing
the same data used to draw Fig. 3.10, we compare Stot , Snum , and Sconf in
Fig. 3.11 and observe that Sconf dominates Snum , i.e., Stot ∼ Sconf .
The perturbation analysis tells us that the growth of Sconf is ascribed to the
suppression of the oﬀ-diagonal elements of ρL caused by the small corrections ofϵα and ϵβ due to interaction. If the number of particles included in the initialstate increases, the suppression of the oﬀ-diagonal terms due to the interaction
is more pronounced. Now, let us focus on the fact that any oﬀ-diagonal element
consists of many components, each of which shows complicated time evolution
due to interaction. Hence any oﬀ-diagonal element decreases in time owing to
the distractive interference among the many components. This phenomenon isreferred to as dephasing. The perturbation analysis shows that a pair of particles
suppresses the oﬀ-diagonal elements of ρL , contributing to the growth of Stot ,28

Figure 3.11: Time evolution of Stot , Snum , and Sconf . We employ the same data
used to draw Fig. 3.9.
when distance x between them is smaller than ξ log(2V t). Hence, the growthof Stot in the many particle case is expressed as
Stot ∝ x(t) = ξ log(2V t).

3.6(3.43)

Non-Hermitain many-body localization

Figure 3.12: Stot as a function of time with L = 12, averaged over various
disorder conﬁgurations in the two cases of h = 2 and 14, where h represents
disorder strength.
The authors of Ref. [36] have studied the localization problem in a HatanoNelson model with many-body interaction. They have reported on level statistics, entanglement entropy of eigenstates, and complex spectrum (cf. Chapter 2).
29

Let us focus on entanglement entropy in the many-body HN model. The authorsof Ref. [36] reported on the time evolution of Stot only in the weak disorder caseof h = 2 and the strong disorder case of h = 14. The behavior of Stot shownin Fig. 3.12 is qualitatively similar to that reported in Ref. [36]. As we shownin Fig. 3.12, Stot exhibits a power-law growth in the weak disorder case and alogarithmic growth in the strong disorder case. These features are observed in
the Hermitian case, as shown in Fig. 3.6; apparently, non-Hermiticity (i.e., nonreciprocal hopping) does not aﬀect Stot in a drastic manner. This observation
leads to one of the two questions in Chapter 1, i.e., to what extent non-reciprocal
hopping qualitatively changes the entanglement behavior. The other question iswhether the relation between a non-reciprocal parameter g and the localization
length still holds in a many-body system (see Chapter 2 for details).
We have conducted a systematic study of Stot (t) and revealed characteristic
disorder-dependence of Stot (t), which is the answer to the ﬁrst question, that is,
non-reciprocal hopping qualitatively changes the entanglement behavior. Additionally, the characteristic behavior of Stot (t) closely related to the second
question. Such a unique behavior of Stot (t) allows estimation of the localization
length.

30

Chapter 4
Non-Hermitian wave packet dynamics
In this chapter, we study how a wave packet spreads in a disordered nonHermitian system, considering the Hatano–Nelson model as a concrete example. Interestingly, a cascade-like wave packet spreading in the Hermitian case issuppressed within the clean limit and at weak disorder, while it revives in thevicinity of the localization–delocalization transition. We explain why the wavepacket dynamics in the non-Hermitian system becomes considerably diﬀerent
from that in the Hermitian case.

4.1

Single-particle case: Hatano-Nelson model and AubryAndré model

Let us consider the Hatano-Nelson model [33] under the periodic boundary condition, which we introduced in Section 2.3:
H = −

L−1 
X

ΓL |j + 1⟩⟨j| + ΓR |j⟩⟨j + 1|



j=0

+

L−1
X

Wj |j⟩⟨j|,

(4.1)

j=0

where |j⟩ represents a single-particle state localized at site j, ΓL = eg Γ0 andΓR = e−g Γ0 with g being a parameter quantifying the degree of non-reciprocity.
g can be regarded as an imaginary vector potential [33]. The ﬁrst two terms
represent the nearest neighbor hopping with non-reciprocal hopping amplitude. 1
The third term is on-site potential. We have chosen quasi-periodic potential as
on-site potential, i.e.,
Wj = W cos(2πθj + θ0 ),(4.2)
which eﬀectively acts as random potential (Aubry-André model [53]), where θis an irrational√ constant, which we choose to be the so-called (inverse) golden
ratio: θ = ( 5 − 1)/2. θ0 ∈ R is a free parameter used to consider a disorder
average; θ0 ’s distributed uniformly in the range of θ0 ∈ [0, 2π) provide a random
ensemble. The θ0 -averaging has been considered over 50 samples, e.g., xG (t) andits ﬂuctuation (Figs. 4.4 and 4.6), Stot , Snum and Sconf .
1

In non-Hermitian quantum mechanics with non-reciprocal hopping, choice of the boundary
condition is a subtle issue in its statics, whereas it is not a central issue in its dynamics (seeAppendix A for details).

31

In the Hermitian case (g = 0), the eigenstates are extended when W is
suﬃciently weak (W < Wc ), while they are localized for W > Wc , where
Wc
= 1,
2Γ0

(4.3)

which provides the critical point (cf. Section 2.2). This is consistent with thedivergence of localization length ξ deﬁned in the localized phase [42, 67]:ξ −1 ≃ log

W
.
2Γ0

(4.4)

The localization length ξ diverges as W approaches the critical value (4.3) fromabove.
In the non-Hermitian case (g ̸= 0), the critical value gc is related to the
localization length ξ as [34, 73] 2
ξ −1 = gc .

(4.5)

Substituting Eq. (4.5) into Eq. (4.4), the delocalization transition point is
Wc = 2Γ0 eg = 2ΓL ,(4.6)

where ΓL > ΓR (g > 0); i.e., Wc in the non-Hermitian case is found simply
by replacing Γ0 in Eq. (4.3) with the left/large hopping amplitude Γ L = eg Γ0 .
Both in the Hermitian and non-Hermitian cases, the location of the transition
point (4.6) does not depend on the energy, indicating the absence of a mobility
edge. When g ̸= 0, the eigenenergy ϵn becomes complex in the extended phase(W < Wc = 2ΓL ) as is also seen in the case of free particle motion described inAppendix C. Contrastingly, the eigenenergy remains real in the localized phase(W > Wc ). Thus, the localization–delocalization transition is accompanied bya real–complex transition of the eigenenergies (see Section 2.3 and Appendix A
for details).
Let us consider how the initially localized wave packet evolves in time following quantum mechanics. Four panels of Figs. 4.1 and 4.2 show examples ofsuch dynamics. We consider the wave packet located in site j = j0 at t = 0:|ψ(t = 0)⟩ = |j0 ⟩.

(4.7)

At time t, the wave packet that follows the Schrödinger equation with timeindependent Hamiltonian evolves asX|ψ(t)⟩ =ψj (t)|j⟩j=X

cn e−iϵn t |n⟩,

(4.8)

n

In the non-Hermitian case: g ̸= 0, g plays the role of an imaginary vector potential thatappears in the wave functions of a localized state so that2

ψ L,R (x) ∼ exp(−|x − xc |∓ g(x − xc )),ξwhere ψ L,R (x) are left and right eigenvectors; xc is its localization center, while ξ represents
the corresponding localization length. If |g| > ξ −1 , ψ L (x) or ψ R (x) diverges; the wave functions ψ L,R (x) no longer represent an exponentially localized state. Therefore, the delocalized
transition point is determined by the condition (4.5). Further details are presented in Section 2.3.

32

where |n⟩ represents the nth single-particle eigenstate of the Hamiltonian withan eigenenergy ϵn , i.e., H|n⟩ = ϵn |n⟩, while cn = ⟨⟨n|ψ(t = 0)⟩. Here, ⟨⟨n|represents the left eigenstate deﬁned as ⟨⟨n|H = ϵn ⟨⟨n|, where ⟨⟨n| ̸= |n⟩† .
Notably, we impose the biorthogonal condition, i.e., ⟨⟨n|m⟩ = δn,m . In the caseof g ̸= 0, the eigenenergy ϵn is typically complex; cf. the free particle case inAppendix C [Eqs. (4.14) and (4.15)], so that the probability amplitude of theeigenstate |n⟩ tends to exponentially grow or decay reﬂecting Im(ϵn ) > 0 orIm(ϵn ) < 0. Consequently, the normalization condition ⟨ψ(t)|ψ(t)⟩ = 1 is notsatisﬁed. In the actual computation, we therefore rescale (renormalize) |ψ(t)⟩ atevery interval ∆t as|ψ(t)⟩|ψ(t)⟩ → |ψ̃(t)⟩ = p,⟨ψ(t)|ψ(t)⟩(4.9)

where ∆t is chosen as ∆t ≃ 10−4 − 10−1 .

Figure 4.1: Single-particle dynamics in the Hermitian case of g = 0. Timeevolution of the initial wave packet: ψj (0) = δj,j0 is visualized with j0 = 300
and L = 601. The amplitude |ψj (t)| [cf. Eq. (4.8)] is shown by a gradationof plot colors indicated in the color bar. The x-axis represents time t, and they-axis represents the site j. Diﬀerent panels (a-d) correspond to diﬀerent valuesof disorder strength W ; W = 0.0, 0.8, 1.6, 2.4, respectively, for panels (a–d). θ0
is ﬁxed at θ0 = 0 (i.e., no disorder averaging). Taken from Fig. 1 of Ref. [74].
©2022 American Physical Society.
Let us ﬁrst consider the Hermitian case. The four panels of Fig. 4.1 show
the distribution of |ψj (t)| in the Hermitian case for diﬀerent values of W . At
site j (the y-axis) and at time t (the x-axis), the amplitude of |ψj (t)| is speciﬁed
33

Figure 4.2: Single-particle dynamics in the non-Hermitian case: g = 1.4. Timeevolution of the initial wave packet under the same condition as in Fig. 4.1
with θ0 = 0, j0 = 580, and L = 601. The values of disorder strength W are
W = 0.4, 3.6, 8.0, and 10.0, respectively, for panels (a–d). Taken from Fig. 2 ofRef. [74]. ©2022 American Physical Society.
by a variation of the plot color indicated in the color bar. In the clean limit[Fig. 4.1(a)], the wave packet spreads symmetrically in two directions. As W
increases (Fig. 4.1(b), (c)), spreading of the wave packet is weakly suppressed.
Finally, beyond the critical disorder strength (Fig. 4.1(d)), the wave packet ceases
to spread. The four panels of Fig. 4.2 show the distribution of |ψj (t)| in the caseof g = 1.4 (L = 601, j0 = 580) for diﬀerent values of W . Unlike the Hermitiancase 3 (Fig. 4.1), the four panels show that the wave packet does not spread atleast in the regime of weak W [cases of panels (a–b)], 4 but rather slides in the
direction imposed by the non-reciprocity g.
In the non-Hermitian case, the wave packet shows unique unidirectional motion in the weakly disordered regime W ≪ Wc [panels (a–b)] [42]; however theHermitian-like (cascade-like) wave packet spreading reappears in the vicinity of
the localization transition point W ∼ Wc [panel (c)]. Beyond the critical dis3

In the Hermitian limit (g = 0) the wave packet spreads with time, unless disorder W is
not too strong (W/J < 2); see e.g., Fig. 2 of Ref. [42]. In a non-Hermitian system (g ̸= 0),
the wave packet does not show the cascade-like spreading seen in the Hermitian case. In any
case, the dynamics becomes very diﬀerent from the Hermitian case. Here, we further clarify
this point.
4
At least in the short time scale; in the long time regime at which the imaginary part Im ϵncomes into play, |ψ(t)⟩ decays into a single eigenstate with a maximal Im ϵn ; see Appendix C
for details. In such a long time scale the wave packet may spread in time, but not in the sense
considered here.

34

order strength W > Wc , the wave packet suddenly ceases to spread, thereby
exhibiting shows localization tendency [panel (d)]. Comparing the three caseson the delocalized side [panels (a-c)], one also notices that the sliding velocityof the wave packet tends to increase as W is increased [42].
To understand why in the non-Hermitian case the wave-packet dynamics
become substantially diﬀerent from that in the standard Hermitian case, theclean limit, W = 0, must√be considered ﬁrst. In this limit, the eigenstates areplane waves ⟨j|k⟩ = eikj / L, so that!
X
X
−iϵk t|ψ(t)⟩ =e|k⟩⟨k|j0 ⟩ ≡ψk (t)|k⟩k

k

1 X X −iϵk t+ik(j0 −j)
= √
e
|j⟩,
L j k

(4.10)

where |ψ(t)⟩ is expressed as a superposition of the plane waves. At site j, eachcontribution to |Ψ(t)⟩ is expressed in the form of a phase factoreiϕ(k) = e−iϵk t+ik(j0 −j) .

(4.11)At t = 0 and j ̸= j0 , such contributions are out of phase and cancel each other,while at j = j0 they add up in phase to form the peak of the initial wave packet.
Similarly, at t > 0, the only non-vanishing contributions [in the summation overk in Eq. (4.10)] are those from the neighborhood of k = k at which the phaseϕ(k) becomes stationary; i.e., ϕ′ (k) = 0, or
2Γ0 sin k t = j − j0 .

(4.12)

Since | sin k| ≤ 1, |j − j0 | = 2Γ0 t ≡ vf t deﬁnes the position of the wave front,
or a “light cone” [42]. In the Hermitian limit the initially localized wave packet
spreads linearly in time as ∆x(t) ∝ t (i.e., ∆x(t) ∼ tσ ) with the exponent σ ≃ 1,
wheresX
(j − j0 )2 |ψj (t)|2(4.13)∆x(t) =
jrepresents the spread of the light cone. In fact, the stationary phase condition
correctly characterizes the free particle motion. Further details are shown inAppendix B. The addition of disorder W suppresses the wave-packet spreading;as shown in Fig. 2 of Ref. [42], the velocity v = ∆x(t)/t characterizing the
spreading of wave packet decreases linearly with W and vanishes at W = Wc .
Upon the addition of non-Hermiticity g ̸= 0, a diﬀerent mechanism or aprinciple governs the wave-packet dynamics of Eq. (4.10), since the eigenenergies
ϵk become complex:
ϵk = −2Γ0 cos(k − ig),
(4.14)
which in the complex energy plane, takes values on an ellipse:
 Re ϵ 2  Im ϵ 2
k
k
+
= 1.
Γ0 cosh g
Γ0 sinh g(4.15)

Among the k values satisfying Eq. (4.15), those with maximal or near maximal
Im ϵk become important in the superposition (4.10). In case of Eq. (4.14), suchvalues of k are found near
k0 = −π/2.
(4.16)
35

Figure 4.3: Single-particle dynamics; proﬁle of the wave function in k-space. The
Fourier transform |ψk (t)| [Eq. (4.10)] of the wave packet is plotted as a functionof k at diﬀerent time slices t = 0, 1, 2, 3, 5, 30, and 100 as indicated in the inset
of panel (a). Panels (a–d) and the inset of panel (d) correspond, respectively, todiﬀerent values of disorder strength W = 0, 0.4, 3.6, 8.0, and 12.0 with g = 1.4
and θ0 = 0. Other settings follow those of Fig. 4.2. Taken from Fig. 3 of Ref. [74].
©2022 American Physical Society.
Thus, in the non-Hermitian free-particle dynamics, the initial state (4.7) dissolves in the course of time evolution (4.10) into a Gaussian wave packet:|ψ(t)⟩ ≃X((j0 − j) + 2(cosh g)t)2)4(sinh g)tjp×e2(sinh g)t / 4(sinh g)t,|j⟩ exp(−(4.17)

which are composed of plane waves with k values found near k0 . As for thederivation of Eq. (4.17), see Eq. (C.2) and related arguments in Appendix C.
The resulting Eq. (4.17) is a wave packet that slides in the direction imposedby g, and its expanse gradually increases as time evolves. The aforementioned
argument is summarized in a guiding principle:
i the survival of Max Im ϵk has priority overii the stationary phase condition [cf. Eq. (4.12) in the Hermitian case].
The non-Hermitian (free-particle) dynamics is fully governed by principle (i),which is replaced with principle (ii) when principle (i) tends to be ineﬀective.
In Fig. 4.3, panel (a) the distribution of |ψk (t)| at certain ﬁxed values of t are36

shown. |ψk (t)| shows a Gaussian-type distribution centered at k = k0 , and its
width tends to become narrower as t evolves [cf. Appendix C].
Panels (b–e) of Fig. 4.3 show how the addition of disorder W aﬀects andeventually destroys this peak structure of ψk . In panel (b), two side peaks
appear at k = k1 and k2 , which reﬂect the quasi-periodic nature of the potential(4.2); Bloch waves of these k values are quasi-commensurate with the potential.
As W is increased, such side peaks multiply [panel (c)], and the system gradually
evolves into the cascade regime represented by panel (d), where the distributionof |ψk (t)| is almost uniform, but still there are plenty of tiny peaks, while in thelocalized regime [inset of panel (d)], the distribution becomes ﬂat and smooth.

Figure 4.4: Single-particle dynamics; velocity and density ﬂuctuation proﬁlesaveraged over 50 samples with diﬀerent θ0 . Panel (a) and (b) represent the sliding
velocity vG = [xG (t)−xG (0)]/t and the density ﬂuctuation σ(t) = ∆x(t)−∆x(0)in the (W, g)-plane by a variation of plot colors indicated in the color bar. These
quantities are evaluated at t = t1 = 20 in the same initial state as in Figs. 4.2
and 4.3. Taken from Fig. 4 of Ref. [74]. ©2022 American Physical Society.
To further quantify features speciﬁc to the non-Hermitian wave packet dynamics, we must focus on
1. how speedily the center of gravity
xG (t) =X

j|ψj (t)|2(4.18)

jof the wave packet moves, and
2. to what extent the wave packet is spread around xG (t).
xG (t) ≃ xG (0) + vG t, so that the velocity of xG (t) is given by vG . In panel (a)of Fig. 4.4, the magnitude of vG is plotted based on the variation of plot color,where vG is determined by evaluating xG (t) at t = t1 5 ) as changing the set ofparameters (W, g), and is indicated by a variation of plot color. The plot showsthat vG is ﬁnite in the extended phase W < Wc , while it practically vanishes
5

The measurement time t1 has been chosen to be a value (t = t1 = 20 in Fig. 4.4) suchthat the wave packet does not travel across the (periodic) boundary.

37

in the localized phase W > Wc . The location of the phase boundary (4.6), or
equivalently,W
g = log(4.19)
2is indicated by a broken curve in the panel. On the side of the extended phase
W < Wc , vG continues to take a relatively large value, until approaching thephase transition; at a ﬁxed value of g, it rather tends to increase as W increases,
until it abruptly falls at the phase transition.
Panel (b) shows a similar plot for the quantity: σ(t) = ∆x(t) − ∆x(0) at a
certain time t = t1 , where ∆x(t) has been redeﬁned assX∆x(t) =(j − xG (t))2 |ψj (t)|2 .
(4.20)j

The quantity σ(t) is expected to measure the extent to which the density ρj (t)is spread around xG (t). The plot shows that similar to the behavior of vG in
panel (a), the spread of the wave packet also exhibits a sharp distinction in theextended (W < Wc ) and localized (W > Wc ) phases. ∆x(t) takes a ﬁnite valueon the side of the extended phase: W < Wc , and the appearance of a peak as W
is increased toward and close to the localization transition (4.19) at a ﬁxed valueof g. We interpret that this enhancement of σ(t) slightly before the localizationtransition reﬂects the cascade-like explosion of the wave packet observed in thedensity proﬁle in panel (c) of Fig. 4.2.

4.2

Case of an interacting system

Here, we consider how the presence of interaction aﬀects the above non-interacting
case. As a concrete model, we have employed the following bosonic tight-binding
model with the nearest neighbor inter-particle interaction V :
H = −

L−1 
X

ΓL b†j bj+1 + ΓR b†j+1 bj



j=0

+

L−1 
X


V n̂j n̂j+1 + Wj n̂j ,(4.21)

j=0

where b†j (bj ) creates (annihilates) a boson at site j, while n̂j = b†j bj counts theparticle number nj at site j. This is the bosonic version of a hybrid consistingof Hatano–Nelson and Aubry–André models. Following Refs. [36] and [37], we
assume that our particles are hard-core bosons: nj = 0, 1. 6 Figure 4.5 shows
examples of multi-particle dynamics in this system. The initial state is preparedin the following domain wall state:|Ψ(t = 0)⟩ = |00 · · · 011 · · · 1⟩,(4.22)

If they are true fermions, anticommutation of bj and b†j may lead to a quantitatively
diﬀerent result under a periodic boundary condition, particularly in the delocalized regime;
see Appendix D for details.
6

38

(i)

(ii)

Figure 4.5: Multi-particle dynamics; evolution of the initial wave packet chosen
to be in a domain wall form expressed in Eq. (4.22). The eight panels show
the evolution of the particle density nj (t) at site j and at time t = t1 basedon the gradation of plot colors indicated in the color bar, where t1 is chosen
as t1 = 2.2 with L = 25, g = 1.4, and θ0 = 0. Panels (a–d) correspond to
W = 0.4, 3.6, 8.0, and 10.0, respectively, for panels (a–d). The four upper panels
in case (i) represent the non-interacting case: V = 0, while those in case (ii)represent an interacting case: V = 2. Taken from Fig. 5 of Ref. [74]. ©2022
American Physical Society.
i.e., bosons occupy the last Nb sites 7 : Nb represents the number of bosons, andhere we have chosen Nb = 3. At time t, the initial state (4.22) evolves asX|Ψ(t)⟩ =cµ e−iEµ t |µ⟩,(4.23)µ

7†
In the actual simulation, we have shifted the domain of occupied sites; i.e., Π j=L
j=L−2 bj |0⟩ →†Πj=L−1j=L−3 bj |0⟩ so that the wave function does not spread across the boundary.

39

where |µ⟩ is the eigenstate of the Hamiltonian expressed in Eq. (4.21) and Eµ
is the corresponding eigenenergy: H|µ⟩ = Eµ |µ⟩. Here, cµ = ⟨⟨µ|Ψ(t = 0)⟩.
Here, ⟨⟨µ| with ⟨⟨µ| representing the left eigenstate deﬁned by ⟨⟨µ|H = Eµ ⟨⟨µ|,
where ⟨⟨µ| ̸= |µ⟩† . We impose the biorthogonal condition, i.e., ⟨⟨µ|ν⟩ = δµ,ν ,analogous to the single-particle case. As the single-particle eigenenergy ϵn iscomplex in general, the many-body eigenenergy Eµ is also complex. As a result,
the time-evolved wave packet |Ψ(t)⟩ as expressed in Eq. (4.23) tends to growexponentially. To avoid this computational diﬃculty, we rescale (renormalize)|Ψ(t)⟩ in the same manner as in Eq. (4.9) at every interval ∆t ≃ 10−4 − 10−1 inthe actual computation. The eight panels of Fig. 4.5 show the evolution of theparticle density
nj (t) = ⟨Ψ(t)|n̂j |Ψ(t)⟩(4.24)
at site j at time t based on color variation; the higher the density, the brighter
the color. The system size L is set as L = 25. The four panels in the upper
case (i) represent the non-interacting case: V = 0, while those in the lower case(ii) represent an interacting case: V = 2. In these panels one can still see thetendency observed in the single-particle dynamics; e.g., the cascade-like feature
in wave packet spreading can be seen in panel (c) [both in (i) and (ii)].
Figure 4.6 represents the distribution of vG and σ(t) = ∆x(t) − ∆x(0) in theparameter space: (W, g); the two ﬁgures correspond, respectively, to the casesof V = 0 and V = 2. vG and σ(t) are calculated similarly in the single-particle
case as follows:X
xG (t) =jnj (t) ≃ xG (0) + vG t,j∆x(t) =sX(j − xG (t))2 nj (t)/jXnj (t).

(4.25)

j

In both the ﬁgures, both vG and σ(t) take a ﬁnite value on the side of the extended phase: W < Wc , while they vanish on the localized side: W > Wc . In
σ(t) [panels (b)], an enhancement is also observed before the localization transition. In single-particle dynamics, as shown in Fig. 4.2, the wave packet spreading
in non-Hermitian systems shows the speciﬁc and unusual features. Figure 4.5
shows that these features found in single-particle dynamics are essentially maintained in multi-particle dynamics though somewhat masked by the inter-particle
interaction V .

40

Figure 4.6: Multi-particle dynamics; velocity and density ﬂuctuation proﬁlesaveraged over 50 samples with diﬀerent θ0 . Panels (a) and (c) show distributionof vG in the (W, g)-plane, while panels (b), (d) represent a similar plot for σ(t) =∆x(t) − ∆x(0). vG and ∆x(t) are evaluated using in Eq. (4.25) in the multiparticle case with the same initial state as in Fig. 4.5. Panels (a–b) represent
the non-interacting case: V = 0, panels (c–d) an interacting case: V = 2. Taken
from Fig. 6 of Ref. [74]. ©2022 American Physical Society.

41

Chapter 5
Non-Hermitian entanglement dynamics
5.1

Entanglement dynamics

The entanglement entropy Stot (t) is deﬁned for a quantum system consisting of
two subsystems so that it can measure how the two subsystems are quantummechanically entangled. This can be rephrased as follows: Stot (t) measures thecomplexity of a given quantum state by using a reduced density matrix for eachsubsystem. Hence, it also characterizes spreading of a wave packet [39, 40, 41],
since such spreading tends to enhance the complexity of the given state. Let us
consider the time evolution of Stot (t) for an initial state. In a non-interacting
system, Stot (t) increases as the corresponding wave function spreads in the system. The initial growth of the number entropy Snum (t) in the quench dynamics
is essentially due to this eﬀect (Fig. 5.1 and Fig. 5.4). In Hermitian systems withreciprocal hopping, an initially localized wave packet spreads symmetrically in
two directions in the clean limit and, after a duration t, the wave function extends to a region of width ∆x ∼ 2vf t, where vf is the velocity of the wave
front: vf ≃ J [42]. Consequently, the entanglement entropy also increases as
Stot (t) ∝ ∆x ∼ 2vf t. In interacting systems, the same argument applies if vf
is replaced with the Lieb-Robinson velocity [75]. The interaction also modiﬁes
the superposition of eigenstates in a nontrivial manner in the time evolutionof a wave packet, leading to the generation of entanglement. In the presenceof disorder W , as its strength is increased, wave packet spreading tends to be
suppression; vf decreases and at a certain critical value W = Wc , vf vanishes.
When W > Wc , the wave function is localized, and the wave packet spreading
is essentially suppressed. As for the behavior of Stot (t), a rapid growth in theextended (delocalized) phase is experienced, while such growth is considerably
suppressed in the localized phase [see Fig. 5.1, panel (a)] [22].
Considering how a wave packet spreads in non-Hermitian systems with nonreciprocal hopping, we proceed to the analysis of how the entanglement entropy
Stot (t) evolves over time in such systems. We employ the same many-body
Hamiltonian (4.21) as in the analysis of wave packet spreading, while the initialstate is prepared as the following density wave form (or Néel form in the pseudospin language [38]):|Ψ(t = 0)⟩ = |101010 · · · ⟩,(5.1)
where on the right hand side of the equation we have employed the computational
basis |ν⟩ = |n1 n2 · · · nL ⟩; nj = 0, 1 represents occupation of the jth site. 1
1

We assume that our particles are hard-core bosons unless otherwise mentioned; cf. Ap-

42

5.2

Entanglement entropy in the non-interacting limit:
wave packet spreading versus entanglement entropy

Figure 5.1: Entanglement dynamics in the non-interacting case V = 0. Threepanels (a), (e), and (i) in the ﬁrst column represent the Hermitian case g = 0,
while the other panels represent non-Hermitian cases g ̸= 0; g = 0.8 for panels(b), (f), and (j) in the second column (Wc ≃ 4.5), g = 1.4 for panels (c), (g),and (k) in the third column (Wc ≃ 8.1), g = 2 for panels (d), (h), and (l) in
the fourth column (Wc ≃ 14.8). Panels (a)–(d) in the ﬁrst row represent theevolution of Stot (t), while panels (e)–(h) in the second row and panels (i)–(l) in
the third row represent Snum (t) and Sconf (t), respectively. They are evaluatedbased on the average over 50 samples with diﬀerent θ0 in the system of L = 12
with the initial state (5.1). Taken from Fig. 7 of Ref. [74]. ©2022 American
Physical Society.
Figure 5.1 shows time evolution of the entanglement entropy for the initialstate (5.1) in non-interacting systems with a variable strength of non-Hermiticity
g. Panels (a), (e), and (i) represent Stot (t), Snum (t), and Sconf (t), respectively,in the Hermitian case g = 0. The characteristic of the Hermitian limit in the
delocalized regime is that particles initially spread in two hopping directions, andthereafter, the system eventually reaches an equilibrium state [76, 77] resulting
from the interference of many plane waves [cf. Eq. (4.10)]. The entanglement
entropies show a rapid initial growth and approach a saturated value in such
weak W regime, while they are strongly suppressed as W is further increased in
the localized regime. In panel (e) we focus on the behavior of Snum (t). In thetwo cases of W = 0.4 and 1.6 in the weak disorder regime, Snum (t) ﬁrst rapidly
grows and approaches the same value subsequently. In any case, particles spreadin the delocalized regime, and the system reaches equilibrium after a suﬃciently
long time although this time lengthens as W increases. Sconf (t) also grows in the
delocalized regime; however, the growth is slightly delayed compared with thatof Snum (t) [see Fig. 5.1 (i)]. As W exceeds the critical value Wc = 2, both thenumber and conﬁguration entropies decrease, but remain ﬁnite. This is because
pendix D.

43

even in the localized phase, the localization length ξ is ﬁnite [78]. This eﬀect
is not negligible particularly if W is not exceedingly large and the system is not
far from the transition point (see Appendix E).
As shown in Chapter 4, the addition of non-Hermiticity g ̸= 0 renders a more
drastic impact on wave packet dynamics; it breaks the symmetrical propagationof particles. In the presence of g ̸= 0, the wave packet ceases to spread but
starts to slide because non-reciprocal hopping strongly suppresses the interference of a wave function (see Appendix C). We have also discussed in the previous
section that as W increases, this sliding velocity often increases, and when W
is further increased and approaches the critical value Wc , the wave packet also
begins to spread, exhibiting a cascade-like feature as in the Hermitian case at
weak W . In the presence of random potential, diﬀerent scattering paths are
expected to interfere despite the presence of non-reciprocal hopping, i.e., quantum interference is partially recovered in this case. In the Hermitian case, theentanglement entropies diminish with an increase in W . In contrast, as g increases, this tendency becomes uncertain and is ﬁnally immediately reversed,
as can be observed in panels (b)–(d) of Fig. 5.1. This result indicates that the
total entanglement entropy is generated by disorder, exhibiting a maximum at
ﬁnite W = W̃c , subsequently decreasing. 2 We argue that this maximum of the

(a)

(b)

Figure 5.2: Distribution of the maximal entanglement entropy, Max Stot (t), inthe (W, g)-plane derived from the data used to draw Figs. 5.1 and 5.4. Max
Stot (t) in the time evolution is plotted in the (a) non-interacting case of V = 0(the corresponding time evolution is plotted in Fig. 5.1) and (b) interacting
case of V = 2 (id. in Fig. 5.1). The location of the delocalization–localizationtransition point in the non-interacting case: g = log W/2 [as given in Eq. (4.19)]
is indicated by a broken curve (in white) as a guide for the eyes. Taken from
Fig. 8 of Ref. [74]. ©2022 American Physical Society with slight modiﬁcation.
entanglement entropies is related to the delocalization–localization transition.
In Fig. 5.2 (a), the maximal value Max Stot (t) 3 of the entanglement entropy isplotted in the (W, g)-plane. The distribution of Max Stot (t) resembles closely tothat of σ(t) = ∆x(t) − ∆x(0) [see Fig. 4.4 (b)], exhibiting a peak close to theboundary of the localized phase expressed by Eq. (4.19), if observed from the2

Number and conﬁguration entropies also show similar behavior [see e.g., panels (f)–(h)and (j)–(l) of Fig. 5.1].
3Here, the maximal refers to maximal in the time evolution for a given g and W .

44

weak W (delocalized) side. That is, “the maximum of Max Stot (t)” coincides
with the regime where a cascade-like expansion of wave packet occurs in thedensity dynamics. In other words, the maximum of Max Stot (t) occurs close to
the delocalization-localization transition point: W = Wc . The unusual behavior
of entanglement entropies in the non-Hermitian non-reciprocal system is thus
revealed to be directly related to a highly speciﬁc methodology through which awave packet spreads or does not spread in this system.

Figure 5.3: Damped oscillatory behavior in the density proﬁle nj (t). The magnitude of nj (t) at site j (abscissa) at time t (ordinate) is expressed by a variation ofplot color indicated in the color bar. The data is obtained for the density–wave
type initial state of Eq. (5.1) with g = 0.8, W = 0.4, θ0 = 0, and L = 12. Taken
from Fig. 9 of Ref. [74]. ©2022 American Physical Society.
Finally, let us comment on the number and conﬁguration entropies. Panels(f)–(h) of Figs. 5.1 show the time evolution of the number entropy. In the delocalized phase, the number entropy rapidly grows in an initial stage in a mannerthat is almost independent of W and subsequently exhibits a damped oscillation. The damped oscillation is caused by the convergence of the initial state
to a single eigenstate |µ0 ⟩ with the maximal imaginary part of eigenenergy Eµ0 ,which also aﬀects the time evolution of the density proﬁle shown in Fig. 5.3.
Note that the state with the maximal imaginary part should be regarded as a
non-equilibrium steady state. This also leads to the suppression of conﬁguration
entropy, as shown in panels (j)–(l) in Fig. 5.1. The convergence to a single destined eigenstate |µ0 ⟩ implies the loss of superposition; the conﬁguration entropy
is signiﬁcantly reduced in this regime of W ≃ 0. The destined eigenstate is delocalized, so that after a certain time, the initial density wave pattern of Eq. (4.22)
is almost erased; (Fig. 5.3), while the number and conﬁguration entropies exhibit
a damped oscillation before converging to a ﬁnial value. Beyond the critical disorder strength Wc , the number and conﬁguration entropies signiﬁcantly ﬂuctuate
as in the Hermitian case.

5.3

Case of V ̸= 0 and g ̸= 0: non-monotonic behavior

Considering the interaction: V ̸= 0. Panels (a), (e), and (i) of Fig. 5.4 show theevolution of the entanglement entropy in the Hermitian limit: g = 0; panels (a),45

Figure 5.4: Entanglement dynamics in the interacting case (V = 2). Threepanels (a), (e), and (i) in the ﬁrst column represent the Hermitian case g = 0,
while the other panels represent non-Hermitian cases g ̸= 0; g = 0.8 for panels(b), (f), and (j) in the second column, g = 1.4 for panels (c), (g), and (k) inthe third column, g = 2 for panels (d), (h), and (l) in the fourth column. The
precise value of Wc is unknown in the interacting case (V ̸= 0); however, it is
not far from (slightly larger than) the non-interacting values given in the main
text and caption of Fig. 5.1. Except for the value of V , the other conditions are
the same as shown in Fig. 5.1. Taken from Fig. 10 of Ref. [74]. ©2022 American
Physical Society.
(e), and (i) represent Stot (t), Snum (t), and Sconf (t), respectively. Comparing thesepanels with the corresponding panels in the non-interacting case [Fig. 5.1 (a),(e), and (i)], the total entanglement entropy Stot (t) [panel (a)] exhibits a speedy
initial growth until reaching approximately 2–5×101 , subsequently the growthdeaccelerates and smoothly crosses over to a linear regime (in the logarithmic
time scale). Comparing these behaviors in panel (a) with the ones in panels (e)and (i), the rapid initial growth arises from the number entropy Snum (t), while
the second slow growth forming a linear regime stems from the conﬁguration
entropy Sconf (t) in panel (i).
As the non-Hermiticity g is also introduced (the remaining panels of Fig. 5.4),
the entanglement dynamics changes its behavior as in the non-interacting case;cf. corresponding panels in Fig. 5.1. In contrast to the non-interacting case, acharacteristic entanglement behavior appears in the interacting case; the second
slow growth of Stot begins to decay after a certain time scale, i.e., a characteristic
non-monotonic behavior appears. The non-monotonic behavior appears in theintermediate regime of W ; however, it is unclear whether it emerges from thedelocalized phase or localized phase. To elucidate this, we employ the maximal
value of the entropies, which are expected to be maximal at 4 Wc close to thedelocalization–localization transition as in the non-interacting case. In panel (b)of Fig. 5.2, the maximal value Max Stot (t) in the time evolution is plotted in the(W, g)-plane, exhibiting a broader maximum in contrast to the non-interacting
4

Especially, the magnitude of conﬁguration entropy Sconf (t) [panel (l)] is sensitive to thechange of W .

46

case [panel (a)], while the location of the peak is slightly shifted to the side oflarger W from the non-interacting value (indicated by a white broken curve).
The location of the phase boundary estimated from the peak of Max Stot (t) in
Fig. 5.2 (b) is consistent with an earlier numerical result: Wc ≃ 6–7 at g = 0.5
in Ref. [37]. Comparing Fig. 5.2 with Fig. 5.4, the non-monotonic behavior
appears in the delocalized side closely near the delocalization–localization transition point Wc .
The non-monotonic evolution of Stot is a consequence of the competition
between two mechanics: (i) dephasing and (ii) convergence of the superposition[Eq. (4.23)] into a single non-equilibrium steady state |µ0 ⟩ with the maximal
imaginary part of the eigenenergy Eµ0 [38]. In the absence of mechanism (ii),the mechanism (i) leads to a slow but unbounded growth of the conﬁguration
entropy, where it is typically logarithmic in the MBL phase [22]. For mechanism(ii) to be operational, the eigenenergy must have a ﬁnite imaginary part Im
Eµ . Panels (j)–(l) of Fig. 5.4 show the evolution of conﬁguration entropy fordiﬀerent sets of parameters. In the localized regime, the non-monotonic behaviorof the entanglement entropy (panels (b)–(d) of Fig. 5.4) stems mainly from the
conﬁguration entropy. Note that dephasing enhances the conﬁguration entropy,
whereas the convergence suppresses it. In the localized regime, these two eﬀects
compete, leading to a non-monotonic entanglement evolution. Contrastingly, nonon-monotonic behavior appears in the delocalized phase because convergence
dominates dephasing. In the non-interacting case of V = 0, the localizationtransition is believed to coincide with the real-complex transition [33, 34, 54],
while in the interacting case of V ̸= 0, whether the superposition (4.23) converges
to a non-equilibrium steady state on the MBL side is a more subtle issue.
Finally, we discuss the behavior of number entropy. In the delocalized phase,the damped oscillatory behavior mentioned earlier is more obvious. As W increases, the damped oscillation disappears and is replaced with a non-monotonic
behavior with a broader maximum. On top of the non-monotonic behavior, aspeedy oscillatory component is also recognized. Such an oscillatory componentis also obvious in the non-interacting case, particularly in the localized regime.
In the non-interacting limit the disappearance of damped oscillation coincides
with the localization transition. Here, we have shown that this is also the casein an interacting system.

5.4

Eﬀect of ﬁnite size and nonequilibrium steady state

Let us comment on the eﬀects of system size. In the three rows of panels in
Fig. 5.5, we compare the evolution of the entanglement entropies: (a)–(d) Stot (t),(e)–(h) Snum (t), and (i)–(l) Sconf (t) for systems of diﬀerent system size L. Notethat the entropies are normalized by the “Page value” [79], given in the case of
NA = NB = L/2 as
1
L
(5.2)
SPage = log 2 − .
22Note that SPage is proportional to the size L of the system, If the entanglement
entropies calculated at diﬀerent system size L are size independent, then it implies that they obeys the volume law over the range of parameters in question.
In the Hermitian case, the magnitude of Stot (t) is size independent [25, 80] at
47

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 5.5: Time evolution of the entanglement entropies obtained by the averageover 50 samples with diﬀerent θ0 in the density–wave type initial state. The
ﬁrst row [panels (a)–(d)], second row [panels(e)–(h)], and third row [panels (i)–(l)] represent Stot (t), Snum (t), and Sconf (t), respectively. The system is both
non-Hermitian g = 0.6 and interacting V = 2. Each column represents dataat W = 1.6, 4.0, 6.4, 8.8 from left to right. Curves plotted in diﬀerent colors
correspond to diﬀerent system sizes as indicated in the inset of panel (a). Taken
from Fig. 11 of Ref. [74]. ©2022 American Physical Society.
the stage of rapid initial growth (area law), while its saturation value after asuﬃciently long time is size-dependent (∝ L), obeying a volume law [21, 22]. In
the ﬁrst row of Fig. 5.5, where diﬀerent panels correspond to diﬀerent strengthsof disorder W with g = 0.6, the total entanglement entropy Stot (t)/SPage is plotted for diﬀerent system sizes L. The magnitude of Stot (t)/SPage clearly decreases
with an increase in L in panel (a), implying an area-law behavior. In panels (b)and (c), it becomes size-independent at its maximum (t ≃ 3–7×101 ), suggestinga volume-law behavior. For W = 4.0 in panel (b) and W = 6.4 in panel (c),both values of W are assumed to be not far from Wc at V = 2. As W is further
increased [panel (d): W = 8.8], the second growth of Stot (t) (a linear growth
region in the logarithmic time scale) lasts longer, and the maximum of Stot (t) is
not really achieved in the time scale shown in the panel.
Comparing panels (a), (e), and (i), the area-law behavior of Stot stems fromthat of Snum , while comparing panels (c), (g), and (k), the volume-law behavior of Max Stot results from an interplay of the number and conﬁguration entropies. The insensitivity of Max Stot to the system size L suggests that the nonmonotonic evolution of the entanglement entropy speciﬁc to the non-Hermitian
many-body system will also occur in case of a larger system size L.
For further clarifying this point in panel (a) of Fig. (5.6), we have plotted
the ratio
fIm = DIm /D,(5.3)
where DIm is the number of eigenenergies with a nonzero (|ImEµ | > 10−13 ) imaginary part Im Eµ , while D is the dimension of the Hilbert space, i.e., the total
number of eigenenergies. The existence of nonzero Im Eµ leads to suppression
48

(a)

(b)

(c)

(d)

Figure 5.6: Size dependence of the (a) fraction of imaginary eigenvalues: fIm[cf. Eq. (5.3)], (b) Max(Im Eµ ), (c) multifractal dimension D2 , and (d) its
ﬂuctuation: ∆D2 . Data acquired for the interacting case: V = 2 (connectedby solid lines) is compared with the non-interacting case: V = 0 (connectedby broken lines). Diﬀerent colors correspond to diﬀerent system sizes: L =10, 12, 14, and 16. The θ0 -average is obtained over 50 samples. Taken from
Fig. 12 of Ref. [74]. ©2022 American Physical Society with slight modiﬁcation.
of the superposition (4.23), leading to the suppression of the entanglement entropies. The delocalized eigenstates are susceptible to the non-reciprocity g ofhopping, generating a ﬁnite imaginary part in the eigenenergy. Thus, the quantity fIm measures the fraction of delocalized eigenstates in the total ensemble ofeigenstates. Panel (a) of Fig. 5.6 exhibits the evolution of the fraction fIm as afunction of W for systems of diﬀerent sizes L. As W is increased, fIm decreases,
ﬁrst gradually and subsequently swiftly: the more drastic this tendency, thelarger the size L. Furthermore, the curves corresponding to a diﬀerent systemsize L practically crosses at the same point at a value of W = Wc ≃ 7 at least in
the interacting case of V = 2 corresponding to the solid curves in panel (a). 5 Ina suﬃciently large system of size L, the curve fIm (W ) tends to become a sharply
5

In the non-interacting case of V = 0 corresponding to dashed curves in panel (a) of Fig. 5.6,
the change of fIm (W ) becomes too drastic, so that one cannot really see the crossing itself,
but the overall behavior of fIm (W ) is similar to this interacting case.

49

edged function: fIm (W ) ≃ 1 for W < Wc , while fIm (W ) → 0 for W > Wc . In
this case, all the eigenstates, including the highly excited states, experience acomplex-to-real transition of eigenvalues at W = Wc .
In panel (b) of Fig. 5.6, we compare the (ensemble-averaged) maximal value of
Im Eµ in the cases of V = 0 and V = 2. As expected, in the non-interacting case:V = 0, the magnitude of Max(Im Eµ ) experiences a sudden fall at the crossing(size-independent) point: W = Wc ≃ 3 − 4. On the two sides of this transition
point, the behavior of Max(Im Eµ ) exhibits a clear change that tends to magnify
as the system size L increases. In particular, in the localized phase (W > Wc ),the magnitude of Im Eµ tends to vanish as L increases. In the interacting case(V ̸= 0) [Fig. 5.6 (b) solid curves], the ensemble-average of Max(Im Eµ ) does
not exhibit a sudden fall; instead, it gradually decays over a broad range of
W . It exhibits no clear signature in the ETH–MBL crossover/transition regime:
W ≃ Wc . Additionally, it exhibits no dependence on the system size L. Basedon these observations, we guess that generically, Im Eµ remains ﬁnite even on
the MBL side: W > Wc ; i.e., the superposition (4.23) converges to a nonequilibrium steady state |µ⟩ = |µ0 ⟩, which has the maximal Im Eµ , although it
may consume a suﬃciently long time. 6 To understand the nature of this steadystate |µ0 ⟩, it is important to understand if it really is in a localized state. This
question arises, because one usually associates the imaginary part of Eµ withan extended state, as explained in Section 2.3. To clarify this point, we haveestimated the multi-fractal dimensionPµ 4log( Dµν=1 |cν | ),(5.4)
D2 = −
log(D)
encoding the (de)localized nature of the eigenstate |µ⟩; D2µ = 1 (= 0) corresponds
to a fully delocalized (localized) eigenstate. cµν ’s is the amplitude of computational basis |ν⟩ = |n1 n2 · · · nL ⟩ of the µth eigenstate, and D is the dimension of
the many-body Hilbert space. In panel (c) of Fig. 5.6, disorder averaged D2µ0
(represented by solid lines) is compared with the disorder and all eigenstates
averaged D2 (represented by broken lines).
The plots show that in the deep MBL regime, the eigenstates |µ0 ⟩ exhibit
a localized tendency, even though its eigenenergy E exhibit a ﬁnite imaginary
part; to be precise, the state |µ0 ⟩ is less localized than the other eigenstates.
Panel (d) of Fig. 5.6 exhibits the variance ∆D2 of the multi-fractal dimension as
a function of W , which suggests a qualitatively diﬀerent behavior in the ETH–
MBL crossover regime for ∆D2 and ∆D2µ0 . To summarize, the superposition(4.23) indeed converges to a non-equilibrium steady state |µ0 ⟩ even in the deep
MBL phase, while the destined state |µ0 ⟩ shows a spatially localized signature.

5.5

Proﬁle of the reduced density matrix

Let us ﬁnally visualize how diﬀerent stages in the time evolution of the entanglement entropy can be understood from the behavior of the reduced density
6

The authors of Ref. [36] have reported that max Im(Eµ ) also shows the real-complex
transition. In fact, in a suﬃciently large disorder strength (W ∼ 10), in panel (b) of Fig. 5.6,
max Im(Eµ ) also weakly shows size-dependent behavior. However, study for max Im(Eµ )remains an underway, so that we have concluded that max Im(Eµ ) is ﬁnite in the numericallyaccessible system size.

50

time=0.002

V=0.0,g=0.0

time=0.002

V=2.0,g=0.0

time=0.002

V=0.0,g=0.8

time=0.5

V=0.0,g=0.0

time=9.5

V=0.0,g=0.0

time=9.5

time=0.5

V=2.0,g=0.0

V=2.0,g=0.0

time=0.5

V=0.0,g=0.8

time=9.5

V=0.0,g=0.8

time=485

V=0.0,g=0.0

time=485

V=2.0,g=0.0

time=485

V=0.0,g=0.8

Figure 5.7: Evolution of the reduced density matrix in the wave-packet dynamics. The ﬁrst, second, and third row, respectively, show the results in the
non-interacting (V = 0) and Hermitian (g = 0) case, those in the interacting(V ̸= 0) and Hermitian (g = 0) case, and those in the non-interacting (V = 0)
and non-Hermitian (g ̸= 0) case. The same initial state as in Fig. 5.1 is used
and the θ0 -average is obtained over 50 samples. Taken from Fig. 13 of Ref. [74].
©2022 American Physical Society.
matrix ΩA (t). In the twelve panels of Fig. 5.7, we have plotted the elements ofΩA (t) at diﬀerent times to visualize its time evolution. We note that there are
4–5 characteristic stages in the evolution of ΩA (t).
In the ﬁrst row of Fig. 5.7, we start with the Hermitian case (g = 0) withno interaction (V = 0). At t = 0, the initial state |Ψ(t)⟩ is chosen to be asimple product state, so that the reduced density matrix Ω A (t = 0) includes asingle ﬁnite element (= 1) in one of the diagonals. As time passes by, the regionof a ﬁnite matrix element (represented by bright spots in the panels) spreads.
However, for a while, most of the amplitudes are limited within the same diagonal
block in certain ΩNj (Nj = 0, 1, · · · , 5, 6) [cf. Eq. (3.27)] 7 as is seen in theﬁrst panel from the left (t = 0.002). From the start of quench dynamics, thesystem is susceptible to the process of wave packet spreading; however, as far
as Nj is kept unchanged, such dynamics does not lead to an immediate increaseof the number entropy. After a certain time, bright spots start to appear in
the neighboring blocks as is seen in the second panel (t = 0.5), and they ﬁnallybecome distributed almost equally in all the blocks as is observed in the third andfourth panels (t = 9.5, 485). It is expected that the number entropy experiencesa rapid growth over this period and becomes saturated afterwards.
Next, we consider interaction (V ̸= 0)). The corresponding results are shown
7

Recall that the reduced density matrix ΩA is block diagonalized as in Eq. (3.27) with ΩNjrepresenting a block in which the number of particles is restricted to Nj .

51

in the second row of Fig. 5.7. In the ﬁrst two panels, the evolution of Ω A (t)appears similar to the non-interacting case, that is, bright spots tend to spreadover all diﬀerent blocks as time passes by, and meanwhile the number entropy
tends to be saturated. In the third and fourth panels, the bright spots are
distributed equally across all the blocks as in the noninteracting case; however,here they converge on the diagonals inside each block ΩNj . 8 This is due todephasing [22]. Due to interaction V ̸= 0, oﬀ-diagonal matrix elements tend toacquire a random phase and vanish on average after certain time. Dephasing
leads to increase of the conﬁguration entropy Sconf (t); thus, in the present caseafter the number entropy is saturated, the total entanglement entropy Stot (t)continues to increase with no systematic bound, except the one due to the sizeof the system.
In the non-interacting (V = 0) and non-Hermitian (g ̸= 0) case (see the third
row of Fig. 5.7), the overall behavior of the evolution of the reduced density
matrix ΩA (t) resembles the Hermitian case in the ﬁrst row except that here thepattern of the matrix elements appears considerably regular inside each blockΩNj , while the pattern appears random in the Hermitian case (g = 0). The
reason is the following: in the regime of weak W , the eigenenergy E is typicallycomplex, so that in the time evolution of a many-body wave packet |Ψ(t)⟩ in
Eq. (4.23), a single eigenstate |µ0 ⟩ with the maximal imaginary part Im Eµtends to predominate in the superposition. This tendency in the evolution of
the entanglement entropy Stot (t) (cf. Fig. 5.1) has already been observed; in theregime of weak W , Stot (t) speedily converges to a ﬁxed value and practicallyshows no ﬂuctuation.

8

In the non-interacting case (ﬁrst row), on contrary, the bright spots spread over all partsof the matrix almost randomly in each block ΩNj .

52

Chapter 6
Concluding remarks
A century has passed, since quantum mechanics ﬁrst appeared in the historyof mankind. In its one-century-long history, quantum mechanics has been applied to the description of sub-atomic objects in various of circumstances. The
discrete quantized spectrum of eigenenergies (the eigenvalues of the Hamiltonian 1 ) for electrons trapped in the atom is a hallmark of quantum mechanics.
The corresponding eigenfunctions take a spatial proﬁle that can be naturallyinterpreted as a bound state. Quantum mechanics has been widely applied tothe description of matter, leading to the birth of the ﬁeld of condensed-matterphysics, wherein matter is analyzed from a microscopic perspective. In thissubﬁeld, roles of periodic crystalline structure, fermionic/bosonic statistics, disorder in the form of a random potential, and many-body eﬀects stemming frominterparticle interaction have been extensively studied.
In this thesis, I have focused on the fact that the (eigen) wave functions tend
to be localized in the presence of a random potential, and how this phenomenon,
also known as Anderson localization, is modiﬁed by the presence of interparticle
interaction. 2 I have also considered the fact that in generic circumstances, oursystem is inevitably inﬂuenced by the environment. Namely, we regard oursystem as an open quantum system, i.e., a closed quantum mechanical system(described by a Hermitian Hamiltonian) coupled to an environment. Following
the standard procedure to determine an eﬀective Hamiltonian, we trace out
the environment, and typically establish a non-Hermitian Hamiltonian as aneﬀective model. Finally, I have focused not only on the eigenstates of such aneﬀective Hamiltonian, but also on quantum dynamics, i.e., the time evolution ofa given initial state. Particular attention is paid to the entanglement entropy
that quantiﬁes the complexity of the state as a function of time.
In Chapter 2, we have reviewed the so-called localization phenomena that
occur in the presence of random potentials. We naturally started with the Hermitian case, and subsequently considered the non-Hermitian case as well. As
an illustrative example, we have discussed in detail the case of the so-called AA
model, in which a quasi-periodic potential mimics the random potential. In thiscase, a duality argument enables us to analytically determine the localization–delocalization point. Naturally, we have numerically checked the validity of thisargument by demonstrating how an eigenstate localizes as disorder strength in1

The Hamiltonian is presumed to be Hermitian [81].
In the interacting case, eigenstate thermalization hypothesis (ETH) and many-body localization (MBL) have been the two competing key paradigms; at least in the Hermitian case.
253

creases. We have subsequently introduced the so-called Hatano–Nelson model(i.e., a one-dimensional tight-binding model with non-reciprocal hopping) as
a prototypical example of a non-Hermitian eﬀective Hamiltonian. We have described how a localization–delocalization transition becomes possible in this spatially one-dimensional 3 but non-Hermitian model. We have also emphasizedthat the wave-packet dynamics in this model is also extremely peculiar.
In Chapter 3, we have reviewed many-body localization, which is the localization phenomenon in an interacting many-body system. A remarkable featureof interacting systems is the appearance of a delocalized phase even in one dimension. A many-body delocalized phase generally satisﬁes ETH that justiﬁes
thermalization in an isolated many-body quantum system. We have introduced
the deﬁnition of thermalization in an isolated many-body system and shownhow the ETH justiﬁes such thermalization. We have also introduced the phenomenological picture of MBL and shown the breaking of the ETH. Thereafter,
we introduced the concrete procedures for calculating Stot and more subdivided
Stot , i.e., Snum and Sconf . We reviewed how Stot characterizes the static anddynamical properties of the ETH and MBL phases. For static property, ETHMBL transition involves the transition of Stot from a volume-law (ETH phase)to an area-law (MBL phase). For dynamical property, Stot exhibits a power-law
growth with time in the ETH phase while a logarithmic growth in the MBL
phase. Based on the behavior of Stot in the Hermitian systems, we reviewed
the dynamical behavior of Stot in the non-Hermitian system with non-reciprocal
hopping and emphasized the open problem which may be overlooked.
In Chapter 4, we have highlighted the nature of the unusual wave packet
spreading in a non-Hermitian system with non-reciprocal hopping (Hatano–
Nelson model) [Eqs. (4.1) and (4.21)]. Disorder has been modeled by a quasiperiodic potential (Aubry–André model) [Eq. (4.2)]. In a Hermitian system, awave packet spreading is gradually suppressed by disorder. Contrastingly, in a
non-Hermitian system, it does not occur even in the clean limit, at least in the
Hermitian way. The wave packet rather slides instead of spreading. The disordertends to localize a wave function, whereas the non-reciprocity tends to delocalize it. Weak disorder exerts negligible eﬀect on this sliding behavior. Whenthe disorder becomes comparable to non-reciprocity, this characteristic behavior
tends to be replaced with a cascade-like wave packet spreading analogous to theHermitian case. We have clearly demonstrated the mechanism [see Eq. (4.17),
and related arguments] why the wave packet slides instead of spreading in oursystem and how this behavior tends to be destroyed by the quasi-periodic potential (cf. Fig. 4.3). In the non-Hermitian case, the fundamental principle thatgoverns the wave packet spreading is diﬀerent from that in the Hermitian case.
In Chapter 5, we have seen how such unusual wave packet spreading in the
non-Hermitian model leads to anomalous behaviors in the entanglement behavior. 4 First, the speciﬁc wave packet spreading mentioned leads to strong suppression of the entanglement entropy, especially, in the delocalized phase as thenon-Hermiticity g increases. Second, in the presence of interaction, a ﬁnite imaginary part of the eigenenergy combined with the logarithmic growth (eﬀect of3

A localization-delocalization transition usually does not occurs under such low dimensionality.
4
As for another anomalous feature in the entanglement dynamics in a non-Hermitian PT
model, see Refs. [82, 83].

54

dephasing) leads to a characteristic non-monotonic behavior of the entanglement
entropy. We have observed that the maximum value of entanglement in the time
evolution becomes maximal at (or at least near) the delocalization–localizationtransition, thus containing information regarding the location of the transition.
This, in turn, signiﬁes that the non-reciprocity g can be used as a probe fordetermining the localization length ξ in the Hermitian system [73], since g is directly related to ξ; i.e., ξ = g −1 [see Eq. (4.5), and related footnote on it]. Notethat identifying the critical localization length ξc is a key to reveal the natureof the ETH–MBL transition [67]. The size dependence of the entanglement dynamics shows that the entanglement entropy Stot (t) at its maximum in the time
evolution [i.e., Max Stot (t)] obeys the volume law, while under other circumstances (typically, in the delocalized phase) Stot (t)/SPage tends to decrease withan increase in L, suggesting an area law. Thus, the entanglement entropy in a
non-Hermitian system exhibits an unusual area–volume–area law type crossover
as a result of the interplay between non-Hermiticity and the interaction.
As shown in Ref. [38], collapse of the superposition in the initial state andconvergence to a single destined eigenstate |µ0 ⟩ with maximal Im Eµ0 play acentral role in the behavior of the entanglement entropy in this non-Hermitiansystem. Here, we have veriﬁed this point through analyses of the entanglement
using diﬀerent indices (e.g., number and conﬁguration entropies) and in a wide
range of parameter regimes. In the localized side, most of the eigenstates are
localized; however, a few eigenstates still include a small but ﬁnite imaginary
part Im E, suﬃcient to compete with the logarithmic growth (dephasing) at least
over an extremely long time scale, leading to the characteristic non-monotonic
behavior of Stot (t).
Finally, extending the analyses performed in this study to systems of larger
size is challenging. In Hermitian systems, Krylov-based time evolution [25] andtensor network techniques [72, 84, 45] are known to be applicable to deal withlarge-sized systems. The extension of these techniques to the non-Hermitian caseis a potential direction of future work.

55

Appendix A
Notes on the choice of the boundary
condition
The non-Hermitian system with non-reciprocal hopping is extremely sensitive
to the choice of the boundary condition. Under the open boundary condition(OBC), all the bulk wave functions tend to localize near a boundary of thesystem. This is referred to as non-Hermitian skin eﬀect [85, 86]. All the eigenvalues are real in this case. Contrastingly, under the periodic boundary condition(PBC), wave functions extend over the entire system (i.e., no non-Hermitian skineﬀect), and eigenvalues become complex.
The sensitivity of the system to the boundary condition has already beenrecognized in the original stusies of Hatano and Nelson [33, 34]. Recent intensive
discussions on non-Hermitian topological insulators have revealed its further
consequences. In topological insulators, the bulk topology under the PBC isin one-to-one correspondence with the appearance/disappearance of topologicalboundary states under the OBC (the bulk-boundary correspondence). In a nonHermitian system that is subjected to the non-Hermitian skin eﬀect, the eigen
wave functions are localized under the OBC, whereas they are extended under
the PBC. This sharp diﬀerence results in the breaking of the bulk-boundary
correspondence [85, 86, 87, 88]. As another remark, the non-Hermitian skineﬀect is proposed to be topological; its occurrence under the OBC is protected bya speciﬁc winding property of the complex energy spectrum under the PBC [89].
Under the PBC, the eigenenergies of the system takes complex values; particularly, the system exhibits a complex spectrum in the clean limit. The complex
nature of the spectrum (Im E ̸= 0) is closely related to the plane wave nature of
the eigen wave function. Under the OBC, this is no longer true even in the clean
limit. In the presence of weak disorder, the eigen wave functions are extended as
far as it is not exceedingly strong and that is suﬃcient for maintaining the complex spectrum. Exceedingly strong disorder causes the eigen wave function to belocalized, pushing the spectrum toward the real axis. Thus, the delocalization–
localization transition in this system inevitably accompanies a spectral transition
from complex to real. In an interacting system V ̸= 0, one can expect that thisstill holds. Indeed, based on the observation that numerically estimated criticalpoints of the ETH–MBL and complex–real transitions are close, the authors ofRef. [36] conjecture that the two transitions actually coincide in the thermodynamic limit. This is also reinforced by certain analytic arguments. In addition
to the conjectured double transition, the authors of Ref. [37] highlighted thethird topological transition, which is described in the last paragraph as at the56

double transition, the winding property of the complex spectrum under the PBC
also changes, which is in one-to-one correspondence with the disappearance ofthe non-Hermitian skin eﬀect under the OBC [37].

57Appendix B
Free particle wave-packet dynamics in a
Hermitian system
The stationary phase approximation is the guiding principle describing the spreading of a wave packet in the Hermitian limit. An explanation of this is given inthis Appendix. The Hamiltonian
X
Hhs =
(B.1)
−J(|j⟩⟨j + 1| + |j + 1⟩⟨j|),j

which is equivalent to Eq. (4.1) at g = 0 and W = 0, is used below. Itseigenenergy and
Peigenvector speciﬁed by a wave number k are ϵk = −2J cos(k)1√and |k⟩ = L j exp(ikj)|j⟩, respectively. Let us consider the time evolutionof the wave packet with an initial state that is localized at site j0 . Its time
evolution is expressed asX|Ψ(t)⟩ =ei2J cos(k)t |k⟩⟨k|Ψ(t = 0)⟩k

1 X i2J cos(k)t ik(j0 −j)
e
e
|j⟩
= √
L k,j
Z
1 X 2π∼dkei2J cos(k)t eikx |x⟩
2π x 0ZX
1 X x π −i2J sin(k)t ikx
ix J˜x (2Jt)|x⟩,=iee dk|x⟩ =
2π x
−π
x

(B.2)

where J˜x (2Jt) is the Bessel function.
Figure B.1(a) shows the time evolution of |Ψ(j, t)| at site j (abscissa) at timet (ordinate) expressed by a variationPof plot colors indicated in the color bar,where Ψ(j, t) is deﬁned as |Ψ(t)⟩ = j Ψ(j, t)|j⟩. One observes that the wave
packet symmetrically spreads in the entire system. This indicates that |Ψ(j0 , t)|approaches 0 in the limit of t → ∞, exhibiting the delocalization tendency. Thatis quantiﬁed by the return probability (RP) deﬁned by
RP = |⟨Ψ(t = 0)|Ψ(t)⟩|2 .

(B.3)

In the limit of t → ∞, RP assumes 0 in the delocalized phase, while it assumes aﬁnite value in the localized phase. Hence, one can characterize the localization–delocalization transition by using RP. In the delocalized phase, the wave packet
58spreads throughout the system, so that |⟨Ψ(t = 0)|Ψ(t)⟩| becomes 0 in the limitof t → ∞. In contrast, in the localized phase, a large portion of the wave packet
remains in the initial state, so that |⟨Ψ(t = 0)|Ψ(t)⟩| remains ﬁnite even in thelimit of t → ∞. The free particle case belongs to the delocalized phase. This can

(a)

(b)

Figure B.1: Free particle dynamics in the Hermitian case: (a)|Ψ(j, t)| and (b)RP as a function of time. Data are obtained in the system of L = 601 with theinitial state localized at j0 = 301. The black dotted line in panel (b) represents
the line of RP= 1.
be seen in the asymptotic behavior of ⟨Ψ(t = 0)|Ψ(t)⟩ by using the stationary
phase approximation. This result is
Z 2π1⟨Ψ(0)|Ψ(t)⟩ =dkei2J cos(k)t
2π 0
Z ∞
1
1
1
2
2
dk(ei2Jt(1+ 2 (k−1) ) + ei2Jt(−1− 2 (k−1) ) )
∼
2π −∞
iπ
iπ
1
= √ (ei2Jt− 4 + e−i2Jt−+ 4 )4t1π= √ cos(2Jt − ),
(B.4)
4
t
where the stationary phase points are k = 0 and k = π. RP is immediately
obtained as
1
RP = |⟨Ψ(0)|Ψ(t)⟩|2 ∼ ,
t

(B.5)

which vanishes in the limit of t → ∞. Figure B.1(b) shows that RP in thefree particle case decreases as time increases. This behavior is consistent withthe asymptotic solution expressed in Eq. (B.5), indicating that the stationary
phase approximation correctly characterizes the wave-packet dynamics in the
Hermitian limit.

59

Appendix C
Free particle wave-packet dynamics in a
non-Hermitian system with Im ϵk ̸= 0

Figure C.1: g dependence of the sliding velocity vG = [xG (t) − xG (0)]/t in thecase of W = 0. From the results shown in Fig. 4.4 (a), data at W = 0 is
extracted and replotted as a function of g. The solid line represents Eq. (C.3).
Data are obtained in the system of L = 601 and θ0 = 0 (no disorder averaging)
and with the initial state localized at j0 = 580. Taken from Fig. 14 of Ref. [74].
©2022 American Physical Society.
Let us consider a free particle motion prescribed by the Hamiltonian (4.1).
For simplicity, we switch oﬀ the quasi-periodic potential (4.2); W = 0. In thisdisorder-free case, the eigenstates of the Hamiltonian (4.1) under the PBC takethe form of a plane wave eik , while the corresponding eigenenergies ϵk becomecomplex; refer to Eqs. (4.14) and (4.15). considering the initial state |ψ(t = 0)⟩
as shown in Eq. (4.7), we consider its time evolution Eq. (4.10). Considering thelimit L → ∞, we replace the summation over k in the last line of Eq. (4.10) by
an integral:X Z 2π1(C.1)|ψ(t)⟩ =|j⟩dk √ e2i cos(k−ig)t+ik(j0 −j) .
2π
0
j
60

The one-body spectrum ϵ = ϵk is complex as in Eq. (4.14), and its imaginary
part Im ϵk becomes maximal at k = k0 = − π2 . Accordingly, in the integral overk in Eq. (C.1), the dominant contributions governing the long time dynamics
stem from those near k = k0 ;|ψ(t)⟩ ≃Xj|j⟩

Z − π +δkπ1dk √ e2i cosh(g)(k+ 2 )t2π
− π2 −δk2(k+ 2 ) )t+i(k+ 2 )(j0 −j)2×e2 sinh(g)(1−
X Z ∞1
dk √ e2i cosh(g)kt
≃
|j⟩
2π
−∞
j
1

π 2

π

k2

×e2 sinh(g)(1− 2 )t+ik(j0 −j)X((j0 − j) + 2 cosh(g)t)2=|j⟩ exp(−)
4 sinh(g)tjp
×e2 sinh(g)t / 4 sinh(g)t.

(C.2)

Based on the last expression, one can read the group velocity of the wave packetasvG = 2 cosh(g),(C.3)
which is consistent with our calculations (Fig. C.1). Note that vG increases with
the increase in g. The last expression takes the form of a Gaussian wave packet
that slides in the direction imposed by g. The expanse ∆x of p
the wave packet
gradually increases in real space in the course of time as ∆x ≃ 2 sinh(g)t, while
the corresponding width ∆k ∼ 1/∆x in the reciprocal space tends to diminish[see Fig. 4.3 (a)]. The above features are also quite manifest in panels (a) and(b) of Fig. 4.2, although the data shown in this ﬁgure is obtained in a weakdisorder case. In time evolution, the state |ψ(t)⟩ as expressed in Eq. (C.1) tends
to be governed by the eigenstates with maximal Im ϵk [see Eq. (4.17)], while theindividual eigenstates take the form of a plane wave.
Note that the form of the Gaussian wave packet (C.2) suggests that it obeys asquare-root scaling typical to classical diﬀusion dynamics. Of course, we consider
the coherent quantum dynamics of the wave function ψ(x, t) that is governedby the Schrödinger equation. However, in the present non-Hermitian setup,
the non-reciprocal nature of the hopping strongly suppresses the interference ofthe complex wave function ψ(x, t), characteristic to the Schrödinger quantum
dynamics. As a result, the wave function ψ(x, t) as expressed in Eq. (C.2)
eﬀectively obeys the classical diﬀusion dynamics.

61

Appendix DEﬀects of anticommutation relation on
entanglement dynamics(a)hard-core boson(b)fermionFigure D.1: Time evolution of the entanglement entropy in the (a) hard-coreboson and (b) fermion models. We employ the same numerical parameters as inFig. 5.4. Taken from Fig. 15 of Ref. [74]. ©2022 American Physical Society.
Thus far we have considered systems of hard-core bosons; cf. Refs. [36] and[37]. In a system of true fermions, the anticommutation of bj and b†j leads to a
subtle sign diﬀerence at the periodic boundary. Here, we examine whether thissign diﬀerence results in a signiﬁcant consequence in entanglement dynamics.
Figure D.1 (a) shows selected plots from Fig. 5.4 (b), which demonstrates time
evolution of the entanglement entropy in the case of g = 0.8 in the hard-coreboson model. Figure D.1 (b) represents the corresponding data in the fermion
62

model under the same condition. In the critical and localized regimes, W > 3–4,
the wave functions are not extended, and the corresponding fermion states are
expected to be immune to the sign diﬀerence. The result of our simulation in
the fermionic case conﬁrms that the entanglement entropy exhibits essentiallythe same behavior in the two models, that is, the anticommutation relation
plays no role in this regime. In the delocalized regime (W < 3–4), the wave
functions are extended over the entire system, so that the resulting fermion state
may be inﬂuenced by the statistics. In this regime, the entanglement entropy
exhibits an oscillatory behavior in the intermediate time scale: a simple damped
oscillation in the case of hard-core boson model. In the fermionic case, one canstill recognize a similar oscillatory pattern; however, it does not take any longer
than a simple damped oscillation and also survives longer than in the hard-coreboson case. Thus, the anticommutation relation inﬂuences the behavior of theentanglement entropy in the delocalized regime at a quantitative level; however,the qualitative statements regarding its behavior in the hard-core boson model
can still be applide to the fermionic case. The behavior of the entanglement
entropy is essentially unchanged in the critical and localized regimes.

63Appendix E
The entanglement entropy in the ETH
and MBL phases (Hermitian case)Let us comment on the relation between the entanglement entropy and the localization length. The multifractal dimension D2 expressed in Eq. (5.4) measuresthe extent to which the eigenstates are localized in Hilbert space; this is related
to the extent to which the wave functions are localized in real space. Thus, inthe regime of strong disorder in the Hermitian case: g = 0, D2 behaves asymptotically as D2 ∼ log(ξ) [67], where ξ represents the localization length in real
space. This implies that the true Fock space-localization D2 = 0 is generally
never achieved [43, 90, 91], since ξ is generally ﬁnite in the localized phase,
except in the limit W → ∞.
Fock-space localization eﬀectively restricts the available Hilbert space, thus
aﬀecting the maximal value of the entanglement entropy; cf. in the free case, it
expressed as Eq. (5.2). In a more generic case with a ﬁnite D2 , it becomes [92]
Stot ∼ DentL
log(2),
2

where Dent is a quantity related to D2 :(1,
D2 ≥ 1/2
Dent =
2D2 , D2 < 1/2.

(E.1)

(E.2)

In the ETH side of D2 > 1/2, the reduction of the multifractal dimension D2
due to a ﬁnite W does not lead to reduction of the entanglement entropy Stot ,while in the MBL side of D2 < 1/2, Stot decreases linearly with a decrease in
D2 . Stot does not exhibit a ﬁnite jump at the ETH–MBL transition point of
D2 = 1/2 and gradually crosses over from the ETH side to the MBL side. In
the time evolution of Stot (t) plotted in Fig. 5.1 (a), its saturated value in thelong time scale remains the same in the delocalized phase of W ≤ 2, while in
the localized phase of W > 2, it decreases continuously from this value with anincrease in W .

64

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