Anderson localization induced by complex potential
R. Wang K. L. Zhang Z. Song




songtc@nankai.edu.cn
School of Physics, Nankai University, Tianjin 300071, China


Uncorrelated disorder potential in one-dimensional lattice definitely
induces Anderson localization, while quasiperiodic potential can lead to
both localized and extended phases, depending on the potential strength. We
investigate the Anderson localization in one-dimensional lattice with
non-Hermitian complex disorder and quasiperiodic potential. We present a
non-Hermitian SSH chain to demonstrate that a nonzero imaginary
disorder on-site potential can induce the standard Anderson localization. We
also show that the non-Hermitian Aubry-Andre (AA) model exhibits a
transition in parameter space, which separates the localization and
delocalization phases and is determined by the self-duality of the model. It
indicates that a pure imaginary quasiperiodic potential takes the same role
as a real quasiperiodic potential in the transition point between
localization and delocalization. Remarkably, a system with complex
quasiperiodic potential exhibits interference-like pattern on the transition
points.


Anderson localization induced by complex potential
    R. Wang, K. L. Zhang, and Z. Song
    December 30, 2023
==================================================







§ INTRODUCTION


The localization phase in quantum systems, which is originally rooted in
condensed matter <cit.>, has recently attracted a lot of theoretical and
experimental interest in a variety of fields, including light waves in
optical random media <cit.>, matter waves in optical potential
<cit.>, sound waves in elastic media <cit.>, and quantum
chaotic systems<cit.>, since the localization of quantum particles could
prevent the transport necessary for equilibration in isolated systems.
Anderson localization<cit.> predicts that single-particle wave functions
become localized in the presence of some uncorrelated disorder, leading to a
metal-insulator transition caused by the quantum interference in the
scattering processes of a particle with random impurities and defects. A
conventional Anderson localization is not controllable in one and
two-dimensional systems. Nevertheless, localization does not require
disorder and fortunately, the Aubry-Andre model <cit.>, which has
quasiperiodic potential, exhibits a transition between a localized and
extended phases. In practice, quasiperiodic potential arises naturally in
optical experiments using lasers with incommensurate wave vectors.
Accordingly, many experiments in such systems have now observed
single-particle localization <cit.>. While any optical
system that includes gain or loss is non-Hermitian by nature, it has been
shown that the presence of the imaginary potential causes many surprising
effects <cit.>.

Motivated by the recent development of non-Hermitian quantum mechanics <cit.> both in theoretical and experimental aspects <cit.>
, we investigate the localization transitions in non-Hermitian regime.
Several pioneer works have been devoted to investigate the tight-binding
system with non-Hermition 𝒫𝒯-symmetric quasiperiodic potential
<cit.>. The extension from real potential to a complex one raises
the question of whether the real and imaginary part of complex potential has
correlated effect on the Anderson localization. Especially, since the
quasiperiodic potential does possess an intrinsic phase, the phase
difference of real and imaginary quasiperiodic potential may influence the
Anderson localization transition. This is the main purpose of the present
work. We investigate the Anderson localization in one-dimensional lattice
with non-Hermitian complex disorder and quasiperiodic potential. We present
a non-Hermitian SSH chain to demonstrate that a nonzero imaginary
disorder on-site potential can induce the standard Anderson localization. We
also show that the non-Hermitian Aubry-Andre (AA) model exhibits a
transition in parameter space, which separates the localization and
delocalization phases and is determined by the self-duality of the model. It
indicates that a pure imaginary quasiperiodic potential takes the same role
as a real quasiperiodic potential in the transition point between
localization and delocalization. Remarkably, a system with complex
quasiperiodic potential exhibits interference-like pattern on the transition
point, i.e., the phase difference between real and imaginary quasiperiodic
potential determines the boundary of transition. Our approach opens a new
way to investigate the interplay of localization and gain/loss in
non-Hermitian system.

This paper is organized as follows. In Sec. <ref>, we present a non-Hermitian SSH chain with
disorder staggered balanced gain and loss. We map this model to an
equivalent Hermitian one and show the existence of Anderson localization.In Sec. <ref>, we show exactly that a
non-Hermitian AA Hamiltonian with imaginary quasi-periodic
potential possesses a self-duality and manifests a localization transition
at the self-dual point. In Sec. <ref>, we demonstrate that the real and imaginary quasiperiodic
potential has interference effect on the Anderson localization transition.
Finally, we give a summary in Section <ref>.



§ LOCALIZATION IN NON-HERMITIAN SSH CHAIN




We consider a non-Hermitian SSH chain with disorder staggered
balanced gain and loss. The simplest tight-binding model with these features
is

    H_SSH   =   (1+δ )∑_j=1^Na_j^†b_j+(1-δ
    )∑_j=1^N-1b_j^†a_j+1
          +H.c.+i∑_j=1^Nγ _j(a_j^†a_j-b_j^†b_j),

where δ and γ _j, are the distortion factor with unit
tunneling constant and the alternating imaginary potential magnitude at
dimmer j, respectively. In this work, we focus on the weak limit of
imaginary potential |γ _j|≪ 1. Here a_l^† and b_l^† are the creation operator of the particle
at the lth site in A and B sub-lattices. The particle can be fermion
or boson, depending on their own commutation relations. (A sketch of the
lattice has been shown in Fig. <ref>.) In the following, we
will show that a nonzero disorder staggered balanced gain and loss can lead
to standard Anderson localization.



We start with the non-Hermitian 𝒫𝒯-symmetric dimmer at position j, with the Hamiltonian

    h_j=(1+δ )(a_j^†b_j+H.c.)+iγ _j(a_j^†a_j-b_j^†b_j).

It can be diagonalized as the form

    h_j=√((1+δ )^2-γ _j^2)(α_jα
    _j-β_jβ _j),

by introducing particle operators <cit.>

    {[     α _j=a_j+e^-iφ _jb_j/1+ie^-iφ _j,        β _j=a_j-e^iφ _jb_j/1+ie^iφ _j; α_j=a_j^†+e^-iφ _jb_j^†/
    1-ie^-iφ _j,     β_j=a_j^†-e^iφ _jb_j^†/1-ie^iφ _j ]
    . ,

with

    tanφ _j=γ _j/√((1+δ )^2-γ _j^2).

Based on the identity

    (1+δ )(a_j^†b_j+b_j^†a_j)+iγ
    _j(a_j^†a_j-b_j^†b_j)  
       =   √((1+δ )^2-γ _j^2)(α_jα _j-
    β_jβ _j),

and the condition

    1+δ≫ 1-δ ,|γ _j|≪ 1,

which lead to cosφ _j≈ e^iφ _j≈ 1, one can
neglect the transition terms between sites with opposite potential, and get
the approximate expression

    H_SSH   ≈   1/2(1-δ )∑_j=1^N-1(
    α_jα _j+1-β_jβ _j+1)
    
          +H.c.+∑_j=1^NV_j(α_jα _j-
    β_jβ _j).

The original system reduces to two independent uniform chains with opposite
real on-site random potential V_j=√((1+δ )^2-γ _j^2)
. Hamiltonian H_SSH is diagonalizable since operators (
α _j,β _j,α_j,β_j)
satisfy the canonical commutation relations. It indicates that H_
SSH can be regarded as a Hermitian model in the context of biorthogonal
inner product. On the other hand, we have shown that if a state

    |ψ⟩ =∑_j(A_jα _j^†|
    0⟩ +B_jβ _j^†| 0⟩ ),

 with A_jB_j=0 for all j, i.e., it has only single-chain component,
{α _j^†| 0⟩} or {β _j^†| 0⟩}, the dynamics of |ψ⟩ is the same as that in a Hermitian chain,
exhibiting the standard Anderson localization. Furthermore, the mapping
between the Hermitian and non-Hermitian Hamiltonian matrices (i.e., the
similar matrix) is a local transformation, which cannot result in
localization transition. In conclusion, we provide an example to demonstrate
that a nonzero imaginary disorder on-site potential can induce the Anderson
localization.

Numerical simulation is performed to demonstrate our conclusion. For
simplicity, we consider a uniform chain with disorder potential. The
Hamiltonian takes the form

    H_And=∑_la_l^†a_l+1+H.c.
    +∑_lμ _la_l^†a_l,

where the potential is simulated by a sequence of random complex variables.According to the theory of Anderson localization, all the eigenstates are
localized for any nonzero real random variables. On the other hand, our
analysis indicates that imaginary random variables may also induce localized
states. We perform the simulation for two cases, (i) μ _l= ran
(-b,b)e^iϕ, with ϕ =0, π /4, and π /2, (ii) μ
_l=μ _0e^iran(-π ,π). Here ran(-b,b) denotes a uniform random number within (-b,b). In
case (i) with ϕ =0, it corresponds to real disorder potential, which is
in the framework of Anderson localization. In case (i) with ϕ =π /2,
it corresponds to imaginary disorder potential, which is the situation of
our prediction. In case (i) with ϕ =π /4, the real and imaginary
disorder potential has the same amplitude. In case (ii), the real and
imaginary part of potential is out of phase but keep an identical norm.

To characterize the localization natures, we employ the inverse
participation ratio (IPR), as a criterion to distinguish the
extended states from the localized ones, which is defined as

    IPR^(n)=∑_l|⟨ψ
    _n| l⟩| ^4/( ∑_l|⟨ψ _n| l⟩| ^2) ^2.

Meanwhile, the average IPR (AIPR) for all energy levels is
defined as the form of AIPR=1/N∑_n=1^NIPR^(n). For spatially extended states, it approaches to zero, whereas it
is finite for localized states. For a non-Hermitian system we introduce the
quantity, dressed energy

    ε̃_n=sgn[real(ε
    _n)]|ε _n|

to specifies the energy in non-Hermitian regime. We plot the result in Fig. 
<ref>, which show that the complex random potential of several types
can induce Anderson localization as our prediction.





§ SELF-DUALITY UNDER IMAGINARY POTENTIAL




Consider a uniform chain with imaginary quasiperiodic potential, which has
the form

    H_Im=∑_l=1^Na_l^†a_l+1+H.c.
    +i2γ∑_l=1^Ncos(ql)a_l^†a_l,

which is an extension from a standard AA model to a non-Hermitian
version by extending γ to iγ, and q=2πβ, β
determines the degree of the quasiperiodicity. The simplicity of H_
Im allows for exact theoretical statements in certain cases. For
example, the 1D disordered Anderson model H_Andwith real potential allows for only localized eigenstates at all energies
independent of how weak the disorder may be. The standard AA model
with the 1D incommensurate potential has either all eigenstates
extended or localized depending on the strength of the potential. It is due
to the self-duality of the Hamiltonian, which concludes that an AA
Hamiltonian with irrational q manifests a localization transition at the
self-dual point

    |γ| =1,

i.e., all eigenstates are extended (localized) when |γ| <1 (|γ| >1). In the following, it
is readily to show that the same thing happens exactly for a non-Hermitian
AA model.

Taking the Fourier transformation

    a_l=1/√(N)∑_n(-1)^le^iqnla_n,

we can obtain

    -iH_Im=γ∑_n(a_n^†a_n+1+H.c.
    )+2i∑_ncos (qn)a_n^†a_n.

Obviously, H_Im and -iH_Im has the
identical complete set of eigenstates and have self-duality with the
self-dual point γ =1. It indicates that the imaginary quasiperiodic
potential takes the same role leading to localization transition. Numerical
simulations are given in Fig. <ref> and Fig. <ref> to demonstrate
our conclusion about the self-duality under the imaginary potential model.



§ INTERFERENCE EFFECT ON LOCALIZATION TRANSITION




We have seen that a real or imaginary quasiperiodic potential has the same
effect on the localization when they show up individually. A natural
question is what happens when the quasiperiodic potential is complex. There
are many variety of complex potential. In this work, we consider the complex
potential with identical frequency

    H_Comp   =   ∑_j=1^N(a_j^†a_j+1+H.c.
    )+2V∑_j=1^Ncos(2πβ j)a_j^†a_j
          +i2γ∑_j=1^Ncos(2πβ j+Φ )a_j^†a_j,

but different phase difference Φ. In the following, we aim at the
effect of Φ on the localization transition point. We consider the
form of Φ in two simple ways. (i) We take the Φ in the range of
[0,2π ]and it isindependent of the values of V and γ.
(ii) Phase difference Φ is taken as a function of V and γ, Φ =mtan ^-1(γ /V), where m is an integer. We note that the
point along V and γ axes of γ -V plane can be exactly
solved as the transition point, i.e., when |γ |<1 (|γ |>1) at
γ axis, all the eigenstates are extended (or localized), the same
as γ axis but for the point |V|<1 (|V|>1) at V axis. The
numerical simulation is performed in the following procedure. For a given
point (V,γ ), we compute the eigenstates by exact diagonalization.
Then the IPR of all the levels and the average IPR 
(AIPR) for all energy levels can be obtained. Meanwhile,
we confirm that the maximal and minimal IPR has not much deviations
from the average, then we can use AIPR to
describe the characters of extended (or localized) eigenstates. In
Figs. <ref> and <ref>, we find that the localization transition
boundary is strongly depends on both the magnitude and the phase of the
complex potential. The AIPR profiles exhibit interference-like
patterns in the parameter space. We also notice that patterns in both
figures are asymmetrically under the switch of V and γ. It is
probably due to the Hermiticity of the hopping term, which breaks the
balance between the real and imaginary potential.





§ CONCLUSION




In summary, we have investigated an extension of localization to
non-Hermitian system. We have obtained three main results. (i) Based on the
equivalence between a disorder non-Hermitian SSH model and a
Hermitian uniform chain with real random on-site potential, we have shown
exactly that the uncorrelated disorder imaginary on-site potential can
induce Anderson localization. (ii) We presented a non-Hermitian AA
model, by replacing a real quasiperiodic potential by an imaginary one. It
has been shown to have self-duality and thus supports the transition from
extension to localization. (iii) Furthermore, we also studied the
interactive effect of real and imaginary quasiperiodic potential on the
localization transition. Numerical simulations indicates that the phase
difference between real and imaginary quasiperiodic potential determines the
boundary of transition, exhibiting an interference-like pattern. Our
approach opens a new way to investigate the interplay of localization and
gain/loss in non-Hermitian system.



§ APPENDIX




In this Appendix, we present a detailed derivation for the diagonalization
of a non-Hermitian SSH chain with Hamiltonian

    H_SSH=(1-δ )∑_j=1^N-1b_j^†a_j+1+H.c.
    +∑_j=1^Nh_j.

Here each h_j describes a non-Hermitian 𝒫𝒯-symmetric dimer

    h_j=(1+δ )(a_j^†b_j+H.c.)+iγ _j(a_j^†a_j-b_j^†b_j),

at position j. It can be diagonalized as the form

    h_j=√((1+δ )^2-γ _j^2)(α_jα
    _j-β_jβ _j),

by introducing canonical particle operators in the aid of biorthogonal inner
product. The particle operators

    {[     α _j=a_j+e^-iφ _jb_j/1+ie^-iφ _j,        β _j=a_j-e^iφ _jb_j/1+ie^iφ _j; α_j=a_j^†+e^-iφ _jb_j^†/
    1-ie^-iφ _j,     β_j=a_j^†-e^iφ _jb_j^†/1-ie^iφ _j ]
    . ,

with

    tanφ _j=γ _j/√((1+δ )^2-γ _j^2),

or inversely

    {[    a_j=( e^iφ _j+i) α _j+(
    e^-iφ _j+i) β _j/2cosφ _j;  b_j=( 1+ie^-iφ _j) α _j-(
    1+ie^iφ _j) β _j/2cosφ _j;   a_j^†=( e^iφ _j-i) α
    _j+( e^-iφ _j-i) β_j/2cosφ
    _j; b_j^†=( 1-ie^-iφ _j) α
    _j-( 1-ie^iφ _j) β_j/2cosφ
    _j ]
    . ,

which satisfy the canonical commutation relations

    [ α _j,α_j^'] _±   =   [
    β _j,β_j^'] _±=δ
    _jj^', 
    [ α _j,α _j^'] _±   =   [ β
    _j,β _j^'] _±=[ α_j,
    α_j^'] _±=[ β_j,
    β_j^'] _±=0,

for the interchain particles,

    [ α _j,β_j^'] _±=[
    α_j,β_j^'] _±=[
    α _j,β _j^'] _±=[ α
    _j,β _j^'] _±=0,

for the intrachain particles. Straightforward derivations show that

    (1+δ )(a_j^†b_j+b_j^†a_j)+iγ
    _j(a_j^†a_j-b_j^†b_j)  
       =   √((1+δ )^2-γ _j^2)(α_jα _j-
    β_jβ _j),

and

    b_j^†a_j+1+a_j+1^†b_j
        =[( e^iφ _j+1+i) ( 1-ie^-iφ _j)
    α_jα _j+1
        +( e^-iφ _j+1+i) ( 1-ie^-iφ _j)
    α_jβ _j+1
        -( e^iφ _j+1+i) ( 1-ie^iφ _j)
    β_jα _j+1
        -( e^-iφ _j+1+i) ( 1-ie^iφ _j)
    β_jβ _j+1]×
        (4cosφ _jcosφ _j+1)^-1+H.c..

Under the condition

    1+δ≫ 1-δ ,|γ _j|≪ 1,

we have

    cosφ _j   =   √((1+δ )^2-γ _j^2)/1+δ
    ≈ 1-γ _j^2/2(1+δ )^2≈ 1, 
    
    e^iφ _j   =   √((1+δ )^2-γ _j^2)+iγ
    _j/1+δ≈ 1+iγ _j/1+δ≈ 1,

and then

    b_j^†a_j+1+a_j+1^†b_j
       ≈   1/2(α_jα _j+1+α
    _jβ _j+1-2β_jα _j+1-β
    _jβ _j+1)  
          +1/2(α _j+1^†α_j^†-α
    _j+1^†β_j^†+β _j+1^†
    α_j^†-β _j+1^†β_j^†) 
       =   1/2(α_jα _j+1+α
    _jβ _j+1-β_jα _j+1-β
    _jβ _j+1)+H.c..

Neglecting the transition terms between sites with opposite potential, we
have

    b_j^†a_j+1+a_j+1^†b_j≈1/2α_jα _j+1-1/2β_jβ _j+1+H.c.
    .

Then we have the approximate expression

    H_SSH   ≈   1/2(1-δ )∑_j=1^N-1(
    α_jα _j+1-β_jβ _j+1)
    
          +H.c.+∑_j=1^NV_j(α_jα _j-
    β_jβ _j),

where real potential V_j=√((1+δ )^2-γ _j^2). On the
other hand, we have

    [ α _j,α _j^'^†] _±   =   
    δ _jj^'/1+sinφ _j,[ β _j,β
    _j^'^†] _±=δ _jj^'/1-sinφ _j, 
    [ α _j,α _j^'] _±   =   [ β
    _j,β _j^'] _±=[ α _j^†,α
    _j^'^†] _±=[ β _j^†,β
    _j^'^†] _±=0,

for the interchain particles,

    [ α _j,β _j^'^†] _±   =   itanφ _jδ _jj^',[ α _j^†,β
    _j^'] _±=-itanφ _jδ _jj^',
    
    [ α _j,β _j^'] _±   =   [ α
    _j^†,β _j^'^†] _±=0,

for the intrachain particles. It indicates that if a state satisfy therelation

    |ψ⟩ =∑_j(A_jα _j^†|
    0⟩ +B_jβ _j^†| 0⟩ ),

 with A_jB_j=0 for all j, i.e., it has only single-chain component,
{α _j^†| 0⟩} or {β _j^†| 0⟩}, the dynamics of |ψ⟩ is the same as that in a Hermitian chain.

We acknowledge the support of NSFC (Grants No. 11874225).

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