Sheng-Lian Jiang(蔣盛蓮), Yanxia Liu(劉彥霞), and Li-Jun Lang(郎利君),3,?

1School of Physics,South China Normal University,Guangzhou 510006,China

2School of Physics and Astronomy,Yunnan University,Kunming 650091,China

3Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,School of Physics,South China Normal University,Guangzhou 510006,China

Keywords: non-Hermitian mosaic model,mosaic-to-non-mosaic mapping,mobility edge,Lyapunov exponent

1.Introduction

The effective non-Hermitian Hamiltonians can be used to describe open quantum systems.[1]Non-Hermiticity can cause intriguing properties, such as exceptional points, non-Hermitian skin effects, non-Hermitian topological phenomena,and the non-Bloch band theory,[2–16]which have no counterparts in Hermitian systems.Meanwhile, non-Hermiticity in disorder or quasiperiodic systems also brings up new phenomena, for example, the mobility edges can emerge separating localized and extended states in the complex plane of spectrum in the Hatano–Nelson model, a prototypical onedimensional(1D)non-Hermitian model characterized by nonreciprocal hopping with random on-site disorder.[17,18]It is well known that an arbitrarily small strength of random disorder leads to the localization of all eigenstates in one-and twodimensional Hermitian models,[19–21]and the effects of disorders in corresponding non-Hermitian models have also been studied recently.[22,23]Besides the non-Hermitian models with random disorders, the non-Hermitian quasiperiodic systems have also sparked a great deal of interest,[24–40]especially the non-Hermitian variants[27–29]of the celebrated Aubry–André model.[41]

Recently,Anderson localization and mobility edges have been regaining much attention in quasiperiodic systems,[42–47]because the critical points of localization in these models can be analytically solved with the aid of self-duality[41]or further Avila’s global theory.[48]Another important reason is that quasicrystals can be experimentally realized for both Hermitian and non-Hermitian versions, which promotes the studies of localization physics and mobility edges.[49–52]The mobility edge is one of the central concepts in condensed matter physics and can be obtained in many Hermitian and non-Hermitian disorder or quasiperiodic systems.[53–56]The search for new typical models that can be exactly solved with mobility edges is still ongoing case by case.

In this work,without arduously dealing with models case by case,we establish a general mapping of 1D non-Hermitian mosaic models to their non-mosaic counterparts of which the critical points of localization or even the Lyapunov exponents(LEs)of localized states have been analytically solved.By the mapping,we can obtain the mobility edges as well as the LEs of the mosaic models.For the purpose of demonstration, we take two examples of non-Hermitian non-mosaic models to explore the localization properties of their mosaic counterparts.One is an Aubry–André-like(AA-like)model with nonreciprocal hopping and complex quasiperiodic potentials.[39]We reconstruct the mobility edges previously obtained by Avila’s global theory,[48]and further derive the LEs.The second example involves the Ganeshan–Pixley–Das Sarma (GPD)model[57]with nonreciprocal hopping.Using the general mapping,we successfully obtain the analytical expressions for the mobility edges and the LEs of its mosaic counterpart, which we validate through numerical calculations.

This general mapping elucidates the mechanism behind the emergence of mobility edges in non-Hermitian mosaic models, as well as the asymmetric localization induced by nonreciprocal hopping, which can be characterized by two LEs.Furthermore,this work extends the mapping in one and two dimensions[58,59]to the non-Hermitian regime.

The rest of the paper is organized as follows.In Section 2,we introduce a class of 1D non-Hermitian mosaic models.Then, we analytically derive a general mapping from this model to its non-mosaic counterpart in Section 3.Subsequently,in Section 4,we apply this mapping to two specific non-Hermitian mosaic models and obtain the mobility edges and the LEs.Finally,we summarize the results and provide a conclusion with some discussions in Section 5.

2.The non-Hermitian mosaic models

We consider a general class of 1D non-Hermitian mosaic models,which can be described by the Hamiltonian

whereJR,L∈C denote the complex strengths of the nonreciprocal hopping between nearest-neighbor sites with the subscripts representing the hopping directions, andVj ∈C is the complex on-site potential at sitejof the form

withλ ∈C being the complex strength of the mosaic potential,κ ≥1 being an integer representing the period of nonmosaic sitesj=κm, andΔj ∈C being a model-dependent function that can induce the localization (e.g., randomly distributed functions, quasi-periodic functions, linear functions,etc.).Every successiveκsites form a so-called quasicell labeled bym, which also denotes themth non-mosaic site that has a nonzero on-site potential.For convenience, in the following we takeNquasicells, i.e.,m=1,...,N, and thus the system sizeL=κN.

By taking|Ψ〉=∑j ψj|j〉,the static Schr¨odinger equation ?H|Ψ〉=E|Ψ〉can be written in terms of the amplitudeψjas

Note that due to the non-Hermiticity of the Hamiltonian ?H,the eigenenergyEcan generally be a complex value.

3.Non-Hermitian mosaic-to-non-mosaic mapping

Since there are(κ-1)sites that are free of on-site potentials between two nearest-neighbor non-mosaic sites,a reasonable inference is that the localization, if occurs, should exist only at the non-mosaic sites.Therefore, one may be curious about the effective coupling between the non-mosaic sites.To this end,we can resort to the transfer matrix.

From the second line of Eq.(3),the system of equations for the mosaic sites between two nearest non-mosaic sites,say themth and the (m+1)th non-mosaic sites, can be given explicitly as follows:

With the aid of the forward transfer matrixF, which is defined as

Eq.(4)can be written as

and thus,the amplitude at mosaic siteψκm+1can be expressed in terms of the amplitudes,ψκmandψκ(m+1), of two nonmosaic sites,yielding

with

Likewise, the relation can also be obtained by considering the backward transfer matrices connecting themth with the(m-1)th non-mosaic sites,and thus the amplitude at mosaic siteψκm-1can be expressed as

The details of derivation for the coefficients can be referred to in Appendix A.

By substituting Eqs.(7) and (9) into the first line of Eq.(3), one can have a system of equations only involving non-mosaic sites,yielding

with

If one regardsmas the effective site index andκjust as a parameter, Eq.(10)is nothing but effectively represents a static Schr¨odinger equation for a non-mosaic tight-binding model withJκL,Rbeing the nonreciprocal hopping,λAκΔκmbeing the on-site potential, andBκbeing the eigenenergy.For convenience,in the following,we will refer to the model(10)as the scaled model,in contrast to the original model(3).

Comparing the scaled model with the non-mosaic model(3)withκ=1,we have the following general mapping:

where the introduction offκreflects the relation between the potential functionsΔκmandΔmin the two models.

This general mapping(12)reveals that if a wave functionψjwith an eigenenergyEis localized at sitejunder the potential functionΔmin the non-mosaic model,then the localization also occurs at the effective sitemunder the potential functionΔκmin the scaled model,described by the wave functionψκmwith the eigenenergyBκ.Moreover, it implies that the wave function is localized at the non-mosaic siteκmin the mosaic model with finiteκ.Therefore, given the critical point of localization determined by a functionf(JL,JR,λ,E)=0 in the non-mosaic model,the critical point in the mosaic model can be determined by

through the general mapping(12).Generally,the critical point of the mosaic model depends onE,which means that there exist mobility edges,even if there are no mobility edges for the non-mosaic model.This is a result ofAκbeing a function ofE.In principle,if one knows the critical point of a non-Hermitian non-mosaic model,the critical point of the corresponding mosaic model can also be obtained using Eq.(13).

Furthermore, LEγ(JL,JR,λ,E)＞0, which characterizes the decaying behavior of a localized state, has also been established for certain non-mosaic models with random or quasiperiodic disorders.In these models, the localized state at sitej0takes the form|ψj|∝e-γ|j-j0|.Using this expression and the general mapping, one can also obtain the LEγ(λAκ fκ,Bκ)＞0 for the scaled model (10) with respect to the “sites” labeled bym.This results in a localized state given by

at the non-mosaic siteκm0,whereγ/κrepresents the inverse of the localization length or the LE[60]of the corresponding mosaic models with respect to the non-mosaic sitesκm.It is worth noting that the critical point can also be obtained from the condition,λAκ fκ,Bκ)=0,which is equivalent to Eq.(13).

In principle,the general mapping(12)is applicable to any non-Hermitian mosaic models.In the following,we will focus on a more specific case where the hopping can be parameterized asJL=te-gandJR=teg(t,g ∈R).Consequently,the scaled model(10)becomes

as a non-Hermitian generalization of the Hermitian case discussed in Ref.[58].Here, we replaceAκandBκin Eq.(10)with the following quantities:

whereaκis dimensionless,εκhas the dimension of energy,and we definea0=0 for the correct expression ofε1.Specifically,

which will be used in the following applications.In terms of the parameters{t,g}∈R,the general mapping(12)becomes

Apparently, wheng=0, this mapping can be reduced to the Hermitian cases studied in Ref.[58].

To demonstrate the validity, we will apply the mapping(18)to two specific non-Hermitian mosaic models in the following.For convenience, we will sett=1 as the energy unit and focus on the cases withg ＞0, which corresponds to a right-biased hopping.

4.Applications

4.1.Non-Hermitian Aubry–André-like mosaic model

As the first application, we take a mosaic version of an AA-like model with nonreciprocal hopping and a complex potential, dubbed non-Hermitian AA-like mosaic model for the convenience of discussion in the following,which is described by Hamiltonian(1)with the potential function in Eq.(2)being

whereβis an arbitrary irrational number and{θ,φ} ∈R represent a complex phase shift.The corresponding non-Hermitian AA-like non-mosaic model undergoes an Anderson localization at

The detailed proof can be referred to in Appendix B.Notably,this critical point of localization is independent of the eigenenergyE;in other words,there are no mobility edges in the non-Hermitian AA-like non-mosaic model.

Based on the critical point Eq.(20), we now apply the mapping(18)to study its mosaic counterpart.Since the difference between two potential functionsΔκm=2cos(2πβκm+θ+iφ)andΔm=2cos(2πβm+θ+iφ)is only between the irrational numbersκβandβ, thefκin the mapping can be taken as 1,because the critical points of Anderson localization for quasiperiodic models are independent of the values of irrational numbers,[61,62]although the eigenstates and eigenvalues under these two potentials are irrelevant in details.Thus, by making the replacementλ →λaκandg →κgin Eq.(20),one can find that the mobility edges,

emerge in the non-Hermitian AA-like mosaic model.This result is identical to that of Ref.[39]obtained by Avila’s global theory,[48]verifying the correctness of the mapping.

Furthermore, one can also obtain the LEs in the non-Hermitian AA-like mosaic model by the mapping.From Ref.[27],we know that two LEsγ±g ＞0 can be used to characterize an asymmetrically localized state in a non-Hermitian model with nonreciprocal hopping,and the smaller oneγ-gdetermines the critical point of Anderson localization,where

is the LE for the corresponding symmetrically localized state in the non-Hermitian AA-like non-mosaic model with reciprocal hopping.

Therefore, with the same replacement, one can get the two LEs of the scaled model(10)as

which depend on the eigenenergyE,leading to an asymmetrically localized state

with the peak being located at the non-mosaic siteκm0.The right-biased asymmetry[i.e.,γ(-)＜γ(+)]of the localized state with respect to the center siteκm0results from the right-biased hopping(i.e.,g ＞0)we set in advance.This can be partially checked by settingγ(-)κ(E)=0, which just gives rise to the critical point in Eq.(21).

For further verification,we compare the numerical results with the analytical ones for the cases ofκ=2 and 3 in Fig.1,where,from Eqs.(21)and(23),the critical points and the LEs are specified as

can be used to characterize the localization of a stateψjmore clearly for a finite system than the commonly used inverse participation number(IPR)[21]

To demonstrate the transition of Anderson localization,of which the critical points are the same under periodic boundary conditions(PBCs)and open boundary conditions(OBCs)due to the blindness about boundary conditions for a bulklocalized state,[27]we use PBCs for numerical calculations in Fig.1 to avoid non-Hermitian skin effect under OBCs,[5]where all bulk states are aggregated together to one boundary(here the right boundary for our settings), leading to similar values of fractal dimension as bulk-localized states.Meanwhile, according to the bulk–bulk correspondence of a non-Hermitian model with nonreciprocal hopping whenφ=0,i.e.,the critical point of Anderson localization under OBCs is also a cut between complex and real spectra under PBCs[27]for Figs.1(a)and 1(c),one can assume the reality ofEand abandon the complex-Esolutions during the numerical calculation of critical points.Due to Im[(E(λ)]=0, the mobility edgesE(λ)are just functions of curves, not surfaces in the Re(E)–Im(E)–λspace.The maximal number 2(κ-1) of mobility edges, as shown in Figs.1(a) and 1(b), can also be expected from Eq.(21), where the highest power inEcan be (κ-1).The two LEs are demonstrated in Fig.1(d)as two slopes from the peak site to the two shoulders.Due to the existence of side peaks on two shoulders beside the main peak, the numerical result can shift occasionally relative to the analytical result for some regions,but the preserved slope still reflects the correctness of the LEs derived from Eq.(25).

Fig.1.The fractal dimension Γ as a function of the real part of the eigenenergy Re(E) and a real potential strength λ, numerically calculated for (a) κ =2 and (b) κ =3 under PBCs with other parameters g=0.2,φ =0,L=F12=144,and β =F11/F12=89/144.(c)The existence or not of the imaginary part of energy with the same parameters as in (a), where (non)zeros of Im(E) with accuracy 10-6 are colored in blue (yellow).The red solid lines in (a)–(c) are the mobility edges calculated analytically by Eq.(25).(d)Comparison of a localized state marked by the red triangle(E=2.5881,λ =2.2)in(b)with the corresponding analytical form Eq.(24), showing that the two LEs can well characterize the asymmetric localization in the non-Hermitian AA-like mosaic model.The dotted line indicates the center site of localization.

4.2.Nonreciprocal Ganeshan–Pixley–Das Sarma mosaic model

The mobility edges in the non-Hermitian AA-like mosaic model emerge from a constant critical point of the non-mosaic counterpart.As another application, we consider the mosaic version of the GPD model(with real potential strengthλ)[57]that incorporates nonreciprocal hopping, dubbed the nonreciprocal GPD mosaic model.Its non-mosaic counterpart already exhibits mobility edges.The Hamiltonian is described by Eq.(1)with the following real potential function:

with the same definition ofβas the non-Hermitian AA-like model and the deformation parameterb ∈(-1,1).

For the non-mosaic case with nonreciprocal hopping,i.e.,κ=1,the mobility edges[34]

can be obtained by Avila’s global theory[48]and the similarity transformation under OBCs.[27]Note hereλ(E)keeps real due to the reality ofEin this model.Wheng=0, it reduces to the reciprocal case with mobility edges being

which is first obtained by a generalized duality transformation,[57]and the LE

is analytically obtained in Refs.[34,36] with the aid of the Avila’s global theory.The conditionγ(E)=0 can also generate the critical point Eq.(30).

With the aid of the similarity transformation under OBCs in Ref.[27], we can readily derive the two LEs,γ(E)±g, in the nonreciprocal GPD non-mosaic model(κ=1 andg/=0),and the validity of the mobility edges given by Eq.(29)can be confirmed by the conditionγ(E)-g=0.

In the nonreciprocal GPD mosaic model,the eigenenergyEmust be a real number for a localized state due to the similarity transformation under OBCs to a Hermitian model.[27]Therefore, we can safely apply the mapping (18) in the real field,and the absolute and squared root operations in Eq.(31)retain their definitions in the real field.

Finally, by the mapping (18), we obtain the two LEs in the nonreciprocal GPD mosaic model as follows:

whereaκandεκdefined in Eq.(16) are real numbers due to the reality ofEfor localized states and the system parameters.The mobility edges can be determined by the condition(E)=0, or directly obtained by the mapping from Eq.(29),yielding

Similarly, we consider the cases ofκ= 1 and 2 for demonstration,as shown in Fig.2,where we use the same setting ofβand PBCs as in the aforementioned non-Hermitian AA-like mosaic model.The mobility edges are also curves that separate real spectra and complex ones, and the number of mobility edges can be maximally 2(κ-1),as expected.

Fig.2.The same meanings and settings as in Fig.1,except for(a),(c)κ=1 and(b),(d)κ=2.The mobility edges[red solid lines in(a)–(c)]are calculated analytically by Eq.(33).The localized state marked by the red triangle in(b)is selected as E=3.7328,λ =2.4.The parameter b=0.5 is chosen for all figures.

5.Discussion and conclusion

We establish a general mapping(12)from non-Hermitian mosaic models to their non-mosaic counterparts.By utilizing the analytical expressions of critical points of localization or even the LEs of localized states in the non-mosaic counterparts, one can derive the analytical expressions of mobility edges and the LEs for non-Hermitian mosaic models.As demonstrations, we take the non-Hermitian AA-like mosaic model and the nonreciprocal GPD mosaic model to test the mapping, successfully yielding analytical expressions for the mobility edges and LEs of these models.

The mapping is applicable not only to quasiperiodic models but also to randomly disordered models and non-disorder models that undergo localization transitions.However,in one dimension, critical points or LEs for non-Hermitian models are less established.Therefore, in this paper, we only focus on the applications of the mapping to the mosaic versions of quasiperiodic models.For instance, in Ref.[58] the mobility edges for the mosaic version of the Wannier–Stark model with reciprocal hopping are obtainable due to the availability of the critical point in the non-mosaic counterpart.However,the absence of the analytical expression of LE prevents us from obtaining the critical point or mobility edges of the mosaic models with nonreciprocal hopping using this mapping.

Appendix A:Derivation of the transfer matrix

For the forward transfer matrixFof Eq.(5) in the main text,defined as

we can diagonalizeFas

where

is the diagonal matrix with

and

is a unitary matrix with its inverse being

Then,one can get

where

Therefore,we obtain the Eq.(7)in the main text:

For the backward transfer matrixB,defined as

we have

Likewise,we can get

and thus Eq.(9)in the main text

It is worth noting that the above derivation should assume that

which corresponds the defectiveness of the transfer matricesFandB.However,we can treat these cases as limiting cases from nondefectiveFandB.

Appendix B: Proof for the critical point of the AA-like model in the complex field

For the conventional AA model,[41]which corresponds to the parameter settingJR=JL=tandVj=2λcos(2πβ j+θ)withβbeing an arbitrary irrational number and{t,λ,θ}∈R in Eq.(1), it is well known that the critical point occurs at|λ/t|=1.However,we cannot directly apply this expression in the mapping(18)because the effective on-site potentialλaκin the scaled model(15)is generally complex.Therefore,it is necessary to determine the critical point in the complex field for the conventional AA model with a complex strengthλ ∈C of the on-site quasiperiodic potential.

The eigenvalue equations for the AA-like model with a complex strengthλ ∈C and a complex phase shiftθ+iφ(θ,φ ∈R)of potential can be written as

which can be reexpressed as

by the transfer matrix(we sett=1 as the unit of energy)

The LE can be defined as follows:[48]

where

and the norm ofTn(E,θ,φ)is defined as

withχ1,2being the two nonnegative eigenvalues ofE,θ,φ)Tn(E,θ,φ).

To find the expression of||Tn(E,θ,φ)||, i.e., the expression of the maximum eigenvalue ofwe resort to the largeφlimit,i.e.,φ →+∞,and the transfer matrix(B3)can be approximated as

Thus,the maximum eigenvalue of(E,θ,φ)Tn(E,θ,φ)can be approximated as|λeφ|2n, and from Eq.(B4) one can get the LE in the largeφlimit:

According to Avila’s global theory,[48]γ(E,φ)defined by Eq.(B4) is a convex, piecewise linear function ofφwith the slope?γ(E,φ)/?φof each piece being an integer.The largeφlimit Eq.(B8) suggests that the slope for anyφ ≥0 is restricted to either 0 or 1.The discontinuity of the slope occurs whenEbecomes an eigenenergyEsof the system except forγ(Es,φ)=0,[63]which represents the extended states.This implies that the LE for an eigenenergy of the system can only be expressed as

Considering that the LE is an even function ofφdue to the propertyTn(E,θ,φ)∈SL(2,C),[48]the above formula can be further modified as

Therefore,the critical point of localization is obtained by setting ln|λ|+|φ|=0,yielding

Note that the LE depends only on the magnitude|λ| of the complex potential strength, independent of its phase.This proof is similar to the scenario whenλ ∈R in Ref.[35].

For the case extended to the nonreciprocal hopping(e.g.,JL=te-gandJR=tegwithg ＞0),one can just use the similarity transformation as described in Ref.[27] to obtain the two LEs

for the asymmetric localized states.

Acknowledgements

Project supported by the National Natural Science Foundation of China (Grant No.12204406), the National Key Research and Development Program of China (Grant No.2022YFA1405304), and the Guangdong Provincial Key Laboratory(Grant No.2020B1212060066).

Data availability statement

The data that support the findings of this study are openly available in Science Data Bank at https://doi.org/10.57760/sciencedb.j00113.00133.