Coexistence of Localized and Extended States in Disordered Systems

Yi-Xin Xiao1, Zhao-Qing Zhang1 and C. T. Chan*1

1Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

It is commonly believed that Anderson localized states and extended states do not coexist atthe same energy. Here we propose a simple mechanism to achieve the coexistence of localized andextended states in a band in a class of disordered quasi-1D and quasi-2D systems. The systemsare partially disordered in a way that a band of extended states always exists, not affected by therandomness, whereas the states in all other bands become localized. The extended states can overlapwith the localized states both in energy and in space, achieving the aforementioned coexistence. Wedemonstrate such coexistence in disordered multi-chain and multi-layer systems.

Disorder plays an important role in many fields ofphysics. It is well known that interference between mul-tiply scattered waves can disrupt the transport of elec-trons and classical waves in disordered media and makethe transport diffusive [1–3]. In 1958, Anderson pre-dicted that disorder can even completely stop diffusion[4]. The phenomenon, now called Anderson localization[1–4], has profound influence on our understanding ofwave transport and has been extensively studied in thepast few decades [5–23]. A widely-accepted feature of An-derson localization is that all states are localized in one-dimensional (1D) and two-dimensional (2D) disorderedsystems due to coherent backscattering effects [1, 24],while three-dimensional (3D) random media have a mo-bility edge [25–31] that separates localized states fromextended states [32, 33]. These features were predictedby both the scaling theory [24] and the self-consistenttheory [34] of Anderson localization. The concept of mo-bility edge was first introduced by Mott who argued thatif a localized state and an extended state are infinitelyclose in energy, they will get hybridized by an infinites-imal change of the system and turn into two extendedstates [25, 26]. Mobility edge has also been found inone-dimensional incommensurate quasi-periodic models[35, 36]. Recently, Anderson localization has also beenobserved in 3D non-interacting cold atoms under the in-fluence of laser speckle potentials [31, 37]. The mobilityedge was found to obey a scaling behavior, independentof the speckle geometry [31, 37].

The argument for the spectral separation of localizedand extended states also applies to a similar questionraised by Von Neumann and Wigner in 1929 [38], namelywhether a bound state can exist in the continuum of ex-tended states (BIC). In the past few years, BICs havebeen achieved by several different mechanisms [39–46] invarious ordered systems. The challenging question is tofind systems which support spectral coexistence of An-derson localized states and extended states. The task isvery different from searching BICs as the media are dis-ordered and Anderson localized states can also form acontinuous spectrum [47]. Until now, there is no guidingprinciple for the creation of the coexistence of Andersonlocalized states and extended states in a band. Some at-

tempts were made in the past to achieve this goal. It wasshown that an inhomogeneous trap combined with a dis-ordered potential can lead to the coexistence of localizedand extended states in energy, but spatially they do notoverlap as the inhomogeneous trap segregates two typesof states and suppresses hybridization [48]. The coexis-tence was also found in Ref. [49], however, complicatedengineering of the hopping parameters and on-site ener-gies is needed. Thus, the challenging problem remains tobe resolved whether it is possible to have simple disor-dered systems that support the coexistence of Andersonlocalized states and extended states in a band.

In this work, we find such coexistence in a class ofquasi-1D and -2D systems. Our mechanism of creatingthe coexistence is to partially randomize the systems ina way that the Hilbert space can be partitioned into twosubspaces so that the states in one subspace are unaf-fected by the presence of randomness and hence they canbe extended, whereas wavefunctions in all other bandsbecome localized by randomness. The coexistence of twotypes of states in a band is naturally achieved. We ex-plicitly demonstrate the coexistence in multi-chain andmulti-layer systems.

We first numerically demonstrate such coexistence in amulti-chain system described by a nearest-neighbor tight-binding Hamiltonian. The system consists of 2N + 1(= 51) coupled chains placed in the x direction, as de-picted in Fig. 1 (a). In the absence of disorder, thesystem is periodic in the x direction and all the chainsare identical. There are 2N + 1 sites per unit cell. Thelattice constant is a = 1, and all on-site energies andhopping parameters are taken to be zero and a constantt, respectively. The band structure of the multi-chainsystem, as shown in Fig. 1(b), comprises 2N + 1 bands.The central band marked by the blue curve has exactlythe same dispersion relation E/t = 2 cos (ka) as that ofan isolated chain. Now we partially randomize the sys-tem by adding random on-site energies εi to every sec-ond chain, namely the red sites in Fig. 1(a). Now Nout of 2N + 1 chains are disordered. To study the An-derson localization properties of such a system, we trun-cate each chain in the x direction to M (= 60) sites andnumerically calculate the eigen-functions and the partic-

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ipation ratio (PR) of each eigenstate, which is defined asPR = (

∑n |cn|2)2/(

∑n |cn|4), where cn are the compo-

nents of an eigenstate |ϕ〉 =∑

n cn|n〉 and |n〉 denotesthe atomic orbitals. For the case of uniform randomnessεi/t ∈ [−5, 5], the calculated energy spectrum and theparticipation ratios of all eigenstates are plotted in in-creasing energy in Fig. 1 (c) and (d), respectively. Fig.1(d) shows clearly a stark contrast of two types of states:Type 1 contains M (= 60) states with large values ofPR (marked by blue dots) and Type 2 contains 2N ×M(= 3000) states with much smaller PR (marked by reddots). These two types of states overlap spectrally in theband of extended states. We will show analytically laterthat Type 1 states are extended states having exactlythe same spectrum as that of a single isolated chain, i.e.,the blue curve in Fig. 1(b), unaffected by the random-ness, and Type 2 states are Anderson localized states. Weshow the wavefunctions of two types of states at the samechosen energy in Fig. 1(e) and (f), which clearly demon-strate their spatial overlap. Thus, we have demonstratedthe coexistence of Anderson localized and extended statesin a band in such systems. It should be stressed thatalthough the extended states constitute an insignificantportion, i.e. 1/(2N + 1) , of the whole space of eigen-states in the thermodynamic limit N →∞, neverthelessthey occupy a significant portion of the entire spectrumas shown in Fig. 1(b) and (c).

The surprising coexistence phenomenon in such a sim-ple disordered system is in fact the consequence of a solidrobust mechanism that can be used to realize the coex-istence of localized and extended states in a class of sys-tems in all dimensions. Basically, such systems consistof multiple identical copies of A subsystem with adja-cent copies separated and coupled by an intermediate Bsubsystem. The A and B subsystems can be chains orlayers. In the absence of randomness, B subsystems arealso identical. A simple example has been shown in Fig.1(a), in which A and B subsystems are identical chainsbut denoted by blue and red, respectively. The Hamil-tonian of a system containing N + 1 copies of A and Ncopies of B can be expressed as

Hsys =

HA 0 · · · 0 T †10 HA · · · 0 T †2...

.... . .

......

0 0 · · · HA T †N+1

T1 T2 · · · TN+1 HB

, (1)

HB = diag(HB , HB , · · · , HB), (2)

where HA and HB are, respectively, the Hamiltonians ofA and B subsystems. HB contains N copies of HB . Ifwe assume that all subsystems are finite, each with Msites, the dimensions of HA and HB are M ×M . Thedimensions of Hsys and HB in Eq. (1) become (2N +1)M × (2N + 1)M and NM × NM , respectively. Theblock matrices Ti denote the couplings between A and

(c) (d)

ka

-2

2

E/t(a) (b)

0 1000 2000 3000

-5

0

5

localizedextended

State number0 1000 2000 3000

0

500

1000

1500extendedlocalized

PR

State number(e) (f)

E/t

x

y

FIG. 1. (a) shows a square lattice comprising multiple chainsextending in the x direction. (b) The band structure withone band in blue color characterized by the same dispersionrelation as that of an isolated chain. (c) The energy spec-trum showing the spectral coexistence of extended states andlocalized states in the presence of onsite disorder on alter-nating (red) chains. (d) The participation ratios of all theeigenstates. (e) and (f) show two states with the same eigen-energy E/t = 1.93, one is extended and the other is localized.

B and have dimensions M ×NM . The explicit form ofHsys in Eq. (1) for small values of N and M can befound in Ref. [50]. It will be shown analytically belowthat the coupled system Hsys always contains a set ofeigenvalues which are the same as those of an isolated Asubsystem, similar to the blue band in Fig. 1(b). Thisimplies that the Hamiltonian can be block diagonalizedto contain one isolated block HA, namely

Q−1HsysQ =

(HA 00 H′

), (3)

where Q is a unitary matrix and H′ is the other blockwith dimensions 2NM×2NM . From Eq. (3), it is easy tosee that the Hilbert space of the system is partitioned intotwo subspaces. Now we introduce randomness into everyB subsystem, which randomizes the block H′. Since thesubspace associated with HA is not affected by the dis-order, all eigenstates in this subspace are extended. The

3

Γ K' M K Γ M

0

3

E

-3

(b) (c)

x

y z

x

y

ε1 ε2 ε3 ε4 εM

(a)

FIG. 2. (a) shows a three-chain system, each chain con-tains M sites and on-site disorder is introduced to the middlechain. (b) shows a multilayer system comprising multiple AA-stacked honeycomb-lattice layers. (c) The band structure ofthe multilayer system.

randomness in the other subspace represented byH′ givesrise to localized states. It is shown in ref. [50] that eachextended eigenstate in HA subspace is a direct sum of alleigenvectors of the N + 1 A subsystems. Since the local-ized states in H′ subspace also involve atomic orbitals ofthe A subsystem, the spatial coexistence of two types ofstates naturally occur. The spectral overlap can alwaysbe achieved by adjusting the randomness. We thus havea very general mechanism that assures the coexistence ofAnderson localized states and extended states in a classof quasi-1D and -2D systems.

To be more explicit, we consider a minimal model ofthree coupled chains, each truncated to contain M sites,where random on-site energies εi are introduced in themiddle chain as shown in Fig. 2(a). The Hamiltonian ofthe system becomes

Hm =

HA 0 T †10 HA T †2T1 T2 HB

, (4)

where all the matrix elements are block matrices withdimensions M ×M . HA and HB denote, respectively,the Hamiltonians of the blue and red chains. The inter-chain couplings are represented by T1 = tIM and T2 =tIM , where IM denotes an identity matrix. We assumethat the single-chain Hamiltonian HA satisfies the eigen-equation HAP = PΛ, where Λ = diag(λ1, λ2, · · · , λM ) isa diagonal matrix and λi’s are the eigenvalues of HA, andP = (ϕ1, ϕ2, · · · , ϕM ) comprises M columns of eigenvec-tors. A similarity transformation HS = X−1HmX with

X = diag(P, P, IM ) can be applied so that

HS =

Λ 0 P−1T †10 Λ P−1T †2

T1P T2P HB

. (5)

We note that when inter-chain couplings are absent,namely T1 = T2 = 0, there are two degenerate sets ofeigenvalues λi, i = 1, · · · ,M . The presence of inter-chaincouplings will split the degeneracy. However, there al-ways exists a set of eigenvalues λi unaffected by the cou-plings. This can be seen as follows. For each degeneratepair of eigenvalue λi of HA, the effects due to HB can bedescribed by a simplified version of Eq. (5), i.e., λi 0 tϕ†i

0 λi tϕ†itϕi tϕi HB

. (6)

Since the rank of the coupling block [tϕi, tϕi] is 1, dueto the rank-nullity theorem [51], λi remains to be aneigenvalue of HS , thus also of Hm, independent of HB .Applying the same argument to all λi, we have shownthat a set of eigenvalues of HA remains and, therefore,Hm can always be block diagonalized into the followingform,

HBD =

(HA 00 H′

)=

HA 0 00 HA R†

0 R HB

, (7)

by a unitary transformation HBD = Q−1HmQ. Con-crete examples with explicit block diagonalization canbe found in ref. [50]. The above block diagonalizationmeans a particular separation of Hilbert space into twosubspaces. We call the subspace spanned by the eigen-vectors corresponding to the invariant eigenvalues λi asthe invariant subspace. For each eigenvalue λi, the asso-ciated eigenvector Φi can be expressed in terms of the di-rect sum of three parts corresponding to the three chains,namely

Φi = − 1√2ϕA1,i ⊕

1√2ϕA2,i ⊕ ϕB , (8)

where ϕA1,i and ϕA

2,i are the normalized eigenvectors ofthe two A (blue) chains and both correspond to λi. AndϕB is a zero vector with M components. That is tosay, the anti-symmetric “combination” of the eigenvec-tors ϕA

1,i and ϕA2,i of the two separate blue chains con-

stitutes an eigenvector Φi for the whole coupled-chainsystem, which has odd parity, conforming to the mirrorsymmetry in the y direction, and vanishes at the sitesin the middle chain. The fact that the degeneracy of λi(in the absence of inter-chain couplings) outnumbers thecoupling channels brought by inter-chain couplings guar-antees λi to be an eigenvalue for the coupled-chain sys-tem. Note such a block diagonalization is valid for moregeneral configurations of inter-chain couplings, such as

4

including next-nearest-neighbor hoppings [50]. Since theinvariant subspace does not involve the sites at the mid-dle chain, the disorder (both diagonal and off-diagonal)introduced into the middle chain will not affect the in-variant subspace. Consequently, states in the invariantsubspace will remain extended, whereas all other eigen-states corresponding to theH′ subspace become localizeddue to the presence of disorder. The spectral coexis-tence can always be achieved by adjusting the random-ness. Since both subspaces involve the atomic orbitalsat the A chains, the extended states and localized stateswill surely also coexist spatially [50].

We can generalize the system from three chains to2N + 1 chains with N + 1 identical A chains separatedand coupled by another N B chains. Following the sim-ilar procedure, it can be shown that the Hilbert spacecan be split into two subspaces with an invariant sub-space spanned by M eigenvectors having the form Φi =(⊕N+1

n=1 cnϕAi,n) ⊕ ϕB (i = 1, · · · ,M), each corresponds

to the eigenvalue λi of the whole coupled-chain system.Here ϕA

i,n is the normalized eigenvector for the n-th A

chain with eigenvalue λi and ϕB is a NM -componentzero vector. The key is that the identical A chains out-numbers the B chains by 1 so that eigenvector compo-nents on each adjacent pair of A chains add destructivelyand vanish at the intermediate B chain [50]. There is al-ways an invariant subspace and all the eigenstates thereinare extended, whatever disorder is added on the B chains.The coexistence of extended states and localized statesshown in Fig. 1 validates the above analysis.

Noticing the generality of the above analysis, it is quiteobvious that the particular separation of Hilbert space isnot limited to quasi-1D systems, but can also be appliedto quasi-2D multilayer systems constructed similarly toachieve the spectral and spatial coexistence in coupled-layer systems. As an example, we consider a system com-prising 2N + 1 (N = 6) AA-stacked honeycomb-latticelayers as shown in Fig. 2(b). Assuming the system is pe-riodic in the x and y directions, we can compute its bandstructure. Since the honeycomb lattice is a two-bandmodel, there should be totally 2(2N + 1) (= 26) bandsas shown in Fig. 2(c), in which two invariant bands aredenoted by blue curves. We now introduce uniform di-agonal disorder εi ∈ [−20, 20] to every B (red) layer andthe system is truncated to contain 420 sites per layer.The energy spectrum and participation ratios are shownin Fig. 3 (a) and (b). They clearly show the spectralcoexistence of the two different types of states in a band.A few eigenstates near zero energy have low participa-tion ratios but are depicted by blue dots (marked as ex-tended states), because they are edge states localized onthe zigzag edges of ordered (A) layers and are actuallyfrom the invariant subspace. To further demonstrate thespectral coexistence, the absolute values of wavefunctions|Ψ(x, y, z)| of two states at the same energy are denotedby colored dots at (x, y, z), as shown in Fig. 3 (c) and (d).

0 2000 4000-20

-10

0

10

20 localizedextended

State number0 2000 4000

0

500

1000

1500

2000 localizedextended

PR

State number

(a) (b)

(c) (d)

E/t

0 max

FIG. 3. (a) shows the coexistence of extended states andlocalized states in the energy spectrum of a system consistingof multiple honeycomb-lattice layers, where diagonal disorderis added to alternating layers. (b) The participation ratiosof all the eigenstates. (c) and (d) show two states with thesame eigen-energy E/t = 2.51, one is extended and the otheris localized.

For better visualization, the sizes of the dots are scaledto be proportional to |Ψ(x, y, z)|. The two states showmarkedly different nature: one is extended, and the otheris localized. Their spatial coexistence is clearly seen.

In short, we proposed a method to create the coex-istence of localized and extended states in a band in aclass of systems, in which the Hilbert space can be par-titioned in a way that the disorder affects only one sub-space causing localization, while the states in the othersubspace remain extended. The coexistence is demon-strated explicitly in both multi-chain and multi-layer sys-tems. We want to emphasize that the method we proposeis very general. It is not limited to the multi-chain andmulti-layer systems demonstrated here. It applies uni-versally to any similar structures as long as the identicalcopies of A subsystem outnumber that of B subsystemso that there is a particular subspace originated from Asubsystems without involving degrees of freedom of theB subsystems, namely sites on the B subsystems. Suchsystems can be experimentally realized using coupled op-tical waveguides [12, 52] and cold atoms [13, 18, 20, 21].For such partially disordered systems, the well-acceptedscaling theory of Anderson localization does not work,neither the concept of mobility edge. Our results implythat any energy stored in the localized states will not becarried away by energy transport in the eigen-channels ofextended states although the energies in these two typesof states overlap both spectrally and spatially.

5

This work is supported by Research Grants CouncilHong Kong (AoE/P-02/12).*Corresponding author: phchan@ust.hk

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