A plane wave study on the localized-extended transitions in the one-dimensional incommensurate systems
AA plane wave study on the localized-extended transitions in theone-dimensional incommensurate systems
Huajie Chen, Aihui Zhou,
2, 3 and Yuzhi Zhou
4, 51
School of Mathematical Sciences,Beijing Normal University, Beijing 100875, China LSEC, Institute of Computational Mathematics and Scientific / Engineering Computing,Academy of Mathematics and Systems Science,Chinese Academy of Sciences, Beijing 100190, China School of Mathematical Sciences,University of Chinese Academy of Sciences, Beijing 100049, China Software Center for High Performance Numerical Simulation,China Academy of Engineering Physics, Beijing 100088, China Institute of Applied Physics and Computational Mathematics, Beijing 100088, China ∗ (Dated: August 19, 2020) a r X i v : . [ phy s i c s . c o m p - ph ] A ug bstract Based on our recently proposed plane wave framework, we theoretically study the localized-extended transition in the one dimensional incommensurate systems with cosine type of potentials,which are in close connection to many recent experiments in the ultracold atom and photonic crys-tal. We formulate a propagator based scattering picture for the transition at the ground state and sin-gle particle mobility edge, in which the deeper connection between the incommensurate potentials,eigenstate compositions and transition mechanism is revealed. We further show that there exists aupper limit of localization length for all localized eigenstates, leading to an fundamental di ff erenceto the Anderson localization. Numerical calculations are presented alongside the analysis to justifyour statements. The theoretical analysis and numerical methods can also be generalized to systemsin higher dimensions, with di ff erent potentials or beyond the single particle regime, which wouldbenefit the future studies in the related fields. . Introduction The localization of quantum waves in the non-periodic potentials has aroused muchresearch interests since Anderson’s seminal paper decades ago [1]. Unlike the fully disor-dered system, the incommensurate system, which consists of two or more periodic com-ponents but lacks overall periodicity, can exhibit localized-extended transition in 1D or2D from the experiments of ultracold atoms [2–4] and photonic crystals [5–7], as well asfrom theoretical studies [7–9]. In the ultracold-atom systems, such transitions can be fur-ther robustly controlled through adjusting the incommensurate potential and interatomicinteraction strength [3], which makes them an ideal platform to simulate quantum manybody e ff ects [10, 11]. Moreover, many salient spectrum and transport properties havebeen observed in the incommensurate systems of 2D materials, for instance the quantumHall e ff ect [12], the greatly enhanced carrier mobility [13], and the unconventional super-conductivity [14]. Their occurrence might deeply relate to the localization of electronsnear the Fermi level [15]. Therefore, a full knowledge of the incommensurate localiza-tion mechanism in the single particle regime is a prerequisite to gain a better control ofthe quantum states in experiments, as well as to understand related quantum many bodye ff ects and novel electronic properties.Given the feasibility of describing the localized states, a majority of the theoreticalstudies on the incommensurate localization are based on the tight-binding model [8, 16–24], which greatly improve our understanding and helps to interpret related experimentalresults. However, one has to be careful in constructing the model Hamiltonian, as the over-simplification might lead to incorrect localization properties. An example is the Aubry-Andr´e (AA) 1D tight binding model [16], which showed that the eigenstates are either alllocalized or all delocalized, depending on the relative strength between the incommensu-rate cosine modulation and the primary lattice. Yet, it has been verified in recent exper-iments that there exists a single-particle mobility edge (SPME) in such incommensurate3ystems [25]. Meanwhile, the existence of mobility edge can be recovered in theoreticalcalculations using the model Hamiltonian with more continuum nature [21, 23, 26], sug-gesting an overlook of high-order hopping e ff ect in the AA model [21, 23]. In addition,some tight binding calculations are performed with finite size or periodic boundary con-dition. This would cause some troubles in distinguishing a truly localized state and anextended state but exhibiting a localized wave packet in the range of system size, whichmight undermine our understanding on the transition mechanism.On the other hand, plane wave basis has several features that would benefit the studyof transition in the incommensurate system. First, it is very convenient in representing theeigenstates of kinetic energy operator and the incommensurate potential, which does notrequire further approximations to describe the Hamiltonian in the single particle regime.Meanwhile, it is naturally compatible with extended systems and one can further circum-vent the periodic boundary condition utilizing the ergodicity as discussed in [9]. Thereforesome systematic errors from the inappropriate boundary conditions can be avoided. Fur-thermore, since the plane waves are generally viewed as the conjugate of the localizedorbitals, one could expect gaining complementary perspectives on the localized-extendedtransition under this representation, which helps to complete our understanding on thesubject. However, previously plane wave studies are limited due to a lack of rigorousmathematical treatment of the corresponding quantum eigenvalue problem.In this paper, we will study the localized-extended transition of the time-independentSchr¨odinger equation for the one dimensional incommensurate systems, utilizing our re-cently developed plane wave framework [9]. Specifically, we formulate a scattering pictureto describe the localized-to-extended transition based on the propagation of plane waves inthe higher dimension reciprocal space, without explicitly solving the eigenvalue problem.Here we mainly study two cases: (a) the ground state transition with increasing potentialstrength, and (b) the transition at the SPME, in which the deeper connection between theincommensurate potentials, the plane wave components in the eigenstates and the transi-4ion mechanism is revealed. We further discuss the existence of a maximum localizationstrength, which implies an intrinsic di ff erence from the Anderson localization. (Other fun-damental di ff erences between the incommensurate localization and Anderson localizationhave also been discussed in recent theoretical studies [23, 27].) Numerical calculationsunder the same framework, which directly solve the eigenvalue problem, are performedto justify the conclusions from the scattering picture. We stress that even though part ofthe conclusions in this paper can be drawn from some revised tight binding models, theplane wave studies provide us unique insights on the mechanism of transition. Also notethat although we restrict our discussions to the incommensurate systems in one dimensionwith cosine-type potentials, our plane wave representation and scattering picture can inprinciple be used to study general incommensurate systems.The rest of this paper is organized as follow. In Section II, we briefly introduce theplane wave framework for incommensurate systems. In Section III, we formulate the scat-tering picture and apply it to study the emergence of localization transition in the groundstate and the localized-to-extended transition at SPME. In Section IV, we discuss the ex-istence of a maximum localization length and compare the incommensurate localizationwith Anderson localization in this context. In Section V, we present some concluding re-marks. Moreover, we discuss the role of incommensurate ratio on the transition based onthe scattering picture in the Appendix. II. Plane wave framework
In this section we briefly introduce the plane wave framework for the simulations ofthe incommensurate systems. We consider the following time-independent Schr¨odingerequation for an one-dimension incommensurate system with two periodic components: (cid:32) − ∂ ∂ x + V ( x ) + V ( x ) (cid:33) Ψ ( x ) = E Ψ ( x ) ∀ x ∈ R , (1)5here V and V are periodic potentials V j ( x + n τ j ) = V j ( x ) with n ∈ Z and τ j the lat-tice constants for j = ,
2. The incommensurateness puts further constraints on the ratiobetween τ j and the corresponding reciprocal lattice G j = π/τ j , that τ τ = G G = β is anirrational number. This leads to the so-called ergodicity (see Section II.2) and is crucial tothe discussions in this work. II.1 The plane wave discretization
Following the discussions in [9], we use the basis functions (cid:8) e i ( mG + nG ) x (cid:9) ( m , n ) ∈I Ecut withindex set I E cut : = (cid:110) ( m , n ) ∈ Z : | mG | + | nG | ≤ E cut (cid:111) (2)to discretize the Schr¨odinger equation (1), where E cut is the energy cuto ff that features theaccuracy and computational cost of this discretization. The ground state wave function Ψ ( x ) is approximated by Ψ E cut ( x ) = (cid:80) ( m , n ) ∈I Ecut u mn e i ( mG + nG ) x with ˆ u = { u mn } ( m , n ) ∈I Ecut theunknown coe ffi cients. Eq. (1) is then discretized into a matrix eigenvalue problem H ˆ u = E ˆ u , (3)where the hamiltonian matrix elements are given by H mn , m (cid:48) n (cid:48) ( k ) = (cid:12)(cid:12)(cid:12) G m + G n (cid:12)(cid:12)(cid:12) δ mm (cid:48) δ nn (cid:48) + V m − m (cid:48) ) δ nn (cid:48) + V n − n (cid:48) ) δ mm (cid:48) ( m , n ) , ( m (cid:48) , n (cid:48) ) ∈ I E cut (4)with V jm the Fourier component of the periodic potential V jm = τ j (cid:82) τ j V j ( x ) e − imG j x d x .To quantitatively describe the extent of localization for a wavefunction Ψ E cut , it is con-venient to use the inverse participation ratio (IPR) [28] to measure the number plane wavescontributing a given eigenstate, which is defined byIPR (cid:0) Ψ E cut (cid:1) : = (cid:88) ( m , n ) ∈I Ecut (cid:12)(cid:12)(cid:12) u mn (cid:12)(cid:12)(cid:12) . (5)6or an extended state, its IPR will scale like O (1) as E cut → ∞ . While for a localized state,the IPR will be approaching 0 with the scaling O (cid:0) E − / cut (cid:1) as E cut → ∞ . The scaling factor1 / II.2 Ergodicity and the higher dimensional interpretation
We will discuss the concept of ergodicity of incommensurate systems particularly inthe higher dimensional representation. We refer to [9] for more details.The ergodicity was originally used to describes the equiprobable access to all states inthe phase space in thermodynamics. The ergodicity in our context is a direct consequencefrom the incommensurateness, and is the root of many unique properties of the incommen-surate systems. It can be stated in the mathematical language as those in Ref. [22, 29, 30].Here we prefer a more direct description: with infinitely large cuto ff s E cut , the coupledwave vectors { mG + nG } m , n ∈ Z will fill the whole reciprocal space R densely, uniformly and unrepeatedly .The one dimensional Schr¨odinger equation (1) can be reformulated in R × R by (cid:32) − (cid:101) D + V ( x ) + V ( x (cid:48) ) (cid:33) ˜ Ψ ( x , x (cid:48) ) = E ˜ Ψ ( x , x (cid:48) ) ∀ ( x , x (cid:48) ) ∈ R × R (6)with the directional derivative (cid:101) D ˜ Ψ ( x , x (cid:48) ) : = (cid:0) ∂∂ x + ∂∂ x (cid:48) (cid:1) ˜ Ψ ( x , x (cid:48) ). Since the potential V ( x ) + V ( x (cid:48) ) is periodic in R × R , the periodicity is restored by this higher dimensionalrepresentation. We note that similar idea has been explored for describing the lattices anddi ff raction patterns of quasi-crystals (see e.g. Ref. [31–34]). It has been shown in [9] that,with an energy cuto ff E cut and the basis set (cid:8) e i ( mG x + nG x (cid:48) ) (cid:9) ( m , n ) ∈I Ecut , the discretization of Eq.(6) leads to the same matrix eigenvalue problem Eq. (3) at Γ point. The solution in Eq. (6)can further be transformed back to that of Eq. (1) by taking the diagonal Ψ ( x ) = ˜ Ψ ( x , x ).We can also observe the ergodicity by the projection in higher dimensional reciprocalspace, as illustrated in the upper right of Fig. 1. When a wave vector ˜ kkk = ( mG , nG ) ( m , n ∈ ) on the two dimensional reciprocal lattice is projected onto line k − k =
0, it givesthe one dimensional wave vector kkk = mG + nG . The ergodicity is reflected by the factthat all the projected points will densely, uniformly and unrepeatedly spread on the line k − k =
0. This observation is crucial to the discussions in the following.
Vanishing potentialSmall potential
FIG. 1. The higher dimension representation of the coupled plane waves. This 2D lattice hasperiodicity β (with the ratio β = √ − ) in k direction and 1 in k direction. Upper right: A 2Dlattice site ˜ kkk = ( mG , nG ) is projected to an 1D wave vector kkk = mG + nG . Lower left: the cosine-type potentials in the Hamiltonian scatter | kkk (cid:105) to its nearest neighbor states. Central: At vanishing V ,the ground eigenstate is mainly composed of the plane waves near the origin (green dotted circle).With increasing potential strength, more plane waves along (cid:104)− , (cid:105) direction are involved to theground state (black dashed ellipsoid). Note there is a factor of √ from the projection on the line k = k in 2D reciprocal space. Be aware that this factor is not present in the transformation from2D to 1D reciprocal space. II. The scattering picture of localized-extended transition
In this section, we will formulate the transition picture within the plane wave frame-work. For simplicity, we will restrict ourselves to the following incommensurate Hamilto-nian in the discussions of localized-extended transition,ˆ H = − ∂ ∂ x + V cos (cid:0) β x (cid:1) + V cos (cid:0) x (cid:1) (7)with β an irrational number and V , V > G = β and G = V ( x ) and V ( x ) being cosine potential. The discussions inthis section can be extended to more general potentials.It is convenient to rewrite the Hamiltonian in the second quantization form:ˆ H = (cid:88) kkk = ( mG , nG ) , m , n ∈ Z | kkk | c † kkk c kkk + (cid:88) kkk (cid:16) V c † kkk c kkk + ( G , + V c † kkk c kkk + (0 , G ) + h . c . (cid:17) , (8)where c + kkk and c kkk are the annihilation and creation operator associated with the plane wavestate | kkk (cid:105) . It is important to note that the norm | · | in (8) for a plane wave kkk = ( mG , nG )is given by | kkk | : = | kkk + kkk | = | mG + nG | , rather than the standard Euclidean norm (cid:0) | nG | + | mG | (cid:1) / . This is essentially a tight-binding Hamiltonian with nearest neighborhopping in the two dimensional reciprocal lattice, as shown in the lower left of Fig. 1.In the following, we first qualitatively investigate the change of plane wave componentsin the ground state as the potential strength grows. Then we formulate a scattering pictureusing the language of propagator, which is further adopted to study the transitions at theground state and SPME. For simplicity we only consider the V = V = V case in thissection. The discussion of V (cid:44) V case is presented in Appendix.9 II.1 Transition at the ground states: a qualitative study
At the vanishing potential strength V , one expects the ground state of ˆ H (defined in(8) with V = V = V ) is mainly composed of the plane waves near the origin. As V increases, more plane waves are mixed into the ground state as the coupling to nearbysites becomes more significant. This is illustrated in Fig. 1, in which the green dottedcircle grows to the black dashed ellipsoid as V increases. The ellipsoid shape in the figurecan be understood by the following arguments. The kinetic (on-site) energy of a site kkk = ( mG , mnG ) is | kkk | = | mG + nG | , which grows much faster along (cid:104) , (cid:105) directionbut remains fluctuated around certain value along (cid:104) , ¯1 (cid:105) direction. Thus we expect lessplane waves along (cid:104) , (cid:105) direction, whose on-site energies are significantly higher than theground state energy, to be mixed into the ground state compared with those along (cid:104) , ¯1 (cid:105) direction. For now, the ground state solution mainly consists of a finite number of planewaves and is an extended wave function in the real space. Consequently, the correspondingIPR value (defined in Eq. (5)) mainly depends on the distribution in the bounded circle orellipsoid and will not decay as the energy cuto ff E cut increases.If V further increases and crosses the critical point, then the ellipsoid of the plane wavecomponents will become a ”stripe” that extends to infinity along (cid:104) , ¯1 (cid:105) direction. Giventhe form of Eq. (8), there must exist some ”scattering” paths that connect the all planewaves close to k + k =
0, through nearest neighbor hopping. As will be quantitativelyformulated later, the most relevant path to the transition is the one with all sites staying atclosest distance to the line k + k = k + k = k + k = k are alsorelevant to the discussion of SPME). In this case, there are infinitely many plane waves,which are connected by the paths extending to infinity, markedly contributing to the groundstate. If we further project all the involved plane waves onto the line k − k =
0, they10orm a continuous (more precisely, densely distributed) band around the origin in onedimensional reciprocal space due to the ergodicity (see also Fig. 2). Now the intervalbetween the k points in the 1D reciprocal space now becomes zero, and the ground stateundergoes a localization transition. This observation is similar to the discussion of quasiparticle lifetime and the localization of Green’s function in the time domain in [36, Chapter3 and Appendix H]. In addition, since the distribution of plane waves is semi 1D along k + k =
0, the IPR value vanishes as O (cid:0) E − / cut (cid:1) , as we have mentioned in Sec. II. Scattering inthe pathScattering offthe pathThe MPD path r.t.
FIG. 2. An MPD path (blue arrows) in the 2D reciprocal space that connects the origin and thelattice sites along (cid:104) , ¯1 (cid:105) direction. The lower half is the same by symmetry. The anti-diagonaldashed lines indicates the boundary of the path. The lattice sites on this path, when transformedback to the 1D reciprocal space (by projecting onto line k = k ), form a continuous band aroundthe origin (red thick bar). The dashed black arrows represent the scattering events to states o ff thepath, whose probability amplitude decays fast away from the boundary. We then verify the above statements by numerical calculations (see [9] for details ofthe algorithm). We take the ratio β = (cid:0) √ − (cid:1) , and simulate two incommensuratesystems with V = .
05 and V = .
3, corresponding to extended state and localized state11espectively (these potential strengths are consistent with the critical strength derived atSec. III.3 ). Their ground state solutions are compared in Fig. 3. With stronger potential,we observe a much more extensive distribution of the occupied plane waves along theanti-diagonal direction, which is consistent with the above analysis.
FIG. 3. The square of ground state solutions in 1D real space and 2D reciprocal space for twosystems with V = .
05 and V = . (cid:104) ¯1 , (cid:105) direction in reciprocalspace. III.2 A propagator-based formulation
The above picture can be translated into a propagator-based formulation that enablesthe quantitative study of the transition, without explicitly solving the eigenvalue problem.For simplicity of notations, we will denote by kkk = ( mG , nG ) the state of a plane wave e i ( mG + nG ) x , and E kkk = | kkk | the corresponding kinetic energy.The propagation of a plane wave kkk i being scattered once by the potential to its neighbor kkk j has the probability amplitude: T ( kkk i → kkk j , E ) = E − E i V E − E j , (9)with E i = E kkk i , E j = E kkk j , and E the frequency of the free propagator. Then the probability12mplitude for an N successive scattering events along a path P = (cid:8) kkk , · · · , kkk N (cid:9) is given by: T ( P , E ) = E − E V E − E V · · · V E − E N . (10)For a MPD path P k respect to k + k = k , we take N → ∞ in Eq. (10), which reads T ( P k , E ) = lim N →∞ N (cid:89) i = VE − E i . (11)Intuitively, T ( P k , E ) can be viewed as a term in the diagram expansion for the Green’sfunction of the Hamiltonian Eq. (8), and one could solve for the Green’s function to re-trieve the properties of the system in principle. This is exactly what Anderson did in hispaper: he adopted the Renormalized Perturbation Expansion (RPE) of the Green’s func-tion [37, 38] to study the localization in the disordered systems [1]. The analysis waslater simplified by Ziman [39], Thouless [40], and further reorganized by Economou andCohen [41, 42]. The Anderson’s original formulation is very complicated, and one couldimagine that the it gets even more complicated in the incommensurate systems since theyare essentially higher dimensional problems.In this paper, we will not go into full details of the Green’s function expansions whenstudying the extended-to-localized transition. Instead, at the edge of the transition, onefinds some specific MPD paths only just connect to the plane waves at infinity, and thecorresponding T ( P , E ) undergoes an abrupt change from 0 to nearly divergence or theother way around. This marks the situation we have mentioned previously: the infinitenumber of plane waves start or cease to markedly mix into the eigenstate, which leadsto the localized-extended transition at the ground state or mobility edge. Therefore, weinvestigate the divergence criterion of T ( P , E ) which represents the abrupt change of theplane wave distribution of the eigenstates during the transition. The reason for choosingthe MPD paths is that they can more e ff ectively reach the plane waves at infinity than otherpaths, thus are most relevant to the transition.13efore we quantitatively describe this situation, one more subtlety needs to be consid-ered. Given the 2D nature of the problem, the plane wave states within the MPD path areinevitably scattered o ff the path into nearby states, as depicted in Fig. 2. This creates othernon-MPD paths to infinity by allowing for short digressions from the MPD path. Thisresults in a reduced probability to reach the infinity for the original MPD path. Since itcould happen at any site within the MPD path, we multiply a factor α ∈ (0 ,
1) for eachscattering event. ( α can also be seen as the average e ff ect from the non-MPD paths in thediagram expansion.) Then the probability amplitude reads: T ( P k , E ) = lim N →∞ N (cid:89) i = α VE − E i . (12)Thus, at the transition point it should fulfill α V | E − E i | ≈ , (13)at given E or V , where | E − E i | indicates the geometric mean over all sites in the path P .The above analysis sacrifices a little bit of rigorousness but facilitates our understandingwith a more direct physical picture. This idea is similar to the analysis proposed by Zimanon the Anderson localization [39]. The conclusions drawn from this scattering picturewill be further checked by the numerical calculations in the same plane wave framework,where the full information of the eigenpairs is obtained. III.3 Transition at the ground states: a quantitative study
Now we adopt the scattering picture to quantitatively describe the localization transi-tion of the ground state. Before the potential strength V reaching the critical point V c , the T ( P , E ) for any MPD path makes negligible contribution to the diagram expansion for anyfrequency E . This is because V < V c makes α V | E − E i | <
1, hence the infinite multiplication14n T ( P , E ) makes it an infinitesimal. This is consistent with previous analysis and calcu-lations that the plane waves infinitely away make inappreciable contribution to the groundstate, which then corresponds to an extended wavefunction.To calculate the critical point V c from Eq. (12), we consider the MPD path P respectto k + k =
0, since this path has the smallest site energies on average thus can minimizethe denominator in Eq. (13). Its site-averaged natural logarithm of T ( P , E ) is given by ∆ ( P , E ) = lim N →∞ N (cid:88) k n ∈P ln α V (cid:12)(cid:12)(cid:12)(cid:12) E − | kkk n | (cid:12)(cid:12)(cid:12)(cid:12) . (14)From the ergodicity, the projected plane waves form a uniform and continuous band cen-tering the origin (see the red thick bar in Fig. 2). Thus we can transform the summation inEq. (14) into an integration: ∆ ( P , E ) = G + G (cid:90) G + G − G + G ln α V (cid:12)(cid:12)(cid:12) E − s (cid:12)(cid:12)(cid:12) d s , (15)where s corresponds to the norm | kkk | . The integral region is determined as follow. Thehopping along P in Fig. 2 proceeds with alternative and balanced upward and leftwardjumps to retain the closest distance to k + k =
0, in which process the maximum distancefrom k + k = G + G . This gives a boundary of [ − G + Q , G + Q ].Now we make a rough estimate of the factor α . First, α is not close to 0 since the planewave energies change quadratically away from the path, thus the major contribution to theground state still comes from the plane waves within the path, which have significantlylower energies. Second, the scattering to the plane waves right near the path is not neg-ligible. They have comparable energies thus can be e ff ectively included into the groundstate by other slightly detouring paths. As a consequence, one would expect a considerabletransfer of the amplitude to the nearest neighbor plane waves of the MPD path. Therefore α can not be close to 1 either. Then α = / V c , we further parameterize E = V c . From the criterion Eq. (13), we have ∆ ( P , = + ln 8 α V c (cid:0) G + G (cid:1) ≈ . (16)Plugging in α = / G = β = (cid:0) √ − (cid:1) and G =
1, we can obtain the critical value V c = .
089 for our exemplified system.To numerically verify the results from the scattering picture, we compute the groundstates with the ratio β = (cid:0) √ − (cid:1) and varying potential strengths V and plot the cor-responding IPR values in Fig. 4. We observe that the slope of IPR changes significantlyaround V = . ∼ .
10, indicating the occurrence of the transition in this region. This isin quantitative agreement with our estimated value V c = . FIG. 4. The IPR of the ground state with varying potential strengths. The predicted critical strength V c = .
089 (indicated by the red dashed line) is in good agreement with the numerical results.
III.4 Transition at the mobility edge
For the transition at SPME, we need to consider the situation with V > V c . Since V hascrossed the critical point, not only P for the ground state satisfies ∆ ( P , E ) ≥
0, there also16xist MPD paths P k and higher frequency ˜ E such that the condition ∆ ( P k , ˜ E ) ≥ ff erent k can exactly overlap with each other bya translational shift (see Fig. 5). This is due to the ergodicity of incommensurate systems:if there is a site kkk in the MPD path respect to k x + k y = k , then due to the ergodicity,there exists another site ˜ k ˜ k ˜ k whose relative position to k x + k y = ˜ k is arbitrarily close to thatbetween kkk and k x + k y = k . Hence shifting kkk to ˜ k ˜ k ˜ k will overlap the two paths. The MPD path r.t.
The MPD path r.t.
FIG. 5. The MPD paths (blue arrows) respect to di ff erent k . These two paths can be translated fromone to the other by a direct shift due to the ergodicity. Despite their same geometries, the MPD paths with respect to larger | k | have higherenergy di ff erences between neighboring sites, which in general reduces the overall prob-ability amplitude. This further results in fewer paths that connect to k points at infinity.Then there exists a critical path P ± kc respect to k + k = ± k c (together with a correspondingfrequency E c ) such that the MPD paths with higher | k | can not connects to infinity, whichindicates the onset of localized-to-extended transition and sets the SPME.We shall estimate the critical energy k c using the scattering picture. For simplicity, weassume the sites of P k c stay on the one side of k + k =
0. Then E c can be parameterized17s (cid:0) k c − G + G (cid:1) to minimize the denominator, which gives largest k c . We can obtain k c bysolving ∆ ( P k c , E c ) = G + G (cid:90) k c + G + G k c − G + G ln α V (cid:12)(cid:12)(cid:12) (cid:0) k c − G + G (cid:1) − s (cid:12)(cid:12)(cid:12) d s = , (17)where the integral region [ k c − G + Q , k c + G + Q ] is derived by similar analysis as that for (15).The estimation of α is also similar to previous section, except now the plane wave energiesincrease quadratically in one direction away from the path while decrease quadratically inthe other. We stress that the E c is not the SPME, but defines the upper and lower boundsfor the plane wave components of the eigenstate near the SPME. This is because E c is thefrequency of the free propagator or loosely regarded as the unperturbed energy, while theSPME is defined respect to the solved eigenspectrum.As an example, we set the potential strength V = . k c using the scattering picture. Plugging in α = / G = β and G = k c = . − k c − G + Q , k c + G + Q ] (note the lower bound is obtainedby the symmetry). In Fig. 6, we present the region | k + k | ≤ k c + G + Q , together with theeigenstate at SPME in the reciprocal space. The eigenstate at SPME is obtained by solvingthe eigenvalue problem Eq. (1) by plane wave methods and searching through the wholeeigenspectrum. We observe in the figure that our theoretical prediction of the boundarymatches well with the numerics.Moreover, the existence of the SPME goes against the prediction from the AA tightbinding model, which states that the eigenstates are either all localized or all delocalized,determined by the ratio between the strengths of the primary and secondary lattices [16].The origin of the discrepancy can be understood by the fact that the AA Hamiltonian isin principle a single band model under extreme tight binding limit and cannot representthe properties of the full spectrum in more general cases. We further emphasis that thelocalization properties are independent of the basis set used to discretize the Hamiltonian.18 IG. 6. The localized state with highest energy in 2D reciprocal space. The dashed lines indicatethe predicted boundary of the reciprocal space distribution.
For this reason, the SPME has also been observed in some revised tight binding models[21, 24], real space calculations [23, 26], and plane wave calculations in this paper.
IV. Comparison with the Anderson localization
One obvious di ff erence between the incommensurate localization and Anderson local-ization is the existence of SPME, which has already been discussed in the existing theoreti-cal and experimental literature [23, 25]. In this paper, we focus on a new implication fromthe scattering picture in the higher dimensional reciprocal lattice, which lead to anotherfundamental di ff erence on the localization length.Assuming negligible inelastic scattering and infinite sample size for simplicity, the lo-calization length in the Anderson localization of 1D disordered system in principle couldbe arbitrarily large. This can be illustrated using Anderson’s analysis [43] based on Lan-dauer’s conductance formula [44]. The key feature in the Anderson localization is thatthe intermediate state between two successive scattering of the conducting particle (gen-erally taken as the plane wave states for simplicity) is random. The average over possible19ntermediate states leads to a linear dependance of ln(1 + S ( L )) ( ∼ γ L ) on L , where S isthe dimensionless resistance for a length L of the sample and γ the linear factor. This, inreturn, gives rise to the exponential decay of the wavefunction in the real space and 1 /γ determines the localization length. It can be further envisioned that by tuning the meanfree path through the defect concentration, one could in principle have arbitrarily small γ ,hence arbitrarily large localization length.While for the incommensurate localization, it is a di ff erent picture: the intermediatestate between two scattering events is fixed. This means the mechanism leading to Ander-son localization will not apply for the incommensurate system: unlike the previous casewhere a continuum of intermediate states are visited by defect-average in a single event ofpropagation, now it is achieved by the infinite number of scattering events along the MPDpaths in the reciprocal space. Consequently, the minimum width of the paths constrainsthe upper limit of localization length in the real space. In other words, a localized wave-function with very large localization length in the real space would require the distributionof the plane waves to behave more or less like “delta function”, where the continuum ofthe k only exists in a very narrow region. However, this is against the scattering picture,where the continuum of projected wave vectors is achieved by some MPD paths that havea minimum width of G + G .We can further illustrate this fact by numerical simulations for the system with V = . π/ (1 + β ), though there exists minor contributionoutside the major localization region.Given this di ff erence, the attempts of using incommensurate systems to simulate theAnderson localization might require further justification. However, we also note that thefinite size e ff ect and noises in the potential might blur the boundary between these twotypes of localization. Distinguishing incommensurate localization and Anderson localiza-20 IG. 7. The ground state, the highest localized state and an extended state in the real space with V = .
0. The vertical lines indicate the maximum localization length from our analysis. tion in experiments thus seems to be an interesting and challenging question for futurestudies.
V. Conclusions
In this paper, we utilize the plane wave framework to study the extended-to-localizedtransitions in the 1D incommensurate systems. A scattering picture has been formulated toquantitatively study the transitions of at the ground states and SPME. Under this picture,we further discuss the fundamental di ff erence between the incommensurate localizationand Anderson localization. The numerical calculations have been conducted alongside tojustify the conclusions from the scattering picture. In principle, the theoretical analysisand numerical methods can be carried over to more general incommensurate systems inhigher dimensions, with more complicated form of potentials and beyond the single parti-cle regime, thus provide theoretical tools to investigate spectrum and transport propertiesof the incommensurate systems in various fields.21 cknowledgements This work was partially supported by National Key Research and Development ofChina under grant 2019YFA0709601. H. Chen’s work was also partially supported bythe Natural Science Foundation of China under grant 11971066. A. Zhou’s work wasalso partially supported by the Key Research Program of Frontier Sciences of the Chi-nese Academy of Sciences under grant QYZDJ-SSW-SYS010, and the National ScienceFoundation of China under grant 11671389.
Appendix. Role of the incommensurate ratio
The ratio β between the periodicity of the periodic components is the key feature of theincommensurate system. But in many existing works on the localization, the role of thisvalue has not been fully explored. In this appendix, we will investigate the extended-to-localized transition with respect to the ratio β . We will not restrict ourselves to the case V = V = V as in Section III, but consider general systems that allow V (cid:44) V .For the general cases, the probability amplitude of a path P at frequency E in Eq. (10)is T ( P , E ) = E − E · V q (1) · E − E · V q (2) · E − E · · · V q ( N ) · E − E N , (18)where q ( i ) = kkk i − → kkk i is parallel to the k direction and q ( i ) = k direction. For the MPD path, the number of the horizontal ( k direction) and vertical ( k direction) jumps has a ratio of 1 /β since the path is extending toinfinity along (cid:104) ¯1 , (cid:105) direction. With this observation, we can rewrite Eq. (15) as ∆ ( E ) = G + G (cid:90) G + G − G + G ln V + β · V β + β · α (cid:12)(cid:12)(cid:12) E − s (cid:12)(cid:12)(cid:12) d s . (19)22or the same critical condition that ∆ ( E ) ≈
0, we have ∆ ( E = = + ln 8 V + β · V β + β · α G (1 + β ) ≈ , (20)where we have replaced G by β G in this equation to better illustrate the role of β . Wesee from Eq. (20) that the incommensurate ratio β influences the transition in two places.First, in the nominator of the propagator, it controls the weight of each periodic compo-nents in the geometric mean of the potential strength. Second, in the denominator of thepropagator, it reflects the energy di ff erences between plane wave states connected by theincommensurate potential. We note Eq. (20) can provide guidelines to the manipulationof the localization transition in experiments of ultracold atoms and photonic crystals.In the following, we demonstrate two simple scenarios of such manipulations throughnumerical calculations. In the first one, we have V = V = V and the incommensurateratio β is varied ranging from 0 . ∼ .
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