Exact Mobility Edges in One-Dimensional Mosaic Lattices Inlaid with Slowly Varying Potentials
aa r X i v : . [ c ond - m a t . d i s - nn ] D ec APS/123-QED
Exact Mobility Edges in One-Dimensional Mosaic Lattices Inlaid with Slowly VaryingPotentials
Longyan Gong ∗ College of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China New Energy Technology Engineering of Jiangsu Province,Nanjing University of Posts and Telecommunications, Nanjing, 210003, China (Dated: today)We propose a family of one-dimensional mosaic models inlaid with a slowly varying potential V n = λ cos( παn ν ), where n is the lattice site index and 0 < ν <
1. Combinating the asymp-totic heuristic argument with the theory of trace map of transfer matrix, mobility edges (MEs)and pseudo-mobility edges (PMEs) in their energy spectra are solved semi-analytically, where MEseparates extended states from weakly localized ones and PME separates weakly localized statesfrom strongly localized ones. The nature of eigenstates in extended, critical, weakly localized andstrongly localized is diagnosed by the local density of states, the Lyapunov exponent, and the local-ization tensor. Numerical calculation results are in excellent quantitative agreement with theoreticalpredictions.
PACS numbers: 72.20.Ee, 72.15.Rn, 71.23.An, 71.30.+h
Introduction. —Anderson localization is one of the mostfocused phenomena in condensed matter physics [1–3].In 1958, Anderson pointed out that disorder can de-struct quantum interference and induce electron localiza-tion in three-dimensional random potential systems [1].As model parameters are varied, such as energy and dis-order strength, a system can undergo phase transitionsfrom the metallic phase with extended states to the in-sulator phase with localized states. When the disorderedpotential strength is below a threshold value, extendedstates are in the middle of band and localized states arenear band edges. In the band, there are critical energies E c , at which states being extended change to being lo-calized. The critical energies are called mobility edges(MEs) [4, 5]. It is known that at zero temperature, theconductivity would be vanish (finite) if the Fermi energylies in a region of localized (extended) states. There-fore, the MEs can mark metal-insulator transitions orlocalization-delocalization transitions.It is important to known what determines MEs. Ac-cording to the scaling theory, all states are localized forone-dimensional (1D) Anderson model and there are noMEs [6]. At the same time, MEs are found in several 1Dinteresting models, e.g., the Soukoulis-Economou modelwith incommensurate potentials [7], short-range corre-lated [8] and long-range correlated [9, 10] disordered po-tential model, and the Anderson model with long-rangehopping[11]. However, there are no analytical resultsabout MEs for these models.On the other hand, the Aubry-Andr´e-Harper (AAH)model may be the most extensively studied 1D quasiperi-odic model [12, 13]. Using a duality transformation, ex-tended (localized) states in position space can be mappedto localized (extended) ones in momentum space. There ∗ Corresponding author. Email address:[email protected] is a self-duality point that related to critical states. Allstates are extended, critical, or localized, which dependon potential strength. In other words, there exists ametal-insulator transition at a critical strength of themodulation potential. However, there are no MEs in itsenergy spectrum. By introducing the exponential short-range hopping terms, a modified AAH model has beenproposed [14]. A family of generalized AAH models hasalso been investigated [15]. Based on generalized dualitytransformations, the analytical critical condition aboutMEs are predicted [14, 15]. As Wang et al. have pointedout, the self duality may be recovered on certain analyti-cally determined energy, where the extended-localizationtransition occurs, while the whole model is not exactlysolvable [16]. Based on the AAH model, they proposea class of 1D mosaic models and multiple MEs are ana-lytically derived through computing the Lyapunov expo-nents (LEs) from Avila’s global theory [16]. It is knownthat the LEs can well characterize the localization prop-erties of exponential decaying states. However, for othertypes of wave function, e.g., power-law wave functions,the sole using of LEs is not sufficient and may lead toerroneous [17]. It will be interesting to exactly obtainMEs beyond the dual transformation and judging by theLEs.At the same time, the slowly varying potential model isanother extensively studied 1D quasiperiodic model [18–20]. Its potential V n = λ cos( παn ν ), where n is thelattice site index and 0 < ν <
1. Extended stateshave been found by the perturbation theory [18] and theWKB approximation [19]. An asymptotic semiclassicalWKB-type theory predicts that two mobility edges areat E c = ± (2 − λ ) for λ ≤
2, extended states in the mid-dle of the band ( | E | < − λ ) and localized states atthe band edge (2 − λ < | E | < λ ) [20, 21]. At thesame time, pseudo-mobility edges (PMEs) are found bya numerically accurate renormalization method [22, 23],which separate weakly localized states from strongly lo-calized ones.It will be highly significant to develop generic modelswith multiple MEs and multiple PMEs. In this letter,inspired by the recent work [16], we propose a family of1D mosaic models inlaid with a slowly varying potential.They have richer phase diagrams than that for the slowlyvarying potential model and the mosaic ones based onAAH model. Using the asymptotic heuristic argumentand the theory of trace map of transfer matrix, multipleMEs and multiple PMEs in their energy spectrum canbe obtained semi-analytically. Further, our theory alsopredicts the localized and extended characterization ofall states in energy spectrum. Model. —We consider an electron moving in 1D mosaiclattice models. The family of such models is describedby H = X n λ n | n ih n | + t X n ( | n ih n + 1 | + | n + 1 ih n | ) (1)and λ n = (cid:26) λ cos( παn ν ) , mod ( n, κ ) = m , m , ..., , otherwise , (2)where λ n is the on-site potential on the n th site, λ > t is the nearest-neighbor hopping integral, which is usedas energy unit. Further, | n i = c † n | i , where c † n is the cre-ation operator of n th site and | i is the vacuum. Modelsare specified by the choice of κ, m , m , ... , and we denoteit by [ κ, m , m , ... ]. It is the uniform potential model for[ κ, m ] = [1 ,
1] and it becomes the slowly varying potentialmodel for [ κ, m ] = [1 , κ = 2 and 3 are [ κ, m ] = [2 ,
0] and [3 , κ = 5 with two m is [ κ, m , m ] = [5 , , κ, m , m , ... ] = [2 , , [3 ,
0] and [5 , , t tt t t t (a) [ ,m]=[2,0] (b) [ ,m]=[3,0](c) [ ,m ,m ]=[5,0,3] FIG. 1: The 1D quasiperiodic mosaic model with[ κ, m , m , ... ] = [2 , , [3 ,
0] and [5 , , t . Semi-analytical method. —Now we provide the semi-analytical results about MEs and PEs by using the asymptotic heuristic argument [20, 21] and the theoryof trace map of transfer matrix [24]. The slowly varyingpotential V n = λ cos( παn ν ) given in Eq.(2) has highlycorrelated disorder feature [10]. It can be written as dV n dn = − λπαn ν − sin( παn ν ) . (3)When n → ∞ and 0 < ν < (cid:12)(cid:12)(cid:12)(cid:12) dV n dn (cid:12)(cid:12)(cid:12)(cid:12) ∼ | sin( παn ν ) | n − ν → . (4)Equivalently, V n +1 − V n ∼ O ( | n | ν − ), which vanishes forlarge n . Therefore, such potential can be taken as a “localconstancy” [20, 21].On the other hand, the Schrd¨oinger equation for theHamiltonian given in Eq.(1) can be written as tφ n − + tφ n +1 + λ n φ n = Eφ n , (5)where φ n is the amplitude of wave at n th site and E isthe corresponding eigenenergy. Eq.(5) can be rewrittenin terms of the transfer matrix M ( n ),Φ n +1 = (cid:18) E − λ n t −
11 0 (cid:19) Φ n ≡ M ( n )Φ n , (6)where Φ n = (cid:18) φ n φ n − (cid:19) . (7)From Eq.(2), V n = λ cos( παn ν ) periodically occurs withinterval κ , so we can introduce a quasicell with the near-est κ lattice sites. In a quasicell, by a successive applica-tion Eq.(6), a recursion relation is given byΦ n +1+ κ = M ( n + κ ) ...M ( n +1)Φ n +1 ≡ M κ ( n )Φ n +1 . (8)If there are j same quasicells, the relation isΦ n +1+ jκ = [ M κ ( n )] j Φ n +1 . (9)Let χ ( E ) = trM κ ( n ) / , (10)then | χ | < V n , V max = λ and V min = − λ , which correspond to the maximum potentialbarrier and potential well. As n → ∞ , V n is a “localconstancy” [20, 21]. Near the maximum potential bar-rier and potential well, we respectively use Eq.(10), andcorresponding V n are approximated by λ and − λ . Forthe two cases, we denote χ ( E ) in Eq.(10) by χ λ ( E ) and χ − λ ( E ), respectively. Our rule to judge the localizationproperties of states is summarized in Table I, which is themain result in this paper. Table I means that an electronwith E is in an extended state if it can penetrate both themaximum potential barrier and the maximum potential TABLE I: State properties judging by χ λ ( E ) and χ − λ ( E ),where E x , W , S and C represent extended, weakly localized,strongly localized and critical, respectively. | χ λ ( E ) | | χ − λ ( E ) | state property < < E x < > W> < W> > S = 1 > , = 1 , < C> , = 1 , < C well. It is in a weakly localized state if it can penetrateonly one. It is in a strongly localized state if it penetratesneither. It is in a critical state if it just penetrates oneor both, therefore MEs and PMEs are determined by thefunctions χ λ ( E ) = ± χ − λ ( E ) = ± Phase diagram. —For model parameters[ κ, m , m , ... ] = [1 , , [1 , , [2 , , [3 ,
0] and [5 , , χ [1 , λ = χ [1 , − λ = E , (11) ( χ [1 , λ = ( E − λ ) ,χ [1 , − λ = ( E + λ ) , (12) ( χ [2 , λ = ( E − λE − ,χ [2 , − λ = ( E + λE − , (13) ( χ [3 , λ = ( E − λE − E + λ ) ,χ [3 , − λ = ( E + λE − E − λ ) , (14)and χ [5 , , λ = [ E − λE + ( λ − E +6 λE − ( λ − − λ ] ,χ [5 , , − λ = [ E + 2 λE + ( λ − E − λE − ( λ −
5) + 2 λ ] . (15)Judged by the rule in Table I, for [ κ, m ] = [1 , | E | < κ, m ] = [1 , | E | < − λ ) at λ ≤
2, localized states at the band edge(2 − λ < | E | < λ ), and two MEs are at E c = ± (2 − λ ),which are just that obtained for the slowly varying po-tential model in Refs. [20, 21].To illustrate phase diagrams for [ κ, m , m , ... ] =[2 , , [3 ,
0] and [5 , , φ β , where β is the index of eigenfunc-tion. The fractal dimensionΓ = − lim N →∞ [ln( IP R ) / ln N ] , (16) -4 -2 0 2 40123 (a) 0.190.40.60.80.93 -4 -2 0 2 40123 (b) FIG. 2: For [ κ, m ] = [2 , E and potential strength λ , where the systemsize N = 500, πα = 0 . ν = 0 .
7, respectively. (b) Phasediagram. The lines in (a) and (b) represent MEs and PMEs,and E x , W , and S respectively represent extended, weaklylocalized, and strongly localized phases. -4 -2 0 2 40123 (a) 0.090.20.40.60.80.96 -4 -2 0 2 40123 (b) FIG. 3: The same as Fig.2, but [ κ, m ] = [3 ,
0] and system size N = 600. -4.7 -2 0 2 4.701234 (a) 0.110.20.40.60.80.94 -4.7 -2 0 2 4.701234 (b) FIG. 4: The same as Fig.2, but [ κ, m , m ] = [5 , , where the inverse participation ratio IP R = P Nn =1 | φ β,n | [25]. For [ κ, m , m , ... ] = [2 , , [3 , , , E and po-tential strength λ are plotted in Figs.2(a), 3(a) and 4(a),respectively. The lines in them represent the MEs andPMEs, which are determined with Eqs.(13)-(15) by theconditions that χ λ ( E ) = ± χ − λ ( E ) = ±
1. Accordingto the rule in Table I, extended ( E x ), weakly localized( W ) and stronger localized ( S ) phases are labelled inFig.2(b), 3(b) and 4(b), respectively. There are MEsseparating extended states from weakly localized states.There also exist PMEs separating weakly localized statesfrom strongly localized states. We numerically find thetwo outermost MEs are beyond the allowed energies,which are denoted by dashed lines. It is known thatΓ → → χ ( E ) is a κ th-orderpolynomial of E . At the same time, MEs and PMEsare determined by the conditions that χ λ ( E ) = ± χ − λ ( E ) = ±
1. Therefore, the number of MEs and PMEs N m ≤ κ , and N m = 4 κ if all roots are different, so thereare N m − n -0.2500.25 n -1.301.3 n -0.2500.25 n -1.41401.414 n -0.2500.25 n -1.501.5 n -0.2500.25 n -1.301.3 n -0.2500.25 n -1.501.5 n -0.2500.25 n -1.41401.4140 2500 5000 n -0.2500.25 n -1.41401.414 -2 -1.414 -1 -0.5 01.31.4141.5 WW SA BC ED GF(h) FIG. 5: For [ κ, m ] = [2 , A − G points in (f) the phasediagram. The potential λ n is also plotted in the red groundof (a-g). Further, the phase diagram can also be directly re-flected by the spatial distribution of wave functions. For[ κ, m ] = [2 , A − G points are shown in Fig.5(h). Fig.5 showsthe extended state spreads over the whole system, thestrongly localized state is more localized than that ofweakly localized one, and critical states [Figs.5 (d) and(f)] at MEs are intermediately distributed in space. Thelocalization property of the “critical” state [Fig.5 (e)] at PME is intermediate between that of strongly localizedand weakly localized states. The critical state [Fig.5 (g)]at the band center is almost extended, which correspondsto that both | χ λ | = 1 and | χ − λ | = 1. Numerical verification. — Now we provide numerical -2.5 -2 -1 0 1 2 2.5-101 (a) =0.5W W W W -4 -2 0 2 4-101 (b) S S =2.5W WW W
FIG. 6: For [ κ, m ] = [2 , χ λ (green line) and χ − λ (magenta line) versus energies E at (a) λ = 0 . λ = 2 .
5, respectively. The vertical lines represent MEs andPMEs, and the horizon lines are for the functions χ = ± verification about MEs and PMEs. Model parameters[ κ, m ] = [2 , λ = 0 . λ = 2 . χ λ and χ − λ with eigenenergies E . Based on the rule in Ta-ble I, extended ( E x ), weakly localized ( W ) and strongerlocalized ( S ) regions are labelled. MEs and PMEs arerepresented by vertical lines. -2.5 -2 -1 0 1 2 2.500.511.5 (a) =0.5 -4 -2 0 2 400.250.50.751 (b) =2.5 FIG. 7: For [ κ, m ] = [2 , D (greencircle) versus eigenenergies E at (a) λ = 0 . λ = 2 . N = 300000. To strengthen our findings, we calculate the density ofstates D ( E ), which is defined by D ( E ) = P Nn =1 δ ( E − E n ). Here E n is the n th eigenenergy. We use a methoddeveloped by Persson to compute all eigenvalues for largesystem with open boundary condition [26]. It has beenfound that the singularity of the DOS can reflect MEs forquasiperiodic systems[20, 21]. Figs.7 (a) and (b) showthat there are sharp peaks in D at MEs and PMEs, sothey support our theoretical results.The energy-depended LE γ ( E ) is often used to char-acterize the electronic localization properties. We use anumerically accurate renormalization scheme to obtainit [22]. The LE is defined by γ ( E ) = − lim N →∞ [ 1 N ln | t eff N ( E ))] , (17) -2.5 -2 -1 0 1 2 2.500.20.40.6 (a) =0.5 -6 -4 -2 0 2 400.40.81.2 (b)W W WWS S=2.5 FIG. 8: For [ κ, m ] = [2 , γ versusenergies E at (a) λ = 0 . λ = 2 .
5, respectively. Insetin (a): Partial enlarger for one of extended regions. Thevertical lines represent MEs and PMEs, and system sizes N =10 (blue dot), 10 (red dot) and 10 (yellow dot). where t eff N ( E ) is the effective hopping integral betweensites 1 and N when all the internal sites between themare properly decimated. In Figs.8 (a) and (b) we plotthe calculated LE as a function of the electron energies E for πα = 0 . ν = 0 . λ = 0 . .
5, respec-tively. It is known that γ = 0 for an extended statewhereas γ > γ approaches zero; as shown in the insetof Fig.8(a), the larger the system size is, the smaller thevalue of γ is. For localized states, γ >
0; the larger thesystem size is, the larger the value of γ is. So the LEcan well distinct extended regions from localized regions.Fig.8 (b) shows there exist inflection points at PMEs,which can reflect the transitions between weakly local-ized phases and strongly localized ones. On the whole,numerical results about γ well agree with our theoreticalpredictions. -2.5 -2 -1 0 1 2 2.500.250.50.751.0 W =0.5(a)W WW -4 -2 0 2 400.250.50.751.0 W W W WS S =2.5(b) FIG. 9: For [ κ, m ] = [2 , ξ r versus eigenenergies E at (a) λ = 0 . λ = 2 .
5, re-spectively. Inset in (a): Partial enlarger for one of extendedregions. The vertical lines represent MEs and PMEs, andsystem sizes N = 500 (blue upward triangle), 1000 (red dia-mond) and 2000 (yellow downward triangle). However, as pointed out by Varga et al. [17], LE is notsufficient to characterize localized states beyond power-law shaped wave functions and may lead to erroneous.In Fig.8(b), LE γ for states in strongly localized regionsis smaller than that for states in weakly localized regions(near band edges), so it can not well reflect the localiza-tion properties of such localized states. Fortunately, for aperiodic system the localization tensor (LT) ξ is found tobe an effective quantity that is able to distinguish metal-lic from insulating states [27, 28]. For a single electronin 1D lattices, the LT is defined by ξ = h ψ | q † q | ψ i − h ψ | q † | ψ ih ψ | q | ψ i , (18)where the complex position q = N πi P Nn =1 [exp( πiN n ) − N is the lattice size and i = √− ξ max = N π for | ψ i = √ N exp( i πjN n ), where j is aninteger [28]. So we define a reduced LE ξ r = ξ/ξ max . (19)The calculations of ξ r are plotted in Fig.9. It showsthat for extended states, ξ r approaches one; as shownin the inset of Fig.9(a), the larger the system size is, thelarger the value of ξ r is. For localized states, ξ r is rel-atively small; the larger the system size is, the smallerthe value of ξ r is. There are drastic changes in ξ r atMEs and PMEs. Therefore, it can well distinguish ex-tended, localized and critical states. More importantly,Fig.9(b) shows on the whole, ξ r is relatively larger forweakly localized states than that for strongly localizedones. Therefore, the numerical results about ξ r are com-pletely consistent with our theoretical predictions. Conclusion. —In this work, we propose a family ofquasiperiodic mosaic lattice model base on a slowly vary-ing potential. They have rich phase diagrams, includ-ing extended, weakly localized, strongly localized phases,mobility edges and pseudo-mobility edges. By using theasymptotic heuristic argument and the theory of tracemap of transfer matrix, we provide semi-analytical solu-tions. The semi-analytical results are in excellent agree-ment with the localization properties characterized bynumerical calculations of the local density of states, theLyapunov exponent, and the localization tensor. We ex-pect our theoretical results to be directly observed inultracold atoms and photonic waveguides. [1] P.W. Anderson, Absence of diffusion in certain randomlattices, Phys. Rev. , 1492 (1958).[2] A. Lagendijk, B. Van Tiggelen, and D.S. Wiersma, Fiftyyears of Anderson localization, Phys. Today, , 24(2009).[3] E. Abrahams (ed.),
50 Years of Anderson Localization (World Scientific, Singapore, 2010). [4] M.H. Cohen, H. Fritzsche and S.R. Ovshinsky, Sim-ple Band Model for Amorphous Semiconducting Alloys,Phys. Rev. Lett. , 1065 (1969).[5] N.F. Mott, Electrons in Disordered Structures, Adv.Phys. , 49 (1967).[6] E. Abrahams, P.W. Anderson, D.C. Licciardello, andT.V. Ramakrishnan, Scaling Theory of Localization: Ab- sence of Quantum Diffusion in Two Dimensions, Phys.Rev. Lett. , 673 (1979).[7] C.M. Soukoulis and E.N. Economou, Localization in One-Dimensional Lattices in the Presence of IncommensuratePotentials, Phys. Rev. Lett. , 1043 (1982).[8] D.H. Dunlap, H-L. Wu, and P.W. Phillips, Absence oflocalization in a random-dimer model, Phys. Rev. Lett. , 88 (1990).[9] F.A.B.F. de Moura and M.L. Lyra, Delocalization in the1D Anderson Model with Long-Range Correlated Disor-der, Phys. Rev. Lett. , 3735 (1998).[10] F.M. Izrailev and A.A. Krokhin, Localization and theMobility Edge in One-Dimensional Potentials with Cor-related Disorder, Phys. Rev. Lett. , 4062 (1999).[11] A. Rodr´ıguez, V.A. Malyshev, G. Sierra, M.A. Mart´ın-Delgado, J. Rodr´ıguez-Laguna and F. Dom´ınguez-Adame, Anderson Transition in Low-Dimensional Disor-dered Systems Driven by Long-Range Nonrandom Hop-ping, Phys. Rev. Lett. , 027404 (2003).[12] S. Aubry and G. Andr´e, Analyticity breaking and An-derson localization in incommensurate lattices, Ann. Isr.Phys. Soc. , 133 (1980).[13] P.G. Harper, Single band motion of conduction electronsin a uniform magnetic field, Proc. Phys. Soc. London,Sect. A , 874 (1955).[14] J. Biddle and S. Das Sarma, Predicted Mobility Edges inOne-Dimensional Incommensurate Optical Lattices: AnExactly Solvable Model of Anderson Localization, Phys.Rev. Lett. , 070601 (2010).[15] S. Ganeshan, J.H. Pixley, and S. Das Sarma, NearestNeighbor Tight Binding Models with an Exact Mobil-ity Edge in One Dimension, Phys.Rev.Lett. , 146601(2015).[16] Y. Wang, X. Xia, L. Zhang, H. Yao, Sh. Chen, J. You,Q. Zhou and X-J. Liu, One-Dimensional QuasiperiodicMosaic Lattice with Exact Mobility Edges, Phys. Rev.Lett. , 196604 (2020). [17] I. Varga, J. Pipek and B. Vasv´ari, Power-law localizationat the metal-insulator transition by a quasiperiodic po-tential in one dimension, Phys. Rev. B , 4978 (1992).[18] M. Griniasty and S. Fishman, Localization by pseudo-random potentials in one dimension, Phys. Rev. Lett. , 1334 (1988).[19] D.J. Thouless, Localization by a potential with slowlyvarying period, Phys. Rev. Lett. , 2141 (1988).[20] S. Das Sarma, S. He, X.C. Xie, Mobility edge in a modelone-dimensional potential, Phys. Rev. Lett. , 2144(1988).[21] S. Das Sarma, S. He, X.C. Xie, Localization, mobil-ity edges, and metal-insulator transition in a class ofone-dimensional slowly varying deterministic potentials,Phys. Rev. B , 5544 (1990).[22] R. Farchioni, G. Grosso and G.P. Parravicini, Electronicstructure in incommensurate potentials obtained using anumerically accurate renormalization scheme, Phys. Rev.B , 6383 (1992).[23] R. Farchioni, G. Grosso and G.P. Parravicini, Renormal-ization and envelope function formalism for incommen-surate systems. IL Nuovo Cimento D, , 375 (1993).[24] M. Kohmoto, L.P. Kadanoff and C. Tang, Localizationproblem in one dimension: Mapping and escape, Phys.Rev. Lett. , 1870 (1983).[25] F. Evers and A. D. Mirlin, Anderson transitions, Rev.Mod. Phys. , 1355 (2008).[26] P.O. Persson, Eigenvalues of tridiagonal matrices inmatlab,http://persson.berkeley.edu/software.html.[27] R. Resta and S. Sorella, Electron localization in the in-sulating state, Phys. Rev. Lett. , 370 (1999).[28] E. Valen¸ca Ferreira de Arag˜ao, D. Moreno, S. Battaglia,G.L. Bendazzoli, S. Evangelisti, T. Leininger, N. Suaudand J.A. Berger, A simple position operator for periodicsystems, Phys. Rev. B99