Localization and delocalization in one-dimensional systems with translation-invariant hopping
aa r X i v : . [ c ond - m a t . d i s - nn ] J a n Localization and delocalization in one-dimensional systems with translation-invarianthopping
Reza Sepehrinia ∗ Department of Physics, University of Tehran, Tehran 14395-547, Iran andSchool of Physics, Institute for Research in Fundamental Sciences, IPM, Tehran 19395-5531, Iran
We present a theory of Anderson localization on a one-dimensional lattice with translation-invariant hopping. We find by analytical calculation, the localization length for arbitrary finite-range hopping in the single propagating channel regime. Then by examining the convergence ofthe localization length, in the limit of infinite hopping range, we revisit the problem of localizationcriteria in this model and investigate the conditions under which it can be violated. Our resultsreveal possibilities of having delocalized states by tuning the long-range hopping.
According to well-known theories of Anderson local-ization [1–3], single-particle wave functions are exponen-tially localized in low dimensional ( d = 1 ,
2) disorderedsystems. Several mechanisms have been identified whichprovide counterexamples to this belief [4]. One route todelocalization is long-range hopping which especially insystems with one-dimensional (1D) geometry is feasiblefor systematic analytical treatment. It has been a usefulmodel to investigate various properties of the Andersontransition and new aspects of it are still being discovered[5].The effect of long-range hopping on localization wasfirst considered by Anderson [1] and subsequently, in theproblem of phonon localization, by Levitov [6]. The fol-lowing picture has emerged: for a hopping amplitude de-caying as 1 /r α with distance r , all states are extended if α < d , whereas for α > d the states are localized. Thiswas well confirmed by the power-law random banded ma-trix model [7] which describes a 1D system with randomlong-range hopping. This model undergoes an Ander-son transition with multifractal eigenstates at α = 1. Itturns out however that the above picture is not universaland does not hold for the models with correlated hopping[5, 8–10]. The latter includes the models with correlatedrandom hopping and non-random hopping with the on-site disorder. It is found that correlated hopping tendsto localize the states even when α < d . Regarding theextended states in these models, there have been few re-ports although the corresponding energies form a set ofnull measure. [11–14].In this paper, we report an analytical study of the lo-calization properties of a class of correlated models char-acterized with translation-invariant hopping and diago-nal disorder. Our approach is to start from the arbitraryfinite-range hopping, for which we are able to obtain thelocalization length, and then take the limit of the infinitehopping range. This leads us to reconsider the criterion oflocalization in systems with long-range hopping and dis-cuss the conditions under which it can be violated. Ourresults reveal possibilities of having delocalized states inthese systems based on the asymptotic behavior of hop-ping. Model . The model under consideration is one-dimensional tight-binding chain, represented by theeigenproblem r X n =1 t n (Ψ i + n + Ψ i − n ) + ǫU i Ψ i = E Ψ i , (1)with hopping range r and weak random potential ǫU i ,where h U i i = 0 and h U i U j i = σ δ ij . Angular bracketsdenote the ensemble average. In the absence of randompotential, the solutions of (1) are plane waves with energy E ( k ) = 2 r X n =1 t n cos nk, (2)where unit lattice spacing is assumed and the wave vector k belongs to the Brillouin zone k ∈ [ − π, π ]. Perturbation theory . The solution of (1), in the pres-ence of the weak random potential, can be treated per-turbatively [15] by rewriting it in terms of variables R i = Ψ i +1 Ψ i , r X n =1 t n n − Y m =0 R i + m + n Y m =1 R i − m ! = E − ǫU i . (3)For an unperturbed plane wave, R i is constant and for aperturbed solution it is assumed to be weakly fluctuat-ing around that constant value, which can be expressedas R i = A exp( B i ǫ + C i ǫ + · · · ). This assumption is validif the unperturbed solution is a single plane wave. Other-wise scattering to other states will produce superpositionof waves with different wavelengths and thus position-dependent R i [16, 17]. Therefore we will be consideringthe single-channel part of the energy band. Since the dis-persion relation (2) is an even function of k , for a givenallowed energy, there are at least, two solutions ± k i.e.one channel of propagation. In order to eliminate one ofthe two wave vectors ± k an infinitesimal imaginary partcan be added to the energy and finally be made to ap-proach zero. From Eq. (3), up to second order in ǫ , wehave r X n =1 t n ( A n + A − n ) = E, (4a) r X n =1 t n A n n − X m =0 B i + m − A − n n X m =1 B i − m ! = − U i , (4b) r X n =1 t n A n n − X m =0 C i + m + 12 n − X m =0 B i + m ! + A − n − n X m =1 C i − m + 12 n X m =1 B i − m ! = 0 . (4c)Equation (4b) can be written in closed form, r X m =1 ( τ m B i + m − − τ ∗ m B i − m ) = − U i , (5)where τ m = P rn = m t n A n and τ ∗ m = P rn = m t n A − n . As E approaches the eigenenrgies of the pure system, A be-comes pure phase and thus A ∗ = A − and therefore τ ∗ m will be complex conjugate of τ m . The Lyapunov expo-nent (LE) and its weak disorder expansion is given by γ ( E ) = lim N →∞ N N X i =1 log R i = h log R i (6)= log A + ǫ h B i + ǫ h C i + · · · . (7)In order to calculate the averages, we take the average ofequations (4b) and (4c), from which we obtain h B i = 0 , (8) h C i = − P rn =1 ̺ n t n ( A n + A − n ) P rn =1 nt n ( A n − A − n ) , (9)where ̺ n = ρ (0) + 2 P nl =1 ( n − l ) ρ ( l ) and ρ ( τ ) is the auto-covariance function h B n + τ B n i . The covariances shouldbe obtained using Eq. (4b). By multiplying B i + j in Eq.(5) for j = − r, − r + 1 , · · · , r − ρ ( l ) = ρ ( − l ) and h B i + j U j i = 0 for j < r − r linear equations r X n =1 [ τ n ρ ( | n − − j | ) − τ ∗ n ρ ( | n + j | )] = − σ τ r δ j,r − ,j = − r, − r + 1 , · · · , r − . (10)This is a linear inhomogeneous system to obtain 2 r un-knowns ρ (0) , ρ (1) , · · · , ρ (2 r − r does not seem to be simple. Withoutexplicitly solving the equations, we were able to construct the numerator in Eq. (9) by linear combination of them.The final result is the closed expression for the average h C i = − σ P rn =1 ( τ n − τ ∗ n )] , (11)and the localization length (inverse LE) follows from it ξ = − σ ǫ " r X n =1 nt n ( A n − A − n ) , (12)= 2 v σ ǫ ; v = − r X n =1 nt n sin nk, (13)where we have used A = e ik . We can see from Eq. (2)that v = ∂E/∂k is the group velocity. At the band edgeswhere the group velocity vanishes the Lyapunov expo-nent diverges which implies the failure of the analyticexpansion in disorder strength [15]. The result (13) im-plies that in the single propagating channel regime thestates will be localized if r X n =1 nt n sin nk < ∞ . (14)As we can see, this condition always holds for the finiterange r .We now consider infinite-range hopping and see if (14)holds in the limit r → ∞ or not. A necessary condi-tion for convergence of the series is nt n → n → ∞ .The first conclusion which can be drawn from this is thatin order to have a localized state the hopping integralsshould necessarily decay faster than n − . This result isindeed the Levitov’s criterion of localization for d = 1.However, the above condition is not a sufficient conditionof convergence. Below we will see the cases for which theabove condition holds, but the series does not converge.Before that, we state a more strict condition of conver-gence. It is known from the theory of trigonometric se-ries [18] that cosine and sine series, in (2) and (13), withmonotonically decreasing coefficients, are convergent ex-cept, perhaps, at k = 0. Therefore, if the hopping de-cays faster than n − but monotonically then the seriesconverges and the states will be localized. We now applythe general result Eq. (13) to specific examples that havebeen studied before by other means. Exponential hopping . First we consider t n = t s n with | s | <
1, we have E ( k ) = 2 t s (cos k − s )1 − s cos k + s , (15) v ( k ) = 2 t s ( s −
1) sin k (1 − s cos k + s ) . (16)From (15) we can see that, for a given energy, there isonly one pair of wavevectors ± k i.e. there is only onepropagating channel and from (16) we can see that for E/t σ ε ξ / t FIG. 1: (Color online) Localization length for exponentialhopping model t n = t s n with s = 1 / all k the localization length is finite and thus the corre-sponding states are localized. This model is studied inRef. [19] by numerical calculation of the inverse partici-pation ratio (IPR). In agreement with their conclusions,our results show weakly localized states at higher energies(see Fig. 1). As the range of hopping becomes shorter( s →
0) the results tend to that of nearest-neighbor hop-ping Anderson model. Random band matrix model withexponential hopping, which is closely related to (1), ex-hibits similar localization properties [20].
Power-law hopping . A more interesting case is t n = t n − α which has been studied in several works [10–13, 19]and shown to exhibit anomalous localization properties.In this model as well the dispersion relation allows a sin-gle channel of propagation, so our results are applicable, E ( k ) = t [Li α ( e ik ) + Li α ( e − ik )]; α > , (17) v ( k ) = it [Li α − ( e ik ) − Li α − ( e − ik )]; α > , (18)where Li α ( z ) = P ∞ n =1 z n n − α . We distinguish three dif-ferent cases:( i ) 0 < α ≤
1. The series E ( k ) converges for all k except k = 0 (band edge) where it diverges to infinity, E ( k → → + ∞ , so the energy spectrum of pure chainis not bounded from above. However, since nt n is notdecreasing, the series v ( k ) does not converge and in factit is oscillating as r → ∞ so the localization length doesnot have a well defined limit. This signals the failure ofthe assumption of exponential localization.( ii ) 1 < α <
2. The series E ( k ) converges everywhere,including k = 0, therefore the energy spectrum of thepure chain is bounded i.e. the bandwidth is finite. Theseries v ( k ) also converges for all k , thus all states havea finite localization length for this range of α . Althoughthe localization length is finite, it increases unboundedly E/t σ ε ξ / t -1 0 1 2 3 4024 α <2 α =2 α >2 FIG. 2: (Color online) Localization length versus energy forpower-law hopping t n = t n − α . Vertical dashed lines showthe upper band edge in each case. close to the upper band edge indicating delocalized states(see Fig. 2). Delocalization of uppermost states has beenpredicted in Refs. [11, 12, 19] and their transition tolocalized states at strong disorder is studied in Ref. [13].However, we do not see a qualitative change of behaviorat α = 3 /
2, as is predicted in Ref. [12] and the power-law localization of states in this power-law hopping model(see Ref. [5]).( iii ) α ≥
2. Both the series E ( k ) and v ( k ) are conver-gent and bounded for all k (see Fig. 2). This confirmsthe numerical results of Ref. [19] where it is found thatthere is a minimum IPR for this range of α . Delocalized states . We now look for the sequences ofhopping integrals for which the localization length di-verges i.e. the condition (14) is not satisfied. The di-vergence of such a series is an old problem in the theoryof trigonometric series [18, 21] and is also related to thetheory of functions with divergent Fourier series.We have already seen that for power-law hopping with1 < α < t n = t n − α sin nk this singular pointcan be shifted into the energy band. This allows us tohave an extended state at a given energy E ( k ) and, inparticular, the band edge can be avoided because the per-turbation theory fails at this point. Such an oscillatinghopping can be induced by RKKY interaction. The di-vergence of the localization length manifests itself in thedispersion curve as an infinite slope i.e. infinite groupvelocity (see Fig. 3). This kind of singularity also occursin the dispersion curve of Hartree-Fock excitations in in-teracting electron system. By superposition of multipleterms, t n = t n − α P i sin nk i , we will have a set of ex- E ( k ) /t -101 k σ ε ξ / t π π /30 π /3 FIG. 3: (Color online) Dispersion relation (top) and Local-ization length (bottom) for modulated power-law hopping t n = t n − α sin nk with 1 < α < k = π/ tended states at given wavevectors. Particularly, this canbe a dense set of energies at any given interval throughthe energy band.In general, for t n = a n sin nk where a n is monoton-ically decreasing but P ∞ n =1 a n = ±∞ , there will be anextended state at k . We note that the single channelcondition on the dispersion relation also needs to be sat-isfied. As an example a n ≥ = t ( n ln n ) − can be con-sidered. The nearest-neighbor hopping should be largeenough such that the dispersion relation satisfies the sin-gle channel condition. Note that hopping decays fasterthan n − but due to nonmonotonicity, condition (14) isnot satisfied at k = k .Finally, we would like to point out the possibility thatextended states form a continuous band rather than aset of isolated energies. It is known that with certain(decreasing) coefficients the trigonometric series in (14)diverges almost everywhere. A suitable example for ourdiscussion is t n = a n sin nq n with certain conditions im-posed on the sequences a n and q n [21, 22]. An explicitchoice is a n ≥ = t ( n ln n ) − and q n ≥ = ln ln n . Againthe nearest-neighbor hopping should be such that the sin-gle channel condition is satisfied. We also note that forthis choice E ( k ) converges. The other interesting casewould be the divergence of the series in a subintervalwhich results in a band of extended states separated bya mobility edge from the localized states; we leave thisto future work. Conclusions . An analytical expression for the local-ization length in a one-dimensional tight-binding modelwith diagonal disorder and arbitrary-range hopping inthe single channel regime is obtained. Finite-range hop-ping always leads to localized states but delocalizedstates emerge in the infinite-range limit. It turns out that for infinite-range hopping, t n . n − is a necessarybut not sufficient condition for localization. We provideexamples which satisfy this condition but violate the con-dition (14) and lead to delocalized states. The additionalrequirement of monotonic decay makes it a sufficient con-dition. Exponential and power-law hoppings were inves-tigated in detail, and a qualitative comparison with pre-vious studies was done. Our results reproduce severalaspects of existing results although we arrive at differ-ent conclusions in some cases. Namely, for power-lawhopping with α ≤ r → ∞ , therefore the assumption of exponentiallocalization seems to be invalid. Also, contrary to thepredicted transition at α = 3 /
2, our results do not indi-cate a qualitative change at this point.
AKNOWLEDGEMENT
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