Distribution of S-matrix poles for one-dimensional disordered wires
I. F. Herrera-González, J. A. Méndez-Bermúdez, F. M. Izrailev
DDistribution of S -matrix poles for one-dimensional disordered wires I. F. Herrera-Gonz´alez, J. A. M´endez-Berm´udez, and F. M. Izrailev , Department of Engineering, Universidad Popular Aut´onoma del Estado de Puebla,21 Sur 1103, Barrio Santiago, Puebla, Pue., Mexico Instituto de F´ısica, Benem´erita Universidad Aut´onoma de Puebla,Apartado Postal J-48, Puebla 72570, Mexico and NSCL and Dept. of Physics and Astronomy, Michigan State University - East Lansing, Michigan 48824-1321, USA (Dated: October 16, 2018)By the use of the effective non-Hermitian Hamiltonian approach to scattering we study the distri-bution of the scattering matrix ( S -matrix) poles in one-dimensional (1D) models with various typesof diagonal disorder. We consider the case of 1D tight-binding wires, with both on-site uncorrelatedand correlated disorder, coupled to the continuum through leads attached to the wire edges. Inparticular, we focus on the location of the S -matrix poles in the complex plane as a function of thecoupling strength and the disorder strength. Specific interest is paid to the super-radiance transitionemerging at the perfect coupling between wire and leads. We also study the effects of correlationsintentionally imposed to the wire disorder. PACS numbers: 05.60.Gg, 46.65.+g, 73.23.-b
I. INTRODUCTION
To date, the paradigmatic one-dimensional (1D) An-derson model with white-noise diagonal disorder has beenrigorously studied in great detail. The main result isthat all eigenstates are exponentially localized in the in-finite geometry, characterized by the amplitude decreasewith the distance from their centers. The characteris-tic scale on which they are effectively localized is knownas the localization length L loc which can be easily com-puted numerically in the frame of the transfer matrixmethod (see Ref. [1] and references therein). This energy-dependent length is of utmost importance due to the sin-gle parameter scaling, according to which all transportproperties of the finite wires are explicitly defined by theratio L loc /N , where N is the length of the wire (see, forinstance, Ref. [2]). Specifically, there are various rigorousapproaches allowing to derive the distribution functionfor the transmission coefficient characterizing the scatter-ing process through finite wires of size N (for references,see [3]).Complimentary to the transfer matrix method, thescattering properties of electromagnetic waves propagat-ing through finite wires can also be studied via the scat-tering matrix ( S -matrix). There is an enormous numberof papers devoted to the theory of the S -matrix in appli-cation to complex physical systems such as heavy nuclei,many-electron atoms, and quantum dots. Assuming aquite complex (chaotic) behavior of the closed (isolated)systems, it was suggested that various types of randommatrices can serve as good mathematical models in de-scribing statistical properties of scattering. In particular,typical models are represented by non-Hermitian matri-ces with a real part in terms of a fully random matrixplus an imaginary part of certain structure absorbingthe details of the coupling to the continuum. In thisway many results both analytical and numerical, havebeen obtained during the last decades (see, for example, Refs. [4–6] and references therein).One important question in scattering theory is aboutthe type of distribution of the widths of resonances,emerging in the transmission coefficient as a function ofthe energy. For sufficiently week coupling to the con-tinuum (slightly open systems) the resonances are wellisolated from each other; a situation termed as isolatedresonances . This happens when the resonance widths aresmaller than the spacings between the locations of reso-nances. A completely different situation arises when thewidths of the resonances are much larger than the spacingbetween them. In this case the resonances are stronglyoverlapped, thus resulting in specific properties of scat-tering. For random matrix models it was shown that thecrossover from isolated to overlapped resonances is quitesharp in dependence on the coupling strength γ , and atthe transition point the mean value of the resonancesdiverges. With the increase of γ towards strongly over-lapped resonances, a remarkable effect emerges, known as superradiance [6]. In this regime a finite number of reso-nances have very large widths while the other resonancesbegin to be more and more narrow with the increase ofthe coupling.The goal of this paper is to study the 1D Andersonmodel with both white-noise disorder and correlated dis-order, focusing on the pole distribution in dependenceof the degree of localization of eigenstates and of thestrength of the coupling to the continuum. Althoughthis model is very different from random matrix models,we expect that some of the properties, such as the super-radiant transition and the divergence of the resonancewidths at the superradiant transition point, develop sim-ilarly to those in random matrix theory (RMT) models.The distinctive property of our model is that by increas-ing the disorder one can change the degree of localizationof eigenstates in the closed wires, therefore, in the wiresattached to the leads the localization effects may greatlyaffect the pole distribution. Another issue to address is a r X i v : . [ c ond - m a t . d i s - nn ] O c t how correlations imposed to the diagonal disorder mod-ify the pole distribution at the mobility edges emergingdue to specific long-range correlations. We hope that ournumerical results can help to develop an analytical ap-proach to the problem of scattering in 1D wires in thecase when a closed system is strongly influenced by lo-calization effects. II. THE MODEL AND SCATTERING SETUP
The 1D Anderson model is defined by the stationarySchr¨odinger equation ψ n +1 + ψ n − + (cid:15) n ψ n = Eψ n . (1)for the electron wave function ψ n of energy E . In whatfollows we consider both uncorrelated and correlated dis-order specified by the site potentials (cid:15) n . The energies E and (cid:15) n are dimensionless quantities measured in units ofthe kinetic electron energy.In the non-disordered case ( (cid:15) n = 0), the solutions ψ n are plane waves with wave number µ defining the disper-sion relation E = 2 cos µ , ≤ µ ≤ π . (2)For the the disordered case we assume that the distri-bution of on-site energies (cid:15) n is characterized by randomvariables with zero mean and variance σ , (cid:104) (cid:15) n (cid:105) = 0 , (cid:104) (cid:15) n (cid:105) = σ . Here (cid:104) . . . (cid:105) stands for the average over disorder realiza-tions. We also assume that the disorder is weak, σ (cid:28) , (3)which is needed to develop a proper perturbation theory.In the case of uncorrelated disorder all electron eigen-states are exponentially localized with the characteristiclength L loc in the limit of infinite wire length ( N → ∞ ).As is known, for white-noise disorder the localizationlength L loc is given by the Thouless expression [7] (seealso Ref. [1]), L − = σ µ = w − E / . (4)Here µ is defined through the dispersion relation (2) andthe second relation is given for a box distribution of (cid:15) n specified by the interval [ − w/ , w/ N with disor-dered on-site potentials (cid:15) n . The first and last sites ofthe disordered lattice are connected to semi-infinite ideal leads through coupling amplitudes √ γ . In this way theleads are considered as a continuum to which the disor-dered wire is coupled according to given boundary con-ditions. The scattering properties of such open system can be formulated in terms of the non-Hermitian Hamil-tonian [4, 6]. The key point of this approach is based onthe projection of the total Hermitian Hamiltonian (dis-ordered wire plus leads) onto the basis defined by theHamiltonian H describing the properties of the closedmodel (the disordered wire only).In our model, the non-Hermitian Hamiltonian has thefollowing form near the band center ( E = 0) [8]: H mn = H mn − i W mn (5)with W mn = 2 π (cid:88) c = L,R A ( c ) m A ( c ) n , (6)where W mn is defined by the coupling amplitudes A L,Ri = (cid:114) γπ (cid:16) δ ( L ) i + δ ( R ) iN (cid:17) . (7)Here, H mn is the Hamiltonian of the 1D Anderson model H nm = (cid:15) n δ mn + δ m,n +1 + δ m,n − , (8)while the non-Hermitian part W is given in terms of thecoupling amplitudes A ( c ) i between the internal states | i (cid:105) and open decay channels c = L, R , where L and R standfor left and right, respectively.With the non-Hermitian Hamiltonian, it is possible toobtain the scattering matrix S in the form, S = 1 − iK iK , (9)where the reaction matrix K is given by K ab = (cid:88) j C ( a ) j C ( b ) j E − E j , C ( c ) j = (cid:88) m A ( c ) m ψ ( j ) m , (10)and ψ ( j ) m is the m component of the j -th eigenstate of theclosed Hamiltonian (8) with eigenvalues E j .One of the quantities studied in this work is the distri-bution of the eigenvaluesΩ k = ω k − i k (11)of the non-Hermitian Hamiltonian (5); here ω k and Γ k are called the position and the width of the k − resonance,respectively. The eigenvalues Ω k can also be treated asthe poles of the S -matrix, since one can write S ( E ) = 1 − πiA T ( E ) 1 E − H A ( E ) , where A ( E ) is the N × A ( c ) i .In general the poles of the S -matrix for the 1D Ander-son model depend on three parameters: the energy E ,the coupling constant γ , and the localization length L loc (given by Eq. (4)). However, without loss of generality,in this work we fix the energy to E = 0. It is known thatdepending on the ratio between the localization lengthand the system size N , there are three different regimesfor the scattering processes: the ballistic regime charac-terized by L loc (cid:29) N , the chaotic regime where L loc ≈ N ,and the localized regime which occurs for L loc (cid:28) N (seefor example Ref. [1]). III. S -MATRIX POLES: WHITE-NOISEDISORDER In this Section we consider the case of uncorrelated dis-order: (cid:104) (cid:15) n (cid:15) n + m (cid:105) = 0, for m (cid:54) = 0. First, in Fig. 1 we reportthe S -matrix poles as the localization length decreases(from left to right) for two different values of the cou-pling strength: γ = 0 . γ = 1 (lowerpanels). In the ballistic regime, L loc (cid:29) N , see Fig. 1(leftpanels), the poles are distributed around the curves cor-responding to the non-disordered case, which are shownas continuous black curves in all panels of Fig. 1. Thisbehavior is easy to understand since the eigenstates ofthe disordered wires in this regime remain close to theeigenstates of wires with zero disorder (plane waves). Inthe chaotic regime, L loc ≈ N , the eigenstates are stillextended (as in the ballistic regime), however, they pro-duce strong fluctuations of Γ values. As one can see bycomparing Fig. 1(left and middle panels), the distribu-tion of poles is quite sensitive to whether the eigenstatesare quasi-regular or chaotic. Finally, for the localizedregime, L loc (cid:28) N , only few eigenstates touch the wireboundaries and, as a consequence, the influence of thecontinuum is reduced producing a distribution of polescloser to the real axis, as Fig. 1(right panels) shows.As the coupling parameter γ increases, we observe thefollowing effect. At zero coupling to the continuum the S -matrix poles are located along the real axis. As thecoupling is turned on the poles acquire an imaginary partand, if the average width (cid:104) Γ (cid:105) is small as compared to thelevel spacing D of the closed system, the cross sections inthe scattering process reveal isolated resonances and thepoles form a single cloud close to the real axis in the com-plex plane. However, with the increase of the couplingparameter γ , a crossover from isolated to overlapping res-onances occurs; this crossover at γ ≈ (cid:28) D ) and the other one to strongly overlappedones (with large Γ; Γ (cid:29) D ). The latter states are termedsuperradiant states since they are short-lived, in contrastwith long-lived states with small Γ. In the literature thissegregation of poles is known as the superradiant transi-tion. As we demonstrate below, the transition betweenisolated and superradiant states is very sharp with re-spect to the change of γ and can be associated with akind of phase transition. Then, in Fig. 2 the aforementioned pole segregationis displayed. The lower cloud in Fig. 2(c) corresponds to( N − N d long-lived states, whereas the upper cloud (la-beled by the small rectangle) represents 2 N d short-livedor superradiant states, being N d the number of realiza-tions of the disorder (the factor 2 here accounts for thenumber of leads connected to the 1D wire). Notice thatin this figure we show the superradiant transition for thechaotic regime L loc ≈ N only, however, the existence ofsuch transition and the value of γ where it takes place donot depend on the degree of localization.It is important to note that the control parameter de-termining the strength of the coupling to the continuumcan be written as [8] κ = 2 πγN D , where D is the mean level spacing at the center of theenergy band of the isolated wire. Note that D can easilybe evaluated if one takes into account the weak disordercondition (3). In this situation, the eigenvalues of theHamiltonian (8) of the disordered wire are practicallyequal to the corresponding eigenvalues of the Hamilto-nian (8) with (cid:15) n = 0. Therefore, the dispersion relation(2) can be used. In addition, if fixed boundary conditionsare imposed, the wave number takes the discrete values µ q = qπ/ ( N + 1), with q = 1 , . . . N , and D is then simplygiven by D = 2 πN . (12)Therefore, near to the band center κ ≈ γ and the super-radiant transition takes place at κ ≈
1. It is quite inter-esting that the same critical value of κ emerges also inother models such as the Gaussian Orthogonal Ensemble(GOE) of random matrices and two-body random inter-action models [9, 10]. The superradiant transition in the1D Anderson model has already been reported in the lit-erature, see for example Refs. [11, 12]. In addition, theinterplay between supperradiance and disorder has beenpreviously established when all the sites of a lattice arecoupled to a common decay channel [13]. Also, the poledistribution, at perfect coupling, for the tree-dimensionalAnderson model has been studied in [14]. IV. MEAN VALUE AND FLUCTUATIONS OFRESONANCE WIDTHS
In the previous Section, we made a qualitative descrip-tion of the superradiant transition by analyzing the dis-tribution of poles of the S -matrix in the complex plane.Now we focus on how the mean value of resonances (moreprecisely, the mean value of the imaginary part of theeigenvalues) depends on the strength of the coupling tocontinuum. This problem has been studied in detail fornon-Hermitian Hamiltonians, see Eq. (5), in which thereal part H is a full random matrix belonging to one of Γ ω (a) Γ ω (b) Γ ω (c) Γ ω (d) Γ ω (e) Γ ω (f) FIG. 1: Imaginary vs. real part of the S -matrix poles Ω [see Eq. (11)] for 1D disordered wires of length N = 800 coupled tothe continuum with strength γ = 0 . γ = 1 (lower panels). The disorder strength was set to σ = 0 . L loc /N = 10 (left panels); σ = 0 . L loc /N = 1 (middle panels); and σ = 0 . L loc /N = 0 . -18 -14 -10 -6 -2 -2 -1 0 1 2 Γ ω (a) -18 -14 -10 -6 -2 -2 -1 0 1 2 Γ ω (b) -18 -14 -10 -6 -2 -2 -1 0 1 2 Γ ω (c) FIG. 2: Imaginary vs. real part of the S -matrix poles Ω [see Eq. (11)] for 1D disordered wires of length N = 800 coupled tothe continuum with (a) γ = 0 .
1, (b) γ = 1, and (c) γ = 1 .
5. The disorder strength was set to σ = 0 .
01 such that L loc /N = 1.Here, 50 wire realizations were used. The small rectangle in (c) encloses the superradiant states. standard ensembles (for example, to the GOE), and theimaginary part W describes the coupling to continuumthrough a finite number of channels according to Eq. (6);see for example [15] and references therein.One important analytical result, in the case of M chan-nels c = 1 , ..., M , is that the mean width of the resonancesreads [16] (cid:104) Γ (cid:105) = − M D π ln (cid:18) τ − τ + 1 (cid:19) , τ = 12 (cid:0) γ + γ − (cid:1) . (13)Here D stands for the mean energy level spacing of theclosed system at the band center E = 0. Relation (13)is known as the Moldauer-Simonius equation which iswidely used in physics [17]. In the case when H is a mem-ber of the Gaussian Unitary Ensemble (GUE) of random matrices (with M equivalent c − channels) the analyticalresult for the whole distribution of individual widths Γ i has been derived in Ref. [18]. The logarithmic diver-gence of (cid:104) Γ (cid:105) at the critical coupling τ = 1 is a directconsequence of the power law decay of large values of Γ i .However this rigorous result refers to an infinite numberof resonances, and for finite N one has to take into ac-count that (cid:104) Γ (cid:105) remains finite for any value of τ , including τ = 1.In the case of the 1D Anderson model, where theHamiltonian H is given by the tridiagonal matrix ofEq. (8) with diagonal disorder, a rigorous expression for (cid:104) Γ (cid:105) is unknown. However, our expectation is that rela-tion (13) may also be applied to the 1D Anderson model.The physical argument for this expectation is that it may N < Γ > γ (a) N=800N=1200 N < Γ > γ (b) N=800N=1200 N < Γ > γ (c) N=800N=1200
FIG. 3: Average resonance width (cid:104) Γ (cid:105) at the band center as a function of the coupling to the continuum γ . Error bars are thecorresponding standard deviation. (a) L loc /N = 10 (ballistic regime), (b) L loc /N = 1 (chaotic regime), and (c) L loc /N = 0 . not be relevant whether a closed system, described by H , is a one-body or a many-body system. We expectwhether this argument is valid only when the eigenstatesof H , describing the closed system, are fully chaotic. Aswe show, strong differences occur for the model withstrong disorder leading to localized eigenstates.With the use of expression (12) for D , we arrive to thefollowing relation for the mean width of resonances: N (cid:104) Γ (cid:105) = − (cid:18) τ − τ + 1 (cid:19) , (14)where M = 2 is explicitly used. Notice that Eq. (14) isinvariant with respect to the change γ → /γ . In fact,one can show that the whole distribution of poles cor-responding to long-lived states is invariant under such achange. Moreover, transport properties in the 1D Ander-son model have been shown to be still symmetric underthe above change [8]. This symmetry has been observedif H mn in Eq. (8) is replaced by full random matrices (seee.g. Refs. [6, 16]).The validity of Eq. (14) is confirmed in Fig. 3 for thethree regimes (ballistic, chaotic, and localized). We ob-serve an excellent agreement between Eq. (14) and thenumerical data except for the points in the vicinity of γ = 1, where differences are due to finite size effects.Surprisingly, the mean average width is practically in-sensitive to the degree of disorder if γ is not too close to γ = 1.In contrast to the mean width, the standard deviation σ Γ for individual widths depends strongly on the valueof the localization length (see error bars in Fig. 3). In-deed, for the ballistic regime, Fig. 3(a), the region with γ ∼ γ = 1) the fluctuations are so strong that theyare of the same size as compared to the mean value ofΓ. This fact is characteristic of the phase transitions wellstudied in statistical mechanics. < Γ > γ No disorderL loc /N=10L loc /N=1L loc /N=0.1
FIG. 4: Mean resonance width (cid:104) Γ (cid:105) of the two superradiantstates as a function of the coupling to the continuum γ in theballistic, chaotic, and localized regimes for disordered wiresof size N = 800. The energy E was set to zero. Here, 50wire realizations were used. The black full line correspondsto the largest eigenvalue of the non-Hermitian matrix of (5)with (cid:15) n = 0. The analysis of the error bars in Figs. 3 shows thatthey are independent of the system size far away fromthe critical coupling γ = 1. This means that the prod-uct N σ Γ is independent of the system size in all regions.Taking into account that neither N (cid:104) Γ (cid:105) depends on thesystem size, one can conclude that the relative fluctua-tions of Γ should not vanish in the thermodynamic limitand Γ can not be considered as a self averaged quantity.Above, we have focused our attention on the polescorresponding to long-lived states. Now, let us look atthe poles that correspond to the superradiant states. InFig. 4 we plot the mean width of the 2 N d largest-widtheigenvalues, where N d is the number of random realiza-tions of the non-Hermitian Hamiltonian matrix (5). For γ > (cid:104) Γ (cid:105) λ E (a) λ E (b)(b) λ E (c) λ E (d)
FIG. 5: Inverse localization length λ ≡ L − loc , see Eq. (15), as a function of the energy for various correlated disorders. Thepower spectrum W ( µ ) of the correlated on-site energies is given by (a) Eq. (18), (b) Eq. (20), (c) Eq. (22), and (d) Eq. (24).In all cases the disorder intensity was set to σ = 0 .
01. Red-dashed curves correspond to numerical data (wires of length 10 were used to compute λ ), while the continuous black curves correspond to Eq. (15). is practically equal to the largest eigenvalue of the non-Hermitian matrix of (5) with (cid:15) n = 0, see the black fullline. The situation is different in the vicinity of the crit-ical coupling γ = 1, since there the disorder plays animportant role. In this region it is observed that theshorter the localization length L loc the larger the meanwidth. A similar effect is observed for the fluctuationsof the widths Γ which acquire their largest value close tothe critical coupling. V. CORRELATED DISORDER
In this Section we consider the case of weak correlateddisorder, for which the Thouless expression (4) is no morevalid and the corresponding localization length gets theform [1] λ ≡ L − = σ µ W ( µ ) , (15) W ( µ ) = 1 + 2 ∞ (cid:88) m =1 K ( m ) cos(2 µm ) . (16)Here, W ( µ ) is the power spectrum of on-site energies (cid:15) n and K ( m ) is the normalized binary correlator defined as K ( m ) = (cid:104) (cid:15) n (cid:15) n + m (cid:105) σ . (17)It is important to stress that Eq. (15) is valid for weakdisorder, and strong deviations from this formula havebeen found near the band center ( µ = π/
2) and the band edges ( µ = 0 , π ), see details in Ref. [1]. Notice that foruncorrelated disorder we have W ( µ ) = 1 and, therefore,Eq. (15) reduces to Eq. (4). Here, λ ≡ L − is also knownas the Lyapunov exponent. In what follows, we considervarious types of correlations imposed to disordered po-tentials. A. Constant localization length
In comparison with Eq. (4), expression (15) containsthe additional energy-dependent term W ( µ ). This factallows one to impose specific correlations for a given en-ergy dependence of the localization length along the en-ergy band, that is of great interest for various applica-tions [1]. Let us start with the simplest, however, non-trivial case of disorder for which the localization lengthdoes not depend on energy. In this case the power spec-trum takes the form, W ( µ ) = 2 sin µ , (18)therefore, the inverse of the localization length is con-stant: L − = σ /
4. The corresponding binary correlatoris given by K ( m ) = δ m, − δ | m | , . (19)This correlator has only three components, K (0) = 1and K ( ±
1) = − /
2, the other components vanish, thusthe correlations are short-range. Figure. 5(a) shows anexcellent agreement between the numerically obtained L − and Eq. (4), except in the vicinity of the band cen-ter where a clear resonant behavior emerges. The re-gion near the band center has been studied in detail (seeRef. [1] and references therein), however in this paper weare interested in the generic properties of the localizationlength, therefore we focus on the energies far enough fromthe band center and band edges.Correspondingly, in Fig. 6 we present the distributionof the S -matrix poles in the complex plane for the corre-lated disorder with the power spectrum of Eq. (18). Thedisorder increases from left to write panels; specifically,left panels correspond to weak disorder while right panelsto strong disorder. The upper panels show the pole dis-tribution for weak coupling ( γ (cid:28)
1) and the low panelsare given for strong coupling ( γ ≈ H are bothdelocalized and chaotic. This situation occurs when thelocalization length is of the order of the system size (seemiddle panels in Fig. 6). B. Inverse localization length proportional to sin µ In the uncorrelated disorder case we have L − ( µ ) ∝ sin − µ , see Eq. (4); that is, L − has a minimum at E = 0and diverges at the energy band edges E = ±
2. Usingcorrelated disorder it is possible to invert this behavior:i.e., making L − to have a maximum at E = 0 and makeit vanishing at E = ±
2. To this end we use the correla-tions defined by the power spectrum W ( µ ) = 83 sin µ , (20)so that L − = σ sin µ/
3. Here Eq. (20) corresponds tothe following binary correlator: K ( m ) = δ m, − δ | m | , + 16 δ | m | , . (21)Thus, the non-vanishing components of K ( m ) corre-sponds to m = 0, m = ±
1, and m = ±
2. The study of our model with the above correlationsmanifests a good correspondence between the analyticalformula for the localization length (15) and numericaldata as shown in Fig. 5(b). Note that a narrow resonanceat the band center persists, indicating a typical deviationfrom Eq. (15). The corresponding pole distributions areshown in Fig. 7. One can see that the inclusion of thiskind of correlated disorder does not change too much theballistic regime picture. Indeed, the poles in Fig. 7 (leftpanels) are distributed pretty much the same as for thenon-disordered case. As the localization length increasesand reaches the chaotic regime (middle panels), the gapat the band center (clearly seen in the case of uncorre-lated disorder) is now practically negligible. Finally, forthe localized regime (right panels) most of the poles arelocated close to the real axis. A distinctive feature forthis type of correlations is that most of the poles areconcentrated in the vicinity of the band edges ( ω = ± C. Mobility edges I
Now we turn our attention to an interesting situationfor which the power spectrum has the form W ( E ) = (cid:26) π/ ( π − µ ) , − E < E < E , , otherwise . (22)In this case the localization length in the correspond-ing isolated system (for γ = 0) is strongly suppressedinside the energy interval [ − E , E ] and enhanced out-side this interval. Thus, the critical values ± E can betreated as the mobility edges (see details and discussionin Ref. [1]). Note that the value of µ is simply deter-mined by the dispersion relation E = 2 cos µ . In ournumerical simulation, we set E = 1. The power spec-trum (22) corresponds to the following binary correlator: K ( m ) = 1 m ( π − µ ) sin(2 mµ ) . (23)This correlator exhibits a power law decay typical of long-range correlated disorder.The prediction of an effective delocalization transition,occurring in the first order approximation with respectto weak disorder, is corroborated in Fig. 5(c). Here, agood agreement between numerical data and the analyt-ical expression (15) is clearly seen. A strong discrepancyemerges around the band center only, where the influenceof the resonant behavior cannot be neglected.The corresponding distributions of S -matrix poles aredisplayed in Fig. 8. In the left panels (ballistic regime)the poles whose real values belong to energy intervalswhere the localization length is infinite are practicallyequal to the corresponding poles of the non-disorderedsystem. We term these energy intervals with λ = 0 aswindows of transparency, since here the transmission ofwaves is practically perfect. Outside of these windows,where the localization length is finite, a clear deviationfrom the non-disordered case is observed. Γ ω (a) Γ ω (b) Γ ω (c) Γ ω (d) Γ ω (e) Γ ω (f) FIG. 6: Imaginary vs. real part of the S -matrix poles Ω [see Eq. (11)] for 1D wires with correlated disorder defined by thepower spectrum (18). Wires of length N = 865 are coupled to the continuum with strengths γ = 0 . γ = 1(lower panels). The disorder strength was set to σ = 0 . L loc /N = 10 (left panels); σ = 0 . L loc /N ≈ σ = 0 . L loc /N = 0 . Γ ω (a) Γ ω (b) Γ ω (c) Γ ω (d) Γ ω (e) Γ ω (f) FIG. 7: Imaginary vs. real part of the S -matrix poles Ω [see Eq. (11)] for 1D wires with correlated disorder defined by thepower spectrum (20). Wires of length N = 934 are coupled to the continuum with strengths γ = 0 . γ = 1(lower panels). The disorder strength was set to σ = 0 . L loc /N = 10 (left panels); σ = 0 . L loc /N ≈ σ = 0 . L loc /N = 0 . As the ratio L loc /N between the localization lengthand the system size decreases, the poles whose real partsbelong to extended eigenstates in the windows of trans-parency begin to spread around the corresponding poles of the non-disorder case. The spreading of these poles isstronger as L loc /N decreases (see middle and right panelsin Fig. 8). With a further decrease of L loc /N , the eigen-states begin to be strongly localized and one can observean accumulation of poles towards the real axis (see rightpanels of Fig. 8). D. Mobility edges II
Here we show that the windows of transparency andthe regions along the band with strongly localized eigen-states can be easily interchanged by the proper choiceof long-range correlations. With respect to the case dis-cussed in previous Subsection, this can be accomplishedby the following power spectrum: W ( E ) = (cid:26) , − E < E < E ,π/ µ , otherwise . (24)With this type of correlations, the eigenstates whoseeigenvalues belong to the energy interval [ E , E ] are ex-pected to be extended, whereas the eigenvalues locatedoutside of this energy window should correspond to lo-calized eigenstates. This prediction is corroborated bythe numerical data as Fig. 5(d) shows. Notice that nowthe ratio L loc /N between the localization length andthe system size evaluated at the band center is infinite.Therefore, strictly speaking, the system is in the ballis-tic regime independently of the system size and disorderintensity. However, as we will see below, the disorder in-tensity still plays an important role in the distribution ofpoles.The binary correlator that results in the power spec-trum of Eq. (24) is given by K ( m ) = sin(2 µ m )2 µ m , (25)which exhibits a power law decay typical of long-rangecorrelated disorder. The distributions of poles of the S -matrix are displayed in Fig. 9. For very weak disor-der, σ < ∼ .
001 (left panels), the poles whose real partbelongs to the extended eigenstates are practically dis-tributed as for the non-disorder case. As we increase thedisorder these poles begin to spread around the corre-sponding poles of the non-disordered wire. In contrast, inthe complementary energy windows with localized eigen-states most of the poles are located close to the real axiswhile the rest acquire a large imaginary part. Therefore,in these localization windows we have two effects. Onthe one hand the number of poles that are close to thereal axis is increased as the disorder becomes stronger;on the other hand, the imaginary part of the rest of polesbecomes larger (see middle and right panels of Fig. 9).If the disorder is strong enough, the distribution of poleswhose real part belongs to the extended eigenstates is nolonger similar to that of the poles of the non-disorderedwire.
VI. SUMMARY
We have studied 1D open tight-binding disorderedwires paying main attention to the distribution of polesof the S -matrix in dependence on the model parameters.The model essentially depends on the system size N , thestrength γ of the coupling to the leads, the square-root-variance σ of weak diagonal disorder, and on the type ofdisorder. In the first part of the paper we have considereduncorrelated disorder and ask the question of how thepole distribution depends on two key parameters. Oneof these parameters is the ratio L loc /N of the localiza-tion length L loc in the corresponding closed system (for γ = 0) to the system size N . It is known that for per-fect coupling ( γ = 1) this ratio determines all transportproperties of the open system, a fact known as the sin-gle parameter scaling in the theory of localization, seefor example Ref. [3]. According to this parameter, weconsider three characteristic situations: extended eigen-states with L loc (cid:29) N (plane waves slightly modified bydisorder), extended chaotic eigenstates with L loc ∼ N ,and localized eigenstates with L loc (cid:28) N .The analysis of the pole distribution in dependence on L loc /N and γ has shown that for the effectively weakdisorder ( L loc (cid:29) N ) the location of poles in the complexplane follows those occurring for a non-disordered poten-tial ( σ = 0 and γ = 0). In the other limit case of rela-tively strong disorder ( L loc (cid:28) N ) it was found that thepole distribution is very different from the previous case.Specifically, the data clearly demonstrates that, when inthe closed wire the eigenstates are strongly localized, inthe open wire the poles are mainly located close to thereal axis. With the increase of coupling to the leads thiseffect is enhanced. It is important to stress that even forweak disorder, the effect of attraction of the poles to thereal axis cannot be neglected (see the data in Figs. 1 and2).Another question is how the mean value of the imag-inary parts of the poles depends on the strength ofthe coupling to the leads. Note that the poles of the S − matrix give the information about the widths ofthe resonances appearing in the transmission of wavesthrough finite disordered wires. These resonances caneasily be observed experimentally, at least for the poleswith a small imaginary part. Our key idea was that thedependence of the mean value of the imaginary parts ofthe poles on the coupling strength is similar to that de-scribed by the famous Moldauer-Simonius relation (13).Although this relation has been derived for random ma-trix models, one can expect that Eq. (13) is also valid forlow-dimensional models such as the model studied here.The numerical data reported in Fig. 3 clearly supportour expectation. Thus our results indicate that the areaof application of the Moldauer-Simonius relation (13) ismuch broader than it was initially expected.The most important feature described by theMoladauer-Simonious relation is that at the criticalpoint, γ = 1, the mean value of the widths Γ diverges.0 Γ ω (a) Γ ω (b) Γ ω (c) Γ ω (d) Γ ω (e) Γ ω (f) FIG. 8: Imaginary vs. real part of the S -matrix poles Ω [see Eq. (11)] for 1D wires with correlated disorder defined by thepower spectrum (22). Wires of length N = 970 are coupled to the continuum with strengths γ = 0 . γ = 1 (lower panels). The disorder strength was set to σ = 0 . L loc /N = 10 (left panels); σ = 0 . L loc /N ≈ σ = 0 . L loc /N = 0 . Γ ω (a) Γ ω (b) Γ ω (c) Γ ω (d) Γ ω (e) Γ ω (f) FIG. 9: Imaginary vs. real part of the S -matrix poles Ω [see Eq. (11)] for 1D wires with correlated disorder defined by thepower spectrum (24). Wires of length N = 800 are coupled to the continuum with strengths γ = 0 . γ = 1 (lower panels). The disorder strength was set to σ = 0 . L loc /N = 10 (left panels); σ = 0 . L loc /N ≈ σ = 0 . L loc /N = 0 . This remarkable fact is well seen in Fig. 3, also demon-strating an increase of fluctuations of individual Γ at thecritical point. Note that these two effects seem to beindependent of the degree of localization, L loc /N . How- ever, the fluctuations themselves are increased with thedecrease of this ratio, therefore, with the increase of thedisorder. It should be stressed that the divergence of themean of widths at γ = 1 can be treated as an indica-1tion of a phase transition, for which the fluctuations areof the order of the mean values. This effect is knownas the superradiance transition well studied in terms offull random matrices in place of the real part H in non-Hermitian Hamiltonians [9].In order to better present the data characterizing thesuperradiant transition, we have plotted the mean valuesof Γ for two poles (for each wire) with the largest valuesof Γ. In the region γ < γ = 1, twoof the N poles have very large values of their imaginaryparts in comparison with all other poles that move backto the real axis with the increase of the coupling. This ef-fect is clearly seen in Fig. 4, where the mean value of thelargest Γ is plotted. As one can see, for γ < γ > N of the wires,together with the windows where the eigenstates arestrongly localized. In this way, one can speak of effectiveband edges. The data in Fig. 5 demonstrate an excel-lent agreement between the numerically found localiza-tion length and the analytical predictions. However, a quite strong discrepancy occurs for energies close to theband center; an effect which is well studied in the litera-ture (see for example the review in Ref. [1]). The originof this discrepancy is the failure of the standard pertur-bation theory used to derive the analytical expression forthe localization length.With the reference to Fig. 5, where the localizationlength is plotted versus energy, we have analyzed the dis-tribution of poles when correlations are imposed into dis-order in the presence of coupling to the leads. In Figs. 6and 7 the pole distribution is shown for correlated dis-order without mobility edges in dependence on the dis-order strength and on the degree of the coupling. Theanalysis shows that, in general, with the increase of dis-order the poles begin to be more scattered in comparisonto the non-disordered wires. Another conclusion is thatfor both weak and strong disorder the poles tend to beconcentrated near the real axis. As for the correlateddisorder resulting in the mobility edges, see Figs. 8 and9, the most important conclusion is that in the presenceof coupling to the leads the distribution of the poles ofthe scattering matrix mainly follows that occurring inthe absence of disorder, provided the coupling parameter γ is not too large. On the other hand, strong couplingto the continuum essentially modifies the distribution ofpoles, however, mainly in those energy windows wherethe eigenstates are strongly localized. VII. ACKNOWLEDGMENTS
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