1

arXiv:cond-mat/0009198v1 [cond-mat.dis-nn] 13 Sep 2000

Transport Properties and Density of States of Quantum Wires with O?-diagonal Disorder

P. W. Brouwerab ? , Christopher Mudryac ? and Akira Furusakid ?

a b c

Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853, USA

Paul Scherrer Institut, CH-5232, Villigen PSI, Switzerland Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

d

Abstract We review recent work on the random hopping problem in a quasi-one-dimensional geometry of N coupled chains (quantum wire with o?-diagonal disorder). Both density of states and conductance show a remarkable dependence on the parity of N . The theory is compared to numerical simulations. PACS: 71.55.Jv, 71.23.-k, 72.15.Rn, 11.30.Rd Keywords: Disordered Systems, Mesoscopic Systems, and Critical Phenomena.

1. Introduction and Results With the realization that the problem of Anderson localization is amenable to a RG analysis came the understanding that some transport and spectral properties of a quantum particle subjected to a weak random potential depend qualitatively only on dimensionality and the symmetries of the random potential [1]. For essentially all types of randomness the dimensionality two of space plays the role of the lower critical dimension: Below two dimensions and for arbitrary weak disorder a quantum particle is always localized (i.e., the wavefunction is insensitive to a change in boundary conditions, or, equivalently, exponential decay of the conductance with sys? Work

at Harvard is supported by NSF grants nos. DMR 94-16910, DMR 96-30064, and DMR 97-14725. ? Supported in part by a grant from the Swiss Nationalfonds. ? Supported in part by Grant-in-Aid for Scienti?c Research (No. 11740199) from the Ministry of Education, Science, Sports and Culture, Japan. The numerical calculations were performed at the Yukawa Institute Computer Facility.

tem size), whereas a ?nite amount of disorder is needed to localize a quantum particle in three dimensions. A notorious exception to this rule is the case of a particle on a single chain with random nearest neighbor hopping which appears in many reincarnations, e.g., random XY spin chains [2] and di?usion in random environments [3]. In all these cases, the cause underlying the di?erent behavior of random systems with o?-diagonal disorder is the existence of a sublattice, or chiral, symmetry, which is absent for diagonal disorder [4]. The purpose of this contribution is to review our recent work (together with Simons and Altland [5]) on the random hopping problem in a quasi-one-dimensional “wire” geometry [5–8]. We consider the density of states (DoS) and conductance, and focus on the di?erences between the cases of o?-diagonal and diagonal disorder in the localized regime. (For o?-diagonal disorder, we assume absence of diagonal disorder.) The quantum wire is the logical intermediate be-

2 tween one and two dimensions.1 The quasi-onedimensional geometry allows us to ?nd exact solutions for the conductance at the band center ε = 0 and the DoS near ε = 0, while, in contrast to purely one-dimensional counterparts, it still shows a crossover from a di?usive to a localized regime, as is the case in two dimensions. In the di?usive regime, di?erences between o?diagonal and diagonal disorder are limited to small quantum corrections to the DoS and conductance. The DoS near ε = 0 follows from the so-called “chiral” random matrix theory (RMT), which was originally proposed in the context of quenched approximations to the QCD Hamiltonian [11]. In chiral RMT, all eigenvalues come in pairs ±ε. For N coupled random hopping chains, level repulsion between the smallest eigenvalue above the band center ε = 0 and its mirror image then leads to a suppression of the DoS near ε = 0,

?1 dN /dε ∝ vF (ε/?)β ?1 , 0 < ε ? ?.

center [12] : dN 1 , ∝ 3 dε ξ |ε ln (εξ/vF )| 0 < εξ/vF ? 1. (2)

Note that, in contrast to the di?usive regime (1), the energy scale that governs the divergence does not depend on the system size, nor does the form of the singularity depend on the symmetry index β . For N coupled chains, Dyson’s result (2), remains valid for odd N ,2 but not for even N [8]: For an even number of chains, the DoS diverges only logarithmically for β = 1, whereas dN /dε shows a pseudogap for β = 2, 4, 1 dN ∝ dε vF εξ vF

β ?1

ln

εξ , vF

0<

εξ ? 1. (3) vF

(1)

Here dN /dε is the DoS per unit volume, vF is the Fermi velocity, ? = 2πvF /N L is the mean level spacing, and the symmetry index β = 1 in the presence of both time-reversal and spin-rotational symmetry and β = 2 (4) if time-reversal symmetry (spin-rotational symmetry) is broken. Several level spacings away from the center of the band, the mirror symmetry of the spectrum plays no important role, and the level statistics were found to agree with those of the standard RMT. Similarly, di?erent quantum corrections to the conductance are obtained for energies close to the band center [6]. On the other hand, beyond the di?usive regime, i.e., for sample lengths L ξ , where ξ is a crossover length scale (to be de?ned after Eq. (4) below), quantum corrections are dominant, and the di?erences between diagonal and o?diagonal disorder can be much more dramatic. While for diagonal disorder, the DoS is constant as ε → 0, Dyson showed that for o?-diagonal disorder dN /dε diverges upon approaching the band

1 Two

The relevant energy scale, however, remains the same as in Eq. (2). The even-odd e?ect in the DoS around ε = 0 is accompanied by an even-odd e?ect for the conductance g at the band center [5– 7]. For odd N , there is no exponential localization; g has a broad distribution, with ?uctuations that are of the same order as the average, ln g = ?4 8(π ? 2)L L , var ln g = , βπξ βπξ (4)

where the crossover length scale ξ = (βN + 2 ? β )?/β , and ? is the mean free path. (For small N there are corrections due to the appearance of a second dimensionless parameter needed to characterize the o?-diagonal disorder, see Ref. [7] for details.) In contrast, for even N , g is exponentially small, its distribution being close to lognormal, ln g = ? 2L + ξ 8L 8(π ? 1)L , var ln g = .(5) βπξ βπξ

dimensional models with o?-diagonal disorder are also studied in the context of strongly interacting electron systems [9]. We refer the reader to Ref. [10] for a review of work done on the two dimensional case.

In this case, ξ can be interpreted as the localization length. Away from the band center and for diagonal disorder, ln g = ?(1/2)var ln g = ?2L/ξ ′ , with ξ ′ = (βN + 2 ? β )?, irrespective of the parity of N , as for quantum wires with diagonal disorder [13]. Note that for even N , despite the similarities, di?erences between o?-diagonal

2 The

proportionality constants in Eqs. (2) and (3) may depend on N .

3 and diagonal disorder persist, though they are more subtle. In the remaining sections of this paper, we review the theory describing these two parity effects. 2. Two microscopic models 2.1. Lattice model The random hopping model is a lattice model with nearest neighbor hopping only, where the hopping amplitude takes a di?erent (random) value between adjacent sites. In our case we consider a two-dimensional square lattice, for which the Schroedinger Equation takes the form Hψi,j =

±

center ε = 0 is a special energy with respect to this transformation. As we shall see below, it is the existence of this extra symmetry that causes the spectral and transport properties of the random hopping model at the band center to be so dramatically di?erent from those of models with on-site disorder. For historical reasons, the sublattice symmetry is referred to as chiral symmetry. 2.2. Continuum model While the lattice version (6) of the random hopping model is the version that is usually considered in the literature, we use a di?erent (continuum) model for our analytical calculations, Hcont ψ (y ) = [iσ3 ?y + σ3 v (y ) + σ2 w(y )] ψ (y ). (7) Here ψ is a 2N component vector (elements of ψ occur in pairs that correspond to left and right movers), v and w are N × N random Hermitean matrices, and the σ? (? = 1, 2, 3) are the Pauli matrices. In Eq. (7) and below we choose our units such that the Fermi velocity vF = 1. The chiral symmetry is now represented by σ1 Hcont σ1 = ?Hcont . For β = 1, w (v ) is (anti-)symmetric; for β = 4, w (v ) is (anti)-self dual. The disorder potentials v and w are chosen independently and Gaussian distributed with zero mean and with variance

? vij (y )vkl (y ′ ) =

(ti,j ;i±1,j ψi±1,j + ti,j ;i,j ±1 ψi,j ±1 ) . (6)

Hermiticity requires ti,j ;i±1,j = (ti±1,j ;i,j )? and ti,j ;i,j ±1 = (ti,j ±1;i,j )? . We consider a quasi-onedimensional geometry (length L of the disordered region much larger than its width N a, where N is the number of chains and a the lattice constant), with open boundary conditions in the transverse direction: t0,j ;1,j = tN,j ;N +1,j = 0. On the left (j < 0) and right (j > L/a), the disordered region is attached to ideal leads (ti,j ;i±1,j = t , ti,j ;i,j ±1 = t⊥ for j < 0 or j > L/a). In the disordered region, the hopping amplitudes show ?uctuations around the average values t and t⊥ , respectively. The ?uctuations are real, complex, or quaternion, for the symmetry classes β = 1, 2, and 4, respectively. The random ?ux model is a special version of the random hopping model for which each plaquette is threaded by a random ?ux. In that case, the hopping amplitudes have magnitude t and a random phase φi,j ;i±1,j , so that the total phase accumulated going around a plaquette equals the phase through the plaquette in units of the ?ux quantum Φ0 . The random hopping model is special because of the existence of a sublattice symmetry: Under a transformation ψi,j → (?1)i+j ψi,j , the Hamiltonian changes sign. Hence, if ε is an eigenvalue of H, then ?ε is so as well. The band

δ (y ? y ′ ) δik δjl ? ξ δ (y ? y ′ ) δik δjl + ξ

2?β β δil δjk

,

? wij (y )wkl (y ′ ) =

2?β β δil δjk

,

respectively. (For small N a slightly more general form of the variance is needed, see [7].) The justi?cation for this change of models is our anticipation that, for weak disorder and for a suf?ciently large system size, only the fundamental symmetries of the microscopic model are relevant. In order to verify this assumption of universality, we compare the theoretical predictions based on Eq. (7) below to numerical simulations of the lattice model (6).

4 3. Fokker-Planck approach 3.1. General method Spectral and transport properties of the continuum model (7) can be studied in a uni?ed way from the statistical distribution of the re?ection matrix r(ε) of the disordered wire. The re?ection matrix is de?ned from the relation between the iL oL amplitudes ψε and ψε of an incoming and an outgoing wavefunction at energy ε on the same (left) side of the disordered region (Fig. 1),

oL iL = r(ε)ψε . ψε

(8)

Transport properties (the conductance G) follow from the eigenvalues of r? r, when the right side of the quantum wire is attached to a lead (Fig. 1a), G= 2 e2 g, g = tr 1 ? r? r , h (9)

while the DoS is obtained from the eigenphases of r when the quantum wire is closed at the right side [14] (Fig. 1b) 1 ? dN = tr r? r . dε 2π iN L ?ε (10)

where V = 0 dy v (y ), and W = 0 dy w(y ). Here we neglected terms that are of order δL2 . Together with the distribution of the disorder potentials w and v , Eqs. (11)–(12) de?ne how the distribution of the re?ection matrix r evolves with the length L of the quantum wire. The RG approach can be represented in terms of a FokkerPlanck equation describing the “Brownian motion” of the eigenvalues of r? r or eigenphases of r in the geometry of Fig. 1a and 1b, respectively. Equations (11)–(12) are exact for the continuum model (7). A di?erent choice for the distribution of v and w, or use of the lattice model (6) instead of Eq. (7), would have led to di?erent statistical properties of the scattering matrix for a thin slice. However, as we have veri?ed numerically, such di?erences are irrelevant in the RG sense, i.e., they disappear for su?ciently long wires (longer than the mean free path ?) and weak disorder (? much larger than the lattice constant a). 3.2. DoS Since the quantum wire is closed at one end, the initial condition at L = 0 for the RG equation (11) is r = 1. The RG ?ow ensures that the re?ection matrix r remains a N × N unitary matrix for all L. Hence, for all L there exists a Hermitean N × N matrix Φ with r = exp(iΦ). The eigenvalues φj of Φ are the eigenphases of r. The change of Φ under the increment (11)–(12) is cot Φ 2 → e?(W +iV ) cot Φ + εδL e?(W ?iV ) 2

δL

δL

The distribution of r is computed using a RG approach: One calculates how the eigenvalues of r? r (in the case of the conductance) or the eigenphases of r (in the case of the DoS) change when a thin slice is added to the disordered region (Fig. 2). If we place the thin slice to the left of the disordered region, the change of the re?ection matrix is given by

′ r → r1 + t′ 1 (1 ? r r1 ) ?1

r t1 ,

(11)

′ where r1 and r1 (t1 and t′ 1 ) are the re?ection (transmission) matrices of the thin slice (Fig. 2). If the length δL of the thin slice is much smaller than a mean free path, these can be computed using the Born approximation,

up to second order in V and W and ?rst order in ε. Taking the length L as a ?ctitious “time”, the eigenphases φj of Φ perform a Brownian motion on the unit circle, which is such that upon increasing L, they move (on average) counterclockwise. Integration of Eq. (10) yields a relation between the φj and the DoS valid in the limit of large L, N (ε′ ) =

0 ε′

r1 t1

′ r1 t′ 1

= = = =

i (V W ? W V ), ?W + 2 1 (V 2 + W 2 ) + 1 + iV ? 2 i (V W ? W V ), W+2 2 2 1 ? iV ? 1 2 (V + W ) +

dε

1 dN = dε 2πN

iεδL, iεδL, (12)

j

?φj . ?L

(13)

[Note that for ε = 0, φj = 0 for all L.] Hence, to ?nd the (average) DoS per unit volume, we compute the (average) “current” of the eigenphases

5 φj moving around the unit circle and di?erentiate to energy. We remark that, in the absence of disorder, the angles φj move around at a constant speed ∝ ε, resulting in a constant DoS. With disorder, their motion acquires a random (Brownian) component, which dramatically a?ects their average speed, and hence the DoS. For a quantitative description a di?erent parameterization of the eigenphases of the re?ection matrix is more convenient, tan(φj /2) = exp(uj ), (14) thus enhancing the current, and hence the DoS [8]. The long-range repulsion Fint , on the other hand, traps an equal number of the particles in two “traps” near u = ln(εξ ) and u = ? ln(εξ )+πi, and thus tends to suppress the DoS. For even N , all particles are trapped (N/2 particles in each trap). Rare events caused by particles that are “thermally” excited out of their traps still allows for a small “current”, resulting in the small DoS in Eq. (3) (Fig. 3a). For odd N , however, only (N ? 1)/2 particles are trapped near each end point, while one particle is left alone, una?ected by the long-range repulsion (Fig. 3b). Motion of this particle is dominated by di?usion, which explains the enhancement of the DoS for odd N . The agreement between the theory and numerical simulation of the lattice random hopping model is excellent as is depicted in Fig. 4. 3.3. Conductance The chiral symmetry results in the symmetry relation r(ε) = r(?ε)? [6]. Hence, at the band center, the re?ection matrix is Hermitean. We parameterize the eigenvalues of r as tanh xj , where the xj are N real numbers, and formulate the RG approach of Sec. 3.1 as a Brownian motion process for the xj . Away from the band center, only the eigenvalues tanh2 xj of the product r? r can be studied in the RG approach: In this case, the sign of the xj has no relevance, since r? r does not change under xj → ?xj . In terms of the xj , the conductance reads g=

j

where the uj are restricted to the two branches Im uj = 0 and Im uj = π in the complex plane (Fig. 3). Starting from the RG equation (11), one then obtains that the joint probability distribution P (u1 , . . . , uN ) for the uj obeys ?P = ?L J =

j<k

j

? ?uj

? ?1 4 J ? 2ε cosh(uj ) P, J βξ ?uj (15)

sinhβ [(uj ? uk )/2].

This Fokker-Planck equation describes the motion of N ?ctitious Brownian “particles” with coordinates uj (j = 1, . . . , N ) and di?usion coe?cient 4/βξ , subject to a driving force Fε = 2ε cosh uj and a long-range repulsive two-body interaction Fint = (2/ξ ) coth[(uj ? uk )/2]. The parameterization (14) is such that a Brownian particle with coordinate uj that vanishes on one of the branches at ±∞ reappears at the opposite branch (dotted arrows in Fig. 3). Analysis of the Brownian motion described by Eq. (15) then leads to the asymptotes (2,3) for the DoS in the random hopping model. A detailed derivation of these results based on an estimate of the steady-state-current supported by Eq. (15) can be found in Ref. [8]. Here we present only a brief sketch, focusing on the origin of the even-odd e?ect: The competition between the driving force Fε and di?usion on the one side, and the long-range repulsive two-body interaction Fint on the other side. The driving force Fε pushes the particles to the right (left) on the lower (upper) branch and thus causes the nonzero steady-state-current. Di?usion keeps the particles mobile where the driving force Fε is small,

(1 ? tanh2 xj ) =

j

cosh?2 xj .

(16)

The RG equation (11) then leads to a FokkerPlanck equation for the L-evolution of the joint probability distribution P (x1 , . . . , xN ) at the band center [5–7], 1 ?P = ?L βξ J=

k>j N j =1

? ? J J ?1 P ?xj ?xj

,

(17)

| sinh(xj ? xk )|β .

Su?ciently far away from the band center, the Levolution of the xj ’s is the same as for a quantum

6 wire with on-site disorder, which is given by the well-known DMPK equation [15] ?P 1 = ′ ?L 2ξ J=

k>j N j =1

? ? J J ?1 P ?xj ?xj

,

(18)

| sinh2 xj ? sinh2 xk |β

k

| sinh(2xj )|,

where ξ ′ = (βN + 2 ? β )?. Both Fokker-Planck equations describe the Brownian motion of N particles with the coordinate xj on the real axis and interacting through a hardcore repulsive two-body potential (Fig. 5). Far away from the band center, a particle at position xj is also repelled by the mirror image at position ?xj as is required by the symmetry of r? r under xj → ?xj . For each xj , the repulsion from the mirror image ?xj ensures that all xj increase linearly with L, thus causing the exponential decay of g with L. On the other hand, at the band center the sign of xj matters since there is no repulsion from mirror images. For odd N there will be no net force on the particle with coordinate x(N +1)/2 (for large L). Hence x(N +1)/2 remains close to the origin, while all other xj increase (or decrease) linearly with L. The coordinate x(N +1)/2 dominates the conductance, its ?uctuations leading to the (L/ξ )1/2 dependence of ln g in Eq. (4). For even N no such “exceptional” particle exists; all xj increase (if they are positive) or decrease (if the are negative) linearly with L, causing the conductance to be exponentially small, cf. Eq. (5). We refer the reader to Refs. [6] for the detailed calculation of the moments of the conductance. The agreement between the theory and numerical simulations of the random ?ux case is excellent as is illustrated in Fig. 6. REFERENCES 1. For a review, see P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57 (1985) 287. 2. B. M. McCoy and T. T. Wu, Phys. Rev. 176 (1968) 631. 3. D. S. Fisher, Phys. Rev. A 30 (1984) 960. 4. A. Altland, B. D. Simons, J. Phys. A32 (1999) L353-L359.

5. P. W. Brouwer, C. Mudry, B. D. Simons, A. Altland, Phys. Rev. Lett. 81 (1998) 862. 6. C. Mudry, P. W. Brouwer, A. Furusaki, Phys. Rev. B 59 (1999) 13221; cond-mat/0004042. 7. P. W. Brouwer, C. Mudry, A. Furusaki, Nucl. Phys. B 565 [FS], 653 (2000). 8. P. W. Brouwer, C. Mudry, A. Furusaki, Phys. Rev. Lett. 84, 2913 (2000). 9. N. Nagaosa and P. A. Lee, Phys. Rev. Lett. 64 (1990) 2450, V. Kalmeyer and S. C. Zhang, Phys. Rev. B 46 (1992) 9889; B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47 (1993) 7312. 10. A. Furusaki, Phys. Rev. Lett. 82 (1999) 604. 11. T. Nagao and K. Slevin, J. Math. Phys. 34 (1993) 2075; J. J. M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70, (1993) 3852; S. Hikami and A. Zee, Nucl. Phys. B 408 (1993) 415; A. V. Andreev, B. D. Simons, and N. Taniguchi, Nucl. Phys. B 432 (1994) 487. 12. F. J. Dyson, Phys. Rev. 92 (1953) 1331. 13. For a review, see C. W. J. Beenakker, Rev. Mod. Phys. 69 (1997) 731. 14. H. Schmidt, Phys. Rev. 105 (1957) 425; M. B¨ uttiker, J. Phys.: Condens. Matter 5 (1993) 9361. 15. O. N. Dorokhov, Pis’ma Zh. Eksp. Teor. Fiz. 36 (1982) 259 [JETP Lett. 36 (1982) 318]; P. A. Mello, P. Pereyra, and N. Kumar, Ann. Phys. (N.Y.) 181 (1988) 290.

7

1

(εdN/dε)

dN/dε

N=1 N=3, β=1 N=3, β=2

?1/3

(a)

40 20

N=1 N=3, β=1 N=3, β=2

(a) →@ @ @ @ @ @ @ @ @ @ @ @ ←@ @ @ @ @ @ @ @ @ @ @ @ FIG. 1.

(b) →@ @ @ @ @ @ @ @ @ @ @ @ @ @ ←@ @ @ @ @ @ @ @ @ @ @ @ @ @

0.5

0 ?7 ?6 ?5 ?4 ?3 10 10 10 10 10

ε

dN/dε

0.4 0.2 0

N=2, β=1 N=2, β=2 N=4, β=1 N=4, β=2

dN/dε

Boundary conditions used for the computation of the conductance (a) and DoS (b).

0

(b)

3 2 1

N=2, β=1 N=4, β=1

0 ?7 ?6 ?5 ?4 ?3 10 10 10 10 10

ε

@ @ t′ → t1 1 ← @ @ @ @ @ @ @ ′ @ r1 ←? @ ?→ r1 @ @ @ @ @ δL

@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ r ←? @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ L

0

0.05

ε

0.1

FIG. 2. A thin disordered slice of length δL is added to the left of a disordered wire of length L.

(a)

Density of states, from numerical simulations for N = 1, 3 (a) and for N = 2, 4 (b). The data shown on the linear scale are computed for L = 200 (L = 500 for N = 1) while the data in the insets are for L = 105 (L = 106 for N = 1). For β = 1 and also for N = 1 the hopping amplitudes are taken from a uniform distribution in the interval [0.5,1.5], while for β = 2 the random ?ux model [6] is used, where the randomness is introduced only via the random phases of the hopping amplitudes. Results of an average over 4 × 104 –106 disorder realizations are shown.

FIG. 4.

sssss

I

R sssss (b) sssss I (a) (b) s (c) FIG. 5.

x N +3 2 u x N +5 2 e ?xN ?1 u

R sssss

FIG. 3. The L-dependence of the eigenphases of r is described in terms of a Brownian motion of ?ctitious particles with coordinate uj where Im u = 0 or π . If N is even, a repulsive interaction (see text) traps all particles near u = ? ln εξ or ln εξ + iπ and prohibits them from moving around (a). If N is odd, one particle can di?use freely around the branches, and thus leads to a signi?cantly higher DoS than for even N (b).

u u u x N +1 xN x N ?1 2 2 2 u u u u x N +3 x N +1 x N ?1 x N ?3 2 2 2 2 e u u ?xN xN xN ?1

For a wire with o?-diagonal disorder, the L-dependence of the eigenvalues tanh2 xj of r ? r is described in terms of a Brownian motion of ?ctitious particles with coordinate xj on the real axis. If N is even, all xj (full circles) repel away from 0 (a), while x(N +1)/2 remains close to 0 for odd N (b). In the case of diagonal disorder, all xj repel from 0 due to repulsion from mirror images (open circles) (c).

8

?(ξ/L) <ln g>

1/2

3

2 1 0 0.0 0.5 1.0

1/2

(ξ/L) var ln g

2 1 0

1.5

(ξ / L)

N=1 N=3, β=1 N=3, β=2

(a)

?(ξ/L)<ln g>

2 1 0 0.0

(ξ/L) var ln g

5

0.5

1.0

1/2

1.5

0

N=2, β=1 N=2, β=2 N=4, β=1 N=4, β=2

(ξ / L)

(b)

0

ξ/L

0.5

1

FIG. 6. Average and variance of ln g versus L/ξ for N = 1, 3 (a) and N = 2, 4 (b). The hopping amplitude t is taken from a uniform distribution in the interval [0.9,1.1], while t⊥ is a real random numbers in [-0.2,0.2] for β = 1 and a complex random numbers with modulus < 0.1 for β = 2. The characteristic length ξ is obtained from a comparison with Eqs. (4) and (5). Results of an average over more than 2 × 104 disorder realizations are shown.