Functionals of the Brownian motion, localization and metric graphs

arXiv:cond-mat/0504513v2 [cond-mat.dis-nn] 7 Oct 2005

Functionals of the Brownian motion, localization and metric graphs Alain Comtet(a,b) , Jean Desbois(a) and Christophe Texier(a,c) July 22, 2005 (a)

Laboratoire de Physique Théorique et Modèles Statistiques, UMR 8626 du CNRS. Université Paris-Sud, Bât. 100, F-91405 Orsay Cedex, France.

(b)

Institut Henri Poincaré, 11 rue Pierre et Marie Curie, F-75005 Paris, France.

(c)

Laboratoire de Physique des Solides, UMR 8502 du CNRS. Université Paris-Sud, Bât. 510, F-91405 Orsay Cedex, France. Abstract

We review several results related to the problem of a quantum particle in a random environment. In an introductory part, we recall how several functionals of the Brownian motion arise in the study of electronic transport in weakly disordered metals (weak localization). Two aspects of the physics of the one-dimensional strong localization are reviewed : some properties of the scattering by a random potential (time delay distribution) and a study of the spectrum of a random potential on a bounded domain (the extreme value statistics of the eigenvalues). Then we mention several results concerning the diffusion on graphs, and more generally the spectral properties of the Schrödinger operator on graphs. The interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated. Finally, we consider a two-dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices. We recall the connection between spectral properties of this model and winding functionals of the planar Brownian motion.

PACS numbers : 72.15.Rn ; 73.20.Fz ; 02.50.-r ; 05.40.Jc.

Contents 1 Introduction 1.1 Weak and strong localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Overview of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 2 4

2 Functionals of the Brownian motion in the context of weak localization 4 2.1 Feynman paths, Brownian motion and weak localization . . . . . . . . . . . . . 4 2.2 Planar Brownian motion : stochastic area, winding and magnetoconductance . . 5 2.3 Dephasing due to electron-electron interaction and functionals of Brownian bridges 6 3 Exponential functionals of Brownian motion and Wigner time delay 3.1 Historical perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Wigner time delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 8 9

4 Extreme value spectral statistics 4.1 Distribution of the n-th eigenvalue : relation with a first exit time distribution .

11 12

1

5 Trace formulae, spectral determinant and diffusion 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 5.2 Description of metric graphs . . . . . . . . . . . . . . 5.3 Trace formulae and zeta functions . . . . . . . . . . 5.4 Spectral determinant . . . . . . . . . . . . . . . . . . 5.5 Quantum chaos on metric graphs . . . . . . . . . . . 5.6 Weak localization and Brownian motion on graphs .

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6 Planar Brownian motion and charged particle in random magnetic field

32

7 Conclusion

36

Bibliography

38

1 1.1

Introduction Weak and strong localization

At low temperature, the electric conductivity σ of metals and weakly disordered semiconductors is determined by the scattering of electrons on impurities. It is given by the Drude formula σ0 =

ne e2 τe , m

(1)

where e and m are the charge and mass of the electron respectively. ne is the electronic density and τe the elastic scattering time1 . This purely classical formula is only valid in a regime where quantum mechanical effects can be neglected. This is the case if the elastic mean free path of electrons ℓe = vF τe is large compared with the De Broglie wave length λF = 2π~/mvF corresponding to the Fermi energy (vF is the Fermi velocity). Strong localization.– When these two length scales are of the same order ℓe ≈ λF (strong disorder) the fact that the electrons are quantum objects must be taken into account and the wave like character of these particles is of primary importance. It is indeed this wave character which is responsible for the localization phenomenon. The multiple scattering on impurities distributed randomly in space creates random phases between these different waves which can interfere destructively. These interference effects reduce the electronic conductivity. In the extreme case of very strong disorder the waves do not propagate anymore and the system becomes insulating. This is the strong localization phenomenon which was conjectured by Anderson in 1958. Weak localization.– In the 80’s it was realized that even far from the strong localization regime the quantum transport is affected by the disorder. Diagrammatic techniques used in the weak disorder limit λF ≪ ℓe , were initiated by the works of Al’tshuler, Aronov, Gor’kov, Khmel’nitzki˘ı, Larkin and Lee [116, 12] (see ref. [13] for an introduction and ref. [4] for a recent presentation). In this regime, called the weak localization regime, the Drude conductivity gets a small sample dependent correction whose average, denoted h∆σi, is called the “weak localization correction”. h· · ·i denotes averaging with respect to the random potential. From the experimental side, this phenomenon is well established and has been the subject of many studies (see refs. [34, 16] for review articles). 1 The elastic scattering time τe is the time characterizing the relaxation of the direction of the momentum of the electron. The conductivity is proportional to τe for isotropic scattering by impurities. For anisotropic scattering the conductivity involves a different time τtr called the “transport” time [17] (see ref. [4] for a discussion within the perturbative approach).

2

Phase coherence and dimensional reduction.– The localization phenomenon comes from the interplay between the wave nature of electronic transport and the disorder. This manifestation of quantum interferences requires that the phase of the electronic wave is well defined, however several mechanisms limit the phase coherence of electrons in metals, among which are the effect of the vibrations of the crystal (electron-phonon interaction) or the electron-electron interaction. We introduce a length scale Lϕ , the phase coherence length, that characterizes the length over which phase breaking phenomenon becomes effective. The lack of phase coherence in real systems is the reason why the strong localization regime has not been observed in experiments on metals. In dimension d = 3 the strong localization regime is only expected to occur for sufficiently strong disorder2 . However, strong localization can also be observed in the weak disorder limit (λF ≪ ℓe ) by reducing the dimensionnality3 . It has been shown in the framework of random matrix theory that the localization length of a weakly disordered quasi-1d wire behaves as λ ∼ Nc ℓe , where Nc is the number of conducting channels4 [79, 80, 166, 30]. Therefore the strong localization regime is expected to occur when coherence is kept at least over a scale λ. At low temperature, in the absence of magnetic impurity, phase breaking mechanisms are dominated by electron-electron interaction, which leads to a divergence of the phase coherence length Lϕ (T ) ∝ (Nc /T )1/3 predicted in ref. [9] and verified in several experiments like [224, 213, 84] (see ref. [180] for recent measurements down to 40mK). Therefore the temperature below which strong localization might be observable is given by Lϕ (T∗ ) ∼ λ, which leads to T∗ ∼ 1/(Nc2 d τe ). The crossover temperature in metals is out of the experimental range, however it becomes reachable in wires etched at the interface of two semiconducting materials, when the number of conducting channels is highly reduced, and the manifestation of the strong localization has been observed in ref. [109]. Strictly one-dimensional case.– In a weakly disordered and coherent quasi-1d wire, the weak localization regime only occurs for length scales intermediate between the elastic mean free path and the localization length ℓe ≪ L ≪ λ. In the strictly one dimensional case, since λ ≃ 4ℓe for weak disorder5 , such a regime does not exist and the system is either ballistic (ℓe ≫ L) or strongly localized (ℓe ≪ L). Anderson localization in one dimension has been studied by mathematicians and mathematical physicists from the view point of spectral analysis and in connection with limit theorems for products of random matrices [45]. A breakthrough was the proof of wave localization in one dimension by Gol’dshtein, Molchanov & Pastur [113]. Although some progress has been made, the multidimensional case is still out of reach and the subject of weak localization has almost not been touched in the mathematical literature. One of the main fields of interest in the last twenty years is the investigation of random Schr¨ odinger operators in the presence of magnetic fields (see for example the refs. [220, 46, 77, 78, 100, 101] ; for recent results on Lifshitz tails with magnetic field see refs. [101, 153]). From the physics side significant advances have been realized. Field theoretical methods based on supersymmetry provide a general framework for 2

Following the scaling ideas initiated by Thouless [156] and Wegner [221, 219] it was shown in ref. [1] that the localization-delocalization transition exists only for dimension d > 3. In d = 1 and d = 2, the fully coherent system is always strongly localized, whatever the strength of the disorder is. 3 The effective dimension of the system is obtained by comparing the sample size with the phase coherence length Lϕ . For example, a long wire of length L and of transverse dimension W is effectively in a 1d regime if W ≪ Lϕ ≪ L. 4 The localization length predicted by the random matrix theory (RMT) can be found in ref. [30] : λRMT = [β(Nc − 1) + 2]ℓe , where β = 1, 2, 4 is the Dyson index describing orthogonal, unitary and symplectic ensembles, respectively. Note that one must be careful with the coefficient involved in this relation since the definitions of λRMT and ℓe differ slightly in RMT and in perturbation theory. 5 The elastic mean free path ℓe = vF τe is given by the self energy 1/(2τe ) = − Im ΣR (E). For example, for a Gaussian disorder with local correlations, hV (x)V (x′ )i = wδ(x − x′ ), we obtain 1/τe ≃ 2πρ0 w for a weak disorder, where ρ0 is the free density of states. In one dimension ρ0 = 1/(πvF ), therefore ℓe ≃ vF2 /(2w), which coincides with the high energy (weak disorder) expansion of the localization length λ ≃ 2vF2 /w [15, 157].

3

disordered systems and also allow establishing some links with quantum chaos [86]. While writing this review, we have tried to collect a large list of references which is however far from being exhaustive. Ref. [157] provides a reference book on strong localization, mostly focused on spectrum and localization properties (see also ref. [158] for a review on 1d discrete models and Lifshitz tails). A recent text about disorder and random matrix theory is ref. [86]. Many excellent reviews have been written on weak localization among which [34, 55] (for the role of disorder and electron-electron interaction, see the book [87] or [152]). A recent reference is [4].

1.2

Overview of the paper

Section 2 shows that several functionals of the Brownian motion arise in the study of electronic transport in weakly disordered metals or semiconductors (weakly localized). The brief presentation of weak localization given in sections 2.1 and 2.2 follows the heuristic discussion of refs. [13, 55], which is based on the picture proposed by Khmel’nitzki˘ı & Larkin. In spite of its heuristic character, it allows drawing suggestive connections with well known functionals of the Brownian motion. The sections 3 and 4 deal with problems of strong localization in one dimension. A powerful approach to handle such problems is the phase formalism6 (presented in refs. [15, 157] for instance). This formalism leads to a broad variety of stochastic processes. Section 3 discusses scattering properties of a random potential and section 4 studies spectral properties of a Schr¨ odinger operator defined on a finite interval. In section 5, we review some results obtained for networks of wires (graphs), which can be viewed as systems of intermediate dimension between one and two. We will put the emphasis on spectral determinants, which appear to be an efficient tool to construct several generating functions characterizing the diffusion on graphs (or its discrete version, the random walk). Finally, in section 6, we show that the physics of a two dimensional quantum particle submitted to the magnetic field of an assembly of randomly distributed magnetic vortices involves fine properties of the planar Brownian motion.

2

Functionals of the Brownian motion in the context of weak localization

2.1

Feynman paths, Brownian motion and weak localization

In the path integral formulation of quantum mechanics, each trajectory is weighted with a phase factor eiS/~ where S is the classical action evaluated along this trajectory. The superposition principle states that the amplitude of propagation of a particle between two different points is given by the sum of amplitudes over all paths connecting these two points. This formulation of quantum mechanics is most useful when there is a small parameter with respect to which one can make a quasiclassical expansion. One encounters a similar situation in the derivation of geometrical optics starting from wave optics. In this case the small parameter is the ratio of the wave length to the typical distances which are involved in the problem and the classical paths are light rays. In the context of weak localization the small parameter is λF /ℓe and the quasiclassical approximation amounts to sum over a certain subset of Brownian paths. The fact that Brownian paths come into the problem is not so surprising if one goes back to our previous physical picture of electrons performing random walk due to the scattering by the impurities. 6

The phase formalism is a continuous version of the Dyson-Schmidt method [83, 188]. A nice presentation can be found in ref. [158].

4

In the continuum limit, describing the physics at length scales much larger than ℓe , this random walk may be described as a Brownian motion. It can be shown that h∆σi is related to the time integrated probability for an electron to come back to its initial position (see ref. [55] for a heuristic derivation or ref. [4] for a recent derivation in space representation) : Z 2e2 D ∞ dt P(~r, t|~r, 0) e−t/τϕ . (2) h∆σi = − π~ τe where D is the diffusion constant and the factor 2 accounts for spin degeneracy. P(~r, t|~r ′ , 0) is the Green’s function of the diffusion equation ∂ − D∆ P(~r, t|~r ′ , 0) = δ(~r − ~r ′ ) δ(t) . (3) ∂t In eq. (2) the exponential damping at large time describes the lack of phase coherence due to inelastic processes. The phase coherence time τϕ can be related to the phase coherence length by L2ϕ = Dτϕ . We have also introduced a cut-off at short time in eq. (2), that takes into account the fact that the diffusion approximation is only valid for times larger than τe .

2.2

Planar Brownian motion : stochastic area, winding and magnetoconductance

Weak localization corrections are directly related to the behaviour of the probability of return to the origin and thus to recurrence properties of the diffusion process. It therefore follows that dimension d = 2 plays a very special role. Consider for instance a thin film whose thickness a is much less than Lϕ . The sample is effectively two dimensional and the correction to the conductivity is given by e2 (4) h∆σi = − 2 ln(Lϕ /ℓe ). π ~ On probabilistic grounds the logarithmic scaling is of course not unexpected here, it is the same logarithm which occurs in asymptotic laws of the planar Brownian motion [181, 151]. The neighbourhood recurrence of the planar Brownian motion favors the quantum interference effects and leads to a reduction of the electrical conductivity. A more striking effect is predicted if one applies a constant and homogeneous magnetic field B over the sample. In this case the classical action contains a coupling to the magnetic field S = eBA where A is a functional of the path given by the line integral Z 1 A= (xdy − ydx) . (5) 2 Properly interpreted, this line integral is nothing but the stochastic area of planar Brownian motion, whose distribution was first computed by P. Lévy before the discovery of the Feynman path integral. The weak localization correction reads Z ∞ i h dt −t/τϕ e2 (6) 1 − E e2ieBA/~ . e h∆σ(B)i − h∆σ(0)i = 2 2π ~ 0 t

The coupling to the magnetic field now appears with an additional factor 2 coming from the fact that the weak localization describes quantum interferences of reversed paths. The expectation E[· · · ] is taken over Brownian loops for a time t. The magnetoconductivity is given by [12, 34, 55] φ0 1 φ0 4 e2 e2 + − ln ≃ h∆σ(B)i − h∆σ(0)i = 2 ψ 2π ~ 2 8πBL2ϕ 8πBL2ϕ 3~ 5

BL2ϕ φ0

!2

,

(7)

where ψ(z) is the digamma fonction and φ0 = h/e is the quantum flux. The r.h.s corresponds to the weak field limit B ≪ φ0 /L2ϕ . This increase of the conductivity with magnetic field is opposite to the behaviour expected classically. This phenomenon is called “positive (or anomalous) magnetoconductance”. Experimentally this effect is of primary importance : the magnetic field dependence allows one to distinguish the weak localization correction from other contributions and permits to extract the phase coherent length Lϕ . This expression fits the experimental results remarkably well7 [34]. Strictly speaking eq. (7) only holds for an infinite sample. In the case of bounded domains the variance depends on the geometry of the system and can be computed explictly for rectangles and strips [76]. The case of inhomogeneous magnetic fields can be treated along the same line. Consider for instance a magnetic vortex carrying a flux φ and threading the sample at point 0. In this case the functional which is involved is not the stochastic area but the index of the Brownian loop with respect to 0 (winding number). The probability distribution of the index has been computed independently by Edwards [85] in the context of polymer physics and Yor [227] in relation with the Hartman-Watson distribution. For a planar Brownian motion started at r and conditioned to hit its starting point at time 1, the distribution of the index when n goes to infinity is 1 2 (8) Proba [Ind = n] ≃ 2 2 K0 (r 2 ) e−r , 2π n where K0 (x) is the modified Bessel function of second kind (MacDonald function). The fact that the even moments of this law are infinite is reflected in the non analytic behaviour of the conductivity h∆σ(φ)i − h∆σ(0)i ∝ |φ| . (9) It is interesting to compare with eq. (7) where the quadratic behaviour is given by the second moment of the stochastic area [169]. The behaviour given by eq. (9) has been observed experimentally [107, 31] and a theoretical interpretation is provided in ref. [182].

2.3

Dephasing due to electron-electron interaction and functionals of Brownian bridges

Dephasing in a wire : relation with the area below a Brownian bridge.– Another interesting example of a non trivial connection between a physical quantity and a functional of the Brownian motion occurs in the study of electron-electron interaction and weak localization correction in a quasi-1d metallic wire. In ref. [9], Al’tshuler, Aronov & Khmel’nitzki˘ı (AAK) proposed to model the effect of the interaction between an electron and its surrounding environment by the interaction with a fluctuating classical field. The starting point is a path integral representation of the probability P(~r, t|~r ′ , 0) in which is included the effect of the fluctuating field that brings a random phase. After averaging over Gaussian fluctuations of the field, given by the fluctuationdissipation theorem, AAK obtained Z Z x(t)=x R R 2 e2 D ∞ ˙ )2 − σe ST 0t dτ |x(τ )−x(t−τ )| − 1 t dτ x(τ −γt 0 w , (10) dt e Dx(τ ) e 4D 0 h∆σi = −2 πSw 0 x(0)=x where T is the temperature (the Planck and Boltzmann constants are set equal to unity ~ = kB = 1), and Sw is the area of the section of the quasi-1d wire. In contrast with eq. (2) where the loss of phase coherence was described phenomenologically by an exponential damping8 with 7

Note that in materials with strong spin orbit scattering, like gold, the effect is reversed and a negative magnetoconductance is observed at small magnetic fields [127, 34]. 8 Note that the introduction of an exponential damping describes rigorously several effects : the loss of phase coherence due to spin-orbit scattering or spin-flip [127], the penetration of a weak perpendicular magnetic field in a quasi 1d wire [8]. In this latter case the effect of the magnetic field is taken into account through γ = 31 (eBW )2 where W is the width of the wire of rectangular section.

6

the parameter τϕ , here the electron-electron interaction affects the weak localization correction through the introduction in the action of the functional of the Brownian bridge9 Z t dτ |x(τ ) − x(t − τ )| . (11) Aet = 0

The additional damping exp −γt in eq. (10) describes the loss of phase coherence due to other phase breaking mechanisms. By using a trick which makes the path integral local in time, AAK have computed explicitely the path integral and obtained : h∆σi =

Ai(γτN ) e2 , LN ′ πSw Ai (γτN )

(12)

where Ai(z) is the Airy function. The Nyquist time τN = ( e2σT0 S√wD )2/3 , gives the time scale over which electron-electron interaction is effective and therefore plays √ the role of a phase coherence time. We have also introduced the corresponding length LN = DτN . Note that the T depen(law) dence of τN ∝ T −2/3 directly reflects the scaling of the area with time : Aet = t3/2 Ae1 . We stress that the AAK theory makes a quantitative prediction for the dependence of the phase coherence length LN as a function of the temperature, which has been verified experimentally for a wide range of parameters : on metallic (Gold) wires [84, 180] (the behaviour of LN ∝ (Sw /T )1/3 was observed before in Aluminium and Silver wires [224]), and on wires etched at the interface of two semiconductors [213].

x (τ)

(a)

x (t − τ )

x (τ)

(b) τ

0

t /2

τ

t

0

t /2

t

Figure 1: Left : The functional At gives the absolute area below a Brownian bridge x(τ ). Right : The functional Aet measures the area between the Brownian bridge x(τ ) and its time reversed counterpart x(t − τ ). The two functionals are equal in law [eq. (14)]. It is interesting to point out that the result of AAK can be interpreted as the Laplace transform of the distribution of the functional (11), E[exp −pAet ]. This functional represents the area between a Brownian bridge and its time reversed counterpart (cf. figure 1b), where E[· · · ] describes averaging over Brownian bridges. The conjugate parameter p is played in (10) by the 2 temperature : p = σe0 STw . The result (12) has also been derived in the probability literature. Let us consider the functional Z t dτ |x(τ )| , (13) At = 0

giving the absolute area below the Brownian bridge starting from the origin, x(0) = x(t) = 0 (cf. figure 1a). The double Laplace transform of the distribution of At has been computed first by Cifarelli and Regazzini [58] in the context of economy and independently by Shepp [195], and led R ∞ dt −γt −√2A √ Ai(γ) t , which is equivalent10 to the result (12). The connection ] = − π Ai to 0 √t e E[e ′ (γ)

9 A Brownian bridge, (x(τ ), 0 6 τ 6 1 | x(0) = x(1)), is a Brownian path conditioned to return to its starting point. √ 10 Note that the 1/ t in the integral computed by Shepp corrects the fact that the averaging over Brownian curves in (10) is not normalized to unity, whereas E[· · · ] is.

7

between the two results is clear from the equality in law11 : (law) At = Aet

which follows from12

(law)

x(τ ) − x(t − τ ) = x(2τ )

(14)

for

τ ∈ [0, t/2] .

(15)

The relation (14) allows us to understand more deeply the trick used by AAK to make the path integral (10) local in time. The distribution of the area At has been also studied by Rice in ref. [183]. Interestingly, the inverse Laplace transform of eq. (12) obtained by Rice has been rederived recently independently in ref. [168] in order to analyze the loss of phase coherence due to electron-electron interaction in a time representation. The study of statistical properties of the absolute area below a Brownian motion was first addressed by Kac [135] for a free Brownian motion, long before the case of the Brownian bridge studied by Cifarelli & Regazzini and by Shepp. Later on, it has been extended to other functionals of excursions and meanders13 [133, 179], which arise in a number of seemingly unrelated problems such as computer science, graph theory [95] and statistical physics [160, 161]. Dephasing in a ring.– Very recently, the question of dephasing due to electron-electron interaction in a weakly disordered metal has been raised again in ref. [159]. In particular it has been shown that the effect of the geometry of the ring and the effect of electron-electron interaction combine in a nontrivial way leading to a behaviour of the harmonics of the magnetoconductance that differs from the one predicted by eq. (2) in ref. [10]. For the ring, the functional describing the effect of electron-electron interaction on weak localization is now given by Z t |x(τ )| (16) dτ |x(τ )| 1 − Rt = L 0 instead of the area defined by eq. (13). x(τ ) is a Brownian path14 on a ring of perimeter L such that x(0) = x(t) = 0. The question has been re-examined in more detail in ref. [211], where the R∞ (nL)2 double Laplace transform of the distribution 0 dt e−γt √1t e− 2t En [e−Rt ] has been derived. En [· · · ] denotes averaging over Brownian bridges defined on a circle, with winding number n. Note that the Laplace transform of the distribution En [e−Rt ] has been also studied in ref. [211].

3

Exponential functionals of Brownian motion and Wigner time delay

3.1

Historical perspective

Exponential functionals of the form, (µ)

At

=

Z

0

t

ds exp −2(B(s) + µs)

(17)

For a given process x(τ ) the two functionals At and Aet are obviously different, however they are distributed according to the same probability distribution. They are said “equal in law”. 12 The relation (15) is easily proved by using that a Brownian bridge (x(τ ), 0 6 τ 6 1 | x(0) = x(1) = 0) can be written in terms of a free Brownian motion (B(τ ), τ > 0 | B(0) = 0) as : x(τ ) = B(τ ) − τ B(1). 13 An excursion is a part of a Brownian path between two consecutive zeros and a meander is the part of the path after the last zero. 14 As for the case of the wire, the functional involved in the study of dephasing in the ring [211] involves the difference x(τ ) − x(t − τ ) instead of the bridge x(τ ). We have used the equality in law (15) which implies the following property : given a Brownian bridge (x(τ ), 0 6 τ 6 t | x(0) = x(t) = 0), for any even function f (x) we Rt (law) R t have 0 dτ f (x(τ ) − x(t − τ )) = 0 dτ f (x(τ )). 11

8

where (B(s), s > 0, B(0) = 0) is an ordinary Brownian motion have been the object of many studies in mathematics [228], mathematical finance and physics [64]. In the physics of classical disordered systems, the starting point was the analysis of the series (Kesten variable) Z = z1 + z1 z2 + z1 z2 z3 + · · ·

(18)

where the zi are independent and identical random variables. Several papers have been devoted to the study of this random variable in the physics [68, 71] and mathematics [138, 217] literature. (µ) It was realized that Z is the discretized version of A∞ defined in eq. (17), which may be interpreted as a trapping time in the context of classical diffusion in a random medium. The tail of the probability distribution P (Z) controls the anomalous diffusive behaviour of a particle (µ) moving in a one-dimensional random force field [42]. The functional At also arises in the study of the transport properties of disordered samples of finite length [170, 173]. In the context of (1) one dimensional localization the fact that the norm of the wave function is distributed15 as A∞ is mentioned in the book of Lifshitz et al [157].

3.2

Wigner time delay

As discussed in the introduction, the localization of quantum states in one dimension is well understood. However since real systems are not infinite, asking about the nature of the states which are not in the bulk is a perfectly legitimate question. It was pointed out by Azbel in ref. [21] that, when a disordered region is connected to a region free of disorder, the localized states acquire a finite lifetime which shows up in transport through sharp resonances, refered as “Azbel resonances”. This picture was used in ref. [132] where it was argued that these resonances could be probed in scattering experiments and lead to an energy dependent random time delay of the incident electronic wave. Motivated by this result and by developments in random matrix theory we consider the scattering problem for the one dimensional Schr¨ odinger equation (in units ~ = 2m = 1) d2 (19) − 2 ψ(x) + V (x)ψ(x) = E ψ(x) dx defined on the half line x > 0 with the Dirichlet boundary conditions ψ(0) = 0. The potential V (x) has its support on the interval [0, L]. Outside of this interval, the scattering state of energy E = k2 is given by 1 −ik(x−L) ψE (x) = √ e + eik(x−L)+iδ , (20) hvE where h = 2π is the Planck constant (in unit ~ = 1) and vE = dE/dk = 2k the group velocity. Eq. (20) represents the superposition of an incoming plane wave incident from the right and a reflected plane wave caracterized by its phase shift δ(E). The Wigner time delay, defined by the relation τ = dδ/dE, can be understood as the time spent by the wave packet of energy E in the disordered region (this interpretation is only valid at high energy ; see refs. [125, 50, 149, 69] for review articles on time delay and traversal times). This representation of the time delay has been used in many papers both in the mathematics and physics literature. Starting from this representation one can derive a system of stochastic differential equations which can be studied in certain limiting cases. However, to understand the universality of the statistical properties of the Wigner time in the high energy limit, we follow a different approach below. 15

(µ)

When µ > 0, At

footnote 11) :

(µ) (law) A∞ =

possesses a limit distribution for t → ∞. Moreover, we have the equality in law (see

1/γ (µ) , where γ (µ) is distributed according to a Γ-law : P (γ) =

9

e

1 γ µ−1 −γ . Γ(µ)

Universality of time delay distribution at high energy Using the expression of the scattering state (20) and the definition τ = dδ/dE, we can obtain the so-called Smith formula [98, 196] relating the time delay to the wave function in the bulk : τ (E) = 2π

Z

L

dx |ψE (x)|2 −

0

1 sin δ(E) . 2E

(21)

Following ref. [15] one can parametrize the wave function in the bulk in terms of its phase and modulus. For this purpose we rewrite the Schr¨ odinger equation as a set of two coupled first order ′ ′ differential equations, dψE /dx = ψE and dψE /dx = (V (x) − E)ψE , and perform the change of ′ (x) = k eξ(x) cos θ(x). One obtains a new set of first variables : ψE (x) = eξ(x) sin θ(x) and ψE 1 V (x) sin 2θ. Up till now order differential equations dθ/dx = k − k1 V (x) sin2 θ and dξ/dx = 2k 16 everything is exact. If we now consider the high energy limit we can neglect the second term on the r.h.s of eq. (21) and integrate out the phase which is a fast variable ; we obtain τ=

1 k

Z

L

dx e2(ξ(x)−ξ(0)) .

(22)

0

This representation of the time delay holds for any realization of the disordered potential. Moreover, one can prove under rather mild conditions on thepcorrelations of the random potential that ξ(x) is a Brownian motion with drift ξ(x) = x/λ + 1/λ B(x) where B(x) is an ordinary Brownian motion and λ the localization length. Using the scaling properties of the Brownian motion gives the following identity in law λ τ = k (law)

Z

L/λ

dx e−2(B(x)+x) =

0

λ (1) , A k L/λ

(23)

(1)

where AL has been defined in eq. (17). This representation of the time delay as an exponential functional of the Brownian motion, first established in ref. [93] by a different method, allows us to obtain a number of interesting results [65, 207] : (i) Existence of a limit distribution for fixed17 τ and L → ∞ P (τ ) =

λ − λ e 2kτ . 2kτ 2

(24)

This result is reminiscent of the random matrix theory prediction in spite of the fact that this theory does not apply to systems that are strictly one-dimensional18 . (ii) Linear divergence of the first moment, hτ i = L/k, and exponential divergence of the higher moments hτ n i ∝ e2n(n−1)L/λ . This divergence reflects a log-normal tail of the distribution for a finite length L. Time delay and density of states The relation (21) also provides another interpretation to these results. In the situation under consideration, the scattering state is directly related to the local density of states (LDoS) by ρ(x; E) = h x |δ(E − H)| x i = |ψE (x)|2 . Therefore the time delay can be interpreted as the DoS 16

To of the correlation function of the disorder R define precisely the high energy limit, let us introduce the integral dx hV (x)V (0)i. The high energy limit corresponds to k ≫ w1/3 . 17 From the remark of the footnote 15, we notice that 1/τ is distributed according to an exponential law. 18 The random matrix theory describes the regime of weak localization or systems whose classical dynamics 1 e−τ0 /τ is chaotic. The distribution of time delay in this framework has the same functional form P (τ ) ∝ τ 2+µ [102, 114, 47, 174] with two differences : (i) the exponent 2 + µ > 2, (ii) the time scale τ0 . w=

10

of the disordered region19 . This establishes a relation between the results given above and the work of Al’tshuler & Prigodin [14] where the distribution of the LDoS was studied for a white noise potential using the method of Berezinski˘ı20 . Time delay for Dirac Hamiltonian at the threshold energy The representation (23) of the time delay as an exponential functional of the Brownian motion only holds in the weak disorder (i.e. high energy) limit. A similar representation can be derived for the random mass Dirac model at the middle of the spectrum. The Dirac Hamiltonian is H D = σ2 i

d + σ1 φ(x) , dx

(25)

where σi are the Pauli matrices. φ(x) can be interpreted as a mass21 . The Dirac equation HD ψ = k ψ possesses a particle-hole symmetry reflected in the symmetry of the spectrum with respect to k = 0 (the dispersion relation of the Dirac equation is linear in absence of the mass term and energy is equal to momentum). The middle of the spectrum is an interesting point where the divergence in the localization length signals the existence of a delocalized state (see footnote 28). These properties should therefore show up in the probability distribution of the Wigner time delay. Since the dispersion relation of the Dirac equation is linear, the time delay is now defined as τ = dδ/dk. The following representation has been derived in ref. [198] : τ =2

Z

L

dx e2

0

Rx 0

φ(y)dy

.

(26)

In contrast with eqs. (22,23), which have been obtained after averaging over the fast phase variable, the representation (26) is exact (for k = 0). If φ(x) is a white noise, then τ=

2 (0) A . g gL

(27)

In this case since the drift vanishes, it is known [170, 228] that there is no limiting distribution when L → ∞. The physical meaning of the lognormal tail P (τ ) ∼

1 2 1 √ e− 8gL ln (gτ ) 2τ 2πgL

(28)

in the limit τ → ∞ is discussed in detail in ref. [198].

4

Extreme value spectral statistics

The density of states (DoS) and the localization length of a one-dimensional Hamiltonian with a random potential can be studied by the phase formalism [15, 157] (see also ref. [158] for discrete 19 The relation between time delay and DoS is sometimes refered as Krein-Friedel relation [97, 144]. This relation has been originally introduced in ref. [37] in the context of statistical physics (see also ref. [67] and §77 of ref. [148]). Note that it has been recently discussed in the context of graphs in refs. [205, 206, 208]. 20 The method of Berezinski˘ı blocks has been introduced to study specifically the case of white noise disordered potential in one dimension [32]. This powerful approach has been widely used and has allowed to derive numbers of important results like in refs. [112, 115] for example. 21 The Hamiltonian (25) with φ(x) a white noise was introduced by Ovchinnikov & Erikmann [176] as a model of one-dimensional semiconductor with a narrow fluctuating gap. It is interesting to point out that the Dirac equation HD ψ = kψ also has an interpretation in the context of superconductivity, as linearized Bogoliubovde Gennes equations for a real random superconducting gap φ(x). Finally it is worth mentioning that more general 1d Dirac Hamiltonians with several kinds of disorder (mass term, potential term and magnetic field) have been studied in refs. [38, 39].

11

models). In the previous section we pointed out that the physics of the strong localization in one-dimension can also be probed from a different angle by considering the scattering of a plane wave by the random potential. This has led to the study of the Wigner time delay. In this section we show that the phase formalism also allows describing finer properties of the spectrum such as the probability distribution of the n-th eigenvalue. This is an instance of an extreme value statistics : given a ranked sequence of N random variables x1 6 x2 6 . . . 6 xN , the problem is to find the distribution of the n-th of these variables in a given interval. In the particular case of uncorrelated random variables, the extreme value distributions have been studied by E. Gumbel [119, 120, 121]. This problem becomes much more complicated when the random variables are correlated, a case which has recently attracted a lot of attention. Such extreme value problems have appeared recently in a variety of problems ranging from disordered systems [44, 54, 70] to certain computer science problems such as growing search trees [162]. In the case of the spectrum of a random Hamiltonian, the eigenvalues are in general correlated variables, apart from the case of strongly localized eigenstates [167]. Below we show that, in the one dimensional case, this problem is related to studying the first exit time distribution of a one-dimensional diffusion process.

4.1

Distribution of the n-th eigenvalue : relation with a first exit time distribution

We consider a Schr¨ odinger equation Hϕ(x) = Eϕ(x) on a finite interval [0, L]. The spectral (Sturm-Liouville) problem is further defined by imposing suitable boundary conditions. We choose the Dirichlet boundary conditions ϕ(0) = ϕ(L) = 0. The spectrum of H is denoted by Spec(H) = {E0 < E1 < E2 < · · · }. Our purpose is to compute the probability Wn (E) = hδ(E − En )i

(29)

for the eigenvalue En to be at energy E (the bracket h·P · ·i means averaging over the random 1 potential). Note that the sum of these distributions L n Wn (E) = ρ(E) is the average DoS per unit length. We now show how the calculation of Wn (E) can be cast into a first exit time problem. Random Schr¨ odinger Hamiltonian We consider the Hamiltonian

d2 + V (x) , (30) dx2 where V (x) is a Gaussian white noise random potential : hV (x)i = 0 and hV (x)V (x′ )i = w δ(x − x′ ). We replace the Sturm-Liouville problem by a Cauchy problem : let ψ(x; E) be the solution of the Schr¨ odinger equation Hψ(x; E) = Eψ(x; E) with the boundary conditions ψ(0; E) = 0 d ψ(0; E) = 1. The boundary condition ψ(L; E) = 0 is fulfilled whenever the energy E and dx coincides with an eigenvalue En of the Hamiltonian. In this case, the wave function ϕn (x) = hR i1/2 L ψ(x; En )/ 0 dx′ ψ(x′ ; En )2 has n nodes in the interval ]0, L[, and two nodes at the boundaries. Let us denote by ℓm (m > 1), the length between two consecutive nodes. We consider the Ricatti variable d ln |ψ(x; E)| , (31) z(x; E) = dx which obeys the following equation: H =−

d z = −E − z 2 + V (x) , dx 12

(32)

ψ ( x ;E )

1 0 0 1 00 1 0 1

E < E0

x L ψ ( x ;E )

E = E0

1 0 00 1 0 1

E0 < E < E1

1 0 0 1 00 1 0 1

..

x L ψ ( x ;E ) L x

..

1

..

2

Figure 2: The probability for the n-th level En of the spectrum to be at E is also the probability for the n + 1-th node of ψ(x; E) to be at L. with initial condition z(0; E) = +∞. This equation may be viewed as a Langevin equation for a particle located at z submitted to a force −∂U (z)/∂z deriving from the unbounded potential U (z) = Ez +

z3 3

(33)

and to a random white noise V (x). Each node of the wave function corresponds to |z(x)| = ∞. At “time” x = 0 the “particle” starts from z(0) = +∞ and eventually ends at z(ℓ1 − 0+ ) = −∞ after a “time” ℓ1 . Just after the first node it then starts again from z(ℓ1 + 0+ ) = +∞, due to the continuity of the wave function. It follows from this picture that the distance ℓm between two consecutive nodes may be viewed as the “time” needed by the particle to go through the interval ] − ∞, +∞[ (the “particle” is emitted from z = +∞ at initial “time” and absorbed when it reaches z = −∞). The distances ℓm are random variables, interpreted as times needed by the process z to go from +∞ to −∞. These random variables are statistically independent because each time the variable z reaches −∞, it loses the memory of its earlier history since it is brought back to the same initial condition and V (x) is δ-correlated. This remark is a crucial point for the derivation of Wn (E). Interest in these random “times” ℓm lies in their relation with the distribution of the eigenvalues. Indeed the probability that the energy En of the n-th excited state is at E is also the probability that the P sum of the n + 1 distances between the nodes is equal to the length of the system: L = n+1 the ℓm are independent m=1 ℓm (this is illustrated hon figure 2). Since i Pn+1 and identically distributed random variables Prob L = m=1 ℓm is readily obtained from the distribution P (ℓ) of one of these variables. We introduce the intermediate variable L(z) giving the time needed by the process starting at z to reach −∞ : therefore we have ℓ = L(+∞). The Laplace transform of the distribution h(α, z) = he−αL | z(0) = z; z(L) = −∞i obeys [105] : Gz h(α, z) = α h(α, z) ,

(34)

where the backward Fokker-Planck generator is Gz = −U ′ (z)∂z + 13

w 2 ∂ . 2 z

(35)

The boundary conditions are ∂z h(α, 0 and h(α, −∞) = 1. By Laplace inversion we R ∞z)|z=+∞ =−αℓ can derive P (ℓ) since h(α, +∞) = 0 dℓ P (ℓ)e . The solution of this problem is given in ref. [204]. Here we only consider the limit E → −∞, which corresponds to the bottom of the spectrum. In this regime the dynamics of the Ricatti variable z can be read off from the shape of the potential (see figure 3).

U ( z) z

E 2 the Brownian motion is known to be transient where as, for d = 2 it is neighbourhood recurrent (in d = 1 the Brownian motion is pointwise recurrent). Therefore we expect the dimension 2 to play a special role. In the large time limit, when t ≫ L2 q √ 2πL √ and t ≫ a2 (a is the lattice spacing), we find an effective length γ Leff ≃ a ln(4/ γa) , that corresponds to a scaling of the winding around the loop nt ∝ (ln t)1/2 .

(91)

This result can be obtained in the same way as for the ring connected to the arm. This time the plane acts as a trap. The distribution of the trapping time is given by the first return probability on a square lattice, which is known to behave at large times like P1 (τ ) ∝ 1/(τ ln2 τ ) [27], from which we can recover eq. (91). Occupation time and local time distribution on a graph Another set of problems concern occupation times : i.e. time spent by a Brownian particle in a given region. An example of such a problem is provided by the famous arc-sine law for the 1d Brownian motion that gives the distribution of the time spent by a Brownian motion (x(τ ), 0 6 τ 6 t | x(0) = 0) on the half line R+ . This result was derived long ago by P. Lévy [155]. It has been extended by Barlow, Pitman & Yor [23] for a particular graph (star graph with arms of infinite lengths) : instead of an infinite line, these authors consider n semi-infinite lines originating from the same point and study the joint distribution of the times spent on each branch. More recently, this problem has been reconsidered in the case of arbitrary graphs [75]. It may be stated as follows : consider a Brownian motion x(τ ) on a graph, starting from a point x0 at time 0 and arriving at a point x1 at time t. Let Tαβ denotes the time spent on the wire Rt (αβ). This functional is defined as Tαβ [x(τ )] = 0 dτ Yαβ (x(τ )) where the function Yαβ (x) is 1 for x ∈ (αβ) and 0 otherwise. Our aim is to compute the Laplace transforms of the joint distribution E Z D P − (αβ) ξαβ Tαβ = dx1 F(x1 , x0 ; t; {ξαβ }) (92) e with

F(x1 , x0 ; t; {ξαβ }) =

Z

x(t)=x1 x(0)=x0

1

Dx(τ ) e− 4

Rt 0

dτ x˙ 2 −

P

(αβ) ξαβ

Tαβ [x(τ )]

.

(93)

The definition (92) takes into account averaging over the final point x1 . A conjugate parameter ξαβ is introduced for each variable Tαβ , i.e. each bond. A closed expression of the Laplace transform of the joint distribution (92) has been derived in ref. [75]. The result is given as a ratio of two determinants, an expression reminiscent of the one of Leuridan [154], although the connection is not completely clear. 30

Similar methods have been also applied in ref. [60] to study the distribution of the local time Rt Tx0 [x(τ )] = 0 dτ δ(x(τ ) − x0 ).

A simplification occurs when the final point coincides with the initial point. Then it is possible to relate the characteristic function to a single spectral determinant. We do not develop the general theory here but instead consider an example close to the one studied in the previous subsection. Let us consider the graph in figure 7 and ask the following question : for a Brownian motion starting from x0 at time 0 and coming back to it at time t, what is the distribution of the time Tarm [x(τ )] spent in the arm if in addition the Brownian motion is constrained to turn n times around the ring ? It is natural to introduce the following function : Z x(t)=x0 Rt 1 2 (94) Dx(τ ) e− 0 dτ ( 4 x˙ +ξ Yarm (x)) δn,N [x(τ )] Fn (x0 , x0 ; t, ξ) = x(0)=x0

where N [x(τ )] is the winding number around the ring. This function is related to the Laplace transform of the distribution of the functional Tarm [x(τ )] E D Fn (x0 , x0 ; t, ξ) e−ξ Tarm [x] = , (95) Cn Fn (x0 , x0 ; t, 0)

where h· · ·iCn denotes averaging over curves of winding n. The denominator ensures normalization. The Laplace transform of (94) is given by a relation similar to eq. (87) Z 2π Z ∞ dθ −inθ d (λ0 ) −γt e (96) ln S (γ) dt Fn (x0 , x0 ; t, ξ) e = 2π dλ λ0 =0 0 0 0

where the appropriate spectral determinant is built as follows : (i) since the starting point is fixed at 0, we introduce mixed boundary conditions at this point, with a parameter λ0 that will be used to extract the “local information”. (ii) A magnetic flux θ is introduced (conjugate to the winding number). (iii) The spectral parameter is shifted in the arm as γ → γ + ξ to introduce the variable ξ conjugate to the time Tarm . We choose the vertex 0 as initial condition and impose the Dirichlet boundary condition (which is achieved by setting λ1 = ∞) at the end of the arm of length b. The spectral determinant is found straightforwardly (for an efficient calculation of S(γ) for a graph with loops, see ref. [177] or appendix C of ref. [3]) : √ √ √ p p sinh γL sinh γ + ξ b √ cosh γL − cos θ (λ0 ) √ 2 γ + λ0 + γ + ξ coth γ + ξ b (97) S (γ) = √ √ sinh γL γ γ+ξ

The terms in the brackets correspond to the matrix37 M (since λ1 = ∞ we can consider only the element M00 ). The first term is the contribution of the loop, the second comes from the boundary condition and the last one comes from the arm. It immediately follows that Z ∞ √ 1 sinh γL −|n|√γLeff (98) dt Fn (0, 0; t, ξ) e−γt = √ e √ 2 γ sinh γLeff 0 with

p √ √ 1p √ γLeff = cosh γL + 1 + ξ/γ sinh γL coth γ + ξ b . (99) 2 Let us consider an infinitely long arm b → ∞. If we are interested on time scales t ≫ τL , √ where τL = L2 is the Thouless time over which the ring is explored, eq. (99) gives γLeff ≃ √ (γ + ξ)1/4 L, therefore from eq. (98) we see that Fn (0, 0; t, ξ) ≃ e−ξt Fn (0, 0; t, 0). The inverse Laplace transform of eq. (95) leads to hδ(T − Tarm [x])iCn ≃ δ(T − t), which means that the Brownian motion spends almost all the time in the arm (this simple result confirms the picture presented to explain the scaling of the winding around the ring of the form nt ∝ t1/4 ). cosh

37

The introduction of the conjugate parameters {ξαβ } corresponds to shift the spectral parameter γ on each √ wire as γ → γ + ξαβ . The matrix to be generalized is not M , given by eq. (59), but M = γM .

31

6

Planar Brownian motion and charged particle in random magnetic field

A model of random magnetic field describing a charged particle moving in a plane and subjected ~ × A(~ ~ r ) due to an ensemble of magnetic Aharonov-Bohm to the random magnetic field B(~r) = ∇ vortices is described by the following Hamiltonian [77, 100] : !2 2 1 X ~uz × (~r − ~ri ) X 1 1 ~ r) + B(~r) = p − A(~ ~ ~p − α H= + πα δ(~r − ~ri ) (100) 2 2 2 (~r − ~ri )2 i

i

where 0 6 α < 1. We have set ~ = e = m = 1. α is the magnetic flux per vortex, in unit of the quantum flux φ0 = h/e : α = φ/φ0 = φ/(2π). The model is periodic in α with a period 1. ~uz is the unit vector perpendicular to the plane. ~ri are the positions of the vortices in the plane : they are uncorrelated random variables (Poisson distribution). The δ interactions (coupling to the magnetic field) are necessary in order to define properly the model [131, 163, 175, 33, 63]. By comparison with scalar impurities discussed in the introduction, the magnetic nature of the scatterers gives rise to rather different properties38 : it was shown in ref. [77] that the spectrum is reminiscent of a Landau spectrum with Landau levels broadened by disorder, in the limit of vanishing magnetic flux α → 0 (see below). This analysis was performed using perturbative arguments supported by numerical simulations. The latter use a relation between the average DoS of this particular model and some winding properties of the Brownian motion. Let Z(t) denotes the partition function for a given distribution of vortices and Z0 (t) the partition function without fluxes. The ratio of partition functions can be written as a ratio of two path integrals Rt 1 2 R R ~r(t)=~a ˙ ~ ˙ E D H d~a ~r(0)=~a D~r(τ ) e 0 (− 2 ~r +iA·~r)dτ Z(t) ~ r i C A·d~ = e , (101) Rt 1 = R R ~r(t)=~a ˙2 Z0 (t) C D~r(τ ) e− 0 2 ~r dτ d~a ~ r (0)=~a

where Z0 (t) = V /(2πt) is the free partition function (V is the (infinite) area of the plane) and h· · ·iC stands for averaging over all closed Brownian curves of the plane. In order to average Z(t) over the Poissonian distribution of vortices, let us consider, for a while, our problem on a square lattice with lattice spacing a. Let Ni be the number of vortices in square i. the magnetic flux through any closed random walk C on this lattice can be written as : I X ~ · d~r = 2παNi ni (102) A C

i

where ni is the number of times the square i has been wound around by C. Averaging Z(t) with the Poisson distribution (µ being the mean density of vortices) P (Ni ) = gives

D

~ r i C A·d~

hZ(t)i{~ri } = Z0 (t) e 38

H

E

C,{~ ri }

(µa2 )Ni −µa2 e Ni ! *

= Z0 (t) exp µ

(103) X n

!+

Sn (e2iπαn − 1)

C

(104)

For scalar impurities, the presence of a magnetic field also strongly affects the spectral properties. The case of a Gaussian disorder projected in the lowest Landau level (LLL) of a strong magnetic field has been studied in ref. [220]. Other disordered potentials have been considered later. In particular it was shown in ref. [46] that for a weak density of impurities, the spectrum can display power law singularities. A physical interpretation of such power law singularities has been discussed by Furtlehner in ref. [100, 101]. The relation between the model of scalar impurities projected in the LLL of a strong magnetic field and the model of magnetic vortices was discussed in refs. [78, 100]. Finally we mention a recent work on Lifshitz tails in the presence of magnetic field [153].

32

where h· · ·i{~ri } denotes averaging over the positions {~ri } of the vortices. The quantity Sn denotes the area of the locus of points around which the curve C has wound n times. This result was derived for a random walk on a lattice but is also obviously valid off lattice as well, for Brownian curves. The winding sectors of a closed curve are displayed in figure 9.

0 2

−1

1 0

1

Figure 9: A closed curve with its n-winding sectors. The label n of a sector is the winding number of any point inside this sector. t The random variable Sn scales like t. In particular, we know [62, 118] its expectation hSn i = 2πn 2. 2 Moreover, it has been shown in [223] that the variable n Sn becomes more and more peaked t when n grows: P (n2 Sn = X) −→ δ(X − 2π ) (the average area of the n = 0-sector has been n→∞ recently studied in ref. [104]). Thus, extracting t, eq. (104) rewrites as Z E D hZ(t)i{~ri } = Z0 (t) dSdA P (S, A) e−µt(S+iA) ≡ Z0 (t) e−µt(S+iA) , (105) C

where P (S, A) is the joint distribution of the rescaled (t independent) variables S and A defined as 2X Sn sin2 (παn) (106) S = t n 1X Sn sin(2παn) . (107) A = t n The averages of these two variables are given by : hSi = πα(1 − α)

hAi = 0 .

(108) (109)

With eq. (105), we observe that V1 hZ(t)i{~ri } has the scaling form F (µt)/t. Thus, its inverse Laplace transform, the average density of states per unit area, is a function of only E/µ and α. Moreover, it is easy to realize that hZ(t)i{~ri } is even in α (each Brownian curve in {C} comes with its time reversed) and periodic in α with period 1. Thus, one can restrict to 0 6 α 6 21 . Let us focus on the two limiting cases α → 0 and α = 1/2. • Limit α → 0 : Landau spectrum. When α → 0, a careful analysis shows that 1

hZ(t)i{~ri } ≃ ZhBi e− 2 hBit 33

(110)

hBit/2 where ZhBi = Z0 (t) sinh(hBit/2) is the partition function of a charged particle in a uniform magnetic field hBi = 2πµα (we recall that the impurity i carries a magnetic field 2παδ(~r −~ri )). The system of random vortices is, thus, equivalent to the uniform average magnetic field, albeit with an additional positive shift in the Landau spectrum : the inverse Laplace transform of the partition function (110) gives the Landau spectrum made of equally spaced infinitely degenerated levels. This corresponds to the oscillating behaviour shown on figure 11. The origin of the shift can be traced back to the presence of the repulsive δ interactions that have been added to the Hamiltonian to define properly the model.

• Half quantum flux vortices (α = 1/2) – The spectral singularity at E = 0. In this case the variable (107) vanishes, A ≡ 0, implying that hZ(t)i{~ri } = Z0 (t)hexp (−µtS)iC where now 2 X Sn . (111) S= t n odd

Performing the inverse Laplace transform, we get the average density of states per unit area: Z E/µ dS P (S) (112) hρ(E)i = ρ0 (E) 0

1 2π

is the free density of states per unit area and P (S) is the probability distri-

/ρ0

1.0

α=0.5

2

1/E 0.0 0

ln(/ρ0)

where ρ0 (E) = bution of S.

0.5

1.0

2.0

3.0

4.0

5.0

6.0

−2

−4

−6

0.0 0.0

−8

0.5

1.0

1.5

2.0

2.5

3.0

E Figure 10: The average density of states at α = 0.5 (the free density of states is constant ρ0 = 1/(2π)). Circles are simulation results. The full line is the fit discussed in the text.

We may use this result to determine the nature of the singularity in the average DoS. As displayed in figure 10 (where we have taken µ = 1), hρ(E)i increases monotonically from 0 to ρ0 (E) with a depletion of states at the bottom of the spectrum. Circles are the result of numerical simulations (on a 2D square lattice, we have generated 10000 closed random walks of 100000 steps each). The full line is a low-energy fit of the quantity hρ(E)i/ρ0 by the function 2 ρfit (E) = a e−b(µ/E) ρ0

(113)

with a = 2.8 and b = 1.4 (see figure 10). It is worth stressing that this behaviour is quite different from the one expected for a disorder due to a scalar potential. The famous Lifshitz 34

argument, applicable to the low energy DoS for a low concentration of scalar impurity, leads instead to39 a behaviour ρLif. (E) ∼ e−c2 µ/E , (114) where c2 is a constant. Our choice for this fit is motivated by a recent numerical work [184] concerning the area A of the outer boundary of planar random loops. In this work, the author suggests that the limit distribution of A is the Airy distribution implying, P for smallPA values, a behaviour of the type exp(−const./A2 )/A2 . Remarking that A = n even Sn + n odd Sn , it is natural to expect for the distribution of the random variable S, eq. (111), a behaviour at small S that is roughly given by exp(−const./S 2 ). Thus, we deduce the form ρfit (E) for the low-energy fit of hρ(E)i. • Transition between α = 0.5 and α → 0. Finally, when α grows from 0 to 0.5, the oscillations in the spectrum must disappear at some critical value αc (see figure 11). Numerical simulations, specific heat considerations and, also, diagrammatic expansions [77] give αc ∼ 0.3. 1.5

2.5

α=0.1

2.0

/ρ0

/ρ0

α=0.5 1.0

0.5

1.5 1.0 0.5

0.0 0.0

0.5

1.0

1.5

0.0

2.0

0

1

2

3

E/ α=0.05

6

7

8

6

7

8

α=0.01

20

4

/ρ0

/ρ0

5

25

5

3 2

15 10 5

1 0

4

E/

0

1

2

3

4

5

6

7

0

8

E/

0

1

2

3

4

5

E/

Figure 11: Average density of states of the Hamiltonian (100) for different values of the flux per tube φ = αφ0 . The average magnetic field reads hBi = 2πµα. We recall that the corresponding Landau spectrum is given by En = hBi (n + 1), for n > 0. From ref. [77]. P The Lifshitz argument applies to the DoS for the random Hamiltonian H = −∆ + i u(~r −~ri ) in dimension d, where u(~r) is a sharply peaked scalar potential. For a low density of impurities µ, the DoS behaves as ρLif. (E) ∼ exp(−cd µ/E d/2 ) at low energy. 39

35

7

Conclusion

A non experienced reader may have the feeling that the two topics which have been covered in this review, namely one-dimensional disordered systems and quantum graphs, are essentially disjoint. In fact there are many interesting links between these two topics, both at a methodological and a conceptual level. The use of metric graphs for modelling quantum phenomena observed in disordered metals goes back to the pioneering work of Shapiro [191] and Chalker & Coddington [56]. These systems have been used to study quantum localization and more recently as a model system for spectral statistics (for a recent review, see ref. [202]). Another set of similar questions is provided by the study of scattering properties of chaotic graphs [143] and disordered systems [103] (see also section 3.2 and ref. [174] for recent developments). We have also seen that apart from its interest to study the spectral properties of metric graphs, the spectral determinant also allows studying several properties of networks of quasi-one-dimensional weakly disordered wires [177, 178, 3]. As a conclusion we would like to mention several open problems : • A study of spectral statistics in the case of graphs with a random Schr¨ odinger operator [there is still a factorised structure but the matrix R is now given by eq. (66)]. • A probabilistic understanding of the star graphs using the tools of excursion theory developed by Barlow et al. [23] in the context of the Brownian spider. A first step would be to recover those probabilistic results (the joint law of the occupation time inside the branches) by a spectral approach. It is however not excluded that the probabilistic approach could provide a key to a deeper understanding of those quantum systems. In the context of classical systems such probabilistic approaches have been very useful, e.g. recent studies of the Stochastic Loewner Equation have made enormous progress in understanding the statistical physics of a class of two-dimensional systems. This calls for new probabilistic techniques for quantum systems as well. • Several functionals of the Brownian motion (13,16) appear when studying the important question of dephasing due to electron-electron interaction in networks of quasi-one-dimensional weakly disordered wires. The fact that such simple functionals appear relies on the translation invariance of the two particular problems studied (an infinite wire [9] and an isolated ring [212]). For a network functional of the Brownian bridge x(τ ) R t with arbitrary topology, the relevant ′ is given by 0 dτ W (x(τ ), x(t − τ )) where W (x, x ) = 21 [Pd (x, x) + Pd (x′ , x′ )] − Pd (x, x′ ) with −∆Pd (x, x′ ) = δ(x − x′ ). It now involves a nonlocal functional in time which is difficult to handle. Progress in this direction would allow clarifying the interplay between the electronelectron interaction and the geometrical effect and help in analyzing recent experimental results [94, 29]. • The question of extreme value spectral statistics was addressed in the framework of random matrix theory [214] and these studies have found several applications in the context of out-ofequilibrium statistical physics (see the review [215]). However this question was first addressed in the context of one-dimensional disordered systems [117]. The study of supersymmetric random Hamiltonian, for which the bottom of the spectrum plays a special role, has emphasized the interest in extreme value spectral statistics [204] (in particular it indicates level correlations). It would be interesting to extend such studies to other models and find some physical situations where these results would be applicable. • A beautiful heuristic argument was provided by Lifshitz to explain the spectral singularity for Hamiltonians with a weak concentration of localized scalar impurities (see for example the book [157]). Other mechanisms should be invoked to explain the nature of the quantum states responsible for the low energy power law behaviour in the case of a uniform strong magnetic field with δ-impurities [100, 101]. For the model of randomly distributed magnetic fluxes, the numerical simulations for α = 1/2 have suggested the new type of singular behaviour (113). It 36

would be interesting to provide heuristic arguments to understand more deeply the origin of this singularity.

Acknowledgments ´ Akkermans, Cyril Furtlehner, The article gives an overview of several joint works involving : Eric Satya Majumdar, Gilles Montambaux, Cécile Monthus, Stéphane Ouvry and Marc Yor. We thank them for fruitful collaborations. Meanwhile we have also benefited from stimulating discussions with Marc Bocquet, Eugène Bogomolny, Oriol Bohigas, Hélène Bouchiat, Markus B¨ uttiker, David Dean, Richard Deblock, Meydi Ferrier, Sophie Guéron, Jean-Marc Luck, Gleb Oshanin, Leonid Pastur and Denis Ullmo. We are pleased to acknowledge Satya Majumdar for his careful reading and valuable remarks.

37

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Functionals of the Brownian motion, localization and metric graphs

Functionals of the Brownian motion, localization and metric graphs

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