A A useful property of winding Brownian trajectories

Quantum oscillations and decoherence due to electron-electron interaction in metallic networks and hollow cylinders


We have studied the quantum oscillations of the conductance for arrays of connected mesoscopic metallic rings, in the presence of an external magnetic field. Several geometries have been considered: a linear array of rings connected with short or long wires compared to the phase coherence length, square networks and hollow cylinders. Compared to the well-known case of the isolated ring, we show that for connected rings, the winding of the Brownian trajectories around the rings is modified, leading to a different harmonics content of the quantum oscillations. We relate this harmonics content to the distribution of winding numbers. We consider the limits where coherence length is small or large compared to the perimeter of each ring constituting the network. In the latter case, the coherent diffusive trajectories explore a region larger than , whence a network dependent harmonics content. Our analysis is based on the calculation of the spectral determinant of the diffusion equation for which we have a simple expression on any network. It is also based on the hypothesis that the time dependence of the dephasing between diffusive trajectories can be described by an exponential decay with a single characteristic time (model A) .

At low temperature, decoherence is limited by electron-electron interaction, and can be modelled in a one-electron picture by the fluctuating electric field created by other electrons (model B). It is described by a functional of the trajectories and thus the dependence on geometry is crucial. Expressions for the magnetoconductance oscillations are derived within this model and compared to the results of model A. It is shown that they involve several temperature-dependent length scales.

73.23.-b ; 73.20.Fz ; 72.15.Rn

I Introduction

Understanding which processes limit phase coherence in electronic transport is an important issue in mesoscopic physics. Such phenomena like weak localization or universal conductance fluctuations are well understood to result from phase coherence effects limited at a given time (or length) scale called phase coherence time (or phase coherent length ). This explains the interest in studying quantum corrections to the classical conductivity : to provide a powerful experimental probe of phase coherence in weakly disordered metals and furnish some informations on the microscopic mechanisms responsible for the limitation of quantum coherence. This limitation originates from the coupling of electrons to external degrees of freedom like magnetic impurities or phonons (1); (2). It also results from the interaction among electrons themselves. The physical origin of this decoherence in weakly disordered metals has been understood in the pioneering paper of Altshuler, Aronov & Khmelnitskii (AAK) (3). In a one electron picture, it is due to the fluctuations of the electric field created by the other electrons. In a quasi-1d wire, these authors have shown that this mechanism leads to the following temperature dependence of the dephasing time . This power-law can be understood qualitatively as follows: the typical dephasing is proportional to the fluctuations of the electric potential, which themselves are known from Nyquist theorem to be proportional to the temperature and to the resistance of the sample. For an infinite wire, the relevant fluctuations are limited to the scale of the coherence length itself. Consequently, the dephasing time has the structure : , where is the dimensionless conductance at the length scale . For a quasi-1d conductor, the conductance is linear in length and the length scales as the square-root of time. Therefore the function scales as , whence the above power law.

More recently, Ludwig & Mirlin (4) and two of the authors (5) have considered the geometry of a ring, and they have shown that the damping of magnetoresistance oscillations could be described with a different temperature dependence of the dephasing time . This new behaviour can be qualitatively understood by considering that the diffusive trajectories encircle the ring and have all a length equal to the perimeter of the ring, so that the relevant resistance is the resistance of the ring itself. As a result : .

In Ref. (5) we have shown how the dephasing on a ring depends on the nature of the diffusive trajectories : the fluctuations of the electric potential affect differently trajectories which encircle the ring and trajectories which do not encircle it. Within this framework, we have analyzed magnetoresistance experiments performed on a square network of quasi-1d wires, and we have found that indeed two characteristic lengths with two different temperature dependence could be extracted from the data (6). These recent considerations have led us to the general conclusion that the dephasing depends on the geometry of the system considered.

The purpose of this paper is to analyze the dephasing process and to calculate the weak localization correction in different geometries, where the decoherence induced by electron-electron interaction may have a more complex structure. In order to address this question, it is important to understand that the weak localization correction depends on two ingredients, one is the probability to have pairs of reversed trajectories, which is related to the return probability for a diffusive particle after a time , the other is the nature of the dephasing process itself. Schematically, the weak localization correction to the conductivity can be written as


where is the average dephasing accumulated along a diffusive trajectory for a time . The return probability has been analyzed in Ref. (7) for various types of networks. Its Laplace transform, the spectral determinant, can be simply calculated from the parameters of the network. More complex is the analysis of the dephasing process itself. A simple and natural ansatz would be to assume an exponential decay . This assumption is correct when the dephasing is due to random magnetic impurities or electron-phonon scattering. For electron-electron interaction the analysis of the AAK result for a wire shows that time dependence is not exponential (8) : . The qualitative reason stands again on the fact that dephasing can be described as due to the fluctuations of the electric potential due to other electrons. Then, one may understand that this dephasing depends on the nature of the trajectories and is not exponential. The main goal of this paper is to describe this dephasing for complex networks and to generalize the known results of the infinite wire and the ring.

The paper is organized as follows : In section II, we recall the physical basis at the origin of this work and in section III we present the general formalism appropriate for our study. In the next sections, we consider successively more and more complex geometries. In section IV, we recall known results for the infinite wire and the ring. In section V, we consider the case of a ring attached to arms and show how the harmonics of the magnetoresistance oscillations are reduced by the existence of the arms. The situation is the same for a chain of rings connected through arms longer than the coherence length. When rings become close to each other the dephasing in one ring is strongly modified by the winding trajectories in the neighboring rings. This is discussed in section VI. The case of an infinite regular network is much more difficult to address since the hierarchy of diffusive trajectories is difficult to analyze, and we have used the limit of the infinite plane as a guideline (section VII). Finally the case of a hollow cylinder (section VIII) is quite interesting since it combines trajectories winding around the axis of the cylinder and two-dimensional trajectories. Throughout the paper, we shall consider two situations, respectively denoted by model A and model B : the case where the dephasing has a simple exponential time dependence, and the case where the dephasing is induced by electron-electron interaction. We shall systematically discuss the analogies and the differences between these two situations.

Ii Background

In a weakly disordered metal, due to elastic scattering by impurities, the classical conductivity reaches a finite value at low temperature, given by the Drude conductivity , where is the electronic density and the elastic scattering time. Quantum interferences are responsible for small quantum corrections to the Drude result. One important contribution, that survives averaging over the disorder (9); (10); (11), comes from interferences of reversed closed electronic trajectories, and therefore diminishes the conductivity. This quantum contribution to the average conductivity is called the weak localization (WL) correction. It has been expressed as (1) where the function describes dephasing and cut off the large time contributions. A simple exponential decay is usually assumed (denoted model A in the present paper). is the phase coherence time, related to the phase coherence length , where is the diffusion constant of electrons in the disordered metal. From eq. (1), we obtain the WL correction in a wire and in a plane (diffusion sets in after a time , whence the lower cutoff in the integrals). In practice, the WL is a small correction to the Drude conductivity and it can be extracted thanks to its sensitivity to a magnetic field. In the presence of a magnetic field, the contribution of a closed diffusive trajectory is multiplied by , where is the magnetic flux through the loop. This phase factor comes from the interference of the two reversed electronic trajectories, whence the factor . After summation over all loops, the additional magnetic phase is responsible for the vanishing of the contributions of loops such that , where is the flux quantum. Therefore the magnetic field provides an additional cutoff at time corresponding to diffusive trajectories encircling one flux quantum. In a narrow wire of width we have and in a thin film (plane) (see Refs. (1); (2)). The two cutoffs are added “à la Matthiessen” (12); (13) as  ; this leads to a smooth dependence of the WL correction as a function of the magnetic field.

The above discussion concerns homogeneous devices (like a wire or a plane). Another experimental setup appropriate to study quantum interferences and extract the phase coherence length is a metallic ring or an array of rings. In this case the topology constrains the magnetic flux intercepted by the rings to be an integer multiple of the flux per ring (we neglect for the moment the penetration of the magnetic field in the wires) : with . This gives rise to Aharonov-Bohm (AB) oscillations of the conductance as a function of the flux with period . Disorder averaging is responsible for the vanishing of these -periodic oscillations : only survive the contributions of the reversed electronic trajectories leading to WL correction oscillations, known as Al’tshuler-Aronov-Spivak (AAS) oscillations (14); (15), with a period half of the flux quantum. It will be convenient to introduce the harmonics of the periodic WL correction. An important motivation for considering the harmonic content in networks, is that it allows to decouple the two effects of the magnetic field (16) : the rapid AAS oscillations () and the penetration of the magnetic field in the wires, responsible for a smooth decrease of the MC at large field (). Since the -th harmonic is given by contributions of loops encircling fluxes we can write


where is the return probability after a time having encircled fluxes. is the cross section of the wires. In an isolated ring of perimeter , this probability reads . Integral (2) gives (14)


where is the phase coherence length. Note that follows from the symmetry under reversing the magnetic field ; in the following we will simply consider . The exponential decay of the harmonics directly originates from the diffusive nature of the winding around the ring : for a time , the typical winding scales as . The AAS oscillations were first observed in narrow metallic hollow cylinders (17); (18) and in large metallic networks (19); (20); (15).

Although the simple behaviour (3) has been used to analyze AAS or AB oscillations (21) in many experiments until recently (see for example Refs. (10); (22)), a realistic description of a network made of connected rings leads to harmonics with a dependence a priori quite different from the simple exponential prediction (3) for two reasons related to the nontrivial topology of the networks.

(i) Winding properties of diffusive loops in networks.– Consider for example the square network of figure 1 made of rings of perimeter . For , an electron unlikely keeps its phase coherence around a ring, therefore AAS oscillations are dominated by trajectories enlacing one ring only and all rings can be considered as independent. In the opposite regime (23) , the interfering electronic trajectories explore regions much larger than the ring perimeter . In this case, winding properties are more complicated (figure 1) and the probability may strongly differ from the one obtained for a single ring . A theory must be developed to account for these topological effects, which leads to an harmonic content quite different from (3). This was done in Refs. (25); (26); (27); (28); (29) for large regular networks. This theory was later extended in Ref. (30) in order to deal with arbitrary networks, properly accounting for their connections to contacts (31).

Figure 1: A square metallic network submitted to a magnetic field. Schematic picture of a closed diffusive trajectory winding a flux is represented ( is the flux per elementary plaquette).

(ii) e-e interaction leads to geometry dependent decoherence.– Not only the winding probability involved in eq. (2) is affected by the nontrivial topology of networks, but also describing the nature of phase coherence relaxation is replaced by a more complex function. When decoherence is due to e-e interaction, the dominant phase-breaking mechanism at low temperature, this relaxation is not described by a simple exponential anymore. This situation will be refered as model B throughout this paper. Such decoherence can be modeled in a one-electron picture by including dephasing due to the fluctuating electromagnetic field created by the other electrons (3). Therefore the pair of reversed interfering trajectories picks up an additional phase that depends on the electric potential . Averaging over the fluctuations of the potential leads to the relaxation of phase coherence. The harmonics present the structure


where averaging is taken over the potential fluctuations and over the loops with winding for time , . In a quasi 1d-wire, the relaxation of phase coherence involves an important length scale, the Nyquist length , characterizing the efficiency of the electron-electron interaction to destroy the phase coherence in the wire. We will see that, in networks also, the Nyquist length is the intrinsic length characterizing decoherence due to e-e interaction. It is given by (3); (33); (34); (36)


where is the cross-section of the wire. We have rewritten the Nyquist length in terms of the thermal length , the elastic mean free path and the number of conducting channels (not including spin degeneracy) ; is a dimensionless constant depending on the dimension ( where is the volume of the unit sphere in dimension (37). In the following we will set . In the infinite wire, the decaying function can only involve the unique length . A precise analysis of the magnetoconductance (MC) of the wire shows that, in this case, the calculation of (2) and (4) for the infinite wire, for which , leads to almost indistinguishable results provided (41); (2) . Therefore the analysis of the MC of the wire suggests that the sophisticated calculation of (4) can be replaced by the simpler one (2) with where is the Nyquist time. However this is a priori not true anymore as soon as we consider networks with a nontrivial topology because electric potential fluctuations depend on the geometry, and therefore the decoherence is geometry-dependent.

Let us formulate this idea more precisely. Being related to the potential as , fluctuations of the phase picked by the two reversed electronic trajectories can be related to the power spectrum of the potential, given by the fluctuation-dissipation theorem (FDT) : where is the resistance of a wire of length . The average is taken over potential fluctuations and closed diffusive trajectories for a time scale . The length is the typical length probed by electronic trajectories. For an infinite wire it scales like therefore where is the Nyquist time. On the other hand, in a ring, diffusion is constrained by the geometry : harmonics of the MC of a ring involve winding trajectories for which the length scale probed is therefore the perimeter , leading to where . Therefore, in a ring, depending on their winding, trajectories probe different length scales : or .

Let us summarize. At the level of eqs. (2,3), is a phenomenological parameter put by hand. The modelization of decoherence due to e-e interaction of AAK shows that, in an infinite wire, the WL correction probes the Nyquist length (the only length scale of the problem). This shows that, in the MC of the infinite wire, the phenomenological parameter must be replaced by . On the other hand the MC of a ring involves a new length scale combination of the Nyquist length and the perimeter. In this case, assuming the simple AAS behaviour , the phenomenological parameter should be substituted by .

Geometry dependent decoherence in ballistic rings.– It is worth pointing that such a geometry dependent decoherence can also be observed in ballistic systems : potential fluctuations responsible for decoherence depend on the precise distribution of currents inside the device, that are affected by the way the current is injected through different contacts (42). Depending whether the measurement is local or nonlocal, different phase coherence lengths have been extracted from the damping of AB oscillations (43). The different are probed by changing the contact configuration (current/voltage probes) (44), whereas in the diffusive ring, the different length scales are probed by considering different harmonics.

Iii General formalism

We first recall the basic formalism and apply precisely the ideas given in the introduction. We will consider the reduced conductivity , defined by


where is the cross-section of the wire. The reduced WL correction has the dimension of a length. As mentioned above, it is a sum of contributions of interfering closed reversed electronic trajectories, which can be conveniently written as a path integral :


is the so-called Cooperon. Summation over diffusive paths for time involves the Wiener measure (we have performed a change of variable so that “time” has now the dimension of a squared length). Each loop receives a phase proportional to the magnetic flux intercepted by the reversed interfering trajectories, where is the vector potential. The factor originates from the fact that the Cooperon measures interference between two closed electronic trajectories undergoing the same sequence of scattering events in a reversed order. Finally we have introduced an additional phase to account for dephasing : dephasing due to penetration of the magnetic field in the wires (12) or decoherence due to electron-electron interaction. In this latter case the phase depends also on the environment dynamics over which one should average.

The dependence of in eq. (7).– In a general network, in the absence of translational invariance, the WL correction to the conductance was shown to be given by an integration of over the network, with some nontrivial weights attributed to the wires. Let us write the classical dimensionless conductance as where the effective length is obtained from addition (Kirchhoff) laws of classical resistances (dimensionless parameter was defined above : , and ). Then the WL correction to the conductance is (30)


where the summation runs over all wires of the networks and is the length of the wire . Eq. (8) was demonstrated in Ref. (30) for the conductance matrix elements of multiterminal networks with arbitrary topology. This result relies on a careful discussion of current conservation (derivation of current conserving quantum corrections can be found in Refs. (45); (46); (47); (48)). This point will play a relatively minor role in the present paper. Eq. (8) may be used in order to calculate geometry dependent prefactors.

Magnetoconductance oscillations and winding properties.– In an array of metallic rings of same perimeter, the magnetic flux is an integer multiple of the flux per ring where is the winding number of the closed trajectory (the number of fluxes encircled). This makes the WL correction a periodic function of the flux , where . The -th harmonic of the MC


involves trajectories with winding number . We can write the harmonics as


where the Kronecker symbol selects only trajectories for a given winding number . Let us introduce the probability for a diffusive particle to return to its starting point after a time , with the condition of winding fluxes


For example, in an isolated ring of perimeter , this probability is simply given by


Then, we can rewrite the harmonics as


which is the structure given in eq. (4). In eq. (13) we have introduced the notation for a closed diffusive path winding times. designates averaging over all such paths, with the measure of the path integral (11). The phase accounts for dephasing and eliminates the contributions of diffusing trajectories at large time. We now discuss two possible modelizations for this function, denoted by “A” and “B”.

iii.1 Model A : Exponential relaxation

The simplest choice is an exponential relaxation, with a dephasing rate  :


This simple prescription correctly describes dephasing due to spin-orbit coupling, magnetic impurities (49); (13), effect of penetration of the magnetic field in the wires (12), or decoherence due to electron-phonon scattering (1); (50). Using (12,13) with this exponential decay yields the familiar result (3) for the isolated ring.

iii.2 Model B : geometry dependent decoherence from electron-electron interaction

It turns out that the simple exponential relaxation does not describe correctly the decoherence due to electron-electron interaction, the physical reason being that this decoherence is due to electromagnetic field fluctuations that depend on the geometry of the system. AAK have proposed a microscopic description (3); (33) that we can rephrase as follows. In eq. (7), the phase picked up by the reversed trajectories depends on the environment (the potential created by the other electrons due to electron-electron interaction) : . Averaging over the Gaussian fluctuations of leads to where the fluctuation-dissipation theorem (written for describing classical fluctuations) gives with (52)


where is the quantum of resistance. The function is related to the diffuson, solution of , by


This function has a physical interpretation discussed in the appendix E : it is proportional to the equivalent resistance between the points and (figure 21). With this remark, we see that eq. (16) can be understood as a local version of the Johnson-Nyquist theorem relating the potential fluctuations to the resistance.

In eqs. (15,16) we have introduced a decoherence rate which depends not only on the time but on the trajectory itself. Therefore the decay of phase coherence is now described by


Within this framework, relaxation of phase coherence is not described by a simple exponential decay like in eq. (14) but is controlled by a functional of the trajectories (54) . Therefore the nature of decoherence depends on the network, through the resistance between and , and on the winding properties of the trajectories.

The central problem of the present paper is to compute the path integral


for the different networks. Such a calculation has been already performed in two cases : the infinite wire (3) and the isolated ring (4); (5).

The logic of the following sections is the following : first we study the winding properties in the network. For that purpose we first compute the WL correction within model A, eq. (13) with (14). The probability can be extracted from an inverse Laplace transform with respect to the parameter . Having fully characterized the winding properties, we use this information in order to study the harmonics within the model B describing decoherence due to electron-electron interaction, eq. (13) with (18).

Iv The wire and the ring

We first recall known results within the framework of model B concerning the simplest geometries that will be useful for the following.

iv.1 Phase coherence relaxation in an infinite wire

The case of an infinite wire was originally solved in Ref. (3). In this case we have and the path integral


can be computed thanks to translational invariance (as pointed in Ref. (61), using the symmetry of the path integral we can perform the substitution , provided that the starting point of the path integral is set to , see appendix A). Combining exponential relaxation (model A) and decoherence due to e-e interaction (model B) allows to extract the function (18) with an inverse Laplace transform of the AAK result (3); (33); (35); (2)


where is the Airy function (62). As mentioned above, this expression is very close to (41); (2) , the result obtained by performing the substitution (see figure 2).

The inverse Laplace transform of (21) was computed in Ref. (8) with residue’s theorem :


where are zeros of . In particular and for . The limiting behaviours are


Note that the short time behaviour can be obtained by expanding . This limit can be simply obtained by noticing that in the wire , therefore where we recover that the Nyquist time scales as .

Figure 2: Comparison between (black continuous line) and (red dashed line). Relative difference (inset) does not exceed 4%.

iv.2 Phase coherence relaxation in the isolated ring

For the isolated ring of perimeter , we have . The path integral (III.2) can be computed exactly (5) (see appendix A). Up to a dimensionless prefactor, we obtain


This result can be simply understood as follows : in the ring, trajectories with finite winding necessarily explore the whole ring. This “ergodicity” implies that and therefore the decoherence rate involves the different time scale , according to the physical argument given in section II. As a consequence Eq. (13) indeed leads to (4)



Figure 3: The relative difference () between the effective length , eq. (33), and its approximation, eq. (36), does not exceed 1.5%.

Phase coherence length : or  ?– Note that the introduction of a new length scale (4) might appear arbitrary since the harmonics may be written uniquely in terms (5) of and . The difference between (27) and (29) is a matter of convention and may be related to the experimental procedure. The usual method extracts the phase coherence length from the analysis of MC harmonics. Then it is natural to see how the winding number scales with the phase coherence length, or more properly how the length scales with and therefore assume the form . From eq. (29) we see that the function is simply the exponential, , with a perimeter dependent phase coherence length . Another procedure may consist in studying the harmonics content as a function of the perimeter , that is for different samples. The experiment is then analyzed with the form . Eq. (27) gives with the geometry independent phase coherence length .

The temperature dependence was first predicted in Ref. (4) using instanton method (with a different pre-exponential dependence) and studied in details in Ref. (5) where the path integral (III.2) was computed exactly for the isolated ring. The effect of the connecting arms was clarified in Ref. (63). The fact that the pre-exponential factor is is related to the fact that the smooth part of the MC, due to the penetration of the field in the wire, probes the same length scale as in the infinite wire.

It is worth pointing the recent work (53) in which the crossover to the 0d limit is studied in a ring weakly connected. In this case the authors get a crossover from (diffusive ring) to (ergodic) for temperature below the Thouless energy. This latter behaviour coincides with the result known for quantum dots in the same regime (64).

iv.3 Penetration of the magnetic field in the wires of the ring

Networks are made of wires of finite width . The penetration of the magnetic field in the wires is responsible for fluctuations of the magnetic flux enclosed by trajectories with the same winding number but different areas. In the weak magnetic field limit, this effect is described by introducing an effective dephasing rate (12)


The question of how to combine the two decoherence mechanisms (models A & B) in the ring was discussed in Ref. (5). It was shown that the WL correction of the ring presents the structure


for , with


where .

Prefactor.– In eq. (32), the pre-exponential factor coincides with the result obtained for an infinite wire (3). The ratio of Airy functions can be approximated as (41); (2) (figure 2). In other terms, we may write the zero harmonic (i.e. the result for the infinite wire) as




This combination expresses that, in a wire of width , the penetration of the magnetic field provides the dominant cutoff when typical trajectories enclose more than one quantum flux  (here for trajectories with winding ).

Exponential damping.– In the exponential of eq. (32), the effective length interpolates between for and for . Its overall behaviour is well approximated by


which differs with (33) by less than 1.5% (figure 3). When is the shortest length, the decay of AAS oscillations can be understood from the fact that modulations of the flux enclosed by trajectories with finite winding become larger than the quantum flux . Eq. (36) was used in the analysis of the recent experiment (6).

Figure 4: Typical shape of the MC curve of a network (here a chain of rings). Rapid oscillations are AAS oscillations. Damping of oscillations over large scale is due to the penetration of the magnetic field in the wires.

These two remarks show that the magnetic length probes two different length scales : in the pre-exponential factor probes the Nyquist length , whereas in the ratio of harmonics , the magnetic length probes the length scale .

iv.4 How to analyze MC experiments in networks

In order to understand the implications of this remark, let us discuss the structure of the typical MC curve of a network. The following discussion applies to the case where (32) holds. Figure 4 represents a typical MC curve, here for a chain of rings. It exhibits rapid AAS oscillations with a period given by , superimposed with a smooth variation over a scale . The phase coherence length can be extracted either from the amplitude of the oscillations or from the decay of the envelope of the MC curve. Which ( or ) is obtained from such a curve ?

Figure 5: Phase coherence length as a function of the temperature obtained from measurements realized on a large square network (6) of lattice spacing m (perimeter m). The length scale has been extracted from the smooth MC while the length has been extracted from ratio of harmonics. (Note that and in Ref. (6) denote eqs. (35) and (36) for ).

According to (32) we see that in the pre-exponential factor, which mostly dominates the smooth envelope, while in the exponential decay, which dominates the damping of the rapid oscillations. In order to decouple the two effects the analysis of the experiments of Ref. (6) have been analyzed as follows (65); (16) : the Fourier transform of the MC curve presents broadened Fourier peaks due to the penetration of the magnetic field in the wires. Integration of Fourier peaks eliminates this effect. Ratio of harmonics involve the length scale . The length was extracted from the smooth envelope for . The temperature dependence of the phase coherence length was extracted in this way in Ref. (6). Results are plotted on figure 5, exhibiting clearly the two length scales in the regime . We see that it is crucial to analyze the experiment in terms of the MC harmonics .

Isolated ring vs ring embedded in a network.– In transport experiments the ring is never isolated : it is at least connected to contacts through which current is injected. Moreover the samples are often made of a large number of loops, in order to realize disorder averaging. The results obtained for the isolated ring are fortunately relevant to describe a more complex network of equivalent rings (Figs. 1 & 9) when the rings can be considered as independent, i.e. when interference phenomena do not involve several rings ; this occurs when (or ), in practice in a high temperature regime. This temperature dependence of harmonics is rather difficult to extract from measurements since harmonics are suppressed exponentially. This has been done only very recently in Ref. (6). Another difficulty is that the “high temperature regime” is in practice quite narrow in these samples due to fact that electron-phonon interaction dominates the decoherence above K (in the sample of Refs. (11); (51) is much larger and when the role of electron-eletron interaction is negligible).

It is an important issue to obtain the expression of the WL correction for a broader temperature range, that is to study the regime . This regime is reached in several experiments (16); (11); (51). In this case diffusive interfering trajectories responsible for AAS harmonics are not constrained to remain inside a unique ring, but explore the surrounding network (see figs. 1, 6 & 9). This affects both the winding properties and the nature of decoherence. The MC oscillations are therefore network dependent. In the following sections we discuss the behaviour of the MC harmonics in the limit (or ) for different networks : a ring connected to long arms, a necklace of rings and a large square network. The case of a long hollow cylinder will also be discussed.

V The connected ring

In this section we consider the case of a single ring connected to two wires supposed much longer than (figure 6). This problem has already been considered in Refs. (7); (5).

Figure 6: A ring connected to reservoirs through long wires and . In the regime , the WL correction is dominated by trajectories exploring the arms, as represented here.

v.1 Model A

Let us consider first the case . The Cooperon is constructed in appendix F.1 (see Ref. (7)) and we obtain


where is any position inside the ring far from the vertices (at distance larger than ).
Now we turn to the regime . The Cooperon is uniform inside the ring and is computed in appendix F.1. We have (7) :


where is any position inside the ring, or in the arms at a distance to the ring smaller than . We emphasize that this behaviour, quite different from (3), is due to the fact that the diffusive trajectories spend most of the time in the wires (7) (the distribution of the time spent by winding trajectories in the arm was analyzed in Ref. (61)).

Winding probability in a ring connected to arms.– We now derive the winding probability for a ring connected to infinitely long arms (figure 7) from the inverse Laplace transform of the Cooperon . At small time, (37) gives


where is inside the ring, far from a vertex (at distance larger than ).

At large time scales, , the arms strongly modify the winding properties around the ring : the time dependence of the typical winding number becomes subdiffusive , to be compared with the behaviour for the isolated ring reflected by eq. (12). For a ring connected to infinite wires, eq. (12) is replaced by the probability (7)


where is the number of arms (as far as is inside the ring or at a distance to the ring smaller than , the Cooperon, or the corresponding probability is almost independent on ). The function , given by (7); (61)


is studied in the appendix B and plotted in the conclusion (figure 17).

Figure 7: A ring connected to wires.

From the conductivity to the conductance.– In the geometry of figure 6, the conductance is not simply related to the conductivity. The classical conductance of the connected ring is given by with where is the length of the wire . is the equivalent length. From eq. (8) :


The Cooperon has been constructed for different positions of the coordinate in appendix F.1. Depending on the ratio , the WL correction is dominated by different terms.
For , eq. (165) shows that harmonics of the Cooperon decay exponentially in the arms (figure 24) ; inside the ring, the Cooperon (167) is almost uniform, apart for small variations near the nodes. Therefore is dominated by integrals and in the ring and we have


where is any position inside the ring far from the vertices (at distance larger than ).
For , using eqs. (165,167) we see that the terms and bring a contribution proportional to the perimeter whereas the terms and bring larger contributions proportional to  : , therefore