Frustration and quantum interference effects in normal and superconducting networks

Physica B 152 (1988) Y-45 North-Holland, Amsterdam

FRUSTRATION AND QUANTUM INTERFERENCE EFFECTS IN NORMAL AND SUPERCONDUCTING NETWORKS R. RAMMAL Centre de Recherches sur les TrCs Basses Tempiratures,

CNRS, BP 166 X, 38042 Grenoble

Ctkfex, France

Recent progress in the physics of superconducting and normal metallic networks is briefly reviewed. Three specific topics will be addressed: i) quasiperiodic and fractal networks (phase diagram, magnetoresistance, . . .), ii) conductance fluctuations in normal metal arrays and iii) magnetization of regular superconducting networks near the phase boundary T,(H). In i) we show how the universal periodic corrections to scaling can appear in the magnetoresistance of fractal networks. In ii) the nonuniversality of conductance fluctuations in normal metals is addressed. In iii) we discuss the possible thermodynamical manifestation of Berry’s topological phase on the magnetization of superconducting networks in the mixed state.

1. Introduction Quantum interference effects have been observed recently in artificial arrays of submicronic conductors at liquid helium temperatures. These interferences have been shown to occur in both superconducting and normal metals as a result of the coherence of the electronic wave functions over macroscopic scales. The study of these systems has brought new insight into basic physical phenomena like frustration, commensurability, weak localization and Landau levels. Thanks to the possibilities offered by the modern microfabrication techniques to built up artificial structures, novel well-characterized topologies are now available to the experiment: periodic, selfsimilar or quasiperiodic networks. In superconducting networks the basic phenomenon is the quantization of the magnetic flux in multiconnected geometries, which is a consequence of the properties of the phase of the superconducting wave function. In extended planar networks where a large number of identical closed loops are adjacent to each other, a new quantization effect takes place: in addition to the fundamental periodic oscillation with flux quantization in the basic cell of the network, new dips appear in T,(H) at subharmonics (+/& +,, = hcl2e = 2.07 x = nlm; n, m = integers; lo-’ G cm*). The physics is the same on: i)

self-similar networks (e.g. Sierpinski gaskets) where dips appear at +/+,, = 4-” (n = integer) and ii) quasiperiodic networks (e.g. Penrose patterns) where sharp dips appear at 4/4,, = m = no (m, n = integers, r = 2 cos(m/5) = golden mean). In networks made of normal metals, quantum interference arises from the coherent backscattering of independent electrons on the impurities. Here coherence takes place over distances (Ld) which exceed several elementary cells of the network. In addition to the oscillations of the magnetoresistance (MR) with period & = hc/2e, coherence effects are also responsible for the anomalous conductance fluctuations at submicronic scales. Section 2 of this paper reviews the main results obtained recently in self-similar superconducting and normal networks with emphasis on the universal periodic corrections to scaling predicted and observed here for the first time. Conductance fluctuations AG in normal metals as will be discussed in section 3, are in general very sensitive to boundary conditions (i.e. measurement procedure). A general framework for the physical content of this phenomenon as well as the calculation of AG are described in the section 3 of this paper. Section 4 of this paper introduces the concept of Berry’s topological phase. The thermodynamical manifestation of this adiabatic phase on the

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38

R. Rammal / Frustration and quantum interference effects in networks

magnetization of superconducting networks is the main object of this part. For a recent review of different topics covered in the talk but not reproduced here, we direct the reader to refs. [1] and [2].

2. Periodic corrections to scaling laws

The magnetoresistance of normal-metal selfsimilar networks (Sierpinski gasket) allowed recently for the first observation of the universal periodic correction to scaling [3]. The basic idea, which is actually a very old one [4], can be summarized as follows. In a physical problem where renormalization-group transformations are exact, periodic oscillating correction to power law behaviors are expected to occur. The magnetoresistance (MR) of a normal-metal selfsimilar network exhibits actually such a behavior in the weak localization regime. In order to highlight these periodic oscillations a regular self-similar network made of submicronic A1 wires: the Sierpinski gasket [5], which is a 2D array of triangles exhibiting a perfect dilation symmetry, has been used. The A1 lines are 0.1 txm thick and 0.3 ~m wide; The elementary triangles, at the lowest length scale, are isosceles triangles of equal height and base: 3.2 ~m. The basic unit corresponds to six stages of iterations 0~
2.1. Periodic corrections: general idea When a renormalization group (RG) transformation holds exactly, there are two basic equations. The first one gives the transformation law of parameters under a scale change (length or time scale, for instance). The second one is simply the transformation equation of a given physical quantity (free energy, Green's function, etc.). For the sake of simplicity let us consider the case of a one-parameter (x) RG and one physical observable F(x). A simple example of a RG transformation is provided by the following functional equation for F:

/ZF(x) = F[~b(x)].

(2.1)

Here F(x) is assumed to be a very well behaved function and /Z denotes a positive real number. The function cb(x), which generates the RG flow describes the qualitative behavior as well as the stability of fixed points and critical exponents. For example if x = 0 denotes a fixed point [~b(0)=0] and ~ b ( x ) = h x + . . . is the corresponding linearized transformation, then w-In/z/In A is the critical exponent describing the power-law solution Fo(x) = x" of eq. (2.1) near this fixed point. Aside from this sort of local analysis of the RG flow near a fixed point, one can be interested in the general solution of eq. (2.1). It turns out that there are actually a large number of simple but nontrivial such solutions: (2.2)

F(x) : Fo(x)p(ln Fo(x)) ,

where p(x) is a In/z-periodic function. This solution can also be written as

F(x) = ~ n,m~O

[

,°x + 0,,,, 1 ,

F,,,,,x 0"÷"cos 2~rm ~

(2.3/ where Fro,., and 0,,., denote constant numbers. The occurrence of these universal oscillations is then a quite general feature of functional equations like eq. (2.1), where th(x) is an analytical function. The MR of self-similar metallic networks provides actually an example of this situation.

2.2. Normal-superconducting boundary line L(H) In the studied sample, the magnetic field which corresponds to one superconducting flux quantum ~b0 = hc/2e in the elementary triangles is ~ 4 . 4 O e . Within the range between 1 mOe and 10 Oe, the critical temperature Tc(H ) of the network has been found to exhibit a very rich structure (To(0) = 1.23 K). In fact five levels of self-similarity of Tc(H) have been obtained. The fine structure was observed down to ~b0/512 and this provides an accurate calibration of the flux quantization at the different hierarchy stages of

R. Rammal / Frustration and quantum interference effects in networks

the Sierpinski gasket. The structure of the line Tc(H) can actually be described up to a very high accuracy, following ref. [6] and recent extension of this work [7]. 2.3. Magnetoresistance oscillations

The fine structure, which was observed on the Tc(H ) line, is no longer present on the magnetoresistance vs. the magnetic flux ~b/~b0. There only remain peaks at integer values of th/~b0, where ~b is the magnetic flux through an elementary triangle. However, close inspection of the MR curve reveals kinks at precise values of the magnetic flux: ~b/~b0 = 4 -n, n = 1, 2 , . . . . This behavior contrasts with the MR behavior on regular Euclidean networks [1], where no structure occurs at tb < ~b0. Furthermore, the logarithmic derivative OIn A R ( H ) / O In H exhibits a very interesting feature (AR = R ( H ) - R(0)). As the field is lowered, this slope increases up to 2 but shows periodic oscillations, the associated period of these oscillations is exactly In 4. The physical origin of the oscillations can be traced back to the fact that the considered structure exhibits a dilation symmetry only under a discrete subgroup of the dilation group (change of length scales by a factor 2). A careful study of the R G flow leads to the following equation for A R / R as a function of ~b/~bo: (2.4)

(ARIR)(~bl~bo) = ~(ARIR)(4d~/qbo).

The factor 4 in eq. (2.4) is just 2 2, i.e. the surface scaling factor, associated to a rescaling of lengths by a factor 2. Eq. (2.4) has the same form as eq. (2.1) and the general solution can be written as

AR(4~) R

~00 =

(4~\~/2/ ~00)

p[2"tr

ln(__~/4~0)~ ln4

(2.5)

/'

where p denotes a periodic function, with period 1. The exponent/3 = d(d - 2 ) / d is the localization exponent as defined in ref. [5] (here a = l n 3 / l n 2 is the fractal dimension and d = 2 In 3/ln 5 is the spectral dimension). In addition to the oscillation of the MR, a power law behavior: A R / R ~ H ~ / 2 has been ob-

39

served at low fields. Such a power law is actually the manifestation of the singular behavior of Landau levels spectrum on the considered structure. It is clear that the existence of a RG transformation for the MR depends of course on the validity of the weak localization theory [8]. Thus the observation of both the predicted power law and the periodic corrections should be viewed as a strong evidence for weak localization theory.

3. Voltage and conductance fluctuations at submicron scales The origin of the interference effects in networks of normal metal is the coherent backscattering of electrons on impurities. This backscattering effect, which is a very well known mechanism for the propagation of linear waves in random media [9], provides the key for understanding the weak localization effects in disordered conductors [10]. A spectacular manifestation of this effect (called also the double passage effect) is the experiment of Sharvin and Sharvin. This experiment is the equivalent of the LittleParks experiment in the sense that it provides a direct evidence of the interference effect in the single loop geometry. The extension of this effect to an extended array of loops is discussed in refs. [1] and [2]. Here, we focus on another spectacular effect predicted by weak localization theories: conductance fluctuations at submicron scales. Before, we describe briefly the basic results of this approach. The basic equation for the quantum correction to the conductivity or of a disordered conductor is [10] 2or0 -Set(r) = ~ C(r, r),

(3.1)

where tr0 is the Drude conductivity, v the electronic density of states and C(r, r) the diagonal component of the Cooperon. The Cooperon is obtained from the solution of the diffusion-like equation

40

R. Rammal / Frustration and quantum interference effects in networks

[(

iV+~ooA

)2 + L ~ 2] C ( r , r ' ) = ~ 1- - ~ ( r - r ' ) . (3.2)

The Cooperon can be interpreted as the classical probability that an electron returns to its original starting point. This probability measures the contribution of closed diffusive trajectories which constitute the key point for understanding the weak localization phenomena. Eq. (3.2) has many similarities with the linearized Ginzburg-Landau equation, excepted a major is replaced by: - L ~ , which corresponds to an "imaginary" coherence length. The phase coherence length L , - ( D % ) 1 / 2 introduces an exponential damping in the Cooperon equation. L~ is defined as the length over which an electron can diffuse before losing the memory of its phase (the phase memory time is %)2 Similar expression for the fluctuation 8o- has also been derived [11], using a summation procedure of the backscattering diagrams.

difference:~2

2

3.1. Random walk interpretation Let us consider first 8o'(r) as given by (3.1), where the average over disorder is already performed. For a system of size L d, the averaged correction can be written as (o"0 = ve2D) 8tr_

o"o

2

1 f

arvh L d drC(r,r),

(3.3)

oro

2 1 f ~vh L d

drA'l(t>>r*)"

(3.4)

Here A n ( t ) is defined as the averaged residence time (up to time t) of a random walk (RW) at its original position r:

An(r) = i dt' P(r, t'[r, O)

(3.5)

0

(D is the diffusion constant).

In eq. (3.5),

P(r, tl0, 0) is the diffusion kernel, the solution of the diffusion equation

(3.6)

subject to the appropriate boundary conditions. The limiting value Au(t>>%) can be extracted from the Laplace transform

P*(r, s) =-f e-Ste(r, tl0, 0) dt 0

by taking the product : sP*(r, s) at s = 0. The counting of closed loops for the RW gives also the fluctuation of 80"2, which now depends on two point r, and r 2. For this we define similarly A 12 as

A12(t )=

i dt~ ti dt 2P(r 2,t21r1,0) 0

0

x P(rl, t,[r2, t2),

(3.7)

which measures the residence time at r I after a visit to r 2 at times t 2 ~< t 1. The meaning of A12(t ) is associated with the possibility of performing loops between two points r 1 and r 2 within the time interval [0, t]. Indeed only this contribution from loops survives after averaging over disorder. Using eq. (3.7), the expression of the averaged fluctuation 8or 2 can be written simply as

f dr2 A,2(t (3.8)

which can also be written as 8or_

DV2p -- r~lp = OP/Ot

A simple calculation shows that eq. (3.8) corresponds exactly to the space-time version of the result of ref. [11]. In this respect eq. (3.8) represents the natural extension of eq. (3.4) to the fluctuation of conductivity. The original argument leading to eq. (3.4) is given in ref. [12]. The interpretation of 8o" and 80 -2 in terms of the residence time statistics of a simple RW allows for a general formalism to calculate 80and 8tr 2 for an arbitrary network geometry [13]. Furthermore, different boundary conditions can be worked out within the same framework. For instance, if we are interested in systems made of wires (cross section S), it is not difficult to derive

R. Rammal / Frustration and quantum interference effects in networks the following results for the resistance correction:

~R/R = r ~ drC(r, r)/L a

(3.9)

[SR2]c/R2=K2 f drl f dr2 C(rl , r2) x C(r2, r])/L 2a for

for (i). For boundary conditions (ii) one has to replace simply sh(...) by c h ( . . . ) in the numerator. Using the notations of ref. [13], this leads to

8R/R = r[coth(L/L,) -T-Lc,/L]/2.

and

(3.12)

From eq. (3.12) one obtains the same result (3.10)

the

resistance fluctuations. Here K 2eELc,/'trhtro S and C(rl, rE) is the diffusion propagator.

3.2. Voltage versus conductance fluctuations It is clear that ~R as well as its fluctuation AR=([~R2]c) 1/2 as given by eqs. (3.9) and (3.10) are integrated quantities and then can be sensitive to boundary conditions (see eq. (3.6), for instance). Indeed, the precise form of C(rl, r2) is very sensitive to the boundary conditions. For the sake of simplicity we shall consider the problem of a single wire. In this case, two situations must be distinguished: - Case (i) or resistance measurement (two-probe configurations): at the boundary with massive contacts, the excess density vanishes and C also. This case corresponds to the usually considered one [11] (absorbing boundary conditions for the RW). - C a s e (ii) or voltage measurement (four-probe configuration): in the absence of flux at the boundary, the normal derivative OC/ax vanishes and not C (reflecting boundary conditions for the RW). In the following, we shall illustrate this difference for the 1D wire: 0 ~< x ~< L. As expected, major differences will appear for L ~< L~, L~ being the only length scale in the problem. Consider first the relative correction to resistance 8R/R. Depending on the considered boundary conditions, C(x) - C(x, x) takes the following form:

C(x) = (L~,/hDS) sh(x/L,) x sh[(L - x ) / L , ] / s h ( L / L , )

41

(3.11)

~R/R ~ t> L6, where ~R/R K/6(Le,/L) for (i) and K(L~,/L) for (ii) are obtained at L ,~ L~. Similarly, the resistance fluctuation is given by L

L

( A N / R ) 2 = KL -2 ~ dXl f dx2 [ s h ( x 1 / L , ) 0 0 x sh[(L - x 2 ) / L J s h ( L / L , ) ] ] 2 (3.13) for boundary conditions (i). The integral in eq. (3.13) leads to the known result (universal fluctuation) [11] K2 (AR/R)2= -~(L/L,) 2 , i.e.

AG =

independent of L at L ~ L~. The boundary conditions (ii) can be worked out similarly with the final result

(AR/ R ) 2 = K2( L~,/ L ) 2 , i.e.

2e2(L~,~ 2 at L ~ L ~ . AG=-~\L/

Translated into the voltage language, one obtains L independent fluctuations of voltage drops, for probes at distance L ~ L , . Recent measurements [14, 15] of voltage-drop fluctuations at submicron scales in metallic wires, confirmed fully the above predictions. For instance, in a four-probe configuration (case (ii)), with constant current leads at the ends of the wire, the voltage drop between two probes has been shown to fluctuate with the probe locations. However, the observed amplitude AV of fluctua-

42

R. Rammal / Frustration and quantum interference effects in networks

tions has been found to be independent of the distance l between probes and this as long as l < L~. Translated into conductance fluctuations this implies AG-1-2. At I>L~, the result AG--1-3/2 has been confirmed in agreement with the classical composition [16] of universal fluctuations of I/L¢~ independent small wires of length L6 each. In addition to the crossover of AG vs. L at L~, further consequences of the analysis above have also been observed: a) the absence of crosscorrelation between two voltage drops with no common probe, b) the occurrence of an anticorrelation in the presence of a common probe. Clearly the strong sensitivity of the conductance fluctuations to boundary conditions is a new evidence for the quantum interference effects at submicron scales. Further investigations in that direction would be the frequency dependence of the resistance fluctuations. This would probably answer basic questions relative to the origin of 1/f noise in conductors.

4. Magnetization of superconducting networks: Berry's phase An unexpected and remarkable phenomenon has recently been discovered by Berry [17] in the context of the adiabatic theorem of quantum mechanics. During an excursion around a closed loop C in the external parameter space, the acquired phase by the adiabatic deformation of the wave function is not simply the familiar dynamic phase. Instead an additional extra phase F(C) may result and depends only on the geometry of the parameter space and the considered loop C. The existence of F(C) can be understood as an Aharanov-Bohm effect in the parameter space and thus is relevant to many areas of quantum physics: condensed matter, molecular physics and particle physics. One of the important consequences of the adiabatic phase F(C) of particular interest here is the modification of the semi-classical Bohr-Sommerreid quantization rule. In what follows we show that the magnetization M(T, H) of a regular superconducting net-

work, in the so-called mixed state, provides an example where the existence of F(C) can be observed on a thermodynamical quantity. In fact the presence of a Berry's phase leads to a rather singular behavior of M close to the normal superconducting phase boundary To(H): at each rational value of the reduced flux, aM/OT exhibits an asymmetric jump. The asymmetry is a consequence of an adiabatic phase, arising in an extended WKB theory.

4.1. Mixed state It is now well established that the mean-field Ginzburg-Landau (GL) theory provides a very accurate description of the phase boundary line T~(H) of extended arrays of superconducting filaments. For instance, novel flux-quantization effects have been predicted and observed: oscillations of T~(H) as well as cusp-like singularities at rational th/~b0. The fine structures of the transition line To(H), very well described by the linearized GL equations, are the signature of a complex organization of the order parameter under the competing effects of the magnetic field (H) and the spatial periodicity of the lattice. In order to describe the ordered phase below Tc(H), the complete nonlinear GL equations must be considered. Of course this calls for an extension of Abrikosov's original approach to the case of the network geometry. As T is lowered below T¢(H), the order parameter increases. However as pointed out by Abrikosov [18], close t o the critical line, the equilibrium solutions must (by continuity) have a strong resemblance to certain solutions of the linearized GL equation: 2~

iV+~A

)2

~b=E~b,

(4.1)

where ~b(r) denotes the order parameter at r. The lowest eigenvalue E(H) of eq. (4.1) is directly related to To(H): Tc(H)/T~(O)= 1 - E(H)~z(O), where ~:(T) = ~:(0) [1 T/T~(O)] -1/2 refers to the temperature-dependent coherence length. The associated degenerate eigenstates correspond to the nucleation of

R. Rammal / Frustration and quantum interference effects in networks

superconductivity in different parts of the sample and have the same energy. The higher order term in GL free energy will eventually favor some particular solutions. An appropriate solution can be built up from a linear combination of the degenerate solutions of eq. (4.1). Proceeding exactly as Abrikosov, one can show [19] that the equilibrium states of the network correspond to the minimum of a generalized Abrikosov's parameter /3A(H ) defined by /3A(H ) = (1~4)/(~2) 2. Here, for a given H, ~b refers to a linear combination of the eigenstates of eq. (4.1) corresponding to min E(H). In general, the physical properties of the mixed state are controlled by the topology of the network through T¢(H) and /3A(H ), which are directly related to the spectrum of eq. (4.1). Explicit expressions for the order parameter (d/~)=(T~(H)-T), the free energy F = (T¢(H) - T) 2 and the magnetization can then be obtained. For instance, the magnetization is given by ~b0 ] T¢(H)- T 2,tr~2(~-~T~(0) flg(2x2--=m=_--~)

47rM= x

dT~(H) - dH

(4.2)

In eq. (4.2), r is the GL parameter and we have defined/~ by ~b0 dE - 2rr d H -

~b0/2~r d T~(H) ~2(0)Tc(0) dH

(4.3)

As a simple consequence of the formalism outlined above, one recovers trivially the known properties of the mixed state of a bulk type II superconductor: Tc(H ) - To(0 ) = H, /z = 1 and /3A - 1.16 (independent of H ) corresponding to a triangular arrangement of vortices. The case of extended networks can be worked out in detail within the same framework [19]. Explicit values of fig(H) have been obtained as well as the order parameter configurations, current patterns, etc., for rational values of the reduced flux. Here we shall focus our discussion on the magnetization behavior only.

43

4.2. Magnetization singularities For the sake of simplicity we limit our discussion to the square lattice case. As pointed out in refs. [20] and [21], the linearized GL equations reduce in this case to a set of finite difference equations for the order parameter ~,~,, at the nodes (m, n) of the network. Using the translation invariance, ~m,, = I~m e-°l" (01 = arbitrary phase factor) one ends up with Harper's equation [20]

q,m+l + ~b,,,_l + 2COS(ylz + O1)~bm= e~bm .

(4.4)

Here y=2"rr~b/~b0, gb=Ha 2 being the flux through an elementary cell (a x a) and e denotes the "tight-binding" energy, defined by e 4cos(aE1/2). The rich spectrum (e, y) of eq. (4.4) has been studied in ref. [22]. Here we are interested only in the lowest edge of this spectrum. In fact M is governed by e ( y ) and de/dy along this edge, the line T¢(H) being related to e(y) through e ( y ) -- 4 cos(a/~(T)). As was shown in refs. [20, 21], T¢(H) as well as e(y) exhibit cusp-like singularities at each rational value of the reduced flux ~b/~b0. For instance, near 3' = 0, e is given [20] very accurately by e = 4 - [ y [ + (~(e-1/Ivl),; leading to a cusp of T¢(H) at zero field and then (eq. (4.2)) to a symmetric jump of OM/OT. More generally, for generic rational flux ¢k/q~o= P/q, OM/OT can be shown to exhibit an asymmetric jump. In order to show this property, one needs an approximate expression of e(y) for y close to rational values of ~b/~b0. It turns out that WKB theory is the appropriate framework to solve this problem. To do this, one can use two different approaches. The first one is an extension of the continuum approximation, used previously in ref. [20] (Bloch picture) and involving a q x q Hermitian eigenvalue equation. The other (Floquet picture) involves considering products of q 2 × 2 transfer matrices. Full solution has been obtained with each method. A detailed account of Bloch picture will be found in ref. [23], here we describe briefly the second approach as outlined first by Wilkinson [24]. Using a transfer matrix formulation [22], eq. (4.4) can

R. Rammal / Frustration and quantum interference effects in networks

44

be written as [ qJm+,

Ie-21osxm

(4.5)

L~m-lJ where x , , = m 7 + 0 1 . For rational flux 3'0= 2~rp/q, this leads to the following dispersion relation: Tr M = 2cos S', where S' is a phase factor and M refers to the string M - T(x + (q 1)3,)"" T ( x + 7)T(x). In this case, e(x, S') is periodic in x (period 2~rp/q) as well as in S' (period 2~r/q) since p and q are assumed to be coprime. For 7 = Y0+ Ay, an adiabatic WKB theory can be implemented, where h-=q A7 plays the role of an adiabatic parameter. The phase space is (x, S'), with x as "position" and S' as "momentum". Within this scheme an interesting feature arises in connection with Bohr-Sommeffeld quantization rule. The eigenvectors of matrix M cannot be given as single-valued and continuous functions of the position in phase space. When they are transported around a closed orbit, they are multiplied by a complex factor when they return to their starting point. The existence of the adiabatic phase F leads to a modification of Bohr-Sommerfeld rule. In particular at the edge of the spectrum, the quantization rule becomes

tq e -

e0

=

--

s

2q 2 +

- -

s

IA

I

x n+½-qSgn(AT)

,

n=integer. (4.6)

The different parameters in eq. (4.6) are defined with M-= I~ oBI in the vicinity of x = 0, S ' = 0 and e = e0 giving the edge of the spectrum at y = %. More precisely, t = q-1 a Tr M/03,, - s = 0 Tr M/Oe and x = [(D - 1)/2C]aC/Ox ½aD/ax, all taken at x = 0 , S ' = 0 and e = %. The adiabatic angle F is given explicitly by F = 2~rK/q and is responsible for the asymmetry of e(y). In fact, from eq. (4.6) one deduces the

values of the slope OelOT to the left (S-) and to the right (S+). For generic p/q, we have Is+l IS-[. The equality is reached only for p / q = 0 and i and this corresponds to an accidental degeneracy, due to a time-reversal symmetry which leads to F = 0. In all studied cases (square and triangular lattices) and for generic rationals p/q, Oe/Oy and then OM/OT exhibit an asymmetric jump. The asymmetry being a consequence of the adiabatic phase, whereas the jumps are due to the infinite extension of the network in both directions. In fact, for semi-infinite [20] networks (ladder, strips . . . . ), Tc(H ) is a differentiable function and OM/OT has no jump. The same conclusion holds for finite networks with the exception of some particular cases as the single loop geometry. Here OM/OT has a sawtooth variation with symmetric jumps at half-integer values of ~b/~b0. Actually Oe/Oy vs. y in the square lattice case relects this sawtooth variation, but with asymmetric jumps, almost everywhere.

4.3. Remarks a) Using

aS/oHIr

Maxwell's

relation

OM/OT[u =

(S = entropy), it is not difficult to trans-

late the asymmetry of OM/OT in terms of entropy variations, close to each rational flux. b) Thanks to a direct and very accurate measurement of aM/OT (dM response to a dT modulation), the predicted behavior of OM/OT is actually accessible to an experimental study on real networks. Preliminary results [25] seem to support the predictions presented here. c) In thermodynamics, Carnot's cycles are examples of holonomy effects governed by inexact differential 1-forms. In this respect, Berry's adiabatic phase is governed by a phase 2-form in parameter space. Therefore, the direct measurement of OM/O T provides the first thermodynamical signature of Berry's phase. For further details, see ref. [25].

5. Conclusion

The study of quantization effects in supercon-

R. Rammal / Frustration and quantum interference effects in networks

ducting and normal metal networks revealed in the past [1, 2] fundamental phenomena at the submicron scales. Clearly these artificial systems appear now as ideal model systems for the study of general physical problems like: frustration, Landau levels, harmonic excitations and coherent effects on novel structures as fractals or quasiperiodic networks. Here we have discussed just three examples: periodic corrections to scaling, voltage and conductance fluctuations and Berry's topological phase. Certainly other examples will be found in the future, either for studying fundamental phenomena in physics or to understand physical properties at the submicron scales.

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