Panayotis Akridas-Morel

Nicolas Cherroret

Dominique Delande
email: dominique.delande@lkb.upmc.fr

Multifractality of the kicked rotor at the critical point of the Anderson transition

We show that quantum wavepackets exhibit a sharp macroscopic peak as they spread in the vicinity of the critical point of the Anderson transition. The peak gives a direct access to the mutifractal properties of the wavefunctions and specifically to the multifractal dimension d2. Our analysis is based on an experimentally realizable setup, the quantum kicked rotor with quasi-periodic temporal driving, an effectively 3-dimensional disordered system recently exploited to explore the physics of the Anderson transition with cold atoms.

I. INTRODUCTION

In the vicinity of the critical point of a continuous phase transition, large fluctuations are observed [START_REF] Landau | Statistical Physics, Course of theoretical physics[END_REF], responsible for dramatic phenomena like e.g. critical opalescence. Beyond mean-field descriptions, renormalization group approaches make it possible to describe critical phenomena at (almost) all scales, and to predict critical exponents [START_REF] Wilson | [END_REF]. Large fluctuations arise as well in quantum phase transitions, where they are usually probed via correlation functions or transport properties. The metal-insulator Anderson transition, taking place in disordered quantum systems, is especially interesting. It separates a metallic phase at weak disorder, where transport is diffusive, and an insulating phase at strong disorder, where transport is inhibited due to interference in multiple scattering from random defects [3,4], a phenomenon known as Anderson localization. The dimensionality of the system is a crucial parameter: Anderson localization is the generic scenario in one-dimensional (1D) systems, while the Anderson transition can be observed in dimension strictly larger than two. In threedimensional (3D) systems, the critical point of the Anderson transition occurs for strong disorder, when the product of the wavenumber k with the mean free path is close to unity, (k ) c ≈ 1 [START_REF] Ioffe | Progress in semiconductors[END_REF]. Although the order parameter for the Anderson transition remains debated [START_REF] Evers | [END_REF], there is nowadays a wide consensus that it is a second order continuous transition, with an algebraic divergence of the localization length on the localized side, ξ ∝ 1/((k ) c -k ) ν , and an algebraic vanishing of the diffusion coefficient on the diffusive side, D ∝ (k -(k ) c ) s . Numerous evidence for these properties have been found in numerical simulations of the standard Anderson model, which has been also used to compute the critical exponents ν = s ≈ 1.57 in dimension d = 3 [7,8]. This value is universal (depending only on the dimension and symmetry properties) and has been confirmed on other models [9]. Numerical studies of the Anderson model have shown that the distribution of conductance at the critical point is universal as well, scale invariant [10] and broad, a clear-cut manifestation of large fluctuations at the critical point. Large fluctuations also show up in the critical eigenstates, which are strongly multifractal, displaying regions where |ψ| 2 is unexpectedly large and regions where it is unexpectedly small [11]. Usually, this property is quantitatively described using the generalized inverse participation ratio (GIPR) [12,13]:

P q = L d d d r |ψ(r)| 2q , ( 1
)
where q is a real number and L is the system size. The multifractality analysis studies how the GIPR averaged over eigenstates and/or disorder realizations scales with L. If the average -denoted by in the following -P q scales like L -τq , τ q is called the multifractal exponent. By construction τ 0 = -d and, by normalization of the wavefunction, τ 1 = 0. One can equivalently use the set of multifractal dimensions d q = τ q /(q -1). A wavefunction delocalized over a set of dimension D (which can be an ordinary or a fractal set) will have d q = D for all q. For multifractal states, finally, d q is a continuous function of q with large positive q values probing the regions of large |ψ| 2 and negative q the regions where |ψ| 2 is vanishingly small. How to experimentally access multifractal dimensions is far from obvious. This in principle requires to measure wavefunctions for various disorder realizations or energies close to the critical point everywhere in space, a tremendously difficult task. Alternatively, one can extract only a part of the information on multifractality by selective, less complete, measurements [14][15][16], for example of the intensity distribution on the exit plane of a disordered slab. In this paper, we show that, at long times, the average density of an initially localized wavepacket spreads, but develops near its initial location a sharp peak giving a direct access to the multifractal properties of the wavefunctions and specifically to the multifractal dimension d 2 . We base our analysis upon an experimentally existing system, the quasi-periodically kicked rotor, which has been shown to display the Anderson metal-insulator transition [17,18], but the mechanism to extract a multifractal dimension from the expansion of a wavepacket is quite general and could be used in other critical systems. Importantly, although the system is 1D, it can be mapped on a 3D disordered system, so that the measured multifractal dimension d 2 is truly the one of 3D critical Anderson-like systems.

The paper is organized as follows. In section II, we describe the quasi-periodically kicked rotor and its connection with quantum disordered systems and the metalinsulator Anderson transition. The theoretical approach explaining the origin of the sharp peak in the average density and its connections with multifractality is developed in section III. This knowledge is used in section IV, showing how to extract the multifractal dimension d 2 . It is demonstrated on the results of numerical simulations, where the values of the parameters are chosen like in the real experiments. In section V, we briefly discuss the experimental perspectives and summarize our results.

II. THE QUASI-PERIODICALLY KICKED ROTOR

A. Hamiltonian

The quasi-periodically kicked rotor (QPKR) is a simple 1D system -a standard rotor -exposed periodically to kicks whose amplitude is modulated quasi-periodically in time. With two quasi-periods in addition to the period of the kicks, the Hamiltonian reads:

H = p 2 2 + K cos x × [1 + ε cos(ω 2 t + ϕ 2 ) cos(ω 3 t + ϕ 3 )] n δ(t -n) , (2)
where the unit of time is the interval between two consecutive kicks. K, ε, ω 2 , ω 3 , ϕ 2 , ϕ 3 (and the Planck's constant governing the quantum evolution) are dimensionless parameters whose roles are discussed below. In the limiting case ε = 0, one recovers the usual kicked rotor [19,20]. As discussed in [21][22][23], the dynamics of the QPKR can be mapped on the dynamics of a 3D periodically kicked pseudo-rotor [24] with Hamiltonian:

H = p 2 1 2 + ω 2 p 2 + ω 3 p 3 +K cos x 1 [1 + ε cos x 2 cos x 3 ] n δ(t -n) . (3)
More precisely, it is shown in [23] that the evolution of

Ψ(x 1 , x 2 , x 3 , t) ≡ ψ(x 1 , t)δ(x 2 -ϕ 2 -ω 2 t)δ(x 3 -ϕ 3 -ω 3 t) (4
) under Hamiltonian H, Eq. (3), is strictly equivalent to the evolution of ψ(x, t) under Hamiltonian H, Eq. ( 2), for any arbitrary initial wavefunction ψ(x, t = 0).

Provided ω 2 , ω 3 , π, are mutually incommensurate real numbers, the 3D periodically kicked pseudo-rotor can itself be mapped [23] on an anisotropic 3D Anderson model, where K controls the disorder strength and ε the anisotropy [25], a fact further confirmed by a low-energy effective field theory [26]. An important consequence of these mappings is that the QPKR is a time-dependent 1D system equivalent to a 3D disordered time-independent system, and thus makes it possible to explore the dynamics of the latter, which may display the metal-insulator Anderson transition.

B. Dynamical localization

Depending on the values of the , K, ε, ω 2 , ω 3 , ϕ 2 , ϕ 3 parameters, the 3D disordered system may be localized (at small disorder) or diffusive (at large disorder). The two regimes are separated by a critical point -usually called mobility edge -where multifractality is expected to play an important role. The mapping of the QPKR to the 3D disordered system is discussed in details in [23]. An important point is that the statistical properties of the 3D disordered system depend only on the parameters , K, ε. Various values of ω 2 , ω 3 , ϕ 2 , ϕ 3 correspond to various realizations of the disorder, all with the same statistical properties. Hence, one can explore the 3D metalinsulator Anderson transition by varying , K, ε with the QPKR.

An important property of the QPKR is that the localized/delocalized dynamics takes place in momentum space, not in position space like the usual Anderson model. Such a localization in momentum space has been dubbed "dynamical localization" [START_REF] Casati | Stochastic behavior of a quantum pendulum under periodic perturbation[END_REF][START_REF] Fishman | [END_REF][29]. Specifically, the system is localized at small K value, i.e. p 2 (t) tends to a constant at long t, and diffusive at large K value, i.e. p 2 (t) ∝ t at long t [where p 2 (t) ≡ ψ(t)|p 2 |ψ(t) denotes the quantum mechanical expectation value]. In between, there is a critical point, whose position can be approximately predicted by a mean-field approach, the selfconsistent theory of localization (SCTL) [25,[START_REF] Vollhardt | Electronic Phase Transitions[END_REF], where the system behaves sub-diffusively: p 2 (t) ∝ t 2/3 at long t. Experimentally, these properties have been confirmed by monitoring the temporal expansion of a wavepacket initially localized in momentum space around p = 0, that is by measuring |ψ(p, t)| 2 at increasing time, with the initial state |ψ(p, 0)| 2 ≈ δ(p).

In the simplest case of the standard kicked rotor where ε = 0, the 3D disordered system reduces to a 2D array of uncoupled 1D disordered systems, which are localized for all K values. This dynamical localization has been observed experimentally using cold atoms in Ref. [START_REF] Moore | [END_REF]. The full 3D Anderson transition with ε = 0 has been later observed and characterized [17,32].

C. Critical point

At the critical point, the spatial fluctuations of the wavepacket have been numerically and theoretically studied [33,34] from an analysis of the GIPRs, Eq. ( 1): they display only very weak multifractal properties. In contrast, we will show in section III that the average den-sity (averaged over disorder realizations) |ψ(p, t)| 2 itself presents a direct, macroscopic signature of the multifractality of the 3D critical model.

Because of sub-diffusion at the critical point, the width of the wavepacket increases like t 1/3 , but its global shape is independent of time, a manifestation of scale invariance. The SCTL makes a definite prediction for this shape [32,[START_REF] Lemarié | Transition d'Anderson avec des ondes de matière atomiques[END_REF]:

|ψ(p, t)| 2 = 3 2 3ρ 3/2 t -1/3 Ai 3ρ 3/2 t -1/3 |p| , (5)
where ρ = Γ(2/3)Λ c /3 is related to the critical quantity

Λ c = lim t→∞ p 2 (t) /t 2/3
, a numerical factor depending on the anisotropy ε [25], with Γ the Gamma function and Ai(x) the Airy function. It is convenient to define scaled variables: P = pt -1/3 , N (P, t) = t 1/3 |ψ(p, t)| 2 so that the SCTL prediction reads:

N (P) = 3 2/3 2ρ 1/2 Ai |P| 3 1/3 ρ 1/2 , (6)
independent of time. This prediction has been found in excellent agreement with the experimental results on the atomic QPKR after few tens of kicks [32], describing both the kink around P = 0 and the tail ∝ exp -α|P| 3/2 .

D. Numerical simulations

In the following, we will use numerical simulations of the QPKR. The structure of the Hamiltonian of the QPKR, Eq. ( 3), makes it very easy to numerically propagate any initial state. The free evolution operator between two consecutive kicks is diagonal in the momentum eigenbasis, while the instantaneous kick operator is diagonal in the position eigenbasis. Because the Hamiltonian is spatially periodic with period 2π, we use the Bloch theorem which makes it possible to restrict to a configuration space x ∈ [0, 2π[ with periodic boundary conditions, changing only the kinetic energy term in the Hamiltonian p 2 /2 to (p + β) 2 /2, where β ∈]-1/2, 1/2] is the Bloch vector. The configuration space, x ∈ [0, 2π[, is discretized in N equidistant points; in momentum space, this corresponds to wavevectors (that is, up to multiplicative factor , momenta) in the ]-N/2, N/2] range. Switching from the configuration space representation of the wavefunction to the momentum space representation involves a Fourier transform of length N, (the dimension of the Hilbert space) which can be done efficiently.

Altogether, the propagation algorithm is thus a series of forward and backward Fourier transforms interleaved with multiplication of each component of the current state by a phase factor. The initial state is chosen as a δ function at the origin ψ(p, t = 0) = δ(p). The quantity |ψ(p, t)| 2 is thus the intensity propagator at time t. The averaging over disorder realizations is performed firstly by averaging over many values of the Bloch vector β, and secondly by averaging over the phases ϕ 2 , ϕ 3 of the quasi-periodic kick amplitude modulation. After averaging |ψ(p, t)| 2 over the disorder realizations, we obtain the disorder-averaged intensity propagator P QPKR (p, t) ≡ |ψ(p, t)| 2 . A simple rescaling of the momentum p to P = pt -1/3 provides us with the quantity N (P, t) = t 1/3 P QPKR (p, t).

The size N of the Hilbert space must be chosen sufficiently large for the momentum distribution to be negligibly small at the maximum momentum |p| = N /2. We used up to N = 49152 for the longest time considered t = 4 × 10 8 . The averaging was performed over 17600 disorder realizations for times up to t = 10 6 , 8800 for t = 10 7 , 1536 for t = 10 8 and 120 for t = 4 × 10 8 .

In Fig. 1, we show the numerically computed N (P) at various times (number of kicks), right at the critical point of the Anderson transition. While the agreement with Eq. ( 6) is excellent at short time (100 kicks, comparable to the duration of the experiment), a sharp peak near p = 0 develops at increasingly long times. The existence and properties of this peak is the central subject of this paper. We show below that this peak -not described by the SCTL -is a manifestation of multifractality at the critical point and is directly related to the multifractal dimension d 2 .

III. MULTIFRACTAL MODEL OF THE SHARP PEAK A. 3D disordered system

To understand the origin of this sharp peak, it is easier to leave the QPKR for a moment and turn back to a standard disordered system such as the Anderson model, where 3D localization takes place in configuration space. The average expansion of a wavepacket with time is described by the disorder-averaged intensity propagator P (r 1 , r 2 , t), which gives the average probability to move from r 1 at t = 0 to r 2 at time t. Its Fourier transform w.r.t. time t, the propagator

P (r 1 , r 2 , ω) = G R (r 1 , r 2 , E +ω/2)G A (r 2 , r 1 , E -ω/2) 2πν (7
) is proportional to the disorder average of the product of retarded and advanced Green's functions at energy E, with ν the density of states per unit volume. Thanks to the statistical translational invariance of the disorder, it depends only on r = r 1 -r 2 . For simplicity of the notations, we write only r and omit the Edependence of quantities. It is convenient to consider the temporal and spatial Fourier transform P (q, ω) = dt d d r P (r, t) e iωt-iq.r . Because the disordered potential is statistically isotropic, it depends only on the modulus |q| and can be written as:

P (q, ω) = 1 -iω + D(q, ω)q 2 . ( 8
) 0 0.2 0.4 0 0.2 0 0.2 0 0.2 -5 0 5 0 0.2 t =10 2 t =10 3 t =10 5 t =10 6 t =10 8 P = pt -1/3 N (P) FIG. 1.
Temporal evolution in momentum space of a wavepacket launched at t = 0. The numerical simulation (black solid curve) is performed for the quasi-periodically kicked rotor at the critical point of the 3D Anderson transition. The global expansion is sub-diffusive in momentum space p 2 (t) ∝ t 2/3 . When plotted vs. dimensionless rescaled coordinate P = pt -1/3 , the density in momentum space N takes a time-independent shape -predicted to be a Airy function, Eq. ( 5) (red dashed curve) -except near the origin where a sharp peak grows with time. This peak is a direct manifestation of multifractality at the critical point. Parameters are K = 8.096, ε = 0.4544, ω2 = 2.67220902067011, ω3 = 2.01719090924681, = 3.54.

Eq. ( 8) defines the momentum and frequency dependent diffusion coefficient D(q, ω). Causality implies that P (r, t) vanishes for negative t, while unitarity of the Hamiltonian evolution implies the conservation of probability d d r P (r, t) = 1 for all t > 0, i.e.:

P (q = 0, ω) = 1 -iω . ( 9
)
Causality also implies that D(q, ω) has no singularity in the upper complex half-plane ω > 0 and cannot diverge more rapidly than 1/q 2 at small q. Because P (r, t) is a real function, its Fourier transform satisfies:

P (q * , ω * ) = P * (q, -ω) (10)
where the * denotes complex conjugation, so that

D(q * , ω * ) = D * (q, -ω). (11)
In particular, D(q, iω) must be real for real q, ω.

Of particular interest is the small ω limit of D(q, ω), which describes long times. In this limit and in a usual diffusive system, D(q, ω) = D 0 = 2 /(3τ ) equals the classical diffusion coefficient, where is the mean free path and τ = m / k the mean free time. In a localized system, D(q, ω) = -iωξ 2 , with ξ the localization length. At the critical point finally, the SCTL predicts D(q, ω) ∝ (-iω) 1/3 [START_REF] Shapiro | [END_REF][START_REF] Müller | Disorder and interference: localization phenomena[END_REF], which yields Eq. ( 5). In turn, deviations from the Airy shape, as visible in Fig. 1, imply that D(q, ω) must deviate from the simple (-iω) 1/3 dependence. Following earlier analyses of the intensity propagator [START_REF] Wegner | [END_REF][START_REF] Wegner | Density correlations near the mobility edge[END_REF], Chalker [START_REF] Chalker | [END_REF] proposed that, at short distance (large q), D(q, ω) acquires a non-trivial q-dependence that we now recall. D(q, ω) must respect the one-parameter scaling law characterizing the Anderson transition at large distance and long time [41]. This scaling law involves the following relevant length scales: the mean free path , 1/q, and L ω = (ωτ ) -1/3 , the mean distance traveled by a particle in time 1/ω at the critical point. The localization length ξ is in general an additional characteristic length, but at the critical point it is infinite and thus irrelevant. In the following, we will only consider the long time limit ωτ 1, so that L ω . The one-parameter scaling law implies that q appears in D(q, ω) only through the qL ω combination. Under this constraint, Chalker's ansatz [42] distinguishes the three following regimes [START_REF] Chalker | [END_REF][START_REF] Brandes | [END_REF]:

(A) When qL ω < 1 (long distance), multifractal correlations have no time to develop and one expects the normal sub-diffusive behavior, i.e. D(q, iω) ∝ D 0 (ωτ ) 1/3 . This is region (ii) in [START_REF] Chalker | [END_REF].

(B) When qL ω > 1 (short distance), but still q < 1, multifractality sets in and D takes a q dependence: D(q, iω) ∝ D 0 (ωτ ) 1/3 (qL ω ) d2-2 . This is region (v) in [START_REF] Chalker | [END_REF].

(C) Finally, at very short distance q > 1, the mean free path sets a non-universal cutoff ensuring that the propagator P (q, ω) falls off sufficiently rapidly at large q, so that no unphysical singularity exists in P (r, t) below the mean free path. While (A) and (B) obey the one-parameter scaling law, regime (C) breaks it at short distance where no (sub)-diffusive behavior makes sense. This is the extreme left part of region (v) in [START_REF] Chalker | [END_REF].

B. Singularity of the propagator near the origin in 3D disordered systems

The behavior of the disorder-averaged intensity propagator (7) near the origin r = 0 is a bit subtle at the critical point. We first consider the non-multifractal case -regime (A) -where the diffusion coefficient D(q, ω) scales like ω 1/3 . We use the mixed momentum-time representation of the intensity propagator: P (q, t) = d d r P (r, t) e -iq.r = dω 2π P (q, ω) e -iωt (12) At very large q, the -iω term in the denominator of Eq. ( 8) can be neglected and the integral over ω computed exactly, e.g. using Eq. 3.761.9 in [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF]. The result is ∝ q -2 t -2/3 . The 1/q 2 behavior at large q converts, after a 3D Fourier transform, into a 1/r divergence in configuration space:

P (r, t) ∝ t -2/3 r -1 . ( 13
)
When the multifractal regime (B) comes into play -that is at short distance r -D(q, ω) scales like q d2-2 ω 1-d2/3 at large q. Again, the -iω term in the denominator of Eq. ( 8) can be neglected and the integral over ω computed exactly. The result is ∝ q -d2 t -d2/3 . After a 3D Fourier transform, this gives:

P multifractal (r, t) ∝ t -d2/3 r d2-3 . (14)
Finally, at very short distance r ≈ -regime (C) -the divergence of Eq. ( 14) is smoothed out. Numerical simulations on the Anderson model [START_REF] Huckestein | [END_REF] and related models [46,47] have confirmed such a behavior and the existence of the regimes (A-C) and the associated scalings, especially non-integer algebraic exponents.

C. Singularity near the origin for the QPKR How can we relate the laws (A-C) to the observed behavior for the QPKR? As discussed in section II A, the dynamics of the 1D QPKR can be mapped onto the one of a 3D disordered system, with a class of specific wavefunctions, Eq. ( 4), perfectly well localized along directions x 2 and x 3 , that is with a uniform density along the conjugate variables p 2 and p 3 at all times, and density scaling like |ψ(p 1 , t)| 2 along the "physical" momentum p = p 1 . Thus, the 1D propagator of the QPKR simply follows from the 3D intensity propagator by summing over the transverse directions p 2 and p 3 . In Fourier space, this means that only q 2 = q 3 = 0 contributes:

P QPKR (p, t) = dω 2π
e -iωt dq 2π e iqp P (q, q 2 = 0, q 3 = 0, ω)

= dω dq 4π 2 e i(qp-ωt) -iω + D(q, ω)q 2 . ( 15
)
Alternatively, the 1D intensity propagator could be obtained by integrating the 3D intensity propagator over the two transverse directions: P (x, y, z, t) dy dz. At this stage, let us point a slight complication: we so far assumed the 3D disordered system to be statistically isotropic. In [48] however, it has been shown that the 3D Anderson system on which the QPKR is mapped is anisotropic so that the disorder-averaged propagator should be strictly speaking written as:

P (q, ω) = 1 -iω + q.D(q, ω).q , ( 16
)
where D is the anisotropic diffusion tensor. Because the 1D propagator of the QPKR involves only q 2 = q 3 = 0 though, the only component of D that matters is D 11 , so that everything boils down to Eq. ( 15). The Fourier transform from ω to t is eventually identical for the QPKR and for a disordered 3D system. Thus, the mixed representation P (q, t), Eq. ( 12), is identical in both cases. It is only the 3D or 1D Fourier transform w.r.t. the q variable which makes a difference.

We have now all the ingredients at hand to infer N (P = pt -1/3 , t) = t 1/3 P QPKR (p, t) for the QPKR. We first look at the non-multifractal contribution, assuming that the 1/q 2 t 2/3 behavior at large q -regime (A) -is valid everywhere. A simple 1D Fourier transform converts it in a |p|/t 2/3 singularity in the intensity propagator P QPKR (p, t). At p = 0, there is a constant contribution scaling like t -1/3 , in accordance with the oneparameter scaling law, finally leading to

N (P) ≈ α -β|P| (17)
at small P = pt -1/3 . In fact, it is possible to perform exactly the full double Fourier transform, see [32,[START_REF] Lemarié | Transition d'Anderson avec des ondes de matière atomiques[END_REF]. The result is Eq. ( 6), which displays explicitly the expected linear singularity near the origin [49].

When the multifractal regime comes into play, D(q, ω) scales like q d2-2 ω 1-d2/3 at large q -regime (B) -resulting in P (q, t) ∝ q -d2 t -d2/3 . The 1D Fourier transform gives a |p| d2-1 /t d2/3 singularity near p = 0, that is P QPKR (p, t) = t -1/3 α -β|pt -1/3 | d2-1 or:

N (P) = α -β|P| d2-1 , (18)
at small P. The α, β constants are not universal and depend on the boundary around qL ω = 1 between the normal subdiffusive and the multifractal regions (A) and (B), but the algebraic dependence |pt -1/3 | d2-1 is universal. Note that because d 2 ≈ 1.24 at the critical point of the 3D Anderson transition [11], the 3D intensity propagator, Eq. ( 14), has an algebraic divergence near r = 0 while the 1D intensity propagator of the QPKR is finite at p = 0, with a non-integer power law singularity.

Whereas the contribution of the mean-field regime (A) to D(q, ω) leads to the kink N (P)-N (0) ∝ |P| at small P, Eq. ( 6), the multifractal law ( 18) is more singular: it is responsible for the small peak near the origin observed in Fig. 1. At short time (say shorter than 100 kicks), this rather weak singularity at the origin is cut at the mean free path, and the normal component (the Airy function) reproduces very well the numerical calculation.

As time grows, the whole wavepacket spreads in size like t 1/3 making the short distance cutoff to act at smaller and smaller P = pt -1/3 . Because d 2 > 1, the algebraic term in Eq. (18) does not diverge at P = 0, only its derivative is infinite.

IV. EXTRACTION OF THE MULTIFRACTAL DIMENSION d2

We can now use the numerically computed wavepackets to extract the value of the multifractal dimension d 2 .

We have used two different methods, see Fig. 2.

A. First fitting procedure

The first fitting procedure uses only the very central part, near P = 0, of the numerically computed disorderaveraged intensity propagator. Indeed, Eq. ( 18) predicts an algebraic cusp at small P clearly visible at long times. We thus fitted the central part of the numerically computed N (P, t) with Eq. ( 18), with three fitting parameters α, β and d 2 . The range of P values used must be not too large, as the fitting expression is expected to be valid only near P = 0. We chose to include points up to |P| = 2.25 -see Fig. 2(c) -but the extracted d 2 value turns out to depend only weakly on the range used. This simple procedure already gives very satisfactory results, with values of d 2 almost independent of time at long time, although a separate fit is done for each time.

At very short p, of the order of the mean free path, the disorder-averaged intensity propagator does not obey the one-parameter scaling law of the Anderson transition (regime (C)), so that the expression (18) is not expected to be valid. In other words, the algebraic cusp at small p is smoothed over one mean free path. The corresponding range in P = pt -1/3 shrinks when t increases, explaining why the peak near p = 0 grows. Such a smoothing affects the quality of the fit. In order to take this fact into account, we have to exclude a small region around P = 0 from the fit. The results are essentially independent of the size of this small region.

B. Second fitting procedure

The second fitting procedure uses the full numerically computed disorder-averaged intensity propagator. It assumes that the momentum-frequency dependent diffusion coefficient follows the Chalker's ansatz [START_REF] Chalker | [END_REF] with the three different regimes presented in the main text. More precisely, we use the following ansatz:

D(q, iω) = 3 2 2/3 D 0 (ωτ ) 1/3 f (qL ω ) ( 19
)
where τ is the mean scattering time, D 0 = 2 /3τ the classical Boltzmann diffusion coefficient ( is the mean free

P = pt -1/3 N (P) FIG. (a) (b) (c) (d)
2. (a) Solid black curve: Numerically computed temporal evolution at 10 6 kicks) in momentum space of a wavepacket launched at t = 0 near momentum p = 0 (see Fig. 1 for the parameter values). The fit by an Airy function, Eq. ( 5), dashed blue curve, which does not take into account multifractality, is obviously bad; The residual (difference between the curve and the best fit) is shown in (b). The central region is very well fitted by an algebraic dependence, Eq. ( 18), (residual shown in (c)) and gives d2 = 1.28 ± 0.03 (note the vertical scale 10 times smaller than in (b)). A fit of the full numerical curve interpolating between the three regimes (A-C) and in particular incorporating the multifractal regime (B) is indistinguishable from the numerical result, and gives d2 = 1.28 ± 0.02; The residual is shown in (d).

path) and L ω = (ωτ ) 1/3 is the mean distance traveled at the critical point in time 1/ω. The fact that the real function f depends only on the product qL ω is a requirement of the one-parameter scaling law. In the non-multifractal regime (A) where qL ω 1, the self-consistent theory of localization predicts that f is constant [25]. The precise constant value of f depends on the cutoffs used in the self-consistent theory [25,[START_REF] Vollhardt | Electronic Phase Transitions[END_REF][START_REF] Müller | Disorder and interference: localization phenomena[END_REF]. If the cutoff is chosen so that the transition takes place at k = 1, the numerical factors in Eq. ( 19) are such that f = 1 in the non-multifractal regime.

In the multifractal regime qL ω > 1, the Chalker's ansatz [START_REF] Chalker | [END_REF] states that D(q, iω) scales like q d2-2 ω 1-d2/3 or, equivalently, f (qL ω ) ∝ (qL ω ) d2-2 . There are of course many possibilities to smoothly connect the f (x) = 1 behavior at small x to the f (x) ∝ x d2-2 decrease at large x. The only requirement is that the transition between the two regimes takes place around x = 1. In order to avoid unphysical Gibbs-like oscillations after Fourier transform, we used the following smooth ansatz:

f (x) = 1 + (x/x 0 ) γ(2-d2) -1/γ ( 20
)
where γ is a positive exponent and x 0 a number of the order of unity characterizing the transition point between the two regimes. This ansatz is of course a bit arbitrary.

We have tried a few other ways of smoothly connecting the two regimes, which give very similar final results. When the parameters D 0 , d 2 , x 0 , γ and τ (or equivalently ) are given, D(q, iω) is entirely specified. In order to compute the disorder-averaged intensity propagator, one has to compute D(q, ω) for real ω, which is rather easy by analytic continuation in the complex plane, as there is no singularity in the upper half-plane ω > 0. The last step is a double Fourier transform from q, ω to p, t to obtain P QPKR (p, t). In order to take into account the nonuniversal behavior at very short distance -regime (C)we convoluted the obtained P QPKR (p, t) by a Gaussian:

g(p) = 1 σ √ 2π exp - p 2 2σ 2 (21)
where σ is a constant of the order of the mean free path .

We used these distributions to fit the numerical results obtained for the kicked rotor. There are 5 different fit parameters: the first one is the "classical" diffusion coefficient D 0 which determines the overall scaling factor of the distribution (or equivalently the value of p 2 (t) /t 2/3 at long time). Taking D 0 as a fit parameter accounts for the somewhat arbitrary numerical prefactor in Eq. (19), not accurately predicted by the self-consistent theory. The second fit parameter is the short distance cutoff σ in Eq. ( 21), of the order of the mean free path. It turns out that the final results are essentially insensitive to the exact value of this parameter. The three left important parameters are d 2 (the figure of merit of our analysis), and x 0 , γ which describe the transition between the normal and multifractal regimes for D(q, ω). We performed three fitting runs:

• In the first run, we fitted all five parameters D 0 , τ, d 2 , x 0 , γ for each time. We observed that the values of x 0 and γ fluctuate in not too large intervals, that is x 0 ∈ [0.24, 0.40] and γ ∈ [2.8, 4.0].

• In a second run, we fixed γ at its most probable value γ = 3.0 and fitted the remaining four parameters.

• In a third run, we additionally fixed x 0 at its most probable value x 0 = 0.3 and fitted the remaining three parameters.

The results of the three fitting runs are very similar. Importantly, the residuals of the fits -deviations between the numerical data and the fitting functions -are very comparable for the three runs, so that they are of almost equal significance. The fluctuations of the fitted values for the three runs give an estimate of the error due the imperfections of the fits. I. Multifractal exponent d2, with estimated uncertainty ∆d2, extracted using the first fitting procedure, that is from fits of the disorder-averaged intensity propagator near momentum p = 0 to Eq. ( 18), for various times t. The uncertainty is not the statistical error of the fit, but rather reflects the fluctuations of the result of the fit when the momentum range and the short-range cutoff are varied. Nevertheless, the result at long times is remarkably stable, proving the robustness of the fitting procedure. II. Multifractal exponent d2, with estimated uncertainty ∆d2, extracted using the second fitting procedure, that is from fits of the full disorder-averaged intensity propagator, for various times t. The uncertainty is the combination of the statistical error of the fit and of the three different values that are obtained when the additional parameters x0, γ are either fitted or fixed. In any case, the smallness of ∆d2 as well as the quality of the fit, see Fig. 2, validates the Chalker's ansatz and proves that it is experimentally possible to measure the multifractal exponent d2.

In Table I, we give the values of d 2 extracted from the numerical data using the first fitting method, for various times. The uncertainties take into account the fluctuations of the results when the range of P used for the fit is varied. It consistently gives a value of d 2 in the [1.24, 1.37] range for a considerably large time interval, between 10 3 and 4×10 8 kicks, in good agreement with the known value 1.24±0.015 for the 3D Anderson model [11]. (a more accurate value d 2 = 1.243 ± 0.006 is given in the unpublished thesis [START_REF] Vasquez | High precision multifractal analysis in the 3D Anderson model of localisation[END_REF]).

With the second fitting method, we found that the fitted d 2 is almost insensitive to the details of the interpolation between the three regimes. The obtained values of d 2 are given in Table II. They are more or less time-independent at long time, which strongly supports the validity of the Chalker's ansatz. They also agree well with the results of Table I and with the known value [11].

For t = 10 6 kicks, the two fitting methods give almost identical results, d 2 = 1.28, and the quality of the fits is excellent, as shown in Fig. 2. The Airy function, in contrast, strongly deviates from the numerical result.

We finally show in Fig. 3 that the same value of d 2 allows us to reproduce almost perfectly the full momentum distribution over a very wide range of times. The fact that a unique form of D(q, ω) reproduces the numerical results over more than 6 orders of magnitude of t is on the one hand a very strong hint that the one-parameter scaling law remains valid for the Anderson transition in the multifractal regime, and on the other hand a confirmation of the validity of the Chalker's ansatz.

V. EXPERIMENTAL PERSPECTIVES AND SUMMARY

We have unveiled the existence of a sharp multifractality peak at the critical point of the Anderson transition. Based on the equivalence between the time-dependent QPKR and the 3D anisotropic Anderson model, we have also shown that the multifractal dimension d 2 of a critical 3D system can be extracted from the peak in the frame of a 1D experimental setup. Although this in principle requires to reach extremely long times, we stress that, even after t = 10 3 kicks -a value already reached in state-of-the-art experiments [START_REF] Manai | [END_REF]-a significant deviation from the Airy shape is already visible. This opens the way to an experimental measure of multifractality properties using the atomic kicked rotor. The method is in no way restricted to the kicked rotor and could be used in other disordered systems [14]. In a full 3D system, the average intensity propagator, Eq. ( 14), is also sensitive to d 2 . If not all three dimensions of space are experimentally accessible, averaging over one or two dimensions still preserves the information on d 2 , although the singularity is somewhat smoothed out.

Note that an apparently similar phenomenon, an enhanced return probability, has been recently experimentally observed on the kicked rotor [52]. It originates from the constructive interference between pairs of timereversed paths for time-reversal invariant systems and is completely different from the "multifractal" peak: it manifests itself on a much shorter spatial scale of the order of the mean free path, that is in regime (C) where the one-parameter scaling law is violated. Moreover, it exists only for periodic driving, as discussed in [52,53] and is thus an unrelated phenomenon.

FIG. 3 .

 3 FIG.3. Black solid lines: numerically computed temporal evolution in momentum space of a wavepacket launched at t = 0 near momentum p = 0 (see Fig.1for the parameter values). Red dashed lines, often hidden by the black ones: prediction taking into account the sub-diffusive dynamics and the multifractality of the eigenstates (regimes (A-C)). The agreement is excellent (residuals are displayed as the lower green curves) over more than 6 decades of time. The same value d2 = 1.26 has been used for all plots.

  Combined with the statistical uncertainty of the fit, they provide a reasonable estimate of the error bars on the determined values of d 2 . Time t 10 3 10 4 10 5 10 6 10 7 10 8 4×10 8 d2 1.37 1.33 1.31 1.28 1.26 1.26 1.26 ∆d2 0.15 0.05 0.04 0.03 0.02 0.02 0.02

	C. Results
	TABLE

  Time t 10 2 10 3 10 4 10 5 10 6 10 7 10 8 4×10 8

	d2	1.19 1.32 1.34 1.33 1.28 1.25 1.24 1.24
	∆d2 0.2 0.2 0.08 0.04 0.02 0.015 0.015 0.01
	TABLE

ACKNOWLEDGMENTS

We thank G. Lemarié, J.C. Garreau, P. Szriftgiser, A. Mirlin and V.E. Kravstov for useful discussions. This work was granted access to the HPC resources of TGCC under the allocation 2013-056089 made by GENCI (Grand Equipement National de Calcul Intensif) and to the HPC resources of The Institute for scientific Computing and Simulation financed by Region Ile de France and the project Equip@Meso (reference ANR-10-EQPX-29-01).


Christian France-Lanord1

Louis Derry1

Annie Michard2

Evolution of the Himalaya since Miocene time: Isotopic and sedimentologic evidence from the Bengal Fan

We report Sr, Nd, O, and H isotopic data and clay mineral abundances for sediments recovered in ODP Leg 116 cores from the Bengal Fan at 1° S. The samples studied cover the period between ca. 17 Ma and the present. We also present new and compiled data on the isotopic compositions of potential source regions for the Bengal Fan sediments. eNd(0) values in the Bengal Fan sediments (all samples) define a narrow range about -16.0. 87Sr/86Sr values (all samples) are also in a narrow range near 0.741. d18O values in quartz separates define a narrow range at +12.8±0.5‰. Coarse biotite-chlorite separates give d18O = 3.6 -5.6‰. Combined d18O values of quartz and biotites indicate a metamorphic source. Clay mineral abundances define two clay facies: an illite-chlorite rich assemblage (IC) and a smectite-kaolinite rich assemblage (SK). d18O in the IC clay fractions is 11.5-15 ‰, while SK clays are 18.2 -22.6 ‰. The narrow range of isotopic values throughout the deposition history implies that the source of the Bengal Fan sediments has not changed since the early Miocene, despite changes in sedimentation rate, sedimentary facies, tectonic history and climactic regime. The difference between d18O in the IC and SK clay fractions represents different alteration histories of the same source material. The SK clays appear to have been altered at low T in the Indo-Gangetic Plain, while the IC clays and coarse fractions preserve metamorphic signatures. The narrow range of the Sr values, despite wide variation in Rb/Sr ratio, also argues for a source that underwent isotopic homogenization shortly before erosion and deposition of the sediment. The source that meets these criteria is the High Himalayan Crystalline series (HHC) or a close analog, although subordinate contributions (probably <20%) from the Lesser Himalaya (LH) and High Himalayan Sedimentary Series (HHSS) are possible.

A model in which the the HHC are exposed to erosion since the early Miocene on the south flank of the orogen by thrusting along the MCT, while the HHSS is simultaneously removed by northward-directed normal faulting satisfies the contraints above. The results of this study require that the Himalaya have been a significant topographic feature since at least the early Miocene. Independent evidence supports this contention. Variations in the sedimentation style in the Bengal Fan since that time appear to represent a combination of factors, including tectonic activity and the coupled effects of climate and sea level changes.

.

The Bengal Fan provides an unparalleled record of Himalayan erosion, which in turn is related to the collision history of India and Asia, and the uplift history of the Himalaya. Erosional processes are also related to climate, thus the Bengal Fan may record information about climate change in the regions that supplied it with sediment. The Bengal Fan has been sampled by underwater drilling, permitting a reconstruction of the erosion history of the Himalaya in Neogene time. We wish to address several questions: What were the sources of sediment to the Bengal Fan, and have these sources varied in time? How can we use the sedimentological record to constrain the erosion history of the source regions? What is the relative importance of physical versus chemical erosion in the denudation of the Himalaya? What role does climate change play in the sedimentation history of the Bengal Fan, and by inference, erosion in the Himalaya?

In this paper, we present sedimentologic and Nd, Sr, O and H isotopic data on samples recovered from the Bengal Fan by the Ocean Drilling Program during Leg 116. These samples yield a record of sedimentation in the Bengal Fan beginning in Early Miocene time. These data, along with sedimentological data from DSDP Leg 22 and new and compiled data on from some potential source regions are used to address the questions listed above. The sedimentological record of the Bengal Fan is an integrated record, sampling the Himalayas across the wide area of the Ganges-Brahmaputra watershed (1.48 million km2, Milliman and Meade, 1983). Thus the Bengal Fan record can yield important constraints on processes in the Himalayan source region unavailable from terrestrial studies. The stratigraphic resolution of the sedimentary record also permits a detailed comparison in time of sedimentary events and possible forcing factors, such as tectonic events or climate change.

Geology of Leg 116 Sites

). The holes are approximately 8 km apart. Hole 717C

penetrated 828 mbsf (meters below sea floor) bottoming in late Miocene sediments, while Hole 718C penetrated 935 mbsf, bottoming in early Miocene sediments. Hole 719B penetrated 466 mbsf, bottoming in late Miocene sediments. The principal lithologies encountered in all holes were silt and mud turbidites, with lesser but variable amounts of biogenic mud turbidites (Shipboard Scientific Party, 1989). Correlations between the holes are based primarily on lithostratigraphy and nannofossil biostratigraphy [START_REF] Stow | Sediment facies and processes on the distal Bengal Fan, Leg 116[END_REF][START_REF] Gartner | Neogene calcareous nannofossil biostratigraphy, Leg 116 (Central Indian Ocean)[END_REF]. A number of potential problems exist with the assignment of biostratigraphic ages to these sediments. Most of the sequence appears to have been deposited near or below the CCD, thus the state of preservation of calcareous fossil material is often poor. Second, it is probable that most of the fossils in the turbidites of the Bengal Fan have been redeposited, and thus fossil-bearing horizons would actually be younger than indicated. However, the succession of highest and lowest occurrences of fauna is both self-consistent and closely comparable with that observed in contemporaneous pelagic sediments, suggesting that redeposition has not seriously disturbed the biostratigraphic record [START_REF] Gartner | Neogene calcareous nannofossil biostratigraphy, Leg 116 (Central Indian Ocean)[END_REF]. [START_REF] Stow | Sediment facies and processes on the distal Bengal Fan, Leg 116[END_REF] have presented a detailed summary of the sedimentology of Sites 717, 718 and 719. Lower to upper Miocene sediments are characterized by silt and mud turbidites. The latest Miocene, Pliocene, and early Pleistocene are characterized by mud turbidites, with interbedded silt turbidites, biogenic turbidites and pelagic clays. In the mid-upper Pleistocene section, sandy and silt-to mud turbidites dominate, except for the last few meters of Holocene pelagic oozes. Variations in sedimentation rates derived from biostratigraphy are consistent with the lithologic variations. Overall, sedimentation rates were high (70-200 m/Ma) during the early and middle Miocene, low (20-70 m/Ma) during the late Miocene-early Pleistocene, and very high (>200 m/Ma) from the early Pleistocene to the Holocene [START_REF] Gartner | Neogene calcareous nannofossil biostratigraphy, Leg 116 (Central Indian Ocean)[END_REF]. A wide variety of detrital minerals have been identified in the Bengal Fan silts, the most common of which are quartz, feldspars, carbonates, micas, amphiboles and epidotes [START_REF] Yokoyama | Mineralogy of silts from the Bengal fan[END_REF]. Two clay facies have been identified in the <2µm Leg 116 sediments: an illite-chlorite rich assemblage (IC) and a smectite-kaolinite rich assemblage (SK) [START_REF] Bouquillon | Sedimentology and isotopic chemistry of the Bengal fan sediments: the denudation of the Himalaya[END_REF][START_REF] Brass | Clay mineralogy of sediments from the Bengal fan[END_REF]. The IC assemblage is dominant from the lower Miocene to upper Miocene, the SK assemblage from the upper Miocene to the lower Pleistocene, and the IC assemblage returns above the lower Pleistocene.

Figure 2 presents a summary of the above data for Hole 717C. Also shown are maximum quartz grain size (MQGS) data obtained from smear slide analysis (Shipboard Scientific Party, 1989). Changes in lithology, clay mineralogy, MQGS and sedimentation rate are correlated. Prior to ca. 11 Ma the sedimentation record in Hole 718C is marked by silt turbidites and coarse MQGS, although the overall sedimentation rate near the bottom of Hole 718C is apparently lower than during other intervals of similar lithology. [START_REF] Cochran | Himalayan uplift, sea level, and the record of Bengal fan sedimentation at the ODP Leg 116 Sites[END_REF] argued that an increase of pelagic facies near the bottom of 718C indicates an early stage of submarine fan development. Thus the lower sedimentation rate below about 650 mbsf may simply reflect less sediment supply to this distal portion of the Fan in its early stages of development. The increase in sedimentation rate after 11 Ma correlates with a sea level drop at about 10.5 Ma, and with the establishment of the "modern" Ganges delta [START_REF] Haq | Chronology of fluctuating sea levels since the Triassic[END_REF][START_REF] Cochran | Himalayan uplift, sea level, and the record of Bengal fan sedimentation at the ODP Leg 116 Sites[END_REF][START_REF] Lindsay | Sequence Stratigraphy and the Evolution of the Ganges-Brahmaputra Delta Complex[END_REF]. The Pliocene section is better developed at Sites 717 and 719, perhaps due to non-deposition or erosion at Site 718.

All holes show a drop in sedimentation rate at ca. 6.8 Ma (Fig. 3). This change is evident in the lithology and the MQGS data. It also corresponds to a marked change in clay mineralogy at all three sites, from the IC assemblage to the SK assemblage (Fig. 2). Coarser clastic deposition recurred for a brief period prior to 3.5 Ma. The sedimentation rate increased markedly at all three Sites at 0.8 Ma, accompanied by a return to coarse clastic sedimentation and the IC clay assemblage. Lithostratigraphic and biostratigraphic correlations between the three Sites are good at this scale of study, providing further confidence in the biostratigraphic ages assignments (e.g. Shipboard Scientific Party, 1989).

DSDP Site 218, Leg 22, is located approximately 1100 km north of the Leg 116 Sites (Fig. 1). Drilling encountered similar lithologies, bottoming at 773 mbsf in Middle Miocene turbidite sediments (Figs. 1, 4) (von der Borsch, Sclater et al., 1974). We correlate Site 218 with Sites 717, 718, and 719 based on lithostratigraphy and biostratigraphy (von der Borsch, Sclater et al., 1974;[START_REF] Gartner | Paleontology synthesis of deep sea drilling results from Leg 22 in the Northeastern Indian Ocean[END_REF]. At Site 218, changes in lithology, grain size, and clay mineralogy similar to those at the Leg 116 Sites support this correlation [START_REF] Pimm | Mineralization and trace element variation in dee-sea pelagic sediments of the Wharton basin, Indian Ocean[END_REF][START_REF] Bouquillon | Sédimentation argileuse au Cénozoïque supérieur dans l'océan Indien Nord-oriental[END_REF].

The occurrence of correlative lithologies, grain size distributions, sedimentation rate and clay mineral assemblages at three different Leg 116 Sites, located near 1°S, and at Site 218, located at 8°N, demonstrates that the important variations in the sedimentation history of the DSDP and ODP sites reflect regional phenomenon. [START_REF] Pimm | Mineralization and trace element variation in dee-sea pelagic sediments of the Wharton basin, Indian Ocean[END_REF] also noted that clastic sediment pulses were correlative over long distances both within the Bengal Fan and into the Nicobar Fan. Such a wide geographic dispersion precludes channel switching or other changes in the location of depocenters as an explanation for the variations in sedimentation rate and lithology. It is also improbable that simply changing the channel network in the Fan, without a change in sediment source, could result in significant changes in the mineralogic composition of the <2mm clay fraction. Finally, studies of other submarine fans suggest that levee construction takes place on a 105 year time scale, but the major variations in sediment composition observed at the Leg 116 sites and Site 218 are of 106 years duration, implying that simple construction of underwater structures is not the process responsible for the sedimentological variations (e.g. [START_REF] Damuth | Anatomy and growth pattern of Amazon deep-sea fan as revealed by long-range side-scan sonar (GLORIA) and high-resolution seismic studies[END_REF][START_REF] Manley | Cyclic sediment deposition within Amazon deep-sea fan[END_REF]. The implications of the basin-wide nature of stratigraphic events in the Bengal Fan are that the sedimentological variations are caused by 1) changes in provenance, and/or 2) changes in transport phenomena in the Ganges watershed or near the head of the Bengal Fan complex.

Analytical Methods

Samples of sediment from ODP collections were selected to provide a range in age and sedimentary facies. Samples were first disaggregated in distilled water. The coarse fraction (> 63 µm) was separated by screening, and the remainder either analyzed as the <63 µm fraction or subject to further size separation. Mono-mineralic separates were prepared from the > 63 µm fraction by hand picking. The preparation of <2 µm and 2-50 µm fractions generally followed the methods described in [START_REF] Bartoli | Disagregation and clay dispersion of oxisols: Na resin, a recommended methodology[END_REF] and [START_REF] Feller | Utilisation des résines sodiques et des ultrasons dans le fractionnement granulométrique de la matière organique des sols[END_REF]. Briefly, the samples (including both those previously screened at 63 µm, or those previously untreated) were disaggregated by agitation with Amberlite cation exchange resin (Na form) for 16 hours. This treatment disperses the fine-grained particles because of the high exchange capacity of the resin. The samples were screened to remove >50 µm particles. The remaining fraction was allowed to sediment for 24 hours, effectively separating the <2µm fraction. The suspended 2µm clays were decanted, and 1 ml of a clean MgCl2 solution (Prolabo Normapur®) was added to flocculate the clays. This sequence was repeated as necessary. The 2-50 µm and <2µm fractions were then separated from solution by centrifugation. The quality of the grain size separations was periodically checked using a microscope-mounted image processing system. In all cases the size separation technique was found to be highly effective. A portion of the <2µm fraction was subject to standard X-ray techniques for clay mineral identification.

Clay fractions were decarbonated by leaching with 10% acetic acid in an ultrasonic bath. Samples for Sr and Nd isotopic analysis were roasted at 700°C in platinum crucibles to remove organic matter. Samples were weighed and spiked, and dissolved in a HF-HNO3-HClO4 mixture. Separation procedures for Rb, Sr, Sm and Nd are described in [START_REF] Alibert | The transition from alkali basalts to kimberlites: isotope and trace element evidence from melilites[END_REF]. Sr and Nd isotopic compositions were measured on a Finnigan-MAT 262 mass spectrometer equipped with an SEM and 6 faraday detectors. Blanks are estimated at ca. 0.5 ng for Nd and 1 ng for Sr (blank 87Sr/86Sr ≈ 0.710). These high blanks result from the complex sample preparation procedure. However, samples sizes were sufficiently large that blank corrections are not necessary.

Oxygen from silicates were extracted by fluorination following the technique of [START_REF] Clayton | The use of bromine pentafluoride in the extraction of oxygen from oxides and silicates for isotopic analysis[END_REF]. CO2 gas was analyzed on VG 602D mass spectrometer. All extractions were duplicated and average reproducibility on clay samples is ±0.3‰.

Results

Mineralogic, isotopic and major element data are listed in Tables 1 and2 for whole rock samples, mineral separates and granulometric fractions. Figures 5 and6 show composite logs of the isotopic compositions of Nd, Sr, O and H as a function of depth/age at Sites 717 and 718 for the different mineral separates or size fractions analyzed. Isotopic analyses of coarse mineral separates and most XRD analyses are from [START_REF] Bouquillon | Sedimentology and isotopic chemistry of the Bengal fan sediments: the denudation of the Himalaya[END_REF]. Isotopic analyses of Sr for the <2µm fraction samples given in [START_REF] Bouquillon | Sedimentology and isotopic chemistry of the Bengal fan sediments: the denudation of the Himalaya[END_REF] were not retained here because these samples were not leached with acetic acid prior to analysis, and are thus not directly comparable.

Neodymium and Strontium

The isotopic compositions of Nd and Sr are remarkably uniform over the whole stratigraphic column. eNd(0) values are between -17 and -14; the entire range is from -13.8 to -19.3 (mean=-16.0, std. dev.=1.2). The two lowest values are from SK samples, however no clear difference is evident overall between SK and IC samples (Figure 7). 87Sr/86Sr ratios for all samples vary between 0.720 and 0.762 (mean=0.741, std. dev.=0.009). For both the whole rock and granulometric fractions 87Sr/86Sr is independent of Sr concentration, which supports the efficiency of the leaching technique in removing carbonates containing non-radiogenic seawater-derived Sr. SK and IC samples have similar 87Sr/86Sr ratios. Not only are 87Sr/86Sr ratios relatively homogeneous with depth, but also across a variety of mineralogies, despite large variations in Rb/Sr ratio.

Stable isotopes d18O values of coarse quartz are remarkably constant with depth at 12.8‰ ± 0.5. Mixtures of coarse biotite and chlorite vary from 7.2 to 9.8‰ and are depleted in 18O with regard to quartz. The d difference between quartz and biotite is 3.6 to 5.6‰, which corresponds to an isotopic equilibrium temperature between 500 and 650°C (e.g. Botinga & Javoy, 1975). Despite the fact that these analyses represent average values of single grains of different origin this average "fractionation" between quartz and biotite is likely to reflect high temperature fractionation in the source rock.

The <2µm fraction samples display two groups of d18O values. IC samples are between 11.5 to 15‰, whereas SK samples are more variable and significantly enriched in 18O at 18.2 to 22.6‰. The values for IC are extremely low for clay minerals, especially in marine sediment.

"Normal" values of marine clays are in the range of 16 to 25‰, with most between 20 and 25‰ [START_REF] Savin | The oxygen and hydrogen isotope geochemistry of ocean sediments and shales[END_REF], Yeh & Savin 1977[START_REF] Perry | Mg, Ca and O18/O16 exchange in the sediment pore water system, Hole 149, DSDP[END_REF]Laurence et al. 1979[START_REF] Taieb | Les isotopes de l'hydrogène, du carbone et de l'oxygène dans les minéraux argileux et les eaux de formation[END_REF]). High values are due to large isotopic fractionation between clay and water at low temperature (e.g. Yeh & Savin, 1977). The IC assemblage clays have not been enriched in 18O by low temperature exchange. Like the coarse micas, the IC clays reflect the high temperature of their source rock, whereas smectite and kaolinite were formed at low temperature.

Discussion

Sources of sediment in the Bengal Fan

The Nd, Sr, O and H isotopic data impose several constraints on the sources of the sediments in the Bengal fan:

-the remarkably constant Nd, Sr and O isotopic values with time are a strong indication that the material has been derived from the same major source or melange of sources since 17 Ma. The isotopic data are consistent with the overall monotony of the mineralogy. The stratigraphic levels rich in SK clays have higher d18O values than elsewhere in the column, but have the same Nd and Sr values as the coarse fraction and samples of the IC clay assemblage.

-the source(s) of this material is old, with a long crustal residence time. Nd model ages are ca. 2.0 Ga in the sediments at all levels, and 87Sr/86Sr cluster about 0.74. The d18O values of quartz and micas are also typical of silicic crystalline crustal sources, such as paragneisses and S-type granites.

-the present-day isotopic composition of Sr is quite homogeneous, despite high but variable Rb/Sr ratios. This requires a recent metamorphic/magmatic event in the source to isotopically homogenize the Rb-Sr system shortly before erosion and deposition of the sediments.

-in the IC levels d18O values are very low for the <2µm clay fraction compared to normal deep sea clays (≈ 13‰). The d18O data reinforce the observation that low temperature alteration minerals are present only in minor abundance in IC levels, which implies that erosion was dominated by physical rather than chemical processes.

-the SK levels are enriched in D and 18O relative to IC levels, but have the same Nd and Sr signature. This implies that smectite and kaolinite are derived from the same material as the illite and chlorite. The SK clays appear to be altered equivalents of the IC material, which preserves the radiogenic isotopic compositions but results in higher d18O and dD values.

We now examine potential sources for the Bengal Fan sediments around the Bengal Basin. Early studies of the Bengal Fan concluded that the Himalaya were the primary source for detrital material [START_REF] Curray | Growth of the Bengal Deep-Sea Fan and Denudation in the Himalayas[END_REF]Shipboard Scientific Party, 1974;[START_REF] Ingersoll | Petrology and provenance of Neogene sand from Nicobar and Bengal Fans, DSDP Sites 211 and 218[END_REF]. Several authors have used mineralogical criteria to identify the presence of several additional components in the Bengal Fan [START_REF] Bouquillon | Sédimentation argileuse au Cénozoïque supérieur dans l'océan Indien Nord-oriental[END_REF][START_REF] Brass | Clay mineralogy of sediments from the Bengal fan[END_REF][START_REF] Stow | Sediment facies and processes on the distal Bengal Fan, Leg 116[END_REF][START_REF] Yokoyama | Mineralogy of silts from the Bengal fan[END_REF]. The most important of these include the Precambrian terranes of south India and the Deccan Traps. Both of these terranes are a potential source of the smectite and kaolinite clays which are common in some Bengal Fan strata. [START_REF] Stow | Sediment facies and processes on the distal Bengal Fan, Leg 116[END_REF] proposed that dark-grey and organic rich turbidites, which correspond to some SK levels drilled on Leg 116, are derived either from the Deccan Traps or the granulite terrains of India and Sri Lanka. This is not supported by the isotopic composition of Sr and Nd in such dark-grey turbidites (e.g. 717C-24X, 66-67 cm and 717C-35X, 28-29 cm) which are indistinguishable from other facies. A component derived from these granulites should be easily resolved isotopically, with eNd(0)<-30 and 87Sr/86Sr ≈ 0.716 (e.g. [START_REF] Peucat | Sr, Nd and Pb isotopic systematics in the Archean low-to-high-grade transition zone of southern India: Syn-accretion granulites[END_REF]. The Deccan Traps are equally distinctive with eNd(0) > 0 and 87Sr/86Sr < 0.708, except for volumetrically minor early flows (e.g. [START_REF] Mahoney | Origin of the Deccan Trap flows at Mahabaleshwar inferred from the Nd and Sr isotopic and chemical evidence[END_REF][START_REF] Lightfoot | Source and differentiation of Deccan Traps Lavas: Implications of Geochemical and Mineral Chemical Variations[END_REF].

Three arguments suggest that non-Himalayan sources are minor. First, the similarity of Nd and Sr isotope ratios between IC and SK levels imply that the smectite and kaolinite have the same source as the illite, chlorite and coarser material. Furthermore, soils from the Ganges delta (Table 2) and the suspended matter of the Ganges and Brahmaputra [START_REF] Goldstein | Decoupled evolution of Nd and Sr isotopes in the continental crust and the mantle[END_REF][START_REF] Goldstein | A Sm-Nd isotopic study of atmospheric dusts and particulates from major river systems[END_REF] have Nd and Sr isotopic compositions identical with those of the Bengal sediments (Fig. 8). Second, the deposition rate at the Leg 116 sites is very high even during periods when the SK assemblage is dominant, and the relief and the rivers from Sri Lanka and eastern India are not likely to supply a significant amount of sediments compared to the Ganges-Brahmaputra system. Finally, the same type of smectite and kaolinite parageneses associated with variable proportions of illite and chlorite are observed (1) in the Tertiary sediments of the Siwaliks (e.g. [START_REF] Mukherjee | Clay minerals in argillaceous sediments of the Himalayan zone[END_REF], (2) in the Ganges delta soils [START_REF] Rahman | Amélioration de la fixation d'azote dans la rhizosphere du riz cultivé sur différents sols du Bangladesh[END_REF], (3) in the Ganges river suspended matter [START_REF] Sarin | Malor ion chemistry of the Ganga-Brahmaputra river system: Weathering processes and fluxes to the Bay of Bengal[END_REF]) and ( 4) in the proximal Bengal Fan [START_REF] Bouquillon | Sédimentation argileuse au Cénozoïque supérieur dans l'océan Indien Nord-oriental[END_REF]). Thus it is not necessary to call on the presence of components derived from the southern Indian shield or Sri Lanka to explain the clay mineral abundances found in the Bengal Fan sediments, as proposed by previous workers [START_REF] Brass | Clay mineralogy of sediments from the Bengal fan[END_REF][START_REF] Stow | Sediment facies and processes on the distal Bengal Fan, Leg 116[END_REF]. While material derived from south India, Sri Lanka and the Deccan Traps is certainly present in the Bengal Fan, the isotopic data demonstrate conclusively that such components are of minor volumetric importance.

We can compare the isotopic compositions of the Bengal Fan sediments with potential Himalayan sources in some detail (Fig. 7). Despite the large area of the Ganges-Brahmaputra watershed, the gross structure of the Himalaya is relatively simple and regular throughout the range (e.g. [START_REF] Gansser | Geology of the Himalayas[END_REF][START_REF] Fort | The Himalayan orogenic segment[END_REF]. The three main Himalayan structural/stratigraphic units have very distinct isotopic features. The Lesser Himalaya (LH), dominated by old Himalayan metasediments (> 1.8 Ga; [START_REF] Deniel | Apport des isotopes du Sr, du Nd et du Pb à la connaissance de l'âge et de l'origine des leucogranites himalayens[END_REF], have very high Sr isotope ratios and eNd(0) around -25, reflecting their long crustal evolution (France-Lanord and Le Fort, 1988;Parrish et al., 1992; Table 2). The High Himalaya Crystalline metasedimentary series (HHC), located above the Main Central Thrust (MCT), have been intensely investigated to examine the genetic relations between the paragneisses and the overlying leucogranites (e.g. [START_REF] Scaillet | Badrinath-Gangotri plutons (Garhwal, India) Petrological and geochemical evidence for fractionation processes in a High Himalayan Leucogranite[END_REF]. Averages of eNd(0) and 87Sr/86Sr are -15 and 0.740 respectively (Fig. 8). The Himalayan leucogranites, which are derived from the HHC, have similar ranges. The High Himalaya Sedimentary Series (HHSS) are Cambrian to Eocene in age, and include limestones, sandstone, shales and some volcanic material. Nd and Sr isotopic data for HHSS sediments are given in Table 2. These data do not provide a comprehensive view of the HHSS but indicate that this unit is isotopically variable, as would be expected of a sedimentary sequence with a wide age range in an evolving tectonic setting. They have higher eNd(0) and lower 87Sr/86Sr than the HHC (Fig. 7), indicative of their younger source ages and, for some, the presence of volcaniclastic components. On histograms of isotopic compositions of Nd, Sr, O and H, the HHC paragneisses appear extraordinarily concordant with the Bengal sediments (Fig. 7). In contrast, the LH and HHSS lie outside the narrow range of values shown by the Bengal Fan. Thus the major source for the detrital material has been an equivalent to the modern HHC, and the LH and the HHSS must be subordinate to the HHC. This is consistent with the present structure of the Himalaya. The HHC is exposed on the southern flank of the high ranges of the Himalaya, which is the region of highest relief and precipitation, hence of most rapid erosion. Conversely, the HHSS is exposed on the north flank of the range, and is undergoing less erosion. The LH contributes relatively little to the Bengal sediments, because it is at a lower elevation, is not glaciated, and is under extensive plant cover.

A more quantitative estimate of the relative importance of the Lesser Himalaya, the High Himalaya Crystalline series, and the High Himalayan Sedimentary Series in supplying Bengal fan sediment may be obtained from isotopic mass balance. Taking average values of each of the three groups, Nd-Sr mass balance suggests that the contribution of LH and HHSS sediments to the Bengal Fan could have varied between 0 and 40%, but was most often closer to the lower of these two estimates. In principle, the Nd and Sr signature of the Bengal Fan could be produced by a simple mixture of LH and HHSS. Two points mitigate against this interpretation. First, the ranges of Nd and Sr values in the Bengal sediments are very narrow at all times. The LH and HHSS lie in structurally and topographically distinct parts of the Himalayan Range. It is rather improbable that a mixture of LH and HHSS would always fall in the same narrow range, despite important changes in sedimentation rate, mineralogy, and tectonic and climactic history. Second, the sediments have highly variable Rb/Sr ratios. This variability would give rise to a wide range in 87Sr/86Sr values in the sediments over the time scale of the age of the HHSS and the LH. In fact, 87Sr/86Sr varies only slightly in the Bengal Fan, implying that the sediment source was isotopically homogeneous on the order of 10-100 Ma before present. This constraint is not satisfied by the HHSS or the LH because the Himalayan metamorphism sensu lato is weak or only locally developed in these units (e.g. Le Fort, 1986). However, the total amount of erosion of the HHSS may be underestimated by the record of clastic sediments of the Bengal Fan. At least 50% of the HHSS is composed of platform carbonates that are mostly dissolved during the erosional process. The abundance of detrital grains of dolomite before ≈ 10Ma in Hole 718C [START_REF] Yokoyama | Mineralogy of silts from the Bengal fan[END_REF]) reflects a higher proportion of carbonate rocks in the source during the early stage of Fan development which could be a by-product of early Miocene erosion of the HHSS.

Implications for Himalayan tectonics since 17 Ma

The first order conclusion from the isotopic data is that an equivalent of the High Himalayan Crystalline series has been the dominant source of sediment supply to the Bengal fan since before 17 Ma. This situation has apparently not varied significantly since that time, despite changes in sedimentation rate in the Fan, alteration history of the sediments, and tectonic and/or climactic changes that may have affected the erosional history of the Himalayan source regions. In a few samples, there is a hint of low eNd(0) material, presumably from either an Indian or LH source. However, this component is always minor, and in most cases not resolvable. The presence of large volumes of sediment derived from the HHC since at least as early as 17 Ma has significant implications for the tectonic history of the Himalaya. Obviously, the sedimentary record requires that the HHC have been exposed at the surface since that time. Furthermore, as argued above, the mineral assemblage (e.g. [START_REF] Yokoyama | Mineralogy of silts from the Bengal fan[END_REF], 87Sr/86Sr and stable isotopic values in Bengal sediments are indicative of a metamorphic source which experienced isotopic homogenization of Sr shortly before erosion and deposition. The majority of single grain 40Ar/39Ar dates of feldspar and muscovite of Leg 116 sediments are in the range of 10 to 50 Ma (Copeland and Harrison, 1990). This evidence is also consistent with a HHC source, and implies that the high temperature metamorphic rocks from the HHC were being eroded at 17 Ma. The large volume of sediment supply to the Bengal Fan requires that the Himalaya were a significant topographic feature throughout this time. Furthermore, the observation that the sources of Bengal sediment have not varied implies that the main lithologies present in the high range were closely analogous to the present HHC. Therefore the general structure of the high range has remained grossly similar since the early Miocene, with the HHC exposed to rapid erosion, while the HHSS and LH have remained in less susceptible positions. A plausible hypothesis is that the HHC have been maintained continuously at the top of the chain by southward-directed thrusting, and that the overlying HHSS have been simultaneously removed by north-directed normal faulting (Fig. 9). In this way the HHSS has been protected from erosion, while the HHC has formed the region of high elevations and relief.

The source of the SK assemblage

The Nd and Sr data show that the IC and SK clay assemblages have the same source, but their O and H isotope compositions are different (Figs. 5,6,7). The IC and SK assemblages therefore represent differences in alteration history of the same material. The d18O values of +20‰ and dD values of -65‰ in the SK samples are consistent with alteration by meteoric water in the Indo-Gangetic Plain at ca. 20°C. It is essentially variation in the smectite/illite ratio that is responsible for the mineralogic difference between the two clay assemblages. Kaolinite is found in both IC and SK samples (Table 1), and in lakes and river sediments of the High Himalayan chain (CRPG unpublished data; [START_REF] Sarin | Malor ion chemistry of the Ganga-Brahmaputra river system: Weathering processes and fluxes to the Bay of Bengal[END_REF]. Conversely, smectite only appears downstream of the Siwaliks in the Indo-Gangetic plain. The ratio of smectite to illite in the Ganges river sediments increases downstream, from zero in the highlands to 6-8 in the Indo-Gangetic Plain [START_REF] Sarin | Malor ion chemistry of the Ganga-Brahmaputra river system: Weathering processes and fluxes to the Bay of Bengal[END_REF]. Estimates of d18O and dD in Bengal smectites correspond with those of clays generated during continental alteration, but are too low to be generated by reaction with cold seawater in the deep ocean [START_REF] Bouquillon | Sedimentology and isotopic chemistry of the Bengal fan sediments: the denudation of the Himalaya[END_REF]. While carbonate precipitation has occured in the section, there is no evidence that diagenetic processes have significantly affected the clastic material in Bengal Fan sediments (e.g. [START_REF] Bouquillon | Sédimentation argileuse au Cénozoïque supérieur dans l'océan Indien Nord-oriental[END_REF]. We conclude that the alteration which has produced the SK assemblage occured during transport in the Ganges-Brahmaputra river system. The intervals rich in SK clays represent periods when the eroded material was altered at low temperature prior to its sedimentation. This process could take place in several ways. Clays could be altered during river transport, they could be deposited in the foreland basin and re-eroded under a warm climate regime, or they could be altered during residence in the Indo-Gangetic plain.

Variation in sedimentation history of the Bengal Fan Between 6.5 and 0.8 Ma, the change in sediment facies, high abundance of smectite, fine grain size, and lower accumulation rates (Fig. 3) indicate a major change in the supply of sediment to the Bengal Fan. Such an event may be related to tectonic evolution of the Himalaya, climatic variations, sea level changes or some combination of these factors.

Major eustatic sea level changes have occured several times since the Miocene [START_REF] Haq | Chronology of fluctuating sea levels since the Triassic[END_REF], and [START_REF] Cochran | Himalayan uplift, sea level, and the record of Bengal fan sedimentation at the ODP Leg 116 Sites[END_REF] proposed that this has been the main control on sedimentation in the distal Fan. During periods of low sea level stand, the delta should be exposed to erosion, and sediments will be redeposited in the deep sea. During periods of high sea level, sediments should be impounded in the delta, and coarse clastic deposition cut off from the distal Fan. A major sea level lowstand at 10.5 Ma has resulted in major erosion of the Ganges-Brahmaputra delta [START_REF] Lindsay | Sequence Stratigraphy and the Evolution of the Ganges-Brahmaputra Delta Complex[END_REF]. The drop in sea level observed at 10.5 Ma and attributed to Antarctic ice sheet growth (Shackelton and Kenneth, 1975;[START_REF] Miller | Tertiary oxygen isotope synthesis, sea level history, and continental margin erosion[END_REF] corresponds to an increase in the accumulation rate in the Hole 718C. However, despite an important rise in sea level after this time, which corresponds to the beginning of growth of the "modern" delta [START_REF] Lindsay | Sequence Stratigraphy and the Evolution of the Ganges-Brahmaputra Delta Complex[END_REF], deposition rates and sediment facies in the distal Bengal Fan remain essentially unchanged (Fig. 4). The sedimentation rate at the Leg 116 sites does begin to drop around 6.8 Ma with the appearance of the SK assemblage, which corresponds to a new rise in sea level (e.g. [START_REF] Cochran | Himalayan uplift, sea level, and the record of Bengal fan sedimentation at the ODP Leg 116 Sites[END_REF]. However, other equally important variations in sea level are not followed by changes in sedimentation, especially between 3 and 1 Ma. The Bengal Fan does show a striking change in sedimentation at about 0.8 Ma. This event apparently correlates with the onset of major glacial cycles in the mid-Pleistocene. It seems unlikely that the prominent stratigraphic break at 0.8 Ma merely reflects a (possibly) somewhat more pronounced drop in sea level. Rather, the data suggest that Himalayan erosion rates increased at 0.8 Ma, perhaps due to increased alpine glaciation, perhaps due to renewed tectonic activity (c.f. [START_REF] Burbank | The la Cenozoic chronologic and stratigraphic development of the Kashmir intermontane basin, Northwestern Himalaya[END_REF]. Furthermore, large Quaternary glacial-interglacial sea level changes are not evident in the distal Fan. [START_REF] Bouquillon | Sédimentation argileuse au Cénozoïque supérieur dans l'océan Indien Nord-oriental[END_REF] find only minor changes in the smectite-illite ratio of clays from the delta and proximal Fan during Pleistocene glacial-interglacial cycles. If the relation between sea level and erosion in the Ganges-Brahmaputra delta is relatively clear, the relation between sea level and sedimentation in the distal Fan is evidently not so simple. Additional evidence against a simple relation between sea level and Fan sedimentation comes from the early and mid-Miocene. Prior to 11 Ma sea level was, on average, higher than during the highstand between 5 and 3 Ma [START_REF] Haq | Chronology of fluctuating sea levels since the Triassic[END_REF] but sedimentation in the Bengal Fan was dominated by coarse and illite-rich sediments. Nor is a major change in the marine d18O record at 14 Ma, the largest in mid-Tertiary time [START_REF] Miller | Tertiary oxygen isotope synthesis, sea level history, and continental margin erosion[END_REF], reflected in the sedimentary record. This implies that other processes are important for controlling sedimentation in the distal Fan, particularly changes in the sediment supply from the river system. It is also important to remember that Neogene sea level changes have been driven by ice volume changes in many instances, therefore they correspond to episodes of climate change (e.g. [START_REF] Miller | Tertiary oxygen isotope synthesis, sea level history, and continental margin erosion[END_REF][START_REF] Crowley | Oxford Monographs on Geology and Geophysics, Paleoclimatology[END_REF]. Climate change in the Himalayan source region of the Bengal Fan sediments can easily result in changes in erosion rate and thus sediment supply.

Studies of both the terrestrial record in Pakistan and the marine record in the Arabian Sea suggest that the monsoonal weather pattern intensified ca. 7 Ma [START_REF] Quade | Develpoment of Asian monsoon revealed by marked ecological shift during the latest Miocene in northern Pakistan[END_REF][START_REF] Kroon | Onset of monsoonal related upwelling in the western Arabian Sea as revealed by planktonic foraminfers[END_REF]. This climactic change appears to be directly reflected in the sedimentation record of the Bengal Fan (Figs. 2,3). The sedimentation rate decreases, while the clay mineralogy shifts to the SK assemblage. However, the relation between the monsoon climate and sedimentation in the Bengal Fan is complex. While the appearance of SK clays may simply be related to a warmer and more humid climate in the Indo-Gangetic Plain, monsoon conditions would be expected to increase erosion rates in the High Himalaya, and thus sedimentation rates in the deep sea. This apparent paradox remains to be resolved. [START_REF] Ruddiman | Northern hemisphere climate regimes during the past 3 Ma: possible tectoinic connections[END_REF] and [START_REF] Ruddiman | Forcing of the late Cenozoic northern hemisphere climate by plateau uplift in southeast Asia ansd the American southwest[END_REF] have proposed that late Neogene uplift of the Himalaya and Tibetan plateau has driven regional and global climate change, including causing the onset of the monsoon. The Bengal Fan records no evidence of renewed uplift and increased erosion coincident with the intensification of the monsoon climate at 7-8 Ma. In fact, the Leg 116 results suggest the opposite -a decrease in sedimentation rate possibly related to a decrease in erosion in the Himalaya. Rather, our results require the presence of the Himalaya as a significant topographic feature since at least 17 Ma. This implies that the late Miocene establishment or enhancement of monsoonal circulation in the Indian region is related to factors other than uplift in the Himalaya.

Tectonic changes are likely to cause important variation in the supply of detrital sediment to the Indo-Gangetic plain. For example, a decrease in the relief and general elevation of the range would directly result in decreased erosion. Episodic variations of the exhumation rate could well be related to motion on the MCT between 20 to 10 Ma, and on the MBT since at least 5 Ma (e.g. [START_REF] Fort | The Himalayan orogenic segment[END_REF]. Direct evidence in favour of this hypothesis is, however, lacking. Various studies have suggested pulses of uplift based on cooling ages or faunal studies (e.g. [START_REF] Gansser | The morphogenic phase of mountain building[END_REF] However, these data have proven very difficult to interpret in a unique way [START_REF] Molnar | Late Cenozoic uplift of mountain ranges and global climate change: chicken or egg?[END_REF][START_REF] England | Surface uplift, uplift of rocks, and exhumation of rocks[END_REF]1990). A simple link between tectonics and sedimentation in the Bengal Fan is therefore not easy to demonstrate because to date, no reliable record of the past elevation history of the Himalaya is available. The Bengal fan itself is probably the best indication that the Himalaya was a highly elevated range since the early Miocene. Independent evidence to support high elevations in the mid-early Miocene comes from two sources. France-Lanord et al. (1988) have shown that rare chloritization of the Manaslu leucogranite was effected by meteoric water (dD values of chlorite are ca. -190‰). This episode of chloritization has been dated at 14.7±0.2 Ma (39Ar/40Ar, Copeland et al., 1990). The dD value of the chloritizing water is estimated at < -140‰, markedly lower than modern meteoric water on the southern flank of the High Himalaya (dD -80‰, France-Lanord et al., 1988), and very similar to meteoric water in Tibet [START_REF] Yu | Oxygen and hydrogen isotopic compositions of meteoric waters in the Eastern part of Xizang[END_REF][START_REF] Grimaud | Chemical study of geothermal waters of Central Tibet (China)[END_REF]. Such extremely low values imply that very strong altitude and temperature effects on the isotopic composition of precipitation were already in place at the tropical latitude of the Himalaya in the mid-Miocene. The second independent indication comes from evidence for large scale normal faulting coaxial with the axis of compression at around 20 Ma. [START_REF] Pêcher | The contact between the Higher Crystallines and the Tibetan Sedimentary Series: Miocene large-scale dextral shearing[END_REF] observed stretching lineations in the upper Manaslu granite which record movement on an overlying normal fault while the granite was still hot, ca. 21 Ma. [START_REF] Hodges | Rapid Early Miocene tectonic unroofing of the metamorphic core of the Himalaya: Evidence from the Qomolangma (Everest) region, Tibet Unpublished abstract of the 7th Himalaya-Karakoram-Tbet workshop[END_REF] find evidence for significant extension along the Qomolangma detachment at the same time. Such coeval and parallel extension and compression suggests gravitational collapse of an already highly elevated region due to limitations in crustal strength [START_REF] Molnar | Some simple physical aspects of the support, structure, and evolution of mountain belts[END_REF] We believe that the Himalayan Range had already achieved an elevation of the same order as the present day by 17 Ma.

Conclusion

Turbidite sedimentation in the distal Bengal Fan began in the early Miocene, before 17 Ma. Sedimentological data from ODP Leg 116 and DSDP Leg 22 define three main intervals of depositional history. From 17 to 6.8 Ma, deposition was characterized by coarse clastic debris, high sedimentation rates, and clay mineralogy dominated by an illite-chlorite assemblage. Between 6.8 and 0.8 Ma, deposition rates decreased, along with grain size, and clays were composed primarily of a smectite-kaolinite assemblage. After 0.8 Ma there was an abrupt return to coarse clastic sedimentation, very high accumulation rates, and the illite-chlorite clay assemblage.

Nd, Sr and O isotopic measurements of Bengal Fan sediments from all stratigraphic levels are very uniform, implying a single dominant source for the entire period since 17 Ma. The Rb-Sr isotopic system in this source was apparently homogenized not long before erosion and deposition of the sediment. Oxygen isotopic systematics on coarse quartz and micas also imply a high temperature metamorphic source, consistent with the observed mineralogy of the detrital sediments (e.g. [START_REF] Yokoyama | Mineralogy of silts from the Bengal fan[END_REF]. A comparison of possible sources for the Bengal Fan sediments shows that the High Himalayan Crystalline series closely satisfies the above requirements, and that alternative sources do not. Isotopic mass balance suggests that only 20% or less of the Bengal Fan sediment was derived from other Himalayan sources such as the Lesser Himalaya or High Himalayan Sedimentary Series, although occasionally this fraction may have been as large as 40%. The isotope data show no resolvable contribution from the Indian subcontinent.

Bengal Fan sediments from IC periods are unaffected by low temperature alteration, implying that physical erosion has been dominant over chemical erosion. The smectite-kaolinite clay mineral assemblage apparently is produced from the same source material by low temperature meteoric alteration in the Indo-Gangetic Plain. The appearance of this assemblage in the Bengal fan appears to correlate with an intensification of the monsoon climate around 7 Ma. Paradoxically, the sedimentation rate and grain size in the Fan both decrease at the same time, implying that other factors are important as well.

The relative lack of LH and HHSS-derived material in the Bengal Fan has implications for the structure of the Himalayan Range since the early Miocene. Southward-directed thrusting on the MCT-MBT system has been accompanied by simultaneous removal of the HHSS by northward-directed normal faulting. The large quantity of Bengal Fan sediment which arrived in the Indian Ocean at 1° S at 17 Ma requires that the Himalaya have been a significant topographic feature at least since that time. Independent evidence supports this hypothesis, and we believe that the Himalaya have been at or near their present elevation since early Miocene time. Evidence for significant uplift in the Himalaya during late Miocene and Pliocene time is absent in the Bengal Fan sedimentary record. [START_REF] Gartner | Neogene calcareous nannofossil biostratigraphy, Leg 116 (Central Indian Ocean)[END_REF], clay mineralogy [START_REF] Bouquillon | Sedimentology and isotopic chemistry of the Bengal fan sediments: the denudation of the Himalaya[END_REF] and maximum grain size (Shipboard Scientific Party., 1989) at Site 717. The period dominated by smectite and kaolinite sedimentation is characterized by lower rate of deposition and smaller grain size. Fig. 4 -Age -depth relations for Holes 218, 717C, 718C, and 719A. Depth is recalculated to correct for the effect of sediment compaction. Biostratigraphic ages are from [START_REF] Gartner | Paleontology synthesis of deep sea drilling results from Leg 22 in the Northeastern Indian Ocean[END_REF] and [START_REF] Gartner | Neogene calcareous nannofossil biostratigraphy, Leg 116 (Central Indian Ocean)[END_REF]. 1 or from [START_REF] Bouquillon | Sedimentology and isotopic chemistry of the Bengal fan sediments: the denudation of the Himalaya[END_REF]. Clay mineralogical profiles are simplified after [START_REF] Bouquillon | Sedimentology and isotopic chemistry of the Bengal fan sediments: the denudation of the Himalaya[END_REF]. [START_REF] Bouquillon | Sedimentology and isotopic chemistry of the Bengal fan sediments: the denudation of the Himalaya[END_REF]. Clay mineralogical profiles are simplified after [START_REF] Bouquillon | Sedimentology and isotopic chemistry of the Bengal fan sediments: the denudation of the Himalaya[END_REF]. 1 and2 and: [START_REF] Blattner | Contrasting 18-O enrichment and origins of High Himalayan and Transhimalayan intrusives[END_REF], [START_REF] Dietrich | The leucogranites of the Bhutan Himalaya (crustal anatexis versus mantle melting)[END_REF], [START_REF] Deniel | Les leucogranites himalayens et leur région source probable : les gneiss de la "Dalle du Tibet[END_REF]& 1987), France-Lanord et al. (1988), France-Lanord and Le Fort, (1988), France-Lanord and Scaillet (1989), [START_REF] Goldstein | Decoupled evolution of Nd and Sr isotopes in the continental crust and the mantle[END_REF], [START_REF] Goldstein | A Sm-Nd isotopic study of atmospheric dusts and particulates from major river systems[END_REF], [START_REF] Parrish | U-Pb geochronology of igneous and metamorphic rocks near the Main Central Thrust in the Langtang area, Central Nepal Himalaya[END_REF], [START_REF] Scaillet | Badrinath-Gangotri plutons (Garhwal, India) Petrological and geochemical evidence for fractionation processes in a High Himalayan Leucogranite[END_REF][START_REF] Trivedi | Rb-Sr ages of granitic rocks within the Lesser Himalayan nappes, Kumaun, India[END_REF], [START_REF] Vidal | Geochemical investigations of the origin of the Manaslu leucogranite (Himalaya, Nepal)[END_REF][START_REF] Vidal | Geochemical comparison between Himalayan and Hercynian leucogranites[END_REF]. Data in italics and most X-ray determinations of clay are from [START_REF] Bouquillon | Sedimentology and isotopic chemistry of the Bengal fan sediments: the denudation of the Himalaya[END_REF].

Figure captions

87Sr/86Sr normalized to 86Sr/88Sr =0.1194. Average value of NBS 987 during the period of analysis was 0.710197 143Nd/144Nd normalized to 146Nd/144Nd=0.7219. Average value of LaJolla Nd during the period of analysis was 0.511847 d18O of NBS 28 = 9.60‰ Le Fort, 1988 and[START_REF] Bouquillon | Sedimentology and isotopic chemistry of the Bengal fan sediments: the denudation of the Himalaya[END_REF] see table 1 for analytical parameters

Fig. 1 -

 1 Fig. 1 -Location of the Leg 116 Sites in the the Bengal Basin.

Fig. 2 -

 2 Fig. 2 -Stratigraphic correlation of the Leg 116 Holes 717C, 718C and 719A and Leg 22 Hole 218. Lithologic logs are after Shipboard Scientific Party (1989) and Von der Borch, Sclater et al. 1974.

Fig. 3 -

 3 Fig.3-Depth versus lithology, chronology[START_REF] Gartner | Neogene calcareous nannofossil biostratigraphy, Leg 116 (Central Indian Ocean)[END_REF], clay mineralogy[START_REF] Bouquillon | Sedimentology and isotopic chemistry of the Bengal fan sediments: the denudation of the Himalaya[END_REF] and maximum grain size (Shipboard Scientific Party., 1989) at Site 717. The period dominated by smectite and kaolinite sedimentation is characterized by lower rate of deposition and smaller grain size.

Fig. 5 -

 5 Fig. 5 -Depth profiles of isotopic composition of Nd and Sr for Holes 717C and bottom of 718C. Data are listed in Table1or from[START_REF] Bouquillon | Sedimentology and isotopic chemistry of the Bengal fan sediments: the denudation of the Himalaya[END_REF]. Clay mineralogical profiles are simplified after[START_REF] Bouquillon | Sedimentology and isotopic chemistry of the Bengal fan sediments: the denudation of the Himalaya[END_REF].

Fig. 6 -

 6 Fig. 6 -Depth profiles of the isotopic composition of O and H for Holes 717C and bottom of 718C. Data are listed in Table1or from[START_REF] Bouquillon | Sedimentology and isotopic chemistry of the Bengal fan sediments: the denudation of the Himalaya[END_REF]. Clay mineralogical profiles are simplified after[START_REF] Bouquillon | Sedimentology and isotopic chemistry of the Bengal fan sediments: the denudation of the Himalaya[END_REF].

Fig. 7 -

 7 Fig. 7 -Histograms of Nd-, O-and Sr-isotope data of the Leg 116 sediments compared to the Himalayan formations. Average values are similar for the HHC and the Leg 116 samples. a) Nd isotope data, B, M represent the values of the coarse biotite+chlorite, and the coarse muscovite separated from sample (116-717C-2H-01,02). b) Oxygene isotope data. c) Sr isotope data, F represents values of coarse feldspar separates. Data are from Table1 and 2 and:[START_REF] Blattner | Contrasting 18-O enrichment and origins of High Himalayan and Transhimalayan intrusives[END_REF],[START_REF] Dietrich | The leucogranites of the Bhutan Himalaya (crustal anatexis versus mantle melting)[END_REF],[START_REF] Deniel | Les leucogranites himalayens et leur région source probable : les gneiss de la "Dalle du Tibet[END_REF][START_REF] Deniel | Isotopic study of the Manaslu granite (Himalaya, Nepal) : Inferences on the age and source of Himalayan leucogranites[END_REF],France-Lanord et al. (1988), France-Lanord andLe Fort, (1988), France-Lanord and Scaillet (1989),[START_REF] Goldstein | Decoupled evolution of Nd and Sr isotopes in the continental crust and the mantle[END_REF],[START_REF] Goldstein | A Sm-Nd isotopic study of atmospheric dusts and particulates from major river systems[END_REF],[START_REF] Parrish | U-Pb geochronology of igneous and metamorphic rocks near the Main Central Thrust in the Langtang area, Central Nepal Himalaya[END_REF],[START_REF] Scaillet | Badrinath-Gangotri plutons (Garhwal, India) Petrological and geochemical evidence for fractionation processes in a High Himalayan Leucogranite[END_REF][START_REF] Trivedi | Rb-Sr ages of granitic rocks within the Lesser Himalayan nappes, Kumaun, India[END_REF],[START_REF] Vidal | Geochemical investigations of the origin of the Manaslu leucogranite (Himalaya, Nepal)[END_REF][START_REF] Vidal | Geochemical comparison between Himalayan and Hercynian leucogranites[END_REF].

Fig. 8 -

 8 Fig. 8 -Schematic cross-section of the Himalaya at 17 Ma and the present day. Thrusting has maintained high elevation in the HHC, while normal faulting has continuously removed the HHSS to the north. This scheme satisfies the requirement that the HHC has been exposed to rapid erosion since the early Miocene, while the HHSS has contributed little to the sediment flux. The black triangle represent a point in the Lesser Himalaya.

Table 1 -

 1 H, O, Sr and Nd isotope data and relative abundances of clay minerals on Leg 116 sediments.

		mbsf Ø dD H2O+ d18O Rb Sr 87Sr/86Sr 87Rb/86Sr 143Nd/144Nd ± eNd(0) Sm	Nd Relative abundance of clay in <2µm
		<øµ ‰ %	‰ ppm ppm					ppm ppm	chlorite	illite int smectite kaolinite
	Leg 116 Hole 717C
	2X-1 (80-81)	15.8 2	14 232 91 0.73723	7.39	0.511824	30 -15.9
		2				0.511774	26 -16.8 6.4 32.5
	6X-2 (27-28)	49.3 2	79	0.7381
	7X-1 (85-86)	56.4 2	13			0.511929	17 -13.8 6.5 32.9	15	41	9	25	10
	15X-2 (49-50)	114.5 2	209 82 0.74048	7.32	0.511791	25 -16.5 4.7 23.5
	19X-CC (5-6)	150.5 2	15 235 87 0.74006	7.81	0.511893	23 -14.5
		2				0.511817	27 -16	7.9 40.7
	20X-1 (76-77)	151.3 2		0.73791		0.511832	28 -15.8 6.5 33.2	16	41	18	20	5
		50	159 251 0.74731	1.83
	21X-3 (34-35)	163.3 50	126 208 0.74064	1.75
		2		0.76224						3	12	13	62	10
	22X-4 (100-101)	175 <63				0.5118	25 -16.4			2	10	13	62	13
	24X-4 (66-67)	193.7 <63				0.511776	12 -16.8			2	8	10	70	10
		2				0.511806	18 -16.2 5.7 27.9
	26X-4 (85-86)	214.4 2 -56 3.4	13 13
	27X-5 (70-71)	223.7 2 -74 5.6	13			0.511855	22 -15.3 3.5 18.4
	28X-6 (19-20)	235.7 2	40 0.72798
	29X-5 (70-71)	242.7 WR -57 5.4	20
	30X-1 (116-117) 246.7 2	23	0.73851						3	5	16	60	16
		50	78 131 0.72791	1.72					3	5	16	60	16
	30X-3 (20-21)	248.7 <63				0.511815	13 -16.1			6	7	14	58	15
	32X-3 (118-119) 268.7 WR -72 6.7	18			0.511898	26 -14.5	5	24

Table 2 -

 2 Sr and Nd isotope data for soil from Bangladesh and rocks from the Lesser Himalaya, High Himalaya Sedimentary series

		Type	Rb	Sr	87Sr/86Sr	87Rb/86Sr	143Nd/144Nd ± Eps 0 Sm Nd TDM
		<øµ	ppm ppm						ppm ppm Ga
	Soil from the Ganges delta
	Batra series	<2µm		32	0.75236	22.18	0.511855	15	-15.3	4.7 23.6
	High Himalaya Sedimentary Series
	NA 178 Jurassic schist	W.R.					0.511795	28	-16	4 20.3	2
	NA 181 Cretaceous (volcaniclastic) W.R.					0.512405	28 -4.5 12.6	59 1.26
	87 28 Cretaceous	W.R.	42	61	0.71227	1.904	0.512093	13 -10.6 9.1 45.5 1.53
	8732 Trias	W.R.	19	45	0.70899	1,19	0,511824	17 -15.9 3.6 18.6 1.93
	LA 194 Indus Margin	W.R.	11 299	0.70981	0.103	0.511883	9 -14.7 5.91 33.2 1.64
	LA 158 Indus Marginn	W.R.	96	50	0.72903	5.289	0.512046	37 -11.6	5 23.2 1.77
	25 1 Suture Ladak	W.R.	7	662	0.70762	0.027	0.512313	34 -6.3 1.6 6.4 1.68
	Suture Ladak	W.R.	44 258	0.70606	0.469	0.51258	24 -1.1	3 13.9 0.88
	Lesser Himalaya
	AP 346Kunchha pelites	W.R.	143 11.1	1.6809	38.07	0.511397	28	-24 8.08	44 2.52
	AP 385Kunchha pelites	W.R.	286 63	1.0301	13.48	0.511331	25	-25 6.58 36.6 2.6
	forData in italic are from France-Lanord and

This study was carried out under the auspices of the French program 'Dynamique et Bilans de la Terre -Fieuve et érosion'. We thank P. Copeland, P. Le Fort, A. Galy and A. Moore for helpful discussion and review. We are grateful to J.J. Tiercelin, G. Mascle and T. Heulin who provided samples for this study. G. Burtin is thanked for his efficient help in clay separation. R. Taieb, D. Dautel and F. Lothe are thanked for their analytical support. CRPG contribution N° 951. CNRS-INSU-DBT contribution N° 490.