The conductance of a ballistic quantum dot (having chaotic classical dynamics and being coupled by ballistic point contacts to two electron reservoirs) is computed on the single assumption that its scattering matrix is a member of Dyson's circular ensemble. General formulas are obtained for the mean and variance of transport properties in the orthogonal (β = 1), unitary (β = 2), and symplectic (β = 4) symmetry class. Applications include universal conductance fluctuations, weak localization, sub-Poissonian shot noise, and normal-metal-superconductor junctions. The complete distribution P (g) of the conductance g is computed for the case that the coupling to the reservoirs occurs via two quantum point contacts with a single transmitted channel. The result P (g) ∝ g −1+β/2 is qualitatively different in the three symmetry classes. The search for signatures of chaotic behavior in quantum mechanical systems [1] has recently been extended to semiconductor nanostructures known as "quantum dots" [2,3]. A quantum dot is essentially a mesoscopic electron billiard, consisting of a ballistic cavity connected by two small holes to two electron reservoirs. An electron which is injected through one of the holes will either return through the same hole, with probability R, or be transmitted through the other hole, with probability T . Classically, the uniform (ergodic) exploration of the boundaries yields T = R, if the two holes are of the same size and sufficiently small that direct transmission (without boundary reflections) can be ignored.For a closed quantum dot (without holes), it is well known that one of the quantum signatures of its classically chaotic character consists in the Wigner-Dyson distribution of the energy levels [4,5]. The WignerDyson distribution was originally derived by randommatrix theory (RMT), and is characterized by a repulsion of nearby levels which depends only on the symmetry of the Hamiltonian. The quantum analog of the ergodic exploration of the dot boundaries by the classical trajectories consists in the Porter-Thomas distribution of the eigenfunctions, as confirmed by numerical studies of their amplitude distribution at the boundaries [6]. The quantum dot with holes is an open, rather than a closed system. Just as the Wigner-Dyson distribution describes the Hamiltonian H of the closed system, Dyson's circular ensemble [7] provides the statistical properties of the scattering matrix S of the open system. To have spectral or scattering properties given by the universal RMT description for H or S can be actually regarded as a precise definition of the somewhat vague concept of "quantum chaos". To what extent a real ballistic cavity is close to this precise universal limit is the subject of the theory of quantum billiards [5,8].Assuming this definition of a quantum chaotic system -and this is our only assumption -we will calculate the statistics of the transmission and reflection eigenvalues of the quantum dot, and hence its transport properties. This allows us to determine the universal quantum si...

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The pair localization length L2 of two interacting electrons in one-dimensional disordered systems is studied numerically. Using two direct approaches, we find L2 ∝ L α 1 , where L1 is the one-electron localization length and α ≈ 1.65. This demonstrates the enhancement effect proposed by Shepelyansky, but the value of α differs from previous estimates (α = 2) in the disorder range considered. We explain this discrepancy using a scaling picture recently introduced by Imry and taking into account a more accurate distribution than previously assumed for the overlap of one-electron wavefunctions.PACS numbers: 71.30, 72.15.R Very recently, Shepelyansky [1] considered the problem of two interacting electrons in a random potential, defined by the Schrödinger equation, U characterizes the on-site interaction and V i is a random potential distributed uniformly in the interval [−W, W ]. The indices n 1 and n 2 denote the positions of the first and the second electron, respectively. Shepelyansky proposed that, as a consequence of the interaction U , certain eigenstates extend over a range L 2 much larger than the one-particle localization length L 1 ∝ W −2 . A key quantity in the derivation of this spectacular result in one dimension is the matrix representation U of the Hubbard interaction in the disorder diagonal basis of localized oneelectron eigenstates. With R n,i the amplitude at site n of the one-particle eigenstate with energy E i we have U ij,lm = U Q ij,lm with Q ij,lm = n R n,i R n,j R n,l R n,m .(The Q ij,lm vanish unless all four eigenstates are roughly localized within the same box of size L 1 . Assuming that R n,i ∝ a n / √ L 1 inside the box, with a n a random number of order unity, and neglecting correlations among the a n at different sites n, one finds Q ≈ 1/L 3/2 1 for a typical nonvanishing matrix element. In [1], this estimate was adopted and used to reduce the original problem to a certain band matrix model, eventually giving L 2 ∝ L 2 1 . Later, Imry [2] employed the Thouless scaling block picture to reinforce, interpret and generalize this result. The key step in this approach involves the pair conductance g 2 = (U Q/δ 2 ) 2 , where δ 2 is the two-particle level spacing in a block of size L 1 and Q is evaluated between adjacent blocks. Using Q ≈ 1/L 3/2 1 as before, Imry finds that g 2 ≈ 1 on the scale L 2 ∝ L 2 1 , in agreement with Shepelyansky. As a second important result both approaches predict that the effect does not depend on the sign of U .In this letter, we confirm the enhancement effect by studying both the original model (1) for finite size samples and an infinite "bag model" with medium-range interaction. However, we find L 2 ∝ L α 1 with α ≈ 1.65 instead of α = 2 in both cases. Moreover, the sign of U is not entirely irrelevant. We suggest that the small value for α is due to a very peculiar distribution of the Q ij,lm .

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An Ansatz is proposed for the joint probability distribution of the eigenvalues of the transfer matrix in the quantum transport problem, based on symmetry arguments and a "maximum entropy" hypothesis. The local statistical behavior of the distribution is predicted to be that of the well-known random-matrix ensembles of Wigner and Dyson; and this result is confirmed by independent numerical calculations. For metals this behavior leads to size-and disorder-independent conductance fluctuations, and this approach suggests an alternative framework for the scaling theory of localization.

For intermediate Coulomb energy to Fermi energy ratios r s , spinless fermions in a random potential form a new quantum phase which is neither a Fermi glass, nor a Wigner crystal. Studying small clusters, we show that this phase gives rise to an ordered flow of enhanced persistent currents for disorder strength and ratios r s , where a metallic phase has been recently observed in two dimensions. PACS numbers: 71.30. + h, 72.15.Rn An important parameter for a system of charged particles is the Coulomb energy to Fermi energy ratio r s . In a disordered two-dimensional system, the ground state is obvious in two limits. For large r s , the charges form a kind of pinned Wigner crystal, the Coulomb repulsion being dominant over the kinetic energy and the disorder. For small r s , the interaction becomes negligible and the ground state is a Fermi glass with localized one electron states. There is no theory for intermediate r s , while many transport measurements following the pioneering works of Kravchenko et al.[1] and made with electron and hole gases give evidence of an intermediate metallic phase in two dimensions, observed [2], for instance, when 6 , r s , 9 for a hole gas in GaAs heterostructures. A simple model of spinless fermions with Coulomb repulsion in small disordered 2D clusters exhibits a new ground state characterized by an ordered flow of enhanced persistent currents for those values of r s . In a given cluster, as we turn on the interaction, the Fermi ground state can be followed from r s 0 up to a first level crossing. A second crossing occurs at a larger threshold after which the ground state can be followed to the limit r s !`. There is then an intermediate state between the two crossings. In small clusters, the location of the crossings depends on the considered potentials, but a study over the statistical ensemble of the currents supported by the ground state gives us two welldefined values r F s and r W s : Mapping the system on a torus threaded by an Aharonov-Bohm flux, we denote, respectively, I l and I t the total longitudinal (direction enclosing the flux) and transverse parts of the driven current. One finds for their typical amplitudes jI t j ഠ exp2͑r s ͞r F s ͒ and I l ഠ exp2͑r s ͞r W s ͒ with r F s , r W s . Below r F s , the flux gives rise to a glass of local currents and the sign of I l can be diamagnetic or paramagnetic, depending on the random potentials. Above r F s , the transverse current is suppressed while an ordered flow of longitudinal currents persists up to r W s , where charge crystallization occurs. The sign of I l can be paramagnetic or diamagnetic, depending on the filling factor (as for the Wigner crystal), but does not depend on the random potentials (in contrast to the Fermi glass). One finds r F s and r W s in agreement with the values delimiting the new metallic phase when 0.3 , k F l , 3, k F and l denoting the Fermi wave vector and the elastic mean free path, respectively. For k F l $ 1, I l is strongly increased between r F s and r W s . This suggests that the...

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The two-probe conductance, g, of a disordered quantum system with N tranverse scattering channels is determined by N real parameters ("levels") {λi} characterizing the transfer matrix M. The appropriate measure for M combined with a "global" maximum entropy hypothesis, leads to a joint distribution for these levels of the same form as the standard random matrix ensembles, except for the occurence of a novel behavior of the level density ρ(λ), which is far from uniform, and depends importantly on the system parameters. We study this density and its implications for conductance fluctuations in detail, showing that the novel behavior of this ensemble stems from the multiplicative composition law for M. For fixed N, as the system length Lz → oo it is the αi ~(1/2Lz) ℓn λ i (which converge to the Lyapunov exponents of M) that have an approximately uniform density. Using this density we develop a Coulomb gas analogy to understand analytically the change in the level and conductance statistics which accompanies the transition from metallic to localized behavior. The metallic regime corresponds to the high-density "phase" of the gas, with statistics similar to standard, logarithmically-correlated ensembles except for the appearance of an "interaction" with image charges near the origin; this leads to a normal distribution p(g) with a universal variance. Possible mechanisms for the occurrence of lognormal tails in the metallic regime are discussed. The localized regime corresponds to a low-density "phase" in which the levels fluctuate independently and the conductance is lognormally distributed with 〈(δℓn g)2〉 ~ 2/g0 where g0 = N ℓ/Lz is the classical conductance and ℓ is the elastic man free path. The consequences of the Coulomb gas analogy for both the conductance and level statistics are confirmed by independent numerical calculations. In our theory, the distribution p(g) depending only on p(A), we study in a second part of this work the finite-size scaling properties of ρ(λ) to address the question of one-parameter scaling. First we generalize for each level α i the scaling law obtained for the disordered chain. Then we show that the convergence of each αi towards their quasi-one dimensional limit (N fixed, Lz → oo) depends only on the average conductance (g) in the range of investigated parameters. However, we cannot rule out a dependence on the index i of the scaling functions which would introduce additional scaling parameters for p(g)