We investigate the localization of two interacting particles in one-dimensional random potential. Our definition of the two-particle localization length, ξ, is the same as that of v. Oppen et al. [Phys. Rev. Lett. 76, 491 (1996)] and ξ's for chains of finite lengths are calculated numerically using the recursive Green's function method for several values of the strength of the disorder, W , and the strength of interaction, U . When U = 0, ξ approaches a value larger than half the single-particle localization length as the system size tends to infinity and behaves as ξ ∼ W −ν 0 for small W with ν0 = 2.1 ± 0.1. When U = 0, we use the finite size scaling ansatz and find the relation ξ ∼ W −ν with ν = 2.9 ± 0.2. Moreover, data show the scaling behavior ξ ∼ W −ν 0 g(b|U |/W ∆ ) with ∆ = 4.0 ± 0.5. PACS number(s): 72.15. Rn, 71.30.+h Recently, there has been intensive attention [1][2][3][4][5][6][7][8][9][10] focused on the problem of the localization of two interacting particles in one-dimensional (1D) random potential. With a few assumptions on the statistical nature of single-particle localized states, Shepelyansky [1] has mapped the problem approximately to a random band matrix model and obtained an expression for the twoparticle localization length, ξ, aswhere U is the on-site interaction in unit of the hopping energy between nearest neighbor pair sites, and ξ 1 the single-particle localization length. This expression is surprising because it implies that ξ can exceed ξ 1 at sufficiently small disorder, i.e. , where c is a constant depending on the statistics of the particles. With the assumption that the level statistics of two interacting particles is described by a Gaussian matrix ensemble, Weinmann and Pichard [4] argued that ξ increases initially as |U | before eventually behaving as U 2 . Moreover, very recently, Römer and Schreiber have claimed the disappearance of the enhancement as the system size grows (see Refs. [7] and [8]).Some of these discrepancies, especially between numerical studies, are due to different definitions for twoparticle localization length between authors and also to lack of careful analysis of the finite size effect of the system size. The system under study is a "quantum mechanical two-body problem" in a sense. Motion of the two particles can be decomposed into the motion of the center of mass (CM) and that of the relative coordinate. We are interested in the CM motion since the wavefunction describing the relative motion would not be different from that arising from the single-particle localization problem in the thermodynamic limit if the interaction is short-ranged. Therefore, in this paper, we use the same definition for ξ as introduced by v. Oppen et al. [5] for the measure for localization length of the CM:Here, G is the Green function and |i, j is a two-particle state in which the particle 1 (2) is localized at a site i (j). The above definition is reasonable for a description of the CM motion as long as U is smaller than or of the order of the hopping energy between ...

Spin relaxation time of conduction electrons through the Elliot-Yafet, D'yakonov-Perel and Bir-Aronov-Pikus mechanisms is calculated theoretically for bulk GaAs, GaSb, InAs and InSb of both n-and p-type. Relative importance of each spin relaxation mechanism is compared and the diagrams showing the dominant mechanism are constructed as a function of temperature and impurity concentrations. Our approach is based upon theoretical calculation of the momentum relaxation rate and allows understanding of the interplay between various factors affecting the spin relaxation over a broad range of temperature and impurity concentration.

We study numerically the imperfection effects in the quantum computing of the kicked rotator model in the regime of quantum chaos. It is shown that there are two types of physical characteristics: for one of them the quantum computation errors grow exponentially with the number of qubits in the computer while for the other the growth is polynomial. Certain similarity between classical and quantum computing errors is also discussed.PACS numbers: 05.45. Mt, 03.67.Lx, 24.10.Cn A great interest to quantum computers has been generated recently by prominent theoretical results and impressive experimental progress which allowed to realise operations with a few qubits (see [1] for a review). The most striking theoretical advantage is an enormous parallelism of quantum computing. Using the Shor algorithm [2] the factorization of large numbers can be done exponentially faster on a quantum computer than by any known algorithm on a classical computer. Also a search of an item in a long list is much faster on a quantum computer as shown by Grover [3]. Experimentally a variety of physical systems is considered for realisation of one qubit, viewed as a two level system, and controlled coupling between a few qubits that forms the basis for realisation of a quantum computer. These systems include ion traps [4,5], nuclear magnetic resonance systems [6], nuclear spins with interaction controlled electronically [7,8] or by laser pulses [9], electrons in quantum dots [10], Cooper pair boxes [11], optical lattices [12] and electrons floating on liquid helium [13]. As a result, a two-qubit gate has been experimentally realized with cold ions [14], and the Grover algorithm has been performed for three qubits made from nuclear spins in a molecule [15].It is clear that in any realistic realisation of a quantum computer a special attention should be paid to the imperfection effects. Indeed, the imperfections are always present and they in principle may seriously modify the computation results comparing to the algorithms based on ideal qubit operations. At present the imperfection effects are tested in the numerical simulations of the quantum Fourier transform (QFT) [4] and the Shor algorithm factorization of 15 [16,17]. The obtained results look to be promising for the quantum computing indicating that a small amount of noise does not change strongly the computations [4] even if in some cases only rather low level of noise is tolerable [17]. However, for different reasons these studies do not allow to obtain analytical estimates of a tolerable imperfection level for a large number of qubits n. Indeed, the Shor algorithm is rather complicated and the capability of nowadays computers become too restrictive [16,17]. Recently the effects of static imperfections on the the stability of quantum computer hardware have been determined for a broad regime of parameters and it has been shown that the quantum hardware is sufficiently robust [18]. However, these results cannot be directly generalised for a specific quantum algorithm operating in ti...

The propagation of an interacting particle pair in a disordered chain is characterized by a set of localization lengths which we define. The localization lengths are computed by a new decimation algorithm and provide a more comprehensive picture of the two-particle propagation. We find that the interaction delocalizes predominantly the center-of-mass motion of the pair and use our approach to propose a consistent interpretation of the discrepancies between previous numerical results. PACS number(s): 72.15. Rn, 71.30.+h The problem of interacting electrons in a disordered potential is one of the important unsolved problems in condensed matter physics. This has been emphasized again by the recent observation [1] of a metal-insulator transition in two dimensional (2d) systems which was theoretically unanticipated. Some time ago, Shepelyansky [2] proposed that it would be worthwhile to consider the simple case of two interacting particles in a random potential. He predicted that unexpectedly, such a particle pair could propagate coherently over distances ξ 2 much larger than the single-particle localization length ξ 1 as long as the two particles are within ξ 1 from each other.Specifically, Shepelyansky obtained for the twoparticle localization lengthwhere U denotes the interaction strength and W the disorder strength. Since ξ 1 ∼ 1/W 2 , Eq. (1) implies an enhancement of the localization length for weak disorder. Shepelyansky's original argument involved several uncontrolled assumptions for the single-particle eigenstates. This led to a number of (mostly numerical) attempts [3][4][5][6][7] to study the problem of two interacting particles more rigorously. Recently, Römer and Schreiber [7] concluded from the TMM that the enhancement effect does not exist. In view of this claim and of the quantitatively different expressions for ξ 2 quoted above, it appears that there are few secured results in this field. Our purpose in this paper is to present a more comprehensive picture of the twoparticle propagation by defining and computing a set of localization lengths. We unambiguously show that the effect exists and propose a resolution of the controversy in the previous works [7,10].It is currently not clear whether these ideas have any relevance to the degenerate finite-density Fermi gas. It appears to be the most promising direction to consider the localization properties of quasiparticle pairs. There have been a number of studies [3,8,9] whether quasiparticle excitations delocalize relative to single-particle ones. While a numerical study for a one-dimensional (1d) system showed delocalization only for unrealistically high excitation energy of the pair (of order of the bandwidth) [8], both arguments [3] and numerical studies [9] in higher dimensions suggest the possibility of a new pair mobility edge close to the ground state.The two-particle problem in one-dimension is described by the Hamiltonianwhere m labels the N sites of the 1d lattice and H 1 is the usual single-particle Anderson Hamiltonianǫ m is a random si...

We study the level spacing statistics P (s) and eigenstate properties of spinless fermions with Coulomb interaction on a two dimensional lattice at constant filling factor and various disorder strength. In the limit of large lattice size, P (s) undergoes a transition from the Poisson to the Wigner-Dyson distribution at a critical total energy independent of the number of fermions. This implies the emergence of quantum ergodicity induced by interaction and delocalization in the Hilbert space at zero temperature.PACS numbers: 71.30.+h, 72.15.Rn, 05.45.Mt The experimental observation of the metal-insulator transition in two dimensions (2D) by Kravchenko et al.[1] has attracted a great interest to interacting fermions in a disordered potential. Indeed, according to the wellestablished result [2], all states are localized for noninteracting particles in 2D. Therefore, in the view of the experimental result [1], a new theory should be developed to understand the interaction effects between the localized fermionic states. However, in spite of various theoretical attempts, a coherent theory for such systems is still not available. While for highly excited states, it has been shown that the repulsive/attractive interaction can induce a delocalization of two interacting particles [3][4][5][6], the properties of low energy states are not understood yet. Recently, in addition to experimental and theoretical investigations, a number of attempts have been made to study these many fermionic systems through numerical simulations [7][8][9]. Even though several interesting features have been reported, the systems studied there are not large enough to observe interaction induced delocalization.In this paper we use another numerical approach based on the analysis of spectral properties of multi-particle fermionic systems. Indeed, Shklovskii et al. have shown that the level spacing statistics is a powerful tool to analyze the Anderson transition in disordered systems [10]. When the states are localized, the levels are not correlated and the statistics is given by the Poisson distribution, P (s) = P P (s) = exp(−s), while in the metallic phase, the states are ergodic and the statistics is close to the Wigner surmise, P W (s) = (πs/2) exp(−πs 2 /4). The critical transition point is characterized by an intermediate statistics which depends on the boundary conditions [11] and the spatial dimension of the system [12]. This approach has also been used to determine the quantum chaos border and the interaction induced thermalization in finite fermionic systems [13] and to detect Anderson transition for two electrons with the Coulomb interaction on 2D disordered lattice [14,15]. All these results demonstrate that the approach developed in [10] allows to investigate efficiently the transition from nonergodic (localized) to ergodic eigenstates.Here we use the above method to study the change of the spectral statistics, P (s), with excitation energy E in a model of spinless fermions with Coulomb interaction on 2D disordered lattice. The Hami...