Exploring Level Statistics from Quantum Chaos to Localization with the Autocorrelation Function of Spectral Determinants

The autocorrelation function of spectral determinants (ASD) is used to characterize the discrete spectrum of a phase coherent quasi- 1- dimensional, disordered wire as a function of its length L in a finite, weak magnetic field. An analytical function is obtained depending only on the dimensionless conductance g= xi/L where xi is the localization length, the scaled frequency x= omega/Delta, where Delta is the average level spacing of the wire, and the global symmetry of the system. A metal- insulator crossover is observed, showing that information on localization is contained in the disorder averaged ASD.Comment: 4 pages, 3 figure

Symmetry Dependence of Localization in Quasi- 1- dimensional Disordered Wires

The crossover in energy level statistics of a quasi-1-dimensional disordered wire as a function of its length L is used, in order to derive its averaged localization length, without magnetic field, in a magnetic field and for moderate spin orbit scattering strength. An analytical function of the magnetic field for the local level spacing is obtained, and found to be in excellent agreement with the magnetic field dependent activation energy, recently measured in low-mobility quasi-one-dimensional wires\cite{khavin}. This formula can be used to extract directly and accurately the localization length from magnetoresistance experiments. In general, the local level spacing is shown to be proportional to the excitation gap of a virtual particle, moving on a compact symmetric space.Comment: 4 pages, 2 Eqs. added, Eperimental Data included in Fig.

Universal Distribution of Kondo Temperatures in Dirty Metals

Kondo screening of diluted magnetic impurities in a disordered host is studied analytically and numerically in one, two and three dimensions. It is shown that in the T_K \to 0 limit the distribution of Kondo temperatures has a universal form, P(T_K) \sim T_K^{-\alpha} that holds in the insulating phase and persists in the metallic phase close to the metal insulator transition. Moreover, the exponent \alpha depends only on the dimensionality. The most important consequence of this result is that the T-dependence of thermodynamic properties is smooth across the metal-insulator transition in three dimensional systems.Comment: 4 pages, 3 figures; added referenc

Localization Length in Anderson Insulator with Kondo Impurities

The localization length, $\xi$, in a 2--dimensional Anderson insulator depends on the electron spin scattering rate by magnetic impurities, $\tau_s^{-1}$. For antiferromagnetic sign of the exchange, %constant, the time $\tau_s$ is {\em itself a function of $\xi$}, due to the Kondo correlations. We demonstrate that the unitary regime of localization is impossible when the concentration of magnetic impurities, $n_{\tiny M}$, is smaller than a critical value, $n_c$. For $n_{\tiny M}>n_c$, the dependence of $\xi$ on the dimensionless conductance, $g$, is {\em reentrant}, crossing over to unitary, and back to orthogonal behavior upon increasing $g$. Sensitivity of Kondo correlations to a weak {\em parallel} magnetic field results in a giant parallel magnetoresistance.Comment: 5 pages, 1 figur

Fluctuations and Ergodicity of the Form Factor of Quantum Propagators and Random Unitary Matrices

We consider the spectral form factor of random unitary matrices as well as of Floquet matrices of kicked tops. For a typical matrix the time dependence of the form factor looks erratic; only after a local time average over a suitably large time window does a systematic time dependence become manifest. For matrices drawn from the circular unitary ensemble we prove ergodicity: In the limits of large matrix dimension and large time window the local time average has vanishingly small ensemble fluctuations and may be identified with the ensemble average. By numerically diagonalizing Floquet matrices of kicked tops with a globally chaotic classical limit we find the same ergodicity. As a byproduct we find that the traces of random matrices from the circular ensembles behave very much like independent Gaussian random numbers. Again, Floquet matrices of chaotic tops share that universal behavior. It becomes clear that the form factor of chaotic dynamical systems can be fully faithful to random-matrix theory, not only in its locally time-averaged systematic time dependence but also in its fluctuations.Comment: 12 pages, RevTEX, 4 figures in eps forma

Information about the Integer Quantum Hall Transition Extracted from the Autocorrelation Function of Spectral Determinants

The Autocorrelation function of spectral determinants (ASD) is used to probe the sensitivity of a two-dimensional disordered electron gas to the system's size L. For weak magnetic fields ASD is shown to depend only trivially on L, which is a strong indication that all states are localized. From nontrivial dependence of ASD on L for infinite L at a Hall conductance of 1/2 e^2/h we deduce the existence of critical wave functions at this point, as long as the disorder strength does not exceed a critical value.Comment: 4 pages, one citation correcte

Dimensional Crossover of Localisation and Delocalisation in a Quantum Hall Bar

The 2-- to 1--dimensional crossover of the localisation length of electrons confined to a disordered quantum wire of finite width $L_y$ is studied in a model of electrons moving in the potential of uncorrelated impurities. An analytical formula for the localisation length is derived, describing the dimensional crossover as function of width $L_y$, conductance $g$ and perpendicular magnetic field $B$ . On the basis of these results, the scaling analysis of the quantum Hall effect in high Landau levels, and the delocalisation transition in a quantum Hall wire are reconsidered.Comment: 12 pages, 7 figure

Nonchiral Edge States at the Chiral Metal Insulator Transition in Disordered Quantum Hall Wires

The quantum phase diagram of disordered wires in a strong magnetic field is studied as a function of wire width and energy. The two-terminal conductance shows zero-temperature discontinuous transitions between exactly integer plateau values and zero. In the vicinity of this transition, the chiral metal-insulator transition (CMIT), states are identified that are superpositions of edge states with opposite chirality. The bulk contribution of such states is found to decrease with increasing wire width. Based on exact diagonalization results for the eigenstates and their participation ratios, we conclude that these states are characteristic for the CMIT, have the appearance of nonchiral edges states, and are thereby distinguishable from other states in the quantum Hall wire, namely, extended edge states, two-dimensionally (2D) localized, quasi-1D localized, and 2D critical states.Comment: replaced with revised versio

Characterization of Quantum Chaos by the Autocorrelation Function of Spectral Determinants

The autocorrelation function of spectral determinants is proposed as a convenient tool for the characterization of spectral statistics in general, and for the study of the intimate link between quantum chaos and random matrix theory, in particular. For this purpose, the correlation functions of spectral determinants are evaluated for various random matrix ensembles, and are compared with the corresponding semiclassical expressions. The method is demonstrated by applying it to the spectra of the quantized Sinai billiards in two and three dimensions.Comment: LaTeX, 32 pages, 6 figure

Magnetolocalization in disordered quantum wires

The magnetic field dependent localization in a disordered quantum wire is considered nonperturbatively. An increase of an averaged localization length with the magnetic field is found, saturating at twice its value without magnetic field. The crossover behavior is shown to be governed both in the weak and strong localization regime by the magnetic diffusion length L_B. This function is derived analytically in closed form as a function of the ratio of the mean free path l, the wire thickness W, and the magnetic length l_B for a two-dimensional wire with specular boundary conditions, as well as for a parabolic wire. The applicability of the analytical formulas to resistance measurements in the strong localization regime is discussed. A comparison with recent experimental results on magnetolocalization is included.Comment: 22 pages, RevTe