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Register a new userAnderson transition on the Bethe lattice: an approach with real energies
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Added by
Francesca Pietracaprina
Publication date
2018
fields
Physics
and research's language is
English
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We study the Anderson model on the Bethe lattice by working directly with propagators at real energies $E$. We introduce a novel criterion for the localization-delocalization transition based on the stability of the population of the propagators, and show that it is consistent with the one obtained through the study of the imaginary part of the self-energy. We present an accurate numerical estimate of the transition point, as well as a concise proof of the asymptotic formula for the critical disorder on lattices of large connectivity, as given in [P.W. Anderson 1958]. We discuss how the forward approximation used in analytic treatments of localization problems fits into this scenario and how one can interpolate between it and the correct asymptotic analysis.
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Classical and quantum conduction on a bond-diluted Bethe lattice is considered. The bond dilution is subject to the constraint that every occupied bond must have at least $k-1$ neighboring occupied bonds, i.e. $k$-core diluted. In the classical case, we find the onset of conduction for $k=2$ is continuous, while for $k=3$, the onset of conduction is discontinuous with the geometric random first-order phase transition driving the conduction transition. In the quantum case, treating each occupied bond as a random scatterer, we find for $k=3$ that the random first-order phase transition in the geometry also drives the onset of quantum conduction giving rise to a new universality class of Anderson localization transitions.
The phase diagram of the random field Ising model on the Bethe lattice with a symmetric dichotomous random field is closely investigated with respect to the transition between the ferromagnetic and paramagnetic regime. Refining arguments of Bleher, Ruiz and Zagrebnov [J. Stat. Phys. 93, 33 (1998)] an exact upper bound for the existence of a unique paramagnetic phase is found which considerably improves the earlier results. Several numerical estimates of transition lines between a ferromagnetic and a paramagnetic regime are presented. The obtained results do not coincide with a lower bound for the onset of ferromagnetism proposed by Bruinsma [Phys. Rev. B 30, 289 (1984)]. If the latter one proves correct this would hint to a region of coexistence of stable ferromagnetic phases and a stable paramagnetic phase.
Using a three-frequency one-dimensional kicked rotor experimentally realized with a cold atomic gas, we study the transport properties at the critical point of the metal-insulator Anderson transition. We accurately measure the time-evolution of an initially localized wavepacket and show that it displays at the critical point a scaling invariance characteristic of this second-order phase transition. The shape of the momentum distribution at the critical point is found to be in excellent agreement with the analytical form deduced from self-consistent theory of localization.
We present a full description of the nonergodic properties of wavefunctions on random graphs without boundary in the localized and critical regimes of the Anderson transition. We find that they are characterized by two critical localization lengths: the largest one describes localization along rare branches and diverges with a critical exponent $ u_parallel=1$ at the transition. The second length, which describes localization along typical branches, reaches at the transition a finite universal value (which depends only on the connectivity of the graph), with a singularity controlled by a new critical exponent $ u_perp=1/2$. We show numerically that these two localization lengths control the finite-size scaling properties of key observables: wavefunction moments, correlation functions and spectral statistics. Our results are identical to the theoretical predictions for the typical localization length in the many-body localization transition, with the same critical exponent. This strongly suggests that the two transitions are in the same universality class and that our techniques could be directly applied in this context.
The boundary condition dependence of the critical behavior for the three dimensional Anderson transition is investigated. A strong dependence of the scaling function and the critical conductance distribution on the boundary conditions is found, while the critical disorder and critical exponent are found to be independent of the boundary conditions.
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