The research will address the effects of disorder on spectra and dynamics of operators which play a role in the dynamics of quantum systems. The incorporation of even weak but extensive disorder is known to result in delocalization of at least some of the normal modes, and in the creation of pure-point spectral regimes, characterized by a dense collection of eigenvalues corresponding to well separated eigenfunctions. The focus of the proposed research is on the less understood question of the existence of extended states in the presence of disorder, and the corresponding Lebesgue absolutely continuous spectra. Also to be investigated is a possible relation of the spectral gap statistics of local operators incorporating disorder with the known eigenvalue statistics of random matrix ensembles. As the effects of randomness are dimension dependent, the initial goal is to clarify some of the issues in high dimensions, where loop effects are in general more controllable. In the converse direction, a new -fluctuation based- method is envisioned for the derivation and analysis of localization in low dimensions, where the effect is most drastic.
Disorder effects play a key role in the conduction properties of quantum systems modeled by random operators, and are of direct relevance for quantum networks. The work will capitalize on recent progress made by the PI with collaborators, and on his past work on the Anderson localization and on other effects of fluctuations, such as the rounding of fist-order phase transitions in low dimensions, in what is known as the Imry-Ma effect. The subject provides the meeting grounds for techniques involving mathematical analysis with ideas drawn from statistical mechanics, where science is made of disorder. The challenge of shedding light on issues of physics enriches also fields of mathematics.
- NSF Grant DMS-0602360 The work has concerned mathematical analysis of topics which are relevant for issues of quantum physics and statistical mechanics. In particular, attention was directed to disorder effects on quantum spectra and dynamics, and to disorder effects on the nature of phase transitions in classical and quantum system. Following are two main areas in which progress was made. 1. New insight on the formation of extended eigenstates under random potential A bit more than 50 years ago Anderson, Mott, Twose, and other physicists, have proposed that the incorporation of random potential in self adjoint operators of condensed matter physics results in a transition in the nature of the eigenstates of a homogeneous operator from extended states, which facilitate conduction, to localized states, at least in certain energy ranges. The transition is accompanied in the reduction of conduction, and it plays also a role in some of the more exotic properties such as the quantum Hall effect. As linear operators play key roles in many fields, myriads of other implications, and other interesting aspects (such as changes in the spectral gap statistics) have since then been noted of this transition. This has led to the mathematical challenge of explaining the spectral and dynamical properties of operators with homogeneous disorder, a task which requires the combination of analysis with probability. The situation which has emerged from the mathematical studies of the ``Anderson localization'', including the PI’s early work sponsored by this project, is that in a number of different contexts we now have robust mathematical tools for proving and explaining localization (including for few particles with local interactions, though not for a systems of positive density). However only limited progress was made in shedding light on the nature of the extended eigenstates (which facilitate conduction) of operators with random potential. In this context, the PI's recent work (which was carried out jointly with S. Warzel) has provided new understanding of the formation of extended states under disorder, for random operators on tree graphs. The work has provided an answer to questions which have been discussed for about 15 years, and also correct in an interesting detail a picture which has been regarded as valid by physicists. Prior to that, the only case for which existence of continuous spectrum, and extended eigenstates, could be established in the presence of random potential (of homogeneous strength) has been the case of models on regular tree graphs. However major gaps have remained between the regime for which extended states have been established and the regimes for which localization was proven. The resent work has closed this gap. It also led to revision of the phase diagram for the case of bounded random potentials, for which it is found that the mobility edge sets in (in the previously envisioned form) only if the disorder is sufficiently large. Of no lesser interest is the mechanism for the formation of extended states, which can be viewed as based on resonant tunneling between what would locally appear to be localized states. For that the disorder actually plays a constructive role, and the exponential increase in the volume is essential. Such exponential increase is found not only on trees but also in the configuration spaces of interacting particles, and work is in progress now on the possible ramifications of this observation for many particle systems. 2. Rounding of Quantum First Order Transitions in low dimensional systems The PI has also been engaged in the study on the effects of disorder on 1st order phase transitions in homogeneous quantum systems. The main result (derived jointly with J.Lebowitz and R. Greenblatt, a graduate student) is that at low dimension (d=2 in general, and up to d=4 dimensions in the presence of continuous symmetry) the introduction of homogeneous but possibly weak disorder results in the rounding of what initially was a first order phase transition. Once again in this area, the mathematical result have clarified issues of physics while these have still been the subject of ongoing debates. Work continues currently on the interesting nature of the phase transition which may remain there at weak disorder (jointly with J. Hanson, a graduate student). In terms of mathematics, the work has involved the combination of functional analysis, operator theory, probability, and techniques which have originated in mathematical physics. Typically, significant results were published in leading mathematics journals or leading journals dedicated to mathematical physics, and when that was deemed appropriate an abbreviated summary of the results was presented in parallel to the physics community through venues such as Phys. Rev. Lett., or Euro. Phys. Lett.
