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We study the effect of localized modes in lattices of size N with parity-time (PT) symmetry. Such modes are arranged in pairs of quasidegenerate levels with splitting delta approximately exp(-N/xi) where xi is their localization length. The level "evolution" with respect to the PT breaking parameter gamma shows a cascade of bifurcations during which a pair of real levels becomes complex. The spontaneous PT symmetry breaking occurs at gammaPT approximately min{delta}, thus resulting in an exponentially narrow exact PT phase. As N/xi decreases, it becomes more robust with gammaPT approximately 1/N2 and the distribution P(gammaPT) changes from log-normal to semi-Gaussian. Our theory can be tested in the frame of optical lattices.

A random matrix ensemble incorporating both GUE and Poisson level statistics while respecting U(N) invariance is proposed and shown to be equivalent to a system of noninteracting, confined, one dimensional fermions at finite temperature.PACS numbers: 05.45+b, 72.15-Rn, 03.65-w (a) Permanent address:

The intensity pattern generated by a monochromatic point source in a random medium is studied. Thc intensity-intensity correlation function ls calculated and lt ls sh0%n that. the intensity, as a function of coordinate, exhibits large fluctuations (the speckle pattern). The sensitivity of this speckle pattern to small changes in the source frequency is also studied. PACS numbers: 42.20.Ji, 71.55.Jv A wave propagating in a random medium undergoes multiple scattering from the inhomogeneities.The scattered waves interfere with each other and, as a result, a certain intensity pattern is formed. In a random medium, as opposed, e.g. , to a crystal, one would, naively, expect an efficient averaging process and therefore a fairly smooth intensity pattern, with only small intensity fluctuations.Instead, however, one finds a highly irregular pattern, with large intensity changes over short distances. The irregularities are not due to noise. Each microscopic realization of the random medium, i.e. , each sample of the statistical ensemble, displays its own pattern -a "fingerprint" which reflects the specific arrangement of the inhomogeneities (impurities) in that sample. This phenomenon is quite familiar in optics where it is termed "a speckle pattern" and usually refers to an intensity pattern formed on a screen by light reflected from a rough surface. Below, this term is used in a somewhat broader sense and refers to an intensity pattern formed in the bulk of a disordered medium when a wave (electromagnetic, acoustic, or an electron wave) propagates through it.There exists a huge literature on the subject. ' 3 In early work, usually certain assumptions were made directly on the statistics of the scattered light (rather than on the statistical properties of the disordered medium). The "first principles" work, i.e. , that which tries to derive properties of the speckle patterns directly from the wave equation, is mostly limited to smooth inhomogeneities (the wavelength much shorter than the characteristic inhomogeneity size). 2~T he subject of light propagation in random media has been recently given a new boost as a result of a number of experiments. These experiments revealed an enhanced backscattering, in combination with large intensity fluctuations6 and high sensitivity of the speckle pattern to relatively small changes of the source frequency. 7 Similar phenomena exist, and are being extensively studied, in the electron transport in disordered systems. The point is that as long as the sample size is smaller than the inelastic scattering length (the mesoseopic regime), an electron propagates coherently through the entire sample and, thus, takes a "fingerprint" of the specifi, for that sample, impurity arrangement.This manifests itself in various interference phenomena and in extreme sensitivity of the conductance to small changes of various factors. s '2 The purpose of the present work is to calculate some properties of speckle patterns, specifically the intensity correlation function, for a scalar field. Such a fiel...

The review deals with the physics of cold atomic gases in the presence of disorder. The emphasis is on the theoretical developments, although several experiments are also briefly discussed. The review is intended to be pedagogical, explaining the basics and, for some of the topics, presenting rather detailed calculations . arXiv:1112.5736v1 [cond-mat.quant-gas]