Many-body localization transition with power-law interactions: Statistics of eigenstates

We study spectral and wavefunction statistics for many-body localization transition in systems with long-range interactions decaying as $1/r^\alpha$ with an exponent $\alpha$ satisfying $d \le \alpha \le 2d$, where $d$ is the spatial dimensionality. We refine earlier arguments and show that the system undergoes a localization transition as a function of the rescaled disorder $W^* = W / L^{2d-\alpha} \ln L$, where $W$ is the disorder strength and $L$ the system size. This transition has much in common with that on random regular graphs. We further perform a detailed analysis of the inverse participation ratio (IPR) of many-body wavefunctions, exploring how ergodic behavior in the delocalized phase switches to fractal one at the critical point and on the localized side of the transition. Our analytical results for the scaling of the critical disorder $W$ with the system size $L$ and for the scaling of IPR in the delocalized and localized phases are supported and corroborated by exact diagonalization of spin chains

Anderson localization on random regular graphs

A numerical study of Anderson transition on random regular graphs (RRG) with diagonal disorder is performed. The problem can be described as a tight-binding model on a lattice with N sites that is locally a tree with constant connectivity. In certain sense, the RRG ensemble can be seen as infinite-dimensional ($d\to\infty$) cousin of Anderson model in d dimensions. We focus on the delocalized side of the transition and stress the importance of finite-size effects. We show that the data can be interpreted in terms of the finite-size crossover from small ($N\ll N_c$) to large ($N\gg N_c$) system, where $N_c$ is the correlation volume diverging exponentially at the transition. A distinct feature of this crossover is a nonmonotonicity of the spectral and wavefunction statistics, which is related to properties of the critical phase in the studied model and renders the finite-size analysis highly non-trivial. Our results support an analytical prediction that states in the delocalized phase (and at $N\gg N_c$) are ergodic in the sense that their inverse participation ratio scales as $1/N$

Multifractality of wave functions on a Cayley tree: From root to leaves

We explore the evolution of wave-function statistics on a finite Bethe lattice (Cayley tree) from the central site ("root") to the boundary ("leaves"). We show that the eigenfunction moments P_q=N \left exhibit a multifractal scaling $P_q\propto N^{-\tau_q}$ with the volume (number of sites) $N$ at $N\to\infty$. The multifractality spectrum $\tau_q$ depends on the strength of disorder and on the parameter $s$ characterizing the position of the observation point $i$ on the lattice. Specifically, $s= r/R$, where $r$ is the distance from the observation point to the root, and $R$ is the "radius" of the lattice. We demonstrate that the exponents $\tau_q$ depend linearly on $s$ and determine the evolution of the spectrum with increasing disorder, from delocalized to the localized phase. Analytical results are obtained for the $n$-orbital model with $n \gg 1$ that can be mapped onto a supersymmetric $\sigma$ model. These results are supported by numerical simulations (exact diagonalization) of the conventional ($n=1$) Anderson tight-binding model

Comparison of clustering properties of observed objects and dark matter halos on different mass and spatial scales

We investigate the large-scale distribution of galaxy clusters taken from several X-ray catalogs. Different statistics of clustering like the conditional correlation function (CCF) and the minimal spanning tree (MST) as well as void statistics were used. Clusters show two distinct regimes of clustering: 1) on scales of superclusters (~40/h Mpc) the CCF is represented by a power law; 2) on larger scales a gradual transition to homogeneity (~100/h Mpc) is observed. We also present the correlation analysis of the galaxy distribution taken from DR6 SDSS main galaxy database. In case of galaxies the limiting scales of the different clustering regimes are 1)10-15/h Mpc; 2) 40-50/h Mpc. The differences in the characteristic scales and scaling exponents of the cluster and galaxy distribution can be naturally explained within the theory of biased structure formation. We compared the density contrasts of inhomogeneities in the cluster and galaxy distributions in the SDSS region. The estimation of the relative cluster-galaxy bias gives the value b = 5 +/- 2. The distribution of real clusters is compared to that of simulated (model) clusters (the MareNostrum Universe simulations). We selected a cluster sample from 500/h Mpc simulation box with WMAP3 cosmological parameters and sigma_8 = 0.8. We found a general agreement between the distribution of observed and simulated clusters. The differences are mainly due to the presences of the Shapley supercluster in the observed sample. On the basis of SDSS galaxy sample we study properties of the power law behavior showed by the CCF on small scales. We show that this phenomenon is quite complex, with significant scatter in scaling properties, and characterized by a non-trivial dependence on galaxy properties and environment.Comment: 15 pages, 7 figures, to be published in Proceedings of the conference "Problems of Practical Cosmology", Saint-Petersburg, June 200

Integral norm discretization and related problems

The problem of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure is discussed in the paper. The above problem is studied for elements of finite dimensional spaces. Also, discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of the multivariate trigonometric polynomials with frequencies from a finite set with fixed cardinality. Both new results and a survey of known results are presented

Perturbed Kitaev model: excitation spectrum and long-ranged spin correlations

We developed general approach to the calculation of power-law infrared asymptotics of spin-spin correlation functions in the Kitaev honeycomb model with different types of perturbations. We have shown that in order to find these correlation functions, one can perform averaging of some bilinear forms composed out of free Majorana fermions, and we presented the method for explicit calculation of these fermionic densities. We demonstrated how to derive an effective Hamiltonian for the Majorana fermions, including the effects of perturbations. For specific application of the general theory, we have studied the effect of the Dzyaloshinskii-Moriya anisotropic spin-spin interaction; we demonstrated that it leads, already in the second order over its relative magnitude $D/K$, to a power-law spin correlation functions, and calculated dynamical spin structure factor of the system. We have shown that an external magnetic field $h$ in presence of the DM interaction, opens a gap in the excitation spectrum of magnitude $\Delta \propto D h$.Comment: 12 pages, 3 figure

Storage and conversion of quantum-statistical properties of light in the resonant quantum memory on tripod atomic configuration

We have considered theoretically the feasibility of the broadband quantum memory based on the resonant tripod-type atomic configuration. In this case, the writing of a signal field is carried out simultaneously into two channels, and characterized by an excitation of two spin waves of the atomic ensemble. With simultaneous read out from both channels quantum properties of the original signal are mapped on the retrieval pulse no worse than in the case of memory based on Lambda-type atomic configuration. At the same time new possibilities are opened up for manipulation of quantum states associated with sequential reading out (and/or sequential writing) of signal pulses. For example, the pulse in squeezed state is converted into two partially entangled pulses with partially squeezed quadratures. Alternatively, two independent signal pulses with orthogonal squeezed quadratures can be converted into two entangled pulses.Comment: 14 pages, 4 figure

Ortho and Para Molecules of Water in Electric Field

Stark effect is calculated by the perturbation theory method separately for the ortho and para water molecules. At room temperature, a 30%-difference in the energy change is found for the two species put in electric field. This implies a sorting of the ortho and para water molecules in non-uniform electric fields. The ortho/para water separation is suggested to occur in the course of steam sorption on a solid surface and of large-scale atmospheric processes.Comment: 4 pages, 2 figure

Resonant supercollisions and electron-phonon heat transfer in graphene

We study effects of strong impurities on the heat transfer in a coupled electron-phonon system in disordered graphene. A detailed analysis of the electron-phonon heat exchange assisted by such an impurity through the 'resonant supercollision' mechanism is presented. We further explore the local modification of heat transfer in a weakly disordered graphene due to a resonant scatterer and determine spatial profiles of the phonon and electron temperature around the scatterer under electrical driving. Our results are consistent with recent experimental findings on imaging resonant dissipation from individual atomic defects

Scaling of Temperature Dependence of Charge Mobility in Molecular Holstein Chains

The temperature dependence of a charge mobility in a model DNA based on Holstein Hamiltonian is calculated for 4 types of homogeneous sequences It has turned out that upon rescaling all 4 types are quite similar. Two types of rescaling, i.e. those for low and intermediate temperatures, are found. The curves obtained are approximated on a logarithmic scale by cubic polynomials. We believe that for model homogeneous biopolymers with parameters close to the designed ones, one can assess the value of the charge mobility without carrying out resource-intensive direct simulation, just by using a suitable approximating function.Comment: 14 pages, 5 figures. Differences from the previous version: part of the text about the second dimensionless system is removed; paragraph about model extensions - dispersion in classical chain and solvating - is adde