On the homomorphic cluster CPA

Among the various extensions of the CPA, the homomorphic cluster CPA (HCPA) was presented recently as the only approximation (i) having the correct analytical properties, (ii) taking multiple scattering into account and (iii) preserving the symmetry of the original lattice. Though very appealing mathematically, we show in this paper that it does not give the right position for the impurity states in the dilute limit with non zero diagonal disorder. Consequently, even if HCPA satisfies points (i) to (iii) and even if it gives the correct dilute limit for purely off-diagonal disorder, it is to be rejected as a method to study systems with non negligible diagonal disorder.     

Systematic approximations in disordered systems

We discuss two systematic approximation schemes for disordered systems (random lattices) by extending the single site CPA. In the first scheme all self energy diagrams up to those includingν scattering centers are summed up. Thisν-center approximation has been discussed before and is here derived using a straightforward projection operator formalism. The second scheme is a “true” cluster-CPA in which the scattering from an arbitrarily large cluster of neighbouring atoms (ν-cluster) embedded in a random crystal is treated just in the same way as in the single site CPA. In contrast to other cluster formulations our theory preserves translational invariance and determines the self energy in a natural way. The two schemes are compared with each other concerning their practical applicability.     

Cluster treatment of disordered systems

We formulate a cluster approximation to treat disordered systems following closely the well-known CPA. In contrast to other cluster theories a minimum of assumptions provides a selfconsistent theory which retains translational invariance and determines the structure of the self-energy uniquely.     

A homomorphism theorem for the Poincaré group

Jordan demonstrated that the group of homogeneous transformations of degree one in ?⁵ is homomorphic to the symmetry group of the Einstein-Maxwell equations in vacuum. It is shown that the Jordan homomorphism theorem is also applicable to the inhomogeneous general linear group. Consequently, the Poincaré group is homomorphic to the group of homogeneous transformations of degree one in a five-dimensional space.     

Cluster coherent potential approximation for disordered photonic crystals using photonic Wannier functions

We present an investigation of disordered photonic crystals (PhCs) based on the combination of photonic Wannier functions with the concept of the coherent potential approximation (CPA). In particular, we provide the theoretical foundation of a real-space cluster CPA that is causal, enforces the proper symmetries of the effective medium, and includes effects of multiple scattering of the same and nearby defects, which is essential for strong defects. Based on this, we present results for the density of states of disordered PhCs for different types of disorder. Our results are thus relevant to such diverse areas as random lasing and the analysis of fabricational imperfections in PhCs.     

Cluster approaches to random alloys: An appraisal

Some of the cluster extensions of the coherent potential approximation (CPA) based on the effective medium theory have been critically studied with respect to the decoupling schemes involved in them. Their computational tractability has been examined and it has been found that theself-consistent calculations in three-dimensional systems are immensely difficult to perform. A self-consistent calculation has been reported for simple cubic lattices with diagonal and off-diagonal disorder using a pair-CPA method. A significant finding of the paper is that it has been shown thatnon-analyticities are a general feature of extensions of CPA within multiple scattering framework. The non-analyticities were reported several times but a general proof of their existence was not noticed. It was also believed that the so-called molecular—CPA is analytic, this has been shown to be wrong here. The density of states results with off-diagonal randomness have been qualitatively understood to yield some information about the influence of off-diagonal randomness on Anderson localisation of an electron.     

Electron correlations and many-body techniques in magnetism

Recent progress in the theory of magnetism and electron correlations is reviewed to clarify the theories developed in the last decade and their mutual relations. A historical development of the theory of magnetism is outlined, and the dynamical coherent potential approximation (CPA) which completely takes account of the dynamical spin and charge fluctuations within the single-site approximation is introduced. Both the dynamical effects on various magnetic properties and the many-body band structure are shown to be explained on the same footing. It is shown that the dynamical CPA is equivalent to the other single-site theories of strongly correlated electrons: the many-body CPA, the dynamical mean-field theory (DMFT), and the projection operator method CPA (PM-CPA). These theories are elucidated with use of a common concept of effective medium or coherent potential. The effects of orbital degeneracy and the realistic calculation scheme are discussed with an emphasis on Hund’s rule coupling. Non-local theories of magnetism and electron correlations which go beyond the single-site approximation are presented. They include the molecular dynamics approach to the magnetic short range order, the dynamical cluster methods as a direct extension of the DMFT, and the self-consistent projection operator approach as an extension of the PM-CPA with use of the incremental cluster expansion. The current problems of their approaches and their future perspective are discussed.     

Assessment of the coherent potential approximation for the absorption spectra of a one-dimensional Frenkel exciton system with a Gaussian distribution of fluctuations in the optical transition frequency

We investigate the accuracy of the coherent potential approximation (CPA) for the optical absorption spectra of a one-dimensional Frenkel exciton system with nearest-neighbor interactions and a Gaussian distribution of fluctuations in the optical transition frequency (diagonal Gaussian disorder). Our earlier studies have established that the CPA gives highly accurate results for the integral of the density of states of this system. In this paper we compare the CPA results for the normalized optical absorption with the finite-array calculations of Schreiber and Toyozawa and Schreiber for the same model. It is found that the CPA results for the absorption are in good agreement with their findings. It is pointed out that an expansion of the density of states in terms of the eigenstates of the ideal (no disorder) array contains a term proportional to the normalized absorption. Since the density of states is accurately approximated by the CPA, the presence of this term explains the success of the CPA in reproducing the absorption spectra. Our findings support the use of the Gaussian disorder model to interpret the absorption spectra of one and quasi-one dimensional exciton systems.     

Towards Secure Big Data Analysis via Fully Homomorphic Encryption Algorithms

Privacy-preserving techniques allow private information to be used without compromising privacy. Most encryption algorithms, such as the Advanced Encryption Standard (AES) algorithm, cannot perform computational operations on encrypted data without first applying the decryption process. Homomorphic encryption algorithms provide innovative solutions to support computations on encrypted data while preserving the content of private information. However, these algorithms have some limitations, such as computational cost as well as the need for modifications for each case study. In this paper, we present a comprehensive overview of various homomorphic encryption tools for Big Data analysis and their applications. We also discuss a security framework for Big Data analysis while preserving privacy using homomorphic encryption algorithms. We highlight the fundamental features and tradeoffs that should be considered when choosing the right approach for Big Data applications in practice. We then present a comparison of popular current homomorphic encryption tools with respect to these identified characteristics. We examine the implementation results of various homomorphic encryption toolkits and compare their performances. Finally, we highlight some important issues and research opportunities. We aim to anticipate how homomorphic encryption technology will be useful for secure Big Data processing, especially to improve the utility and performance of privacy-preserving machine learning.     

Temporal contrast improvement in chirped pulse amplification systems by a four-grating compressor and by spectral modifications

The quest for higher peak focused intensities and better temporal contrast drives one to improve the design of all possible stages of the chirped pulse amplification (CPA) system. In this paper, we have analyzed the role of dispersion and spectral profile on the temporal shape and contrast ratio of the output pulse of a CPA system. The simulations indicate that an initial sech² or a Gaussian pulse in the CPA system is best for a good contrast ratio. Incorporating a four-grating based pulse compressor in the CPA system improves the contrast as well as provides the flexibility to compensate the dispersion upto the fourth order. The simulations also detail the effect of spectral profile tailoring on the contrast ratio and peak power.     

Short range order and the metal-insulator transition in disordered systems

The effect of short range order on the electronic density of states, the conductivity, and localization is calculated for a model binary alloy in the simple cubic configuration using a cluster extension of CPA. In the presence of short range order a metal insulator transition takes place as the concentration passes through a critical range.     

Influence of long range order on electrical resistivity in alloys

We have established the CPA expression of electrical conductivity of a one orbital tight binding band model alloy in terms of the long range order parameter. We investigate its variation with temperature from the completely ordered alloy at T = 0 K till the order-disorder transition in terms of band parameters: scattering potential and filling of the band.     

Two-Party Privacy-Preserving Set Intersection with FHE

A two-party private set intersection allows two parties, the client and the server, to compute an intersection over their private sets, without revealing any information beyond the intersecting elements. We present a novel private set intersection protocol based on Shuhong Gao’s fully homomorphic encryption scheme and prove the security of the protocol in the semi-honest model. We also present a variant of the protocol which is a completely novel construction for computing the intersection based on Bloom filter and fully homomorphic encryption, and the protocol’s complexity is independent of the set size of the client. The security of the protocols relies on the learning with errors and ring learning with error problems. Furthermore, in the cloud with malicious adversaries, the computation of the private set intersection can be outsourced to the cloud service provider without revealing any private information.     

Phonon density of states of ann-component alloy

We have generalized the coherent potential approximation (CPA) of Tripathi and Behera to the case of ann-component alloy. It is seen that then-component CPA density of states reproduces the binary, ternary quartenary alloys etc when the appropriate limits are adopted.     

Random-Cluster Representation for the Blume–Capel Model

We present in this paper a way to perform the mapping of the spin-1 Blume–Capel model into a random-cluster model, and analyze thermodynamic properties of the former model in terms of geometric properties of clusters generated in the random-cluster representation. It is shown that there are two different relevant types of cluster, and that one of them is the exact analogue of the type of cluster generated in the Ising model. We use this result to derive expressions for thermodynamical properties on the second-order transition line which are equivalent to the ones found in the Ising model. The other type of cluster is responsible for the first-order transitions, and we may see the tricritical point as a point where both types of cluster compete on the same footing.     

One-particle properties of compositionally modulated alloys in the coherent potential approximation

We propose a simple generalization of the original single-site coherent potential approximation (CPA) applicable to compositionally modulated alloys. Although the original single-site CPA gives an almost structureless density of states, we find that for strong modulation and relatively strong scattering our generalization of the CPA yields densities of states which exhibit considerable structure and which represent the exact densities of states remarkably well. In this sense our generalization is more accurate than the original CPA from which it was obtained.     

Effect Algebras Are Not Adequate Models for Quantum Mechanics

We show that an effect algebra E possess an order-determining set of states if and only if E is semiclassical; that is, E is essentially a classical effect algebra. We also show that if E possesses at least one state, then E admits hidden variables in the sense that E is homomorphic to an MV-algebra that reproduces the states of E. Both of these results indicate that we cannot distinguish between a quantum mechanical effect algebra and a classical one. Hereditary properties of sharpness and coexistence are discussed and the existence of {0,1} and dispersion-free states are considered. We then discuss a stronger structure called a sequential effect algebra (SEA) that we believe overcomes some of the inadequacies of an effect algebra. We show that a SEA is semiclassical if and only if it possesses an order-determining set of dispersion-free states.     

Localization in disordered alloys: Single site CPA

A criterion for localization first put forward by Anderson is evaluated using the single site CPA. Using phase space arguments, it is shown that in one and two dimensions single site CPA predicts that all states are localized. In three dimensions the prediction is that all states are extended. Other localization criteria appropriate to CPA are discussed and explicitly compared.     

Electron localisation in systems with purely off-diagonal randomness

A probability distribution for the off-diagonal matrix elements v_nm of the tight binding Hamiltonian is assumed to be $\text{P(v}{}_{\mathrm{nm}}\text{) =}\text{1}\text{Wv}{}_{\mathrm{nm}}$ for $\text{e}{}^{\mathrm{<mtext>?</mtext><mtext>W</mtext><mtext>2</mtext>}}\text{≤ v}{}_{\mathrm{nm}}\text{/v}{}_0\text{≤ e}{}^{\mathrm{<mtext>?</mtext><mtext>W</mtext><mtext>2</mtext>}}$ and P(v_nm = 0 otherwise. A homomorphic cluster CPA with the L(E) criterion is used to study localisation in a simple cubic lattice and a computer simulation is used to study a square lattice with the participation-ratio criterion. It is found, in both cases, that Anderson's transition takes place for a critical degree of disorder.     

High order dispersion control for femtosecond CPA lasers

Taking advantage of the temperature dependence of the refractive index, an arrangement is proposed for thermal control of dispersion of a chirped pulse amplification (CPA) laser. A glass slab, inserted into the Fourier plane of a stretcher or a compressor, having a spatially varying temperature profile across the beam ensures continuous variation of the spectral phase shift of the pulses. Model calculations are carried out to investigate the feasibility of the arrangement. As a demonstration, simple temperature profiles are created which compensate for the material dispersion of the thermal slab. In a proof of principle experiment it is proved that changes of spectral phase of femtosecond pulses follow the spatially varying temperature profile of a BK7 slab inserted into the compressor of a CPA system. Such a thermal slab is lossless, has a large spectral range, introduces no pixellation and exhibits a high damage threshold. Since it is easy to build into either the stretcher or the compressor of existing CPA lasers, it may become a promising candidate for high order dispersion compensation of high-power femtosecond laser systems. PACS 42.65.Re; 42.79.-e; 42.60.By