612,483 research outputs found
Crystal Structure Representations for Machine Learning Models of Formation Energies
We introduce and evaluate a set of feature vector representations of crystal
structures for machine learning (ML) models of formation energies of solids. ML
models of atomization energies of organic molecules have been successful using
a Coulomb matrix representation of the molecule. We consider three ways to
generalize such representations to periodic systems: (i) a matrix where each
element is related to the Ewald sum of the electrostatic interaction between
two different atoms in the unit cell repeated over the lattice; (ii) an
extended Coulomb-like matrix that takes into account a number of neighboring
unit cells; and (iii) an Ansatz that mimics the periodicity and the basic
features of the elements in the Ewald sum matrix by using a sine function of
the crystal coordinates of the atoms. The representations are compared for a
Laplacian kernel with Manhattan norm, trained to reproduce formation energies
using a data set of 3938 crystal structures obtained from the Materials
Project. For training sets consisting of 3000 crystals, the generalization
error in predicting formation energies of new structures corresponds to (i)
0.49, (ii) 0.64, and (iii) 0.37 eV/atom for the respective representations
Quantum Chaos and Thermalization in Isolated Systems of Interacting Particles
This review is devoted to the problem of thermalization in a small isolated
conglomerate of interacting constituents. A variety of physically important
systems of intensive current interest belong to this category: complex atoms,
molecules (including biological molecules), nuclei, small devices of condensed
matter and quantum optics on nano- and micro-scale, cold atoms in optical
lattices, ion traps. Physical implementations of quantum computers, where there
are many interacting qubits, also fall into this group. Statistical
regularities come into play through inter-particle interactions, which have two
fundamental components: mean field, that along with external conditions, forms
the regular component of the dynamics, and residual interactions responsible
for the complex structure of the actual stationary states. At sufficiently high
level density, the stationary states become exceedingly complicated
superpositions of simple quasiparticle excitations. At this stage, regularities
typical of quantum chaos emerge and bring in signatures of thermalization. We
describe all the stages and the results of the processes leading to
thermalization, using analytical and massive numerical examples for realistic
atomic, nuclear, and spin systems, as well as for models with random
parameters. The structure of stationary states, strength functions of simple
configurations, and concepts of entropy and temperature in application to
isolated mesoscopic systems are discussed in detail. We conclude with a
schematic discussion of the time evolution of such systems to equilibrium.Comment: 69 pages, 31 figure
Qubism: self-similar visualization of many-body wavefunctions
A visualization scheme for quantum many-body wavefunctions is described,
which we have termed qubism. Its main property is its recursivity: increasing
the number of qubits reflects in an increase in the image resolution. Thus, the
plots are typically fractal. As examples, we provide images for the ground
states of commonly used Hamiltonians in condensed matter and cold atom physics,
such as Heisenberg or ITF. Many features of the wavefunction, such as
magnetization, correlations and criticality, can be visualized as properties of
the images. In particular, factorizability can be easily spotted, and a way to
estimate the entanglement entropy from the image is provided
A Unifying Model for Representing Time-Varying Graphs
Graph-based models form a fundamental aspect of data representation in Data
Sciences and play a key role in modeling complex networked systems. In
particular, recently there is an ever-increasing interest in modeling dynamic
complex networks, i.e. networks in which the topological structure (nodes and
edges) may vary over time. In this context, we propose a novel model for
representing finite discrete Time-Varying Graphs (TVGs), which are typically
used to model dynamic complex networked systems. We analyze the data structures
built from our proposed model and demonstrate that, for most practical cases,
the asymptotic memory complexity of our model is in the order of the
cardinality of the set of edges. Further, we show that our proposal is an
unifying model that can represent several previous (classes of) models for
dynamic networks found in the recent literature, which in general are unable to
represent each other. In contrast to previous models, our proposal is also able
to intrinsically model cyclic (i.e. periodic) behavior in dynamic networks.
These representation capabilities attest the expressive power of our proposed
unifying model for TVGs. We thus believe our unifying model for TVGs is a step
forward in the theoretical foundations for data analysis of complex networked
systems.Comment: Also appears in the Proc. of the IEEE International Conference on
Data Science and Advanced Analytics (IEEE DSAA'2015
Full Spin and Spatial Symmetry Adapted Technique for Correlated Electronic Hamiltonians: Application to an Icosahedral Cluster
One of the long standing problems in quantum chemistry had been the inability
to exploit full spatial and spin symmetry of an electronic Hamiltonian
belonging to a non-Abelian point group. Here we present a general technique
which can utilize all the symmetries of an electronic (magnetic) Hamiltonian to
obtain its full eigenvalue spectrum. This is a hybrid method based on Valence
Bond basis and the basis of constant z-component of the total spin. This
technique is applicable to systems with any point group symmetry and is easy to
implement on a computer. We illustrate the power of the method by applying it
to a model icosahedral half-filled electronic system. This model spans a huge
Hilbert space (dimension 1,778,966) and in the largest non-Abelian point group.
The molecule has this symmetry and hence our calculation throw light
on the higher energy excited states of the bucky ball. This method can also be
utilized to study finite temperature properties of strongly correlated systems
within an exact diagonalization approach.Comment: 21 pages, 7 figures, abstract rewritten, a few changes in text, to
appear in International Journal of Quantum Chemistr
Multilayer Networks in a Nutshell
Complex systems are characterized by many interacting units that give rise to
emergent behavior. A particularly advantageous way to study these systems is
through the analysis of the networks that encode the interactions among the
system's constituents. During the last two decades, network science has
provided many insights in natural, social, biological and technological
systems. However, real systems are more often than not interconnected, with
many interdependencies that are not properly captured by single layer networks.
To account for this source of complexity, a more general framework, in which
different networks evolve or interact with each other, is needed. These are
known as multilayer networks. Here we provide an overview of the basic
methodology used to describe multilayer systems as well as of some
representative dynamical processes that take place on top of them. We round off
the review with a summary of several applications in diverse fields of science.Comment: 16 pages and 3 figures. Submitted for publicatio
Model tests of cluster separability in relativistic quantum mechanics
A relativistically invariant quantum theory first advanced by Bakamjian and
Thomas has proven very useful in modeling few-body systems. For three particles
or more, this approach is known formally to fail the constraint of cluster
separability, whereby symmetries and conservation laws that hold for a system
of particles also hold for isolated subsystems. Cluster separability can be
restored by means of a recursive construction using unitary transformations,
but implementation is difficult in practice, and the quantitative extent to
which the Bakamjian-Thomas approach violates cluster separability has never
been tested. This paper provides such a test by means of a model of a scalar
probe in a three-particle system for which (1) it is simple enough that there
is a straightforward solution that satisfies Poincar\'e invariance and cluster
separability, and (2) one can also apply the Bakamjian-Thomas approach. The
difference between these calculations provides a measure of the size of the
corrections from the Sokolov construction that are needed to restore cluster
properties. Our estimates suggest that, in models based on nucleon degrees of
freedom, the corrections that restore cluster properties are too small to
effect calculations of observables.Comment: 13 pages, 15 figure
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