130,148 research outputs found
Efficient Symmetry Reduction and the Use of State Symmetries for Symbolic Model Checking
One technique to reduce the state-space explosion problem in temporal logic
model checking is symmetry reduction. The combination of symmetry reduction and
symbolic model checking by using BDDs suffered a long time from the
prohibitively large BDD for the orbit relation. Dynamic symmetry reduction
calculates representatives of equivalence classes of states dynamically and
thus avoids the construction of the orbit relation. In this paper, we present a
new efficient model checking algorithm based on dynamic symmetry reduction. Our
experiments show that the algorithm is very fast and allows the verification of
larger systems. We additionally implemented the use of state symmetries for
symbolic symmetry reduction. To our knowledge we are the first who investigated
state symmetries in combination with BDD based symbolic model checking
Exploring corner transfer matrices and corner tensors for the classical simulation of quantum lattice systems
In this paper we explore the practical use of the corner transfer matrix and
its higher-dimensional generalization, the corner tensor, to develop tensor
network algorithms for the classical simulation of quantum lattice systems of
infinite size. This exploration is done mainly in one and two spatial
dimensions (1d and 2d). We describe a number of numerical algorithms based on
corner matri- ces and tensors to approximate different ground state properties
of these systems. The proposed methods make also use of matrix product
operators and projected entangled pair operators, and naturally preserve
spatial symmetries of the system such as translation invariance. In order to
assess the validity of our algorithms, we provide preliminary benchmarking
calculations for the spin-1/2 quantum Ising model in a transverse field in both
1d and 2d. Our methods are a plausible alternative to other well-established
tensor network approaches such as iDMRG and iTEBD in 1d, and iPEPS and TERG in
2d. The computational complexity of the proposed algorithms is also considered
and, in 2d, important differences are found depending on the chosen simulation
scheme. We also discuss further possibilities, such as 3d quantum lattice
systems, periodic boundary conditions, and real time evolution. This discussion
leads us to reinterpret the standard iTEBD and iPEPS algorithms in terms of
corner transfer matrices and corner tensors. Our paper also offers a
perspective on many properties of the corner transfer matrix and its
higher-dimensional generalizations in the light of novel tensor network
methods.Comment: 25 pages, 32 figures, 2 tables. Revised version. Technical details on
some of the algorithms have been moved to appendices. To appear in PR
Generating derivative structures: Algorithm and applications
We present an algorithm for generating all derivative superstructures--for
arbitrary parent structures and for any number of atom types. This algorithm
enumerates superlattices and atomic configurations in a geometry-independent
way. The key concept is to use the quotient group associated with each
superlattice to determine all unique atomic configurations. The run time of the
algorithm scales linearly with the number of unique structures found. We show
several applications demonstrating how the algorithm can be used in materials
design problems. We predict an altogether new crystal structure in Cd-Pt and
Pd-Pt, and several new ground states in Pd-rich and Pt-rich binary systems
Fusion process of Lennard-Jones clusters: global minima and magic numbers formation
We present a new theoretical framework for modelling the fusion process of
Lennard-Jones (LJ) clusters. Starting from the initial tetrahedral cluster
configuration, adding new atoms to the system and absorbing its energy at each
step, we find cluster growing paths up to the cluster sizes of up to 150 atoms.
We demonstrate that in this way all known global minima structures of the
LJ-clusters can be found. Our method provides an efficient tool for the
calculation and analysis of atomic cluster structure. With its use we justify
the magic number sequence for the clusters of noble gas atoms and compare it
with experimental observations. We report the striking correspondence of the
peaks in the dependence on cluster size of the second derivative of the binding
energy per atom calculated for the chain of LJ-clusters based on the
icosahedral symmetry with the peaks in the abundance mass spectra
experimentally measured for the clusters of noble gas atoms. Our method serves
an efficient alternative to the global optimization techniques based on the
Monte-Carlo simulations and it can be applied for the solution of a broad
variety of problems in which atomic cluster structure is important.Comment: 47 pages, MikTeX, 17 figure
- …