437 research outputs found
The Density Matrix Renormalization Group for finite Fermi systems
The Density Matrix Renormalization Group (DMRG) was introduced by Steven
White in 1992 as a method for accurately describing the properties of
one-dimensional quantum lattices. The method, as originally introduced, was
based on the iterative inclusion of sites on a real-space lattice. Based on its
enormous success in that domain, it was subsequently proposed that the DMRG
could be modified for use on finite Fermi systems, through the replacement of
real-space lattice sites by an appropriately ordered set of single-particle
levels. Since then, there has been an enormous amount of work on the subject,
ranging from efforts to clarify the optimal means of implementing the algorithm
to extensive applications in a variety of fields. In this article, we review
these recent developments. Following a description of the real-space DMRG
method, we discuss the key steps that were undertaken to modify it for use on
finite Fermi systems and then describe its applications to Quantum Chemistry,
ultrasmall superconducting grains, finite nuclei and two-dimensional electron
systems. We also describe a recent development which permits symmetries to be
taken into account consistently throughout the DMRG algorithm. We close with an
outlook for future applications of the method.Comment: 48 pages, 17 figures Corrections made to equation 19 and table
Cylindrical algebraic decomposition with equational constraints
Cylindrical Algebraic Decomposition (CAD) has long been one of the most
important algorithms within Symbolic Computation, as a tool to perform
quantifier elimination in first order logic over the reals. More recently it is
finding prominence in the Satisfiability Checking community as a tool to
identify satisfying solutions of problems in nonlinear real arithmetic.
The original algorithm produces decompositions according to the signs of
polynomials, when what is usually required is a decomposition according to the
truth of a formula containing those polynomials. One approach to achieve that
coarser (but hopefully cheaper) decomposition is to reduce the polynomials
identified in the CAD to reflect a logical structure which reduces the solution
space dimension: the presence of Equational Constraints (ECs).
This paper may act as a tutorial for the use of CAD with ECs: we describe all
necessary background and the current state of the art. In particular, we
present recent work on how McCallum's theory of reduced projection may be
leveraged to make further savings in the lifting phase: both to the polynomials
we lift with and the cells lifted over. We give a new complexity analysis to
demonstrate that the double exponent in the worst case complexity bound for CAD
reduces in line with the number of ECs. We show that the reduction can apply to
both the number of polynomials produced and their degree.Comment: Accepted into the Journal of Symbolic Computation. arXiv admin note:
text overlap with arXiv:1501.0446
Entanglement renormalization and topological order
The multi-scale entanglement renormalisation ansatz (MERA) is argued to
provide a natural description for topological states of matter. The case of
Kitaev's toric code is analyzed in detail and shown to possess a remarkably
simple MERA description leading to distillation of the topological degrees of
freedom at the top of the tensor network. Kitaev states on an infinite lattice
are also shown to be a fixed point of the RG flow associated with entanglement
renormalization. All these results generalize to arbitrary quantum double
models.Comment: 6 pages, 17 eps files. v2: References updated, typos corrected.
Includes appendix not in published versio
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