1,497 research outputs found
Improved recursive computation of clebsch-Gordan coefficients
Fast, accurate, and stable computation of the Clebsch-Gordan (C-G) coefficients is always desirable, for example, in light scattering simulations, the translation of the multipole fields, quantum physics and chemistry. Current recursive methods for computing the C-G coefficients are often unstable for large quantum numbers due to numerical overflow or underflow. In this paper, we present an improved method, called the sign-exponent recurrence, for the recursive computation of C-G coefficients. The result shows that the proposed method can significantly improve the stability of the computation without losing its efficiency, producing accurate values for the C-G coefficients even with very large quantum numbers. (C) 2020 The Author. Published by Elsevier Ltd.Peer reviewe
On the Computation of Clebsch-Gordan Coefficients and the Dilation Effect
We investigate the problem of computing tensor product multiplicities for
complex semisimple Lie algebras. Even though computing these numbers is #P-hard
in general, we show that if the rank of the Lie algebra is assumed fixed, then
there is a polynomial time algorithm, based on counting the lattice points in
polytopes. In fact, for Lie algebras of type A_r, there is an algorithm, based
on the ellipsoid algorithm, to decide when the coefficients are nonzero in
polynomial time for arbitrary rank. Our experiments show that the lattice point
algorithm is superior in practice to the standard techniques for computing
multiplicities when the weights have large entries but small rank. Using an
implementation of this algorithm, we provide experimental evidence for
conjectured generalizations of the saturation property of
Littlewood--Richardson coefficients. One of these conjectures seems to be valid
for types B_n, C_n, and D_n.Comment: 21 pages, 6 table
Pseudo-Character Expansions for U(N)-Invariant Spin Models on CP^{N-1}
We define a set of orthogonal functions on the complex projective space
CP^{N-1}, and compute their Clebsch-Gordan coefficients as well as a large
class of 6-j symbols. We also provide all the needed formulae for the
generation of high-temperature expansions for U(N)-invariant spin models
defined on CP^{N-1}.Comment: 24 pages, no figure
Universal Factorization of Symbols of the First and Second Kinds for SU(2) Group and Their Direct and Exact Calculation and Tabulation
We show that general symbols of the first kind and the second
kind for the group SU(2) can be reformulated in terms of binomial coefficients.
The proof is based on the graphical technique established by Yutsis, et al. and
through a definition of a reduced symbol. The resulting symbols
thereby take a combinatorial form which is simply the product of two factors.
The one is an integer or polynomial which is the single sum over the products
of reduced symbols. They are in the form of summing over the products of
binomial coefficients. The other is a multiplication of all the triangle
relations appearing in the symbols, which can also be rewritten using binomial
coefficients. The new formulation indicates that the intrinsic structure for
the general recoupling coefficients is much nicer and simpler, which might
serves as a bridge for the study with other fields. Along with our newly
developed algorithms, this also provides a basis for a direct, exact and
efficient calculation or tabulation of all the symbols of the SU(2)
group for all range of quantum angular momentum arguments. As an illustration,
we present teh results for the symbols of the first kind.Comment: Add tables and reference
Tensor network states and algorithms in the presence of a global SU(2) symmetry
The benefits of exploiting the presence of symmetries in tensor network
algorithms have been extensively demonstrated in the context of matrix product
states (MPSs). These include the ability to select a specific symmetry sector
(e.g. with a given particle number or spin), to ensure the exact preservation
of total charge, and to significantly reduce computational costs. Compared to
the case of a generic tensor network, the practical implementation of
symmetries in the MPS is simplified by the fact that tensors only have three
indices (they are trivalent, just as the Clebsch-Gordan coefficients of the
symmetry group) and are organized as a one-dimensional array of tensors,
without closed loops. Instead, a more complex tensor network, one where tensors
have a larger number of indices and/or a more elaborate network structure,
requires a more general treatment. In two recent papers, namely (i) [Phys. Rev.
A 82, 050301 (2010)] and (ii) [Phys. Rev. B 83, 115125 (2011)], we described
how to incorporate a global internal symmetry into a generic tensor network
algorithm based on decomposing and manipulating tensors that are invariant
under the symmetry. In (i) we considered a generic symmetry group G that is
compact, completely reducible and multiplicity free, acting as a global
internal symmetry. Then in (ii) we described the practical implementation of
Abelian group symmetries. In this paper we describe the implementation of
non-Abelian group symmetries in great detail and for concreteness consider an
SU(2) symmetry. Our formalism can be readily extended to more exotic symmetries
associated with conservation of total fermionic or anyonic charge. As a
practical demonstration, we describe the SU(2)-invariant version of the
multi-scale entanglement renormalization ansatz and apply it to study the low
energy spectrum of a quantum spin chain with a global SU(2) symmetry.Comment: 32 pages, 37 figure
Full configuration interaction approach to the few-electron problem in artificial atoms
We present a new high-performance configuration interaction code optimally
designed for the calculation of the lowest energy eigenstates of a few
electrons in semiconductor quantum dots (also called artificial atoms) in the
strong interaction regime. The implementation relies on a single-particle
representation, but it is independent of the choice of the single-particle
basis and, therefore, of the details of the device and configuration of
external fields. Assuming no truncation of the Fock space of Slater
determinants generated from the chosen single-particle basis, the code may
tackle regimes where Coulomb interaction very effectively mixes many
determinants. Typical strongly correlated systems lead to very large
diagonalization problems; in our implementation, the secular equation is
reduced to its minimal rank by exploiting the symmetry of the effective-mass
interacting Hamiltonian, including square total spin. The resulting Hamiltonian
is diagonalized via parallel implementation of the Lanczos algorithm. The code
gives access to both wave functions and energies of first excited states.
Excellent code scalability in a parallel environment is demonstrated; accuracy
is tested for the case of up to eight electrons confined in a two-dimensional
harmonic trap as the density is progressively diluted and correlation becomes
dominant. Comparison with previous Quantum Monte Carlo simulations in the
Wigner regime demonstrates power and flexibility of the method.Comment: RevTeX 4.0, 18 pages, 6 tables, 9 postscript b/w figures. Final
version with new material. Section 6 on the excitation spectrum has been
added. Some material has been moved to two appendices, which appear in the
EPAPS web depository in the published versio
Phase diagram and structural properties of a simple model for one-patch particles
We study the thermodynamic and structural properties of a simple, one-patch
fluid model using the reference hypernetted-chain (RHNC) integral equation and
specialized Monte Carlo simulations. In this model, the interacting particles
are hard spheres, each of which carries a single identical,
arbitrarily-oriented, attractive circular patch on its surface; two spheres
attract via a simple square-well potential only if the two patches on the
spheres face each other within a specific angular range dictated by the size of
the patch. For a ratio of attractive to repulsive surface of 0.8, we construct
the RHNC fluid-fluid separation curve and compare with that obtained by Gibbs
ensemble and grand canonical Monte Carlo simulations. We find that RHNC
provides a quick and highly reliable estimate for the position of the
fluid-fluid critical line. In addition, it gives a detailed (though
approximate) description of all structural properties and their dependence on
patch size.Comment: 27 pages, 10 figures, J. Chem. Phys. in pres
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