919 research outputs found
Tensor-Structured Coupled Cluster Theory
We derive and implement a new way of solving coupled cluster equations with
lower computational scaling. Our method is based on decomposition of both
amplitudes and two electron integrals, using a combination of tensor
hypercontraction and canonical polyadic decomposition. While the original
theory scales as with respect to the number of basis functions, we
demonstrate numerically that we achieve sub-millihartree difference from the
original theory with scaling. This is accomplished by solving directly
for the factors that decompose the cluster operator. The proposed scheme is
quite general and can be easily extended to other many-body methods
A formally exact field theory for classical systems at equilibrium
We propose a formally exact statistical field theory for describing classical
fluids with ingredients similar to those introduced in quantum field theory. We
consider the following essential and related problems : i) how to find the
correct field functional (Hamiltonian) which determines the partition function,
ii) how to introduce in a field theory the equivalent of the indiscernibility
of particles, iii) how to test the validity of this approach. We can use a
simple Hamiltonian in which a local functional transposes, in terms of fields,
the equivalent of the indiscernibility of particles. The diagrammatic expansion
and the renormalization of this term is presented. This corresponds to a non
standard problem in Feynman expansion and requires a careful investigation.
Then a non-local term associated with an interaction pair potential is
introduced in the Hamiltonian. It has been shown that there exists a mapping
between this approach and the standard statistical mechanics given in terms of
Mayer function expansion. We show on three properties (the chemical potential,
the so-called contact theorem and the interfacial properties) that in the field
theory the correlations are shifted on non usual quantities. Some perspectives
of the theory are given.Comment: 20 pages, 8 figure
Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach
Using large arguments, we propose a scheme for calculating the two-point
eigenvector correlation function for non-normal random matrices in the large
limit. The setting generalizes the quaternionic extension of free
probability to two-point functions. In the particular case of biunitarily
invariant random matrices, we obtain a simple, general expression for the
two-point eigenvector correlation function, which can be viewed as a further
generalization of the single ring theorem. This construction has some striking
similarities to the freeness of the second kind known for the Hermitian
ensembles in large . On the basis of several solved examples, we conjecture
two kinds of microscopic universality of the eigenvectors - one in the bulk,
and one at the rim. The form of the conjectured bulk universality agrees with
the scaling limit found by Chalker and Mehlig [JT Chalker, B Mehlig, PRL,
\textbf{81}, 3367 (1998)] in the case of the complex Ginibre ensemble.Comment: 20 pages + 4 pages of references, 12 figs; v2: typos corrected, refs
added; v3: more explanator
Obstructions to combinatorial formulas for plethysm
Motivated by questions of Mulmuley and Stanley we investigate
quasi-polynomials arising in formulas for plethysm. We demonstrate, on the
examples of and , that these need not be counting
functions of inhomogeneous polytopes of dimension equal to the degree of the
quasi-polynomial. It follows that these functions are not, in general, counting
functions of lattice points in any scaled convex bodies, even when restricted
to single rays. Our results also apply to special rectangular Kronecker
coefficients.Comment: 7 pages; v2: Improved version with further reaching counterexamples;
v3: final version as in Electronic Journal of Combinatoric
The Saxl Conjecture and the Dominance Order
In 2012 Jan Saxl conjectured that all irreducible representations of the
symmetric group occur in the decomposition of the tensor square of the
irreducible representation corresponding to the staircase partition. We make
progress on this conjecture by proving the occurrence of all those irreducibles
which correspond to partitions that are comparable to the staircase partition
in the dominance order. Moreover, we use our result to show the occurrence of
all irreducibles corresponding to hook partitions. This generalizes results by
Pak, Panova, and Vallejo from 2013.Comment: 11 page
Screened-interaction expansion for the Hubbard model and determination of the quantum Monte Carlo Fermi surface
We develop a systematic self-consistent perturbative expansion for the self
energy of Hubbard-like models. The interaction lines in the Feynman diagrams
are dynamically screened by the charge fluctuations in the system. Although the
formal expansion is exact-assuming that the model under the study is
perturbative-only if diagrams to all orders are included, it is shown that for
large-on-site-Coulomb-repulsion-U systems weak-coupling expansions to a few
orders may already converge. We show that the screened interaction for the
large-U system can be vanishingly small at a certain intermediate electron
filling; and it is found that our approximation for the imaginary part of the
one-particle self energy agrees well with the QMC results in the low energy
scales at this particular filling. But, the usefulness of the approximation is
hindered by the fact that it has the incorrect filling dependence when the
filling deviates from this value. We also calculate the exact QMC Fermi
surfaces for the two-dimensional (2-D) Hubbard model for several fillings. Our
results near half filling show extreme violation of the concepts of the band
theory; in fact, instead of growing, Fermi surface vanishes when doped toward
the half-filled Mott-Hubbard insulator. Sufficiently away from half filling,
noninteracting-like Fermi surfaces are recovered. These results combined with
the Luttinger theorem might show that diagrammatic expansions for the
nearly-half-filled Hubbard model are unlikely to be possible; however, the
nonperturbative part of the solution seems to be less important as the filling
gradually moves away from one half. Results for the 2-D one-band Hubbard model
for several hole dopings are presented. Implications of this study for the
high-temperature superconductors are also discussed.Comment: 11 pages, 12 eps figures embedded, REVTeX, submitted to Phys. Rev. B;
(v2) minor revisions, scheduled for publication on November 1
Spectra of large time-lagged correlation matrices from Random Matrix Theory
We analyze the spectral properties of large, time-lagged correlation matrices
using the tools of random matrix theory. We compare predictions of the
one-dimensional spectra, based on approaches already proposed in the
literature. Employing the methods of free random variables and diagrammatic
techniques, we solve a general random matrix problem, namely the spectrum of a
matrix , where is an Gaussian random
matrix and is \textit{any} , not necessarily symmetric
(Hermitian) matrix. As a particular application, we present the spectral
features of the large lagged correlation matrices as a function of the depth of
the time-lag. We also analyze the properties of left and right eigenvector
correlations for the time-lagged matrices. We positively verify our results by
the numerical simulations.Comment: 44 pages, 11 figures; v2 typos corrected, final versio
Stability of Kronecker coefficients via discrete tomography
In this paper we give a new sufficient condition for a general stability of
Kronecker coefficients, which we call it additive stability. It was motivated
by a recent talk of J. Stembridge at the conference in honor of Richard P.
Stanley's 70th birthday, and it is based on work of the author on discrete
tomography along the years. The main contribution of this paper is the
discovery of the connection between additivity of integer matrices and
stability of Kronecker coefficients. Additivity, in our context, is a concept
from discrete tomography. Its advantage is that it is very easy to produce lots
of examples of additive matrices and therefore of new instances of stability
properties. We also show that Stembridge's hypothesis and additivity are
closely related, and prove that all stability properties of Kronecker
coefficients discovered before fit into additive stability.Comment: 22 page
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