110 research outputs found
The Hopf algebra of Feynman graphs in QED
We report on the Hopf algebraic description of renormalization theory of
quantum electrodynamics. The Ward-Takahashi identities are implemented as
linear relations on the (commutative) Hopf algebra of Feynman graphs of QED.
Compatibility of these relations with the Hopf algebra structure is the
mathematical formulation of the physical fact that WT-identities are compatible
with renormalization. As a result, the counterterms and the renormalized
Feynman amplitudes automatically satisfy the WT-identities, which leads in
particular to the well-known identity .Comment: 13 pages. Latex, uses feynmp. Minor corrections; to appear in LM
Improved lower bounds for the ground-state energy of many-body systems
New lower bounds for the binding energy of a quantum-mechanical system of
interacting particles are presented. The new bounds are expressed in terms of
two-particle quantities and improve the conventional bounds of the Hall-Post
type. They are constructed by considering not only the energy in the
two-particle system, but also the structure of the pair wave function. We apply
the formal results to various numerical examples, and show that in some cases
dramatic improvement over the existing bounds is reached.Comment: 29 pages, 5 figures, to be published in Phys. Rev.
New Lower Bound on Fermion Binding Energies
We derive a new lower bound for the ground state energy of N
fermions with total spin S in terms of binding energies of (N-1) fermions. Numerical examples are provided for some simple
short-range or confining potentials.Comment: 4 pages, 1 eps figur
Criticality of the Mean-Field Spin-Boson Model: Boson State Truncation and Its Scaling Analysis
The spin-boson model has nontrivial quantum phase transitions at zero
temperature induced by the spin-boson coupling. The bosonic numerical
renormalization group (BNRG) study of the critical exponents and
of this model is hampered by the effects of boson Hilbert space
truncation. Here we analyze the mean-field spin boson model to figure out the
scaling behavior of magnetization under the cutoff of boson states . We
find that the truncation is a strong relevant operator with respect to the
Gaussian fixed point in and incurs the deviation of the exponents
from the classical values. The magnetization at zero bias near the critical
point is described by a generalized homogeneous function (GHF) of two variables
and . The universal function has a
double-power form and the powers are obtained analytically as well as
numerically. Similarly, is found to be a GHF of
and . In the regime , the truncation produces no effect.
Implications of these findings to the BNRG study are discussed.Comment: 9 pages, 7 figure
Moments of vicious walkers and M\"obius graph expansions
A system of Brownian motions in one-dimension all started from the origin and
conditioned never to collide with each other in a given finite time-interval
is studied. The spatial distribution of such vicious walkers can be
described by using the repulsive eigenvalue-statistics of random Hermitian
matrices and it was shown that the present vicious walker model exhibits a
transition from the Gaussian unitary ensemble (GUE) statistics to the Gaussian
orthogonal ensemble (GOE) statistics as the time is going on from 0 to .
In the present paper, we characterize this GUE-to-GOE transition by presenting
the graphical expansion formula for the moments of positions of vicious
walkers. In the GUE limit , only the ribbon graphs contribute and the
problem is reduced to the classification of orientable surfaces by genus.
Following the time evolution of the vicious walkers, however, the graphs with
twisted ribbons, called M\"obius graphs, increase their contribution to our
expansion formula, and we have to deal with the topology of non-orientable
surfaces. Application of the recent exact result of dynamical correlation
functions yields closed expressions for the coefficients in the M\"obius
expansion using the Stirling numbers of the first kind.Comment: REVTeX4, 11 pages, 1 figure. v.2: calculations of the Green function
and references added. v.3: minor additions and corrections made for
publication in Phys.Rev.
Monotonicity of quantum ground state energies: Bosonic atoms and stars
The N-dependence of the non-relativistic bosonic ground state energy is
studied for quantum N-body systems with either Coulomb or Newton interactions.
The Coulomb systems are "bosonic atoms," with their nucleus fixed, and the
Newton systems are "bosonic stars". In either case there exists some third
order polynomial in N such that the ratio of the ground state energy to the
respective polynomial grows monotonically in N. Some applications of these new
monotonicity results are discussed
Two-dimensional SIR epidemics with long range infection
We extend a recent study of susceptible-infected-removed epidemic processes
with long range infection (referred to as I in the following) from
1-dimensional lattices to lattices in two dimensions. As in I we use hashing to
simulate very large lattices for which finite size effects can be neglected, in
spite of the assumed power law for the
probability that a site can infect another site a distance vector
apart. As in I we present detailed results for the critical case, for the
supercritical case with , and for the supercritical case with . For the latter we verify the stretched exponential growth of the
infected cluster with time predicted by M. Biskup. For we find
generic power laws with dependent exponents in the supercritical
phase, but no Kosterlitz-Thouless (KT) like critical point as in 1-d. Instead
of diverging exponentially with the distance from the critical point, the
correlation length increases with an inverse power, as in an ordinary critical
point. Finally we study the dependence of the critical exponents on in
the regime , and compare with field theoretic predictions. In
particular we discuss in detail whether the critical behavior for
slightly less than 2 is in the short range universality class, as conjectured
recently by F. Linder {\it et al.}. As in I we also consider a modified version
of the model where only some of the contacts are long range, the others being
between nearest neighbors. If the number of the latter reaches the percolation
threshold, the critical behavior is changed but the supercritical behavior
stays qualitatively the same.Comment: 14 pages, including 29 figure
Cluster Monte Carlo and dynamical scaling for long-range interactions
Many spin systems affected by critical slowing down can be efficiently
simulated using cluster algorithms. Where such systems have long-range
interactions, suitable formulations can additionally bring down the
computational effort for each update from O() to O() or even
O(), thus promising an even more dramatic computational speed-up. Here, we
review the available algorithms and propose a new and particularly efficient
single-cluster variant. The efficiency and dynamical scaling of the available
algorithms are investigated for the Ising model with power-law decaying
interactions.Comment: submitted to Eur. Phys. J Spec. Topic
Weakly-Bound Three-Body Systems with No Bound Subsystems
We investigate the domain of coupling constants which achieve binding for a
3-body system, while none of the 2-body subsystems is bound. We derive some
general properties of the shape of the domain, and rigorous upper bounds on its
size, using a Hall--Post decomposition of the Hamiltonian. Numerical
illustrations are provided in the case of a Yukawa potential, using a simple
variational method.Comment: gzipped ps with 11 figures included. To appear in Phys. Rev.
Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov-deGennes random matrices
Spatially and temporally inhomogeneous evolution of one-dimensional vicious
walkers with wall restriction is studied. We show that its continuum version is
equivalent with a noncolliding system of stochastic processes called Brownian
meanders. Here the Brownian meander is a temporally inhomogeneous process
introduced by Yor as a transform of the Bessel process that is a motion of
radial coordinate of the three-dimensional Brownian motion represented in the
spherical coordinates. It is proved that the spatial distribution of vicious
walkers with a wall at the origin can be described by the eigenvalue-statistics
of Gaussian ensembles of Bogoliubov-deGennes Hamiltonians of the mean-field
theory of superconductivity, which have the particle-hole symmetry. We report
that the time evolution of the present stochastic process is fully
characterized by the change of symmetry classes from the type to the type
I in the nonstandard classes of random matrix theory of Altland and
Zirnbauer. The relation between the non-colliding systems of the generalized
meanders of Yor, which are associated with the even-dimensional Bessel
processes, and the chiral random matrix theory is also clarified.Comment: REVTeX4, 16 pages, 4 figures. v2: some additions and correction
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