1,355 research outputs found
A Convergent Iterative Solution of the Quantum Double-well Potential
We present a new convergent iterative solution for the two lowest quantum
wave functions and of the Hamiltonian with a quartic
double well potential in one dimension. By starting from a trial function,
which is by itself the exact lowest even or odd eigenstate of a different
Hamiltonian with a modified potential , we construct the Green's
function for the modified potential. The true wave functions, or
, then satisfies a linear inhomogeneous integral equation, in which
the inhomogeneous term is the trial function, and the kernel is the product of
the Green's function times the sum of , the potential difference, and
the corresponding energy shift. By iterating this equation we obtain successive
approximations to the true wave function; furthermore, the approximate energy
shift is also adjusted at each iteration so that the approximate wave function
is well behaved everywhere. We are able to prove that this iterative procedure
converges for both the energy and the wave function at all .Comment: 76 pages, Latex, no figure, 1 tabl
Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials
We prove higher rank analogues of the Razumov--Stroganov sum rule for the
groundstate of the O(1) loop model on a semi-infinite cylinder: we show that a
weighted sum of components of the groundstate of the A_{k-1} IRF model yields
integers that generalize the numbers of alternating sign matrices. This is done
by constructing minimal polynomial solutions of the level 1 U_q(\hat{sl(k)})
quantum Knizhnik--Zamolodchikov equations, which may also be interpreted as
quantum incompressible q-deformations of fractional quantum Hall effect wave
functions at filling fraction nu=1/k. In addition to the generalized
Razumov--Stroganov point q=-e^{i pi/k+1}, another combinatorially interesting
point is reached in the rational limit q -> -1, where we identify the solution
with extended Joseph polynomials associated to the geometry of upper triangular
matrices with vanishing k-th power.Comment: v3: misprint fixed in eq (2.1
Higher-Order Corrections to Instantons
The energy levels of the double-well potential receive, beyond perturbation
theory, contributions which are non-analytic in the coupling strength; these
are related to instanton effects. For example, the separation between the
energies of odd- and even-parity states is given at leading order by the
one-instanton contribution. However to determine the energies more accurately
multi-instanton configurations have also to be taken into account. We
investigate here the two-instanton contributions. First we calculate
analytically higher-order corrections to multi-instanton effects. We then
verify that the difference betweeen numerically determined energy eigenvalues,
and the generalized Borel sum of the perturbation series can be described to
very high accuracy by two-instanton contributions. We also calculate
higher-order corrections to the leading factorial growth of the perturbative
coefficients and show that these are consistent with analytic results for the
two-instanton effect and with exact data for the first 200 perturbative
coefficients.Comment: 7 pages, LaTe
On the sign of kurtosis near the QCD critical point
We point out that the quartic cumulant (and kurtosis) of the order parameter
fluctuations is universally negative when the critical point is approached on
the crossover side of the phase separation line. As a consequence, the kurtosis
of a fluctuating observable, such as, e.g., proton multiplicity, may become
smaller than the value given by independent Poisson statistics. We discuss
implications for the Beam Energy Scan program at RHIC.Comment: 4 pages, 2 figure
External Momentum, Volume Effects, and the Nucleon Magnetic Moment
We analyze the determination of volume effects for correlation functions that
depend on an external momentum. As a specific example, we consider finite
volume nucleon current correlators, and focus on the nucleon magnetic moment.
Because the multipole decomposition relies on SO(3) rotational invariance, the
structure of such finite volume corrections is unrelated to infinite volume
multipole form factors. One can deduce volume corrections to the magnetic
moment only when a zero-mode photon coupling vanishes, as occurs at
next-to-leading order in heavy baryon chiral perturbation theory. To deduce
such finite volume corrections, however, one must assume continuous momentum
transfer. In practice, volume corrections with momentum transfer dependence are
required to address the extraction of the magnetic moment, or other observables
that arise in momentum dependent correlation functions. Additionally we shed
some light on a puzzle concerning differences in lattice form factor data at
equal values of momentum transfer squared.Comment: 21 pages, 5 figures; discussion in Sect. IV C expanded, Figs. now B&W
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Absence of vortex condensation in a two dimensional fermionic XY model
Motivated by a puzzle in the study of two dimensional lattice Quantum
Electrodynamics with staggered fermions, we construct a two dimensional
fermionic model with a global U(1) symmetry. Our model can be mapped into a
model of closed packed dimers and plaquettes. Although the model has the same
symmetries as the XY model, we show numerically that the model lacks the well
known Kosterlitz-Thouless phase transition. The model is always in the gapless
phase showing the absence of a phase with vortex condensation. In other words
the low energy physics is described by a non-compact U(1) field theory. We show
that by introducing an even number of layers one can introduce vortex
condensation within the model and thus also induce a KT transition.Comment: 5 pages, 5 figure
The Renormalization Group and the Superconducting Susceptibility of a Fermi Liquid
A free Fermi gas has, famously, a superconducting susceptibility that
diverges logarithmically at zero temperature. In this paper we ask whether this
is still true for a Fermi liquid and find that the answer is that it does {\it
not}. From the perspective of the renormalization group for interacting
fermions, the question arises because a repulsive interaction in the Cooper
channel is a marginally irrelevant operator at the Fermi liquid fixed point and
thus is also expected to infect various physical quantities with logarithms.
Somewhat surprisingly, at least from the renormalization group viewpoint, the
result for the superconducting susceptibility is that two logarithms are not
better than one. In the course of this investigation we derive a
Callan-Symanzik equation for the repulsive Fermi liquid using the
momentum-shell renormalization group, and use it to compute the long-wavelength
behavior of the superconducting correlation function in the emergent low-energy
theory. We expect this technique to be of broader interest.Comment: 9 pages, 2 figure
The arctic curve of the domain-wall six-vertex model in its anti-ferroelectric regime
An explicit expression for the spatial curve separating the region of
ferroelectric order (`frozen' zone) from the disordered one (`temperate' zone)
in the six-vertex model with domain wall boundary conditions in its
anti-ferroelectric regime is obtained.Comment: 12 pages, 1 figur
A renormalized large-n solution of the U(n) x U(n) linear sigma model in the broken symmetry phase
Dyson-Schwinger equations for the U(n) x U(n) symmetric matrix sigma model
reformulated with two auxiliary fields in a background breaking the symmetry to
U(n) are studied in the so-called bare vertex approximation. A large n solution
is constructed under the supplementary assumption so that the scalar components
are much heavier than the pseudoscalars. The renormalizability of the solution
is investigated by explicit construction of the counterterms.Comment: RevTeX4, 14 pages, 2 figures. Version published in Phys. Rev.
Conformal invariance in three-dimensional rotating turbulence
We examine three--dimensional turbulent flows in the presence of solid-body
rotation and helical forcing in the framework of stochastic Schramm-L\"owner
evolution curves (SLE). The data stems from a run on a grid of points,
with Reynolds and Rossby numbers of respectively 5100 and 0.06. We average the
parallel component of the vorticity in the direction parallel to that of
rotation, and examine the resulting field for
scaling properties of its zero-value contours. We find for the first time for
three-dimensional fluid turbulence evidence of nodal curves being conformal
invariant, belonging to a SLE class with associated Brownian diffusivity
. SLE behavior is related to the self-similarity of the
direct cascade of energy to small scales in this flow, and to the partial
bi-dimensionalization of the flow because of rotation. We recover the value of
with a heuristic argument and show that this value is consistent with
several non-trivial SLE predictions.Comment: 4 pages, 3 figures, submitted to PR
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