48 research outputs found
Dyson instability for 2D nonlinear O(N) sigma models
For lattice models with compact field integration (nonlinear sigma models
over compact manifolds and gauge theories with compact groups) and satisfying
some discrete symmetry, the change of sign of the bare coupling g_0^2 at zero
results in a mere discontinuity in the average energy rather than the
catastrophic instability occurring in theories with integration over
arbitrarily large fields. This indicates that the large order of perturbative
series and the non-perturbative contributions should have unexpected features.
Using the large-N limit of 2-dimensional nonlinear O(N) sigma model, we discuss
the complex singularities of the average energy for complex 't Hooft coupling
lambda= g_0^2N. A striking difference with the usual situation is the absence
of cut along the negative real axis. We show that the zeros of the partition
function can only be inside a clover shape region of the complex lambda plane.
We calculate the density of states and use the result to verify numerically the
statement about the zeros. We propose dispersive representations of the
derivatives of the average energy for an approximate expression of the
discontinuity. The discontinuity is purely non-perturbative and contributions
at small negative coupling in one dispersive representation are essential to
guarantee that the derivatives become exponentially small when lambda -> 0^+ We
discuss the implications for gauge theories.Comment: 10 pages, 10 figures uses revte
Asymptotically Universal Crossover in Perturbation Theory with a Field Cutoff
We discuss the crossover between the small and large field cutoff (denoted
x_{max}) limits of the perturbative coefficients for a simple integral and the
anharmonic oscillator. We show that in the limit where the order k of the
perturbative coefficient a_k(x_{max}) becomes large and for x_{max} in the
crossover region, a_k(x_{max}) is proportional to the integral from -infinity
to x_{max} of e^{-A(x-x_0(k))^2}dx. The constant A and the function x_0(k) are
determined empirically and compared with exact (for the integral) and
approximate (for the anharmonic oscillator) calculations. We discuss how this
approach could be relevant for the question of interpolation between
renormalization group fixed points.Comment: 15 pages, 11 figs., improved and expanded version of hep-th/050304
Improved Conformal Mapping of the Borel Plane
The conformal mapping of the Borel plane can be utilized for the analytic
continuation of the Borel transform to the entire positive real semi-axis and
is thus helpful in the resummation of divergent perturbation series in quantum
field theory. We observe that the rate of convergence can be improved by the
application of Pad\'{e} approximants to the Borel transform expressed as a
function of the conformal variable, i.e. by a combination of the analytic
continuation via conformal mapping and a subsequent numerical approximation by
rational approximants. The method is primarily useful in those cases where the
leading (but not sub-leading) large-order asymptotics of the perturbative
coefficients are known.Comment: 6 pages, LaTeX, 2 tables; certain numerical examples adde
The non-perturbative part of the plaquette in quenched QCD
We define the non-perturbative part of a quantity as the difference between
its numerical value and the perturbative series truncated by dropping the order
of minimal contribution and the higher orders. For the anharmonic oscillator,
the double-well potential and the single plaquette gauge theory, the
non-perturbative part can be parametrized as A (lambda)^B exp{-C/lambda} and
the coefficients can be calculated analytically. For lattice QCD in the
quenched approximation, the perturbative series for the average plaquette is
dominated at low order by a singularity in the complex coupling plane and the
asymptotic behavior can only be reached by using extrapolations of the existing
series. We discuss two extrapolations that provide a consistent description of
the series up to order 20-25. These extrapolations favor the idea that the
non-perturbative part scales like (a/r_0)^4 with a/r_0 defined with the force
method. We discuss the large uncertainties associated with this statement. We
propose a parametrization of ln((a/r_0)) as the two-loop universal terms plus a
constant and exponential corrections. These corrections are consistent with
a_{1-loop}^2 and play an important role when beta<6. We briefly discuss the
possibility of calculating them semi-classically at large beta.Comment: 13 pages, 16 figures, uses revtex, contains a new section with the
uncertainties on the extrapolations, refs. adde
Emergence of hexatic and long-range herringbone order in two-dimensional smectic liquid crystals : A Monte Carlo study
Using a high resolution Monte Carlo simulation technique based on
multi-histogram method and cluster-algorithm, we have investigated critical
properties of a coupled XY model, consists of a six-fold symmetric hexatic and
a three-fold symmetric herringbone field, in two dimensions. The simulation
results demonstrate a series of novel continues transitions, in which both
long-range hexatic and herringbone orderings are established simultaneously. It
is found that the specific-heat anomaly exponents for some regions in coupling
constants space are in excellent agreement with the experimentally measured
exponents extracted from heat-capacity data near the smecticA-hexaticB
transition of two-layer free standing film
Experimental observation of two-dimensional fluctuation magnetization in the vicinity of T_c for low values of the magnetic field in deoxygenated YBa_2Cu_3O_{7-x}
We measured isofield magnetization curves as a function of temperature in two
single crystal of deoxygenated YBaCuO with T_c = 52 and 41.5 K. Isofield MvsT
were obtained for fields running from 0.05 to 4 kOe. The reversible region of
the magnetization curves was analyzed in terms of a scaling proposed by Prange,
but searching for the best exponent . The scaling analysis carried
out for each data sample set with =0.669, which corresponds to the
3D-xy exponent, did not produced a collapsing of curves when applied to MvsT
curves data obtained for the lowest fields. The resulting analysis for the Y123
crystal with T_c = 41.5 K, shows that lower field curves collapse over the
entire reversible region following the Prange's scaling with =1,
suggesting a two-dimensional behavior. It is shown that the same data obeying
the Prange's scaling with =1 for crystal with T_c = 41.5 K, as well
low field data for crystal with = 52 K, obey the known two-dimensional
scaling law obtained in the lowest-Landau-level approximation.Comment: 4 pages, 3 figure
Generating Bounds for the Ground State Energy of the Infinite Quantum Lens Potential
Moment based methods have produced efficient multiscale quantization
algorithms for solving singular perturbation/strong coupling problems. One of
these, the Eigenvalue Moment Method (EMM), developed by Handy et al (Phys. Rev.
Lett.{\bf 55}, 931 (1985); ibid, {\bf 60}, 253 (1988b)), generates converging
lower and upper bounds to a specific discrete state energy, once the signature
property of the associated wavefunction is known. This method is particularly
effective for multidimensional, bosonic ground state problems, since the
corresponding wavefunction must be of uniform signature, and can be taken to be
positive. Despite this, the vast majority of problems studied have been on
unbounded domains. The important problem of an electron in an infinite quantum
lens potential defines a challenging extension of EMM to systems defined on a
compact domain. We investigate this here, and introduce novel modifications to
the conventional EMM formalism that facilitate its adaptability to the required
boundary conditions.Comment: Submitted to J. Phys.
Fisher's zeros as boundary of renormalization group flows in complex coupling spaces
We propose new methods to extend the renormalization group transformation to
complex coupling spaces. We argue that the Fisher's zeros are located at the
boundary of the complex basin of attraction of infra-red fixed points. We
support this picture with numerical calculations at finite volume for
two-dimensional O(N) models in the large-N limit and the hierarchical Ising
model. We present numerical evidence that, as the volume increases, the
Fisher's zeros of 4-dimensional pure gauge SU(2) lattice gauge theory with a
Wilson action, stabilize at a distance larger than 0.15 from the real axis in
the complex beta=4/g^2 plane. We discuss the implications for proofs of
confinement and searches for nontrivial infra-red fixed points in models beyond
the standard model.Comment: 4 pages, 3 fig
On the Dominance of Trivial Knots among SAPs on a Cubic Lattice
The knotting probability is defined by the probability with which an -step
self-avoiding polygon (SAP) with a fixed type of knot appears in the
configuration space. We evaluate these probabilities for some knot types on a
simple cubic lattice. For the trivial knot, we find that the knotting
probability decays much slower for the SAP on the cubic lattice than for
continuum models of the SAP as a function of . In particular the
characteristic length of the trivial knot that corresponds to a `half-life' of
the knotting probability is estimated to be on the cubic
lattice.Comment: LaTeX2e, 21 pages, 8 figur
Disorder-Induced Critical Phenomena in Hysteresis: Numerical Scaling in Three and Higher Dimensions
We present numerical simulations of avalanches and critical phenomena
associated with hysteresis loops, modeled using the zero-temperature
random-field Ising model. We study the transition between smooth hysteresis
loops and loops with a sharp jump in the magnetization, as the disorder in our
model is decreased. In a large region near the critical point, we find scaling
and critical phenomena, which are well described by the results of an epsilon
expansion about six dimensions. We present the results of simulations in 3, 4,
and 5 dimensions, with systems with up to a billion spins (1000^3).Comment: Condensed and updated version of cond-mat/9609072,``Disorder-Induced
Critical Phenomena in Hysteresis: A Numerical Scaling Analysis'