156 research outputs found
On Size and Shape of the Average Meson Fields in the Semibosonized Nambu & Jona-Lasinio Model
We consider a two-flavor Nambu \& Jona-Lasinio model in Hartree approximation
involving scalar-isoscalar and pseudoscalar-isovector quark-quark interactions.
Average meson fields are defined by minimizing the effective Euklidean action.
The fermionic part of the action, which contains the full Dirac sea, is
regularized within Schwinger's proper-time scheme. The meson fields are
restricted to the chiral circle and to hedgehog configurations. The only
parameter of the model is the constituent quark mass which simultaneously
controls the regularization. We evaluate meson and quark fields
self-consistently in dependence on the constituent quark mass. It is shown that
the self-consistent fields do practically not depend on the constituent quark
mass. This allows us to define a properly parameterized reference field which
for physically relevant constituent masses can be used as a good approximation
to the exactly calculated one. The reference field is chosen to have correct
behaviour for small and large radii. To test the agreement between
self-consistent and reference fields we calculate several observables like
nucleon energy, mean square radius, axial-vector constant and delta-nucleon
mass splitting in dependence on the constituent quark mass. The agreement is
found to be very well. Figures available on request.Comment: 12 pages (LATEX), 3 figures available on request, report FZR 93-1
The repulsive lattice gas, the independent-set polynomial, and the Lov\'asz local lemma
We elucidate the close connection between the repulsive lattice gas in
equilibrium statistical mechanics and the Lovasz local lemma in probabilistic
combinatorics. We show that the conclusion of the Lovasz local lemma holds for
dependency graph G and probabilities {p_x} if and only if the independent-set
polynomial for G is nonvanishing in the polydisc of radii {p_x}. Furthermore,
we show that the usual proof of the Lovasz local lemma -- which provides a
sufficient condition for this to occur -- corresponds to a simple inductive
argument for the nonvanishing of the independent-set polynomial in a polydisc,
which was discovered implicitly by Shearer and explicitly by Dobrushin. We also
present some refinements and extensions of both arguments, including a
generalization of the Lovasz local lemma that allows for "soft" dependencies.
In addition, we prove some general properties of the partition function of a
repulsive lattice gas, most of which are consequences of the alternating-sign
property for the Mayer coefficients. We conclude with a brief discussion of the
repulsive lattice gas on countably infinite graphs.Comment: LaTex2e, 97 pages. Version 2 makes slight changes to improve clarity.
To be published in J. Stat. Phy
Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations
We reconsider the conceptual foundations of the renormalization-group (RG)
formalism, and prove some rigorous theorems on the regularity properties and
possible pathologies of the RG map. Regarding regularity, we show that the RG
map, defined on a suitable space of interactions (= formal Hamiltonians), is
always single-valued and Lipschitz continuous on its domain of definition. This
rules out a recently proposed scenario for the RG description of first-order
phase transitions. On the pathological side, we make rigorous some arguments of
Griffiths, Pearce and Israel, and prove in several cases that the renormalized
measure is not a Gibbs measure for any reasonable interaction. This means that
the RG map is ill-defined, and that the conventional RG description of
first-order phase transitions is not universally valid. For decimation or
Kadanoff transformations applied to the Ising model in dimension ,
these pathologies occur in a full neighborhood of the low-temperature part of the first-order
phase-transition surface. For block-averaging transformations applied to the
Ising model in dimension , the pathologies occur at low temperatures
for arbitrary magnetic-field strength. Pathologies may also occur in the
critical region for Ising models in dimension . We discuss in detail
the distinction between Gibbsian and non-Gibbsian measures, and give a rather
complete catalogue of the known examples. Finally, we discuss the heuristic and
numerical evidence on RG pathologies in the light of our rigorous theorems.Comment: 273 pages including 14 figures, Postscript, See also
ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.
- …