123 research outputs found
Nonasymptotic critical behavior from field theory
The obtention (up to five or six loop orders) of nonasymptotic critical
behavior, above and below Tc, from the field theoretical framework is presented
and discussed.Comment: 9 page
Field-Theoretic Techniques in the Study of Critical Phenomena
We shortly illustrate how the field-theoretic approach to critical phenomena
takes place in the more complete Wilson theory of renormalization and
qualitatively discuss its domain of validity. By the way, we suggest that the
differential renormalization functions (like the beta-function) of the
perturbative scalar theory in four dimensions should be Borel summable provided
they are calculated within a minimal subtraction scheme.Comment: 32 pages, LaTeX, 9 figures, to appear in Journal of Physical Studie
Renormalization group domains of the scalar Hamiltonian
Using the local potential approximation of the exact renormalization group
(RG) equation, we show the various domains of values of the parameters of the
O(1)-symmetric scalar Hamiltonian. In three dimensions, in addition to the
usual critical surface (attraction domain of the Wilson-Fisher fixed
point), we explicitly show the existence of a first-order phase transition
domain separated from by the tricritical surface
(attraction domain of the Gaussian fixed point). and are two
distinct domains of repulsion for the Gaussian fixed point, but is not
the basin of attraction of a fixed point. is characterized by an
endless renormalized trajectory lying entirely in the domain of negative values
of the -coupling. This renormalized trajectory exists also in four
dimensions making the Gaussian fixed point ultra-violet stable (and the
renormalized field theory asymptotically free but with a wrong
sign of the perfect action). We also show that very retarded classical-to-Ising
crossover may exist in three dimensions (in fact below four dimensions). This
could be an explanation of the unexpected classical critical behavior observed
in some ionic systems.Comment: 13 pages, 6 figures, to appear in Cond. Matt. Phys, some minor
correction
Exact renormalization group equation for the Lifshitz critical point
An exact renormalization equation (ERGE) accounting for an anisotropic
scaling is derived. The critical and tricritical Lifshitz points are then
studied at leading order of the derivative expansion which is shown to involve
two differential equations. The resulting estimates of the Lifshitz critical
exponents compare well with the calculations. In the case
of the Lifshitz tricritical point, it is shown that a marginally relevant
coupling defies the perturbative approach since it actually makes the fixed
point referred to in the previous perturbative calculations
finally unstable.Comment: Final versio
The Wilson exact renormalization group equation and the anomalous dimension parameter
The non-linear way the anomalous dimension parameter has been introduced in
the historic first version of the exact renormalization group equation is
compared to current practice. A simple expression for the exactly marginal
redundant operator proceeds from this non-linearity, whereas in the linear
case, first order differential equations must be solved to get it. The role of
this operator in the construction of the flow equation is highlighted.Comment: 10 page
Peculiarity of the Coulombic criticality ?
International audienceWe study the Coulombic criticality of ionic fluids within the restricted primitive model (RPM). We indicate that for the RPM, analysed in terms of the field of charge density, the corresponding Landau-Ginzburg-Wilson effective Hamiltonian has a negative -coefficient. In that case, solving the ``exact'' renormalization group equation in the local potential approximation, we show that close initial Hamiltonians may lead either to a first order transition or to an Ising-like critical behavior, the partition being formed by the tri-critical surface. This situation apparently illustrates the theoretical wavering encountered in the literature concerning the nature of the Coulombic criticality. Nevertheless, it is most probable that, in terms of the field considered, the model does not display any criticality
Renormalization group domains of the scalar Hamiltonian
Using the local potential approximation of the exact renormalization group
(RG) equation, we show various domains of values of the parameters of
the O(1) -symmetric scalar Hamiltonian. In three dimensions, in addition to
the usual critical surface Sc (attraction domain of the Wilson-Fisher fixed
point), we explicitly show the existence of a first-order phase transition domain Sf separated from Sc by the tricritical surface St (attraction domain
of the Gaussian fixed point). Sf and Sc are two distinct domains of repulsion for the Gaussian fixed point, but Sf is not the basin of attraction
of a fixed point. Sf is characterized by an endless renormalized trajectory
lying entirely in the domain of negative values of the ϕ⁴ -coupling. This
renormalized trajectory also exists in four dimensions making the Gaussian fixed point ultra-violet stable (and the ϕ⁴₄ renormalized field theory
asymptotically free but with a wrong sign of the perfect action). We also
show that a very retarded classical-to-Ising crossover may exist in three
dimensions (in fact below four dimensions). This could be an explanation
of the unexpected classical critical behaviour observed in some ionic systems.Використовуючи наближення локального потенціалу точного рівняння ренормалізаційної групи (РГ), ми показуємо різні області значень
параметрів O(1) симетричного скалярного гамільтоніану. У трьох вимірах додатково до звичайної критичної поверхні Sc (область притягання фіксованої точки Вільсона-Фішера), ми явно показуємо існування області фазового переходу першого ряду Sf , відокремленої від Sc трикритичною поверхнею Sf (область притягання гаусової фіксованої точки). Sf і Sc є дві різні області відштовхування для
гаусової фіксованої точки, а Sf не є в ділянці притягання фіксованої
точки. Sf характеризується нескінченою ренормалізованою траєкторією, яка повністю лежить в області негативних значень констант
взаємодії ϕ⁴ . Ця ренормалізована траєкторія також існує в чотирьох
вимірах, роблячи гаусову фіксовану точку в ультрафіолетовій області
стабільною (і ренормалізовану теорію поля ϕ⁴ асимптотично вільною, але з неправильним знаком ідеальної дії). Ми також показуємо,
що дуже запізнений кросовер від класичної до ізінгівської поведінки може існувати у трьох вимірах (фактично нижче чотирьох вимірів).
Це може бути поясненням для неочікуваної класичної критичної поведінки, яка спостерігається в деяких іонних системах
Analytic continuation of Taylor series and the two-point boundary value problems of some nonlinear ordinary differential equations
We compare and discuss the respective efficiency of three methods (with two
variants for each of them), based respectively on Taylor (Maclaurin) series,
Pad\'{e} approximants and conformal mappings, for solving quasi-analytically a
two-point boundary value problem of a nonlinear ordinary differential equation
(ODE). Six configurations of ODE and boundary conditions are successively
considered according to the increasing difficulties that they present. After
having indicated that the Taylor series method almost always requires the
recourse to analytical continuation procedures to be efficient, we use the
complementarity of the two remaining methods (Pad\'{e} and conformal mapping)
to illustrate their respective advantages and limitations. We emphasize the
importance of the existence of solutions with movable singularities for the
efficiency of the methods, particularly for the so-called Pad\'{e}-Hankel
method. (We show that this latter method is equivalent to pushing a movable
pole to infinity.) For each configuration, we determine the singularity
distribution (in the complex plane of the independent variable) of the solution
sought and show how this distribution controls the efficiency of the two
methods. In general the method based on Pad\'{e} approximants is easy to use
and robust but may be awkward in some circumstances whereas the conformal
mapping method is a very fine method which should be used when high accuracy is
required.Comment: Final versio
The Wilson-Polchinski exact renormalization group equation
The critical exponent is not well accounted for in the Polchinski
exact formulation of the renormalization group (RG). With a particular emphasis
laid on the introduction of the critical exponent , I re-establish
(after Golner, hep-th/9801124) the explicit relation between the early Wilson
exact RG equation, constructed with the incomplete integration as cutoff
procedure, and the formulation with an arbitrary cutoff function proposed later
on by Polchinski. I (re)-do the analysis of the Wilson-Polchinski equation
expanded up to the next to leading order of the derivative expansion. I finally
specify a criterion for choosing the ``best'' value of to this order.
This paper will help in using more systematically the exact RG equation in
various studies.Comment: Some minor changes, a reference added, typos correcte
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