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 $S_{c}$ (attraction domain of the Wilson-Fisher fixed point), we explicitly show the existence of a first-order phase transition domain $S_{f}$ separated from $S_{c}$ by the tricritical surface $S_{t}$ (attraction domain of the Gaussian fixed point). $S_{f}$ and $S_{c}$ are two distinct domains of repulsion for the Gaussian fixed point, but $S_{f}$ is not the basin of attraction of a fixed point. $S_{f}$ is characterized by an endless renormalized trajectory lying entirely in the domain of negative values of the $\phi ^{4}$-coupling. This renormalized trajectory exists also in four dimensions making the Gaussian fixed point ultra-violet stable (and the $\phi_{4}^{4}$ 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

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

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 $O(\epsilon ^{2})$ 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 $O(\epsilon)$ finally unstable.Comment: Final versio

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 $\varphi ^{4}$-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

Addendum-erratum to: ``Nonasymptotic critical behavior from field theory at d=3. II. The ordered-phase case. Phys. Rev. B35, 3585 (1987)

This note is intended to emphasize the existence of estimated Feynman integrals in three dimensions for the free energy of the O(1) scalar theory up to five loops which may be useful for other work. We also correct some misprints of the published paper.Comment: One figure and one table added, some additions in the tex

An analytical approximation scheme to two point boundary value problems of ordinary differential equations

A new (algebraic) approximation scheme to find {\sl global} solutions of two point boundary value problems of ordinary differential equations (ODE's) is presented. The method is applicable for both linear and nonlinear (coupled) ODE's whose solutions are analytic near one of the boundary points. It is based on replacing the original ODE's by a sequence of auxiliary first order polynomial ODE's with constant coefficients. The coefficients in the auxiliary ODE's are uniquely determined from the local behaviour of the solution in the neighbourhood of one of the boundary points. To obtain the parameters of the global (connecting) solutions analytic at one of the boundary points, reduces to find the appropriate zeros of algebraic equations. The power of the method is illustrated by computing the approximate values of the ``connecting parameters'' for a number of nonlinear ODE's arising in various problems in field theory. We treat in particular the static and rotationally symmetric global vortex, the skyrmion, the Nielsen-Olesen vortex, as well as the 't Hooft-Polyakov magnetic monopole. The total energy of the skyrmion and of the monopole is also computed by the new method. We also consider some ODE's coming from the exact renormalization group. The ground state energy level of the anharmonic oscillator is also computed for arbitrary coupling strengths with good precision.Comment: 5 pages, 3 tables, Late