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

Universality and quantum effects in one-component critical fluids

Non-universal scale transformations of the physical fields are extended to pure quantum fluids and used to calculate susceptibility, specific heat and the order parameter along the critical isochore of He3 near its liquid-vapor critical point. Within the so-called preasymptotic domain, where the Wegner expansion restricted to the first term of confluent corrections to scaling is expected valid, the results show agreement with the experimental measurements and recent predictions, either based on the minimal-substraction renormalization and the massive renormalization schemes within the $\Phi\_{d=3}^{4}(n=1)$-model, or based on the crossover parametric equation of state for Ising-like systems

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

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

Classical-to-critical crossovers from field theory

We extent the previous determinations of nonasymptotic critical behavior of Phys. Rev B32, 7209 (1985) and B35, 3585 (1987) to accurate expressions of the complete classical-to-critical crossover (in the 3-d field theory) in terms of the temperature-like scaling field (i.e., along the critical isochore) for : 1) the correlation length, the susceptibility and the specific heat in the homogeneous phase for the n-vector model (n=1 to 3) and 2) for the spontaneous magnetization (coexistence curve), the susceptibility and the specific heat in the inhomogeneous phase for the Ising model (n=1). The present calculations include the seventh loop order of Murray and Nickel (1991) and closely account for the up-to-date estimates of universal asymptotic critical quantities (exponents and amplitude combinations) provided by Guida and Zinn-Justin [J. Phys. A31, 8103 (1998)].Comment: 4 figs, 4 program documents in appendix, some corrections adde

Crossover scaling from classical to nonclassical critical behavior

We study the crossover between classical and nonclassical critical behaviors. The critical crossover limit is driven by the Ginzburg number G. The corresponding scaling functions are universal with respect to any possible microscopic mechanism which can vary G, such as changing the range or the strength of the interactions. The critical crossover describes the unique flow from the unstable Gaussian to the stable nonclassical fixed point. The scaling functions are related to the continuum renormalization-group functions. We show these features explicitly in the large-N limit of the O(N) phi^4 model. We also show that the effective susceptibility exponent is nonmonotonic in the low-temperature phase of the three-dimensional Ising model.Comment: 5 pages, final version to appear in Phys. Rev.

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

Is the mean-field approximation so bad? A simple generalization yelding realistic critical indices for 3D Ising-class systems

Modification of the renormalization-group approach, invoking Stratonovich transformation at each step, is proposed to describe phase transitions in 3D Ising-class systems. The proposed method is closely related to the mean-field approximation. The low-order scheme works well for a wide thermal range, is consistent with a scaling hypothesis and predicts very reasonable values of critical indices.Comment: 4 page

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