CRITICAL PHENOMENA OF FLUIDS: LANDAU-GINZBURG-WILSON MODEL

C. Di Castro, Roberto Raimondi

Subir Sachdev A

Jyoti Sharma

F. Fonseca, Pedro D. Fonseca, Sede Bogot

This work explains the method of Wilson renormalization applied to the Ginzburg-Landau model. The calculation shows in detail how to perform the renormalization of the partition function, and the deduction of the renormalization group equations. The fundamental purpose of the manuscript is to provide a guidance to graduate students in physics, in the area of statistical mechanics.

Andrea Pelissetto C, Ettore Vicari

A fundamental issue in the renormalization-group (RG) theory of critical phenomena concerns the allowed values of critical exponents that are consistent with the continuous nature of a phase transition. Here we conjecture a lower bound for the length-scale exponent $ν$, which should hold for the large class of continuous transitions associated with $d$-dimensional Landau-Ginzburg-Wilson (LGW) $Φ^4$ theories with a multicomponent scalar field $φ$ and a unique $φ\cdot φ$ quadratic term (including some extensions with fermionic and gauge fields), describing many universality classes of critical phenomena. If $Δ_φ=(d-2+η)/2$ is the dimension of the order-parameter field $φ$, and $Δ_\varepsilon=d-1/ν$ is the RG dimension of the energy operator $\varepsilon$, which can be identified with $[φ\cdot φ]$ (the squared field with a proper subtraction of the mixing with the identity), we conjecture the inequality $Δ_\varepsilon \ge 2 Δ_φ$, which implies $ν\ge (2-η)^{-1}$ and $γ= (2-η)ν\ge 1$. These inequalities are supported by general arguments for ferromagnetic lattice models, by $ε$-expansion results for generic LGW $Φ^4$ theories close to four dimensions, exact relations for two-dimensional minimal conformal field theories, and are consistent with all further known (numerical, perturbative, and exact) results for LGW $Φ^4$ theories. In particular, since unitarity requires $η\ge 0$, the above inequality implies $ν\ge 1/2$ for unitary theories. This lower bound is more restrictive than $ν> 1/d$, derived by noting that $ν=1/d$ characterizes the singular finite-size behavior at first-order transitions.