Critical exponents describe the behavior of physical quantities near continuous phase transitions.
Analytical results can be theoretically achieved in mean field theory in high dimensions or when exact solutions are known such as the two-dimensional Ising model.
The theoretical treatment in generic dimensions requires the renormalization group approach or, for systems at thermal equilibrium, the conformal bootstrap techniques.
The critical dimension above which mean field exponents are valid varies with the systems and can even be infinite.
The control parameter that drives phase transitions is often temperature but can also be other macroscopic variables like pressure or an external magnetic field.
For simplicity, the following discussion works in terms of temperature; the translation to another control parameter is straightforward.
More generally one might expect Let us assume that the system at thermal equilibrium has two different phases characterized by an order parameter Ψ, which vanishes at and above Tc.
The critical exponents can be derived from the specific free energy f(J,T) as a function of the source and temperature.
[1] The classical Landau theory (also known as mean field theory) values of the critical exponents for a scalar field (of which the Ising model is the prototypical example) are given by If we add derivative terms turning it into a mean field Ginzburg–Landau theory, we get One of the major discoveries in the study of critical phenomena is that mean field theory of critical points is only correct when the space dimension of the system is higher than a certain dimension called the upper critical dimension which excludes the physical dimensions 1, 2 or 3 in most cases.
The problem with mean field theory is that the critical exponents do not depend on the space dimension.
It can even lead to a qualitative discrepancy at low space dimension, where a critical point in fact can no longer exist, even though mean field theory still predicts there is one.
[2] This value is in a significant disagreement with the most precise theoretical determinations[3][4][5] coming from high temperature expansion techniques, Monte Carlo methods and the conformal bootstrap.
The accuracy of this first principle method depends on the available computational resources, which determine the ability to go to the infinite volume limit and to reduce statistical errors.
The conformal bootstrap is a more recently developed technique, which has achieved unsurpassed accuracy for the Ising critical exponents.
Close enough to the critical point, everything can be reexpressed in terms of certain ratios of the powers of the reduced quantities.
In a sufficiently small neighborhood of the critical point, we may linearize the action of the renormalization group.
They fall into universality classes and obey the scaling and hyperscaling relations These equations imply that there are only two independent exponents, e.g., ν and η.
[10] More complex behavior may occur at multicritical points, at the border or on intersections of critical manifolds.