RG fixed point

A fixed point of a renormalization-group transformation and its linear stability data, which determine scaling and critical exponents.

RG fixed point

Definition (RG fixed point)

Let $R_b$ be a renormalization-group (RG) map implementing coarse-graining by a scale factor $b>1$ on a space of Hamiltonians or coupling parameters $g$ (e.g., $g=(K,h,\ldots)$ ). An RG fixed point is a point $g^\star$ such that

R_b(g^\star)=g^\star \quad \text{for a given } b>1 \text{ (equivalently, for all } b \text{ in a consistent RG scheme).}

A fixed point represents a scale-invariant effective description.

Linearization and relevant/irrelevant directions

Linearize $R_b$ at $g^\star$ :

g' - g^\star = DR_b(g^\star)\,(g-g^\star) + \cdots

Choose eigen-directions $u_i$ of the Jacobian with eigenvalues $\lambda_i(b)$ :

DR_b(g^\star)u_i=\lambda_i(b)\,u_i,\qquad \lambda_i(b)=b^{y_i}.

The numbers $y_i$ are scaling dimensions (RG eigenvalues):

relevant if $y_i>0$ (perturbations grow under coarse-graining),
irrelevant if $y_i<0$ (perturbations die out),
marginal if $y_i=0$ (need higher-order analysis).

A fixed point is (infrared) stable if it has no relevant directions except those forced by tuning parameters (e.g., temperature-like and field-like).

Fixed points and critical behavior

Near a continuous phase transition, long-distance behavior is governed by a stable fixed point. If $y_t$ is the temperature-like relevant exponent, then the correlation-length exponent is

\nu = \frac{1}{y_t},

and other critical exponents follow from RG eigenvalues and scaling relations.

Definition (RG fixed point)

Linearization and relevant/irrelevant directions

Fixed points and critical behavior

Prerequisites / cross-links