Renormalization-group (RG) transformations

Coarse-graining maps of models to effective models at larger scales; fixed points, relevance/irrelevance, and extraction of critical exponents.

Renormalization-group (RG) transformations

A renormalization-group (RG) transformation is a procedure that (i) coarse-grains microscopic degrees of freedom and (ii) rescales lengths so the system can be compared across scales. Iterating this transformation produces a flow in the space of Hamiltonians/couplings, organizing phases and critical behavior.

Definition (conceptual)

An RG map at scale factor $b>1$ is a transformation

R_b:\ \text{(microscopic model)}\ \longrightarrow\ \text{(effective model)}

so that long-distance observables (correlations, free energy density, etc.) are approximately preserved after:

Blocking / coarse-graining (integrate out or average short-scale degrees of freedom),
Rescaling of lengths to restore the original lattice spacing,
Parameter read-off (express the coarse-grained model in the same family with new couplings).

In coupling coordinates $K=(K_1,K_2,\dots)$ , one writes

K' = R_b(K).

Fixed points and universality

A point $K^\star$ with

R_b(K^\star)=K^\star

is an RG fixed point . Fixed points control scaling limits:

Stable fixed points describe phases (attractive under iteration).
Critical fixed points separate phases and control universal singularities.

This is the RG basis for universality classes .

Linearization and critical exponents

Linearize near a fixed point:

K' - K^\star \approx DR_b(K^\star)\,(K-K^\star).

If $v_i$ is an eigenvector with eigenvalue $\lambda_i$ , write $\lambda_i=b^{y_i}$ :

$y_i>0$ : relevant direction (grows under coarse-graining),
$y_i<0$ : irrelevant direction (decays),
$y_i=0$ : marginal (needs higher-order analysis).

A standard identification is $\nu=1/y_t$ , where $y_t$ is the eigenvalue exponent associated with the thermal perturbation (moving away from critical temperature). This links RG to critical exponents and to scaling relations .

Concrete example: block-spin RG (Ising-type intuition)

For the Ising model on $\mathbb{Z}^d$ , a block-spin map partitions the lattice into $b^d$ -site blocks and defines a coarse spin (e.g., majority rule). Integrating out microscopic spins inside blocks produces effective couplings between block spins, generating the flow $K\mapsto K'$ .

Definition (conceptual)

Fixed points and universality

Linearization and critical exponents

Concrete example: block-spin RG (Ising-type intuition)

Prerequisites